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ONS Wee Nea SIS Res we | STU eweeweeee Webbed CSCC Se al ei See Nha che asad re ODS SO Nh SIS high MA | SSA Ate DIDI S ‘3 Ad J Nd Ned aed Na \ ad Na J edele | | Nass Ved a Neg Veale Na PPP SPSS SS Beth YF Na ee a a AAS) heh hey, { Sew Lz —, IS es IIIS SWC SCS ewe eee weve BOSSE | WO rin, batted j\—~ Rad PFWwZ Sy SA= | ) ee y ISIS ISLS II SSS belch ASE enw J Sade) S Me | x © f ia iv one i ati, : Are ni rm Te eu tier ta = - oF SY cuss§ MITHSONIAN CONTRIBUTIONS TO KNOWLEDGE. Vins Xolex- EVERY MAN IS A VALUABLE MEMBER OF SOCIETY, WHO, BY HIS OBSERVATIONS, RESEAROHES, AND EXPERIMENTS, PROCURES KNOWLEDGE FOR MEN.—SMITHSON. Cisse TOW WAS at EN GE ON: PUBLISHED BY THE SMITHSONIAN INSTITUTION. MDCCCLXXIY. : - : . 4 fh ae J : ered TM Sf ‘ ah rue) 4 La PP Veet BRIN A Th eo prey ’ ae a, ; COLLINS, PRINTER, 795 JAYNE sv. ADVERTISEMENT. Tuts volume forms the nineteenth of a series, composed of original memoirs on different branches of knowledge, published at the expense, and under the direction, of the 8» ‘hsonian Institution. The publication of this series forms part of a general plan adopted for carrying into effect the benevolent intentions of JAMES SMITHSON, Esq., of England. 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It is however impossible, in most cases, to verify the state: ments of an author; and, therefore, neither the Commission nor the Institution can be responsible for more than the general character of a memoir. The following rules have been adopted for the distribution of the quarto volumes of the Smithsonian Contributions :— 1. They are to be presented to all learned societies which publish Transactions, and give copies of these, in exchange, to the Institution. 2. Also, to all foreign libraries of the first class, provided they give in exchange their catalogues or other publications, or an equivalent from their duplicate volumes. 3. To all the colleges in actual operation in this country, provided they furnish, in return, meteorological observations, catalogues of their libraries and of their students, and all other publications issued by them relative to their organization and history. 4. To all States and Territories, provided there be given, in return, copies of all documents published under their authority. 5. To all incorporated public libraries in this country, not included in any of the foregoing classes, now containing more than 10,000 volumes; and to smaller libraries, where a whole State or large district would be otherwise unsupplied. OBRETCERS OF THE SMITHSONIAN INSTITUTION. THE PRESIDENT OF THE UNITED STATES, Ex-officio PRESIDING OFFICER OF THE INSTITUTION. THE VICE-PRESIDENT OF THE UNITED STATES, Ex-officio SECOND PRESIDING OFFICER. MORRISON R. WAITE, CHANCELLOR OF THE INSTITUTION. JOSEPH HENRY, SECRETARY OF THE INSTITUTION. SPENCER F. BAIRD, ASSISTANT SECRETARY. PETER PARKER, JOHN MACLEAN, EXeEcuTIvVE CoMMITTEE. WILLIAM T. SHERMAN, Henry WILSON, Morrison R. WaAITrE, HANNIBAL HAMLIN, Joun W. STEVENSON, AARON A. SARGENT, SAMUEL S. Cox, EBENEZER R. Hoar, Gerry W. Haz_eton, JoHN MACLEAN, PETER PARKER, WiirAmM T. SHERMAN, ASA GRay, . J. D. Dana, Henry Coprée, REGENTS. Vice-President of the United States. Chief Justice of the United States. Member of the Senate of the United States. oe 6 iss ce 66 66 cc 5 ce co 66 oe Member of the House of Representatives (DEUS: 6 ce 6c Ge ae ee Citizen of New Jersey. “ of Washington. ee 66 “of Massachusetts. “of Connecticut. ee of Pennsylvania. MEMBERS EX-OFFICIO OF THE INSTITUTION. Utysses 8S. Grant, . . . . . . President of the United States. Henry Witson, . . . . . . . Vice-President of the United States, Hamitton Fish, . . . . . . . Secretary of State. B. H. Bristow, ..-« .. . . Secretary of the Treasury. Wr \WeebEEKNAPS 2. . . wseeretary of War. Grorce M. Roseson, . . . . . Secretary of the Navy. J. A.J. CRESWELL, . .. . . . # SPostmaster-General. Grorce H. Wituiams, . . . . . Attorney-General. Morrison R. Waite, . . . . . Chief Justice of the United States. M. D. Leggatt, . . . . . . . Commissioner of Patents. HONORARY MEMBER. Cotumsus DeLAno. ; The Secretary of the Interior, TABLE OF CONTENTS.’ PAGE ARTICLE JI. Inrropuction. Pp. 16. P Advertisement : : > . ° iii List of Officers of the Srartieonien Tnstitution 3 : : : ix ARTICLE II. (No. 240.) PropiemMs or Rotary MOTION PRESENTED BY THE GYROSCOPE, THE PRECESSION OF THE WQUINOXES, AND THE PENDULUM. By Brevet Maj.-General J. G. Barnarp, Colonel of Engineers, U. S. A., A.M., LL.D., Member of National Academy of Sciences. 1871-1873. 4to. pp. 74. The Precession of the Equinoxes and Nutation as resulting from the pay of the Gyroscope 1 On the Motions of Freely Suspended and Gyroscope penanlacie and on the Pendulum and Gyroscope as exhibiting the Rotation of the Earth . 15 On the Internal Structure of the Earth considered as affecting the Phe- nomena of Precession and Nutation : < New Addendum . : ns : . : : : 42 ARTICLE III. (No. 241.) A Contrisurion To THE History or THE FresH-WaTER ALG™ or North America. By Horatio C. Woop, Jr., M.D., Professor of Botany, and Clinical Lecturer on Diseases of the Nervous System in the University of Pennsylvania; Physician to the Philadelphia Hospital, ete. October, 1872. 4to. pp. 274. Twenty-one colored plates. Preface Vv Introduction : 1 Fresh-Water Alge of the United States 9 Class Phycochromophycee : : > : : 9 Order Cystiphoree 3 : ‘ : : : 10 Family Chroococcaceze . : : : : : 10 Order Nematogenese : : - F : : 15 Family Oscillariaceze ; : : : . ¢ 16 Family Nostochacez : : : . 5 s 23 Family Rivulariaces : ; F : ; . 43 Family Scytonemacez : : : ‘ ‘ : 55 Family Sirosiphonacee : : : 2 : c 67 Class Chlorophyllacez : : : : ; 17 Order Coccophycez : c ; : < 5 78 Family Palmellacee . i : : : : 78 Family Protococlaceze : é é : : ‘ 85 Family Volvocinee ‘ : : : : : 98 Order Zygophycer F : : : : . 100 Family Desmidiacee : 5 ; : : 5 OD Family Zygnemacee : . : . é 5G) ' Each memoir is separately paged and indexed. xi¥ ARTICLE IV. TABLE OF CONTENTS. Order Siphophycex Family Hydrogastree Family Vaucheriacee Family Ulvacee Family Confervacee Family Gidogoniacexe Family Chroolepidex Family Chetophoracee Class Rhodophycee Family Porphyracexe Family Chantransiacex ; i ; a : 0 Family Batrachospermacexe Family Lemaneacee Supplement 6 : . ¢ Geographical List of Species : : . ; . . Bibliography : Index c Explanation of the Plates (No. 262.) AN INVESTIGATION OF THE ORBIT OF URANUS, WITH GENERAL TABLES OF 1s Motion. By Simon Newcoms, Professor of Mathematics, United States Navy. October, 1873. 4to. pp. 296. INTRODUCTION . : : 6 : : : Cuaprer I. Method of Determining the Perturbations of the Longitude, Radius Vector, and Latitude of a Planet by direct Integration. Notation and general differential formule Formation of the required derivatives of the Spumamiine fametion Correction of these derivatives for terms of the second order Integration formule for perturbations of radius vector Development of functions of rectangular co-ordinates Integration of perturbations of radius vector Formule for perturbations of longitude to terms of the eee orden Motion of the orbital planes Perturbations of the second order Henendne on the stiien of the Breit planes . 5 Reduction of the onside to the ecliptic Expressions for the latitude Cuarter II. Application of the Preceding Method to the Computation of the Perturbations of Uranus by Saturn. Data of computation : Numerical expressions for R aad its derivatives ; Perturbations of radius vector Perturbations of longitude Perturbations of latitude Cuarter IIT. Perturbations of Uranus produced by Neptune and Jupiter Adopted elements of Neptune Development of # and its derivatives fa the echca of Neptael The term of long period between Neptune and Uranus . Perturbations of the longitude produced by Neptune Perturbations of the radius vector produced by Neptune Perturbations of the latitude produced by Neptune Perturbations produced by Jupiter PAGE 174 175 176 182 186 188 203 205 213 214 215 217 221 225 229 235 249 253 10 12 13 14 17 22 24 25 27 31 34 44 49 51 53 54 55 58 60 61 62 TABLE OF CONTENTS. xV PAGE Cuarter IV. Terms of the Second Order due to the Action of Saturn. Preliminary investigation of the orbit of Saturn 5 : é 65 Perturbations of Saturn and Uranus : 5 : 68 Formation of the expressions for the terms of the aeéotid order : 69 Perturbations depending on the square of the mass of Saturn . 76 Perturbations depending on the product of the masses of Jupiter and Saturn : 3 C : : : : ; 17 Cuaprer V. Collection and Transformation of the preceding Perturba- tions of Uranus. Terms independent of the position of the disturbing planet : 3 79 Secular variations : - : 80 Auxiliary expressions on which the aExtoators depend : : 81 Reduced expressions for the latitude of Uranus . 5 ; : 93 Positions of Uranus resulting from the preceding theory ° 5 98 Elements III of Uranus . : : : F ; 3 99 Cuaprer VI. Reduction of the Observations of Uranus, and their Com- parison with the preceding Theory. Reduction of the ancient observations . : : : OG) Their comparison with the provisional theory. . : 5 UNO) Discussion of the modern observations . c > ¢ 5 UIT Reduction of the Results to a uniform system. : : o JULI Adopted positions of fundamental stars . A 5 5 U8} Discussion of corrections to reduce the different Sreereatigns to a homo- geneous system : : : : : elt Table of these corrections : é 0 . 120 Results of the observations from lil to 1830 : : : . 123 Observations from 1830 to 1872 . . C . 126 Table to convert errors of right ascension and declination of Uranus into errors of longitude and latitude ; ¢ : ee Loi Tabular summary of results of observations, 1830 to 1872 : - Il Corrections to be applied to the positions of Uranus in the Berlin Jahr- buch and the Nautical Almanac to reduce them to positions from the provisional theory : . : C : : oil CuarrerR VII. Formation and Solution of the Equations of Condition Resulting from the preceding Comparisons. Expressions of the observed corrections to the longitudes of the provi- sional theory in terms of the corrections to the heliocentric co-ordinates 158 Expressions of the same quantities in terms of the corrections to the ele- ments of Uranus and the mass of Neptune. 161 Table to express errors of heliocentric co-ordinates as errors of elements 162 Discussions and solutions of the equations thus formed . : GS Concluded corrections to the elements of longitude : : 5 Clie Corrections to the inclination and node of Uranus ° : 5 kiss Cuaprer VIII. Completion and Arrangement of the Theory to fit it for Permanent Use. Correction of the coefficients of the long inequality between Uranus and Neptune for the terms of the second order. : é les Concluded elements, or elements IV of Uranus . : ‘ ; 181 Long-period and secular perturbations of the elements . : . 182 Table of these perturbations from A.D. 1000 antil A D 2200 . - 184 Mean elements of Uranus ; , : : etsy! Expressions for the concluded theory of Ur: anus . : - iS TABLE OF CONTENTS. Cuaprer IX. General Tables of Uranus. Enumeration of the quantities contained in the several tables Precepts for the use of the tables Examples of the use of the tables Tables of Uranus Subsidiary tables SMITHSONIAN CONTRIBUTIONS TO KNOWLEDGE. - 240 —— PROBLEMS OF ROTARY MOTION PRESENTED BY THE GYROSCOPE, THE PRECESSION OF THE EQUINOXES, AND THE PENDULUM. BY Brevet Mas.-Gen. J. G. BARNARD, EE .S. A., A.M., LL.D., MEMBER OF NATIONAL ACADEMY OF SCIENCES, ‘iL OF ENGINEERS U [ACCEPTED FOR PUBLICATION, 0¢ CTOBER, 187i.] ADVERTISEMENT. Tue three following papers were read at intervals before the National Academy of Sciences, and subsequently presented to the Smithsonian Institution for publi- cation. JOSEPH HENRY, Secretary Smithsonian Institution. ( iii ) . 1 a . . f f r . ‘ . . — ‘ i . , ‘ . ‘ ¥ THE PRECESSION OF THE EQUINOXES AND NUTATION AS RESULTING FROM THE THEORY OF THE GYROSCOPE. In a paper published in the American Journal of Science, in 1857, and in Bar- nard’s American Journal of Education’ [No. 9] of the same year, I remarked :— “The analogy between the minute motions of the gyroscope and that grand phenomenon exhibited in the heavens, the ‘precession of the equinoxes,’ is often remarked. In an ultimate analysis, the phenomena, doubtless, are identical,” &c. It is the object of the present paper to deduce the analytical expressions of this phenomenon directly from the theory of the gyroscope. A brief summary of the processes used and results arrived at in the paper referred to is necessary as a preliminary. Let A, B, C, D (Fig. 1) be a solid body of any shape, retained by the fixed point O (within or without its mass). Ox, Oy, and Oz are the three co-ordinate axes, Fig. 1. fixed in space, to which the motion of the body is referred. Ox,, Oy;, Oz, are the three principal axes belonging to the point O, and which, of course, partake of the 1 “The Phenomena of the Gyroscope Analytically Examined.” 1 October 1871. ( 1 ) 2 PRECESSION OF THE EQUINOXES AND NUTATION body’s motion. ‘The position of the body at any instant of time is determined by se of the moving axes. Ae eae of determining the positions of the axes Oxy, Oy,, and Oz,, with reference to the (fixed in space) axes Ox, Oy, Oz, three auxiliary angles are ee If we suppose the moving plane of 2, y,, at the instant considers: to intersect the fixed plane of ay in the line NN’ and call the angle eON=y, and the angle between the planes xy and «, 7, (or the angle 40%); and the angle NOx,=$¢ (in the figure these angles are supposed acute at the instant taken), these three angles idetermine the positions of the axes Ox,, Oy,, Oz (and hence of the body) at any instant, and will themselves be functions of the time. and the rotary velocities about the axes of 215 Yy, and z, may be expressed in terms of them and of their differential coefficients. When a body is a solid of revolution, revolving with an angular velocity , about its axis of figure, and acted upon by the accelerating force of gravity (the fixed point O being in the axis of figure), the general equations of rotary motion (by processes fully developed in the paper referred to)! take the form sin? ote mae (cos 0—cos w) 1 i ~epdy , di? 2Mgy ae sin? 6 de + dz= a - (cos 0—Cos w) dp = ndt+-cos 6dy In which M is the mass of the body. A its moment of inertia about an equatorial axis through O. OR ar ob “its axis of figure. g the force of gravity. y the distance OG from centre of gravity to the point of support. @ the initial value of 6, or its value at the instant when the body has no other motion than the rotation m about its axis of figure. Eliminating aN between the first two equations (1), and putting ¢ A C*n? 262 2. —-=A, and —__ — “¥ wea Wy anc DAG a we get 3 sin? 2 _ 29 [sin’ §—2,3* (cos §—cos «)] (cos 0—Ccos «) : ; ale eS oC aie and the first equation (1) becomes 4, in? V9 Ig — sin 7 A (cos 6—cos w) Cn |r Cn The quantity B= Ape UE, 5 “AY } yt 24NGg 27 A Myy may be very great in consequence of the rotary velocity, , being great, or ( being small) in consequence of the ratio * The analysis therein used was mostly taken from P oisson as far as equations (of that paper) (9), (10), (11), corresponding to (7), (8), (9) of this ; but the subsequent developments were original. RESULTING FROM THE THEORY OF THE GYROSCOPE. g being very great. In the pl f the gyroscope, the firs li Dy/ Alyy’ eing very great. In the phenomena of the gyroscope, the first condi- tion obtains ; in the case of the earth, attracted by the sun or moon (x being small), it is easy to show that the alternative condition is fulfilled." 3 Putting a equal to zero in equation (3) we get =o for the maximum of 6, and a for the minimum, the equation, cos 6=—?-+ y/ 1+2)3? cos +34 in which, if 3 is very great, the value of cos 6 differs but slightly-from that of cos o. Hence by introducing a new variable uv, equal to «—, and deducing the values of dj and (by development) of sin? @ and cos @ (neglecting the higher powers of w) and substituting in (3) and (4), they become (omitting, as relatively small, cos © in the factor cos o +46), 5 a du , ry V 2usino—457u? TM _oA(9\3_& oe Uae) sin @ 1 F.quation (5) gives by integration and putting @ (Z)?=n. es . re sin w sin 7At which substituted in (6) gives 8, dy _1 ke sin *ht dt p? 1 Te es a: t= 95 kt— Te sin 2ht If we make ©=90°, sin o=1, in equation (6), deduce the value of dt, and sub- stitute in (5) we get, 0: C——— udu uw—U 23? the differential equation of the cycloid, generated by a circle of which the diameter : ‘ 7 iS =>5, and having a chord>- DG ¢ 5D 267 1 For the earth the moments of inertia, A and C, with reference to principal axes through the centre, differ very little. The value of 8 may therefore be approximately written Z| a and the denominator Mg is to be replaced (17) by a Substitute the value of Z (19) and put, for the sun, 2 —n; (25), sin 6 ti and the value of 6 becomes , Which is very large. io cos 0 The value as depending on the moon’s attraction is (28a), = | AD ., of the same order 2nwN3(C—A) cos 6 of magnitude as before. 4 PRECESSION OF THE EQUINOXES AND INU) AVE OBN If the value of w is not 90°, ~ ome ; j ees the diameter of this circle will be OE sinw; but the quantity ——, then measures an angle of an arc of a small circle having a radius =sin a; ane the chord of the curve is reduced in the same proportion as its sagitta, and the curve is still a cycloid. The axis of figure gyrates therefore about the vertical through Oas if it was attached to the circumference of a small circle of the minute diameter specified, the centre of which circles moves with a uniform horizontal velocity, which velocity is the mean rate of gyration; or the motion may be compared to that of a cone having its axis of figure constituting an vertex at O and diameter of bas t element of this cone) rolling upon a fixed conical surface, all the elements of which make, with the vertical, the angle w; but this imaginary cone ts not fixed in the body (save in the exceptional case of the moments of inertia A and C'being equal). For the rotary velocity of the body is , while that of the cone is 23 Since the rotary velocities of the body and cone are different, the instantaneous axis cannot move along the chord of the cycloid, nor with uniform velocity. The common methods of investigating the Precession of the Equinoxes, founded upon the incipient rate of motion of the instantancous axis, involve this error, which does not become apparent, simply because the moments of inertia A and Care, for the earth, so nearly equal. ‘The instantaneous axis will describe a prolate cycloid having the same chord as nm : 1 the common one (, -\ and a sagitta — —_sin w ( ai 23° 203? The mean rate of gyration is given by the coefficient of ¢ in equation (9); it is 1 P35} 11. ay Cn Thus far I have supposed that at the origin of time, or at the moment when the accelerating force commenced to act, the body had no other motion than a rotation, n, about its axis of figure. It remains to prove, that if, at this instant, there are small (compared to n) velocities about either or both the other principal axes, the rate of gyration will be the same. The solution will be perfectly general if we suppose at this instant a velocity, m, about the axis of x only, and assume at same moment I=, ~=90°, we should get, instead of equations (3) and (4), the two following — 2 in? 9 2 2My 2Cmn .. Orn? 112) sin? () Po 4 Y sin? @_~ oe sin o— ah (cost eens —m* (cos -+-cos 0) | (cos 0—cos @) 1 (Z)*, or substituting values (2) for @ and 2; Y. dy On = (cos 0—cos w)+m sin o sin? @ dt RESULTING FROM THE THEORY OF THE GYROSCOPE. 5 Substituting in these w—w for 0, rejecting all small quantities of the second order to} b) J Do (among which is m”), and introducing 3 and 4 (see equations 2 13. 2 Gi =u ( sin 023, |2mn) m)—43°u* Gg we 14. ay ey Ue ae oe aired sil w The integral of (13) (using / with its already given value) in (7) is 5 982 ed (sin om) sin” kt fy Ii, w— a3" Substitute in (14) and integrate x 1 1 m ‘ 16. ] = t— —.- Qhet ie 237 tie ( 467 2ksn =) te : \ arse : : . M, : The coefficient ¢ in (16) is identical with that of equation (9), = iL showing 'n that although the character of the gyratory motion is altered, and the axis of figure, instead of moving on a common cycloid (which forms cusps) and coming periodically to rest, moves along a prolate cycloid or even without undulation, yet the rate of gyration is unchanged. Y If o=90° and ie wu and _ become zero for all values of ¢, and the body gyrates horizontally without nutation.' In all that precedes, the revolving body has been supposed retained by a fixed point in its axis of figure, but not at its centre of gravity, while the accelerating force, being gravity itself, acts through that centre. If, instead, the fixed point by which the body is retained is the centre of gravity, and the accelerating or disturbing forces any other whatever (provided their direc- tion is invariable and their resultant acts through a fixed point of the axis), the 1 This is the case referred to in the preceding paragraph in which the moment of the accelerating force (or couple) is equal to that of (what I have styled in the work before referred to) the “deflect- Crm My That this case should arise, a determinate relation between m, n, and g, expressed by the equation mn My . ee =——=—-, is necessary. g When this relation exists, the movement may be represented by the rolling of a conical surface (the locus, in the body, of the instantaneous axis), described about the axis of figure with the angle ing force,” which has for its value the expression (approximately) equal to ae. upon another, all of the elements of which make, with the vertical, n Mgy On?’ ( the angle — when w is not 90°, the centrifugal as well as the deflecting force affects the relation between m, n, and g). But no such relation is essential to the gyration expressed by (11); and, in the case of the preces- sion of the equinoxes, the supposition of rolling cones is not realized. There are, probably, no two instants of time at which the precessional movements of the axis are identically the same. 6 PRECESSION OF T WE EQUINOXES AND NUTATION ly the same, the moments of inertia A and C, referring to ; centre of gravity, and Mg expressing the intensity of the distance from the centre of gravity of the point equations will be precise principal axes through the resultant of the forces, and y the through which the resultant acts. ; In Gepression (11) Mgy is the moment of the force with respect to the point O, divided by the sine of the angle (#) which its direction makes with the axis of ficure. Denote that moment by Z. ‘Then the expression for the velocity of gyra- tion (11) becomes, Pa. L ite Cn sin 0 If the body in question, like the earth, is acted upon by forces, the resultant of which does not pass through its centre of gravity, its movements about that centre are precisely the same as if that centre were fixed; in other words, it will gyrate about the line connecting its centre and the origin of the force with a velocity denoted by expression (11). In the case of the earth, however, the direction of the disturb- ing force and its moment are constantly changing, and I have to assume something not proved in what foregoes, viz., that the elementary gyration at each moment of time will be likewise expressed by (11); an assumption xot (probably) strictly true, since, when the forces are constant in direction and intensity, equation (14) shows (the value of w, equation (15) being substituted) that the gyratory velocity, though its mean is always expressed by (11), varies at cach instant unless the value of m has a certain relation to that of &. Since the integral of these varying elementary displacements shows, under all circumstances of constantly directed force (though these elementary motions of the axis exhibit all possible directions with regard to that of the force), a mean rate of gyration expressed by (11), we may assume that the fact will hold good though the direction and moment of the force change.! In the case of the earth there is probably no instant of time at which it is revolv- ing exactly about its axis of figure; the quantity m has, for it, in all cases, a finite (though exceedingly small) value; neither observation nor (scarcely) analysis can detect the minute diurnal (nearly) nutations which belong to the diurnal cycloidal movement; and hence the presumption that the gyration is at all instants perpendicu- lar, or nearly so, to the direction of the force, and hence that even its elementary values vary little from expression (11).? Such an assumption is made in all the Investigations not, like Laplace’s, purely analytical, with- out always giving the true grounds on which it should be based. * In reality, if the moment L remains the same for different values of" 6, the elementary displace- ment produced by the gyration is independent of 9, for, though the expression 2 varies inversely Cn sin 6 on which the displacement takes place increases in like Again, that a revolving body should gyrate around a given axis it is not necessary that the accelerating force should be always parallel in direction to that axis, but that it should remain in the moving plane through the axis of figure and the given axis. The general equations of rotation would be the same. ; as sin 6, yet the radius of the small circle proportion. RESULTING FROM THE THEORY OF THE GYROSCOPE. 7 Let a, be the equatorial diameter of the earth. 6, be the polar diameter. p, its variable density. C, its moment of inertia about the polar axis. A, its moment of inertia about an equatorial one. —b are cece —*~— earth’s ellipticity. S, the absolute attractive force of the sun, or its attraction upon a unit of mass at a unit’s distance. r, the mean distance of centres of sun and earth. 2, y,z being rectangular co-ordinates of any element of the earth’s mass, dm; the origin being the earth’s centre, the axis of z the polar one, of # an equatorial one in a plane passing through the sun’s centre, of y an equatorial one per- pendicular to this plane. The moment of the sun’s attractive force upon the earth is shown in various works on precession (vide Mr. Airy’s “ Figure of the Earth,” Encyc. Metropolitana) to be (@ being the angle of earth’s axis with line drawn to sun). 18. e SSf (a’—z") dx dy dz sin 0) cos 0 the integral being taken through the spheroid. The quantity under the signs of integration may be written p (x+-y") dm—p (y’+-2") dm, the integral of the first term of which is C, and of the second A. Hence the moment of the sun’s force (18) 19. (CA) sin § cos (=L Hence the gyration produced upon the earth by the sun’s force avout the line of its direction is (17) 20. 3S C-A cos #; and in time df, Pio GO 38 C—A Die = ie cos: dé Fig. 2. Let EST be a great circle in the plane of the ecliptic, EE’ an equatorial one, PE'TP’ a great circle through the tropics, PSP’ one through the sun in any position, C the centre of the earth, and PCP’ its axis. SCP’ is the angle @. If the sun moves in the ecliptic from £ (the equinox) towards 7’ with an angular velocity m,, mt will be the value of the are BS. In the spherical triangle P’ST, right-angled at 7, we have cos P'S (or cos 6)=cos P’T cos TS=sin TE’ sin SE. TE’ is the inclination of the equator to the ecliptic ; call this Z Then cos #=sin J sin nt 8 PRECESSION OF THE EQUINOXES AND NUTATION The elementary gyration about the line SC will be therefore, (21) O( sae / : 22 ae es sin I sin nt de. 3 nr CU If this rotation about SC is decomposed into components about the lines 7C' and EC, they will be } 23 3» : mc sin I sin? nt dt. ni } 3S CA |. . 2 acts S ~ sin [sin nt cos nyt dt. 24. zy 1 pe G. The component (23) represents a rotation of the pole about TC, the radius of its motion being PB, or cos £ To obtain the actual value as an are of a great circle, of this minute displacement, it must be multiplied by cos 7; and to refer this to the pole of the ecliptic as angular motion, it must be divided by sin J” Performing these operations, integrating, and remembering that by Kepler’s laws S 47 —- — Die ( 7 —number of units of time in one year) 5 we get 3 for prece SSI mn 3 7 2 > r 2 fy: 3n~ C—A 3n,C—A : 25. Jey Ce cos Lj ——— = === cos simeznit 2 VC 4 n And for nutation . 3 n, C—A .. 26. et = ssi COS 2730. 4n C oa DiNee 5 ‘ Dt Se Che first term of (25) is the mean solar precession; making ¢ —~~, it gives for the ny annual solar precession = OA 27. 3 2 — cos fi n C Expression (26) is the solar nutation, and the ‘second term of (25) gives the equation of the equinoxes in longitude, or the fluctuating term of the precession cor- responding to the nutation. ‘These expressions correspond to those obtained by the ordinary solutions. They differ from most of them, however, in having Cin the denominator instead of A, an error of those solutions I have alluded to before, which, however real, analyti- cally, exerts no important influence on the result. By the above method the precession is the integral of the components of gyration about a solstitial diameter of the ecliptic, which line itself, by the pro- cess of precession, has an angular motion equal to that precession, the real effect Pp ae ae | In the spherical triangle PP’’P in which P’” is the pole of the ecliptic SSB and PP an are of a great circle through which the pole P has moved (equal to (23) x cos J), and elementary precession) see sin I the sides P/’P are = J, the angle PP’/’/P (or the RESULTING FROM THE THEORY OF THE GYROSCOPR. 9 being a revolution about an axis (the pole of the ecliptic) perpendicular to the plane of that angular motion. In other words, if we integrate directly equation . ul (23) and make t— ae , and yan, we shall get ny 28. Balt et Se n C ; and this will be the total angular motion of the pole P about the solstitial line 7 in one revolution of the sua; but by this very motion of the pole the equinoxes have moved an angle measured by this displacement referred to the pole of the ecliptic —that is, by the angle expressed by (27)—and the solstitial line 7’C has of course, undergone the same movement, and the next annual gyration will be about the consecutive line 7” C, and so on; producing a continuous motion of the pole P about the pole of the ecliptic P”. To obtain the precession due to the moon, it is necessary to substitute in (19) for St POY in which Jf" is the attractive force of the moon and (7) its mean distance. rT io x 7 8 3 : eae 2n(7r)? 2a(r)? 27 But 7? (time of moon’s revolution) is, by Kepler’s laws, =. mid TS) Fr as (calling the mean angular velocity of the moon n, and the ratio of earth’s mass to that of moon’s mass, 7) (28a); hence AE eS (7? 1+-y If 7 is the inclination of the moon’s orbit to the equator during any one revolu- tion (regarded as constant for that time), we should obtain for the precession and nutation, referred to the pole of the moon’s orbit, expressions analogous to (25) and (26). Although the moon’s disturbing effect. as above expressed, is almost exactly double that of the sun, yet the larger divisor »,, introduced by integration, renders the value of (26) and of the fluctuating term of (25) very small for the moon—say about {th the corresponding values for the sun. Hence these terms are usually disregarded in the lunar expressions. The elementary precession due to the moon about the pole of its own orbit would be by (25) 99, 3 n. C=A : Soe ee COS TOE: 2n(l+y) C€ From this, by the usual methods, can be deduced the real precession and nuta- tion. But it will be more in harmony with the object of this paper, and indeed more elegant, to reduce the gyration produced by the moon directly to precession and nutation. If we substitute for »,, _ , and sin ¢ for sin J in (28) we shail get, for the 4 total gyration about the line of greatest declination, produced by one revolution of the moon in its orbit, the expression : 30. Ms Os Sadan = Sin 2. 2 December, 1871. 10 PRECESSION OF THE EQUINOXES AND NU TAD TON In which the line of greatest declination is regarded as stationary during the single volution, and taking a consecutive position for the next; but it will be in har- i 5 . . 2) fact, and allowable, to regard the line as in continuous motion and mony with the -oyration to be uniformly spread over the time : ge. of the revo- the above amount of gyration to be vas Nn lution, producing thus an elementary gyration, in the time dé, of : Bae C—A dl. 9 n(1-tn) i Let 82 be a great cirele in the plane of the ecliptic; 9” 2 the line of equinoxes, NON tlte line of moon’s nodes, */’4 the equator, and Nm'M WN’ the moon’s orbit crossing the equa- tor at m. ‘The line of the moon’s maximum declination, OJ/, will be 90° from the line Om’. The pole # of the earth is supposed to undergo a displacement by gyration about OJ/represented by HE’; the precession produced will be the angle HOLE’; the nutation, the angle Fa EH’. In the spherical triangle Nm’ the angle at NV is = /’, the inclination of moon’s orbit to ecliptic ; the angle at ¥ is the supplement of Z (inclination of the equator) and the angle at m’ is 7 (or the variable inclination of the moon’s orbit to the equator), and the side *N is =n, (calling the angular velocity of the moon’s node »,); therefore, sin 7 dt. 32. cos i = cos J’ cos J+ sin I’ sin I cos 1st and 33. tang m’r — Sia ee. In the spherical triangle mm’ B34. tang m* = cos J tang m'y OS is the line of maximum declination of the sun, or the solstitial diameter of the ecliptic about which the annual gyration produced by the sun is made. As the inclination I’ of the moon’s orbit is small, the are MM’, drawn through J, is approximately equal to m2, and the angle Os differs immaterially from the com- plement of me; hence by (33) and (34) 39. tang MIM’ — —_ cos Lsin net ___ _—os Fsin f’ sin nt =a sin Icot l’—cos cos nt sin cos ’—cos sin I’ cos Tg If the gyration ; Sis de sade , 1 gyration about OM (31) is decomposed into components about OM and O04, we shall have for the first (calling the coefficient of sin 7.dt RK) of sin 7.dé, 36. K sin i cos Mid, and for the second 7 K sini sin Midt. RE SU LN GE RIO M LH TH HORRY OFTHE GYROSCOPE: ~ Ii] To refer the displacement expressed by (36) to the pole of the ecliptic O, as angular motion, we must (see note page 8) multiply by cos M/’#' or (from which it differs but slightly) cos 7 and divide by sin /; we thus obtain for the elementary precession 38. as cos JJM’ sin i cos 7 dt, sin and for nutation 39. Ksin MM sin i dt. The maximum value of the arc WI’ is about 11° 56’; the line WO describing during an entire revolution of the moon’s nodes an elliptical cone about OS of which the minor semi-diameter (SA/’) is 5° 84’ (about), and the semi-major 11° 56’, By conceiving the elementary motion of the pole of the earth (or its gyration) as at each instant about the line J/O, as it makes its conical revolution, the undulating nature of that motion, or the “‘nutation,” is easily conceived. Approximate values of sine and cosine of Jf may be determined from (35); which, substituted with those of ¢ (32) in (38) and (39), will enable us to integrate and obtain very accurate expressions for the lunar precession and nutation.” But these expressions, nearly free from errors of approximation, may be more elegantly determined as follows: When the angle of the moon’s orbit with the equator is minimum, the angle i=/—/’; when maximum i=/-+T' (epochs corresponding to ng=0 and n,f—=7); the angle MM’ is zero, and the corresponding rates of pre- cession are by (38) on OY LF (a) jee 2d L’) for nt=0. 2 sin ae : (0) j quia a) for n,t=z. 2 sin When n,t=$z2, we have sao TAY cs eg) ee V 1—cos? I cos? I’ V cos? Ficos? Lf cos WM’ F=cos I cos? I’, cos i=cos J cos I’, sin i=V 1—cos? I cos? I’ (the three first being the residuals of exact analytical expressions after omission of quantities of inappreciable magnitude). Hence, by (38), the rate of precession for n,f=$7 is (c) K cos I cos? I’ * : Assume the formula for precession to be K(Pt+ P’ sin njt+P” sin 2n;t); * When the mocn’s orbit intersects the ecliptic in a solstitial line, the elementary precession K cos 2 (29) about its own pole is reduced, with but slight error, to the same about the ecliptic pole, by simple multiplication by cos 7’: a result coinciding with the above. ® See Additional Notes, p. 51. 12 PRECESSION OF THE EQUINOXES AND NUTATION the rates of precession will be (a’) K (P+ 1,P!+-2nsP") for n= 0 (0) K(P— n,P'+2n,P") “ ni 2 (¢) K (P—2n,P") “« nt—tn Equating (a) to (a’), &c., we deduce P=} cos I (cos’ I'+-cos 21’)=cos Gee sin’ J’) sin 21’ cos 2 1 . 1 sin2/’cos2l__E oos Tsin 27 cot QE Qn. a1bs P=— : sin [ Ns P= 2 cos I (cos ’—cos WL)=— 3%, cos J sin? 77 Ns Hence the formula for precession may be written 44. amet ae Ose cos iaa2 sin*/’) — sin 2/' cot 2/sin né 2n(l+y) C 2 Ns fae sin’ J’ sin 2n,t i 4n., Similarly we would get for the nutation € 2 Y te 45. 3 te C= A cos 7 L COS Nt. 2n(1+7) Ns The ratio of actual Imnar precession to what it would be were the moon’s orbit in the ecliptic, is therefore expressed by es sin? J’=0.99 (very nearly)." The third term of (44) indicates a slight periodical variation from the true elliptic motion referred to in the next paragraph. There should be a corresponding term in (45) which may be obtained by the same process, but they are both too minute to enter into computations. * It is worthy of remark that the formule of Laplace [3100] and [3101] (Bowditch) contain no such coefficient qualifying the mean lunar precession, though one is found in all the more popular solutions ; neither do they contain the term (quite minute) in 2n,t of (44), but, on the other hand contain terms in 2n,v (corresponding to the terms in 2n,f of 25 and 26) Gin referred to in the fourth par. (page 9), are generally omitted as inappreciable. RESULTING FROM THE THEORY @F THE GYROSCOPE. 13 If we multiply the coefficient of sin n,f in (44) by sin J, we shall have the value as an arc of a great circle of this fluctuating displacement called the equation of the equinoxes in longitude. The coefficient thus modified will represent the minor semi- diameter, and that of the nutation proper (45), the major semi-diameter of the ellipse of nutation; they have the ratio cos 2/: cos J nearly. ‘This ellipse has its major axis (equal to about 18” of arc) directed towards the pole of the ecliptic. The period of its description on is that of the revolution of the moon’s nodes. At Ns the same time a minute ellipse of semi-annual nutation due to the sun is super- imposed upon this. It has its longer axis (about one second of arc) likewise directed to the pole of the ecliptic. ‘The smaller axis is to the major as cos 7: 1. It is easy to show that the precession caused by the sun and moon is equal (with slight difference due to the ratio we have just been considering) to what it would be if those bodies were uniformly distributed in solid rings over circles (in the plane of the ecliptic) about the centre of the earth, having radii equal to their mean dis- tances. In this case, unless there was a particular relation between the couple producing: the initial rotation of the earth and that arising from the attraction of the two rings, there would be an extremely minute nutation, of which the period 22 in A would be (see equation 9) =n O ; which is almost identical with the siderial 2k 4 5 aor : aay i the ratio being a : 1. If we suppose the primitive rotation of the earth n U to be that alone, about its axis of figure m, then the nutation will exhibit the com- mon cycloidal motion of equation (10); but its total amount would be but about zi; second of arc. An explanation of the deviation of rifled projectiles will be found in what is said (pages 8 and 9) in reference to the conversion of gyration about a shifting axis into precession about an axis perpendicular to the plane of motion of the first. Elongated rifled projectiles, while they maintain almost unaltered their “angle of elevation,” are found to deviate with great uniformity from the vertical plane of projection, in a direction corresponding to the twist of the gun, while spherical projectiles (fired from rifled guns) having precisely the same rotary motion, do not so deviate; showing that the cause of the phenomenon is something else than the direct action of friction or pressure of the air. In the remarks made in the paragraphs just referred to, it is explained how the consecutive small annual gyrations about the line from the centre of the earth to the sun (in the tropics) become, in their integral, the movement which we call precession, about an axis perpendicular to the plane of motion of that line, inasmuch as each small primary gyration causes a corresponding shifting of the line about which it takes place. Something very similar occurs to produce the deviation of elongated projectiles. In issuing from the gun, the resultant of the atmospheric resistance (denoted by the arrow /2) coincides with the axis of the projectile, and it has no other effect than to retard the motion of translation; but the action of gravity causes the trajectory to curve downwards, and the direction of the atmo- spheric resistance becomes oblique to the axis of the projectile (at @), and (in 14 PRECESSION OF THE EQUINOXES AND NU ACORN almost all forms of projectiles) passes above, not through, the centre of gravity We have then the essential conditions of gyration, viz., a solid of revolution q. “ A . 2 9 1 > > 1 sx) 16 cd rapidly about its axis, and a dynamic “couple” (7. e., the inertia of the revolving projectile’s motion of translation acting through its centre of gravity, and the resistance of the air acting through a point of the axis more er less distant from q) tending to turn the projectile upwards about a horizontal (or “ equatorial”) axis through g, and there is in fact, at each instant, an elementary gyration about a. line through yg, parallel to R (the atmospheric resistance). If this line retained an invariable direction, the integral effect of these elementary gyrations would be to revolve down the axis of the projectile, and we should ultimately find it assuming horizontal and even sub-horizontal directions. But such cannot be the case; the direction of the axis is no sooner deviated, laterally, from its original direction, than a (nearly) corresponding change takes place in the direction of the resistance 2 (since from the elongated form of the projectile, the direction of its motion follows pretty nearly that of its axis) and in that of the line (parallel to &) about which gyration takes place. ‘The integral of such a series of elementary gyrations, accom- panied by a corresponding horizontal angular motion of the line about which they take place, is angular motion about a line perpendicular to the plane in which that line shifts direction, that is, about a vertical. Hence the vertical direction (or “elevation”’) of the axis of the projectile remains constant, or nearly so, while its horizontal direction undergoes a progressive angular precession (if I may so term it), and the deviation of rifled projectiles is thus seen to have analogy with the pre- cession of the equinoxes.} In what precedes I do not profess to throw new light on a subject so thoroughly studied as the Precession of the Equinoxes; my object has been rather to make evident the analogy that exists between “the minute motions of the gyroscope and that grand phenomenon exhibited in the heavens,” and to show how a common analysis applies to both. * It is quite probable that there are other causes of deviation, the friction of the air being (in case of long ranges) one. Experimental facts are needed for a full discussion of this subject. ON THE MOTIONS OF FREELY SUSPENDED AND GYROSCOPIC PENDULUMS, AND ON THE PENDULUM AND GYROSCOPE AS EXHIBITING THE ROTATION OF THE EARTH. Ler the point of suspension of the pendulum be taken as the origin of rectangu- lar co-ordinates, the axis of z vertical (downwards), and those of a and y in the plane of the horizon, the former directed to the east, the latter to the north. The forces which act on the pendulum (considered as concentrated in its centre of oscillation) are Ist. Gravity, the resultant of the earth’s attraction, and of the centrifugal force of its rotation. 2d. The tension of the pendulum cord. 3d. The force of inertia, the components of which are represented by the differ- ER di di Vat 4th. The disturbing forces arising from the earth’s rotation. 5th. The resistance of the air. If we represent the length of the string by /, its tension by N, and the force of ential cocffiicient gravity by g, and neglect the forces named in the 4th and 5th categories, we shall have the three equations: aa Nx ee ie Ny 1 OY AA © dt l dz Nz ls eee a The forces due to the earth’s rotation which disturb the relative motions of a projectile, or any material particle moving near the earth’s surface, have been expressed by Poisson (Journal de l’Ecole Polytechnique Cahier, 26) as follows :—" 1 These expressions are perfectly general and applicable to a// problems which involve relative motions near the earth’s surface, whether of solids or fluids. They (or their equivalent) appear in the work of Laplace (Mee. Cel., Vol. IV.), as well as of Poisson (from whom I have quoted them), in the investigations of the motions of projectiles. It is somewhat extraordinary that both of these great analysts should have failed to perceive their remarkable application in the motions of the gyroscope and pendulum, to the exhibition of the earth’s rotation, although the latter has seemed to desire such an exhibition in the sentence: “Quoique la rotation de la terre soit maintenant établie (15 ) THE PENDULUM AND GYROSCOPE 16 x=2n(4Y sin ee COs A ) (2) Y =—2n = sin 2 Zo =—2n = COs A. In which 2 is the latitude and n the angular velocity of rotation of the earth, These analytical expressions make their appearance in the transformations of the equations of motion, near the earth’s surface, expressed in co-ordinates referring to fixed axes, into others referring to the moving axes which are used in this analysis. But these forces can be obtained and their origin better understood by the following considerations. The centrifugal force of a material point at rest on the earth’s surface at the 9 9 9 . . . nr COS A : . . given latitude will be “ GAcOae (r being the carth’s radius), If it has a small 7 COSA 1: 2 (n 7 COS a+e daz . i relative velocity ,, to the east, the centrifugal force will become ——WH— * dt 2 7 COS A aie a Subtracting the former expression from the latter (omitting Fi 7) we get for the centrifugal force arising from the relative velocity i the expression ane ay Ae ; : : 5 aha ae ‘The component of this in the direction of the axis of y will be —2n sin 47,, which corresponds to the value of Y (equations 2) of Poisson, and is to be added to the second member of the second of equations (1). The component of the force just calculated in the direction of the axis of Z is —2n cos pl dt This force corresponding to Poisson’s value of Z is to be added to the second member of the third of equations (1). A body moving on a meridian of the earth’s surface from south to north will have the moment of its quantity of motion, with reference to the earth’s axis, diminished ; in virtue of which it will press with a certain force towards the east avec toute la certitude que les sciences physiques comportent, cependant une preuve directe de ce phénoméne doit interesser les géoméetres ct les astronomes.” The former, in making a partial appli- eation to the pendulum, of his investigations, absorbed, apparently, in the single object of proving that the accuracy of the instrument as a measure of time was not affected, has inadvertently assumed that the disturbing force normal to the plane of oscillation is “trop petite pour écarter sensiblement le pendule de son plan et avoir aucune influence appreciable sur son mouvement.” (Journal de I’Ecole Poly. Cahier, 26, p. 24.) It is true, indeed, that the force he mentions, even if permitted free action, will have but an inappreciable influence upon the time, and none whatever when, as in the chronometer, the plane is constrained to fixedness ; but the effect is ewmulative in changing the azimuth of the freely suspended pendulum. These same disturbing forces, introduced along with the attractions of the sun, moon, and earth, into the general equations of equilibrium of fluids, produce in a very simple manner the differential equations for the tidal motions. (Vide American Journal of Science, 1860.) AS EXHIBITING THE ROTATIONS OF THE EARTH. 17 and the moment of the force thereby developed is equal to ke (the moment of its quantity of motion). This moment for the pendulum in any latitude 2, is nr? cos? A, of which the differential coefficient, taken with reference to 2 as a function of ¢, is —2nr? sin 2, cos eS which is the moment of the force required. Dividing by the d}. radius of rotation, 7 cos 2, we have +-2nr sin 2- dt for the expression of the force which (acting positively in the direction of the axis of «) is to be taken with the plus sign. In the system of rectangular co-ordinates which I am using corresponds to the = € : did aon on : : velocity expressed by r dt? and substituting it therefor, we have for a disturbing force in the direction of the axis of x the expression 2n sin aly dt In almost precisely the same way it may be shown that a body falling towards the centre of the earth with a velocity wi will have the moment of its quantity of dt ee RCS ee motion diminished by 2n7 cos” Aq Siving rise to the force _dz 2n cos A ; “dU The sum of these two expressions constitutes the disturbing force Y of Poisson, and is to be added to the second member of the first of equations (1), and these equations become! ie Nv dz t= —— SiN ie TA9, cos Aa ay , Ny ads (3) apr | =— 2n sin ATE G2) Nz Tx aE + yp aes 22 Cos Ae 1 There are really other disturbing forees (comparatively slight indeed) than the X Y and 7 of Poisson (equation 2), as appears from the following considerations :— Draw a line through the origin of co-ordinates parallel to the axis of the earth, and project the moving body on the plane of yz. The distance of the projection from the line will be y sin a+z cosa, the distance of the body from the plane of yz being 2: hence there will be a centrifugal force relatively to this line, due to the earth’s rotation, tending to increase the ordinates # y z by its components n? x n? sin a (y sin a-+z cos a) n? cosa (y Sin a+z cos a) With these expressions added, respectively, to the second members of equations (3), they correspond to those found in Carmichael (Calcul. of Operations), who quotes from Galbraith and Houghton (Proc. R. Trish Acad., 1851). They express forces of the second order in minuteness, compared with those expressed by equations (2), and, insensible in their effects, are neglected in all discussions. They are noticed here only to recognize their existence and to show their origin, 3 January, 1872. 18 THE PENDULUM AND GYROSCOPE Since a?+y?+2=?, we have x dx+-y dy+zdz=O. Hence multiplying equa- tions (3), respectively, by dx, dy, and dz, and adding, we have ee ae at de dy--dz._. (3); Cue sa eo pl This expression is independent of n, and the velocity at any point of the path depends, in the same way as does that of a pendulum vibrating over a motionless earth, wpon the height of fall. ‘The plane of vibration of the chronometer pendu- lum is maintained in a fixed relative position, thereby differing from a “freely sus- pended” pendulum. It will be seen hereafter, in treating of the gyroscope pendu- lum, that the forces which maintain this relative fixedness are equivalent to a force varying directly as the angular velocity, applied at the centre of gravity, normally to the path. Such a force will have no influence upon the velocity. Hence the time of vibration of the chronometer pendulum is not affected by the earth’s rota- tion, nor by the azimuth angle of the plane of vibration.’ Multiplying the first of equations (3) by y, and the second by a, and adding, we get :— x CY dy. de dz ee FE ay) Ep) 2n Cos Ay == Yae "ae" (y i a )+ Vat Integrating :— (4) yea =n sin A (a?+7?)+ C+2n cos a fyaz The above (4) expresses that the moment of the quantity of motion about the axis of z is equal to a constant C (depending upon any arbitrarily given initial value) increased by what is due to the constant angular motion m sin 4, and by the area 24 ydz (im the case of ordinary plane vibration this is the projection on * This conclusion is not invalidated by the introduction of the disturbing forces of the order n? referred to in note to p. 17, for, since the are of vibration of the chronometer pendulum is exceedingly small, z may be considered as equal to J, the pendulum’s length, and.y as very minute. Those forces will thence be nx 3 nl sin Qa n* Ll cos? a The third of these is an increment to gravity, and the first tends to prolong vibrations in the prime vertical. The second is null in its effects, since, being always positive, it retards the vibration in one direction as much as it accelerates it in the other. But they are all ‘nappreciably minute, the last being, for the seconds pendulum, an increment to the force of gravity at the equator of about 1 1800.000 000” decreasing the time of vibration by about oo The first has the con- ade in a prime vertical (or any other plane ciable even when (as in the prime vertical) it is a trary tendency to increase the time of vibrations if m than a meridian), but its effect is equally inappre maximum, AS EXHIBITING THE ROTATIONOF THE EARTH. 19 the plane of yz of the circular segment included between the are and chord of vibration), multiplied by x cos 2. ‘This multiplier, which is the component of the earth’s rotation about a diameter of the earth normal to that passing through the locality, indicates that the term 2n cos aS y dz expresses a disturbance produced by this complementary component. As this term for vibratory motion is small and periodic, passing through nearly equal positive and negative values in the course of a double vibration, it follows that » sin 2 expresses the mean increment of angu- mortion, or, in other words, that, to the plane or spherical vibrations exhibited by the pendulum over a motionless earth, there is, superadded, in consequence of this rotation, a uniform azimuthal motion measured by the earth’s rotating velocity mul- tiplied by the sine of the latitude. This is the material fact or peculiar feature of the freely suspended pendulum, and we see that it is exhibited by equation (4) generally for all ordinary vibrations, whether plane or spherical. We shall see hereafter, however, that the disturbing term of equation (4) 2n cos afy dz expresses a tendency to a like motion about the complementary axis, and that, on the sup- position of an infinite velocity, this tendency may be realized, and the plane of motion, by the joint effect of the two components, turn around a parallel to the earth’s axis, with an angular velocity equal and contrary in direction to n. In the case of very small deviations from the vertical, the equations (3) may be solved as follows: ‘The variations of z then become of the second order of minute- ness compared with those of a and y, and omitting them we have between x and y and their differentials the relations ae Nx ~ AF F Get 7 2h sin 4g, =0 (: dy di N, . de +- 7 +2n sin ig =0 the integrals of which are (Gregory Examp. p. 390), < xc—=+% (D cos Gt-+-F£ sin (it) 2) y=— (D sin Bt—E cos Pt) in which to 2 is given both the values obtained from the quadratic equation Ay — Ay =a; B te in which = a,=—2n sin A, @—=1; hence solving the quadratic WG Bay Gat aty(4) Substituting the values of a,, &c., and omitting the second term under the radi- cal as inappreciably small, we have * This equation, = ?—= 2n sina B, can be got by substituting the integrals —C cos (Bl—e), y=C sin (sf—e) in the given equations. THE PENDULUM AND GYROSCOPE B=—n sin at 9 : e ; Representing these two values by 3, and /,, equations (5) may be put in the form (by writing for D, and E,, C, cos , and C, sin ¢,, &c., and reducing) a= O, cos (Git—a)+ C, cos (Axt—es) y=—C, sin (3,f—a,)— C, sin (G,t—e) Assuming for (=0, «=0 and ea it will give ¢,=«—=4 2, and the above € become gC, sin 3,t-+C, sin Bot y=C, cos Byt+-C, cos Bat Instead of the arbitrary constants C, and C, we may write }(A-+B) and 3(A—B), at the same time substituting the values of @, and 8, and developing; by which the preceding equations become (putting ” sin A=n') 6 —JAWCOS ae ésin wv (+B sin, |? i cos n't (6) as = Vea cos | t cos n' t—B sin |? tsin n't If we transfer the co-ordinates now referring to (relatively) fired axes, to others moving With the relative angular velocity 7’, that is, if we transfer to axes making at any instant the angle msinA¢ with the fixed ones, the new co-ordinates. will have the values a'=x cos n’t—y sin nit y =z sin n’'t+-y cos nit or, substituting values of a and y, x =B sin Ae t aj —YANGOS NE From which we may obtain AP of? 1 B? y?—= A? B? which is the equation of an ellipse, having A and B for semi-transverse and semi- conjugate axes. If B—0 the ellipse becomes a right line, hence the earth’s rota- tion causes an azimuthal motion of this line, or of the axes of the ellipse if the motion is elliptical, equal to the component of that rotation about the local axis and in the reverse direction. ‘The motions of the “gyroscope pendulum,” which is but the ordinary gyroscope with an exceedingly long arm (or distance y, of my analysis, from the point of ps in poe axis to the centre of gravity), are indicated by equations precisely simular to the above, deduced from an identical analysis - 1 i the solutions just given, that the ares of ee ce ae veri eee ] NSE eH ee EEN G LE ROWAT TORO Min BAR TH: pal may be disregarded. To prove this, I refer to my expression (1) and the context, in my analysis of the ‘“ Gyroscope.”* Cn («) yi & for the deflecting force, as I call it (a force due to the rotation of the disk with angular velocity n, and acting, at the centre of gravity, normally to the plane of angular motion of the disk-axis, or of the arm of the gyroscope pendulum), in which J is the mass, C its moment of inertia about the disk-axis, y the distance from its centre of gravity to point of suspension, and v, the angular velocity. Disregarding the vertical motions represented by a on account of the smallness KG 1 dy of the ares, oF ae ad — 7 GE al substitute 7 for the y mentioned above) would repre- sent very nearly the components of angular velocity of the centre of gravity. Substituting these for 7, we shall get the components of the “deflecting force,” and the equations of motion will be, a ewes _ Cn dy dt PM dt ay Cn dx ae ‘ pte ai a2 dt a These equations are identical in all pat the value of the coefficients with (3), when transformed to (3)., under the same license. Of course the motions of the gyroscope pendulum would have the same solutions, the mean azimuthal motion of Y, the nodes of its orbit being expressed by half the coefficient of a =e the moment of inertia A, of the gyroscope, with reference to a principal axis through the point of support, is (7 being supposed to be very large compared to the dimensions of the disk) very nearly ?J/, the mean azimuthal motion is more simply , or, since expressed by ae This may be more generally proved as follows: The first of the general differ- ential equations (equations 4 of my analysis) of gyroscopic motion is dy Cn dim sin? @ a os —e) in which 6, counted from the inferior vertical, is the variable inclination, and ¥ the azimuth angle of the disk-axis or pendulum arm, and ¢ a constant depending on dy initial values of @ and - i a 1 See American Journal of Science, 1857, and Barnard’s American Journal of Education, 1857. THE PENDULUM AND GYROSCOPE 22 Develop cos 6—=—(1—sin* 0)! and we get dy Cn ab pyar ame sin? O—2 sin* 0—&c.) dime 2A. sin’ 6 For ordinary ranges of pendulum vibration the terms involving positive powers of sin @ (which express an excess of nodal motion for large excursions) may be omitted, and we have dy Cn , Cn 1—e di 0 2A AS sin? 0 Thus we see that for small vibrations, whether spherical or plane, the azimuthal ion i : ifor (O2TeSSI f the nodes Cn and a fluctuating motion is made up of a uniform progression of the nodes oe ing term which represents the angular velocity in the orbit. Indeed we have, in the second term, the motions of the spherical pendulum. If we suppose the pendulum to have been propelled from a state of rest in the } . . . . vertical, d voust have a finite value when 0 is indefinitely small, and ¢ must hence dt , 2 ne Cn b : be unity. Hence we sce that at the very outset the initial value Ov must be attri- d buted to eo and that the pendulum reacquires it at every return excursion, that dt is, whenever @ diminishes indefinitely. Hence the pendulum continues to pass through the vertical at every return. The horizontal pro- jection of the curve would be a series of loops radiating from a common centre. For each complete vibration : Cn . the integral of oat would represent the entire angular o —__ a motion of the nodal axis (much exaggerated in the dia- me gram) from A to A’, &c,, and the integral of the remain- x He # ing terms should be 27. These loops are in fact but the path a pencil attached to a common pendulum would trace upon a paper beneath, turning with uniform angular velocity about the projection of the point of suspension. eee sae) becomes zero for the case just con- A sin’ 0) sidered, it is evident that, at the moment of passing through the vertical, the limiting value must be considered infinity, and that the integral through the infinitely short time of passage must be 2; for the azimuthal position undergoes, at that instant, an increment (or decrement) of a semi-circumference. There is an identical case in the spherical pendulum. Regarding plane as the final limit of narrowing spheri- cal vibrations, it is evident that the azimuthal velocity of passage by the vertical becomes very great and has its limit infinity when they pass through the vertical. Though the numerator of the fraction at Sa eee ey . , his expression 5 (equal to aj? nearly) is a very different thing from the 1 ; “mee ssion”’ ¢ avroscoe Ig Ten ; i 1ean precession” of the gyroscope, Q3a),7 given in my analysis, The latter is AUS HeXeHe Bil ISN (GH ER OVA ONO FE Tn) BARE. 23 the mean azimuthal motion of the body itself, the former that of the nodes of its orbit. ‘The latter is strictly true only for very great values of 8; the former, rightly interpreted, is always true, though it has no special applicability except for (as in the gyroscope pendulum) small values of 6. The harmony of the two expressions is easily shown. Practically the gyroscope is made up of not only a rotating disk, but a non- rotating frame. In estimating the “deflecting force,” therefore, in the expression (a) C should apply to the disk alone, and W/ and y to the entire mass of disk frame and stem. The solutions that have been given of equations (4), for the freely suspended pendulum are restricted to very small motions; the following is general. Transfer equations (4) to polar co-ordinates by substituting for x, y, z the values! x—l sin ¢} sin 6 y=l cos } sin é z=1 cos 0 in which @ denotes the azimuth of the pendulum measured from the north, and @ its deviation from the vertical, and we get & SCOR ens @ sin’ 6 d 6 sin? sin? @ | (7) Pn sin As . : : ; . 0 oA0 d in which C is a constant depending on arbitrary initial values of 7 the final term a corresponding to the last term of (4). At the equator we have ~=0, and the azimuthal velocity expressed by the third » 2 ae ; : eee term of (7) becomes —— 5d cos @ sin’ 6 d#, which being periodic, produces but sin very minute change in the plane of vibration. If the pendulum is propelled, from a state of rest in the vertical, in the direction measured by the angle @ from the meridian, this angle will be but very slightly affected by the minute values of the above expression during the outward excursion, and the increment Soa. Gl ; : ; ; 5 ; which - u receives will be almost exactly neutralized (quite so if cos @ were abso- a lutely invariable) during the return, and the angular velocity due to the term will again become zero; which cannot happen unless the pendulum again pass through the vertical on its return, in which case ¢ will be as little varied during the return ; (otherwise @ will, during the return, pass through all possible values from 0 to $7, and integration is impracticable). Hence we may assume ¢ as constant, and, as in any other latitude, the term in question is, multiplied by cos 2, the same as at the equator, we may generally integrate that term for plane vibrations, considering @ constant, and putting C—O. Equation 7 thus becomes, (8) on sin A—n cos jee Se! 1 The following analysis, as far as equation (9), is modified from Galbraith and Houghton. Proc. R. L. Acad. 24 THE PENDULUM AND GYROSCOPE If the amplitudes of the vibration are very minute, so that @ may be substituted for sin @ in (7), the integral becomes dp i 21 cos 2 Cos @ 6 (9) di =n sin A— 1 COS 2 p expressing the precession in azimuth, n sin A, precisely as it results from the former analysis (equation 6). The slight periodic disturbance expressed by the second focmaot (9) escaped that analysis, however, owing to the omission of the terms involving dz. In the above integrals the angle @ must be taken at 180° greater or less on one l saree side of the vertical than on the other, and the parts of expressed by it will have contrary signs. Hence the curve described in each complete excursion, disregard- ing the superadded uniform azimuthal motion expressed by the first term of the second member of (9), will have the form of an excessively attenuated leminiscate, or figure of 8.” For greater amplitudes equations (8) will apply until 6 becomes nearly equal to 180°; if @ equals or exceeds 180°, it cannot be assumed that the pendulum will pass through the zenith (the condition for @ to remain nearly constant), and the integral becomes inapplicable and erroneous. The foregoing integrals involve the condition that the pendulum shall pass through the vertical, and imply that vibration is induced by propulsion from a state of rest in the vertical. But, in the usual form of the experiment for exhibit- ing the rotation of the earth, the pendulum starts from a state of relative rest at the extremity of the initial vibratory arc. If we disregard the symbolic integral of (7), as may be done, since the minute periodic disturbance it measures has no influence upon the permanent azimuthal motion, that equation will become (a) doy, sin ii ee dt P sin’ 6 The second term of the second member is identically the equation of the “ spheri- cal pendulum.” ‘The latter, we know, exhibits an azimuthal motion of the apsides of its orbit, very minute when ( is small, but incomparably greater than the horary azimuthal motion when, C being large, the conjugate dimensions of the orbit approaches equality to the transverse, and of which the limit corresponding to per- fect equality of these dimensions is for one vibration, 2 ©, Een ees COS” Oy as may be deduced from the expression for U,, par. 731, Peirce, Analyt. Mech., or from expression [79] and [83] of Méc. Cél. (Bowditch), by making a=b and deter- mining the corresponding values of ¢ and dé, In the case under consideration the value of C will be determined by making 10 for dh > moti a ic commencement of motion, (i =").. hence Y > ° * 9 C=—n P sin 4 sin? 6, ” See Additional Notes, Parole ACS EGER DING) DHE RORATTOM OF THE HAR TH. 25 The pendulum will not move in a plane passing through the vertical, but on a conical surface differing slightly from such a plane, and there will ensue a slight apsidal motion reverse to, and diminishing, the apparent horary motion. (4) ‘The equation oP _ = ae is usually solved by the aid of elliptic in- tegrals (vide Prof. Peirce’s Analyt. Mech., p. 418); but for present objects the ordinary processes of integration are preferable. From equations (3), and (4) may easily be deduced, neglecting terms containing n, and bearing in mind that v+y=l—2, « dx-+-y dy=—z dz, (c) Bee an ldz ~~ £Y (P—#) (e+ 292z)—C# in which the upper or lower sign of the radical is to be taken according as dz is positive or negative, that is, as the pendulum is descending or ascending. ‘The quantity under the radical may be put in the form (vide Mec. Celeste, Bowditch, Viole pco2)). 29 (a—z) (z—5b) (+4) in which f _Ptab 20m ath > 2g (F—a’) C—O’) dl 2 @ Oe a+b 9 0B dV) a+b a and 4 being the greatest and least values of 2. If now we transfer the origin of co-ordinates to the lowest point of the spherical surface by substituting for z, a, and b, 1—u, /—a,l—(, Pe replace C' and dt in (0) by the values above found, we shall have (putting ¢+_> 29 =, we € =) (ES) ldu y ath u(2i—u) [(p—w) (u—a) (3B—u)}* the varying sign of the radical being understood. If we develop the two factors (2d—u)“ and (p—w)~™, and multiply the results, we shall have _ GS2G —b°) ldu i 1 ace a= aa u[—a, Hat u—u ail ay aipl ty ye i Ut 3 fale Ape ! a) oe Gta a 6p! tte] Strike out the common factor 7, and remove the factor 2p' into the denominator of the first radical factor (which factor then becomes V (—a) (Ib) =V a3), wd the above integral becomes (vide Hirsch, Integral ‘Tables, pp. 160-164), w riting U for —a 3+(a+3) u—w’, 4 January, 18732, do= THE PENDULUM AND GYROSCOPE 26 oft VO ee p= COs arapcea )45Vo8(; a V (a+8)'—4a8 (eae Gees aap + ava teVa8lety, 1 35? Je g A Japa (e) 3 1 1 See = Veet rot oy Tre u+4(e+8) )y T+ 1 x 2 U ; ? (2(0+8)— a/3 Joos = Vv —____ || &c.,--cons't. 5 2 V (a4(3)’—4a3 In order that the ares in the above expression should continually increase with the time, the positive or negative sign must be applied to the radical 4/ U accord- ing as w is increasing or diminishing: taken from w=a to u=£ (or the converse), the ares all become =z, and the non-circular functions vanish. Hence the azimuthal angle passed over by the pendulum in its motion from a lowest to a highest point of its orbit (or the converse) is expressed by oly elvale iy2 — ples St pe VICE Se as OD Sor Bate toys op) gas ) 1 7, 1. 8 15 , 105\5@-Cerseeeaa) ae + 6VA(q apt ane teyet aay) ee The sum of the terms after unity included in the brackets is the ratio by which the azimuth angle exceeds a quadrant; or, if the integral is taken through an entire revolution (relatively to the apsides), it, multiplied by 27, is angle of advance of the apsides per revolution. For motion nearly oscillatory, of whatever amplitude (¢.e., a being small and 3 arbitrarily large), or for spherical motions of considerable amplitude (a and 3 taken within limits not exceeding say one-third of /, corresponding to a swing of over 90°), p, always greater than 2/, differs but slightly from that magnitude. Giving p that value, and taking the angle @ for a complete vibration, or a semi-apsidal revolution, we have the formula, aA 3 af , 1.3.5 7/a8.4(a+6) q) oe ; ied fed SDS AY - li gy o=n[ 145 a +124 (uy? +&e. ] Ifa and 3 are both small and nearly equal, and @ the angle of which they are versed sine then VCs ats ; the versed sine, then J 7 =}$ sin’ 6, and the apsidal motion corresponding to the second term of the above (the following terms neglected) becomes 27 sin’ 0; agree- ing with the expression (a), on p. 24, when developed for the same case. If the pendulum moves nearly horizontally in a great circle, that is, if a+8=2/ A 72 sees : = : 3 . ares and a @=l (nearly), then p, C, and é are each infinitely great, and (f) becomes oe ee ieee ree Pim an 14 9 } { { 167 39 | &e. Jan AUS) HexXe Eo B PONG DHE ROWATLON OF THE EARTH. QT which denotes that the line of apsides moves through a quadrant while the pendu- lum is passing, through 180° azimuth, from a highest to a lowest point; in other words, that the highest and lowest points are diametrically opposite, and the apsides are apparently stationary. ‘This theorem is true (as shown by Prof. Peirce), what- ever be the inclination of the great circle, though it cannot be generally made evi- dent by the above formule. If in (c) we make transformations and substitutions already described, develop the. factor (p—u)~, integrate between the limits w=a, w=, and double the result, we shall have for the time of one vibration, or one semi-orbital revolution, (h) TP Hee + (aa) ) 1.3 ap aia ea (a+b p° 2.4 2p” 2.4.6 ie il 5 303 a 7 246 +&e.] and if a0, that is, if the motion is a oscillatory, then p=2/, and this becomes : ii g ©) Pant [1+0y art 3a) (gy) +8] and more generally for any value of « and 3 not exceeding the versed sine of thirty or forty degrees: PotD +0) a a4 Gy Hod) Mr) £8] When a and ( are both small, as in most pendulum experiments, the terms after the first in the brackets may be omitted, and we have the ordinary expression for the time z | Ds g In the expression (g), omitting all the terms in brackets after the second, it is evident that that term will measure the apsidal motion for the time 2 A L ; and hence a that the total integral of equation (a) taken through that time will be, [~ sin alt ee zs and if we denote by @’ the angle of azimuth of the apsidal line measured from its initial direction, we shall have, at any time ¢, substituting for a and 3, / (1—cos 6,) and 7 (1—cos @,), (x) g=[n sin A +1 [faces 6,) (1—cos 0.) | or, since @, is always small, k’) g=[n sin A the y ae! Le sin | The angle 9, being given, 6, is determined for the ordinary pendulum experiment by the consideration that the constant C, of which the value is found p. 24, is the moment of the quantity of motion. As ¢ is very minute, the actual velocity at the 28 THE PENDULUM AND GYROSCOPE 9,), and hence the moment at that point lowest point will be, sensibly, 1/ 2g/(1—cos sin 2, sin? §,, from which deducing the will be 7 sin 6, } 2ql (1—cos 6,)=C=—nP value of sin 0, and substituting in (/’), we have ‘ ; ou Bos (1) g=n sin (1—gsin 6.)¢ The second term in brackets is the retardation from the true horary motion; it being implied, of course, that the amplitude of oscillation is preserved unimpaired, ‘Though-small, this retardation would be sensible, especially if 6, had a considerable magnitude, say eight or ten degrees, though inappreciable for very small oscillations. Practically, the resistance of the air constantly diminishes the value of 6,, and failure to procure a perfect state of rest to the pendulum before it is set free, or currents of air, may give quite different values to C’ and ec, and determine the cha- racter of the orbital motion to be progressive instead of retrograde; and it is gene- rally observed that the conjugate dimension of the orbit increases (probably owing to the resistance of the air) as @, diminishes. In this way the apsidal motion due to the orbit may acquire a value quite considerable compared to the proper horary motion, which will apparently be sensibly retarded or accelerated. By observing, at any period of the experiment, the value of 6, and @, and the direction of the orbital motion, the coefficient of ¢ in the formula (A’) will give the theoretical rate of azimuthal motion at that instant. If the orbital motion is retrograde and reer ae Dy Se A | 3 g(1—cos 6.) the line of the apsides would be stationary. For Columbia College, where n sin 4=00004747, with a pendulum of 26 feet in length, and a value of 6, of 6°, this would give §,—3/,3, the actual semi-axes of the projection of the orbit being about two feet nine inches and one-third of an inch. It would generally te sufficiently accurate to substitute for 1—cos 6,, 3 sin? 6,, by which formula (4) would become more simply, (x) g'=[n sin 2+ ae sin 6, sin , | (m) sin 0,— in which it is seen that the deviation from the proper horary motion is proportional to the area of the projection of the orbit. The above, or (4), expresses the azimuthal motion as it would be were there no other forces acting than those included in the investigation, in which case 6, and @, would be invariable. In point of fact there are practically numerous disturbing forces, of which, however, the resistance of the air is the most considerable, and through which 6, and (), are incessantly changing, and it is not improbable that the change of shape of the orbit may, in itself, cause some variation of direction of the line of apsides:! a matter which cannot be decided until the problem of the spheri- cal pendulum is solved with the resistance taken into account. ; om er wees : An investigation of the simpler case of the plane elliptical motion of « central force proportional to the distance, in thee aiceeity soa a medium which resists either directly or as the square AS ESXSHEB EDOING DHE ROTA TION, OF THE! EAR TH. 29 To get a clearer idea of what is expressed by the periodic term of (7), —2n cos % JS cos g sin° 0d@ (which corresponds to the integral f y dz of (4)), we must revert to the latter equation. Conceive the pendulum propelled from a state of rest in the vertical, with a very great angular velocity, denoted by v, in the plane of the meridian. Were the earth motionless, it would continue to whirl in this plane, passing through the zenith at every revolution. Introduce the element of the earth’s rotation, and the two terms of equation (4) containing n take effect, by the first of which the plane of revolution moves in azimuth with the angular velocity nsin 2. ‘The second expressing that there will be an increase of the moment of the quantity of motion about the vertical after a time 7’ proportionate to the area generated in that time fy dz. Under these conditions this area is cumulative, and at the end of one revolution expresses the area of the circle of radius 7. Let us suppose that the plane of motion turns about a line parallel to the complementary terrestrial axis with an angular velocity n cos 2. At the end of the time 7’ (sup- posed very small) the plane will make with the meridian the angle n cos 27, and as the quantity of motion in its own plane is v/, its moment referred to a vertical axis will, from zero, have become v/’ sin (” cos 27’), or, substituting the small are for its sine, of n cos AT stadt Qn But 7, for one revolution, is expressed by = hence the above becomes U 2n cos Ax 0 The area of the circle which is generated in the same time is 77” and is expressed by the integral if y dz, and it is easy to show for each successive revolution that the area fy dz multiplied by 2n cos 2 corresponds to an increment of the moment of quantity of motion about the vertical which it would receive from a turning of the plane about the complementary axis through the angle ncos 2 ¢. Hence, for the particular case under consideration, the second term of second member of equation 4 expresses an angular motion about the complementary axis of which x cos 4 is the velocity. The resultant of this, and the azimuthal com- ponent, is rotation about an axis parallel to that of the earth, and opposite in direction to the earth’s rotation. The above theorem can be analytically demonstrated. ‘The quantity N, expres- sive of the tension of the cord, is made up of the centrifugal force due to the pen- dulum’s relative angular motion and of the variable component of the force of gravity (neglecting, as we have done, quantities of the order n*). If this centri- : : Nee fugal force is so great that the component of gravity may be neglected, an will of the velocity, indicates no apsidal motion accompanying the decrease of parameters of the orbit. Neither, however, does it indicate the enlargement of the minor axis ‘initially very small) so uni- versally observed in tne pendulum experiments. 30 THE PENDULUM AND GYROSCOPE (~ being the impressed angular velocity of pendulum reduce to the constant v dar end and third of equations (3) will yield the equation movement), and the second wags cos A—z sin 4)=—v" (y cos A—z sin A)—g sim 2 dt? the integral of which is : g sin A — < y cos A—z sin Nes - “1 A cos vt+-B sin vt The above is independent of n; y cos A—z sin 2 is the value of the new co-ordi- nate of y/ when the axis of y is changed to parallelism to that of the earth. Hence, as thus transformed, this co-ordinate is unaffected by the earth’s rotation, the plane of pendulum motion must turn (if it turns at all) about such an axis. : di 1 p The assumption for ‘=0, of 7=0; z=, opal, =: gives A= a= 7,)sin my 2 5 —lnCOSen (10) hence y cos A—z sin A=— J sin A (1—cos vt) sin (vt—A) a The second term of the second member of the above gives precisely the value which the first member would have, were the plane of pendulum motion stationary or were it turning with any angular velocity about an axis parallel to the earth’s. The first term is, owing to the assumed high value of v, very minute, and is periodic, the period being equal to ~~, the time of the pendulum’s revolution in its circular a orbit. Owing to its minuteness and periodicity, this term may be neglected, and equation (10) becomes y COS A—z sin A= sin (vt—A) to satisfy which, and at the same time the three differential equations (3) (omitting g, as we have found reason to do in (10), and also omitting terms containing ,, in the developments), requires the following values for the co-ordinates :— xl sin nt cos (vt—A) y=! [cos % sin (vt—A)-+sin A cos nt cos (vt—aA)] z—=I [—sin A sin (vt—A)-+-cos A cos nt cos (vt—A)] Changing the axes of y and z by turning them through the angle 2, we should have for new co-ordinates, a’=/ sin nt cos (vwt—a) yl sin (vt—A) Z—I cos nt cos (vt—2) If we now change the plane of yz by moving it through the variable angle né about the axis of 7’, we get, a a |) sin (wt—2) 2—I cos (wt—’) yt AS EX HEB IDLING THE ROTATIONSOFR THE BAR TH. 31 These values show that the plane of pendulum revolution turns about an axis parallel to the earth’s with the relative angular velocity n; or, in other words, that the plane preserves its parallelism to itself in space. If we had a succession of pendulums rigidly connected with each other, the dis- turbing effect of gravity would be eliminated. Such a succession would be simply a gyroscope, and the gyroscope, mounted in gimbals and set running in a meridian plane, would exhibit the apparent rotation of its disk around an axis parallel to the earth’s axis equal and contrary to that cf the earth; which is simply saying that as the earth revolves the plane of the disk maintains its parallelism to itself, and if we suppose its axis directed at a star in the plane of the equator, it would follow that star so long as the rotation of the disk is sustained.* The pendulum experiment, in its ordinary form, exhibits not the whole rotation of the earth, but only one component of it; the component which belongs to an axis passing through the locality. It is perhaps quite as interesting and important, as being the only experimental demonstration we can have of a principle difficult of comprehension, but as fundamental to mechanics, since its enunciation by Euler, as the corresponding one of the decomposition of linear velocities, viz., that of the decomposition into distinct components, of rotary velocities. The plane of the pen- dulum appears to turn relatively to the surface of the earth simply because the earth turns just so much underneath it, the earth really revolving about the local axis with a certain calculable component of velocity. The earth turns at the same time with another component of velocity about another axis (the complementary one), and the joint effect, or the resultant of the two components, is the rotation about the polar axis. ‘The second component, very great as we approach the equator, where the first vanishes entirely, is not exhibited by the pendulum, and is only detected by analysis as a slight disturbance. Convert the pendulum into the gyro- scope, however, and this second component appears equally with the first. ! Owing to the friction of the gimbals, there would be, practically, besides the motion above described, a motion of the axis in the plane of the meridian, north or south, according to the direc- tion of the disk rotation; this angular motion might be greater or less than the equatorial motion, but would be, with a well-constructed apparatus, independent of it, at least for the brief time during which a gyroscope experiment would last. * “ 2 ON THE INTERNAL STRUCTURE OF THE EARTH CONSIDERED AS AFFECTING THE PHENOMENA OF PRECESSION AND NUTATION. ‘Tue equations of precession and nutation are, as is well known, entirely indepen- dent of any particular law of density, and are functions only of the absolute values of the moments of inertia about the equatorial and polar axes A and C;,' and are independent indeed of the figure of the earth, except so far as it affects the values of these moments. Moreover, if the earth, instead of being solid throughout, is (as supposed by most geologists) a solid shell inclosing a fluid nucleus, it is only necessary (leaving out of consideration the pressure that may be exerted on the interior surface by the fluid) that the shell should have these moments of inertia. My. Poinsot” has obtained as the results of calculation for a homogeneous spheroid, values of precession and nutation identical with those of observation, by taking the ellipticity at and 1 1 308.65 7 (the ratio of mass of moon to that of the earth), at ae We have, assuming a uniform density, indicating by a and b the equatorial and axial radii, and by e the ellipticity :— ee re cee E C= {57 a‘ b=(approx.) 5 mb’ (14-42) ae Nn eee 8 2b 2173 oa A= 5" @ +a*b )=penl (1-+3e) C—A e 46 = = =—e— 2 (46) gi (awa e—4e : : 1 C—A 1 ae If ¢ is taken at —_—__ then ~_~— _____. _ But all meridian measurements of 308.65 C 32.7 1 T assume, of course, the equality of all moments of inertia, A, about the equatorial axes, and overlook all questions as to the non-symmetry of the earth with respect to its axis of figure or to the equator; for, in fact, neither the rotation of the earth nor any observable celestial phenomena reveal it. 7 Connaissance des temps, 1858. § January, 1872. ( 33 ) 34 PRECESSION OF THE EQUINOXES AND NUTATION i o- : the earth indicate an ellipticity greater rather than less than ——;* and the latest 1 Hoe C—A Il =) ~ - ~~ - r =? = T 7 > le ; oa € T —_ z determination makes it 995°) by giving which value to e we obtam ~~~ 999 Therefore the observed precession is to that which would result in a homogeneous spheroid, from the formulas, with the latest determined value of e introduced, as - ’ 1 299 : 311.7, provided the relative mass of the moon be but 35° y =) . ‘The value of ~~ would be the same for a homogencous shell of which the inte- rior surface had the same ellipticity, e, as the exterior; or it would be the same for a shell of which all the elementary strata had the same ellipticity, in which the density, constant through each stratum, should vary according to any law, from : 3 24 stratum to stratum. The ratio of 299: 312.7, so nearly unity (= 0.96 =e nearly), while the ratio of mean to surface density of the earth is so high, indicates nearly uniform ellipticity of stratification, and hence fluidity of origin ; while, on the other hand, the considerable inequalities in the equatorial axes indicated in the note below are incompatible with the hypothesis of actual fluidity beneath a thin crust, and are, to the measure of their probability, a disproof of it. ‘The effect upon the axial movements of such a shell which would result from the pressures of an internal fluid has been made the subject of an elegant mathematical investigation by W. Hopkins, '.R.S., in the Philosophical Transactions of 1839— 40-42. On the supposition of a uniform density of shell and fluid, and the same ellipticity for inner and outer surfaces of the shell, the precession will be the same * Airy, “Figure of the Earth,” Encye. Metrop.; Guillemin: Madler, Am. Journ. of Science, Vol. 30, 1860, makes the polar compression of greatest meridian — Ba 292.109 of smallest meridian — ot : 302.004 (Article translated by C. A. Schott, U.S. Coast Survey, from Prof. Heis’ “ Astronomie, Météorologie et Géographie,” Nos. 51, 52. 1859.) . y¥ Appendix “Figure of the Earth” to the “Comparisons of Standards of Length,” published 1866 by the British Ordnance Survey, gives for a “spheroid of revolution,” Cee oe ae ? of three axes, “—° _ 1 b—¢ 1 a—b_ 1 ¢ 295 ¢ 285197 We. 43 position to the former being 154 : 138. for a spheroid 13.38’ ce 32495: [be probabilities of the latter sup- { There is yet great uncertainty as to the relative mass of the moon, and as long as that point is unsettled, so is also the ratio of observed to calculated precession. Laplace, from observations of the tides at Brest, fixed it at —_, which number is adopted by Pontécoulant. Former determinations io . : 1 1 from the observed nutations make it 80.753 but 37 Was the determination from the coefficient of ot} ] » Voit ; a . . . nutation of Lindenau. Guillemin gives gg and these two last numbers coincide nearly with that > >Pm . ,cn “Oy , 7. Ty 7 ue d by F oinsot. A discussion by Mr, Wm. Ferrell, member of National Academy of Sciences, of tidal observations made for a series of years at the port of Boston, as well as those at Brest, gives results contirmatory of the larger ratio of Laplace. Serret (Annales de l’Observatoire Imp. 1859) assumes C—A 1 and deduces — — 327 Thes ios j i = id deduce a ca 0.00327. These ratios are adopted by Thomson and Tait, §§ 803, 828. Archdeacon Pratt (“Figure of the Earth,” 4th ed. 1871) adheres to Laplace’s determination. IN RELATION TO THE EARTH'S INTHRNAL STRUCTURE. 35 and the nutation essentially the same as for a homogeneous spheroid; but for the actual case of a heterogeneous fluid contained in a heterogeneous shell he finds that the ellipticity of the inner surface of the shell must be less than that of the exterior in the proportion of the observed precession to the precession of a homoge- neous spheroid of same external ellipticity, a proportion which he assumes to be & t in accordance to the then received numbers for ellipticity, and for the mass of the moon. The fulfilment of the condition, in the actual constitution of the earth, is improb- able; for the isothermal surfaces are in all probability (and indeed by his own mathematical conclusions) of progressively greater ellipticity from the surface in- wards. He finds, however, the requisite decrease of ellipticity m the fluid surfaces of equal density, assuming the well-known hypothetical law of density of Laplace sin gb a sau SS ae 8 LOMA know not the influence which pressure has upon solidification, b and it seems probable that the interior surface of the shell would conform nearly to the surfaces of equal temperature. ‘The demand which he makes for 800 or 1000 miles thickness of shell is therefore a minimum (for the data used), while the more probable result is a much greater thickness or even entire solidity. But however elegant may be Mr. Hopkins’ analysis, the basis of the structure is but slender, and those results have not been generally accepted as fully decisive of the question. Sir Wm. Thomson, F.R.S., brings forward (Philosophical Transactions, 1863; also Thomson’s and ‘Tait’s Treatise on Natural Philosophy, 1867) arguments against the popular theory of a thin crust, which are more forcible. A thin crust would itself undergo tidal distortion, and the height of the apparent tides of the ocean be thereby much reduced, while the-actual precession would be diminished in the same ratio; that is, the differential forces of the sun and moon would expend themselves in producing these solid tides instead of producing precession. From a theoretical investigation (given in a separate paper in the same volume)! of the deformation experienced by a homogeneous elastic spheroid under the influ- ence of any external attracting force, he arrives at the result that, if the earth had no greater rigidity than steel or iron, it would yield about 2 as much to tide-produc- ing influences as if it had no rigidity—more than ? as mnch if its rigidity did not exceed that of glass. Moreover, the apparent ocean tides (or difference of high and low water level) would be (if H# is the measure for a perfectly rigid earth) 0.59 H, if the earth had the rigidity of iron or steel only; a Hi if it had that of glass. Z, As to precession, the centrifugal force of the crowns of the tidal elongation would balance 7 of the dynamic couple resulting from the sun’s or moon’s attraction if the earth had only the rigidity of glass, and 2 if it had only that of steel. “That the effective tidal rigidity, and what we call the precessional effective ' See also ‘‘ Treatise on Natural Philos,,” § 832 ed seq. 86 PRECESSION OF THE EQUINOXES AND N UAVAT Om rividitv of the earth, may be several times as much as that of iron (which would ' Jhenomena, both of the tides and precession, sensibly the same as if the earth were perfectly rigid), it is enough that the actual rigidity should be several e actual rigidity of iron throughout 2000 or more miles thick- make the } times as great as the ness of crust.” A theorem fundamental to the establishment of the above propositions is, that a revolving spheroid destitute of rigidity, a homogeneous fluid one, for instance, would have no precession. Sir W. Thomson does not mathematically demonstrate this theorem, but by use of an hypothesis gives an elegant illustration of its truth, for which, though ‘to me it is convincing, I prefer to substitute the following demonstration. Such a spheroid, all the particles of which revolve about an axis with a common angular velocity , and attract each other by the law of universal gravitation, would have the form of an ellipsoid of revolution, the ellipticity of its meridional section being : ae (See “Figure of the Earth,” Encyc. Metrop., par. 33, by Prof. ] Airy.) Attracted by the sun, its tides would be expressed by the terms of [2316] Méc. Cél., Book IV (Bowditch). Of these three terms, the first (a function of the declination only of the attracting body) and the third (the semi-diurnal oscilla- tion) express tidal elevations symmetrically distributed on each side of the equator, which would, hence, exert no influence through the centrifugal forces of their masses, upon precession. The second therefore, or the diurnal tide, is alone to be considered, Conceive a meridian plane passed through the sun at any declination, the “couple” exerted by its attraction would be exerted wholly to turn the spheroid about an equatorial axis normal to this plane. We have therefore to investigate what dynamic couple, with reference to this same axis, will be exerted by the cen- trifugal force of the diurnal tidal protuberance. As the calculation involves the state of things at but a single instant of time, the angle, nt-+-a—y, may be written @ and counted from the meridian of the sun: p, the uniform density of the fluid, taken as unity. The height, y, of the diurnal tide will be expressed for all parts of the spheroid by Patel (47) y= aus sin 0 cos 0 sin” cos 2 cos af alg in which 4 is the polar distance or complement of the latitude of the locality, and ( the declination of the sun. If, with Laplace, we put cos A=, and sin A=V 1— 1’, the mass of the elementary column of height y will be ydu da, and its centrifugal * g being the force of gravity at the equator of the hypothetical spheroid. + The expression, in the original, for the diurnal oscillation, is 3L 3 3 \ Sin V cos V sin 6 cos 6 cos (nt-+e—q) (2) ( Dp The notation of my paper on the of sae re precession of the equinoxes is substituted, and the assumed value i p introduced, IN RELATION TO THE EARTH'S INTERNAL STRUCTURE. 37 force (the radius of the spheroid being taken at unity, and the variation, assumed slight, due to ellipticity, disregarded) n* ydu daV 1—,.. The component of this tending to tilt the spheroid about the axis in question is n°y du daV 1—w cos a, and its moment n’y du da uV 1—n? 60s a. Substituting the value of y (47), the above becomes 158 .. 2 2 2 n?.,— sin 6 cos 6 du da w’ (1—xz") cos’ a 27°y Integrating, first with reference to 4 from »-=—1 to .=-+1, then with reference to a from 0 to 2 2, we get, as the expression for the couple due to “the centrifugal force of the crowns of the tidal elongation,” resisting the sun’s action, . IS sc (48) Qz7'* “ sin 0 cos 0 aa We have found (19) for the moment of the sun’s force, producing precession, the expression 38 (C_A) sin 6 cos 6 9 and (46), (CA) =n0 (6 being taken at unity) and e, as already stated, is for a homogeneous fluid spheroid ae Making these substitutions, the above ex- pression becomes identical with (48). The precessional force of the sun is, there- fore, exactly neutralized by the centrifugal force of the tidal swelling. The theorem could, doubtless, be demonstrated for a revolving fluid spheroid in equilibrium, of which the density of the strata varies. Without extending any further the mathematical analysis, it will be sufficient to remark that the calculation of the tidal elevations is, identically, that of equilibrium of form of the revolving body subjected to a foreign attraction, and in the calculation the motion of rotation is disregarded, and the centrifugal force, which expresses its entire effect upon the form, alone considered. Under this point of view, equilibrium of form is, necessarily, equilibrium (or stability) of position. For if any effective turning force exists, it must, in order not to interfere with equilibrium of form, either be so distributed as to give each individual particle of the spheroid its proper relative quantity of tyrn- ing motion, or it must be a distorting force. The first alternative cannot be admit- ted; the second is excluded by the hypothesis of equilibrium. Hence, there can be no turning (or precessional) force. The accuracy of the foregoing analysis is complete, except that the consideration of relative motion of the particles is excluded. But Laplace shows (p. 604, Vol. IT, Bowditch) that as the depth of the ocean increases, the expressions for the tidal 1 There are slight errors of approximation : 1st, in the tidal expression (47) itself; 2d, in the above integration which disregards the variation of the radius; and, 3d, in the value of C—A. They neutra- lize each other in the final result. 38 PRECESSION OF THE EQUINOXES AND INU IAt TORN) oscillations given by the dynamic theory approximate rapidly to those of the «equilibrium theory,” with which, when the depth is very great, or the spheroid wholly fluid, they are essentially identical. Moreover, he shows (p. 219, Vol. I) that the vertical motions of the particles, when the depth is small, may be disre- garded. When the spheroid is wholly fluid, a// the relative motions of the particles are of the same order as the vertical ones and exceedingly minute; and the forces of inertia thereby developed are insensible compared with those we have been eon- sidering.’ - : By parity of reasoning the truth of Sir W. Thomson’s propositions concerning a solid but yielding spheroid is made evident; for exactly in the same ratio to the tides of a fluid spheroid that the solid tidal elevations are produced (the actual ellipticity of the earth being nearly that of equilibrium with the centrifugal forces), will the precessional couple due to the tide-producing attraction be neutralized by their centrifugal action.2 That a thin solid crust, such as geologists generally assume, would yield and exhibit tidal elongations, seems without calculation very probable; but if Sir W. Thomson is correct as to the rigidity required in even a wholly solid earth, the hypothesis of a thin crust must be abandened, and it would seem indecd that rigidity several times as great as the actual rigidity of iron throughout 2000 or more miles thickness of crust would be incompatible with a very high internal temperature. Without having recourse to Sir W. Thomson’s profound analysis, the necessity, in order that there shall be no sensible solid tidal wave, of a very high rigidity 1 The foregoing demonstration does not conflict with Laplace’s theorem that ocean tides do not affect the precession; for his theorem applies only to a shallow ocean over a rigid nucleus, of which ovean the precessional couple, by altered attractions, pressures, and centrifugal forces due to generation of living forces in the fluid, is transferred to the nucleus. I have already alluded to the minuteness of the motions of the particles of a fluid spheroid. The remarks apply, @ fortiori, to those of an elastic solid. Vibratory motions, properly speaking, cannot exist, for the elastic forces extremely minute are always held (sensibly) in equilibrium hy the distorting forces. The solid surface would oscillate in the same sense that the ocean tides oscillate, ¢. e., by a “forced” tide-wave. : * “It is interesting to remark,” say Thomson and Tait (§ 848, ‘Treatise, &c.”), “that the popular geological hypothesis of a thin shell of solid material, having a hollow space within it filled with liquid, involves two effects of deviation from perfect rigidity which would influence in opposite ways the amount of precession. The comparatively easy yielding of the shell must render the effective moving couple due to sun and moon much smaller than it would be if the whole interior were solid, and, on this account, must tend to diminish the amount of precession and nutation. But the effective moment of inertia of a thin solid shell, containing fluid in its interior, would be much less than that of the whole mass if solid throughout; and the tendency would be to much greater amounts of pre- cession and nutation on this account.” The co-efficient of precession of the “thin solid shell” would be (p. 34) the same, nearly, as that of the spheroid of which the homogeneous Strata have the same ellipticity. Its precession-resisting couple (48) due to tidal distortion would be just what is necessary to develop its proportional influ- ence upon the precession of that shell, upon which the fluid contents can exert influence only through This is identically Prof. Hopkins’ problem. The thin shell of popular geological hypothesis would, however, be subject to tidal distortions scarcely inferior in magnitude to those of a wholly fluid spheroid; by which, as we have seen, the sun and moon’s “moving-couple” is wholly neutralized throughout the whole spheroid. their pressure. IN RELATION TO THE EARTH’S INTERNAL STRUCTURE. 39 for the earth, may be made evident from the following considerations: A rod of steel extending towards the sun from the centre to the surface of the earth, would be elongated by the differential force of the sun’s attraction 0°.975, or one foot, nearly. The height of the solar tide of a homogeneous fluid spheroid is 1°.355 ; but the mutual attraction of the elevated particles produces 0.793 of this, and the remaining 0.542 is the proper measure of the direct action of the solar force. In the case of the rod the elastic forces of the steel alone are considered; in the spheroid gravitation is the sole binding force. ‘The maximum extension of the rod per unit of length would be expressed by the decimal .000000055 corre- sponding to a tensile force of 1.87 lbs. (taking the coefficient of elasticity at 34 millions Ibs.) per square inch.” ‘The necessity of the extreme rigidity demanded by Sir W. Thomson is recognized when it is seen how excessively minute would be the elastic forces developed in the production of distortion, in a rigid earth spheroid, commensurable with fluid tide-waves.* In a paper “On the Secular Cooling of the Earth” (Trans. R.S.E., 1862, and Appendix to “ Treatise, &c.”), Sir W. Thomson applies a solution of Fourier to the determination of the interior temperature and its rate of increase downwards, ® See Additional Notes, p. 51. 1M. Delaunay, President of the French Academy, after quoting (Comptes rendus 1868) from the paper of Sir W. Thomson to which I have already referred, the results of Hopkins and some corro- borating remarks from Sir W. Thomson’s paper (referred to above), says: “Ainsi, on le voit, l’ob- jection mise en avant par M. Hopkins, contre les idées genéralement admises par les geologues sur la fluidité interieure du globe terrestre, est rezardée par plusieurs savants anglais comme parfaitement fondée. Je suis d’un avis diametralement opposé: je crois que |’ objection de M. Hopkins ne repose sur aucun fondement réel.” M. Delaunay then refers to an experiment made under his direction with a glass vase 0™ 24 in diameter, as furnishing decisive proof that the ‘“‘viscosity” of a liquid as per- fectly fluid as water even, is sufficient to cause it to take up the rotary motions of its enveloping shell, provided that those motions are relatively slow, as are those which constitute the precession and nutation of the earth; and he goes on to say: “Hence it does not appear to me possible to admit that the effect of the perturbing forees to which precession and nutation are due extend only to a portion of the mass of the terrestrial globe; the entire mass ought to be carried along (entraince) by the perturbing actions, whatever may be the magnitude attributed to the interior fluid portion, and consequently the consideration of the phenomena of precession and nutation can furnish no datum for estimating the greater or less thickness of the solid crust of the globe.” M. Delaunay seems to be unaware that Sir W. Thomson coincides with Prof. Hopkins only in this (as the sequel of the very paper quoted shows), that he demands a great thickness of crust, and, moreover, that the interior, to the depth of this crust, shall be not merely “solid,” but possessing a rigidity ‘‘several times as great as that of iron.” I have endeavored to show that Sir W. Thomson’s argument is irrefragable; but, based upon wholly different considerations, it is certain that no degree of “viscosity” assigned to an internal liquid will refute it. I have remarked, at the outset of this discussion, that Prof. Hopkins’ results “have not been genc- rally accepted as decisive ;” but I cannot admit that, as a test of their tenability, the experiment of M. Delaunay possesses the crucial character which he attributes to it. Viscosity, considered as an accelerating force tending to impart to a fluid the rotary motions of an enveloping shell, is directly proportional to the surface of contact, and inversely to the mass of contained liquid; in other words, it varies inversely as the diameter of the envéloping shell. The effect of viscosity of the fluid con- tents of the earth compared to those contained in a similar spherical envelope of only ten inches diameter, would be express var 7 racti ; xpressed (nearly enough) by the fraction oaarens? 4() PRECESSION OF THE EQUINOXES AND NUTATION: assuming a uniform primitive melting temperature of 7000° Fah., and a lapse of 100 millions of years since the cooling process commenced. «The rate of increase of temperature from the surface downwards would be sen- sibly ,1, of a degree per foot for the first 100,000 feet or 80. Below that depth the rate of increase per foot would begin to diminish sensibly. At 400,000 feet it would have diminished to about ;4; of a degree per foot. At 800,000 feet it would have diminished to less than ;1, of its initial value, that is to say, to less than 5.5 of a degree per foot; and so on, rapidly diminishing. Such is, on the whole, the most probable representation of the earth’s present temperature, at depths of from 100 feet, where the annual variations cease to be sensible, to 100 miles, below which the whole mass, or all except a nucleus cool from the beginning, is (whether liquid or solid) probably at, or very nearly at, the proper melting temperature for the pressure at each depth.” The high rigidity demanded is difficult to conceive of in connection with a temperature in the solidified mass “at or near” that of melting, extending down- wards indefinitely towards the centre of the earth. Hence Poisson’s reason- ing, which results in showing that the earth to have become thoroughly cooled and to have been subsequently reheated, superficially, harmonizes better with the demand for rigidity than that of Leibnitz, which supposes it to be now cooling from a (throughout) incandescent liquid state. In the latter case the law of actual temperature as deduced from Fouricr’s formule (as expressed above by Sir W. Thomson) would extend to the centre; in the former case only to the unmelted portion or to the “nucleus cool from the beginning.” Referring to the increase of temperature, with depth observed in mines, &c., Poisson remarks: ‘Fourier et ensuite Laplace ont attribué ce phénoméne A la chaleur dorigine que la terre conserverait 4 l’époque actuelle et qui croitrait en allant de la surface au centre, de tel sort qu’elle fait excessivement élévée vers le centre * * * en vertu de cette chaleur initiale la temperature serait aujourd’hui de plus de 2000 degrés 4 une distance de la surface égale seulement au centiéme du rayon; au centre elle surpasserait 200,000 degrés. * * * Mais quoique cette explication ait été généralement adoptée, jai exposé, dans mon ouvrage, les difficultés qwelle presente, et qui m’ont paru la rendre inadmissible, * * * je crois avoir demontré que la chaleur developpée par la solidification de la terre a du se dissiper pendant la durée de ce phénoméne, et que depuis longtemps il wen subsiste plus aucune trace.” (Théorie de la Chaleur.) Poisson, as is well known, attributes the increase of temperature, with depth, observed in the earth’s crust, to the passage, at a remote period, of the solar system through hotter stellar regions, the temperature of which, he argues, should differ from place to place. Even a hypothetical case of the illustrious author, conform- ing to his theory, of an increased superficial temperature, 5000 centuries ago, of 200° C, diminishing by cooling (by transition to cooler stellar regions ) to 5°, 500 centuries ago, and, subsequently, to the actual mean temperature, would scarcely meet the present demands for time, of paleontology; while the determinations of the conductivity of the earth’s crust made near Edinburgh show, according to Sir W. Thomson, a necessity for increments of temperature of 25°, 50°, and 100° DN] RE RATEON LO OTHE HART ES [INDE RENAL STRUCTURE. 4} (Fah.), for past periods of only 1250, 5000, and 20,000 years, and authorize him to pronounce Poisson’s hypothesis impossible, without destruction of life, or relega- tion of the date to so remote an era as to demand an intensely heated stellar region. A reheating after solidification which should again fuse the surface to great depths would be, it seems to me, as “inadmissible” for the origin of observed sub- terranean temperatures, as the heat originally “developed by the solidification of the earth.” Hence, the hypothesis of a reheating to fusion of the surface by impact of meteoric bodies would be likewise excluded by Poisson’s theory of in- ternal temperatures. In the paper referred to (p. 39), Sa: Wm. Thomson discusses the probable circum- stances of solidification of the earth, assuming the known crust-materials (granite, &c.), in a molten state, as the constituent, and reasons that in consequence of the great condensation of granite in freezing, solidification must commence at the centre, and that “there could be no complete permanent incrustation all round the surface “till the globe is solid, with, possibly, the exception of irregular, comparatively “small spaces of liquid ;” such separation of constituents in the process of crystal- lization taking place in all liquids composed of heterogeneous materials, and, indeed, is observahle in the lava of modern volcanoes. He infers from the probable phenomena developed, into the discussion of which he goes at some length, “ re- “sults sufficiently great and various to account for all that we see at present, and “all that we learn from geological investigation, of earthquakes, of upheavals, and *‘subsidences of solid, and of eruptions of melted rock.” Still we would, if possible, find reason to attribute a lower than “the proper melting temperature’’-to the solidified interior. Ice, indeed, preserves its rigidity unimpaired up to the point of fusion, and there may be a few other substances that have the like property; but it seems to be an exceptional one. The known constitu- ents of the earth’s crust certainly do not possess it, at least under ordinary pressures. If, as suggested by Prof. Joseph Le Conte (Am, Journal of Science, Nov. Dec. 1872), the “conductivity” be increased by pressure and condensation, such diminished temperatures may obtain. 6 January, 1872 42 NEW ADDENDUM. NEW ADDENDUM. A pew words are in place here concerning the results of the late Prof. Hop- kins’ investigation (against which M. Delaunay’s objections, (note, page 38,) are especially directed), briefly stated, pages 34, 39. They are as follows: First For HOMOGENEOUSNESS. ‘Supposing the earth to consist of a homogeneous spheroidal shell (the ellipticities of the outer and inner surfaces being the same) filled with a fluid mass of the same uniform density as the shell;” then, “the precession will be the same, whatever be the thickness of the shell, as if the whole earth were homogeneous and solid.” SECOND, FOR HETEROGENEOUSNESS, his result may be thus expressed : (a) na-P=R\' (14 eR ) 3 Ga “where P, denotes the precession of a solid homogeneous spheroid of which the ellipticity =e,, that of the earth’s exterior surface, and P’ the precession of the earth, supposing it to consist of an interior heterogeneous fluid contained in a heterogeneous spheroidal shell, of which the interior and exterior ellipticities are respectively ¢ and ¢,, the transition being immediate from the entire solidity of the shell to the perfect fluidity of the interior mass.” In the multiplier of “, second member of (a), q is the ratio of external to internal ey polar radius of the shell; s depends on the varying ellipticity and density of the strata of equal density of the shell; 2 depends on the density of the fluid interior. For a thin crust the coeffisient in question is unity nearly; for a thick one it will be somewhat greater if ¢ be less than a. It cannot fail to be observed that, under the conditions just before expressed for homogeneousness—i. e., equality of external and internal ellipticities—we get from the formula (s becoming zero) the same result, i. e, P’ = P,, as for that case. In accordance with rational hypothesis as to the internal condition of the earth, equalities of ellipticities for the surfaces of a thin crust (and corresponding equality of densities), or closely approximate equalities would be expected. The necessity for a thick crust arises, therefore, from the alleged discrepancy between the observed and calculated annual precessions (50 seconds and 57 seconds), which, according U — == §, nearly, assuming the moon’s mass ;1,, and i to Prof. Hopkins, makes fs the earth’s ellipticity sto: (The real discrepancy is probably very much less. See page 34, et sequentia.) lnm =o . ; he original ADDENDUM, hurried] in what follows, somewhat modified a y written while the work was in the printer’s hands, has been, nd amplified. NEW ADDENDUM. 43 If, in applying the expression (@), the symbolic fraction, for a first approximation, be omitted, we have, according to above assumption of discrepancy, ¢ = Je. This value of ¢ will be in excess; hence, the thickness of crust deduced from it will err the other way, and a determination on this basis will give a thickness which must, in fact, be exceeded. The limit of solidity, proceeding inwards, may and probably does depend upon both temperature and pressure. Isothermal surfaces Prof. Hopkins finds to have increasing ellipticities. Surfaces of equal pressure, deduced from the hypothetical law of density, =, fo. have diminishing ellipticities, and if gb, = 150° the above law agrees sufficiently well with the actual ellipticity and ratio of surface to mean density of the earth. This law for e = Ze, demands a thickness of crust of 4 the radius, or 1000 miles. This is a minimwm, since the actual surface of solidifica- tion (lying between this and the corresponding isothermal surface) would have greater (and hence too great) ellipticity. Before commenting upon this application, and upon the real meaning of the formula, I return to the case of homogeneousness. Some of the results arrived at by the analysis of Prof. Hopkins may be illustrated by the following considera- tions: The fluid spheroid, treated of p. 36, is subjected, by the attraction of the sun, to the distortion expressed by (47). ‘This distortion, as shown by the form of the expression, is equivalent to an exceedingly slight rotational displacement* of figure about an equatorial axis, such as would be caused by displacing through a still more minute angle the planes of diurnal rotation. It is one of the beautiful results of the analysis to show that the change in the direction of the centrifugal force due to this slight obliquity of the planes of rotation is equivalent to turning forces at all points of the fluid exactly proportional to their distances from the equatorial axis. Let now a rigid shell, exactly conforming internally to the external surface of the fluid, be applied, and the whole turned back until the planes of rotation are restored to perpendicularity to their axis; the precessional effect of the attracting body now operates upon the whole mass; for there are no longer counteracting tidal protuberances. If we take that part of (47) which is due to the direct action of the sun, viz., —— sin @ cos @ sin 4 cos 2 cos a (for, the protuberances being repressed 7g by the shell, the pressures on its interior which replace them will arise only from the direct action), and estimate it as a pressure and calculate the elementary couples for an internal ellipticity, e, we shall find the integral couple (and this corresponds with Prof. Hopkins’ result) to be identical with (48) viz., exactly that due to the 1 The required angle of the displacement is the height of the tidal wave (47), for « —0, divided by = for an ellipse of ellipticity, e, (2e sin 2 cosa). Prof. Hopkins shows that the corresponding divergence of the planes from perpendicularity develops a couple = as x ne multiplied by the sine 15 f twice this ar =o . ; : nn ES Sis: F of twice this are (or twice the are itself). Performing the operations we get, aE ne —— sin 6 cos 8, in 0 if which we have the solar couple (19) and (48), which causes the displacement, since e = (C—A.) a tt NEW ADDENDUM. . s 11 couple which the sun would exert on the fluid mass considered as a solid.’ It would 5 2 F G . ssional force of the shell in the ratio — 4 1 of the analysis. By qd —_— virtue of this pressure the fluid tends to transform its own precession into an increase the prece augmented precession of the shell. ; It requires, however, but an extremely minute angular separation of the axes of the shell and fluid to generate counter-pressures equivalent to those which caused the separation.?_ The divergence cannot, therefore, be progressive, but is simply a minute oscillation of the two axes, or a rotation around each other. In the latter form it appears in the analysis which, otherwise, gives to the internal fluid mass a precession identical with that of the enveloping shell. Prof. Hopkins confines his analysis for the case of homogeneousness to equal ellipticities for the bounding surfaces of the shell. Excepting the case of sphe- ricity for the inner surface, the result would be the same—viz., an unchanged pre- cession, however the ellipticities might differ. I now return to the formula (@) and remark, that it is an inaccurate expression for a slight difference (P,—P’) due to the fact that the spheroid is heterogeneous— that it is not capable of being made a test of internal fluidity, or a measure of thick- ness of crust. I have already shown that for homogencousness the couple due to pressure on the inner surface of the shell is identical with the sun-couple upon the fluid mass solidified, a result approximately true (as will be shown hereafter) if the density of the fluid strata vary. Hence, if we take the sum of the sun-couple exerted on a shell of interior and exterior ellipticities, ¢« and ¢, and of the pressure-couple developed in the fluid,* and divide by the moment of inertia of the entire mass and by @, we shall have the rate of gyration of the entire mass considered as a solid. Referring to Prof. Hopkins’ analysis and symbolism, the quotient will bet a, ,d (a é a (e p Ba e) da’ + 2a’ « ws pada © mp (v) CRUE ee Oe D) 43 : 5 2 7°70 Gk, Oke oO (a) =f a p Gi da’ Multiply the above by this arm, by g, by the elementary surface dud, and, again, by cos «, and we get the elementary component tending to tilt the shell. The integral, with proper substitutions, is equivalent again to (19) or (48). j * There is another process which may take effect in neutralizing internal pressure. I have remarked (last par. p. 6), that, considered as a perfectly rigid body, the precessional motions of the earth cannot be precisely those assumed. In fact, our imperfect integrals of the conditional differential equations present the anomaly of a varied motion in which the generating force does no work ; no yielding to the tilting couple having place. There are necessarily some, too minute to be detected, ‘ The lever arm is also 2e sin x cos a. nutational movements. In case the precessional force were augmented by so large a ratio as = i g—l1 Ww > for m she S$) i i i ould be for a thin shell, these nutational movements would surpass in magnitude those necessary to generate the required counteracting pressures. * T use provisionally Pr anleend : 7 s. 5 > : provisionally Prof. Hopkins’ computations for this, involving ie pada’; its erroneous- ness will appear hereafter. am > au ~ The symbols p, a, , correspond to S, 6, n, of p. 7 for which ¢’ is the radius of the shell. a ; pis the density of stratum, solid or fluid, ellipticity, and a’ the polar radius ; a, is the external, and a the interna peler NEW ADDENDUM. 45 Denote the moment of inertia of the entire spheroid by 1 = =76 (a,) o ee ee ay « 7 =~ n{o(a,)—o(a)] 15 «“ ‘“ ‘“ ‘ ‘“ “nucleus Cy 8 xo (a) 15 Then (4) = zs (o (4) —6 («)) and the above expression, reduced to precession, will become ik h , a eee > (1) () a ate) Prof. Hopkins gets for the precession of the same spheroid considered as fluid within the shell, (his symbolic abbreviations used in both cases) | (v2) ( na) ? +Gy\' tT? * gi) In this last expression (y,) and (y,) denote coéfficients of gyration which one and the same couple (7. e. the centrifugal force, by pressure on the shell and by reaction on the fluid mass—the assumption being made that the latter, having its proportionate force on each particle, gyrates as a solid) produce upon the shell and fluid mass respectively. ‘They should be therefore inversely proportional to the respective moments of inertia of the shell and nucleus, rendering the expressions (a) and (vy) identical. But this apparent identity is brought about by asswming that Prof. Hopkins’ expression for the pressure-couple on the shell arising from the sun’s attraction on the fluid to be identical (or at least approximately so) with that which would be exerted on the same heterogeneous fluid solidified ; by which assumption I introduce in (x), att r (which is Prof. Hopkins’ symbolic abbreviation of q — anew OO Pe =, oa puda i : Je ee ee da re aes Gp : instead o 2 [a (ay) o(a) ] which latter expression belongs to the case just specified, of the solidified fluid. Now in the case of nature—. e. the earth with the received hypothetical laws of density and ellipticity, the two expressions differ in a ratio (about 4:3) so greatly exceeding unity, as to forbid the assumption of approximate equality. ‘The error of the first expression will be better appreciated by referring to the quan- * The interpretation of (#) and (y) is obvious. P is the coefficient of precession for a homoge- neous shell of uniform ellipticity »; instead thereof let the shell be heterogeneous with same internal surface, but of an external ellipticity «,. P for such a shell will have a fractional increment denoted by the ratio s. By the pressure of the internal fluid the precessional coefficient of this shell will be still further increased by a ratio denoted (a result of the analysis) by Jey 7 But the shell is con- g— strained to carry along and to take up a common precession with the nucleus, and the coefficient will a be thereby diminished in a ratio | of (x) or (according to Prof. Hopkins) the corresponding ex- 1 pression of (y). Since PP, =, expression (a) is readily deducible from (y). 1 46 NEW ADDENDUM. tities (y,) and (y.) in (y). The brief account given, p. 43, will show how, as re- sulting from the analysis, a rotating homogeneous fluid enveloped and confined by a shell reacts, with a practical rigidity conferred by rotation, against the shell, (when the respective axes of rotation are slightly separated) and thereby receives an angular motion “ precisely as if it were solid,” (Phil. Trans., 1839, p. 394). It is clear that, through this interaction, the shell, likewise, must receive an angular motion, and that these several angular motions must be in inverse ratio to the respective movements of inertia of shell and nucleus; and so, for homogeneous- ness, the analysis makes them. When we come to heterogeneousness, the same modus operandi is (and rightly) attributed to the fluid; and again, most clearly, the relative angular motions of shell and fluid should be in above-mentioned in- h of which verse ratio; whereas their ratio is quite differently computed to be — q the value has just been given. ‘The error (for there is clearly one) is in the com- putation of the pressure-couple developed in the fluid and exerted upon the shell by centrifugal force, when their axes of rotation are slightly separated. The same error of computation (exhibiting itself by the identical symbol, — Lm g—l enters into the expression for the pressure-couple developed by solar attraction, and introduces agsin the above symbol as the third term of the second factor of (y). Without going into lengthy discussion, it is sufficient to remark that both the centrifugal foice and the foreign attraction produce in the strata of equal density of a heterogeneous fluid, special configurations, and expend themselves in so doing; and, moreover, that the prior establishment of these forms of equilibrium is assumed, and necessarily assumed, in the analysis. It is, therefore, unwarrantable to integrate (as is done) through the fluid mass these forces, as free forces, to get its pressure upon the shell.! Ihave shown, I think, that in the expression (7) the first factor should be corrected 7 to be, as it is in (x), da® a, da ro pa inkee dda da 1 The correction consists in substituting for (vi) in the first factor.on @), ——————— (72) G (a,)—s (a) h : 2 7 : : = : i [or its value, p. 45]. The same correction for . p introduced I a . . p f 5 into the second factor, would require e f° p ce da’ to be substituted for the second Aa instead of term of numerator of (v). But that (v) should belong to entire solidity we require Gk CGY = (Go he : ; ie a da’. Now these quantities differ Inappreciably, for taking the entire * It is obvious that the taking account of the internal motions by which the configurations pro- duced by foreign attraction adapt themselves to the diurnal rotation, does not meet the point made. * Although the errors in the two cases have the same expression, it does not follow that the cor- rections should be zdentical. The configurations of strata of like density due to foreign attraction are superinduced on the previously established configurations due to the centrifugal force, which have the varying ellipticity «. The correction should involve this ellipticity, and it is probable that the above symbol is, if we disregard the slight inlernal motions, the true one. Indeed, I think T may venture to affirm, that, given a heterogeneous fluid wholly enveloped by a rigid bounding surface, NEW ADDENDUM. j 47 integrals from zero to a, and multiplying by a a, the last becomes (Thomson and Tait, § 825), : Ma,’ («, — +m) =C —YA, for the earth as actually constituted. And the first (deduced from Hopkins, Phil. Trans., 1840, pp. 203 and 204), is (nearly) : Ma,’ e, = C—A, for same spheroid with wniform internal ellipticities. (JZ= mass of the earth). [The value of the first, using the constants of density of Archdeacon Pratt, «Figure of the Earth,” 4th ed., p. 113, is somewhat less than this last expression. ] Now m= 51, (ratio of centrifugal force to gravity) and hence 2 (e,— 4m) is very little less than ¢,. Fora fluid nucleus, the inequality would be still less. Hence it appears that both («#) and (y) (when corrected) express very nearly the precession of the solidified earth; and, moreover, that the effect upon preces- sion due to the variation of internal ellipticity is very small, the precession of the earth considered as rigid being, essentially, that corresponding to uniform ellipticity ; or what is the same thing, that of a homogeneous spheroid of its external form. This also appears in the comparison of the value of ooe , as established from observation, and the resulting calculated ellipticity. he first is .00327 and the second 1, (Thomson and Tait, § 828). Now a homogeneous spheroid of the latter ellipticity would have for — a value (e — 4e*) of .00332; a difference of about ;4. Variations in the constants which enter into the expressions for internal density give rise to variations in the ca/culated ellipticity—and, of course, in the resulting precession; but if the external ellipticity is defined by a rigid shell, the effect of internal variation is, in the case in hand, almost nil. Hence, had the hypothetical consolidation of the earth, of p. 43, been carried to the very centre, no material approximation to the desired correction of { in the calculated precession would have been found.’ In fact, the problem for heterogeneousness subjected to a foreign attraction, and a condition of static equilibrium assumed, the pressure-couple exerted by the fluid on the shell cannot differ from that which the attraction would exert on the solidified fluid. In the case of homogeneousness, I have arrived (p. 43: the results, though based on an ellipticity corresponding to fluid equilibrium, hold good for any small ellipticity) at the exact expression for the pressure-couple, from the function expressing the tidal protuberance due to the foreign attraction. The tidal configuration of the heterogeneous earth, wholly liquefied, would result from the transcen- dental analysis of Hopkins, pp. 203, 204, or of Thomson and Tait, § 822-824, and the maximum height would be one foot, very nearly; but the pressure function cannot be readily deduced. It would depend on gravity and [§ 825] “the value of C— A may be determined solely from a knowledge of surface or external gravity, or from the figure of the sea level without any data regard- ing the internal distribution of density.” * It is curious, to say the least, that there should be ground for the remark that the expressions (a) s and (y), which latter, with its author’s valuation of (ny) | may be writen ; =U h Ue give, 48 NEW ADDENDUM. was practically solved under Prof. Hopkins’ treatment of it when it was shown that, for homogeneousness, the precession for internal fluidity was the same as for solidity, and when it appeared that the analysis applied would exhibit the action of the heterogencous fluid as if disposed in strata of like ellipticity with that of the inner shell surface. ‘The erroneous computation of pressures has merely the effect of exaggerating in a like ratio the densities of all the fluid strata. Hence, and hence only, a resulting precession differing materially from that which would result from solidity. In conclusion, I remark, 1st. The analysis of Prof. Hopkins, in its application to a homogeneous fluid and shell, seems to establish (and the result is confirmed by its harmony with tidal phenomena as developed in p. 43) that the rotation im- parts to the fluid a practical rigidity’ by which it reacts upon the shell as if it were with decreasing internal ellipticities, for fluidity of nucleus less precession than would belong to soli- ” dity. This is obvious since his le is greater than the ratio ae (nearly) which should take its place on the latter hypothesis. at t ' T do not concur with Sir William Thomson in the opinions quoted in note, p. 38, from Thomson and Tait, and expressed in his letter to Mr. G. Poulett Scrope (“ Nature,” February Ist, 1872), so far as regards fluidity, or imperfect rigidity, within an infinitely rigid envelope. I do not think the rate of precession would be affected. That no increase arises from fluidity I have endeavored to show; and it is unquestionably a corollary of Prof. Hopkins’ investigations. As regards imperfect rigidity, Sir William Thomson bases his argument upon the assumption that ‘the whole would not rotate as a rigid body round one ‘instantaneous axis’ at each instant, but the rotation would take place internally, round axes deviating from the axes of external figure, by angles to be measured in the plane through it and the line perpendicular to the ecliptic in the direction towards the latter line. These angular deviations would be greater and greater the more near we come to the earth’s centre. * * * * * Hence the moment of momentum round the solsticial line would be sensibly less than if the whole mass rotated round the axis of figure.” If I do not misunderstand his language, Sir William Thomson assumes that the same bending distortion which would ensue from the application of a couple to the external portions of a non- rotating spheroid, would, equally and zdentically, take place in a rotating one: thus causing the angle made by the planes of the external rings of matter and the solsticial line to be increased ; with a corresponding diminution of the component of On about this line. In the case specified by him (an extreme one) while sensible and important nutational movements would ensue, the mean precession would be insensibly affected; but I do not think precisely such elastic yielding would take place. As an extreme case of an infinitely rigid and infinitely hin shell containing matter completely destitute of rigidity, take the fluid spheroid of p. 36, and conceive it enveloped by such a shell. It is still, as shown, p. 43, susceptible (and susceptible only) of the extremely minute deflections of its planes of rotation by which precession is completely annihilated. Confer now upon the con- tents of the shell rigidity, uniform, or varying from surface to centre, continuously or discontinn- ously, in any arbitrary manner, and you have every possible ease of imperfectly rigid matter con- tained within a perfectly rigid crust. I can attribute no other effect to the conferred rigidity than a restoration of the lost precession—in whole or in part; nor can I suppose the shell enveloping imperfectly rigid matter to change its obliquity more than that which contains the fluid; regard being had to conditions of equilibrium without reference to living forces generated. I must remark that this hypothetical case, though as admissible for argument as any other form of “preternaturally rigid” crust, is exceptional. With a shell of Jinile moment of inertia, having some comparable relation to that of the fluid contents, the precession, instead of being annihilated, would be that due to the entire mass NEW ADDENDUM. 49 a solid mass, while its pressure imparts to the shell the requisite couple to preserve the precession unchanged. 2d. The same practical rigidity is; with entire reason, attributed to the heteroge- neous fluid by which (leaving out of view minute relative oscillations which do not affect the mean resultant in other natural phenomena and should not in this) the shell and fluid take a common precession. 3d. The two masses retaining their configurations, mutual relations, and rotary velocities, essentially unaltered by the hypothesis of internal fluidity, it would be a violation of fundamental mechanical principles were the resulting precession not identical with that due to the entire mass considered as solid. 4th. The common and identical precession of fluid and shell resulting from the analysis, is indispensable to any conception of precession for the earth as composed of thin shell and fluid; for otherwise internal equilibrium would be destroyed and the “figure of the earth” cease to have any assignable expression. The entire mass, fluid and solid must (without invoking the aid of “ viscosity”), be “carried along in the precessional motion of the earth.” ‘The analysis I have examined de- monstrates the possibility and exhibits the rationale of such a community of pre- cession, but fails in the attempt to exhibit a test of the existence or absence of internal fluidity. 5th. The powerful pressures that would be exerted upon a thin and rigid shell would probably produce in it noticeable nutational movements;! while if the shell be not of a rigidity far surpassing that of the constituents of the cognizable crust, the “ precessional motion of the earth” would, owing to the neutralizing effect of tidal protuberances, scarcely be observable. ‘ Vide p. 44, and note 2: without reference to conventional ‘“ Nutation” which is but a form of precession. In connection with these relative motions of shell and fluid, it is in place to allude to the “Vindication of Mr. Hopkins’ method against the strictures of M. Delaunay,” by the late Archdeacon Pratt (‘‘ Figure of the Earth,” 4th ed., p. 132). He reasons, that, if at any moment, the crust and fluid be arranged as to density, “exactly as if they had been hitherto one solid mass and be moving alike, this state cannot possibly continue.” For the shell will be acted upon not only by the foreign attraction but by the fluid pressure, and will “begin to move quicker,” with a precession due to both the thence arising couples. That this should not occur requires, he esti- mates, the counteracting centrifugal force of a tidal protuberance, in a crust supposed 100 miles thick, of seventy-four feet. The writer does not seem to be aware that the author whom he vindicates finds no such relative acceleration of the shell (vide p. 44, § 2, of this Addendum) as resulting from the pressure; and, strangely, for an authority on the “ Figure of the Harth,” fails to recognize that “an clevation of the outer surface of the crust”—that is a tidal distortion—of a@ single foot, would relieve the shell from all pressure. This is, perhaps, a natural result of the use of an expression (Prof. Hopkins’) for the pressure which disregards the influence of “ Figure.” 7 March, 1873, ) M. 50 ADDENDU ADDENDUM TO NOTE 1, Pacer 38. Tux apparent antagonism between the theorem of the text and that of Laplace suggests a few additional words. The theorem of Laplace is that “in whatever manner the waters of the ocean act upon the earth, either by their attraction, their pressure, their friction, or by the various resist- ances which they suffer, they communicate to the axis of the earth a motion which is very nearly equal to that it would acquire from the action of the sun and moon upon the sea, if it form a solid mass with the earth.” (Méc. Cél., Bowditch [3345 ].) The theorem is demonstrated in two distinct, quite different, manners. The last demonstration is founded upon the principle of the “conservation of areas;” and as the result of éhis demonstration the proposition is stated in the above quoted words. The first demonstration is purely analytical, and, after stating that “ this fluid” (7. e. of the ocean) “acts upon the terrestrial spheroid by its pressure and by its attraction,” Laplace proceeds to find the analytical expressions for the precession and nutation-producing couples due to this pressure and to this attraction as they are modified by the attraction of the sun and moon upon the fluid. He then proceeds to caleulate these couples for the material substance of the ocean, considered as rigidly connected (or forming a solid mass) with the earth. He finds the couples, so calculated, respectively, identical in the two cases, and epitomizes the result as follows: “the phenomena of the precession of the equinoxes and the nutation of the earth’s axis ave exactly the same as if the sea form a solid mass with the spheroid which it covers.” [3287. ] But this demonstration is limited by the assumption that ‘the sea wholly covers the terrestrial spheroid or nucleus, that is of a regular depth, and suffers no resistance from the nucleus;” and bolh demonstrations imply an ocean of (relatively) small depth. Under the last mentioned treatment of the subject the proposition of Laplace and that which I demonstrate are but the extreme phases exhibited by the solution of a problem, according as the datum be that the depth of the sea is minute (in which case its entire precession-producing couple, not effectively exerted upon its own mass, is almost wholly transferred to the solid nucleus); or that the nucleus is very small, in which case the lost precession-producing couple of the fluid is but in small part transferred to the nucleuas—or wholly disappears with the vanishing of the latter. I think, however, that the last-mentioned (first in point of order) demonstration of Laplace is not as general as the language quoted [3287] would indicate. The omission of variations of the radius- vector, #, in all the integrations gives rise to errors which do not seem to me to be identical in the two processes by which the couples are calculated, when the variation of depth is very small. An apt illustration of the above remarks is derived from the supposition—as admissible as any other—that, the depth, y, is constant. In this case, whatever be the ellipticity of the solid nucleus, the value of y (the height of the diurnal or precession-affecting tide, see [2253] [3333] and also p. 36) is zero, and, of course, the couples [3272] and [3273] become zero—as will be found by performing the integrations in those equations. So, of course, do expressions [3284] and [3285] become zero, with y made constant. But these last should not be zero except for the case of sphericity of the nucleus.'| The expressions do, not seem to me capable of sustaining the inference which I have aes ene when the depth of the sea is uniform; a case which most naturally presents itself to the mind. A shell of slight internal ellipticity and small uniform thie kness has a precessional coéficient of four-fifths the value of that of a shell bounded by surfaces of equal elli : er pticity. Hence in general the variations of R (or the ellipticity) produce effects insensible compared to those of the depth. ‘ ; - ADDITIONAL NOTES. 51 Mr. Airy (‘Tides and Waves, Art. 127) bases his demonstration of the theorem exclusively upon the principle of the conservation of areas, remarking at the outset, ‘‘if the earth and sea were so entirely disconnected that one of them could revolve for any length of time with any velocity, in- creasing or diminishing in any manner, while the other could revolve with any other velocity changing in any other manner, we could pronounce nothing as to the effect of the fluctuation” (tidal) “upon precession.” A spheroidal nucleus wholly covered by an ocean of uniform depth, suffering no resistance, does not seem to me to lack much for fulfilling the above conditions. If velocities are generated in the waters of the ocean by solar (or lunar) attraction, the centrifugal forces due to them might be looked to (though not alluded to by Laplace) as agents for transferring, from the fluid to the nucleus, the precession-producing couples due to the fluid mass, especially in the above hypothetical case. It will be found, however, by reference to the expressions [2260], that they give rise to no couple, and are, moreover, very minute. The motion which the displacements [2260] [2261] indicate is a slight oscillation of the axis of the fluid envelope, moving as a solid, about the axis of the nucleus, the angular distance between these axes being slightly less than 2 seconds: it is, I presume, that which a non-rotating shell would have were the attracting body, with constant distance and declination, to move, with angular velocity n, in right ascension. In the case in hand it is the fluid shell which revolves and, suffering no change of form, would be itself affected by its proper precessional couple to the exclusion of the oscillation above described. PACD SD setae OO NGA NeOLRaBES: Nore To PAGE 11. ® The process indicated is a more legitimate carrying out of the methods peculiar to this paper than what follows in the text. The tangent of MM (35) may be (approx.) taken for the sine, and the cosine taken constant at unity, as may also be the cos J’. From (32) we may calculate by developing and neglecting terms in which sin? 7’ enters sin t = (1 — cos? 2) = sin J—sin I’ cos I cos ngf sin 7 cos 7 = sin I cos [—sin I’ cos2I cos n,t—} sin*J’ sin2T cos?n,t Introducing these values in (38) and (39), and integrating we get expressions identical with 44 and 45, except a (practically) immaterial difference in the coefficient of ¢ in the first which becomes 1—4sin’l’ instead of 1 — 3 sin7/’. Nove To PAGE 24. © The foregoing interpretation of the symbolic integral in (7), adopted with hesitation from authors cited, is based on assumed constancy of the angle ¢; but this angle necessarily varies, slowly indeed, but progressively, by the azimuthal motion measured by n sina. The conditions for the formation of a leminiscate are not, therefore, rigidly fulfilled. It will be found, however, taking into account a complete excursion, that the slight increment which will enure to the moment of the quantity of motion, sints from this cause on one side of the vertical, will be neutralized on the other, in consequence of thé opposing signs of cos 9, in opposite azimuths; or, at least, the resultant increment or decrement will be a quantity of the second order in minuteness, and hence, affecting only in the same degree the azimuthal motion a Nore T0 PAGE 39. © The differential attraction of the sun on any length dy of the rod, at distance y from the earth’s S r —z)? centre is ( = =) dz, r being sun’s distance. Integrate from y¥—y to y= ‘the earth’s re radius). 52 ADDITIONAL NOTES 1 R 1 x ) © ee rere oe The above divided by the coefficient of elasticity Z will give the elongation per unit of length at any point. Multiply by dz and integrate from z—0 to z= Kh, and the total elongation is Seiacie as lo (i—4)}. BE U—E = eat oee rl) Since log (1 — = =— fi = aS —) es &e., the foregoing will reduce, approximately to “ r r Amie 3 Tr 2S 8Er If 7 = earth’s mass, g = gravity at its surface, and 1 the ratio of J7to S, and m the ratio of is n length of rod of weight Z (per square inch section) to R, we shall have: S=ngk?, H=mgR, and the above expression for total elongation becomes: an R 9 2n ph (2) 3m Vine Take “© — 4000 1. = 4000 x 5280 feet; n= 316000; m—=.479 (the latter value 7 92000000 23000’ based on # — 384 mill’ns Ibs. per square inch, and a steel rod of that section to weigh 3.4 lbs. per foot length) and the total elongation (2) becomes 0.975. The maximum extension per unit of length is at the centre, and is found by putting y—0 in (1) and dividing by #. It is oe a Be = LB} ip i F -000000055, indicating a strain of 1.87 per square inch. The ratio of the fofal elongation (2) to the total length of the rod 2 is two-thirds of the above, indicating about that ratio for the ellipticities of superficial and central strata of a steel globe distorted by the sun’s attraction ; a result thus rudely calculated which differs little from that given in Thomson and Tait, § 837. SMITHSONIAN CONTRIBUTIONS TO KNOWLEDGE. OAL — A CONTRIBUTION HISTORY OF THE FRESH-WATER ALG NOT hy AM Ei LC x’. BY HORATIO C. WOOD, Jr., M.D., PROFESSOR OF BOTANY, AND CLINICAL LECTURER ON DISEASES OF THE NERVOUS SYSTEM IN THE [ACCEPTED FOR PUBLICATION, FEBRUARY, 1872.) ADVERTISEMENT. Tue following memoir was referred for examination to Dr. John Torrey and Dr. F. A. P. Barnard, of Columbia College, New York. They recommended its pub- lication provided certain changes were made in the manuscript. ‘These having been made by the author, the work is published as a part of the series of “ Smith- sonian Contributions to Knowledge.” JOSEPH HENRY, Secretary, S. L. WASHINGTON, October, 1872. (iii ) PREFACE. OF all the various branches of Natural History, none has been more enthusias- tically and more successfully prosecuted in the United States than Botany. The whole field has been most thoroughly occupied, save only as regards certain of the lower cryptogams, and amongst the latter, it is the fresh-water Algew which alone can be said to have been almost totally neglected. In this fact lies my apology for offering to the scientific public the following memoir. In doing this, so far from thinking that the work contains no error, I hasten to disarm criticism, and to ask with solicitude for a favorable reception, in view of the difficulties of the investigation, which I have conducted alone, and almost unaided. The investigation was first undertaken in connection with my elementary studies of Materia Medica and Therapeutics, and has since been prosecuted at intervals amidst the distractions of medical teachings and practice, and in some cases with- out immediate access to authorities. The field covered is so wide that it is almost impossible to exhaust it, and, if it were not for rapidly increasing professional engagements, I would gladly devote more time to the subject ; but, as it is, I must leave to others to carry on the work thus begun. While saying this, it is but just to state that nothing here published has been done hastily, but that all is the result of arduous and conscientious investigation. A very large part of my material has been of my own gathering, and was studied whilst fresh; but I am indebted to several persons for aid by collections. First of all, I desire to offer my thanks to Dr. J. 8. Billings, U.S. A., and to Professor Ravenel, of South Carolina; to the former for assistance in various ways, and for collections made near Washington City; to the latter for very large collections made in ‘Texas, South Carolina, and Georgia. I am also indebted to Mr. C. F. Austin for a large collection gathered in Northern New Jersey, to Mr. William Canby for some beautiful specimens obtained in Florida, to Professor Sereno Watson for Rocky Mountain plants, and to Dr. Frank Lewis for a number of White Mountain desmids. These various collections were partly dried and partly preserved in a watery solution of carbolic acid or of acetate of alumina, both of which I have found more or less satisfactory preservatives. The present investigations embrace all families of the fresh-water alge except the Diatomacee, which, as every one knows, are so numerous as to constitute in Cv) vi PREFACE. themselves a special study. As I have paid no attention to these plants, they are of course not included in this memoir. In the synonymy I have generally followed Prof. Rabenhorst. ‘The original de- scriptions of the forms, especially those of the older authorities, are very frequently so meagre and obscure, that the species cannot be recognized by them with any cer- tainty, Prof, Rabenhorst has gone over the ground most carefully, with access to the whole literature of the subject and probably to all extant type specimens, and his decisions are, no doubt, as accurate as the circumstances will allow. To attempt to differ from them, to go behind his work to the original sources and make fresh interpretations, would cause endless confusion. I have, therefore, nearly always contented myself with his dictum, and have referred to him as the authority for the names used, ‘The following references were omitted through a misunderstanding from the first portion of the text. Page 14. Calospheerium dubium, GruNNow. Rapennorst, Flora Europ. Algarum, Sect. I. p. 55. “ 15. Merismopedia convoluta, Briésisson. Rasennorst, Flora Europ. Algarum, Sect. I. p. 58. “18. Oscillaria chlorina, Kurzinc. RasBenuorst, Flora Europ. Algarum, Sect. I. p. 97. “ 18. O. Fréhlichii, Kivzina. Rasennorst, Flora Europ. Algarum, Sect. I. p. 109. “ 19. O. nigra, VaucuER. Rasennorst, Flora Hurop. Algarum, Sect. I. p. 107. “ 19. O. limosa, AGARDH. WRasenuorst, Flora Europ. Algarum, Sect. I. p. 104. «21. Chthonoblastus repens, Ktrzinc. Rapennorst, Flora Europ. Algarum, Sect. I. p. 132. “ 22. Lyngbya muralis, AGArpu. Harvey, Nereis Boreali-Americana, pt. III. p. 104. In the text after the “ Habitat,” a name is quoted as the authority therefor; if such a name be in brackets, it signifies that the specimens were simply collected by such individual, but that the identification was made by some one else; when there is not aname wninclosed in brackets, it is meant that the identification was made by the author of this memoir, Since the present memoir has gone to press, I have received from the author a copy of “ Algwe Rhodiacew. A list of Rhode Island Algw, collected and prepared by Stephen 'T. Olney, in the years 1846-1848, now distributed from his own her- barium.” In the introduction to this list, Mr. Olney says: “Of the fresh-water species, I have few for distribution. These were obtained mainly in the environs of this city, and were placed in twenty-seven small vials in Goadsby’s solution, and sent to Prof. Harvey, who submitted them to the judgment of the most learned Eng- lish botanist in this particular department, G. H. K, Thwaites, Esq., then of Bris- tol, England. The large number of species found in this collection, in so limited a range, and collected within a very short period, is surprising, and shows what more persistent collections will develop. I have not time to collate the numerous publications of the lamented Prof. Bailey, or I might have made the list of this portion of Rhode Island plants more complete.” Che chlorosperms of this list are as follows — Porphyra vulgaris, Aa.-Harv. Ner Bor. Am. 3. 53. Newport Bangia fuscopurpurea, Lynes.-Harv. Ner. Bor. Am. 3.54. Southern Rhode Island PREFACE. vii Enteromorpha intestinalis, LyNGB.-Hary. Ner. Bors Am. 3. 56. Providence to Newport. Enteromorpha compressa, GREV.-HAary. Ner. Bor. Am. 3. 56. Southern Rhode Island. Enteromorpha tlathrata, Grry.-Hary. Ner. Bor. Am. 3.56. Newport. Ulva latissima, L.-Hary. Ner. Bor. Am. 3. 59. Providence. Ulva lactuca, L.-Harv. Ner. Bor. Am. 3. 60. Providence. Tetraspora lacunosa, CHAvv.-Hary. Ner. Bor. Am. 3.61. 7. perforata, Batuny Mss. Providence. Tetraspora lubrica, AG. Providence. Batrachospermum pulcherrimum, Hass. Providence. Batrachospermum moniliforme, Rortu.-Hary. Ner. Bor. Am. 38. 63. Providence. Chetophora endiveefolia, AG.-Harv. Ner. Bor. Am. 3. 69. Providence. Draparnidia glomerata, AG.-Hary. Ner. Bor. Am. 3. 72. Previdence. Stigeoclonium minutum, Kurz. Providence. Cladophora rupestris, L.-Harv. Ner. Bor. Am. 3. 74. Newport. Cladophora glaucescens, Grirv.-Hary. Ner. Bor. Am. 3. 77. Rhode Island. Cladophora refracta, Rotu.-Hary. Ner. Bor. Am. 3.79. Southern Rhode Island. Cladophora Rudolphiana, Ac.-Hary. Ner. Bor. Am. 3. 80. Providence. Cladophora gracilis, Grirk.-Hary. Ner. Bor. Am. 3. 81. Little Compton. Cladophora fracta, Harv. Ner. Bor. Am. 3. 82. Rhode Island, Bailey. Chetomorpha xrea, Dituw.-Har. Ner. Bor. Am. 3. 86. Newport, ete. Chetomorpha Olneyi, Hany. Ner. Bor. Am. 3. 86. Little Compton. Chetomorpha longiarticulata, Harv. Ner. Bor. Am. 3. 86. Little Compton. var. crassior, Hary. Ner. Bor. Am. 3. 86. Little Compton. Chetomorpha sutoria, Berx.-Hary. Ner. Bor. Am. 3.87. Newport. Zygnema malformatum, Hass. 1. 147. Providence. Zygnema cateneforme, Hass. 1. 147. Providence. Zygnema Thwaitesti, OLNEY, n. s. Near Z subventricosum, Providence. Zygnema longatum, Hass. 1. 151. Providence. Zygnema striata, OLNEY, n. s. “ Cells evidently striated,” Thwaites. Providence. Tyndaridea bicornis ? Hass. 1. 162. Providence. Tyndaridea insignis ? Hass. 1. 163. Providence. Mesocarpus parvulus, Hass. 1. 169. Providence. Mougeotia genuflexra, Ac.-Hass. 1. 173. Providence. Vesiculifera concatenata, Hass. 1. 201. Providence. Vesiculifera xqualis, Hass. 1. 205. Providence. Vestculifera bombycina, Hass. 1. 208. Providence. Vesiculifera Candollii, Hass. 1. 208. Providence. Bulbochexte Thwaitesii, OLNEY, n. s. Providence. Lyngbya majuscula, Harv. Bor. Am. 3. 101. Providence. Spheroplea virescens, BERK. Providence. Spheroplea punctalis, BERK. Providence. Tolypothrix distorta, Ktrz.-Hass. 1. 240. Calothrix confervicola, Ac.-Harv. Ner. Bor. Am. 3. 105. Providence. Calothrix scopulorum, Ac.-Harv. Ner. Bor. Am. 3.105. Providence. Hyalotheca dissiliens, Brey.-Raurs. Des. 51. (Gloeoprium.) Providence. Hyalotheca mucosa, Euru.-RAurs. Des. 53. Providence. Didymoprium Grevillit, Kurz.-Raurs. Des. 61. Rhode Island, Bailey. Didymoprium Borreri, Raurs. Des. 58. Rhode Island, Bailey. Desmidium Swartzii, Ac.-Ratrs. Des. 61. Throughout United States, Bailey. Aptogonum Baileyi, RatFs Des 209. Worden’s Pond, Rhode Island, Bailey. Micrasterias rotata, Raurs. Des. 71. Providence. Micrasterias radiosa, AG.-Raurs. Des. 72. Maine to Virginia, Bailey. Micrasterias furcata, Raurs. Des. 73. Worden’s Pond, Rhode Island, Bailey. Micrasterias Orux-Melitensis, Raurs. Des. 73. Maine to Virginia, Bailey. Micrasterias truncata, Bres.-Ratrs. Des. 75. United States, Bailey. Vill PREFACE. Micrasterias foliacea, BArLEY-IXALFs. Desm. 210. Worden’s Pond, Rhode Island, Bailey. Micrasterias Bayleyi, Raves. Desm. 211. thode Island, Bailey. Euastrum oblongum, Raurs. Des. 80. Rhode Island, Bailey. Euastrum crassum, Kirz.-Raurs. Des. 81. Rhode Island, Bailey. Euastrum ansatum, Euru.-Raurs. Des. 85. £. binale Knrz. Providence. Euastrum elegans, Kirz.-Rars. Des. 89. Providence. Euastrum binale, Raurs. Desm. 91. Providence. Cosmarium cucumis, Corps.-RALFs. Desm. 93. United States, Bailey. Cosmarium bioculatum, Raurs. Des. 99. Providence. Cosmarium Meneghinii, Bres.-RArs. Des. 96. United States, Bailey. Cosmarium crenatum, Raurs. Des. 96. Providence. Cosmarium amenum, Bres.-Ratrs. Des. 102: Providence. Cosmarium ornatum, Raurs. Des. 104. Providence. Cosmarium connatum, BReB.-Raurs. Des. 108. Providence. Cosmarium Cucurbita, Raurs. Des. 109. Providence. Cosmarium grandituberculatum, OLNEY, 0. 8. ; “near C. cucumis, but with large tubercles on the frond.” Providence. Staurastrum orbiculare, Raters. Des. 125. Providence. Staurastrum hirsutum, Raurs. Des. 127. Providence. Staurastrum Hystriz, RAurs. Des. 128. Providence. Staurastrum gracile, Raurs. Des. 136. Providence. Staurastrum tetracerum, Rates. Des. 137. United States, Bailey. Staurastrum cyrtocerum, BRres.-Raurs. Des. 139. Providence. Tetmemoras Brébissoni, Raurs. Des. 145. Providence. Tetmemoras granulatus, RAuFs. Des. 146. Providence. Penium margaritaceum, Bres.-Raurs. Des. (Closterium Eur.) Providence. Penium Digitus, Bres.-Raurs. Des. 151. (Closteriwm lamellosum.) Docidium nodulosum, Bres.-Raurs. Des. 155. Maine to Virginia, Bailey. Docidium Baculum, Bres.-Rates. Des. 158. United States. Bailey. Docidium nodosum, BattBy-Raurs. Des. 218. United States, Bailey. Docidium constrictum, BatLEy-Ratrs. Des. 218. Worden’s Pond, Bailey. Docidium verrucosum, BAtLEY-Ratrs. Des. 218. Rhode Island, Bailey. Docidium verticillatum, BAtLEY-RAurs. Des. 218. Worden’s Pond, Bailey. Closterium Lunula, Huru.-Raurs. Des. 163. New England, Bailey. Closterium moniliferum, Huru.-Raurs. Des. 163. New England, Bailey. Closterium striolatum, Euru.-Raurs. Des. 173. New England, Bailey. Closterium cuspidatum, Battey-Ratrs. Des. 219. Worden’s Pond, Bailey. Pediastrum tetras, Raurs. Des. 182. New England, Bailey. Pediastrum heptactis, Raurs. Des. 183. Providence. Pediastrum Boryanum, MeneGcu.-Raurs. Des. 187. Maine to xi ailey. Pediastrum ellipticum, Hass.-Raurs. Des. 188. Maine to Tings oe Scenedesmus quadricauda, BrEB.-Raurs. Des. 190. Maine t irgini iley. Scenedesmus obtusus, MEYEN.-Raurs. Des. 193. Maine to vi INTRODUCTION. AxtHoucH beset with difficulties in the outset, no branch of natural science offers more attractions, when once the study is fairly entered upon, than the fresh- water alge. The enthusiasm of the student will soon be kindled by the varicty and beauty of their forms and wonderful life processes, and be kept alive by their abundance and accessibility at all seasons of the year; for unlike other plants, the winter with them is not a period of counterfeited death, but all seasons, spring, summer, autumn, and winter alike, have their own peculiar species. They have been found in healthy life in the middle of an icicle, and in the heated waters of the boiling spring; they are the last of life alike in the eternal snow of the moun- tain summit and the superheated basin of the lowland geyser. In their investigation, too, the physiologist can come nearer than in almost any other study to life in its simplest forms, watching its processes, measuring its forces, and approximating to its mysteries. Sometimes, when my microscope has revealed a new world of restless activity and beauty, and some scene of especial interest, as the impregnation of an cedogonium, has presented itself to me, I confess the. enthusiastic pleasure produced has been tempered with a feeling of awe. To any on whom through the want of a definite pursuit the hours hang heavy, to the physiologist who desires to know cell-life, to any student of nature, I can commend most heartily this study as one well worthy of any pains that may be spent on it. An aquarium will often, in the winter time, give origin to numerous interesting forms, but it is not a necessity to the fresh-water algologist; besides his microscope and its appliances, all that he absolutely needs is a few glass jars or bottles and the fields and meadows of his neighborhood. — The great drawback to the investigation of these plants has been the want of accessible books upon them. In the English language there is no general work of value, and the various original memoirs are separated so far and wide in the Continental and English journals, as to be of but little use to most American readers. The Flora Europewum Algarum Aque Dulcis et Submarine, of Prof. Ra- benhorst, has done much to facilitate the study, and its cheapness brings it within the reach of all. It merely gives, however, brief diagnoses of the various species, but with the present memoir will, I trust, suffice for the American student, at least until he is very far advanced in his researches. il November, 1871. ( 1 ) INTRODUCTION. Cs) A certain amount of experience and knowledge of the subject greatly facilitates the collection of these plants, but scarcely so much as in other departments of eryp- togamic botany, since most of the species are so small that the most experienced algologist does not know how great the reward of the day’s toil may be until he places its results under the object glass of his compound microscope. In order to aid those desirous of collecting and studying these plants, I do not think I can do better than give the following hints as to when and where to find, and how to preserve them. There are three or four distinct classes of localities, in each of which a different set of forms may be looked for. These are: stagnant ditches and pools; springs, rivulets, large rivers, and other bodies of pure water ; dripping rocks in ravines, &e.; trunks of old trees, boards, branches and twigs of living trees, and other localities. In regard to the first—stagnant waters—in these the most conspicuous forms are oscillatori and zygnemacee. The oscillatorie may almost always be recog- nized at once, by their forming dense, slimy strata, floating or attached. gene- rally with very fine rays extending from the mass like a long, delicate fringe. The stratum is rarely of a bright green color, but is mostly dark; dull greenish, blackish, purplish, blue, &c. The oscillatoriz are equally valuable as specimens at all times and seasons, as their fruit is not known, and the characters defining the species do not depend upon the sexual organs. The zygnemas are the bright green, evidently filamentous, slimy masses, which float on ditches, or lie in them, entangled amongst the water plants, sticks, twigs, &c. They are only of scientific value when in fruit, as it is only at such times that they can be determined. Excepting in the case of one or two very large forms, it is impossible to tell with the naked eye with certainty whether a zygnema is in fruit or not; but there are -one or two practical points, the remembrance of which will very greatly enhance the probable yield of an afternoon’s search. In the first place, the fruiting season is in the spring and early summer, the latter part of March, May, and June being the months when the collector will be best repaid for looking for this family. Again, when these plants are fruiting they lose their bright green color and become dingy, often yellowish and very dirty looking—just such specimens as the tyro would pass by. ‘The fine, bright, green, handsome masses of these alge are rarely worth carrying home. After all, however, much must be left to chance; the best way is to gather small quantities from numerous localities, keeping them separate until they can be examined. Adhering to the various larger plants, to floating matters, twigs, stones, &c., in ditches, will often be found filamentous alge, which make fine filmy fringes around the stems, or on the edges of the leaves; or perchance one may meet with rivulariz or nostocs, &c., forming little green or brownish balls, or indefinite protuberances attached to small stems and leaves. These latter forms are to be looked for especially late in the season, and whenever seen should be secured. In the latter part of summer, there is often a brownish, gelatinous scum to be seen floating on ditches. Portions of this should be preserved, as it frequently con- tains Interesting nostocs and other plants. TEN DRO) DUCA lr ONE 3 vo In regard to large rivers, the time of year in which I have been most successful in such localities is the latter summer months. Springs and small bodies of clear water may be searched with a hope of reward at any time of the year when they are not actually frozen up. I have found some exceedingly beautiful and rare alge in such places as early as March, and in open seasons they may be col- lected even earlier than this. ‘The desmids are most abundant in the spring, and possibly most beautiful then. They, however, rarely conjugate at that time, and the most valuable specimens are therefore to be obtained later—during the summer and autumn months; at least, so it is said; and the experience I have had with this family seems to confirm it. ivulets should be watched especially in early spring, and during the summer months. From the time when the weather first grows cool in the autumn, on until the cold weather has fairly set in, and the reign of ice and snow commences, is the period during which the alge hunter should search carefully all wet, dripping rocks, for specimens. Amongst the stems of wet mosses—in dark, damp crevices, and little grottos beneath shelving rocks—is the alge harvest to be reaped at this season. Nostocs, palmellas, conjugating desmids, sirosiphons, various unicellular alge, then flourish in such localities. My experience has been, that late in the autumn, ravines, railroad cuttings, rocky river-banks, &c., reward time and labor better than any other localities. The vaucherias, which grow frequently on wet ground, as well as submerged, fruit in the early spring and summer in this latitude, and are therefore to be col- lected at such times, since they are only worth preserving when in fruit. In regard to alge which grow on trees, I have found but a single species, and do not think they are at all abundant in this latitude. Farther south, if one may judge by Professor Ravenel’s collections, they are the most abundant forms. Although perhaps of but little interest to the distant collector, yet for the sake of those living nearer, I will occupy a few lines with an account of the places around Philadelphia which will best repay a search for fresh-water alge. As is well known, below the city, there is what is known as the “‘ Neck,” a perfectly level extent of ground lying in the fork between the rapidly approaching rivers, Schuyl- kill and Delaware. ‘This is traversed by numerous large ditches, and, especially just beyond the city confines, has yielded to me an abundant harvest. My favorite route is by the Fifth Street cars to their terminus, then across the country a little to the east of south until the large stone barn, known as “Girard’s Barn,” is reached. A large ditch lies here on each side of the road, which is to be followed until it crosses the Pennsylvania Railroad, then along this to the west, until the continuation of Tenth Street crosses it. Here the ditches cease, and the steps are to be turned homeward. From Girard’s barn to the crossing just alluded to, ditches great and small lie all along and about the route, ditches which have often most abundantly rewarded my search, and enabled me to return home richly laden. The best season for collecting here is from March to July, and again in October, when some of the nostocs may be looked for. Crossing the river Delaware to the low country below and above the city of Camden, the collector will find himself in a region similar to that just described, 4 INTRODUCTION. and like it cut up by numerous ditches, in which are pretty much the same forms as in the “Neck.” But by taking the Camden and Atlantic cars for twenty to forty miles into New Jersey to what is known as the “ Pines,” he will get into a very different country; low, marshy, sandy grounds, with innumerable pools, and streams whose dark waters, amber-colored from the hemlock roots over which they pass, flow sluggishly along. I have been.somewhat disappointed in my collections in such localities. Fresh-water alge do not appear to flourish in infusion of hem- lock, and consequently the streams are very bare of low vegetable life. On the other hand, in pools in the more open places, my search has been repaid by find- ing some very curious and interesting forms, which apparently are peculiar. North of Philadelphia are several places, which at certain seasons will richly reward the microscopist. Along the Delaware River, there is a similar country and flora to that of the “Neck.” But back from the river things are quite dif- ferent. The North Pennsylvania Railroad passes near Chelten Hills, some eight miles or so from the city, through some deep rock cuttings, which are kept con- stantly dripping by numerous minute springs bursting from between the strata. At the proper season, these will yield an abundant harvest. Besides these, there is also a stream of water with ponds running along by the road, which should be looked into. I have seldom had more fruitful trips than some made very early in the spring to this locality; but then it was in little pools in the woods, and espe- cially in a wooded marsh or meadow to the left of the road, some distance beyond the station, that I found the most interesting forms. The Schuylkill River and its banks have afforded materials for many hours of pleasant work. In the river itself a few very interesting forms have been found; but it is especially along its high banks that the harvest has been gathered. The dripping rocks and little wood pools in the City Park are well worth visiting; but the best locality is the western bank, along the-Reading Railroad, above Mana- yunk, between it and the upper end of Flat Rock tunnel. Down near the river, at the lower end of the latter, will be found a number of beautiful, shaded rocky pools, which, in the late summer, are full of Chaetophora and other alge. Along the west rocks of the river side of the bluff, through which the tunnel passes, are to be found, late in the fall, numerous alge. It is here that the Palmella Jessenii grows in such abundance. West of the city, in Delaware and Chester Counties, is a well wooded and watered, hilly country, in which, here and there, numerous fresh-water algee may be picked up. As to the preservation of the alge—most of the submerged species are spoiled by drying. Studies of them should always, when practicable, be made whilst fresh. Circumstances, however, will often prevent this, and I have found that they may be preserved for a certain period, say three or four months, without very much change, in a strong solution of acetate of alumina. An even better preservative, however, and one much more easily obtained, is earbolie acid, for I have studied desmids with great satisfaction, which had been preserved for five or six years in a watery solution of this substance. In regard to the strength of the solution I have no fixed rule, Always simply shaking up INTRODUCTION. 5 a few drops of the acid with the water, until the latter is very decidedly impreg- nated with it, as indicated by the senses of smell and taste. Almost all species of alge which are firm and semi-cartilaginous, or almost woody in consistency, are best preserved by simply drying them, and keeping them in the ordinary manner for small plants. ‘The fresh-water alge which bear this treatment well belong to the Phycochromophycee, such as the Nostocs, Seytonema, &c., the true confervas not enduring such treatment at all. When dried plants are to be studied, fragments of them should be soaked for a few minutes in warm, or for a longer time in cold water. The only satisfactory way that alge can be finally prepared for the cabinet is by mounting them whole or in portions, according to size, for the microscope. Of the best methods of doing this, the present is hardly the time to speak; but a word as to the way of cleaning them will not be out of place. Many of them, especially the larger filamentous ones, may be washed by holding them fast upon an ordinary microscope slide, with a bent needle or a pair of forceps, and allowing water to flow or slop over them freely, whilst they are rubbed with a stiffish camel’s-hair pencil or brush. In other cases, the best plan is to put a mass of the specimens in a bottle half full of water, and shake the whole violently; drawing off the water from the plants in some way, and repeating the process with fresh additions of water, until the plants are well scoured. At first sight, this process would seem exceedingly rough, and liable to spoil the specimens, but I have never seen bad results from it, at least when practised with judgment. The water seems so to envelop and protect the little plants that they are not injured. After all, in many instances it appears impossible to clean these algz without utterly ruining and destroying them—the dirt. often seeming to be almost an inte- grant portion of them; so that he who despises and rejects mounted specimens, simply because they are dirty and unsightly, will often reject that which, scienti- fically speaking, is most valuable and attractive. In finally mounting these plants, the only proper way is to place them in some preservative solution within a cell on a slide. After trial of solution of acetate of alumina and various other preservative fluids, I have settled upon a very weak solution of carbolic acid, as the best possible liquid to mount these plants in. Acetate of alumina would be very satisfactory were it not for the very great tendency of the solution to deposit minute granules, and thus spoil the specimens. As every one knows, the great difficulty in preserving microscopic objects in the moist way is the perverse tendency of the cells to leak, and consequently slowly to allow entrance to the air and spoil the specimen. As I have frequently found to my great chagrin, the fact that a slide has re- mained unchanged for six months, or even a year, is no guarantee that it will remain so indefinitely. It becomes, therefore, exceedingly important to find some way of putting up microscopic objects that can be relied on for their preservation. Where carbolated glycerine jelly or Canada balsam can be used, the solid coating which they form around the specimens constitutes the best known protection. Except in the case of the diatoms, however, these substances so shrivel and distort the fresh- water alge immersed in them as to utterly ruin them. I lost so many specimens 6 INTRODUCTION. by the old ways of mounting, that, becoming disheartened, I gave ep aul idea of making a permanent cabinet, until a new cement, invented by Dr. J. G. Hunt, of this city, was brought to my notice. This is prepared as follows:— “Take damar gum, any quantity, and dissolve it in benzole; the solution may be hastened by heat. After obtaining a solution just thick enough to drop readily from the brush, add enough of the finest dry oxide of zinc—previously triturated in a mortar with a small quantity of benzole—until the solution becomes white when thoroughly stirred. If not too much zine has been added, the solution will drop quickly from the brush, flow readily, and dry quickly enough for convenient work, It will adhere, if worked properly, when the cell-cover is pressed down, even when glycerine is used for the preservative medium. Keep in an alcohol- lamp bottle with a tight lid, and secure the brush for applying the cement in the lid of the bottle.” Its advantages lie in the circumstance, that the glass cover can be placed upon the ring of it whilst still fresh and soft, and that in drying, it adheres to both cover and slide, so as to form a joint between them of the width of the ring of cement, and not, as with asphaltum, gold size, &c., simply at the edge and upon the outside of the cover. It is readily to be seen how much less liability to leakage must result from this. ‘The method of mounting with it is as follows: A ring of any desired size is made, by means of an ordinary Shadbolt’s turn-table, upon a slide, which is then placed to one side to dry. When required for use, the specimen, cover, &c., being all prepared and ready, the slide is again placed upon the turn- table and a new ring of cement put directly upon the old one. ‘The specimen is immediately placed within the cell thus formed, and the requisite quantity of the carbolated water placed upon it. The cover, which must be large enough to entirely or nearly cover the cement ring, is now picked up with the forceps, the under side being moistened by the breath to prevent adhesion of air-bubbles, and placed care- fully in position. It is now to be carefully and equably pressed down with some force. By this, any superfluous water is squeezed out and the cover is forced down into the cement which rises as a little ring around its edge. The pressure is best made with a stiff needle, at first on the centre and then upon the edges of the cover, which may finally be made slowly to revolve underneath the needle point. The slide may then be put aside to dry; or, better, an outside ring of the cement thrown over its edge in the usual manner. Where a deep cell is required, several coats of the cement should be placed one over the other, each being allowed to dry in turn. If time be an object, and only a shallow cell be necessary, the first ring of cement may be dispensed with, and the whole mounting of the specimen be done in a few minutes. Even with this cement and the utmost care in mounting, the cabinet should be occasionally inspected, for there will always be some slides into which air will penetrate. When such are found, efforts may be made to stop the leak by new rings of cement overlaid upon the old, but very often entire remount- ing of the specimen is the only satisfactory cure. The classification which I have adopted in this memoir is that of Professor Ra- benhorst. I have finally selected it, not as being absolutely natural, but as conve- nient, and as rarely doing much violence to the natural relations of the various species. INTRODUCTION. 7 Our knowledge of the life-history of the alge must make very many advances before the true system can be developed, and abstinence from adding to the present numerous Classifications is an exhibition of self-control not very common. There are, however, certain great groups, which are already plainly foreshadowed, and which no doubt will be prominent points in the perfected classification. Amongst these are the Conjugate, or those plants in which sexual reproduction occurs by the union of two similar cells. In the present paper all the plants of this family described are together, since the diatoms are not noticed ; but in Raben- horst’s work the latter plants are very widely separated from their fellows, and this seems to me the weak point of the Professor’s system. é i \ ¥ « : i 1 ‘A. in . : i 5 7 a i - ; , a P _ f 7 f i ‘ 7 7 ita i : ” ve | - ad i i i , i yy By T q : <; 7 " 1 — ~ : ip r ; ) at ' ; i AY 1 i ' = ‘ vp ‘ TABLE OF CONTENTS. Advertisement . A . Preface Introduction ‘ , f Class PHYCcOCHROMOPHYCEE Order CysTIPHOR& Family CHRoococcAcE£ Order NEMATOGENE® Family OscILLARIACE Family NosrocHacE® Family RIivuLARIACE® Family ScyroNEMACER Family SrrostpHonacEs® Class CHLOROPHYLLACEE Order CoccopHycEs& Family PALMELLACE® Family PRoTococLaAcEs Family VoLvocinEx Supplement : ‘ ; Geographical List of Species . Bibliography . Index : : Explanation of the Plates, PAGE 10 10 17 78 78 85 98 Order ZYGoPHYcEz Family DesMIDIACE® Family ZYGNEMACE Order S1pHopHYCEs Family HypRroGasTREa Family VAUCHERIACER Family ULvAcExZ . Family CoNFERVACE”: Family G2poGoNnraAcE® Family CHRrooLEPIDEs Family CH@®TOPHORACE . Class RnopopnycEe® Family PorpHyRAce® Family CHANTRANSIACES . Family BATRACHOSPERMACE Family LEMANEACE® ( ix ) PAGE iii 100 100 159 174 175 176 182 186 188 203 205 213 214 215 217 221 FRESH-WATER ALGA OF THE UNITED STATES. Crass PH YCOCHROMOPHYCEZ. Plante wni- vel multicellulares, in aqua vigentes vel extra aquam in muco matricali nidulantes, plerumque familias per cellularum generationes successivas ortas formantes. Cytioderma non siliceum, combustibile. Cytioplasma phycochromate coloratum, nucleo destitutum, granulis amylaceis plerumque nullis. Propagatio divisione vegetativa, gonidiis immobilibus vel sporis tran- quillis. Unicellular or multicellular plants living in water, or incased in a mater- nal jelly out of it, mostly in families formed from successive generations of cells. Cytioderm not siliceous, combustible. Cytioplasm an endochrome, brown, olivaceous, fuscous, &ec., destitute of nucleus, mostly without starch granules. Propagation by vegetative division, by immovable gonidia or tranquil spores. The phycochroms are plants at the very bottom of the scale, distinguished by the simplicity of their structure and the color of their protoplasm, which, instead of being of the beautiful green that marks chlorophyll, is fuscous, or yellowish, bluish, brownish, or sometimes particolored, and rarely greenish, but of a shade very distinct from the chlorophyll green, more lurid, bluish or yellowish, or oliva- ceous in its hue. The nucleus appears to be always wanting. The cell wall is oftentimes distinct and sharply defined, but in many instances it is not so, the walls of different cells being fused together into a common jelly in which they are imbedded. In a large suborder the wall is replaced by a sheath, which in some genera surrounds cells with distinct walls, in others, cells without distinct walls, and in still others, a long cylindrical mass of endochrome, which may be looked upon as a single cell. Many of the phycochroms are unicellular plants in the strictest sense of the word, but more often the cells are conjoined, so as to form little families, each cell of which is in a sense a distinct individual capable of separate life, yet the whole bound together into a composite individual. Rarely the phycochrom is a multi- 2 January, 1872. ( 9 ) 10 FRESH-WATER ALG& OF THE UNITED STATES. cellular plant in the stricter use of the term. Increase takes place by the multipli- cation of cells by division, and also by the formation of enlarged thick-walled cells, to which the name of spores has been given, although it is entirely uncertain whether they are or are not the result of sexual action. There are numerous peculiar forms of cell multiplication by division occurring in these plants, the dis- cussion of which will be found scattered through the remarks on the various families and genera. The method of reproduction, and in fact the life history in general, of the phy- cochroms, is still involved in such mystery, that I am not aware that absolute sexual generation has been demonstrated in any of them. This being the case, it is not to be wondered at that many have conjectured as possible, and some have roundly asserted as true, that the phycochroms are merely stages in the life history of higher plants ; that they are not species, and, consequently, that any attempt at describing such is little more than a busy idleness. In regard to some of them it has certainly been rendered very probable that they are merely fixed stages of higher plants, On the other hand, in the great bulk of the forms, no proof whatever has been given that they are such. ‘They all certainly have fixed, definite characters, capa- ble of being expressed and compared, so that the different forms can be defined, recognized, and distinguished. If, therefore, future discoveries should degrade them as subordinate forms, names will still be required, and definitions still be necessary to distinguish them one from the other, so long as they are common objects to the microscopist. If Nostoc commune, for example, were proven to be a peculiar state or develop- ment of Polytricwm commune, I conceive it would be still known as Nostoe commune. But, as previously stated, no proof whatever has as yet been furnished for the vast majority of the plants of this family, to show that they bear any such relation to higher plants; and until some such proof is forthcoming, certainly the only scien- tific way to act, is to treat them as distinct species. Orver Cystiphore. Plante unicellulares. Cellule singule vel plures in familias consociate. Unicellular plants. Cells single or consociated in families. In this order the cells are oblong, cylindrical, spherical, or angular. They are sometimes single, or more commonly are united by a common jelly into families, which sometimes are surrounded by distinct coats. The mucus or jelly, in which the cells are imbedded, is mostly, but not always, colorless, and varies in firmness from semifluid to cartilaginous. The division of the cells may take place either in one, two, or three directions or planes. Famity CHROOCOCCACEA. Character idem ac ordine. Characters those of the order. FRESH-WATER ALG OF THE UNITED STATES. ii Genus CHROOCOCCUS, Nxcent. Cellule globose ovales vel a pressione mutua plus minus angulose, solitarize vel in familias con- sociata, liberx (a vesica matricali non involute) ; cytiodermate achromatico, homogeneo, seepe in muco plus minus firmo confluente; cytioplasmate eruginoso vel pallide cxruleo-viridi, non rare luteolo vel aurantiaco, interdum purpurascente. Generationum successivarum divisio alternatim ad directiones tres. Syn.—Protococcus, Aa. et K1z., &., ex parte. Pleurococcus, MENGH. Globulinz et Protosphxrix, TURPIN, ex part. Cells globose, oval, or from mutual pressure more or less angular, solitary, or consociated in free families (not involved in a maternal vesicle); Cytioderm achromatic, homogeneous, often confluent into a more or less firm mucus; cytioplasm ruginous or pale bluish-green, not rarely yellowish or orange, sometimes purplish. Successive generations arising by alternate division in three directions. C. refractus, Woop. C. cellulis in familias solidas arcte consociatis, plerumque subquadratis, sepius triangularibus, rare angulosis; familiis sepius lobatis; cytiodermate tenui, vix visibile, achroo; cytioplas- mate subtiliter granulato, subfusco vel subluteo vel olivaceo, valde refrangente. Diam.—Cell gp59/’—saoo0 ’, Tare in cellulis singulis gg55/’; famil. ydyg’”—y49/”. Syn.—C. refractus, Woop, Prodromus, Proc. Amer. Philos. Soc., 1869, 122. Hab.—In rupibus irroratis prope Philadelphia. Cells closely associated together into solid families, mostly subquadrate, very often triangular, rarely multiangular; families often lobed; cytioderm thin, scarcely perceptible, transparent ; cytioplasm finely granular, brownish, olivaceous, or yellowish, highly refractive. Remarks.—The color of this species varies from a marked almost fuscous brown to a light yellowish-brown, the lighter tints being the most common. The cells are remarkable for their powerful refraction of the light, resembling often oil as seen under the microscope, especially if they be the least out of the focus. They are very closely joined together to form the families, many of which are composed only of four cells. Often, however, a large number of the cells are fused together into a large, irregular, more or less lobate family, and these sometimes are closely joined together into great irregular masses. I have occasionally seen large single cells with very thick coats, whose protoplasm was evidently undergoing division. Are such a sort of resting spore? The color of the protoplasm varies. Perhaps the more common hue is a sort of clay tint. Bluish-oliye and a very faint yellowish- brown are not rarely seen. The species grows abundantly on the wet rocks along the Reading Railroad between Manayunk and the Flat Rock tunnel. Fig. 5, pl. 5, represents different forms of this species; those marked a, magnified 750 diameters; 6, 470 diameters; c, 950 diameters. C. multicoloratus, Woop. C. in strato mucoso inter algas varias sparsus; cellulis singulis et sphericis, vel 2—4 (rare 8) aut angulis aut semisphericis aut abnormibus in familias oblongas eonsociatis ; cytiodermate crasso, hyalino, haud lamelloso; tegumentis plerumque nullis, interdum subnullis; cytioplasmate ple- Tumque homogeneo, interdum subtiliter granulato, vel luteo-viride vel cwruleo-viride vel luteo vel subnigro, vel brunneo, vel saturate aurantiaco, sepe ostro tincto. Diam.—Cell., sing. sine tegm., 5755/7 cum teg. y255/’; cell. in famil. sing. 3255"—ae0-” 7 tt 10 Vr. 6 A 7 vt Fam. long. g350//—a580''5 lat. ax00"—as00”- 12 FRESH-WATER ALG OF THE UNITED STATES: Syn.—G. multicoloratus, Woop, Prodromus, Proc. Amer. Philos. Soc., 1869, 122. Hab.—in rupibus humidis prope Philadelphia. C. occurring scattered in a mucous stratum with other alge ; cells spherical and single, or else angular semi-spherical or irregular and associated together in oblong families of from 2-4 (rarely 8); inner coat thick, hyaline, not lamellate; outer oes generally, wanting, sometimes indistinctly present; endochrome mostly homogeneous, sometimes minutely granular, either a yellowish-green or bluish-green, or yellowish or brown, or blackish, sometimes tinged with bright lake. The cells of this species do not appear to have any tendency to unite to form large masses or fronds. On the contrary they are generally very distinct. ‘Their color varies very much, in a larger number of instances they were a decided yellow- ish-green, tinged at some point or other with a beautiful lake. When several cells are formed by division out of one cell, a similar division of the very thick surround- ing gelatinous coat follows separating them finally entirely one from the other. I have seen a single cell which appears to be an encysted form of this, of which I give a drawing. Fig. 6, pl. 5, represents different forms of this species magnified 260 diameters. Cc. thermophilus, Woop. C. cellulis singulis aut geminis vel quadrigeminis et in familias consociatis, oblongis vel sub- globosis, interdum angulosis, haud stratum mucosum formantibus; tegumento crassissimo, achroo, haud lamelloso, homogeneo; cytioplasmate viride, interdum subtiliter granulato, inter- dum homogeneo. Diam.—Cellule singule sine tegumento longitudo maximay,yy”’, latitudo maxima 3545y/’. Syn.—C. thermophilus, Woop, American Journal Science and Arts, 1869. Hab.—Benton Springs, Owen Co., California (Mrs. Parz.). Cells single, geminate, or quadrigeminate and consociated into families, oblong or subglobose, sometimes angular, not forming a mucous stratum; tegument very thick, transparent, not lamellate, homogeneous ; cytioplasm green, sometimes minutely granulate, sometimes homo- geneous. Remarks.—Remarks upon this species will be found under the head of Nostoc calidarium, W ood. Genus GLOEOCAPSA, Krz. “ Cellule spherice aut singule aut numerose in familias consociate ; singule tegumento vesiculi- forme (cytiodermate tumido) incluse, post divisionem spontaneam in cellulas duas filiales factam utraque tegumento se induit, dam ambe tegumento matricali involute remanent; cellularum harum filialium iterum in duas cellulas divisione continuo repetita, tegumentum atavie restat et sese exten- dens familiam totam circumvelat. Cytioderma crassum, sepe crassissimum, cellule lumen crassitie gquans vel superans, achromaticum vel coloratum, plerumque lamellosum; lamella vel strata non raro discedentia. Cytioplasma eruginosum, ceruleo-viride, chalybeum, rufescens, luteo-fusecum, &c. Cellularum divisio directione ad tres dimensiones alternante. Cellule generationum ultimarum minores quam priorum sunt.” (Rab.) Syn.—Globulina et Bichatia, TURPIN, ex part. Gloeocapsa, K1z., ex part. Microcystis, MENEGH., ex part. Alle ° . “ : ° ° 5 . Cells spherical, either single or associated in numbers into families; the single cell included a vesiculiform tegument (the tumid cytioderm); this cell then undergoing division into two daughter- FRESH-WATER ALGAZA OF THE UNITED STATES. 13 tells, each of which has a distinct tegument, the whole being surrounded with that of the old mother- cell. This process of division is then repeated again and again, the original cell-wall remaining and surrounding the family thus formed. Cytioderm thick, often very thick, equalling, or exceeding in diameter the cavity of the cell, achromatic or colored, mostly lamellated, lamellie or strata not rarely separating. Cytioplasm of various colors, xruginous, bluish-green, chalybeate, reddish, yellowish- fuscous, &c. Division of the cells occurring in three directions. The last generation of cells smaller than the earlier ones. G. sparsa, Woop. G. in strato mucoso sociis algis variis sparsa; cellulis sphericis, vel oblongis vel ovatis, 2-8 in familias consociatis; familiis subglobosis vel subovatis, interdum numeroso-ageregatis ; tegu- mentis internis aureofuscis, firmis, rarissime coloris expertibus, homogeneis, vel lamellosis; tegumentis externis achromaticis, rare subachromaticis, plerumque vix visibilibus ; cytioplas- mate homogeneo. Diam.—Max. cell. oblong. sine tegum. long., s55/’; lat., 309’; cell. glob., sine tegum., We LAP ut sooo 3 Cum tegum., yy75/’; fam., 715’. Syn.—G. sparsa, Woon, Prodromus, Proce. Amer. Philos. Soc., 1869, 123. Hab.—In rupibus irroratis prope Philadelphia. G. scattered in a mucous stratum composed of various alg; cells spherical, or oblong, or ovate, associated together in families of from 2-8; families subglobose or subovate, sometimes aggre- gated together in large numbers ; inner tegument yellowish-brown, firm, rarely colorless, homo- geneous or lamellate; external tegument achromatic, rarely subachromatic, generally scarcely visible. Reemarks.—This species was found in a rather firm, grumous or gelatinous coat- ing of a light brown color, growing on the rocks at Fairmount Water Works, chiefly composed of a very minute nostochaceous plant, but contained numerous other alge. The color of the tegument is yellowish-brown, sometimes with some red in it, sometimes with something of a greenish tint. This inner colored coat is not generally more than once or twice lamellate, often it is not at all so. This species seems somewhat allied to G. styophila, but differs slightly in the form of the cell, and more especially in not having a distinct thallus, and in the families being small and containing but few cells. Fig. 7, pl. 8, represents this species, magnified 750 diameters. Genus CELOSPH RIUM, N#GE11. Thallus parvus, e cellulis minimis in familias periphericas consociatis vel in stratum periphericum simplex et in muco tegumentis celerrime confluentibus formato nidulantibus compositus. Cellu- larum divisio, initio generationum serierum, in omnem fit directionem, tum denique alternatim ad superficiei spheric utramque directionem. Thallus small, composed of very small cells consociated into peripheral families, or in a simple peripheral layer, inclosed in their quickly confluent teguments. Division of the cells at first in all directions, afterwards only in each direction on the surface of the sphere. Cc. dubium, Grvv. ? C. thallo microscopico, subgloboso vel enorme, natante, congregato; cellulis globosis aut sub- globosis; eytioplasmate pallide erugineo, subtiliter granulato. Diam.—Cell. plerumque gqs5q/’ = -00016’’; rare ggg = .00025; fam. yoh°op/—qa4s5”” = -00083/’—. 0033’. 14 FRESH-WATER ALG# OF THE UNITED STATES. Hab.—In aquis stagnis, prope Philadelphia. Thallus microscopic, subglobose or irregular, floating, aggregated in great numbers; cells glo- bose or subglobose; cytioplasm finely granulate, pale wruginous green. Remarks.—I found this beautiful little plant forming a dense scum on a stag- nant brick-pond, below the city, in the month of July. The scum was of the “color of pea-soup,” and so thick was it, that I think a quart of the plants might have been readily gathered. The fronds were of various sizes, and many of them were apparently undergoing division—some of them seemed to have little fronds in their interior. 'They were composed of an exceedingly transparent firm jelly, in which the cells were placed, often so as to leave the central parts of the frond empty, merely forming a sort of filament-like layer around the edge. Rarely they were in such numbers as to be crowded together over the whole surface of the frond. In some of the younger fronds the cells formed a little ball within the jelly, instead of being scattered through its outer portion. I have seen some large single cells three or four times the size of the ordinary frond cell, swimming amongst the plants, of which they are apparently the reproductive gonidia. ‘Their cell-coats are very firm and thick. The fronds themselves are often closely aggregated together into little masses, and I think it probable that there is a state of the plant, in which the jelly becomes softened and the fronds more or less fused together in protococcus-like masses. ‘This plant appears to be the same as the European C. dubium, but differs from ‘the description in the fronds not attaining to anything like the size. It is very prabable, however, that this depends upon age or circum- stances of growth, and that American plants may be found as large as the European. Genus MERISMOPEDITA, Meryen. Cellule globose, aut oblonge, aut ovales, tegumentis confluentibus, 4, 8, 16, 32, 64, 128 in fami- lias tabulatas, unistratas consociate. Thallus planus, tenuis, plus minus quadratus, in aqua libere natans. Cellularum divisio in planitiei utramque directionem. Cells globose, oblong, or oval, joined together by their confluent coats into tabular families of 4, 8, 16, 32, 64, 128. Thallus, a more or less quadrate plane, swimming free in the water. Division of the cells occurring in all directions in the one plane. M. nova, Woop. M. thallo membranacco, distinete limitato, cellulis numerosissimis composito; cellulis ovalibus, arcte approximatis, 16 in familias consociatis, dilute czruleo-viridibus, interdum medio con- strictis; thalli marginibus rectis, integris. Syn.—. nova, Woon, Prodromus, Proc. Amer. Philos. Soc., 1869, 123. Diam.—Cell. ad. p55/’ = 0.0025”. Hab.—In flumine Schuylkill, prope Philadelphia. Thallus membranaceous, distinctly limited, composed of very numerous cells; cells oval, closely approximated, consociate in families of 16, light bluish-green, sometimes constricted in the middle; margin of the thallus straight and entire. temark:s—Vhe only specimens I have ever seen of this species were found grow- ing in the Schuylkill River adherent to, or entangled in, a lot of filamentous algze. FRESHWATER ALGA OF THE UNITED STATES. 15 The frond is very sharply defined, and, under a low power, is of a uniform bluish- green tint. The cells are associated in primary families of 16, of a number of which the thallus is composed. ‘The species appears to be most closely allied to M. mediterranea, Neg., from which it differs very essentially in the size of the fronds, and perhaps even more closely to JL glauca, the only character separating it from which is the straight margin. I have myself some doubts whether it ought not to be considered as merely a form of I. glauca. Fig. 8, pl. 8, represents this species, magnified 400 diameters. M. convoluta, Bres. M. thallo membranaceo, oculis nudis visibili, plus minus convoluto; familiis e cellulis geminis et in subfamilias dispositis, 256 compositis, interdum familiis duabus in familia gemina conjunc- tis; cellulis sphricis aut oblongis; cytioplasmate homogeneo, viridi. Diam.—Cell. gp5y'’ = 0.00017’; fam. long. ,35’' = .06’'; lat. 35/7 = 0.04’. Hab.—In aquis quietis prope Philadelphia. Thallus membranous, visible to the naked eye, more or less folded; families composed of 256 geminate cells, arranged in subfamilies, sometimes two of these families conjoined with a com- posite family ; cells spherical or oblong; cytioplasm homogeneous, green. Remarks.—When my Prodromus was published, the only specimens of this plant which I had seen were contained in a mounted slide given me by my friend Dr. J. Gibbons Hunt, of this city. Since then I have found it growing in a very shallow, quiet, but fresh, sweet pool at Spring Mills, making a distinct green layer upon the mud many feet inextent. Of course, there were millions of specimens in this layer. The fronds are irregular in shape, often somewhat ovate, sometimes subquadrate, variously torn, and not rarely somewhat lobate. Their edges are frequently very sharply defined and rendered firm and prominent by several rows of cells being crowded closely together along them. The cells in the body of the frond are arranged in large parallelogrammatic families, composed of 256 cells. There are 16 cells on each side, the families being parallelogrammatic rather than square, owing to the oblong shape of the cells. This cell family is composed of four subfamilies, each containing 64 cells. These are again subdivisible into four more or less distinct groups of 16 cells each. The cells are, finally, generally closely geminate, each pair being very distinctly separated from its neighbors. In certain stages of growth, as immediately after a general division of the cells, two of the large cell-families spoken of are often temporarily joined together to form a huge family of 512 cells, but soon separate one from the other. Orver Nematogenex. Plante multicellulares vel pseudo-multicellulares. Cellule filum (trichoma) formantes et ple- rumque vagina tubulosa homogenea vel lamellosa incluse. Trichomata aut simplicia aut ramificata, Plants multicellular or pseudo-multicellular. Cclls forming a filament, and generally included in a tubular lameHate or homogeneous sheath. Filaments either simple or branched. 16 FRESH-WATER ALGAZA OF THE UNITED STATES. Famity OSCILLARIACEA. Trichomata simplicia haud vero multicellularia, sed distincte articulata, plerumque vaginata, mo- tionibus variis preedita. Filaments simple, not strictly multicellular, but distinctly articulate, mostly vaginate, moving in various ways. Genus OSCILLARIA, Bosc. Trichomata simplicia, plerumque distincte articulata, rigida, recte vel parum curvata, rarius cir- cinata vel spiraliter convoluta, plerumque lete colorata, motu triplici pzdita, in muco matricali nidulantia vel vaginula tubulosa angustissima utroque fine aperta inclusa; articuli fronte disciformes. (R.) Filaments simple, mostly distinctly articulate, rigid, straight, or somewhat curved, very rarely circinate or spirally convolute, capable of three motions, floating in a maternal jelly, and shut up in a fine tubular sheath, open at both ends ; joints from the front disciform. The oscillaria are very peculiar plants, which flourish almost in every situation in which fresh water is to be found. ‘The purest springs are not always free from their presence, although they occur most abundantly in stagnant pools and ditches, where animal or vegetable matters are undergoing decay. When viewed in mass, floating upon some foul pool, few objects in the vegetable world are better calcu- lated to excite disgust. A dark, slimy scum reeking with its putrescent surround- ings, they seem to offer nothing of pleasure or interest. But, when brought home to the table of the microscopist and placed beneath his object-glass, they startle the observer by the wonders of their life-history. Living rods, writhing, twisting, bending, curling, creeping, gliding hither and thither; incessant, apparently cause- less, motion, occurring too in what to most minds is the very type of fixity and passivity—a plant. No marvel, then, that they are so famous. The structure of an oscillatoria is about as simple as it can be. An outside colorless cellulose sheath, which is probably in the uninjured filament closed at the end, although, as seen by the microscope, violence and age have often torn it open. Within is a long rod of variously colored endochrome, distinctly articulated by, at great or less intervals, breaks in the color, which appear as dark lines under a low power, but, under a higher objective, are revealed as narrow linear portions of protoplasm lighter and more transparent than the rest. Frequently at the joints there is a marked tendency to separation between the successive articles, and avery decided contraction of the endochrome on each side, so as to leave a little gutter, or dividing trench. The endochrome is sometimes homogeneous, sometimes con- tains numerous granules, which are, however, never ail in their nature. The color of the endochrome varies very greatly in the different species. Slate color, blue, greenish, olivaceous, are among the most common hues. According to Dr, Ferdinand Cohn (Botan. Zeitung, 1867, p. 38; Sitzung, 15th Dec. 1866, der Schlesischen Gesellschaft fiir Pupibadhicre Gin. the eens matter of the oscillatoria consists of true chlorophyll, and a substance which he calls Pha ycocyan, but which he states to be different from Phykokyan of Kiitzing, the Phychochrom of Negeli, and also from Phycocyan of the latter authority. ‘The chlorophyll is, FRESH-WATER ALG#Z OF THE UNITED STATES. 17 of course, soluble in ether and alcohol but not in water; but the Phycocyan (Cohn) is insoluble in alcohol and ether, but soluble in water after the death of the oscil- latoria. Itis precipitated out of its solution by acids, alcohol, and metallic salts, as a blue jelly, but potash and ammonia throw it down as in a colorless, gelatinous mass. I have myself frequently noticed that oscillatoria after death will yield a bluish coloring matter to water, but thought that such coloring matter was the result of a partial decomposition, and I think that Professor Cohn has by no means established as a fact that his Phycocyan exists in the oscillatoria during life. As to the method of reproduction of these plants, we are as yet almost entirely in the dark. Individuals do multiply by the breaking up of the internal endo- chrome into masses or sections through a separation at the joints. ‘These little masses frequently grow immediately into new individuals. Sometimes, however, they roll themselves into a ball, but whether they then have the power of coating themselves with a protective wall and passing into a sort of resting spore or not, I cannot say. The specific characters of the oscillatoria are derived from the color, form, mode, and place of growth, &c., of the large common mass, its thickness, consistency, the absence or presence of radii, &c. Descending to the individual filament, the characters are drawn from the size, the color, the length of the articulations, and the shape of the uninjured ends. Thus, it is to be noted, whether the latter are gradually narrowed (attenuated), or preserve their size to the very point, whether they are acutish or obtuse, rounded or truncate, whether they are straight or con- stantly curled. The activity and modes of motion are also to be remarked. Some species merely glide across the field of the microscope, some are constantly curling and uncurling at their ends, some bending to and fro almost like a pendulum, some are very sluggish, others very active and restless. After all, however, it must be confessed that the specific characters are very un- satisfactory, much more so than in any other phycochroms which I have studied. A very large number of European forms have been described, some few of which I have been able to recognize. I have also ventured to name a few forms appa- rently distinct, but have refrained from going farther into their specific study, because I have found it so unenticing, and my time has been so limited. Professor Bailey, in Silliman’s Journal, N. S., vol. i1., states that he has identified a few species of this family, although with great hesitation and doubt. At the time he wrote there were really no known grounds upon which specific unity could be predicated in these plants, and I therefore think that his identifications are of but little value, although holding the most profound respect for his abilities as a naturalist. The list he gives is as follows :— : O. tenuissima, Ag. Warm Springs of Washita. O. tenuis, Ag. Providence, Rhode Island. West Point, New York. Culpepper County, Virginia. O. decorticans, Gener. Common everywhere on pumps, &c. O. muscorum, Ag. West Point, New York. O. nigra, Vauch. West Point, New York. O. corium, Ag. 3 February, 1872. 18 FRESH-WATER ALG# OF THE UNITED STATES. O. chlorima, Kurzina. O. interdum in strato sordide viridi natante, interdum in aqua diffusa; trichomatibus rectis, vivide moventibus, vel articulatis et cum cytioplasmate granulato, vel inarticulatis et cum cytioplasmate haud granulato; cytioplasmate hyalino, interdum coloris fere expertibus, m dilutissime viride; apiculo haud attenuato, obtuse rotundato, recto; articulis dia- interdu metro subequalibus. Diam.—zeyq''—sdo0'’ = -00014"—.0001"". Hab.—In stagnis prope Philadelphia. Sometimes swimming on the water as a dirty-greenish stratum, sometimes diffused in the water; filaments straight, actively moving, either articulated and having the cytioplasm filled with blackish granules, or else neither articulate nor granulate, cytioplasm hyaline, almost colorless, or with a faint greenish tint; ends of the filaments not attenuate, straight, obtusely rounded ; joints about equal to the diameter. Remarks.—1 found this species in the month of August, 1869, in one of the stagnant brick-ponds below the city. It occurred as a sort of floating scum, or ‘else diffused through the water, which was then opaque and greenish. It resembled so a protococcus in gross appearance that I did not think of its being an oscilla- toria until I placed it under the microscope. ‘The filaments are almost colorless, and, in most instances, are very distinctly granulate and articulate. The dissepi- ments are in such cases clear and transparent, perfectly free from granules. This form is very close to the descriptions of the European O. chlorina, Ktz., but differs somewhat from descriptions, chiefly in habit of growth. The filaments, when in mass, are often seen to be curved under the restraining force of use glass cover, but when free I think always straighten themselves. Fig. 1, pl. 1, represents a single filament, magnified 750 diameters. 0. Frohlichii, Kz. ? O. strato indefinito, tenue, viride; trichomatibus late viridibus, subrectis, vivide oscillantibus, ad genicula nonnihil pellucidis et leviter contractis et rarissime granulatis; articulis diametro 2, 3,4 plo brevioribus ; cytioplasmate obscure aut distincte minutissime granulate ; apiculo haud attenuato, late rotundato. Diam.—z 855''—Tes5'’ = 0.00066’’—0.0004. Hab.—In flumine Schuylkill. Stratum indefinite, thin, green; filaments bright green, straightish, vividly oscillating, some- what pellucid at the joints, where they are slightly contracted and very rarely granulate ; articles 2, 3, 4 times shorter than the diameter, cytioplasm obscurely or distinctly very mi- nutely granulate; apex not attenuate, broadly rounded. Remarks.—I found this species growing upon the bottoms of the shallows in the Schuylkill River and its larger tributaries, forming a somewhat badly defined stratum, rather, indeed, a coating on the mud than a definite stratum. 'The motion is exceedingly active, the filaments bending and gliding, and their apices con- stantly curling and extending in all directions. The apices are very blunt. The filaments are not often seen woven and twisted together into a mass composed simply of themselves, but are stuck together loosely, each filament remaining straightish, with numerous little masses of mud between them, I have not hee FRESH-WATER ALGA OF THE UNITED STATES. 19 able to identify the species positively, but have referred it with doubt to O. Frdhli- chi. Fig. 2, pl. 1, represents the end of a filament. ©. nigra, Vaucu. O. strato plus minus compacto, amplo, plerumque natante, atro-viride, cum radiis longis ; tricho- matibus plerumque flexuosis; apice obtuse rotundato; articulis diametro 2 ¢ plo brevioribus; dissepimentis distincte granulatis; cytioplasmate pallide cwxsio. a Diam.—z 255 '—zGo0 Hab.—In fossis stagnis prope Philadelphia. Stratum more or less compact, ample, broad, mostly floating, blackish-green, with long radii; filaments mostly flexuous; apices obtusely rounded; joints ? shorter than broad; dissepi- ments distinctly granulate; cytioplasm pale-grayish. Remarks—This species is found in thick, rather loose strata, floating, especially when old, on stagnant waters, or adhering to plants, &c., or the muddy shores and bottom of ditches, foul aquaria, &c. The color of the stratum is a very dark blackish-green, with a peculiar, glossy, repulsive appearance. ‘The single filaments are of a pale-bluish neutral tint, sometimes a little greenish, very much curved and entangled, or more rarely straightish. ‘Their motion is active. ‘The measure- ments do not quite equal those given by European authorities, but otherwise the plant agrees well with their descriptions. Fig. 3a, pl. 1, represents the mass of the plant as seen with the noted eye; fig. 3b, alow: a number of filaments slightly magnified; fig. 1c, a broken portion of a filament magnified 260 diameters, with the sheath projecting beyond the endo- chrome; fig. 1d, the end of a filament still more highly magnified. @O. limosa, Acarpn. O. trichomatibus subrigidis et subrectis, vivide oscillantibus, exeruleo-viridibus, in stratum muco- sum lete saturate viride et modice longe radians et natans collectis et intertextis, distinete articulatis ; articulis diametro subequalibus, interdum duplo brevioribus (post divisionem), ad genicula distinete constrictis; dissepimentis haud granulatis; apiculo obtuso, haud atten- uato, interdum recto, interdum curvato ; cytioplasmate granulato. wr , 1 Diam.—z355 Hab.—In stagnis prope Camden, New Jersey. Filaments straightish and somewhat rigid, vividly oscillating, bluish-green, interwoven into a bluish-green, floating stratum, with moderately long radii, distinctly articulate; articles about equal to the diameter, or after division one-half shorter, at the joints distinetly constricted ; dissepiments not granulate; apices obtuse, not attenuate, sometimes straight, sometimes curved; cytioplasm granulate. Remarks.—I have found this species floating on foul ditches near Kaighn’s Point, New Jersey, in the month of May. ‘The color of the stratum is a very pure deep-green; the single filaments vary from arather bright deep-green to a pale blue- green, according to the power under which they are seen. The apices are not at all attenuate. The constriction at the articles is scarcely visible with a lower power than ith. The stratum is rather thin, with a good deal of dirt adhering to its bottom, 20 FRESH-WATER ALG& OF THE UNITED STATES When grown in a bottle, the plant appears as a very thin stratum growing up the sides. ‘The agreement of this plant with the descriptions of the European O. limosa is very close, so that I do not think it can be separated from it, although in O. limosa the dissepiments are said to be distinctly granular. Fig. 4, pl. 1, represents a filament of the American plant magnified 1250 dia- meters. ‘The aaloe and form are closely counterfeited, but the chaiactenene sepa- ration of the endochrome into parts at the joints is decidedly exaggerated. 0. neglecta, Woop. O. trichomatibus modice brevibus, aut dilute purpuraceo-plumbeis aut plumbeo-cinereis, pler- umque rectis, aut stratum mucosum atro-purpureum haud distinete radiante formantibus, aut in strato gelatinoso haud radiante subplumbeo dispersis et cum algis aliis intermixtis, rare _ oscillantibus sed lente sese moventibus; articulis diametro fere 4 plo brevioribus; dissepi- mentis plerumque haud granulosis, rare indistincte granulosis; apiculo obtuse rotundato, interdum breviter nonnihil attenuato. Syn.—O. neglecta, Woop, Prodromus, Proc. Amer. Philos. Soc., 1869, 124. Diam.—yz355'' = -0066. Hab,—In stagnis prope Philadelphia. Filaments rather short, of a dilute purplish-lead color, or leaden-gray, generally straight, either forming a mucous, blackish-purple stratum without marked rays, or diffused with other alge in a gelatinous mass, rarely oscillating but gliding; articles about four times shorter than broad; joints for the most part not granulate, rarely indistinctly granulate; ends obtusely rounded, occasionally short, somewhat attenuate. Remarks.—I have found this plant in the shallow ditches along the track of the Norristown Railroad above Manayunk, growing in two different ways. In the one it forms a distinct, soft, gelatinous, floating stratum of a very dark purplish color, consisting of nothing but interwoven filaments, and provided with long rays. In the other, the plant is largely mixed with diatoms and other alge into a thick, gelatinous stratum without rays, whose color is a dirty slaty tint, which, however, is not all distinctive, and often varies as the proportion of the different constituents varies. The color of the single filaments is a slaty, almost neutral tint. The cytioplasm is remarkable for the numerous very minute spots more transparent and with less color than the surrounding parts. The ends of the filaments are often abruptly obtuse, frequently however there is a very short taper. Motion does not appear to be very active, and seems especially to be gliding, rather than a bend- ing to and fro of filaments. Fig. 5a, pl. 2, is an outline drawing of a filament magnified 450 diameters; 5d is a portion of a filament. ©. imperator, Woop. O. in strato mucoso, plerumque natante, olivaceo- atro, longe radiante; trichomatibus rectis aut subrectis, tranquillis, dilute viridibus vel saturate olivaceis, hand cueillantibeay sed ambulan- tibus; apiculis nonnihil attenuatis, late rotundatis vel subtruncatis, curvatis; articulis diame- tro eo 12 plo brevioribus, ad genicula indistincte contractis ; cyhicpieante homogeneo, olivaceo-viride; vaginis firmis, ad genicula distinete transverse si atis. Syn.—O. imperator, Woon, Prodromus, Proc. Amer. Philos. Soc., 1869, 124. Diam.—.002'’. FRESH-WATER ALG OF THE UNITED STATES. 21 Hab.—In stagnis prope Philadelphia. O. occurring in an olive-black, mucous stratum, mostly swimming and with long rays; filaments straight or straightish, light-green or deep-olive, tranquil, not oscillating, but moving with a gliding motion; ends somewhat attenuate, broadly rounded or subtruncate, curved; articles 5-12 times shorter than broad, slightly contracted at the joints; cytioplasm homogeneous, olive-green; sheaths firm, distinctly transversely grooved at the joints. Remarks.—The strata of this species are often of great extent, and resemble more masses of spirogyra than of the ordinary oscillatoria. They are very loose in texture and are very slimy, whilst their edges are fringed by the long tranquil rays. In certain conditions of growth, the endochrome of the filaments is so dense -as to render them very opaque and the articulations very obscure. ‘The sheaths when emptied show the marks of the joints very distinctly; but, at times, when gorged with cytioplasm, scarcely can the sheath itself be seen. The color of the filament is also affected by the state of the protoplasm, so that it varies from a lightish-green with an olive tint to a very decided dark olive. This species seems to be closely allied to the European O. princeps, from which, however, it differs in its motion, which is always very slow and merely gliding, its color, the distance of the dissepiments, and the much longer curvature of the ends. It grows everywhere in the ditches around the city; when mature, generally floating upon the surface with an adherent under-stratum of dirt, but, in its earlier history, often adhering to the bottom. Fig. 6a, pl. 1, is a drawing of the end of a filament; fig. 65, represents a small fragment of a filament, showing the tendency to take a roundish or barrel shape; much of the endochrome has been squeezed out by the injury which has broken the filaments. Genus CHTHONOBLASTUS, Krz. Phormidii trichomata fasciatim congesta et vagina communi mucosa apice clausa vel aperta inclusa. Tales fasciculi numerosi in stratum (quasi thallum) gelatinosum, passim amoso-divisum aggregati. Vagine communes achromatice, spe lamellose, plus minus ampliate, rarius indistinct et subnulle, evacuate, plerumque valde intumescentes. Trichomata Phormidii modo oscillantia, articulata et vaginata, rigida, recta vel parum curvula, in fasciculos funiformes plus minus dense contorta, apice soluta et divaricata. Cellulas propagatorias observare mihi contigit. (R.) Filaments fasciately placed together and included in a common mucous sheath with open or shut apex. A number of these fasciculi aggregated in a gelatinous stratum (pseudothallus), which is gelatinous, and here and there ramosely divaricate. Common sheath colorless, often lamellate, more or less enlarged, rarely indistinct and nearly wanting, when empty mostly markedly intumescent. Filaments oscillating like to those of Phormidium, articulate and vaginate, rigid, straight, or a little curved, more or less densely entangled into cord-like fasciculi, with the apex dissolved and dis- severed. Ch. repens, Krz. Ch. terrestris, strato plus minus expanso, saturate wrugineo-chalybeo aut olivaceo-fuscescente, mucoso-membranaceo; trichomatibus squalibus in fasciculos filiformes, swepe valde elon- gatos, e vagina communis apertura penicillatim exsertos congestis; articulis diametro wquali- bus dissepimentis granulatis, apiculo obtuse recto. (R.) Species, mihi ignota. 22 FRESH-WATER ALG OF THE UNITED STATES: Hab.—Common on damp earth. West Point, New York; Bingham, Massachusetts; Provi- dence, Rhode Island; Baily, Silliman’s Journ., N. S., vol. iii. Terrestrial, stratum more or less expanded, deep sruginous chalybeate, or olivaceous fuscous, mucous membranaceous; filaments equal, in filiform fasciculi, which are often much elongate and penicillately exserted from the open common sheath ; joints as long as broad, the dissepi- ments granulate ; the apex obtuse, straight. Genus LYNGBYA, AGARDH. Trichomata inarticulata vel breve articulata, cellulis perdurantibus instructa. Vagine sepe colo- rate, crass, sepe lamellone. Filaments not articulate, or shortly so, furnished with heterocysts. Sheaths often colored, thick, often lamellate. “T,, muralis, Ac. Filaments somewhat rigid, thickish, tortuous, very long, interwoven in a bright, grass-green stratum; annuli strongly defined. Ag. Syst., p. 74; Harv. Man. Ed., p. 160; Conf. muralis. Dillw., tab. 7, E. Bot. t. 1554. p. aquatica, Hab.—Var. 8. in pools of fresh water, Whalefish Island, Davis Straits. Dr. Lyall. The specimens are mixed with turfy soil. Except in the submerged habitat, this agrees with the ordinary form. Intermixed with threads of the usual size and structure are others cohering in pairs, as in LZ. copulata, Harv., which is obvionsly only a state of this widely dispersed species. I have not received specimens of the ordinary L. mwralis from America ; but no doubt it is common on damp walls, &e., as in Hurope generally.” I have never identified this species, and have simply copied Harvey’s account of it from the Nereis Boreali Americana, pt. III. p. 104. L. bicolor, Woop. L. trichomatibus simplicibus, in cxspites nigro-virides vel cwruleo-virides dense intricatis, varie curvatis, plerumque inarticulatis, interdum breviter articulatis et ad genicula contractis; cytio- plasmate dilute ceruleo-viride, plerumque copiose granulato, spe interrupto; cellulis perdu- rantibus cylindricis, sepe elongatis, saturate brunneis, sparsissimis; vaginis firmis, achrois, in trichomata matura modice crassis. Syn.—L. bicolor, Woop, Prodromus, Proc. Amer. Philos. Soc., 1869, 124. y ale id? Diam.—yz7'5y""- Hab.—In flumine Schuylkill prope Philadelphia. L. with the filaments closely interwoven into a blackish or bluish-green mat ; filaments variously curved, simple, mostly inarticulate, sometimes shortly articulate with the joints contracted; endochrome light bluish-green, mostly very granulate, often interrupted ; heteroeysts cylin- drical, often elongate, deep brown, yery few; sheaths firm, transparent, in old filaments moderately thick. Remarks.—This species is abundant in the shallow water of the Schaylkiil River, near Spring Mills, where it forms dark waving tufts a half inch or more in height, which are adherent either to the bottom of the stream or tosome firm sup- port, such as large growing plants, sticks fixed in the mud, &c. When examined with the microscope, these tufts are seen to be composed of innumerable, very long, motionless, greatly curved filaments. They do not seem to be attached to their support, but in the denser parts are woven into a very thick mat, which apparently adheres en masse to the fixed body. These filaments are very rarely articulate, FRESH-WATER ALG OF THE UNITED STATES. 98 but, when they are, the joints are shorter than broad. The endochrome is mostly very granulate ; sometimes, however, it is much more homogeneous. The sheaths in the old filaments are rather thick, and frequently partially empty; the exterior of such sheaths has often a rough, ragged look. The larger cells are very few in number. ‘They are elongated cylinders with concave ends. I have found this plant in the Schuylkill River, just above Fairmount dam, in a younger state, and appa- rently without heterocysts. The threads near their ends had their endochrome distinctly articulate, like an oscillatoria, but elsewhere the protoplasm was continu- ous. It often contains numerous large granules resembling minute starch grains, which however fail.to exhibit the reaction with iodine. Fig. 7, a, pl. 1, represents a portion of the filament slightly magnified; fig. 7, b, a heterocyst from the same specimen more magnified ; fig. 7, ¢ and d, are drawings from another specimen from the same locality, each magnified 800 diameters; fig. 8, pl. 1, represents the form alluded to in the text as having been found in the Schuylkill River just above the dam. Famity NOSTOCHACEA. Trichomata simplicia, e cellulis distinctis composita, interdum vaginata, articulata, in gelatina immersa, cellulis perdurantibus, et interdum sporis porro instructa. Filaments simple, composed of distinct cells, sometimes vaginate, imbedded in jelly ; furnished with heterocysts and sometimes with spores also. Remarks,—The nostochacee are plants of simple construction, consisting of a more or less firm jelly in which are imbedded serpentine filaments, composed of numerous cells. ‘These cells are mostly more or less globose, especially in the true nostocs, so that the filament has a moniliform aspect. ‘They have not distinct walls, or at least any that can be distinctly seen by ordinary powers of the microscope, and are sometimes closely connected, sometimes rather widely separated. No nuclei are usually discernible ; I have, however, seen in some instances central spots, which were possibly of that nature. The filaments themselves are of various length, almost always tortuous, sometimes widely separated, sometimes closely in- terwoven. The gelatinous portions of the fronds are of various consistence—some- times semifluid, sometimes very firm, almost cartilaginous. The order is divisible into two families—the Nostocs proper and the Spermosiree. In the former, the outer portion of the frond is condensed and firm, forming a sort of outer coat or epidermis, which is sometimes quite distinct, but in other instances can scarcely be said to exist. In the filaments of a true nostoc are placed at irregular intervals cells, which are mostly larger than the others, and have thick, distinct walls. ‘These cells con- tain very little or no chlorophyllous protoplasm. They are often, but by no means always, provided with numerous exceedingly attenuated, hair-like processes, or quiescent cilia. These bodies were supposed by Kiitzing to have some sexual value, and received from him the name of Spermatia. But, as their functions are entirely unknown, the name of heterocysts, first applied by M. Allman, is prefera- ble. They are the ‘connecting cells” of Thwaites. No one has as yet demon- 24 FRESH-WATER ALGZ OF THE UNITED STATES. strated the existence of anything indicating sexuality in the nostoes proper, or shown any body at all worthy to be looked upon as a spore. Their ordinary method of reproduction is simply a slight modification of that of growth. If a fragment of an actively growing nostoc is placed under the micro- scope, the filaments of it are seen to be irregular and distorted, thicker in one place than another, the cells misshapen, and sometimes apparently lumped and fused together. The formation of new filaments is taking place in such cases by the simultaneous growth and longitudinal segmentation of the cells of the old, and this may occur through the whole or in only a portion of the length of the latter. (Pl. 2, fig. 10.) The filament of a nostoc is, in other words, capable of a double growth or de- velopment, the result in one instance being increase in its length, in the other the production of a new form like itself. he first of these is brought about by a transverse division of the cells, so that out of each single cell two are formed, placed end to end, each daughter-cell at first only half the size of their parent, but soon attaining to its full stature. In the other case great increase in the size of the cell occurs almost consentaneously with a longitudinal or lateral segmentation, the cell dividing in the direction of its length, instead of transversely, so as to form two cells lying side by side instead of end to end. ‘The misshapen filaments alluded to simply represent different stages of this change, which goes on until two perfect filaments lie side by side, to be finally more or less widely separated by the jelly which they secrete around themselves. This process of growth continues until the plant has arrived at its mature size, when it ceases. During this time the inner portion of the frond has been be- coming more and more liquid, and finally the outer epidermis bursts and the thoroughly softened inner portion is discharged. In this way, innumerable filaments are set free, which are endowed with a power of motion similar to, but much less active than, the gliding of the oscillatoria, by means of which they are diffused in the water. Scattered in this way, carried hither and thither by currents, each minute thread, fixing itself to some object, at last becomes the centre from which a new plant is formed in a manner similar to that already described. In the second division of the Nostochacee, the jelly is always much less firm than in the true nostocs, and is not condensed in the outer portions. The fronds are therefore soft, almost diffluent, and entirely shapeless. The filaments them- selves also differ from those of the true nostocs. There are no fixed differences in the vegetable cells or heterocysts, however, although the former are apt to be- come more cylindrical and the filament consequently less moniliform. It is espe- cially in the possession of distinct reproductive sporangial cells that the differences are to be found. These are much larger than the ordinary cells, from which, in their first appearance, they are not distinguishable ; but, when the frond has attained a certain age, the spore-cells begin to enlarge both in diameter and length, and finally assume a form and size apparently fixed within narrow bounds for each species, and surround themselves with distinct, often quite thick coats. It is very possible that the production of new individuals may take place by a detachment of portions of the frond and subsequent growth, as described in the Nostocs proper, FRESH-WATER ALG OF THE UNITED STATES. 26 but increase of the species does certainly occur by means of these so-called spores. The growth of the plant takes place in the same way as in the true nostocs, The filaments increase in length by transverse division and consequent multiplica- tion of the cells, whilst new filaments are formed by the consentaneous longitudinal division of all the cells of a filament. The spores of a Cylindrospermum have the power of germinating after prolonged desiccation, they having been successfully cultivated even from specimens long preserved in the herbarium. Their development has been carefully and success- fully studied by M. Thuret. According to this authority the first change consists in an elongation of the spore, which ruptures the wall of the sporangium, pushing a portion of it before it. Directly after this the spore undergoes division, so that out of it is formed a little torulose filament, composed of four or five cells. Growth takes place at both ends, but more rapidly at the free one. The new cells formed are smaller than those which arise directly from the spore, but, finally, all the arti- cles assimilate. The wall of the sporangium remains attached for a long time to the end of the filament forming a little cap to it. The heterocysts, according to Thuret, at first are indistinguishable from the ordinary cells, but after awhile the granules in them begin to disappear, the color to pale, the outer wall to become apparent and grow thicker, until at last a perfect “connecting cell” is educed. I have, myself, carefully watched the early development of the spores of a cylin- drospermum, and can confirm, in all essential particulars, the description Thuret has given of the process. Fig. 10, pl. 2, represents a partially formed filament, to which the empty sporangium is still attached. As no sexual reproduction has as yet been shown to exist among the Nostocha- cer, it is very evident that their whole life-history is not comprised within the changes which have been detailed. It has long been known that the gonidia of many lichens have the power of independent existence, i.e. that when they are discharged from their thallus they can continue*to live and multiply, if circum- stances favor them, without giving origin to a new thallus. This, and the great similarity of structure between the nostocs and the lichen genus Collema, has suggested a possibly close relation between the two. ‘The first observer, I believe, who asserted that they were different stages of the same plant was Dr. Hermann Itzigsohn. His observations are, however, rendered of so little value by his own statements that it is not necessary to review them here. ‘Thus, he says, that after seven years’ observation he had yet to See a true one called alge, that the Desmidia are, at least, two-celled, &c. &c. The most weighty observations upon this subject are those of Professor Julius Sachs and of J. Baranetzky—the former published in the Botanische Zeitung for 1855, the other in the Bulletin of the St. Petersburg Academy for 1867. Professor Sachs states that he watched a whole bed of Nostoc commune deve- loping into Collema bulbosum. He says that the peculiar Collemoid threads first appeared as little lateral offshoots or prolongations from the cells of the nostoc filarent, and rapidly developed into well-formed collemoid filaments. | Every possible stage from the typical nostoc to the typical collema was seen repeatedly. 4 February, 1872. 26 FRESH-WATER ALGH OF THE UNITED STATES. The development of the distinguishing threads of the collema out of the ordinary nostoc-cell has never been confirmed by any other observer ; but 1f seems to me that it must be at least provisionally accepted, although De Bary expresses some doubt of it. (Morphol. und Physiol. der Pilze, Flechten, &c., p. 290:) The researches of M. Baranetzky were directed to the developing of a nostoe out of a collema. Hicks and other observers had previously stated that they had seen this, but none of them had given sufficient details as to the method of their observations, to be fully convincing. M. Baranetzky placed sections of actively growing fronds of Collema pulposum, Ach, upon smooth, damp earth, using all proper precautions to prevent external influence. After some days the sections became less transparent and intensely green from the crowding of the gonidia, which were now arranged in curved rows closely rolled together into balls. Upon the upper surface of the section appeared little gelatinous balls or warts, which contained gonidia in rows, and gradually developed typical nostoc forms, whilst on the edges of the sections appeared little colorless wart-like masses of jelly, in which, after some time, appeared gonidia, some of which developed into the typical nestoc form, others into true collemoid plants. Mr. Baranetzky further states that he watched the body of the section gradually change by the continual growth and increase of the rows of gonidia, before alluded to, and by the disappearance of the collemoid threads, until at last the whole mass of the tissue of the lichen had been converted into a true nostoc, which was finally identified as Nostoc vesicarium, D, C.1 I have no observations of my own to offer upon this subject; but think enough has been done to show not only that the nostocs proper have very close relations with the collemoid lichens, but that they are probably a peculiar phase in their life-history. This being the case, it may seem a perfectly superfluous work to indicate species amongst the nostocs. ‘To any one who has given much study to the fresh-water alge, the reply td this will immediately suggest itself; namely, that in the present state of the science it seems impossible to avoid it; they are so commonly thrust at one by collectors, amateurs, &c., are so distinct, are so often the subject of tongue and pen, that they must have aname. ‘The idea that at- taches to the term species is at present not a very definite one; that there are, however, amongst the nostocs fixed forms, which do not change into one another, and can readily be distinguished, I have no doubt. Such forms are herein de- scribed. If they be only life stages of lichens, I have no doubt that it will finally be found that each so-called species of nostoc has its own peculiar so-called species of lichen, from which it alone springs, and into which alone it can develop. It seems to me, then, that as yet no cause for abandoning the specific names of the * In order to aid any one desirous of going over this subject more thoroughly, a list of papers is appended :— Ventenab und Cassini Opuscula Phytolog., 1817, vol. ii. p. 361. Dr. Hermann Itzigsohn. Botanische Zeitung, 1854, De p2le Prof. Julius Sachs. Botanische Zeitung, 1855, p. 1. Bayrhoffer. Botanische Zeitung, 1857. Hicks. Journal of Microscopical Science, 1861, p. 90. Baranetzky. Bulletin de la Société des Sciences Nat., St. Petersburg, vol. xii. p. 418. FRESH-WATER ALG OF THE UNITED STATES. Q7 nostocs has been shown, but only reason to study also their relations with the vari- ous collema. In regard to the Spermosirea, there is as yet no direct proof whatever connect- ing them with lichens. It is very possible that they are not so closely related to the true nostocs as is generally believed, so that the probabilities of their being lichens are at present so remote, that for the systematist to refuse to take note of their distinct forms, seems to me most unwarrantable. SuBpraAMity NOSTOCE. Thallus peridermate plus minus distineto instructus, sporis destitutus. Thallus provided with a more or less distinct integument, and destitute of spores. Genus NOSTOC, Vaucuer, (1803.) Thallus gelatinosus, varie coloratus, aut globosus vel subglobosus aut foliaceo-membranaceus et irregulariter expansus, sepe bullatus. Trichomata plus minus moniliformia. Cellule perdurantes exacte sphwricx vel rare oblong. Thallus gelatinous, variously colored, either globose or subglobose, or foliaceously membranous and indefinitely expanded, often a bulla. Filaments more or less moniliforme. Heterocysts exactly spherical or rarely oblong. a. Thallus globosus vel subglobosus, vel disciformis. Thallus globose, subglobose or discoid. N. Austinii, Woop, (sp. nov.) N. subglobosum, parvum, plernmque magnitudine ovorum piscium, rare ad 2/’, fuscescente, vel nigrescente, interdum durum interdum submolle, superficie sepe corrugata; tricho- matibus varie curvatis, dense intricatis vel distantibus et laxissime intricatis, viridibus, fuscescentibus, subplumbeis vel luteo-brunneis, in thallis minoribus swpe distincte vaginatis, in thallis majoribus haud vel indistincte vaginatis ; articulis maturis globosis, sepe didymis, crasse granulatis; cellulis perdurantibus articulorum diametro cequalibus vel paulo majoribus, globosis, interjectis vel terminalibus, plerumque sparsis. Diam.—Cell. Veg , 7255" —a13300 = -0026”—00033"; cell. perdurant, .00033”. Hab.—in rupibus irroratis, New Jersey. (Austin.) Subglobose, small, mostly the size of fish-eggs, but reaching the diameter of nearly two lines, fuscous or blackish, sometimes very hard, sometimes much softer; surface often corrugated ; filaments variously curved, densely intricate or distantly and loosely interwoven, greenish, fuscous, subplumbeis or yellowish-brown, in the smaller fronds often distinctly vaginate, in the larger indistinctly or not all vaginate; mature joints globose; often didymous, coarsely granulate; heterocysts equal to the diameter of the other joints or a little larger, globose interspersed or terminal. Remarks.—The fronds of this distinct species vary greatly in appearance; the larger of them are often almost colorless, and, when viewed with the microscope, are seen to be composed of a transparent colorless jelly, with remarkably large filaments scattered through it. These filaments are generally without sheaths, though occasionally a sheath can be faintly traced. The smaller fronds are much firmer than the larger and are more decidedly colored. Some of them are entirely opaque, looking simply black when viewed by transmitted light under the micro- scope. In these the filaments are densely crowded together, often misshapen and 28 FRESH-WATER ALG£# OF THE UNITED STATES. provided with distinct broad brownish sheaths: every gradation exists between these forms and the first described fronds. The heterocysts are quite uniform in size, agreeing in diameter with the largest vegetative cells, they are always single. This species is most nearly allied to N. ichthyoon, RaBenu.; from which it is separated by the differences in the sheaths, the greater size of the filaments, and the single heterocysts. It gives me great pleasure to dedicate the species to Mr. Austin, by whom it was collected near Gloucester, New Jersey, growing amidst mosses on rocks. N. pruniforme, (Rorn,) Agh. N. magnum, gregarium, noncoherens, globosum, magnitudine pisi, pruni majoris et ultra, oliva- ceum vel saturate zrugineum, etate provecta fusco-nigrescens, haud raro cavum, levissimum, intus aquosum, peridermate coriaceo subachroo ; trichomatibus subequalibus, hic illic tumidis, laxe intricatis; articulis globosis, plerumque compressis, sepe didymis, arcte connexis; cel- lulis perdurantibus articulis duplo majoribus, plerumque terminalibus, rarius interjectis. R. Species mihi ignota. Diam.—Artic. 0.00024”—0.0003” ; cell. perdur. 0.0003—0.00045”. (R.) Syn.—N. pruniforme, (Roru,) Ac. Rasenuorst, Flora Europ. Algarum, Sect. II. p 168. Hab.—Maine. Leidy. Large, gregarious, not cohering, globose, varying from the size of a pea to a large plum, or even beyond this, olivaceous or deep eerugineous, in old age blackish fuscous, often hollow, very smooth, within watery, periderm coriaceous, somewhat transparent; filaments subequal, here and there swollen, laxly intricate; articles globose, mostly compressed, often twofold, closely connected; heterocysts twice the size of the vegetative cells, mostly terminal, rarely interspersed. Remarks.—1 have never found this species; but some years since some speci- mens, sent to the Academy of Natural Sciences of Philadelphia from Maine, were identified by Professor Joseph Leidy as belonging to it. N. verrucosum, (Liny.) Vaucu. N. magnum, gregarium, bipollicare et ultra, subglobosum, sepe lobatum, verruculosum, irrora- tum, initio solidum, postremo cavum, vesiciforme, saturate brunneo-viride; peridermate mem- branaceo-coriaceo, olivaceo-fuscescente; trichomatibus varie curvatis, centralibus parcioribus et laxissime intricatis, periphericis densius intricatis; articulis oblongis, rare globosis, arcte connexis, crasse granulatis; cellulis perdurantibus interstitialibus vel terminalibus, sphericis, articulorum diametro duplo majoribus. Diam.—Cell. vegetativ. .000166"; cell. perdurant. .000233”. Syn.—N. verrucosum, (LInN.) Vaucn. Rasenuorst, Flora Europ. Algarum, Sect. II. p. 176. Hab.—In fonte, Centre County, Pennsylvania. Large, subglobose, often lobed, warty ; gregarious, two inches in diameter, growing under water, fixed, in the beginning solid, afterwards hollow, bladder-shaped; periderm membranaceous, coriaceous, olivaceous-fuscous; filaments variously curved, centrally fewer, and laxly intricate, towards the outside much more close; articles oblong, rarely globose, closely connected, coarsely granulate; heterocysts interstitial or terminal, spherical, twice the size of the other joints. Remarks.—In the summer of 1869, I found a nostoc growing in great abundance in a very cold, large, limestone spring in Centre County, Pennsylvania, which I FRESH-WATER ALG OF THE UNITED STATES 99 have referred to N. verrucosum with some little hesitation. Some of the fronds were smoothish, others very decidedly warty. |My specimens are old plants, which have become hollow by the discharge of their internal contents. It is pos- sibly on this account I have not been able to verify the minute description given by Professor Rabenhorst. As the latter may not be accessible to some of those who consult these pages, I append the latter part of it, which differs from that given by myself from the American plants. “'Trichomatibus flexuoso-curvatis, quasi triplici ordine ; centralibus parcioribus, laxissime implicatis, apices versus plus minus attenuatis, articulis oblongis, sub- distantibus, periphericis densius spe densissime intricatis, basi haud raro cellulis biseriatis, articulis globosis, arcte connexis, extremis (nonnisi in thallo vetusto occurrunt) subflagelliformibus, articulis oblongis, cylindraceis sphericisque simul immixtis, distantibus; cellulis perdurantibus sphericis interjectis terminalibusque, nonnunquam pluribus simul seriatis articulorum diametro duplo triplove majoribus.” According to Professor Harvey (Nereis Bor. Amer., part iii. p. 114), this species has been collected by Dr. Lyall in pools of fresh water, Isle of Disco, and at Beechey Island, Arctic Regions; also by Mr. Fendler at Sante Fé, New Mexico. N. alpinum, K7z. N. rupestre, immersum ; thallo suborbiculare, erecto, membranaceo, ad i=? unciam lato, ad lineas duas vel tres crasso, tenaci, saturate olivaceo-fusco, levi, seepe rugoso-plicato, cam mar- gine integro et plerumque incrassato; trichomatibus varie curvatis, laxe vel nonnihil dense implicatis; articulis fuscis vel dilute wrugineis plerumque globosis, seepe subtiliter granulatis, arcte connexis; cellulis perdurantibus sphericis plerumque articulorum diametro paulo ma- joribus, interdum subzqualibus, interjectis vel terminalibus. Diam.—Artic. vegetativ. .00016//—00023”; cell. perd. .00026. Syn.— N. alpinum, Krz. Phycol. General., p. 206, No. 10.” Rapennorst, Flora Europ. Algarum, vol. ii. p. 174. “ N. Sutherlandi, Dicktr.” Harvey, Nereis Boreali Americana, part iii. p. 114. “ N. cristatum, BAILEY.” Harvey, Nereis Boreali Americana, part iii., 1857, p. 114. Growing attached by its margin to the-rocks in running water; thallus suborbicular, erect, membranaceous }—#? an inch high and 1—3 lines thick, very tenacious, deep olive-green, smooth, often rugosely plicate especially at the base, with the margin entire, rounded, and mostly thickened; filaments variously curved, laxly or somewhat densely interwoven; arti- cles fuscous or greenish, mostly globose, often finely granulate, closely connected; heterocysts spherical, generally a little larger than the ordinary cells, sometimes about equal to them, interspersed and terminal. Remarks.—This interesting little plant was found in the mountain rivulets near West Point, New York, by the late Prof. Bailey, and received from him the specific name cristatum, first published in Harvey’s work on the North American Alge. Ihave myself seen it growing in very great abundance in rapid mountain streams in the central portions of this State. It is doubtless, therefore, an inhabi- tant of the whole Alleghany range. In the low country, east or west of these mountains and their outlying hills, I do not know of its having been found. I have very recently received specimens of a nostoc from Sereno Watson, Esq., undoubt- edly belonging to this species, which were collected by himself, in cold streams in the Clover Mountains, Nevada, at an altitude of 11,000 feet. Under the name of J. 30 FRESH-WATER ALGAZ OF THE UNITED STATES. Sutherlandii a nostoc has been described by Mr. Dickie, which was collected in the neighborhood of Baflin’s Bay, and must be referred to this species, although the description given of it is very imperfect. Again, N. alpinum, K1rz., appears to be in all respects similar to the North American forms. So that this cosmopolitan little plant seems only to ask for a cold shelter, and it flourishes. The Alps, the Alle- ghanies, the Rocky Mountains, and the cold North are its homes. ‘To those who believe in a single centre for a species, the suggestion that it has spread across the globe, through the arctic regions, and followed our mountain chains southward, will of course present itself. As I have seen it, the plant is very abundant where it grows, five, six, twelve, or more of the little fronds adhering to a single pebble. The frond is generally longer than broad, the margin sometimes sinuous but never, as I have seen it, lobate or incised. It appears finally to burst and discharge its inner portion, whilst the outer cortical portion, now a little vesicle containing a globule of air, is set free and floats down the stream. N. depressum, Woop, (sp. nov.) N. enormiter suborbiculare, minutum, gregarium et interdum aggregatum muscos immersos adherens, mangitudine seminis sinapeos vel parvius, durum, elasticum, subnigris; peridermate firme, achroo; trichomatibus plerumque laxe intricatis, haud vaginatis; articulis globosis, plerumque modice arcte connexis, rare distantibus; cellulis perdurantibus globosis, ceteris paulo majoribus. Diam.—Artic. veget. max: .0002’; cell. perdurant. max .00029. Hab.—In rivulis, New Jersey (Prof. Austin). , “ Trregularly suborbicular, gregarious and sometimes aggregated, elastic, blackish, about the size of a mustard-seed, or smaller, adhering to immersed mosses ; periderm firm, translucent; fila- ments not vaginate, mostly loosely interwoven; joints globose, generally rather closely con- nected, rarely distant ; heterocysts rather larger than the other. Remarks.—This plant was found by Prof. Austin attached to a brook-moss (Dichelyma), growing in a rapid rivulet in Northern New Jersey. ‘he minute fronds sometimes are so thin and spread out as to be almost folia- ceous. ‘The species I take to be most nearly allied to N. lichenoides of Europe, from which it is, however, apparently distinct. In the American plant the fila- ments and heterocysts are a little larger, and the frequent elliptical cells of the European plant are wanting. ‘The frond also apparently does not grow so large as the European, and is further distinguished by its flat, discoid form. In many of the specimens examined the filaments are very thick, irregular, and contorted, the cells being fused together. In other instances, a filament will be plainly double, and every grade between these conditions is present. This is plainly owing to a process of growth, to the cells enlarging and dividing laterally so as to form new filaments. N. sphzericum, (Porrer,) Vaucu. N. globosum, interdum oblongum vel ovale, gregarium, sepius aggregatum, raro tamen conflu- ente, durum, elasticum (in etate provecta intus molle et subaquosum?), olivaceum, magnitudine seminis sinapeos, ad cerasi parvi ; peridermate firmo, pellucido; trichomatibus intricatis, luteo- lis, aut prasinatis aut dilute ceruleis; articulis plerumque subquadratis, interdum transverse FRESH-WATER ALG OF THE UNITED STATES. 831 subovalibus, arcte connexis; cytioplasmate granulato; cellulis perdurantibus interjectis terminalibusque, sphericis. Diam.—Artie. diam. long. gag” = -000125"; transv. gy559” = .00017"; cell. perdurant. TT 00020Ks 3400 Syn.—wN. sphericum (Porrer.) Vaucu. Rapenuorst, Flora Europ. Algarum, Sect. II. p. 167. Hab.—In fontibus, prope Philadelphia. Globose, sometimes oblong or oval, gregarious, but rarely confluent, hard, elastic (in advanced age within soft and watery ?), olivaceous, varying from the size of a mustard-seed to that of asmall cherry; periderm firm, pellucid; filaments intricate, yellowish, greenish or bluish; articles mostly subquadrate, sometimes transversely suboval, closely connected; cytioplasm granular; heterocysts interspersed or terminal, spherical. Remarks.—The specimens from which the above diagnosis was prepared were found at Spring Mills, adhering to mosses and twigs in the water. The fronds were remarkable for their firmness and elasticity. The color was a dull, rather greenish, olive; that of the filaments varied from a decided greenish to a marked yellowish, or sometimes an almost silvery bluish tint. The heterocysts were rather few in number, and were either terminal or interstitial, sometimes they were with- out, sometimes with evident endochrome. The length of the general articulations varied a good deal, it was, however, mostly less than their breadth, which seems quite constant. When kept in water in the house, this species softens, and the periderm as it were peels off, allowing the interior to disperse itself as it gradu- ally becomes more and more diffluent. Most of the fronds afforded ample evidence of their method of growth by the presence of filaments in every stage of division. Fig. 10, pl. 2, represents filaments of this species. N. czeruleum, Lynas. N, minimum, seepe microscopicum, enormiter globosum vel subglobosum, affixum, gregarium, sejunctum vel aggregatum ; trichomatibus valde inequalibus; articulis elongato-cylindraceis, vel acute ellipticis, vel perfecte ellipticis, vel globosis, vel subglobosis, vel subquadrangulis, sejunctis et nonnihil distantibus vel arcte connectis aut confluentibus; cellulis perdurantibus globosis, passim interjectis terminalibusque, ceteris duplo vel subduplo majoribus. Diam.—Cell. perdurant, .000303; cell. vegetat. plerumque .00012—000166/’; rarius .0001— 00021. Syn.—N. ceruleum, Lynas. Rapenunorst, Flora Europ. Algarum, Sect. II. p. 167. Hab.—Inter muscos, New Jersey (Prof. Austin). Very small, often microscopic, irregularly globose or subglobose, affixed, gregarious, separate or ageregated ; filaments very unequal; articles elongate-cylindrical, or acutely elliptical or perfectly so, or subglobose, or globose, or subquadrangular, separate and somewhat distant or closely connected or, confluent; heterocysts globose, interspersed or terminal, double or about double the size of the other cells. Remarks.—I am indebted to Mr. Austin for specimens of this species collected by him in Northern New Jersey. The fronds grow attached to moss and are very mi- nute, the largest I have seen being not more than half a line in diameter. The filaments are remarkable for their inequality, which is often very perceptible in different parts of the same filament. Ihave referred my specimens to NV. eeruleum— 32 FRESH-WATER ALGZ OF THE UNITED STATES. the only differences between them and the European plant are that they are not so large, and do not agree in color, many of them being browner; but these are certainly insufficient grounds for separating them. Prof. Rabenhorst speaks of observing the contents of heterocysts dividing up so as to form a little colony of cells, which finally break through the maternal wall. I have only studied mounted specimens, but have seen very clearly heterocysts in which this process was taking place. N. punctatum, Woop, (sp. nov.) ki N. terrestre; thallo expanso orbiculare vel nonnihil irregulare, tenuissimo, erugineo, parvo membranaceo, pellucidulo ; trichomatibus laxe intricatis, varie curvatis, articulis globosis vel sepius ellipticis, plerumque medio pellucidulis, laxe connexis; cellulis perdurantibus termi- nalibus vel interjectis. Diam.—Cell. vegetat. ys25y” = 000166; cell. perdur. yogoq” = -00083. Hab.—In terrestre, New Jersey, (Prof. Austin.) Terrestrial ; thallus expanded, irregular or orbicular, very thin, eruginous, small, membranous, pellucid; filaments loosely interwoven, variously curved, joints globose or often elliptical, mostly pellucid in the centre, loosely connected; heterocysts terminal or interspersed. Remarks.—My. Austin has kindly sent me the only specimens I have seen of this species; they are labelled “Damp Ground, Sept.” The fronds, which are often aggregated, are very small and exceedingly thin, especially in their central por- tions, where they are quite translucent; in form they are often circular, some- times quadrangular, sometimes quite irregular. As to size, most of them are not more than two lines in diameter, some three, or possibly five lines. ‘The margins are often reflexed and thickened, especially in the smallest fronds. Two kinds of filaments are visible; Ist, those which I take to be in a perfected quiescent state; 2d, those which are in active growth. The former are composed of globose, or more commonly elliptical joints, which are remarkable for the possession of a central translucent, almost colorless spot, the endochrome apparently being arranged in a ring around the outer part of the cell. This is, however, occasionally want- ing. ‘The filaments, which are in active growth, are very irregular in form, often much broader than the others; their cells very irregular and sometimes fused together into one mass. The measurements given in the diagnosis were taken from the filaments of the first kind. b. Thallus indefinite expansus. Thallus indefinitely expanded. N. Cesatii, Bats. N. terrestre; thallo longe lateque expanso, gelatinoso-membranaceo, viridi-flavescente ; tricho- matibus flexuoso-curvatis, sublaxe implicatis, pallide wrugineis; articulis sphericis, laxe vel arctius connexis; cellulis perdurantibus sphericis, et interjectis et terminalibus. Diam.—Artic. .00016—.0002; cell. perdur.—.00033”. Syn.—N. Cesatii, Baus. Rapennorst, Flora Europ. Sect. II. p. 175. Hab.—In terrestre, Kansas (Prof. Parry); Texas (Prof. Ravenel). Terrestrial; thallus broadly and indefinitely expanded, gelatinous-membranaceous, yellowish- green; filaments flexuously curved, rather laxly implicate, pale-greenish; articles spherical, laxly or more closely connected ; heterocysts spherical, both interstitial and terminalibus. FRESH-WATER ALGA OF THE UNITED STATES. 33 Remarks.—This plant was sent to me by Dr. C. C. Parry, from whose letter the following is extracted: “I send enclosed specimens of a singular land Alga? which I met with in this vicinity; lightly attached to bare patches of soil interspersed with buffalo grass. In the adjoining bluffs are cretaceous shales full of seams and layers of selenite, from the decomposition of which the bottom soil becomes strongly impregnated with various saline matters. The present season has been characterized by unusual quantities of rain, causing extensive floods over what is usually a dry, arid district.” The agreement between the mature forms is essentially perfect. There can be scarcely any doubt as to the identification, although I have not seen the Ameri- can plant in its young state. The fronds appeared to be 1—2 lines in thickness, with its surface smooth, or sometimes with close subparallel ridges or wrinkles. According to Rabenhorst, the young European N. cesatii is in the beginning globose, and pale golden-yellow ; soon, however, bursting and spreading out into an indefinitely expanded thallus. Among the algze collected by Prof. Ravenel in Texas is a Nostoc, labelled “On Mud Flats, Cedar Bayou, Harris Co.,” which comes so close to N. cesatii, that I think it must be referred to it. It differs only in being more olivaceous, some- what firmer and in the size of the heterocysts—the largest of the latter which I have examined, attaining the size only of .00027”. ‘The largest vegetative cells are .00017 in diameter. N. calcicola? Ac. N. thallo irregulariter expanso, enormiter sublobato, tenue, membranaceo, cartilagineo, elastico, pellucido, aut laete viride, vel brunneo, vel dilute viride, irregulariter undulato plicato vel bullatoo; peridermate plerumque subnullo; trichomatibus cum filis leptothrichoideis ramosis intermixtis, flexuosis, plerumque distantibus, rarissime e cellulis biseriatis compositis ; cellulis subglobosis, oblongis, ovalibus, cum ceteris ellipticis intermixtis, plerumque laxe connexis; cellulis perdurantibus spnzricis, interjectis et terminalibus. Diam.—Art. x50 —10bo00 = -0001"4—.0001"; cell. perdur. yo355’—gabo” = -0003"— 0002”. Syn.—N. calcicola, AG. Rasenuorst, Flora Europ., Sect. II. p. 174. Hab.—In rupibus, Georgia. (Prof. Ravenel.) Thallus irregularly expanded, membranaceous, thin, cartilaginous, elastic, pellucid, bright green, pale green or brownish, thin, irregularly undulately plicate or bullate; periderm mostly scarcely distinguishable; filaments intermixed with branched leptothrix filaments, flexuous, mostly distant, very rarely composed of biseriate cells; cells subglobose, oblong, oval, inter- mingled with elliptical ones, mostly loosely connected ; heterocysts spherical, interspersed or terminal. Remarks.—This species is one of those sent me by Dr. Billings. It was collected near Catoosa Springs, Georgia, by Prof. H. W. Ravenel. In the dried state it is of a dirty olive-green, and very much wrinkled and irregular on its surface. The largest specimens are about an inch long. There is no very distinct periderm, although in some places the filaments are placed more closely together on the outer portions of the frond. ‘This plant seems to agree with the descriptions of the 4 March, 1872. 34. FRESH-WATER ALG& OF THE UNITED STATES. European S. calcicola, from which it differs somewhat, however, in having its hete rocysts both terminal and among the cells, and also somewhat in their size. N. calidariuma, Woop. N. thallo maximo, indefinite expanso, aut membranaceo-coriaceo vel membranaceo-gelatinoso vel membranaceo, aut lete virdi vel sordide olivaceo-viridi vel olivaceo-brunneo, irregulariter profunde laciniato-sinuato, ultimo eleganter laciniato; trichomatibus inequalibus, interdum flexuoso-curvatis, plerumque subrectis et arcte conjunctis, in formis duabus occurrentibus ; forma altera parva, viridi, articulis cylindricis, cum cellulis perdurantibus hic illic interjectis, vaginis interdum obsoletis, sepius diffluentibus; forma altera maxima, articulis globosis vel oblongis, aurantiaco-brunnea, cellulis perdurantibus ab articulis ceteris haud diversis. Diam.—Forme prime articuli maximi yyjy5 une.; cellule perdurantis gq'gp unc. Forme tee ; 1 ‘areal jal 1 secunde articuli long. 5,455 to zg Une, lat. gqo5 tO selon articuli globosi 5 5'5q to zy'gg Une. Syn.—N. calidarium, Woop, American Journal of Science and Arts, 1869. Hab.— Benton Springs, Owen’s Valley, California” (Mrs. Partz). Thallus very large, indefinitely expanded, either membrano-coriaceous or membrano-gelatinous or membranaceous, either bright green or dirty olive-green or olive-brown, irregularly pro- foundly Jaciniately sinuate, finally elegantly laciniate; filaments unequal, sometimes flexu- ously curved, but mostly straightish and closely conjoined, occurring in two forms; the one small, green, with cylindrical joints, the heterocysts scattered here and there, the sheaths sometimes absent, often diffluent; the other form very large, with globose or oblong: articles, orange-brown, the heterocysts not different from the other cells. Remarks.—Numerous specimens of this species were received from Mrs. Partz, who collected them in Benton’s Spring, a thermal water situated in the extreme northern point of Owen’s Valley, California, sixty miles southwest from the town of Aurora. ‘The following extract from a letter of Mrs. Partz describes the place and mode of their growth more minutely. ““T send you a few samples of the singular vegetation developed in the hot springs of our valley. ‘These springs rise from the earth in an area of about eighty square feet, which forms a basin or pond that pours its hot waters into a narrow creek. In the basin are produced the first forms, partly at a temperature of 124°—135° Fahr. Gradually in the creek and to a distance of 100 yards from the springs are developed, at a temperature of 110°—120° Fahr., the Alga, some growing to a length of over two feet, and looking like bunches of waving hair of the most beau- tiful green. Below 100° Fahr., these plants cease to grow, and give way to a slimy fungus growth, though likewise of a beautiful green, which, finally, as the tempera- ture of the water decreases, also disappears. ‘They are very difficult to preserve, being of so soft and pulpy a nature as not to bear the least handling, and must be carried in their native hot water to the house, very few at a time, and floated upon paper. After being taken from the water and allowed to cool they become a black pulpy mass. But more strange than the vegetable are the animal organ- izations, whose germs, probably through modifications of successive generations, have finally become indigenous to these strange precincts. Mr. Partz and myself saw in the clear water of the basin a very sprightly spider-like creature running nimbly over the ground, where the water was 124° Fahr., and on another occasion dipped out two tiny red worms.” FRESH-WATER ALGA OF THE UNITED STATES. 35 In regard to the temperatures given, and the observation as to the presence of animal life in the thermal waters, Mr. William Gabb, of the State Geological Survey, states that he has visited the locality, knows Mrs, Partz very well, and that whatever she says may be relied on as accurate. The color of the dried specimen varies from a very elegant bluish-green to dirty- greenish and fuscous-brown. After somewhat prolonged soaking in hot water, the specimens regained apparently their original form and dimensions, and were found to be in very good condition for microscopical study. The plant in its earliest stages appears to consist simply of cylindrical filaments, which are so small that they are resolved with some difficulty into the component cells by a first-class one-fifth objective. Fronds composed entirely of filaments of this description were received. Some of these were marked as “first forms,” and as having grown in water at a temperature of 160° Fahr. Probably these were collected immediately over the spot where the heated water bubbled up. At this temperature, if the collection made is to be relied on as the means of judging, the plant does not perfect itself. ‘To the naked eye these ‘first forms’ were simply membranous expansions, of a vivid green color and indefinite size and shape, scarcely as thick as writing-paper, with their edges very deeply cut and running out into a long, waving, hair-like fringe. Other specimens, which grew at a much lower temperature, exactly simulated those just described, both in general appear- ance and microscopical characters. These, I believe, were the immature plant. The matured fronds, as obtained by the method of soaking above described, were “oelatinous membranous,” of a dirty-greenish or fuscous-brown at their bases, and bright green at their marginal portions, where they were deeply incised and finally split up into innumerable hair-like processes. Proximally they were one, or even two, lines in thickness, distally they were scarcely as thick as tissue paper. ‘Their bases were especially gelatinous, sometimes somewhat translucent, and under the microscope were found to have in them only a few distant filaments. Two sets of filaments were very readily distinguished in the adult plant. The most abundant of these, and that especially found in the distal portions of the fronds, were composed of uniform cylindrical cells, often enclosed in a gelatinous sheath. The diameter of such filaments varies greatly; in the larger the sheaths are generally apparent, in the smaller they are frequently indistinguishable. In certain places these filaments are more or less parallel side by side, and are glued together in a sort of membrane. It is only in these cylindrical filaments that I have been able to detect heterocysts, which are not very different from the other cells; they are about one-third or one-half broader, and are not vesicular, but have contents similar to those of the other cells. In one instance only was I able to detect hairs upon these heterocysts. The larger filaments are found especially near the base and in the other older portions of the frond. Their cells are generally irregularly elliptical or globose, rarely are they cylindrical. They are mostly of an orange-brown color; and there exists a particular gelatinous coating to each cell rather than a common gelatinous 36 FRESH-WATER ALG&Z OF THE UNITED STATES. sheath to the filament. These larger threads are apparently produced from the smaller filaments by a process of growth. Near the base and in the under portions of the fronds, these filaments are scat- tered in the homogeneous jelly in which they run infinitely diverse courses. In the upper portions of the frond, and at some little distance from the base, the ad- joining cells are very close to one another, and pursue more or less parallel courses, with enough firm jelly between to unite them into a sort of membrane. This plant certainly belongs to the Nostochacee, and seems a sort of connecting link between the genera Hormosiphon of Kiutzing and Nostoc. The best algologists now refuse to recognize the former group as generically distinct; and the characters presented by this plant seem to corroborate that view. Adherent to, and often more or less imbedded in, the fronds of the Nostoc, were scattered frustules of several species of diatoms, none of which was I able to iden- tify. In some of the fronds there were numerous unicellular Alge, all of them representatives of a single species belonging to the genus Chroococcus, Nageli. This genus contains the very lowest known organisms—simple cells without nuclei, multiplying, as far as known, only by cell-division. These cells are found single or associated in small families; and in certain species these families are united to form a sort of indeterminate gelatinous stratum. In these species the families are com- posed of but very few cells, surrounded by a very large, more or less globular or elliptical mass of transparent, firm jelly. The species is very closely allied to Chroococcus turgidus, vay. thermalis, Rabenh., from which it differs in the outer jelly not being lamellated. The technical description of this plant will be found in the proper place. Fig. 2a, pl. 2, represents the most mature and largest filament; Fig. 26, a small filament from the same frond, each magnified 800 diameters. Fig. 2 c, represents portions of the upper surfaces of fronds. N. comminutum, Krz. N. thallo indefinite expanso, gelatinoso, natante, modo viride, plerumque sordide ferrugineo; trichomatibus flexuosis, plerumque subdense intricatis; articulis globosis (ante divisionem factam subeylindricis), subtiliter granulatis, interdum lete viridibus, plerumque ferrugineis aut luteo-fuscescentibus aut fuscis; cellulis perdurantibus globosis, articulorum diametro duplo majoribus, interjectis aut terminalibus. Diam.—Artic. goo5”"; cell. perdur. x55”. Syn —N. comminutum, Krz. Rasennorst, Flora Europ. Algarum, Sect. IT. p. 179. Hab.—In fossis natante, prope Philadelphia. Thallus indefinitely expanded, gelatinous, floating, mostly sordidly ferruginous, sometimes greenish; trichomata flexuous, mostly subdensely intricate; joints globose (before division subcylindrical), minutely granulate, sometimes bright green, sometimes ferruginous, yellow- ish-fuscous, or fuscous; heterocysts globose, about twice as long as ordinary joints, both interspersed and terminal. Remarks.—This species is to be found floating on the surface of the ditches below the city in the latter part of August and September, forming a repulsive, ferruginous, slimy scum. The periderm is not very apparent, and indeed the sepa- FRESH-WATER ALG OF THE UNITED STATES. 37 rate fronds are not distinct. The filaments are very long, mostly closely intricate, very much curved; in some places they are more sparse. ‘Their color is mostly a sort of yellowish ferruginous-green, sometimes they are, decidedly, almost purely ferruginous, more rarely a bright green. ‘This plant agrees pretty well with the descriptions of the European Nostoc comminutuwm, and I believe is the same species; if, however, N. lacustre of IKutzing is distinct from N. comminutum, this is also; but I incline to the opinion that they are all different forms of one plant. Fig. 3, pl. 2, represents a single filament magnified 800 diameters. N. commune, VAvcu. N. terrestre, thallo irregulariter expanso, difformi, undulato-plicato, tremulo, intus aquose gela- tinoso, «tate provecta plerumque excavato, peridermate subcoriaceo firmo, olivaceo, luteo- fuscescente vel luteo-fusco cincto; trichomatibus flexuoso-curvatis, pallide srugineis, laxe implicatis, equalibus vel subequalibus, haud raro a basi ad medium usque cellulis biseriatis compositis; articulis sphzricis vel e mutua pressione subquadrangularibus, laxe connexis, passim distantibus, puncto centrali turbato preditis; cellulis perdurantibus globosis, articu- lorum diametro duplo majoribus, interstialibus terminalibusque. Diam.—Cell. vegetat. .00012”—.00016”; cell. perdurant. .00025”—.00033”. Syn.—WN. commune, Vaucu. RaBenuorst, Flora Europ. Algarum, Sect. II. p. 175. Hab.—In terrestre, New Jersey. (Austin.) “ Rio Bravo. Schott.” Harvey. Terrestrial ; thallus irregularly expanded, shapeless, undulate-plicate, tremulous, within of the consistence of thin jelly, in advanced age mostly hollow ; periderm subcoriaceous, firm, oliva- ceous, yellowish-fuscous ; filaments flexuously curved, pale green, laxly implicate, equal or subequal, not rarely composed of a double series of cells from their base to their middle ; articles spherical or subquadrangular from mutual pressure, loosely connected, here and there distant, furnished with a central spot; heterocysts globose, twice as large as the vegetative articles, interstitial and terminal. Remarks.—The only specimens I have seen of this species are very old ones, which have burst and discharged their central portions. I have consequently pre- ferred to copy the diagnosis of Prof. Rabenhorst. My specimens agree pretty closely with it. The filaments, and also the single cells, are closer together than his words would seem to indicate. My measurements of the heterocysts, as given above, are larger than those of Prof. Rabenhorst. ‘They agree, however, with his text, which his own measurements do not. I am indebted to Prof. Austin for specimens of this species, which he collected in Northern New Jersey. According to Professor Harvey this plant was collected by Dr. Schott along the Rio Bravo, where it is common on dry flats after rains. SuspraMILy SPERMOSIRE. Thallus sine peridermate, interdum nullus. Trichomata sporis instructa. Thallus without any periderm, sometimes absent. Filaments furnished with spores. Genus ANAB/ENA, Bory. Trichomata moniliformia, evaginata; sporis sphericis, aureis vel aureo-fuscis, plerumque singulis, cum cellulis vegetativis vel perdurantibus conjunctis. 38 FRESH-WATER ALG OF THE UNITED STATES. Filaments moniliform, without sheaths; spores spherical, yellow or yellowish-fuscous, mostly sin- gle, variously placed as to the heterocysts and ordinary cells. Remaris.—The characters which I have given are somewhat different and less exacting than those of Prof. Rabenhorst, otherwise our American species would hardly be covered by the diagnosis. Professor Harvey in his Phycologia Britan- nica states that A. Jussieu had preoccupied the name, Anabaena, by applying it to a genus of Euphorbiacee. ‘The date of Bory’s name is, however, 1823, whilst that of Jussieu is 1824. Hence, it is the latter which must be changed. A. gelatimosa, Woop. A. thallo mucoso gelatinoso, indefinite expanso, dilutissime brunneo, nonnihil pellucido ; tricho- matibus haud vaginatis, leviter flexuoso-curvatis, nonnihil distantibus, haud intricatis, aut. dilute aureis aut dilute ceruleo-viridibus; articulis globosis, homogeneis; cellulis perdu- rantibus articulorum diametro fere «equalibus, globosis, vel rare oblongis; sporis terminalibus, singulis, globosis (fusco-brunneis ?). Syn.—A. gelatinosa, Woop, Prodromus, Proce. Amer. Philos. Soc., 1869, 126. Hab.—Prope Philadelphia. Thallus gelatinous, mucous, indefinitely expanded, somewhat. pellucid, with a brownish tinge ; filaments not vaginate, somewhat curved, rather distant, not intricate, either a light golden- yellow or light bluish-green; joints globose, homogeneous; heterocysts about equal to the filaments in diameter, globose or rarely oblong; spores terminal, globose. Remarks.—The color of the shapeless mass of jelly of which the frond is com- posed is a light,brown with, in places, a decided reddish or flesh-colored tint. ‘The heterocysts are either interstitial or terminal, no hairs were detected on them; they are mostly globose and only occasionally are they oblong. Fig. 4, pl. 2, represents a filament of this species magnified 750 diameters; the color of the endochrome of the large spore was possibly due to its being dead. A. flos aque, (Lynos.) Krz. A libere natans, submembranacea, wruginea; trichomatibus plus minus curvatis, sepius circi- natis; articulis sphericis vel e mutua pressione modo ellipticis modo oblongo-quadratis; cel- lulis perdurantibus ellipticis singulis vel geminis; cytioplasmate pallide «rugineo granulato turbato ; sporis exacte globosis aureo-fulvis lucidis, singulis interjectis, articulorum diametro subduplo majoribus. R. Species mihi ignota. Diam.—Artic. 0.00017”—0,00025"; diam. long cell. perd. 0.00048”—0.00053"; — spor. 0.00032"—0.0004". Syn.—A. flos aque, (Lynas.) Krz. Rapenuorst, Flora Europ. Algarum, Sect. II. p. 182. Hab.—“ Round Pond, West Point, New York.” Prof. Bailey. Silliman’s Journal, N. S., vol. iil. 18 Swimming free, submembranaceous, eruginous; filaments more or less curved, very often cir- cinnate; articles spherical, or, from mutual pressure, elliptical or oblong quadrate ; heterocysts elliptical, single or geminate ; cytioplasm pale eruginous, granulate ; spores exactly globular, golden-fulvous, bright, singly interspersed, nearly twice the diameter of the joints. A. gigantea, Woon. A. thallo nullo, trichomatibus singulis et numeroso-consociatis, natantibus, rectis, in «tate Juveni spiraliter convolutis ; articulis pleramque subglobosis, arcte connexis, granulosis ; cel- FRESH-WATER ALGA OF THE UNITED STATES. 39 lulis perdurantibus interjectis, articulis vegetativis subequalibus, utroque polo punctiforme incrassatis, subsphericis; sporis subspheericis. Syn.—A. gigantea, Woop, Prodromus, Proce. Amer. Philos. Soc., 1869, 145. Hab.—In stagnis natante, prope Philadelphia. Diam.—Artic. vegetat. max. g755- Heterocysts go = .0005. Spor. lat. y3j55 = Long. tooo = 001. Thallus wanting ; filaments occurring floating singly on water or in great numbers, straight, but in the young state often spirally convolute ; articles mostly subglobose, closely connected, granular, heterocysts subspherical, interstitial, a very little larger than the vegetative cells, thickened at each end in a punctiform manner; spore subspherical. Remarks.—This plant was found by myself, late in the summer, floating upon a brick-pond below the city, forming a part of a thick, dirty-green, “ pea-soup colored,” almost pulverulent scum. The filaments, though occasionally in great numbers, were never, that I saw, joined together by any jelly so as to form a frond. Fig. 5, pl. 3, represents a short filament of this species magnified 750 diameters. Genus CYLINDROSPERMUM, Krz. Spore ante cellulam terminalem orte. Spore developing from the next to the terminal cell. Cc. minutuma, Woop. C. trichomatibus dilute serugineis, plerumque flexuoso-curvatis et intricatis, interdum subrectis ; articulis cylindricis, ad genicula plus minus constrictis, homogeneis vel granulatis; cellulis perdurantibus terminalibus, hirsutis, globosis; sporis ellipticis, diametro 2—3 plo longioribus, subtilissime granulatis. Syn.—C. minutum, Woop, Prodromus, Proc. Amer. Philos. Soc., 1869, 126. Diam.—Artic. s¢59"; spor. long. yggy"; tramsv. goyq”- Hab.—In stagnis prope Philadelphia. Filaments light zeruginous-green, generally curved and intricate, sometimes straightish; articles £ 5 fo} Lit = g ? cylindrical, more or less constricted at the joints, homogeneous or granulate; heterocysts terminal, hirsute, globose; spores elliptical, 2—3 times longer than broad, very minutely granulate. Remarks.—This species was found by myself at Spring Garden, New Jersey. With a number of other alge it formed a ferruginous-brown gelatinous mass, growing in a deep, shaded, very stagnant pool. In most instances the filaments were closely interwoven, and sometimes formed minute greenish balls, just large enough to be visible to the unassisted eye. In other instances they were mixed up with various alge in little indefinite masses. There is apparently a stage in the life of the plant, when it consists of a single filament enclosed in a little cap- sule, for mixed in with the rest of the gelatinous scum were little microscopic, sub- globose masses, with a firm outer periderm and a single filament coiled up in the centre. The color of the filaments was generally a faint bluish-green, sometimes, however, with a yellowish tint. The spores were decidedly yellowish. Fig. 6, pl. 2, represents a fragment of a filament with the spore magnified 800 diameters. 40 FRESH-WATER ALG&Z OF THE UNITED STATES. C. flexuosum, (AG.) RaBenu. C. strato gelatinoso, saturate viride, indefinite expanso; trichomatibus equalibus, pallide vel saturate cwruleo-viridibus, plerumque valde flexuosis et intricatis, sepius circinatim vel fasciatim convolutis, interdum subrectis, et fasciatim contextis; articulis oblongis, ad geni- cula plus minus contractis, homogeneis vel granulatis, distinetis; cellulis perdurantibus terminalibus, subglobosis, rare hirsutis, nonnunquam in trichomatis utroque fine; sporis oblongo-cylindricis, diametro 2—3 plo longioribus, distincte granulatis. Diam.—Spor. y53gy = -000416"; cell. veget. aoe, = 000LGEY Syn.—C. flecuosum, (AG.) Raxennorst, Flora Europ. Algarum, Sect. IT. p. 188. Hab.—In locis irroratis, prope Philadelphia. Stratum gelatinous, deep green, indefinitely expanded ; filaments equal, pale or deep bluish- green, mostly very flexuous and interwoven, often circinnately or fasciately convolute ; some- times straightish and in bundles ; articles oblong, more or less contracted at the joints, homo- geneous or granulate, distinct; heterocysts terminal, subglobose, rarely hirsute, sometimes at both ends of the filament; spores oblong-cylindrical, 2 or 3 times longer than broad, dis- tinctly granulate. Remarks,——The color of the filaments in young specimens is deeper than in the older, which, however, grew in a much darker locality. The young spores are a yellowish-green, afterwards they are of a sort of yellowish reddish-brown. In one instance two spores were seen closely conjoined together at the end of a filament. In some filaments one or more heterocysts occur interstitially. Often one or more filaments will be seen coiled together like a rope. On the banks of the Schuyl- kill River I have found this species in two localities in the latter part of Sep- tember. In the one instance it grew along the Reading Railroad, just above the Flat Rock tunnel, in a dark little grotto, formed by shelving rocks. In the other case, it was on wet ground by a horse-trough very near the west end of the upper bridge at Manayunk. Fig. la, pl. 3, represents a filament, magnified 450 diameters. Fig. 1b, a portion of a filament, magnified 800 diameters. C. macrospermum, K7z. C. trichomatibus curvatis vel subrectis, pallide erugineis; articulis cylindricis vel subcylin- dricis (in forma Kuropxa “globosis vel ellipticis”), ad genicula plus minus constrictis, passim confluentibus; cellulis terminalibus plerumque ellipticis vel ovatis, diametro paulo vel subduplo longioribus; sporis elliptico-oblongis vel oblongo-cylindraceis, viridibus (in formam Europxam maturam “saturate fuscis”), subtiliter granulosis, diametro duplo lon- gioribus. Diam —Trich. cell. transv. gg55” = .00003”; spor. .00046”—.00054”. Syn.—C. macrospermum, Krz. Rapenuorst, Flora Europ. Algarum, Sect. II. p. 186. Hab.—In rivulis, South Carolina. (Prof. Ravenel.) Filaments curved or straightish, pale ruginous ; articles cylindrical or subeylindrical (in European species “‘ globose or elliptical”), more or less constricted at the joints, here and there confluent; terminal cells mostly elliptical or ovate, a little longer or about twice as long as broad; spores elliptical-oblong or oblong cylindrical, greenish (in mature European Specimens deep fuscous), finely granular, about twice as long as broad. Remarks.—I have received this species from Professor Ravenel, who collected it near Aiken, South Carolina, in the month of September ; with it was the follow- FRESH-WATER ALG#H OF THE UNITED STATES. 41 ing note: ‘In bottom of shallow, slowly running streams, adhering to ground or fallen leaves, &c., gelatinous green.” ‘The specimens agree well with the descrip- tion of the European form, except that I have never seen the joints globose or ellip- tical, but always cylindrical, as they are said to be sometimes in the typical speci- mens. ‘The color of the spores also is not “fuscous,” but that probably depends upon their not being fully mature. Fig. 7, pl. 2, represents the spore of this species with the neighboring hetero- cyst, magnified 750 diameters. C. comatum, Woop (sp. nov.) C. terrestre, stratum gelatinosum erugineum interdum brunneo tinctum, formans; trichomati- bus flexuosis, intricatis, haud spiralibus, equalibus; articulis breve cylindraceis, diametro equalibus ad plus duplo longioribus, plerumque sejunctis, pallide erugineis, obscure granu- latis ; cellulis terminalibus subglobosis ; sporis oblongo-cylindricis, diametro fere duplo longi- oribus, granulatis, luteo-brunneis ; membrana crassa, distincte granulata. Diam.—Spor. transv. ysooy” = .00042”. Long. ,asoy” = .00092”. Artic. .0001”. Hab.—In terra uda; Niagara, Canada. Growing on the ground, forming a gelatinous stratum of an eruginous color, sometimes tinged on edges with brown; filaments flexuous, equal, intricate, not spiral; joints shortly cylin- drical, equal to or more than twice as long as the diameter, mostly separated, pale eruginous, obscurely granulate, terminal cells subglobose; spores oblong-cylindrical, about twice as long as broad, granulate, yellowish-brown; membrane thick, distinctly granulate. Remarks.—I1 found this Cylindrospermum growing upon the ground in the marshes which border the Niagara River just above the Canadian Falls. It formed a bright, eruginous, gelatinous, but firmish, almost membranous, stratum. The filaments are often quite long, and are composed of short, cylindrical cells, mostly placed rather far apart. ‘The terminal cells are remarkable for being abun- dantly provided with long, flexible, hair-like processes, upon the ends of which are minute lobular bodies (cells?). These appendages are so minute as to make it diffi- cult to determine their structure, and although I have studied them with a ,'.th immersion lens, giving a power of nearly 2500 diameters, there are some points about them still undetermined. I do not know whether they or the little globules are hollow or not. I do feel pretty certain, however, that the little globules are distinct bodies, and that they finally drop off, leaving the naked hair behind. Is it possible that they have any sexual significance? The spore-wall is thick, and under a high power is seen to be distinctly granulate. The granules are of course small, but in the perfected spore can plainly be seen with an eighth objective pro- jecting out from the margin. Fig. 8, pl. 2, represents the spore-end of a filament, magnified 1375 diameters. 2 Genus DOLICHOSPERMUM, TuwailrtEs. Spore elliptice, oblonge vel cylindracee, inter cellulas vegetativas orte, sepe in seriebus con- nexe, a cellulis perdurantibus disjuncte. Spores elliptical, oblong, or cylindrical, occurring amidst the vegetative cells, often connected in series, separated from the heterocysts. 6 April, 1872. 42 FRESH-WATER ALG& OF THE UNITED STATES. Syn.—Spherozyga, (AUCTORES, partim.) Dolichospermum, Tuwaire’s MSS. Mr. J. RALFs on the Nostochinez, Ann. Mag. Nat. Hist. 1850, p. 335. Remarks.—This genus differs from Spherozyga in that the spores have no rela- tion, in regard to position, with the heterocysts. Professor Rabenhorst, in his Flora, does not acknowledge it; but it is very evident that he has neither seen the original paper of Mr. Ralfs, nor the species upon which the genus was founded, for he mentions none of the latter, either as good species or synonyms, and the memoir itself is not included in his bibliographical list. The generic characters given by myself are essentially those of the original description, with the excep- tion that the filaments in the latter are said to be aggregated into a stratum, which is not true of the American forms herein described. D. subrigidum, Woop. S. natans; trichomatibus singulis, rectis aut subrectis, minimis, dilute viridibus: articulis cylindraceis aut subglobosis, distinctis; sporis cylindraceis, in medio gradatim nonnihil constrictis, singulis aut duplicis, sine cellulis perdurantibus inter se; cellulis perdurantibus breve cylindraceis, singulis, distinctis. Syn.—Spheroziga subrigidum, Woop, Prodromus, Proc. Amer. Philos. Soc., 1869, p. 123. Diam.—Cell. veg. trans. go'5y” = .00016”; spor. transv. g3y5/—aeyn” = 00023” —.00022”; long. yz'5y9’ = -00066”; cell. perd. transv. 4,55 = .00022/. Hab.—In stagnis prope Philadelphia. S. Floating; filaments single, straight or straightish, very small, light green; articles cylin- drical or subglobose, distinct; spores single or double, in the middle gradually a little con- stricted, not having a heterocyst between them; heterocysts shortly cylindrical, single, distinct. Remarks.—I have found this species growing in the scum floating upon the ditches below the city. The filaments are always, as I have seen them, scattered. They seem always to be nearly straight, or entirely so, and indeed preserve their straightness so constantly as to suggest the name given the species. The spores are very distinct, and all that I have seen were greenish, cylindrical, and constricted in the middle, so that their sides are concave. Their position does not seem to be uniform, any further than that they are amongst the ordinary cells. The heterocysts are large, almost equalling the spores in diameter; I have never detected hairs on them. ‘This species appears to be most nearly allied to D. Thwaitesii of Ralfs, from which it differs in not forming a stratum, and in the great proportionate diameter of the heterocysts. I have never seen any measurements of D. Thwaitesit. Fig. 2, pl. 3, is a filament, magnified 975 diameters. D. polysperma, (K7z.) 8. Pee plerumque subsolitariis, sed interdum consociatis et intricatis, dilute cseruleo- Hise NRE races aut varie curvatis et flexuosis; articulis aut subsphericis aut breve cylin- aricis; i q =H eee oe AEC A . i ; 7 u 13 perdurantibus globosis aut Jatissime ellipticis, articulorum diametro paulo vel duplo majoribus; sporis plus minus elongatis, cylindraceis—in «tate immatura, sparse granulatis, dilute cxruleo-viridibus, et eum membrana haud distincta,—in wetate matura dense granulatis et cum membrana subcrassa. FRESH-WATER ALGA OF THE UNITED STATES. 43 Diam.—Artic. spon’ = -00016”; spor. y5455'—r5 505 = -00026"—.00033”. Syn.—S. Carmichelii, Harvey, Phycol. Brittanica, T. exiii. S. polysperma, (Krz.) Rasenuorst, Flora Europ. Algarum, Sect. II. p. 192. Hab.—In stagnis, New Jersey. S. filaments mostly subsolitary, but sometimes associated and interwoven together, light bluish- green, straightish, or variously curved and flexuous; articles either subspherical or shortly cylindrical; heterocysts globose or very broadly elliptic, a little larger to twice as large in diameter as the ordinary joints ; spores more or less elongate, cylindrical, in the uncertain condition sparsely granulate, light bluish-green, with the membrane not distinct, in the mature state densely granulate, and with a thickish membrane. Remarks.—I found this species growing in a brownish jelly, with various other alge, in a. pool east of Camden, New Jersey. The filaments were mostly scattered, but in some places numbers of them were collected in little masses. In some fila- “ments almost all the cells were developed into spores, so that a single thread con- tained ten or even more spores. In by far the larger number of such cases there was between each pair of spores a heterocyst ; sometimes, however, the latter was wanting, and the spores would be attached to one another. My specimens differ somewhat from the European form, but are too close to separate from them. ‘They equally resemble, however, S. Carmichelii. Indeed, I cannot see any sufficient reason for separating the species. S. Carmichelii is, to be sure, a salt-water plant. I have, however, received specimens collected by Dr. Lewis, near Stonington, which I believe grew in salt water, and which agree in every respect with my fresh-water specimens. Fig. 3, pl. 3, represents a portion of a filament, magnified 750 diameters. Famity RIVULARIACEA. Thallus gelatinosus, mollis vel induratus, vel crustaceus, interdum calce impletus, subglobosus velamorphus. ‘Trichomata ad oscillarium morem articulata, vaginata, sed interdum tate provecta cum yaginis in gelatinam matricalem confluentibus et haud visibilibus, simplicia vel pseudoramosa, superne attenuata, sepius in apicem piliformem longe producta, parallela vel radiatim disposita, cellula basale hyalina globosa et interdum cellulis interstialibus instructa. Spore (manubria, Krz.), singule plerumque inter cellulam perdurantem basilarem et cellulas vegetativas posite, sepe per- magne, cylindrice, plerumque pachydermatice. Vegetatio terminalis. Propogatio sporis tranquillis. Thallus gelatinous, soft, or indurated, or crustaceous, sometimes filled with lime, subglobose or amorphous. Trichomata. articulated like an oscillatoria, vaginate, but sometimes, when old, with the sheaths confluent in the maternal jelly and not visible, simple or pseudoramose, attenuated above, often with the apex prolonged into a long hair, parallel or radiately disposed, furnished with globose hyaline, thick-walled basal cells, and sometimes with interstitial cells. Spores cylindrical, generally placed between the basal and vegetative cells, often very large, mostly with thick coats. Vegetation tranquil. Propagation by means of tranquil spores. Remarks.—In the Rivulariacee the thallus is always small; but is most gene- rally in the various species somewhat definite in form and size. Its consistency in our North American forms varies from that of an exceedingly soft, formless jelly to that of a gristly mass. The maternal jelly is usually colorless, sometimes brown- ish or yellowish. There is never any condensation of the outer portion of the 44 FRESH-WATER ALGZ OF THE UNITED STATES. frond into a periderm. ‘The filaments commonly radiate from the centre to the circumference ; sometimes, especially in the softer fronds, they are simply parallel with one another. ‘The sheaths vary in their breadth, firmness, and distinctness. These little plants grow chiefly in the water; some species are said to live in the air in exceedingly damp places, but I have not as yet met with any such. ‘They appear to prefer cold climates, although I have received specimens from South Carolina. With us, I have only found them in the late autumn and winter months. As to their life-history very little appears to be known; I have not been able to make any observations myself upon this point, nor to obtain access to the papers' by De Bary, almost the only sources of such information, and therefore pass by the subject. Genus NOSTOCHOPSIS, Woop. Trichomata ramosa cum cellulis perdurantibus aut in lateribus sessilibus aut in ramulorum brevissi- morum apicibus dispositis. Vagine nulle. Thallus definitus. Syn.—WNostochopsis, Woop, Prodromus, Proc. Amer. Philos. Soc., 1869. Thallus definite; filament branched ; heterocysts sessile upon the sides of the filaments, or raised upon the apices of short branches; sheaths none. Remarks.—The curious plant upon which this genus is founded has the habit of anostoc. The outer portion of the frond is condensed, so as to give the appear- ance of a periderm; but there is, in reality, no true periderm. ‘The consistence of the thallus is that of a firm, gelatinous mass. ‘The trichomata or filaments radiate from the inner part of the frond towards the outer surface, but many of them take their origin in the outer portions of the thallus. In most places they are distinctly articulated, and, indeed, the joints being swollen and moniliform, in some parts they almost seem to be composed of globular cells, resembling some- what the filaments of a nostoc ; on the other hand, in certain portions they are not at all articulated, and this for long distances. No sheaths are anywhere visible. The heterocysts are, strangely enough, never placed in the continuity of the fila- ments. Sometimes they are sessile immediately upon the latter, sometimes they are raised upon very short branches. They are globose, with rather thick walls. Possibly, however, I am mistaken in believing these bodies to be heterocysts, for they may be rather of the nature of spores, as is somewhat indicated by their thick walls, and often apparently dense contents. Their round shape, and the absence of anything else representing heterocysts, has induced me, however, so to consider them. In my Prodromus I placed this plant provisionally amongst the nostocs ; but the radiation of the filaments from within outwards, and especially their being branched, on second thought seem to me to indicate a closer relation with the Rivulariacee. The genus appears to be a sort of connecting link be- tween the two families. 4S Blora,?? 11863: FRESH-WATER ALG OF THE UNITED STATES. 45 N. lobatus, Woop. N. thallo vivide viride aut luteo-viride, cavo, enormiter lobato, natante, modice magno, firmo gelatinoso ; trichomatibus plerumque longis, flexuosis, dilute viridibus, plerumque articulatis, partim inarticulatis, cylindricis aut sub-moniliformibus, sparse granulatis. Diam.—Trichom. yzp90" = -00006"—7 355" = .00013" 5; cell perdum. 5725” = .00026”. Syn.—N. lobatus, Woop, Prodromus, Proc. Amer. Philos. Soc., 1869. Hab.—In Schuylkill Flumine, prope Philadelphia. Thallus bright green or yellowish-green, hollow, irregularly lobed, floating, moderately large, firm, gelatinous; filaments mostly long, flexuous, dilute green, mostly articulate, partly inar- ticulate, cylindrical or somewhat moniliform, sparsely granulate. Remarks.—I found this plant floating upon the Schuylkill River just above Manayunk. ‘The hollow frond was buoyed up by a bubble of gas contained within it. It was an irregular, flattened, somewhat globose mass, of a bright green color and about half an inch in diameter. It seems very probable that in its earlier condition, it was a solid attached frond. The long slender filaments are often very tortuous, but run a pretty direct general course towards the outer surface. Fig. 6a, pl. 3, represents a section of the frond slightly magnified; a, b, c, por- tions of filaments magnified 800 diameters. Genus GLOIOTRICHIA, J. Ac. (1842.) Trichomata e planitie orta pseudoramosa, distincte vaginata; vagine ample, basi plerumque saccate, transverse undulato-plicat, plus minus constricte, apice aperte, non laciniate. Spore magne cylindrice. Filaments springing from a plane, pseudoramose, distinctly vaginate; sheath ample, mostly saccate at the base, transversely undulately plicate, more or less constricted, open at the apex, not laciniate. Spores large, cylindrical. Remarks.—This genus was, I believe, first indicated by Professor Agardh in his Alge Maris Mediterranei et Adriatici, a work to which I have not access. On account of this, and also because I have not seen any of the typical species of the genus, I have preferred simply copying the generic characters given by Professor Rabenhorst. If my understanding of “e planitie orta” is cor- rect, I do not think it true. Professor Rabenhorst’s own figure of Rivularia shows that the filaments do not all arise on one plane; although he asserts the character equally for that genus. In our American species the filaments do not all arise on one plane, nor can they be spoken of as “ pseudoramosa.” G. incrustata, Woop. G. globosa vel subovalis, firma, solida, ad pisi minimi magnitudinem, dilute viridis, erystallo- phora; trichomatibus rectis aut leviter curvatis, in pilum productis, viridibus aut flavescen- tibus, seepe infra lete viridibus sed supra flavescentibus, haud ordinatim articulatis ; articulis inferioribus in trichomatibus maturis brevibus, plerumque compressis ; pilo apicale recto aut leviter curvato, plerumque indistincte articulato, sepe interrupto; vaginis amplis, achrois, saccatis, interdum valde constrictis ; sporis cylindricis, sepe curvatis, diametro ad 9 plo lon- gioribus ; cellulis perdurantibus sphericis. 46 FRESH-WATER ALG#® OF THE UNITED STATES. . - en Z ” 9 2 - Sy Quali. Tal, Diam.—Trichom. cum vag. 7295"—7250'5 Sporis max. 7550" —7zs'o0 5 cell. perd. g550- Syn.—G. incrustata, Woop, Prodromus, Proc. Amer. Philos. Soc., 1869, p. 128. Hab.—Schuylkill River, plantas aquaticas adherens. Frond globose or suboval, firm, solid, about the size of a very small pea, light green, crystal bearing; filaments straight or slightly curved, produced into long hairs, green or yellowish, sometimes bright green in their proximal portions but yellowish above, not regularly articu- late; lower articles in the mature filament short, and generally compressed ; apical seta straight or slightly curved, mostly indistinctly articulate, frequently interrupted; sheath ample, transparent, saccate, sometimes strongly constricted; spores cylindrical, frequently curved, about 9 times as long as broad. Remarks.—I found this species growing attached to some little plants, either in the Schuylkill near Spring Mills, or else in the spring itself, I do not know which. The roundish fronds varied in size from a mustard-seed to that of a half-grown pea. They were of a decided green color, but appeared grayish from the amount of carbonate of lime in and upon them. ‘The larger balls, when cut in two, were distinctly separable into a central and cortical part. ‘The former was more gela- tinous and contained fewer of the filaments than the latter. The filaments mostly arose in sets together, 7.e. there were one or more zones or planes in which the bases of the filaments were placed together. This, however, was not strictly the case, as there were almost always some scattered trichomata. ‘The matured fila- ments are very distinct. ‘Their sheaths are very large, and often saccate, with wavy, loose-looking margins ; sometimes they are suddenly transversely constricted, once or more in their length; sometimes they look as if a tight spiral band were wound around; sometimes they are entirely tree from any constrictions. These sheaths are open above, appearing as though they had been melted away. The spore is long and cylindrical, and is highly granular. ‘The endochrome is gene- rally articulated below, the joints are often so nearly globular in the lower portions as to give a moniliform appearance ; sometimes the articles are compressed. ‘The upper portion of the trichoma is frequently interrupted, and if at all articulated is very irregularly and indistinctly so. The younger filaments have their endo- chrome variously and irregularly interrupted. The basal cells are globular. I believe the formation of new filaments and the consequent growth of the frond take place by distal portions of the projecting endochrome separating from the parent filament, then forming a basal cell, and lastly a sheath. (See Plate 00.) ‘The carbonate of lime does not exist as a definite incrustation, but in the form of semi-crystalline masses scattered through the frond. This species seems to come closer to G@. boryana than any described species, from the description of which it differs, in the color of thallus, in the latter being always solid (at least so I have found it late in the fall, when the spores were fully perfected), in its habit of in- closing crystals of carbonate of lime, in the curved spores; and, doubtless, a com- parison of the specimens would show still more important differences. Fig. 4a, pl. 3, represents a section of a frond moderately magnified ; fig. 4, the basal end of a filament magnified 460 diameters ; fig. 8c, filaments magnified 260 diameters. FRESH-WATER ALG&A OF THE UNITED STATES. 47 G. angulosa, (Rorn.) J. Acu. G. globoso-angulosa, cava, viridi-fuscescens, ad cerasi magnitudinem; trichomatibus strictis, torulosis, superne leviter flexuosis, passim interruptis; articulis inferioribus plus minus compressis, diametro duplo triplove longioribus ; vaginis amplis, achrois hie illic leviter con- strictis; sporis plus minus elongatis, oblongo-ovatis vel ellipsoideo-cylindricis, diametro 3—6— 10 plo longioribus, zrugineo-fuscescentibus, nonnunquam leviter curvatis, eytioplasmate sub- tiliter granuloso, turbato. (R.) Species mihi ignota. Diam.—Cell. perd. 0.00036”—0.0005". Spor. max. 0.00059”. (R.) Syn.—G. angulosa, (Rotu.), J. AGARDH., RABENHORST, Flora Europ. Algarum, Sect. II. p. 201. Hab.—Hudson River prope West Point. (Bailey.) Globose angular, hollow, greenish-fuscous, attaining the size of a cherry ; filaments strict, toru- lose, above somewhat flexuose, here and there interrupted; inferior joints more or less com- pressed, 2-3 times longer than their diameter ; sheath ample, colorless, here and there slightly constricted; spores more or less elongate, oblong-ovate or ellipsoidal-cylindrical, 3-6-10 times longer than the diameter, eruginous-fuscous, sometimes slightly curved, cytioplasm very minutely granulate. Genus RIVULARIA, (Roru.) Agu. Thallus et trichomata eadem que Gloiotricha, sed vagine arctissime, sepe in gelatinam matri- calem confluentes, quasi nulle. Thallus and filaments similar to those of Gloiotricha, but the sheaths very close, often confluent in the gelatinous matrix and apparently wanting. Remarks.—TVhe characters given above are those of Professor Rabenhorst. Flora Europ. Algarum, Sect. II. p. 206 R. cartilaginea, Woop. R. subglobosa, parva, cartilaginea, saturate brunnea vel subatra, solitaria in plantis aquaticis :— trichomatibus maturis-sterilibus, rectis aut subrectis, cylindricis, elongatis, haud articulatis ; cytioplasmate seepe interrupto; vaginis arctis et distinctis; cellulis perdurantibus globosis, diametro subequalibus :—trichomatibus fertilibus—rectis aut subrectis, supra spora cellulis 8-9 instructis; sporis elongatis, rectis, eylindricis; vaginis nonnihil crassis, arctis:—trichomatibus immaturis breve articulatis; vaginis subamplis. Diam.—Trich, cum vag. s59”; spor sayy”: Syn.—R cartilaginea, Woop, Proc. Am. Philos. Soc., 1869, p. 128. Hab.—In palude, Northern Michigan. Frond subglobose, small, cartilaginous, deep brown or blackish, solitary upon aquatic plants; mature sterile filaments, cylindrical, elongated, not articulated, their cytioplasm frequently interrupted, their sheaths close and distinct, their heterocysts globose and about equal to them in diameter; fertile filaments straight or nearly so, above the spores furnished with 8 or 9 cells; spores elongate, straight, cylindrical; sheaths rather thick, close; immature filaments shortly articulate, their sheaths rather large. Remarks.—The frond of this species grows attached to the leaves of water-plants, and has its under side markedly flattened so that it is somewhat semi-globose. The filaments which compose the mass of the very firm frond are elongated, cylindrical, and of nearly or entirely uniform diameter throughout. The sheaths are close, distinct, rather thin, open above, and, in many instances, almost or even entirely empty. Scattered amongst such filaments are the fertile ones. These have at their base an elongated cell, in which is the long cylindrical spore, which varies 48 FRESH-WATER ALG& OF THE UNITED STATES. very greatly in length in the various filaments, but is almost always shorter than the cell containing it. Just beyond the spore is a series of distinct, variously shaped cells, about seven in number, which are, as I have seen them, empty. In the outer portions of the frond occur what I believe to be young filaments. These are distinguished by their rapidly decreasing in diameter towards their distal end, by their being distinctly articulated, by their basal cell not being distinctly sepa- rated as in the older filaments, and by their sheaths being more ample. These various filaments composing the fronds do not arise from any one place, but commence at very different distances from the centre, and pursue a more or less straight course to the circumference of the frond, from which they often project. Fig. 9, pl. 2, represents a section of the frond moderately magnified ; fig. 9 0, is a drawing of the basal part of a filament magnified 800 diameters. Genus ZONOTRICHA. Thalli pulvinato-hemispherici, sepe confluentes, calee pregnantes, plus minus indurati, basi plani affixa, etate provecta plerumque excavati, intus zonati; zonis concentricis, variegatis; trichomata pseudoramosa, gracilia, inequalia, apice hyalina et plus minus longe cuspidata vel in pilum producta. Vagine firme, homogenee vel longitudinaliter plicato-fibrillose, apice integre vel dilatate et in fibrillas solute. Spore ignote. Thalli pulvinately hemispherical, often confluent, impregnated with lime, and more or less indu- rated, fixed by the flattened base, in advanced age mostly excavated, zoned within; zones concentric variegated ; filaments pseudoramose, slender, unequal, their apices hyaline and more or less cuspid- ate or prolonged into a hair; sheaths firm, homogeneous, or longitudinally plicately fibrillose, their apices entire or dilated and dissolved in fibrilla. Spores unknown. Z. mollis, Woop (sp. nov.) Z. interdum subhemispherica sed gregaria et in stratum nonnibhil mammillosum confluens, submollis, cinerea vel griseo-carnea, parcezonata; trichomatibus longissimis, angustis, flexuosis; vaginis arctis, decoloratis, non fibrosis, firmis; trichomatibus internis articulatis, sepe interruptis; articulis disjunctis, diametro wqualibus ad 4 plo longioribus; cellulis perdurantibus singulis globosis. Diam.—Trich. ¢. v. ys$oy" =-00017". Sine vag y5355// = 000084”. Hab.—In saxis irroratis, ‘Cave of the Winds,” Niagara, Wood. Z. sometimes subsemispherical but gregarious and confluent into a somewhat mammillate, rather soft stratum, ashy or grayish flesh-colored, sparsely distinctly zoned; filaments very long, narrow, flexuous; sheaths close, colorless, not fibrillose, firm; internal filament articulated, often interrupted ; joints separated, equal to 4 times longer than the diameter; heterocysts single globose. Remarks.—Kvery American tourist is familiar with that most wonderful spot, the so-called “Cave of the Winds,” at Niagara. It is simply a place where it is possible to go underneath a portion of the great cataract, and then round upon the rocky debris outside of it. Growing upon these rocks, eternally wet and glistening with foam and spray, I found this and the following species. The present form was much the most abundant, making a slippery, grayish, or grayish flesh-colored coat- ing to many of the rocks, dotted here and there with the rigid, blackish fronds of FRESH-WATER ALGA OF THE UNITED STATES. 49 its fellow. This coating was not at all uniform, but was covered with mammillated masses, and consequently varied from two to six lines in thickness. Internally, it was striated or radiated, but not so evidently as the following species, and presented several distinct variegated zones. It was quite soft to the touch, as well as readily broken or crushed, and under the microscope was seen to contain very little lime salt. When dried it has a pronounced sebaceous appearance. The filaments com- posing it are remarkable for their great length, often apparently running from the bottom to the top of the frond. They are rarely if ever branched, and appear never to be furnished with any heterocysts save at their enlarged base. I have never seen any distinct hairs terminating them, their ends always appearing broken and open, They are often quite flexuous or even tortuous. ‘The internal filament is remarkable for having its articles so distinctly separated. It is often very much interrupted, and in specimens preserved in carbolic-acid water is of an orange-brown color. Fig. 3, pl. 4, represents a single filament magnified 260 diameters. Z. parcezonata, Woop, (sp. nov.) Z. nigro-viridis, enormiter semiovalis, ad 6” longa, dura, lubrica, non fragilis, calee pregnans, intus a basi distincte radiata, parce et sepe obsolete zonata; trichomatibus modice longis, subrectis; trichomatibus internis cylindricis inarticulatis vel articulatis, et interdum monili- formibus ; articulis longis et cylindricis vel brevibus et globosis; vaginis amplis, fibrillosis; cellulis perdurantibus basalibus et interjectis, his oblongis vel cylindricis, illis globosis et sepe geminis. Diam.—Cell. perd. basal. gy59” =.00017"; trichom. cum vag. g¢59’—s;359 =. 00025" —.00037”. Sine vag. .00006”—.00008”. Hab.—In saxis irroratis. ‘‘ Cave of the Winds,” Niagara. J Var.—Z. cinerea. Blackish green, irregularly semioval, to 6 lines long, hard, slippery, not fragile, impregnated with lime, internally distinctly radiate, sparsely and often obsoletely zoned; filaments moderately long, straightish ; internal filament cylindrical, not articulated or articulated, sometimes monili- form; joints long and cylindrical, or short and subglobose; sheath ample, fibrillose ; heterocysts basal and interposed in the body of the filament; the former globose, often geminate; the latter oblong or cylindrical. Var.—Cineritious in color. Remarks.—I found this plant growing on rocks as glossy, blackish, very hard and slippery fronds or masses, which varied in size from that of very small shot to nearly half an inch in length. The larger ones were not nearly so high as long, and presented irregular, almost bossellated upper surfaces. The filaments are often very evidently and frequently pseudoramose. ‘The external surface of the broad sheath is covered with numerous fibrille, which envelop and seem sometimes to wrap it round and round. ‘The color of the frond internally, when broken, is mostly a dark chocolate, and the surface presents a radiated appearance, with but two or three zones at most, and, in the very dark specimens, even these are not evident. No signs of spores have been found. Certain specimens which I ob- tained growing with the others, instead of being blackish in color, are grayish, but 7 April, 1872. 50 FRESH-WATER ALG OF THE UNITED STATES. agree in all other respects with their fellows. This gray color depends, I believe, upon the deposit of an immense quantity of lime salts, which in such specimens constitute by far the larger portion of the frond. Fig. 4, pl. 4, represents a section of frond, slightly magnified. It is cither this, or the preceding species, which is referred to by Professor Bailey in Silliman’s Journal, vol. iii, under the name of Rivularia calcarea, Sm. The present form may possibly be that plant, but not having been able to find any description sufficiently well made out to make identification possible, 1 have de- scribed both species as new. Z minutula, Woon, (sp. nov.) Z. minutissima, nigro-viridis, subglobosa, haud distinete zonata, nonnihil mollis, muscicola, calce non pregnans; trichomatibus internis, breve articulatis, distinctissime fasciculatim psendora- mosis; vaginis crassis, amplis, sepe dilute aurantiaco-brunneis, apice plerumque coloris expertibus fissis et apertis; cellulis perdurantibus ovato-globosis. Diam.—Trich. intern. .00012"—.00021” ; cell. perd. .00025.” Hab.—In lacu, “ Clear Pond,” muscis affixa, Adirondack Mountains. Very small, blackish-green, subglobose, not distinctly zoned, rather soft, growing on mosses, not impregnated with lime ; internal filaments shortly articulate, very distinctly fasciculately pseu- doramose; sheaths thick, ample, often pale orange-brown, with their apices mostly colorless, torn and open; heterocysts ovately globose. Remarks.—TVhe locality in which I found this plant is in the heart pf the Adi- rondack wilderness. ‘The little frond in none of my specimens is larger than a mustard-seed, and is not distinctly zoned. The plants were collected in the begin- ning of July, and very possibly are not fully grown, as the season of general growth opens very late in its parent lake. Very possibly, later in the year, it may be found larger and distinctly zoned. ‘The general appearance of the plant, the character of its sheath, and the marked branching habit of the filaments have caused me to place it in this genus. Genus DASYACTIS, Krz. Thallus gelatinosus, mollis, non zonatus. Trichomata matura sepe haud vaginata. Spore nulle. Thallus gelatinous, soft, homogeneous, not zoned. Mature filaments often not vaginate. Spores absent. D. mollis, Woop. D. parva, ad magnitudinem pisi minimi, enormiter subglobosa, mollis, gelatinosa, dilute viridis; trichomatibus plerumque subrectis, partim distincte, partim indistincte articulatis; vaginis, in trichomatibus maturis haud visibilibus, in trichomatibus juvenibus supra subamplis; cel- lalis perdurantibus sub-globosis, globosis, vel ellipticis, diametro duplo majoribus, plerumque singulis sed interdum bi vel triseriatis. Diam.—Trich. go55"—azl50" 3 Cell. perd. yaya”. Syn.—D. mollis, Woop, Prodromus, Proce. Amer. Philos. Soc., 1869, p. 128. Hab.—In palude plantas aquaticas adherens, Northern Michigan. FRESH-WATER ALG&H OF THE UNITED STATES. 5 ol Frond small, about the size of a small pea, irregularly subglobose, soft, gelatinous, light green, filaments generally straightish, partly distinctly, partly indistinetly articulate ; sheaths in the mature filament not perceptible; in the young filaments rather large in the upper portion ; heterocysts subglobose or globose or elliptic, twice as large as the filament, generally single but sometimes bi or tri-seriate. Remarks.—I found this species growing attached to the little leaves of various minute cryptogamic and phaneerogamic water-plants, in a small bog, near the mouth of Carp River, in Northern Michigan. ‘The frond is somewhat translucent, with a slightly greenish tint, and has a soft, gelatinous consistency. The matured trichoma or filaments are more or less radiating, very long, generally nearly straight and parallel. Their joints or articles are long, mostly not very distinctly separated, and often are entirely wanting. The sheaths are entirely lost, no traces of them being perceptible. They seem to be altogether melted down into the homoge- neous jelly, in which the filaments are imbedded. ‘The basal cell is large, mostly globular, and very prominent. On the edges of the frond may frequently be seen small, evidently immature filaments, which have no distinct basal cell. Around the basal portion of these young trichoma there is a well-marked close sheath, which near the apex is wanting. In their immature filaments the joints are mostly very short, rather distinctly separated, almost globular. Fig. 5, pl. 4. Genus MASTIGONEMA, Scuwase. Trichomata articulata, sursum flagelliformia vel subulata, simplicia vel pseudoramosa (nonnunquam fasciculatim pseudoramosa), procumbentia vel erecta, in thallo indistincto caspitoso-agegregata; vagine arctze et homogenex vel ample et plus minus distincte lamellose, apice plerumque aperte, interdum laciniate. Filaments articulate, superiorly flagelliform or subulate, simple, or falsely branched, sometimes fasciculately so, procumbent or erect, cxspitosely aggregated into a sort of thallus; sheaths close and homogeneous or ample, and more or less distinctly lamellate, the apex for the most part open, sometimes laciniate. M. fertile, Woop, (sp. noy.) M. cespitosum, cum algis alteris intermixtum; trichomatibus simplicibus, elongatis, flexuoso- curvatis, apice truncatis; trichomatibus internis viridibus, seepe interruptis, interdum dis- tincte articulatis interdum inarticulatis; articulis diametro 8-5 plo longioribus; vaginis modice arctis, firmis, achrois, crassis, coloris expertibus, apice truncatis et apertis ; sporis cylindricis, sparsis, in filamento unico sepe pluribus, in cellulis inclusis; cellulis perdurantibus globosis, interdum compressis trichomatis diametro fere equalibus. Diam.—Filam. 357” = -00033”" 5 spor. ggop” = .000166". Hab.—In stagnis. Alleghany Mountains, Centre County, Pennsylvania. Cexspitose, intermixed with other algx; filaments simple, elongate, flexuously curved, trun- cate at the apex; internal filament green, often interrupted, sometimes articulated, some- times not articulate; joints 2-3 times longer than their diameter; sheath moderately close, thick, firm, transparent, and colorless, truncate and open at the apex; spores cylindrical, scattered, each contained in a cell, frequently several in a filament; heterocysts globose, sometimes compressed, about equal in diameter to the filament. 59 FRESH-WATER ALGZ OF THE UNITED STATES. Remarks.—I found this plant in a stagnant pool in “ Bear Meadows,” forming a filamentous, felty mass with Gidogonium echinatum and other alge. ‘The variously curved and interlaced flexible filaments are always simple and of uniform, or nearly uniform, diameter through their whole length; excepting that in some instances there are small, local, bulbous enlargements of the sheath. Though the ends of the filaments in all the specimens I have seen are abruptly truncate, it is very possible that in the young trichoma the apex is prolonged into a long hair as in most of the Mastigonema. The inner filament is sometimes very distinctly arti- culated, often, however, it is not at all so. ‘The sheaths are firm, not at all lamel- late, and generally project beyond the inner trichoma. ‘The spores are cylindrical, yellowish, with a pretty distinct, although very close coat. ‘They are always in- closed in distinct cells, and are mostly several in a filament, placed at intervals in its length. This is the first instance, at least that I know of, in which a species of this genus has been found in fruit, and it is interesting to note the resemblance of the spores to those of the more commonly fruiting rivularias. At the same time the peculiar arrangement of the spores is remarkable, and if the other species of MJas- tigonema should be found to have the more common exclusively basal arrangement of spores, I think it would afford good ground for considering MV. fertile as the type of a new genus. Moreover, the filaments are not united into a distinct thallus, and also want the apical hair of MJastigothrix, so that it is very probable that they represent an undescribed genus. Until, however, the fructification of the European species is elucidated, it seems best to forbear multiplying names. Fig. 1, pl. 4, represents a single filament of this species. M. halos, Woon, (sp. nov.) M. cxspitulis; trichomatibus simplicibus, in etate matura valde elongatis et cum vaginis trun- catis et apertis,—in «tate immatura modice brevibus et in setam modice longam achroam productis; trichomatibus internis breve articulatis, subtiliter granulatis continuis vel varie interruptis ; vaginis firmis, modice crassis, sepe distincte lamellosis, coloris expertibus ; cel- lulis perdurantibus subglobosis Diam.—Sine vag. = .0003”; cum vag.= .0005”. Hab.—In xstuario, Stonington, Conn. (Dr. F. Lewis.) In little tufts; filaments simple, in mature state greatly elongate, and with the sheath truncate and open,—in the young condition shorter and often ending in a rather short seta; internal filament shortly articulate, minutely granular, continuous or variously interrupted; sheath firm, rather thick, often distinctly lamellated, colorless ; heterocysts subglobose. Remarks.—This species is an inhabitant of salt, or at least brackish water, having been collected in Stonington Inlet by Dr. Frank Lewis. The filaments are very long and always simple; forming apparent exceptions to this, I have seen once or twice a number of young filaments so united as to give the appearance of having been produced from one old one, and in other cases young filaments growing from the side of an old one; but I believe those are always set free so soon as they attain a certain size. In one instance there were large, globular cells, with very thick walls, produced, and lying free, in the sheath. Are these spores? They are FRESH-WATER ALG#H OF THE UNITED STATES 53 well shown in figure 26, pl.5. Associated with them were a number of similar cells which had not obtained as yet the outer thick wall. ‘The color of the filaments is in my specimens of a rich golden brown; but, as they have been preserved in car- bolic acid water, I cannot speak positively as to the original tint. The heterocysts are subglobose, sometimes compressed, sometimes somewhat triangular. ‘They about equal in diameter the internal filament. Fig. 2, pl. 5, represents a small cluster of youngish filaments of this species. Mi. sejunctum, W000, (sp. nov.) M. thallo cespitulo, molle, parasitico ; trichomatibus simplicibus, plerumque inarticulatis, sed, interdum breve, interdum longe, articulatis, continuis, rarius interruptis, apice attenuatis, flayo-olivaceis aut viridibus, sparse granulatis; vaginis plerumque amplis et distinctis, hya- linis, seepius valde undulatis, apice plerumque valde amplificatis et in fibrillas solutis; cellulis perdurantibus diametro subzqualibus ; sporis nullis. Diam.—Trichom gyyq” = .00016”; cum vag. sq55” = .0005”. Hab.—In plantarum aquaticarum foliis, Carp River, Michigan. Thallus somewhat czspitose, soft, parasitic; filaments simple, mostly inarticulate, but some- times shortly sometimes long articulate, continuous or more rarely interrupted, attenuate at the apex, yellowish-olive or greenish, sparsely granulate; sheaths mostly ample and distinct hyaline, often strongly undulate, the apex mostly much amplified and dissolved into fibrille ; heteroeysts about equal to the filament in diameter; spores wanting. Remarks.—This species was found in the Carp River bog, growing on the edges of minute leaves, so as to form little prominences or thickenings of the margin. The trichomata are quite distinct from one another, and can scarcely be said to be united into a frond, although they all appear to radiate from the base, where they are consolidated into a dense mass. The sheaths are generally quite distinct, much broader than the cytioplasm, and are not sensibly dilated below. In most speci- mens they are very distinctly alternately dilated and contracted, or in other words, undulated. This is especially the case when the sheaths are quite wide. Above, they are rapidly and widely dilated, are distinctly fibrillose, and appear to gradu- ally melt away. ‘The cytioplasm is rarely articulated, and, when it is so, the joints are scarcely longer than broad, and are most generally confined to the distal end of the filament. The species appears to be most nearly allied to IM Bauerianum, Grun., from which, however, it is quite distinct. Fig. 2a, pl. 4, represents this species magnified 250 diameters ; fig. 2 6, a single filament magnified 800 diameters. M. clongatum, Woop. M. initio subglobosum, postea sepe nonnihil fusinum, nigro-viride, Iubricum, firme ; trichoma- tibus erugineis, valde elongatis, flagelliformibus, interdum inarticulatis sed sxpius breve articulatis, interdum ad genicula valde constrictis, apice interdum truncatis sed plerumque in pilum, longum, achroum, flexuosum, productis; vaginis achrois, arctis, sepe apice trun- ~ ceatis; cellulis perdurantibus globosis vel subglobosis. Diam —z 255" = .00026.” Syn.—M. elongatum, Woop, Prodromus, Proc. Amer. Philos. Soc., 1869, p. 128. Hab.—In aquario. FRESH-WATER ALG#& OF THE UNITED STATES. Thallus at first subglobose, afterwards frequently fusiform, blackish-green, slippery, firm ; fila- ments eruginous, very elongate, sometimes not articulated, but more generally shortly arti- times strongly contracted at the joints; apices sometimes truncate but generally culated, some a long, flexuous, translucent hair; sheath transparent, close, frequently trun- produced into cate at the apex; heterocysts globose or subglobose. Remari:s.—This species grew in my aquarium on some brook-moss, which I obtained from a spring above Manayunk. It forms little nodules of the size of a pin’s head upon the wire-like stems, or sometimes longer fusiform masses, which are apparently produced by the coalescence of a number of the little globes. The color of these fronds, which are very firm, is a blackish-green. ‘The filaments radiate from the base in all directions, and at the apex are tipped with a very long hair-like flexu- ous point, or they are truncate, apparently from the breaking off of this terminal seta. The endochrome is not unfrequently interrupted within the sheath. When it is articulated, the joints are usually about as long as broad, and frequently are distinctly separated from one another. ‘The sheath is sometimes quite apparent and distinctly truncate and open above, but in other instances is with difficulty perceived anywhere, and above is lost in the long hyaline point. At the points of attachment of the frond the filaments are so densely crowded as almost to appear to be coalescent, though I believe they are never really so; yet it is often almost impossible to separate them one from another by pressure on the glass cover, without entirely mashing and distorting the filaments. Fig. 1 a, pl. 5, represents a section of a frond of this species slightly magnified ; fig. 1 b, a single filament magnified 460 diameters. Genus MASTIGOTHRIX, Krz. Trichomata singula, plerumque sparsa, parasitica intra thallum Chetophorarum aliarumque algarum, flagelliformia, in apicem piliformem achroum hyalinum cuspidata, distincte articulata, arcte vaginata, basi cellula perdurante instructa. (R.) Filament single, mostly scattered, parasitic within the thallus of Chetophora or other alge, flagelliform, with the apex produced in a. hyaline hair, distinetly vaginate, furnished with a basal heterocyst. Remarks.—1 have simply copied the generic description of Prof. Rabenhorst, although it seems tome mare than doubtful whether the place of growth is any generic character whatever. I have relied more on the long hyaline apical hair, although our American form does grow in a gelatinous palmella like jelly. Mi. fibrosa, Woop. M. dilute vel ceruleo-viridis, vel olivaceo-viridis, vel sub-eruginea, infra haud articulata, sursum spe breve articulata, apice in trichomata matura in setam hyalinam, distincte articulatam, longam, producta; vaginis achroois—in filamento immaturo, supra distinctis, latis, hyalinis, infra modice crassis, arctis,—in trichomata matura infra arctis, indistinctis, supra in fibrillis dissolutis, apice absentibus; cellulis perdurantibus globosis, interdum geminis. ; Kas Nis Diam.—Trichone s3'55"; cell. perdur. Ieeoe —r1s¢00 Syn.—M. fibrosa, Woon, Prodromus, Proce. Amer. Philos. Soc., 1869, p. 129. Hab.—P rope Philadelphia. FRESH-WATER ALG@A OF THE UNITED STATES. 55 Light bluish-green, or olivaceous-green, apex in the mature filament prolonged into a long, distinctly articulated hyaline seta; sheath transparent—in the immature filament distally, broad, and distinct although hyaline, below rather thick and close; in the mature filament below close, indistinct, above dissolved in fibrille and wanting at the apex; heterocysts globose, sometimes geminate. Remarks.—This plant was found growing with other low alge in a thick jelly, which clothed some wet, dripping rocks near Manayunk. In the young filaments the sheath is produced above into a broad, thick, gelatinous-looking portion, the cavity of which is often scarcely apparent. The cytioplasm in such filaments is mostly of a light bluish-green color, is granular and not very apparent. In older filaments, the trichoma above is prolonged into a long, curved hyaline point, and the sheath just below the base of this is split into a number of fibrilla. No spores were perceived. ‘The increase of the species appears to take place in the follow- ing manner: Near the middle of the filament a tumid swelling forms, in the centre of which appears after awhile a constriction, and this increases until at last there are shaped out the bases of two filaments. ‘Then the heterocysts appear, and finally the two halves of the original trichoma separate—each a perfect filament. (Fig. 3, pl. 5.) Sometimes, instead of a pair of filaments being thus formed, but a single base is shaped out at the place of swelling, and the original filaments split, as it were, thus giving origin to a second trichoma, which for awhile appears as a branch of the former, but is soon detached from it. In some specimens there are two heterocysts, unless the proximal of these, which is a light orange-clay color, represents a spore. Fig. 3, pl. 5, represents different forms of this species. Famity SCYTONEMACEA. Trichomata articulata, sepe moniliformia vel submoniliformia, vaginata, pseudoramosa, cellulis limataneis, ad pseudoramulorum basin, vel interstitialibus, plerumque pachydermaticis instructa. Vagine e stratis pluribus (etsi non semper distinctis) formite, superficie leves, corrugate vel exasperate, crustate, nonumquam stratis exterioribus in fibrillas discedentibus, haud raro passim intumescentes vel ocreate. Vegetatio non terminalis; cellularum vegetativarum divisio ad unam directionem, initio in tricho- matibus medio, postea in utroque fine spe alternans. Cellulw limitanex ad utrumqne polum locello lucido instructe. Propagatio gonidiis pleramque exultima generatione ortis. Gonidia plerumque numerosa seriata e vagina se exserunt tumque in singula secedunt. (R.) Filaments equal, articulate, often moniliform or submoniliform, vaginate, pseudoramose, furnished with heterocysts which are either interstitial or at the base of the branches, and are mostly thick- walled. Sheaths formed of numerous strata (not always distinct), their surface smooth, corrugate, or roughened, the exterior stratum sometimes breaking up into fibrille, not rarely intumescent or ochreate. Vegetation not terminal; division of the cells occurring in one direction, in the beginning in the middle of the trichoma, afterwards often alternately at each end. Heterocysts furnished with a trans- lucent spot at each end. Propagation mostly by gonidia arising from the last generation Gonidia mostly numerously seriate, passing out of the sheaths and then separating one from the other. 56 FRESH-WATER ALG& OF THE UNITED STATES. R-marks.—TVhe Scytonemacee are simple or branched filamentous plants, which grow in water, or in the air, upon tree-trunks, rocks, fences, &c., in moist localities. A number of individuals of one or more species are almost always associated to form on the ground little mats, or in the water attached or floating masses of vary- ing color and characters according to the species. ‘The individual filaments are composed of two distinct parts, the inner the protoplasmic matter, the outer the cellulose sheath. The former of these is a long cylindrical mass, which is occa- sionally interrupted by a distinct thick-walled cell, spoken of in this memoir as the heterocyst, or “cellulis perdurantibus.” The inner filament is composed of colored protoplasm, which is sometimes homogeneous, but in other cases is dis- tinctly granular. It is most generally articulated after the manner of an oseilla- toria, but occasionally it is continuous for a great portion of its length, and in one species, which is here described, although very possibly not belonging in the family, there are, at regular intervals, partitions running across from one side to the other of the sheath, so that the inner filament may be said to be made up of a number Of cells. The heterocysts are of various shapes, globular, compressed, cylindrical, oblong, &e, &c. They are mostly provided with a bright colorless spot at each end. Their number varies according to the species. Sometimes they are single, in other cases there are several of them arranged in series. ‘They are placed either at the origin of the branches, or are scattered apparently without definite arrangement in the length of the filament. In the one case, they are known as “ basal,” in the other as “interstitial.” In any species, either of these methods, or both of them, may prevail; but a certain amount of specific value attaches to the situation of the heterocysts. Their function is totally unknown, although some have imagined them to have a sexual significance and even to be spermatozoids, but there is no proof whatever of the truth of such suppositions, and it is, I think, very certain that these heterocysts are not of the nature of spores. The sheath of the Scytonema is composed of one or more strata, which are often very distinct from one another, but are more often, perhaps, not so. It is opaque or translucent, aud has its outer surface smooth, or tubercular, fibrillate or roughened in some way. The specific characters in this family can best be commented upon under two heads—namely, those which are discoverable with the unaided eye, and those which the microscope alone can reveal. ‘The points to be observed under the first of these are as follows: The place of growth of the plant, whether in the air or in the water, and, if it live in the air, to what it is attached—whether to stones, dead wood, or living trees, and it is possible that in some cases it may be found that certain species of Scytonema inhabit only certain species of trees. If the plant be in water, it must be noted whether it be attached or floating. Then the habit of growth must be looked at, including in this the size and thickness of the masses of filaments, whether they be flocculent, turfy, crustaceous, membranous-gelatinous, &c., their softness or rigidity, their color, as well as the arrangement in them of the filaments. ‘To discover the latter, it will generally be necessary to use a low power of the microscope, and at the same time the mode and profusion of branch- ing of the individual plant should be studied. FRESH-WATER ALG# OF THE UNITED STATES. 57 The second class of characters are those discoverable only with the higher powers. They are divisible into two sets; those afforded by the inner filament and those derived from the sheaths. In the first of these the points to be noted are, the diameter of the filament; its color, whether it be or be not articulated, and if it be the length of the joints; whether it is uniform or moniliform; whether it be homogeneous or granulate; then the heterocysts should be examined as to their size, position, arrangement, shape, number, and color. The diameter of the sheath, its homogeneousness, its color, firmness, and the condition of its outer surface are to be included in the specific study. Genus SCYTONEMA. Trichomata cxspitoso-congregata vel fasciculata, plus minus pseudoramosa, cellulis interstitialibus instructa; vagine gelatinoso-membranacew, e stratis (interdum obsoletis) pluribus eylindraceis com- posite; cellulis perdurantibus singulis. Filaments cespitosely-congregate or fasciculate, more or less pseudoramose; furnished with in- terstitial cells; sheaths gelatinous-membranaceous, composed of many cylindrical, sometimes ob- solete, strata; heterocysts single. a. Terrestres vel aquatice. a. Terrestrial or aquatic. 8S. sim:plice, Woop, (sp. nov.) S. in strato modice crasso, subtomentoso, nigro-viride ; trichomatibus valde elongatis, flexuoso- curvatis, parcissime pseudoramosis vel seepe sine pseudoramulis; pseudoramulis geminis vel singulis, plerumque elongatis ; trichomatibus internis modo distincte articulatis, modo inar- ticulatis, apice interdum brevissime articulatis, granulosis, pallide viridibus, ad genicula sepe nodosis vel disjunctis, articulis plerumque diametro equalibus ad 7 plo longioribus; vaginis plerumque supra truncatis et apertis, pellucidis, sepe coloris expertibus, interdum dilute aureo-brunneis; cellulis perdurantibus cylindricis, interjectis, diametro 2-5 plo longioribus. ” 1 Diam.—Trich. cum vag. 3-59’—1e59 = -0004”—.00066"; sine vag. 72405’—7259 = 00013" —.00026”. Hab.—In lignis irroratis, South Carolina. (Ravenel.) S. in a moderately thick, somewhat tomentose, blackish-green stratum; trichomata very elongate, flexuously curved, very sparsely branched or frequently without branches ; branches geminate or single, mostly elongate; internal filament partly distinctly articulate, partly inarticulate, granular, pale-greenish, in its apex sometimes very shortly articulate, sparsely granular, often nodose or disjoined at the joints; articles mostly from equal to to 7 times longer than the diameter; sheaths thick, transparent, often colorless, sometimes pale yellowish-brown, mostly open and truncate at apex; heterocysts cylindrical, interspersed, 2-5 times longer than their diameter. Remarks.—I am indebted to Professor Ravenel for specimens of this species. They are preserved in solution of acetate of alumina and accompanied by the fol- lowing label: “Adhering to the wet sides of a wooden gutter, leading water from a spring, September 29, 1869: Aiken, South Carolina.” The filaments are remarkable for the fewness of their branches. Generally, indeed, there are no 8 May, 1872. 58 FRESH-WATER ALG OF THE UNITED STATES: branches whatever, and I have never seen more than a single pair, or, at most, three branches to a filament. ‘The mass of filaments is blackish-green, somewhat tomentose and quite shiny in appearance. ‘The articles are often very long, and the internal filament is frequently in such cases enlarged into a sort of globular node at the joint. Not at all rarely there is a very decided break in the endo- chrome at the joints. ; This species is very close to S. Austinii, from which, however, I think it suffi- ciently distinct. The points of difference are in the much firmer, much more colored and opaque, and rougher sheath of that species; in the swollen ends of the internal filament of S Austinii, and its shorter articles, with the absence of nodes or distinct interruption of the endochrome at the jomts. The heterocysts are also quite different in the two forms, whilst the filaments of WS. simplice are much the longer. Ss. Austinii, Woop, (sp. nov.) S. rupicola, strato tomentoso, cxspitoso, crasso, fuseco-nigro; trichomatibus adscendentibus, curyatis, pleramque simplicibus; trichomatibus internis wrugineis vel fuscescentibus, articu- latis vel inarticulatis, fine spe valde incrassatis; articulis diametro plerumque multo bre- vioribus, interdum longioribus; yaginis rubido- vel aureo-fuscescentibus, seepe sub-opacis, firmis, indistincte lamellosis, in apice plerumque achrois et coloris fere expertibus, superficie subrugosa et hirta; cellulis pedrurantibus breviter cylindricis, vel subquadratis vel subglo- bosis, interdum valde compressis et diametro multo brevioribus. Diam.—Fil. cum. vag. .0006”—.0008"; sine vag. .00016”—.0004”. Hab.—In rupibus, “ Little Falls, New Jersey.” (Austin.) S. growing on rocks, stratum tomentose, and somewhat turfy, brownish-black ; trichomata ascending, mostly simple, curved ; internal filament sruginous or fuscous, articulate or inar- ticulate, often very much thickened at the ends; articles much shorter to longer than their diameter ; sheaths reddish or yellowish-fuscous, at the apex colorless and transparent, firm, indistinetly lamellate ; surface rough; heterocysts shortly cylindrical, subquadrate or sub- globose, sometimes strongly compressed and much shorter than broad. Remarks.—This plant occurs as a blackish stratum of one or two lines in thick- ness, forming a sort of miniature turfy cushion upon the rock. When examined with the hand-glass, this layer is seen to be composed of a great number of ascend- ing curved filaments whose color, in some specimens, is a reddish-brown; in others, apparently younger, yellowish-brown. Under the compound microscope the sheaths in the older filaments are seen to be much roughened externally and irregular in outline. The young sheaths are smooth. ‘The filaments are mostly simple, since I have not seen more than a half dozen having even a single branch. The heterocysts are scattered at irregular intervals, and are remarkably irregular in form—sometimes much shorter than broad, sometimes several times as long. As the ends of the filaments are approached the internal filament suddenly swells out and inereases sometimes to twice the diameter it has in the central part of the filament. In the filament proper it rarely attains a diameter of more than .0003”, and is commonly about .00025’’, whereas at the ends it very generally approaches the maximum .00042”, FRESH-WATER ALG#A OF THE UNITED STATES. 59 S. immmersum, Woo, (sp. nov.) S. immersum cum algis alteris intermixtum et plantas aquaticas adherens; trichomatibus elongatis ; pseudoramulis plus minus distantibus, plerumque geminis, et e basi divergenter adscendentibus, brevibus aut elongatis; trichomatibus internis lete wrugineis, interdum dis- tincte articulatis, interdum inarticulatis, apice obtuse rotundato, ewrugineo; articulis diametro subequalibus vel brevioribus ; vaginis amplis, hyalinis, coloris expertibus; cellulis perdu- rantibus distinctis, singulis, interjectis, subcylindricis, diametro interdum fere duplo breviori- bus, interdum duplo longioribus. Diam.—Sin. vag. ysion’ = -000415". Cum vag. y3259/’ = .00075”. Hab.—In aquis quietis, Cumberland County, New Jersey. 8. immersed, intermixed with other alge and adhering to aquatic plants; filaments elongate; branches mostly geminate, more or less distant, short or elongate; internal filaments bright xruginous, sometimes distinctly articulate, at others not so, apex obtusely rounded eruginous ; joints about equal to the diameter or shorter; sheath ample hyaline, colorless; heterocysts distinct, single, interjected, subcylindrical, sometimes about half as long as broad, sometimes nearly twice as long. Remarks.—I found this plant in September, 1869, in Shepherd’s Mill Pond, near Greenwich, Cumberland County, New Jersey, forming, with other alge, a floccu- lent, greenish-black, slimy coating to the stems and finely dissected leaves of Ranunculus aquatilis. ‘The branches are very few in number in most specimens, and when they are more plentiful are apt to be short and abortive. ‘Their apices do not differ materially from their other portions. Fig. 9, pl. 2a, represents a portion of a filament of this specimen magnified 750 diameters ; fig. 2 6 a whole filament magnified 260 diameters. S. Naegelii, Krz. (’) S. exsptoso-floceosum, bryophilum, nigro-viride; trichomatibus, plerumque sparse pseudora- mosis, pseudoramulisque elongatis et intricatis; trichomatibus internis breviter articulatis, seepe interruptis, sepe nonnihil moniliformibus, viridibus aut in etate provecta brunneis; articulis swepe sejunctis, diametro plerumque brevioribus, subtiliter granulatis ; pseudoramulis plerumque singulis ; vaginis modice arctis, interdum subamplis, haud distincte lamellosis, modice crassis, hyalinis, coloris expertibus ant in «tate provecta dilute fusco-brunneis; cellulis perduranti- bus nonnihil reniformibus, pleramque nullis, basilaribus. Diam.—Fil. cum vag. plerumque 7,%55’—max. gy'yq” 5 Sine vag. pysoo Cell. perdurant. lab. tz000 —long. papo0 - Syn.—S. Negelii (Krz.), RABENHORST, Flora Europ. Algarum, Sect. II. p. 252. Hab.—In fonte, prope Belvidere, Centre County, Pennsylvania. Growing in small, blackish-green woolly mats attached to mosses; filaments mostly sparsely branched, with the branches elongate and intricate; internal filament shortly articulate, often somewhat moniliform, often interrupted, green, or, in mature state, brownish; joints often disjoined, mostly shorter than the diameter, finely granulate ; branches mostly single; sheaths moderately close, sometimes ample, not distinctly lamellate, rather thick, hyaline, colorless, or, in old age, light fuscous brown; heterocysts mostly wanting. Remarks.—I found this plant in the large spring that supplies Bellefonte with water, growing attached to mosses, so as to form little dark-green mats around 60 FRESH-WATER ALG#& OF THE UNITED STATES. their stems and branches. These mats never exceeded an inch in length in any specimens that came under my notice. The filaments themselves are apparently not much branched and are densely interwoven. ‘The sheaths are close, rather thick, not lamellate, of uniform diameter, except in that they are occasionally locally swollen, and are truncate and open at the end. The internal filaments are frequently much interrupted, and in the younger plants are of adeep green. The joints are in many instances much separated, and in most cases very distinct. The filaments. indeed show a remarkable tendency to break up at the joints, so as to form a series of dish-like gonidia, so that the articles, or endochrome masses, may be generally described as strongly compressed spheres. In all the specimens that I have examined, I have seen but a single heterocyst. This was at the base of a branch, was somewhat reniform, and about three-fifths as long as broad. I have referred this species, doubtfully, to S. Naegelit, Ktz., the only account of which that I have met with, or know of, is a brief diagnosis in Rabenhorst’s Flora, in which many of the essential characters are omitted. Fig. 6, pl. 8, represents a portion of a filament of this species. S. thermale, Krz. S. strato tenue, nigrescente; trichomatibus flexuoso-curvatis, intricatis, parce pseudoramosis, internis pallide wrugineis, sepe coloris fere expertibus, passim interruptis, plerumque inar- ticulatis sed seepe indistincte et interdum distincte articulatis, granulosis ; articulis diametro brevioribus vel subequalibus ; pseudoramulis plerumque brevibus, geminis, in diametro tri- chomatibus equalibus vel subequalibus et interdum usque ad medium conjunctis, basi coa- litis, sepe e basi divergentibus; vaginis crassis, indistincte lamellosis, vel luteo-fuscis vel fuscis, sed passim fere coloris expertibus, plerumque vix pellucidulis, in ramulorum apice sepe hyalinis et coloris fere expertibus; cellulis perdurantibus, subquadratis vel cylindricis, singulis, interjectis. Diam.—Tr. cum vag. yx$o9'—rz500" = -00042”—.00058; sine vag. gop” = -000166"— az000 = -00025. Syn.—S thermale, Krz., RaBENuHORST, Flora Europ. Algarum, Sect II. p. 250. Hab.—In terra argillacea, South Carolina. (Ravenel.) Stratum thin, blackish; filaments flexuously curved, intricate, sparingly branched; internal filament pale-greenish, often almost colorless, here and there interrupted, mostly inarticulate, but often indistinctly and sometimes distinctly articulate, granular; joints shorter or about as long as broad; branches geminate, mostly short, equal or subequal to the filament in diameter, coalescent at the bases, rarely so even to their middle, mostly divergent from the base ; sheath thick, indistinctly lamellate, yellowish-fuscous, and scarcely semitransparent, but here and there nearly colorless and pellucid, generally so in the apices of the branches; heterocysts subquadrate or cylindrical, single, interspersed. Remarks.—I am indebted to Professor Ravenel for specimens of this species pre- served in solution of acetate of alumina. ‘The label reads, “Damp surface of hard clay, Sept. 29, 1869.” The sheaths are quite thick and scarcely translucent, so that the color of the inner filament seen through them is that of themselves. Curiously enough, one of these dark sheaths will for a space lose its color and be very trans- parent, in such places and in the apices of the branches, the inner filament is often a decided pale-green ; at other times it is almost colorless. The end of the sheaths are mostly closed, but I have seen them open, with the inner filament project- FRESH-WATER ALGA OF THE UNITED STATES. 61 ing. The branches are nearly always short, and divergent from their united bases. The heterocysts have frequently one of their ends rounded; and are quite numerous. ‘This species corresponds too closely to Rabenhorst’s description of Scytonema thermale to be separated, but it is possible a comparison of specimens might show decided differences—the description of the European form is not very full. The American plants seem to approximate most closely the Var. intextum. I have seen a single branch given off only in one instance. Fig. 1, pl. 6, represents a filament of this species magnified 260 diameters; fig. 1 b, the outline of a heterocyst magnified 750 diameters. S. Myochrous, Ac. S. strato tenui, pannoso-tomentoso, obscure fusco (nonnunquam subsericeo); trichomatibus validissimis, fuscis, lucidis, leniter curvatis, adscendentibus, internis erugineis, apice (articul. term. 5-6) rubellis, distincte articulatis; pseudoramulis plerumque geminis, sepe longissimis flaccido-erectis, trichomate dimidio circiter tenuoribus; trichomatis vaginis crassis, distincte lamellosis, firmis, pulchre luteo-fuscis, superficie levissimis, ramulorum semper pallidioribus (luteis, rarius achrois), apice spe achrois, clausis et obtuso-rotundatis; cellulis perdu- rantibus oblongis vel subcylindricis, achrois, trichomatis interni diametro subzqualibus. (R.) Strato-obscure olivaceo, trichomatibus parce pseudoramosis, ad 2.” crassis ; pseudo- rannulis singulis, vaginis achrois vel luteolis; vag. trich. luteo-fuscis. (R.) Species mihi ignota. Diam.—Trichom 0.0011”—0.0014"; ramulorum ad 0.00068”. (R.) Syn.—S. Myochrous, AGarpu; Var. Contextum, CARMICHAEL. RABENHORST, Flora Europ. Algarum, Sect. IT. p. 254. Hab.—‘ Foot of Crow’s-nest, West Point.” Bailey. Silliman’s Journal, N.S. vol. iii. Strato thin, pannosely tomentose, obscurely fuscous (sometimes somewhat silky) ; filaments very strong, fuscous, bright, slightly curved, ascending ; the internal wruginous, distinetly articu- late with the apex (terminal 5—6 joints) reddish; branches mostly geminate, often very long, flaccidly erect, about one-half thinner than the filament; sheath of the filament thick, dis- tinctly lamellate, firm, beautifully yellowish-fuscous, surface very smooth; sheath of the branches always paler (luteous or rarely colorless) with the apex colorless, short and obtusely rounded; heterocysts about equal in diameter to the internal filament. Stratum obscurely olivaceous, filaments sparsely branched, about ,';" thick; branches single, with the sheaths transparent or yellowish; sheath of the trichoma luteo-fuscous. S. calotrichoides. Kvurzic(2). S. cespitosum, mucosum, plerumque cum algis variis intermixtum; trichomatibus plus minus curvatis; pseudoramulis plerumque geminis, varie curvatis, simplicibus, elongatis; tricho- matibus internis modo distincte articulatis, modo inarticulatis, interdum moniliformibus, luteo-viridibus vel werugineis, granulosis; articulis plerumque diametro brevioribus sed in- terdum permulto longioribus, haud rare vel subglobosis vel valde compressis ; cellulis per- durantibus singulis, subeylindricis; vaginis plerumque pellucidulis, distincte lamellosis, in trichomatibus plerumque rubido-vel luteo brunneis sed interdum coloris expertibus, in pscu- doramulis hyalinis, coloris expertibus vel dilutissime luteis vel dilute luteo-brunneis. 6 . : ” ‘ Diam.—Cum vag. max. 37%55/’=.00075"; plerumque 4.55” =-00045"; sine vag. 7255" —a0 00 3 pseudoram. 3755” =.0005”. Syn.—S. calotrichoides, Ktz. Rasenuorst, Flora Europ. Algarum, Sect. II. p. 252. Hab.—South Carolina. (Ravenel.) FRESH-WATER ALG OF THE UNITED STATES. Cespitose, mucous, mostly intermixed with various alge; filaments more or less curved; branches mostly in pairs, elongate, simple, variously curved ; internal filament partly dis- tinctly articulate, partly not articulate, sometimes moniliform, yellowish-green or zruginous, granular; joints mostly shorter than the diameter, sometimes much longer, sometimes sub- xlobose or strongly compressed; heterocysts single, subcylindrical ; sheaths distinctly lamel- late, mostly reddish or yellowish-brown, but sometimes colorless, in branches hyaline, color- less, or with a very faint yellowish tint, or sometimes brownish. Remarks.—The specimens, from which the above description was drawn up, were sent me by Professor Ravenel from South Carolina. ‘The extremities of the sheaths are either closed, or open. ‘The branches are almost always in pairs, and sometimes three or four are given off together, but this is not common. They are often nearly or quite colorless; the main filament is generally a sort of brown—sometimes quite bright from the predominance of the yellow hue. Although my specimens do not precisely agree with the descriptions of the European §. calotrichoides, yet the disagreement does not seem sufficient or sufficiently constant to separate specifically the two forms; the most important of the differences is in the coloration of the sheaths and heterocysts, which in the American plant are commonly, but not universally, respectively brownish and greenish. The label, which Professor Ravenel has attached to some of the specimens, reads, “In wet, boggy places, on rotten pine boards, Sept. 25, 1869.” Fig. 2, pl. 6, represents a filament of this plant magnified 250 diameters. S. cataracta, Woop. S. rupicola, cxspitosum, fusco-atrum, longe et late expansum; trichomatibus flexuosis, flexili- bus, fere 0.25” longibus, vage pseudoramosissimis, superficie lavibus; pseudoramis elongatis, singulis, rarissime geminis, liberis, interdum fuscis, seepius hyalinis, apice plerumque truncatis et rare nonnihil attenuatis et seepe barbais sed hand rubellis; trichomatibus internis erugi- neis, tenuissimis, plerumque distincte articulatis; articulis diametro plerumque brevioribus, sed interdum longioribus, spe sejunctis, sepe subglobosis ; vaginis crassis et firmis; cellulis perdurantibus et basilaribus et interjectis, singulis, rarissime geminis. Diam.—Trich. cum vag. plerumque .00045”; max. .0011”; sine vag. max. .00013”. Syn.—S. cataracta, Woop, Prodromus, Proc. Am. Phil. Soe., p. 129, 1869. Hab.—In flumine Niagara prope cataractam. S. forming on rocks an extended turf-like stratum of a brownish-black color ; filaments flexuous > i} flexible, almost 0.2 long, irregularly branched, their surface smooth; branches elongate, single, rarely in pairs, free, sometimes fuscous, frequently hyaline, their apices generally truncate, rarely somewhat attenuate, frequently provided with enlargements, never reddish ; cytioplasm xruginous, very thin, generally distinetly articulate; articles mostly shorter than broad, but sometimes longer, frequently disjoined, often subglobose; sheaths thick and firm; heterocysts both basal and interjected, single, extremely rarely geminate. Remarks.—This species grows abundantly in the Niagara River, on the rocks below the great cataract. It is really in little tufts, but these are in many cases placed so closely as to form a broad turflike coating to the stones. Often, however, the tufts are in smaller patches, and are of sufficient length to wave with the eddies and currents in the water. The branches are almost always given off FRESH-WATER ALGA& OF THE UNDMTED STATES. 63 singly since I have examined some hundreds of specimens, and have only in one instance detected them in pairs. ‘The apices of the branches, and indeed of the main filaments, are beautifully colorless and hyaline, and not unfrequently a branch will have this hyaline sheath for a long distance. The extreme ends are mostly truncate and open, and, often near them, the sheaths will have marked swellings ; a condition which, for want of a better term, I have spoken of as being barbate. Sometimes near the end of the filament the diameter of the sheath will be suddenly lessened. ‘The large cells are both interstitial and placed at the bases of the branches; they are more or less oblong or quadrangular, sometimes being scarcely longer than broad, but in other cases several times longer. At their posi- tion there is very generally a sort of globular enlargement of the filament. The sheath is sometimes very obscurely lamellate. The color of the older filaments is a dark, almost chocolate-brown. ‘This is apparently the species referred to by Professor Bailey as being Scytonema ocellatum of Harvey, in Silliman’s Journal, vol. il. N. S., although that plant, according to Professor Rabenhorst, belongs to the genus Sirosiphon. Fig. 1a, pl. 7, represents a portion of a filament, magnified 280 diameters ; fig. 1 6, a whole filament slightly magnified. S. dubiuma, Woop (sp. nov.) S. immersum, in floccis mucoso-tomentosis olivaceo-nigris plantas aquaticas adherens, vel in strato mucoso et nonnihil tomentoso dispositum; trichomatibus valde elongatis et arcte in- tricatis, varie curvatis, plerumque sparse pseudoramosis; pseudoramulis plerumque singulis, et plus minus distantibus et modice brevibus, vel interdum brevissimis et abortivis et nonnihil confertis; trichomatibus internis sepe in pseudocellulis distinctis contentis, interdum con- tinuis et indistincte articulatis vel inarticulatis, plerumque dilute ceruleo-viridibus sed inter- dum lete erugineis, subtiliter granulatis; vaginis arctis plerumque modice crassis et firmis, hyalinis, coloris expertibus; cellulis perdurantibus cylindricis, diametro 2-6 plo longioribus. Diam.—Cum vag. yx355’—tsb 55 = -00025”—.0004”. Hab.—In aquis quietis, Cumberland County, New Jersey. Immersed, adhering to water plants in olive-black tomentose flocculent masses, or arranged in a mucous and somewhat tomentose stratum ; trichomata very long and closely interwoven, variously curved, mostly sparsely branched; branches generally single, more or less distant, and moderately short, sometimes very short, abortive, and somewhat crowded ; internal fila- ment often contained in distinct cell-like apartments, sometimes continuous and indistinctly articulate, or not at all articulate, finely granulate, mostly a pale bluish-green, sometimes a bright zruginous color; sheath close, mostly rather thick and firm, hyaline colorless; hetero- cysts cylindrical, 2-6 times longer than broad. Remarks.—\ found this plant, September, 1869, in Shepherd’s Mill Pond, near Greenwich, Cumberland County, New Jersey. It formed dark, ugly, somewhat slimy, tomentose flocculi adhering to, and binding together, the finely-dissected leaves of Ranunculus aquatilis. The filaments are very long, slender, and sparsely branched. ‘The branches are given off at right angles, or nearly so, but are fre- quently sharply bent just above their origin. They are often, but not always, rather short. The most remarkable character that the plant possesses is that in many filaments there are very distinct regular partitions stretching across from 64 FRESH-WATER ALG OF THE UNITED STATES. side to side, so that the interior is divided, as it were, into successive cell-like chambers, in which the colored protoplasm is contained, This character seems almost to separate the plant from the genus Scytonema, but I have deemed it insufficient grounds for indicating a new genus. Since writing the preceding remarks, I have received specimens of this species from Professor Ravenel, who collected them in South Carolina, near the town of Aiken. They agree in all respects, except that they form a dark, mucous, somewhat tomentose coating to pieces of wood. : Fig. 3 a represents the outline of a series of the cells alluded to, magnified 750 diameters, and figs. 34 and 3¢, portions of filaments magnified 460 diameters, b. Arboricole. b. Growing on trees. S. cortex, Woop. S. minutissimum, stratum tenue submembranaceum formante; trichomatibus sparse pseudoramu- losis, pseudoramulisque repentibus et plus minus coneretis, viridibus aut dilute fuscis, varie curvatis, haud rigidis; ecytioplasmate viride, articulato, rare distincte granuloso; articulis diametro longioribus aut brevioribus; vaginis arctis, nonnihil tenuibus, plerumque coloris expertibus, sed interdum dilute fuscis; cellulis perdurantibus et singulis et geminis, et basa- libus et interjectis, globosis vel subglobosis. Diam.—Trich. cum vag. 7259" —7250"- Syn.—Scytonema cortex, Woop, Prodromus, Proc. Am. Philos. Soe., 1869, p. 130. Hab.—South Carolina. S. very minute, forming a thin, submembranaceous stratum ; filaments sparsely branched, toge- ther with the branches, creeping and more or less concreted together by their sides, green or light brown, variously curved, not rigid; cytioplasm (internal filament) articulate, rarely distinctly granulate; joints longer or shorter than broad; sheaths close, rather thin, trans- parent, generally colorless but sometimes light brown; heterocysts globular or subglobular, single or in pairs, basal or otherwise. Remarks.—I have specimens of this species collected in South Carolina by Pro- fessor Ravenel, who found it growing on the bark of Platanus occidentalis. The thin, almost membranous stratum which it forms, is of a dark olive-black, and has to the eye a sort of minutely warty appearance. ‘The filaments are so involved and so adherent, one to the other, that I have not been able to separate any length of them, nor are the branches distinguishable from the main filaments. The sheaths are rather thin, and often not very apparent. Fig. 4, pl. 6, represents this species, S. Ravenelii, Woop. S. lignicola, breve cxespitosum, viride-nigrum; trichomatibus plerumque repentibus, vel fusco- olivaceis vel aureo-fuscis, modice pseudoramosis ; ramis ascendentibus, rigidis, flexuosis rare pseudoramulosis, vel fusco-olivaceis vel aureo-fuscis, rarissime cum apicibus subachrois; tri- chomatibus internis coloris expertibus, granulosis, sepe vagina erympentibus, plerumque articulatis ; articulis diametro longioribus aut brevioribus; vaginis arctis, crassibus, fusco- olivaceis vel aureo-fuscis, pleramque supra truncatis et apertis, superficie nonnunquam irregu- laribus; cellulis perdurantibus subquadratis vel subglobosis singulis aut rare geminis, inter- jectis ; in stato juvene trichomatibus internis erugineis, vaginis tenuibus. ° : QAI, (YS ’ * . Diam.—Trich. cum vag. 7g50 —zy00 5 TAM cum vag. 7;4,5’—7;%;y"; trich. sine vag. sa's0” — 2.0005. FRESH-WATER ALG OF THE UNITED STATES. 65 Syn.—S. Ravenelii, Woop, Prodromus, Proce. Am. Philos. Soc., 1869, p. 130. Hab.—In cortice, South Carolina. S. Forming little turfy spots of a greenish color, on bark; filaments mostly creeping, either brownish-olive or yellowish-brown, moderately branched ; branches ascending, rigid, flexu- ous, very rarely provided with secondary branchlets, either brownish-olive or yellowish- brown, rarely subtransparent at the apex; cytioplasm colorless, granular, often extending out beyond the sheaths, generally articulate; joints longer or shorter than broad; sheaths close, thick, brownish-olive or yellowish-brown, for the most part truncate at their ends and open, their surface sometimes irregular; heterocysts subquadrate, single, interstitial. Remarks.—I am indebted to Prof. H. W. Ravenel for specimens of this very distinct species. Some of these are labelled as having grown on the twigs of a celtis in South Carolina, other specimens are on the bark of a willow. ‘The branches, which are mostly shortish, simple, and variously curved, are sent up in great numbers by the creeping stems, and, like the stems themselves, are mostly free, but not unfrequently are closely adherent by their edges. The internal trichoma or cytioplasm, owing to the great thickness of the sheaths, is not very apparent within these latter, but not unfrequently projects for a dis- tance beyond them, when it is seen to be colorless, very granular, and mostly, but not always, distinctly articulated. In the young plant the filaments are bright- green, often not more than ;-4,, of an inch in thickness, and have the sheath very thin, or may be almost imperceptible. It affords me great pleasure to dedicate this species to Professor Ravenel, not as an acknowledgment merely of his aid in my studies of this hitherto neglected branch of the North American Flora, but rather of the great services he has rendered science in some of its kindred branches. Fig. 4, pl. 5, represents the end of a filament of this species magnified some 450 diameters. Genus TOLYPOTHRIX, Krz. Trichoma scytonemacea cum cellulis perdurantibus seriatis. Filament similar to that of secytonema, but with the heterocysts seriate. T. distorta, (Miter) Kurz. T. cespitoso-floccosa, lete et pulchre viridis; trichomatibus intertextis, lete viridibus, modo distincte articulatis modo inarticulatis; articulis diametro brevioribus spe aut sub-nullis aut nullis; pseudoramulis singulis; vaginis arctis, homogeneis, vitreis ; cellulis perdurantibus basilaribus et interdum interjectis, pachydermaticis, plerumque in parallelogramme enormis forma, plerumque 4-seriatis, subachrois, interdum sparsissime granulatis. Diam.—y350 —s000 - Syn.—T. distorta, (Mtitnr) Krz. Rasenworst, Flora Europ., Algarum, Sect. II. p. 275. Hab.—In aquario, Philadelphia, Wood. Rhode Island (Olney) Thwaites. Warden’s Pond, Rhode Island; Reservoir Pond, West Point ; Fourth Lake, Madison, Wisconsin, Bailey. Flocculent cespitose, bright, beautiful green; filaments interwoyen, bright green, partly dis- tinctly articulate, partly continuous ; articles shorter than long, often very indistinct, some- times absent; branches single; sheaths close, homogeneous, glassy; heterocysts basilar, 9 May, 1872. 66 FRESH-WATER ALG OF THE UNITED STATES. sometimes interspersed, thick-walled, mostly irregularly parallelogrammatic, mostly 4-seriate, semitransparent, sometimes very sparsely granulate. Remarks.—This species grew spontaneously in the aquarium of my friend Dr. Frické, to whom I am indebted for specimens of it, forming little, bright-green balls adherent to the various aquatic plants. It approaches so very closely the European J. distorta, that I have considered it as a mere variety of it, although it differs in having the heterocysts mostly arranged in fours, and also apparently in their shape—they being in our plant mostly parallelogrammatic. Fig. 1 a, pl. 8, represents a section of heterocysts magnified 800 diameters; fig. 1 4, a portion of filament magnified 800 diameters. Genus PETALONEMA, Berk. (1833.) Scytonematis trichomata vaginis crassissimis e stratis numerossissimis brevioribus, infandibuli- formi dilatatis, imbricatis et plerumque dilutissime coloratis compositis. (R.) Syn.—Arthrosiphon, Krz. (1845.) “ Pilaments stratified, decumbent, free, simple, or branched. Tube or sheath very wide, flat- tened, longitudinally and transversely striate and crenulate at the edge; endochrome oliva- ceous annulated, here and there interrupted by a heterocyst. Branches issuing in pairs, formed by the division and protrusion of the endochrome of the original filament. When placed under the microscope the filaments present the appearance of a cylindrical cen- tral column, containing annulated, olive-colored endochrome, and a wide wing-like border at each side of the column. This border or sheath is obliquely striate, the striae running in an arch from the margin toward the centre, where they become parallel, and are then continued longitudinally downward along the medullary column, till lost in the density. The margin of the wing is closely crenulate and in age transversely striate at the crenatures as though jointed. Such is the apparent structure; the real structure seems to be, that an annu- lated central filament is inclosed within a number of compressed, trumpet-mouthed gelatino- membranaceous tubular sheaths, one arising within the other, and successively developed as the growth proceeds. These sheaths, thus concentrically arranged, are indicated by arching longitudinal striz; and the mouths of the younger sheaths, projecting slightly beyond those of the older, form the crenatures of the margin.” Harvey. ‘ P. alatum, Ber«. A. pulvinato-erustaceus, rupicola, varie coloratus; trichomatibus internis rugineis, curvatis, parce pseudoramosis, modo continuis, modo torulosis, submoniliformibus, apice plerumque paulum incrassatis, sepe roseolis, rotundatis; articulis distinctis, granulosis, diametro sub- wequalibus vel paulo brevioribus; vaginis stratis internis, aureis vel aureo-fuscentibus, externis achrois, vitreis; cellulis perdurantibus interjectis et ad pseudoramulorum basin, plerumque solitariis, subglobosis vel oblongis, dilute fuscis. (R.) Species mihi ignota. Diam.—Trich. intern. 0.00016”—0.00032" ; vag. 0.00377”. (R.) Syn.—Arthrosiphon alatus, (Grey.) RaBenn. Flora Europ. Algarum, Sect. II. p. 265. Petalonema alatum, Brrxetey. Harvey, Nereis Boreis Americana, part iii. p. 99, Smithsonian Contributions, 1846. Hab.—* On dripping rocks under Biddle Stairs, Niagara Falls.” (Harvey.) “This forms strata of a dark chestnut-brown color and of indefinite extent on the surface of rocks or soil exposed to the constant drip of water. The filaments are decumbent, lying without order in the gelatinous matrix in which they are developed, and which forms the FRESH-WATER ALGA OF THE UNITED STATES. 67 groundwork of the stratum. They appear to be unattached .to the soil, and each filament may be about half an inch in length; but they are commonly found broken off at the inferior end, or the lower part decays whilst the upper continues to grow. They are slightly curved, in serpent-like fashion, never quite straight; at first they are simple, but now and then emit lateral branches, which issue at considerable angles and generally in pairs. When a filament is about to branch, a rupture takes place in the side of the sheath, and the endochrome issues in two portions, one connected with the upper, the other with the lower half of the filament; these form the nuclei or medullary portion of two new branches and become duly invested with a membranous sheath, and gradually put on the aspect of the adult filament. The endo- chrome is granular, dark-brown, and annulated at short intervals, the transverse rings being placed very close together in the youngest portions, and less closely in the older, where they are distant from each other about twice the diameter of the column. This annulated endo- chrome is interrupted at certain fixed places, where an ellipsoidal cell is formed, separating the endochrome of the lower from that of the upper portions.” Harvery. Remarks.—I have never seen either the genus or species, and therefore am forced to copy the descriptions of both from Rabenhorst and Harvey. Famy SIROSIPHON ACE. Thallus ramosus, e cellulis pachydermaticis ant uni vel pluri seratis et in vagina ampla inclusis formatus, interdum cellulis perdurantibus instructus. Ramificatio vera fit cellularum vegetativarum quarundam divisione in axis longitudinalis directionem, qua ex re cellule dn sororie gignuntur; cellula inferior in trichomatis continuitate permanet, superior divisione continua repetita in eandem directionem se ad ramum explicat. Propagatio adhue ignota. Frond branched, formed of thick-walled cells in an ample sheath, sometimes furnished with hete- rocysts. Cells uni- or multi-seriate. Branches formed by a longitudinal division of certain cells, so as to form two sister cells; the inferior of which remains as part of the trichoma, whilst the other, by repeated divisions, grows into a branch. Propagation not known. Remarks.—The Sirosiphonacee are the most complex in their organization of all the Phycochromophycew, in so far as the protoplasm within the sheaths is every- where broken up into a number of distinct cells, each of which is provided with a thick coat or wall as well as in the circumstance of the frond having more perfect branching. The so-called pseudo-branches in the other families are more truly comparable to distinct fronds or thalli remaining attached to the parent thallus than to distinct branches, whilst among the sirosiphons the branches really belong to the original thallus. The heterocysts are much more frequently absent than present, only one of the known American species being furnished with them. The sheaths are generally not so distinctly sheaths as among the oscillatoria, &c., for, instead of being distinct tubes, they appear rather in most cases as masses of firm jelly, the outer portion of which is hardened almost into a periderm, and in the inner part of which the cells are imbedded. Their color varies from the transparent colorlessness of glass to a dark opaque-brown. Their surface is per- haps most frequently smooth, but at times is tuberculate or otherwise roughened. I have never seen anything like spores about them. 63 FRESH-WATER ALG& OF THE UNITED STATES. These plants grow in the majority of cases in the air, in such situation as on the face of dripping rocks, on the trunks and branches of trees, on moist ground, &c.; but some of the species are found in the water, either attached or floating. They generally form little mats of indefinite extent, but occasionally the filaments are united more closely into an almost membranaceous stratum. The species are, I think, in most instances readily distinguished, the characters being partly discoverable with the unaided eye and partly microscopic. The points to be attended to in the first category are the size, color, form, and consistency of the mats of fronds, and the place of growth. In the second are included the general shape of the frond and its size and method of branching; the general shape, color, and size of the cells, the thickness of their walls and the method of their arrangement, both in the main thallus and the branches, also the form, &c., of the end cells of the branches; the heterocysts, their absence, or, if present, their frequency, size, shape, color, and position; the sheaths, their color and firmness, and the character of their surface. Genus SIROSIPHON, Krz. Trichomata torulosa, vaginata, plerumque ramossissima et aureo- vel olivaceo-fusca, e cellulis pachydermaticis 1-2-3 vel pluri-seriatis formata et cellulis interstitialibus (seepe nullis) subglobosis vel oblongis coloratis instructa. Vagina plerumque crassissima, firma, pulchre aureo-fusca, lutea vel olivacea, in apicem obtusum plus minus attenuata. Filament torulose, sheathed, mostly very much branched, yellowish, or olivaceous-fuscous, formed of thick-walled 1-2-3 or many seriate cells and furnished with interstitial cells (often wanting) which are globose or oblong and colored. Sheaths mostly very thick, firm, beautiful golden fus- cous, clay-colored or olivaceous, more or less attenuate at the obtuse apex. a. Cellula in trichomatibus plerumque in serie simplice vel duplici ordinata. a. Cells mostly arranged in a simple or double series in the filament. S. scytenematoides, Woop. S. strato submembranaceo, nigro-viride, sepe interrupto, cum superficie inequale; trichoma- tibus spe arcte intricatis, flexuosis aut varie curvatis, haud rigidis, plerumque vix ramosis; cellulis uniseriatis, interdum interruptis, arctis, irregulare quadrangulis, diametro subequa- libus aut 1-3 plo brevioribus, haud distinete granulatis, eeruleo-viridibus; vaginis amplis, haud distincte lamellosis, superficie enormiter corrugatis et hirtis, plerumque coloris experti- bus sed interdum dilute brunneis. Diam.—Sine vag. max. 755” = .00066”; cum vag. max. 719,” = .0013”. Syn.—S. scytenematoides, Woon, Prodromus, Proc. Amer. Philos. Soc., 1869, p. 134. Hab.—South Carolina. (Ravenel.) ¥ . . . . S. In a submembranaceous, blackish-green, frequently interrupted stratum, with an uneven surface ; filaments often closely intricate, flexuous or variously eurved, not rigid, mostly sparsely branched; cells uni-seriate, sometimes interrupted, close, irregularly quadrangular, about equal in length to their diameter, or about 1-3 times shorter, not distinctly granulate, bluish-green; sheaths ample, not distinctly lamellate, their surface rough and corrugate, transparent, mostly colorless, sometimes light-brown. Remarks.—This species was collected in South Carolina by Prof. Ravenel, who found it in the month of February growing on the limbs of Myrica eerifera, The FRESH-WATER ALGA OF THE UNITED STATES. 69 blackish-green layer, which it makes upon the bark is very peculiar, being almost membranaceous, and especially in the dried state, presenting a rough, somewhat warty surface. ‘The trichomata have the sheath more distinctly in the form of a hollow cylinder, or, in other words, more plainly a sheath, than any other species IT have seen of the genus; the cells are also without any apparent walls, and are placed very closely together, so that the whole filament looks very like a scyto- nema. Fig. 1, pl. 9, represents a portion of a frond magnified 260 diameters. S. pellucidulus, Woop. S. immersus; trichomatibus ramossissimis, solitariis vel subsolitariis; ramis plerumque unila- teralibus, ramulosis; ramulorum apicibus late rotundatis, haud attenuatis; cellulis in serie- bus simplicibus dispositis, in trichomatibus nonnihil rotundatis, in ramulis sepe angularibus, plerumque compressis, diametro xqualibus—4 plo brevioribus; terminalibus eylindricis et obscure articulatis; cellulis interstitialibus nullis; vaginis arctis, hyalinis, haud lamellosis ; cytioplasmate zrugineo vel brunneo, minute granulato. Diam.—Trich. cum vag. 78,5” = .00106”; sine vag. = .0008”. Syn.—S. pellucidulus, Woop, Prodromus, Proc. Amer. Philos. Soc., 1869, p. 133. Hab.—In stagnis, prope Hibernia, Florida, (M. W. Canby.) S. immersed ; filaments very much branched, solitary or subsolitary ; branches mostly unilateral, branched; apices of the branches not attenuate, broadly rounded ; cells disposed in a simple series, in the trichoma somewhat rounded, in the branches frequently angular, mostly com- pressed, from equal to 4 times shorter than the diameter ; terminal cell cylindrical, obscurely articulate ; interstitial cells none; sheath close, hyaline, not lamellate; cytioplasm erugin- ous or brown, minutely granulate. Remarks.—This species was collected by Mr. William Canby in a little marsh pool near Hibernia, Florida. ‘The branches are given off in abundance, mostly in a unilateral manner, are often very long, and about equal in diameter to the main filament, and_ give origin to numerous branchlets. The sheaths are very trans- parent and very close. I have never seen them in any way lamellate or fibrous, or of any color. The cells do not have very apparent walls. In the main fila- ment and branches they are globose, or, more commonly, very much compressed, but in the newer branches, and sometimes in the older, they are very angular. The few cells near the end of the branches are so shaped as to remind one of the phalanges of the fingers. The last cell is cylindrical and has a number of cells indicated in it. The color of the young cells varies from a deep bluish-green to a ferruginous-brown—that of the older from a light bluish-green to ferruginous- brown. Fig. 2a, and 2, pl. 8, represent portions of filaments of this species. S. compactus, (Ac.) Krz. S. strato expanso, tomentoso, fusco-nigro; trichomatibus elongatis ramulisque adscendentibus, apice interdum paullum attenuatis sed sepe clavatis, obtusis ; trichomatibus internis e cellu- larum serie simplici formatis, et pleruamque moniliformibus; cellulis diametro subequalibus vel brevioribus, subglobosis vel subquadratis, spe compressis ; cytioplasmate dilute cxruleo- viride, subtiliter granulatis ; cellulis apicalibus cylindricis et oscillarium modo, sepe indistincte, 70 FRESH-WATER ALG OF THE UNITED STATES. articulatis ; vaginis firmis, aureo- vel rubido-fuscis, in ramulis sepe subluteis, haud distinete lamellosis; cellulis perdurantibus plerumque modice numerosis, singulis, subglobosis, swepe valde compressis, dilute fuscentibus. Diam.—Plerumque reso’ 1280” = -0008"—.001"; max. 3; = -0013”; cell. perdurant. ron. = .00058”. Syn.—Scytonema compactum, Aaarnu. Syst. p. 38, N. 3. Harvey's Manual, p. 154. Hassalia compacta, Hassat, Fresh-water Algw, p. 232, t. Ixvill. f. 3. Sirosiphon compactus, (AG.) Krz. RaBennorst, Flora Algarum, Sect. II. p. 287. Hab.—In rupibus caleareis, New Jersey. (Austin.) Prope Salem, Mass. (Russel.) Stratum expanded, tomentose, fuscous-black; filaments and branches ascending, with their obtuse ends sometimes slightly attenuate but often clavate; internal filaments composed of a single series of cells, mostly moniliform; cells shorter than or nearly as long as broad, subglobose or subquadrate, often compressed; apical cell cylindrical and articulate somewhat like an oscillatoria; cytioplasm light bluish-green, finely granulate; sheath firm, reddish or yellowish-brown, yellowish in the branches and near the ends; heterocysts mostly rather numerous, single, subglobose, brownish. Remarks.—The specimens from which the above description was drawn up were received from Messrs. Austin and Russell, and have been considered as identical with the European S. compactus, although not in absolute agreement with the descriptions thereof. The most important of the differences are in the matter of size, the measurements given by Prof. Rabenhorst not equalling those attained to by the American plant. The differences, however, do not seem sufficient to separate the forms, and, in the absence of European specimens, the two have been considered one species. The sheaths in the older portions of the filaments are nearly opaque, but in the branches and younger portions they are quite translucent. The heterocysts some- times are truncate at one end. The internal cells are rarely arranged in a double series, such arrangement is, however, much more common in the specimens re- ceived from near Salem, than in those found in Northern New Jersey. Mr. Rus- sell’s specimens are labelled as growing on shaded and moist rocks in patches two or three inches wide. ) Fig. 3 a, pl. 8, represents the end of a filament of this magnified 150 diameters; 3b, a fragment magnified 250 diameters; 3c, a heterocyst magnified 860 dia- meters. S. Crameri, Brvca. S. cxspitibus, tomentosis, spatiose expansis, fusco-nigris; trichomatibus vage ramosis; ramis plerumque singulis, spe elongatis, spe clavatis; cellulis internis uniseriatis, diametro sub- zequalibus vel brevioribus, interdum subglobosis, seepe subquadratis, in wtate provecta sepe € pressione mutua valde compressis et transverse oblongis, aureo-fulvis vel in state juvene interdum erugineis; cellulis terminalibus in massam subeylindricam coalescentibus; cellulis perdurantibus nullis; vaginis aureo-fuscis in state provecta plus minus subopacis et distincte lamellosis, in wtate juvene plus minus pellucidis et seepe coloris expertibus. Diam —Trich. eum vag. max plerumque .002” ; interdum 00225”; ram. .0015”—.0025” ; trich. sine vag. .00083”. Syn.—S. Crameri, Briaa. RasBennorst, Flora Europ. Algar., Sect. II. p. 288. FRESH-WATER ALGA&A OF THE UNITED STATES. 71 Hab.—In rupibus irroratis inter muscis minutis. Mount Tahawus (vulgo Mount Marcy),! alt. 5000 feet. : Forming a blackish, widely expanded, tomentose turfy covering to rocks; filament with scat- tered branches; branches mostly single, often elongate and clavate; cells uniseriate, about equal, or shorter than long, sometimes subglobose, often subquadrate ; in advanced age often strongly compressed and transversely oblong from mutual pressure, yellowish, or sometimes, when young, greenish ; the apical cells coalescent into an irregularly cylindrical mass ; hete- rocysts wanting ; sheaths yellowish-brown ; at maturity more or less subopaque, and distinctly lamellate ; in youth more or less transparent, and sometimes colorless. Remarks.—Near the top of Mount Tahawus, in the Adirondack Mountains, there is, at ‘an altitude of about five thousand feet, a steep slope of bare rock, the bed of an old landslide, over portions of which water is continually drip- ping. In such places the plant under consideration flourishes, forming with some very minute mosses a blackish, turfy coating to the rock of many feet, or even yards, in extent. The specimens agree well with the descriptions of the European plant, which also grows at about the same altitude as the American. They have, however, one peculiarity not noted in description of the European form, namely, that oftentimes the sheath of a branch widens out until it is actually much larger than the main filament. The color of the cells in the European form is said to be eruginous; but I conceive this depends somewhat upon the age of the specimens and is scarcely of primary value. The only other difference worth noticing is that my measurements exceed somewhat those given of the European plant. Ido not think, however, there is any good ground for separating the forms as distinct species. The finding of an Alpine plant growing on a mountain half way across the world from its first discovered home, at practically the same altitude, is a matter worth noting as a fact in Botanical Geography. S. neglectus, Woop. S. immersus; trichomatibus subsolitariis, longis usque ad lineas quatuor, cylindricis, ramossis- simis ; ramulis singulis ; cytioplasmate interdum wrugineo, plerumque aureo-brunneo; cellulis uniseriatis rarissime biseriatis, subglobosis, interdum sejunctis sed plerumque arcte connectis et moniliformibus, modo confluentibus, haud distincte pachydermaticis ; cellulis terminalibus elongato-cylindricis, spe nonnihil oscilatorium modo articulatis; cellulis interstitialibus nullis ; vaginis interdum brunneis, plerumque coloris expertibus. Diam.—Trichom. cum vag. ¢},” =.0017”; sine vag. yyy” = .001”. Syn.—S. neglectus, Woop, Prodromus, Proc. Amer. Philos. Soc., 1869, p. 133. Hab.—In stagnis, New Jersey. S. immersed, subsolitary, attaining a length of 4 lines, cylindrical, very much branched ; branches single ; cytioplasm eruginous, mostly yellowish-brown ; cells uniseriate, very rarely biseriate, subglobose, sometimes separate but more frequently closely united and moniliform; terminal cell an elongated cylinder, often articulate somewhat like an oscillatoria; interstitial cells wanting ; sheaths transparent, sometimes brown, mostly colorless. 1 “ Tahawus,” cloud splitter. The Indian names of the American mountains ought to be retained, in spite of the fact that some vulgar land surveyor has defiled the Adirondacks with the names of politicians, through whose influence he hoped for patronage. 72 FRESH-WATER ALG OF THE UNITED STATES. Remarks.—This plant was found in a very stagnant pool, forming, with various other species of alge, a gelatinous, eruginous-brown stratum, through which the single plants were thickly scattered, without anywhere forming the major portion of the mass. ‘The plants themselves are large enough to be distinguished by the unaided eye. Under the microscope the sheaths are seen to be exceedingly trans- parent and colorless, except in the older part of the filament, where they are often dark brown and opaque; but even in such ease, the edges are translucent and lighter colored. The internal cells or globose masses rarely have distinct coats, and even when ~ such were apparent, as in the older portions of the plants, there appeared to be a communication between the cells. The original main stem is rather short, shorter often than numerous branches into which it breaks up. Very often the apices of the branches are colorless and entirely empty, consisting simply of sheath; often, however, they are occupied by a cylinder of protoplasm, which is sometimes arti- culated more or less distinctly like an oscillatoria, Fig. 4, pl. 8, represents a fragment of a filament with a small branch. S. lignicola, Woop. S. strato expanso, tomentoso, atro; trichomatibus ramossissimis, arcte intertextis; ramulis ab- breviatis vel elongatis, subrectis aut varie curvatis, apicibus obtuse rotundatis vel subacumi- natis ; trichomatum et ramulorum cellulis uni- vel biseriatis, rare in trichomatibus maturis mul- tiseriatis, plerumque pachydermaticis, dilute vel saturate xrugineis, enormibus, plerumque homogeneis ; cellulis terminalibus in trichomatibus immaturis elongatis, cylindricis, seepius nonnihil oscillatorium modo articulatis, granulosis; vaginis sat amplis, haud achrois, vel luteo-brunneis vel fuscentibus vel ferrugineis. Diam.—Trich. cum vag. max. ;355” = .00066”. Syn.—S. lignicola, Woop, Prodromus, Proce. Amer. Philos. Soe., 1869, p. 133. Hab,—South Carolina. (Ravenel.) Occurring in an expanded, tomentose, black stratum; filaments very much branched, closely interwoven, branches abbreviate or elongate, nearly straight or variously curved, their apices obtusely rounded or subacuminate; cells 1-2 seriate, mostly thick-walled, light or deep eruginous, irregular, mostly homogeneous; terminal cells elongate, cylindrical, frequently articulate somewhat like an oscillatoria, granulate; sheaths somewhat ample, not transparent, light bright, fuscous or ferruginous. Remarks.—I have seen dried specimens only of this plant, which were collected by Prof. H. W. Ravenel, in South Carolina, It is said to grow on old boards, and appears to be a very distinct species. There are frequently two or three very short, stubby branches arising together. The apices of the filaments and branches are in some cases filled with endochrome to the end, and are broadly rounded at the apex. In other cases the sheath of the filament extends a distance beyond the endochrome, and is finally rapidly diminished to a point. The cells within the filaments are of various shapes, sometimes globular, sometimes quad- rangular, more often irregular. The original specimens from which this descrip- tion was written were collected in April. I do not know whether they grew immersed, or merely on boards exposed to the weather. I have since received FRESH-WATER ALGA OF THE UNITED STATES. 73 specimens collected in the month of August, which grew on boards over which spring water was constantly running. ‘These specimens agree perfectly with the others, except that the filaments are larger and the elongated apical cell is wanting; differences which I believe to be due to the specimens collected in August being older than those first received. Fig. 2 a and 2d, pl. 9, were taken from the types, whilst fig. 2 c, pl. 9, from the August specimens. a. Cellule plerumque in serie duplict vel multiplici. a. Cells generally in double series, or multiple series. S. argillaceus, Woop, (sp. nov.) S. strato tenui, expanso, subnigro, submembranaceo ; trichomatibus brevibus, dense intricatis et sxepe nonnihil concretis, ramosis, irregularibus; pseudoramulis brevibus, varie curvatis, nonnihil rigidis, plerumque ascendentibus, apice nonnihil attenuatis; cellulis subglobosis, spe compressis, plerumque in serie simplici sed interdum in serie duplici, vel rare multiplici; cellulis apicalibus valde elongatis, cylindricis, scytoneme trichomatibus internis similibus; vaginis crassis, firmis, in trichomatibus maturis saturate rubido-brunneis, in ramulis seepe luteo- brunneis et in apice hyalinis et fere coloris expertibus; cellulis perdurantibus nullis. : Diam.—zeF55" = -000833". Hab.—In palude argillacea, South Carolina. (Ravenel.) Stratum thin, expanded, blackish, submembranaceous; filaments short, densely intricate, and frequently somewhat concreted, giving ovigin to numerous branches, irregular; branches short, variously curved, somewhat rigid, mostly ascending, apex somewhat attenuate; cells sub- globose, often compressed, mostly in simple series, sometimes in double, rarely even in multi- ple; apical cells elongate, cylindrical, resembling the inner filament of a scytonema; sheath thick, firm, in the mature filament deep reddish-brown, in the branches yellowish-brown, at the apices of the branches nearly colorless and transparent ; heterocysts absent. Remarks.—1 am indebted to Prof. Ravenel for this plant, which was found by him on a moist clay bank near Aiken, South Carolina, August, 1869. It forms a thin, somewhat membranous, dark stratum, the filaments of which are so closely united that it is almost impossible to tease them apart with needles. Neighboring filaments are often united at the edges so as to form distinct bundles, and even the branches are sometimes concreted, although, generally, as seen under the microscope, they project from the mass in all directions. The surface of the fila- ments is mostly rough and ragged with fibrille and membranous projections. In the older filaments the cells are often entirely absent. They are mostly single, but sometimes multiple in the filaments; in the branches they are often partially double. The ends of the older branches are often broken and empty, whilst those of the younger are rounded. ‘The color of the cells, as I have seen it, does not strikingly differ from that of the sheaths. Fig. 3 a, pl. 9, represents a portion of an old frond magnified 460 diameters, and fig. 3 6, the end of a younger branch. No. 79. Collection of Ravenel, Aug. 1869. S. guttula, Woop. S. in maculis subnigris, parvis, tenuibus, plerumque rotundatis, interdum enormibus, dispositus; trichomatibus arcte intertextis, ramossissimis, rigidis, inequalibus, subeylindricis, nonnihil 10 May, 1872. 74 FRESH-WATER ALG OF THE UNITED STATES. contortis; ramulis abbreviatis vel nonnihil elongatis, apice obtuse rotundatis; ramulorum et trichomatum cellulis tri-multiseriatis, plerumque pachydermaticis, ferrugineo-fuscis, enormiter globosis, homogeneis; cellulis apicalibus interdum breve eylindricis, haud articulatis; vaginis sat amplis, luteo-brunneis vel dilute ferrugineo-brunneis. Diam.—Max. trich. cum vag. 735” =.0013”. Syn.—S. guttula, Woon, Prodromus, Proc. Amer. Philos. Soc., 1869, p. 132. Hab.—South Carolina, in Taxodium distichum. (Prof. Ravenel.) Arranged in small, thin, black spots, which are generally round, but sometimes irregular : fila- ments closely interwoven, very much branched, rigid, unequal, subeylindrical, somewhat con- torted; branches abbreviate or somewhat elongate, apex obtusely rounded; cells of the trichoma and. branches 3 to many seriate, mostly with thick coats, ferruginous-fuscous, irregu- larly globose, homogeneous; apical cells sometimes shortly cylindrical, not articulate, sheaths ample, yellowish-brown. Remarks.—This species was found growing on the bark of Taxodium distichum, by Prof. H. W. Ravenel, in South Carolina, and by him given to Dr. Billings, U.S. A., to whom I am indebted for specimens. It forms on the bark minute roundish, blackish, dot-like spots of about a line in diameter, or sometimes, appa- rently, by the coalescence of two or more of these spots, larger irregular patches. The habit of the plant is a rigid one. ‘The main stem is often irregular in size, variously bent and rebent, and mostly gives off a number of branches, which fre- quently nearly equal the main filament in size, and like it are bent in various directions. They also frequently give origin to numerous short branches. In some instances, there is a distinct apical cell, which is cylindrical, but only two or three times longer than broad; in many cases, however, this cylinder being want- ing, the ordinary cells extend to the extreme apex. Fig. 4 a, pl. 8, represents a filament, and fig. 4 6, the end of a branch magnified 460 diameters. S. acervatus, Woop. 8. in guttulis minutissimis, suberustaceis, nigris, in strato subcontinuo sepe ageregatis; tricho- matibus parvis et brevibus, rigidis, admodum inqualibus, prostratis, tuberculis, arcte et dense ramossissimis, viridibus aut aureis aut brunneis ; ramulis brevibus, plerumque haud ramulosis, erectis aut ascendentibus, sepe abbreviatis et papilliformibus, obtusis, seepe lateraliter connatis ; cellularum serie in trichomatibus multiplici in ramulis plerumque simplici ; cellulis subglobosis vel subangularibus, viridibus, haud distincte granulosis, in ramulorum apice seepe breve eylin- dricis et interdum obsolete articulatis ; vaginis aureis, nonnibil hyalinis. r ic . 15". » " Diam.—Trich. max. 7355”; ram. 7e00 —Te50" Syn.—S. acervatus, Woop, Prodromus, Proc. Amer. Philos. Soc., 1869, p. 132. Hab.—South Carolina, in cortice (Ilex opaca). (Prof. H. W. Ravenel.) Arranged in drops, which are very minute, suberustaceous, black, and frequently aggregate into a subcontinuous stratum ; filaments small and short, prostrate, rigid, somewhat unequal, tuber- culate, densely and closely branched, green or golden or brown; branches short, for the most part not branched, erect or ascending, frequently abbreviate, and papilliform, obtuse; series of cell multiple in trichoma, mostly simple in the branches; cells subglobose or subangular, green, not distinetly granulate, in the apices of the branches frequently shortly cylindrical and sometimes obsoletely articulate ; sheaths: golden, somewhat hyaline. . FRESH-WATER ALGA OF THE UNITED STATES. 15 Remarks.—This species was found in winter by Prof. H. W. Ravenel in South Carolina, growing upon the bark of Llexv oj, aca, forming minute, firm, crustaceous, roundish dots or masses, much smaller than a mustard-seed, but in some cases so closely aggregated as almost to make a continuous stratum. When one of these dots is placed under the microscope, the branches are seen presenting their ends upon all sides, reminding one of some varieties of coral, and between these are blackish matters, which prevent the whole dot from being seen. These branches are frequently placed very close to one another, and cohere by their edges so as to make a sort of membrane or a solid mass. ‘The filaments themselves ure mostly obscured in the dense mass of branches which clothe them. ‘This species seems to be closely allied to S. coralloides, and I am not certain whether it is distinct or not. It is certainly very much smaller. S. pulvinatus, Brés. S. pulvinatus, humectatns, saturate olivaceo-niger, ad tres lineas crassus; trichomatibus crassis- simis, ramossissimis, fuscescentibus, enormiter curvatis; ramulis polymorphis pro etate cras- sitie magnitudineque variis, apice plerumque obtuse rotundatis ; trichomatum cellularum serie multiplici, ramulorum 2-4 plici; vaginis crassis, luteo-fuscis ad saturate-fuscis, vel pellucidis vel non pellucidis, interdum rugoso-tuberculis. Diam.—Trich. cum. vag. max. .0042”. Syn.—S. pulvinatus, (BrEB.) RABENHORST, Flora Europ. Algar., Sect. II. p. 290. Hab.—In rupibus prope Philadelphia. Wood. In moist, deep olive-black cushion-like masses of two or three lines thick ; filaments very thick, much branched, brownish, irregularly curved; branches polymorphous, varying in thickness and size, mostly with their apices obtuse; cells of the filament many seriate, of the branches two to four seriate; sheaths thick, yellowish-fuscous to deep fuscous, pellucid or opaque, sometimes rugose-tuberculate. Remarls.—I have received specimens of this species found by Mr. Austin in Northern New Jersey, growing on the exposed face of rocks. The size attained to exceeds that given by Mr. Rabenhorst for the European form. The color of the cytioplasm varies from an almost verdigris-green to fuscous. Besides these specimens, Dr. I. Gibbons Hunt has given me fresh ones of a Sirosiphon which he found growing on the face of dripping rocks along the Wissahickon Creek, near this city. These are much smaller in every way than their more northern brethren, and differ in other respects, I think, sufficiently for a distinct variety. ‘The filaments and branches are much flatter than in Mr. Austin’s specimens. I append a description. (Var. parvus.) S. trichomatibus in cespite saturate olivaceo-nigro arcte intertextis; trichomatibus crassissimis, enormiter ramosissimis, luteo-fuscescentibus, varie curvatis; ramulis polymorphis, apice plerumque obtuse rotundatis; trichomatum cellularum serie multiplici, ramulorum 1-4 plici; cytioplasmate granulato, plerumque saturate fuscescente, interdum lite viride ; vaginis crassis, dilute luteo fuscescentibus, interdum achrois. Diam.—Trichom. eum. vag. max. 33,” = .03” 76 FRESH-WATER ALG OF THE UNITED STATES: Filaments closely interwoven into a deep olive-black turfy mass, very thick, irregularly and fre- quently branched, yellowish-fuscous, variously curved ; branches polymorphous, their apices mostly obtusely rounded ; series of cell in filament multifold, in branches 14 fold ; eytio- plasm granulate, mostly deep brown, sometimes bright green; sheaths thick, light yellowish- brown, sometimes transparent. Remarks.—The fronds are very irregular in form and size, much branched, and so closely interwoven that they mostly cannot be separated without breaking. The branches are sometimes short and stumpy, sometimes they are very long. The color of the cells approaches somewhat to a chocolate, at times with a little red in it so as to give something of a mahogany tint. The walls of the cells are mostly very thick, but they are often lost in the general mass of the frond. In the branches, the cells are often so closely crowded as to almost obliterate their walls. In a few specimens I have found the cells to be of a bright green color, instead of that just mentioned. ‘The exact meaning of this I do not know; it would scarcely seem to indicate immaturity, for I have found it in the oldest portion of large fronds, whose other parts were of the normal color. Fig. 1, pl. 10 represents a filament of this variety magnified 160 diameters. I have received from Prof. Ravenel certain dried alge, labelled Stigonema Ravenelli, BERKELEY, which appear to me to belong to this genus. In what place Berkeley described them, if ever, I do not know, nor why he placed them in the genus Stigonema, ‘The following is a description of the species :— S. strato sub-nigro; trichomatibus arcte intertextis, ramossissimis, enormibus, varie curvatis ; ramulis brevibus et sublongis, varie curvatis, latis, apice nonnihil attenuatis et obtusis; tri- chomatum et ramulorum cellulis arctis, enormibus, in serie duo-multiplici enormiter dispositis ; cytio-plasmate Lomogeneo, lete viride; vaginis aureis, lucidis. Diam.—Max. trich. cum vag. 7145". When dried blackish; filaments closely interwoven, very'much branched, irregular, and variously curved ; branches short or largish, variously curved, broad, their apices somewhat attenuated and obtuse; cells of the filament and its branches very close, irregular, irregularly arranged in a twofold or multiple series; endochrome homogeneous, bright green; sheath yellow, semitranslucent. Remarks.—This plant was collected by Prof. Ravenel on the now famous Look- out Mountain. It is of a thick, bushy habit, and appears to form turf-like mats of a line or two in thickness and of a blackish color. The filaments throw off in all directions very numerous branches, some of which are short and stumpy, others quite long, and are themselves the parents of numerous secondary branches. The longer branches often rival the main filament in size, and like it vary continually, in being irregularly expanded and contracted. There is never a long, articulated cell, not even in the apices of the branches. The apices are often somewhat attenuated, and are always more or less obtuse. The cells are of a bright green color, are very irregular in form, and are often very irregularly arranged in rows of from two to five, both on the main filament and branches. ‘The base of the filament often gives origin to several small, cylindrical, root-like processes. FRESH-WATER ALG& OF THE UNPTED STATES. "I Fig. 4 a, pl. 9, represents a frond of this plant magnified 125 diameters; fig. 4 , a fragment magnified 460 diameters. Professor Bailey, in American Journal of Sciences, vol. iii, new series, states that he has found two species of the genus Stigonema, namely, St. atrovirens, AG. and St. mammillosum, AG.; the former growing on wet rocks at Indian Falls, Putnam County, New York; the latter at Round Pond, near West Point. I have no personal knowledge of the genus, but, according to authorities, it belongs to the lichens rather than the alge, apothecia having been detected in various species, Crass CHLOROPHY LLACEA.' Plantule aquatic vel acre, uni-, bi-, vel multicellulares, aut singulee aut consociate, familias formantes. Vegetatio terminalis vel non terminalis. Ramificatio aut nulla aut vera, sed cellularum non divisione, potius prolificatione. Cytioderma non siliceum, combustibile, seepius e stratis successivis compositum, substantiam gelatinosam plerumque liquidam exsudans. Cytioplasma chlorophyllosum, chlorophyll loco nonnunquam erythrino vel substantia oleosa coccinea, carnea aut rufescente coloratum, nucleo (centrali vel laterali) plerumque praeditum, granulis amylaceis rarissime carens. Multiplicatio fit cellularum divisione vegetativa. Fcecundatio ple- rumque sexualis. Propagatio fit aut oosporis vel zygosporis aut gonidiis tranquillis vel agilibus. Aquatic or aerial uni-, bi-, or multicellular plants occurring singly, or consociated in families. Vegetation terminal or not so. Branches either wanting, or if present, true branches, although formed rather by a process of proliferation than division of the cells. Cytioderm not siliceous, combustible, often composed of successive strata. Cytioplasm chlorophyllous, sometimes colored by an oily crimson, flesh- colored or yellowish-red substance, in the place of the chlorophyl, gene- rally furnished with a nucleus (either lateral or central), very rarely without starch granules. Growth occurring by the division of the cells. Fecundation generally sexual. Propagation taking place by oospores or zygospores, or by tranquil or motile gonidias. ‘ The description of this Class and Order is that of Prof. Rabenhorst. 78 FRESH-WATER ALG& OF THE UNITED STATES. Orver Coccophycer. Alege unicellulares. Cellule aut singule (plerumque perfecte segregate) aut plures in familias consociate, tegumentis involute vel nude, aut ramificatione aut vegetatione terminali destitute. Propagatio fit aut cellularum divisione aut zoogonidiis. Unicellular algew. Cells either single (mostly entirely segregate), or mostly consociated in fami- lies, walled or clothed with teguments, destitute of branches or terminal vegetation. Propagation by means of zoospores, or by the division of the cells. . Famity PALMELLACEA. Alge unicellulares sensu latiori. Cellule aut singule aut numerose, familias constituentes, in muco matricali plus minus firmo, stratum gelatinosum amorphum, sepius figuratum, tubulosum (Hormospora) varie divisum et perforatum (Tetraspora), quasi ramificatum (Hydrurus) formante nidulantes, vel nullo (Rhaphidium, Dactylococeus). Cytioderma plerumque tenue, sepius tegumento gelatinoso aut homogeneco aut lamelloso preditum. Cytioplasma homogeneum, «tate provecta ple- rumque distincte granulosum, viride, aut rubescens aut fuscescens, vesicula chlorophyllosa semper instructum (excepto Rhaphidio). Multiplicatio fit cellularum divisione vegetativa, propagatio gonidiis ex ultima cellularum gene- ratione transitoria cytioplasmatis divisione varia ortis. Gonidia tegumentis liberata, polo autico cillis vulgo binis plerumque instructa et alacriter cireumvagantia. (R.) Alge unicellularin a broad sense. Cells either single or numerous, constituting families, inibedded in a jelly to form a gelatinous stratum which is amorphous or shaped, as tubular (Hormospora), variously divided and perforate (Tetraspora), falsely branched (Hydrurus), or sometimes is wanting (Rhaphidium, Dactylococcus). Cytioderm mostly thin, often furnished with a gelatinous or homo- geneous or lamellate tegument. Cytioplasm homogeneous, mostly at maturity distinctly granular, green-reddish or fuscous, always furnished with a chlorophyllous vesicle (except Rhaphidium). Multiplication taking place by a vegetative division of the cells, propagation by transitory gonidia arising by various divisions of the protoplasm from the last vegetative generation. Gonidia with- out integument, mostly furnished with two cilia at the anterior end, and moving about actively. Genus PLEUROCOCCUS, Mencu. (RABENH.) Cellule globose vel e mutua pressione angulose, plerumque nucleo instructe, tum singule tum in familias consociate. Cytioderma firmum, seepe crassum, leve, hyalinum; cytioplasma homoge- neum viride vel oleosum rubrum. Multiplicatio cellularum vegetativarum divisione in directionem ad omnes dimensiones alternantem. Propagatio fit gonidiis intra sporangia ortis. Cells globose or angular from mutual pressure, mostly furnished with a nucleus, sometimes single, sometimes aggregated into families. Cytioderm firm, often thick, smooth, hyaline; cytioplasm homogeneous-green or oleaginous-red. Multiplication occurring by a vegetative division of the cells alternately in three directions. Propagation by means of gonidia, formed within sporangia. P. seriatus, Woop, (sp. nov.) > eres 86 FRESH-WATER ALG OF THE UNITED STATES. Propagatio fit gonidiis, que intra cellulam matricalem cytiogenesi libera oriuntur et duplicis indolis sunt; altera majora, que macrogonidia, altera minora que microgonidia dicuntur ; illa oblonga, polo antico plerumque rostelliformi-producta, pallidiora, ciliis vibratoriis preedita, polo postico truncato- rotundata, obscure viridia, individuum propagant ; hee forma similii, itidem mobilia, brevi postea in statum quiescentem transeunt, druique in sporas perdurantes (Hypnosporas, Braun) transmu- tantur. (R.) Unicellular alge, in the strictest sense of the word, chlorophyllous, without terminal growth or true branching, without a vegetative generation of cells. They live either single, segregate, or con- sociated into families. The cells of these families, either indefinitely increasing in number (then families in the true sense of the term), or of definite number (then forming a ccenobium). Propagation by means of gonidia arising within the mother-cell by free cell-formation ; gonidia of two kinds; the one larger, macrogonidia—the other smaller, microgonidia; the former oblong, mostly produced into a pale bicilate beak anteriorly, rounded and greenish at their hinder end, developing into the individual plant ; the microgonidia similar to these and also motile, but passing after a short time into a quiescent state, and at last into resting spores or hypnospores. Genus PROTOCOCCUS, Ag. 1824. Cellule spheroidex, segregate, cytiodermate tenui, hyalino, absque tegumentis, libere natantes vel extra aquam in stratum tenue pulvereum cumulate. Cytioplasma initio homogeneum, denique granulosum, viride vel rubellum. Spheroidal cells, segregate, cytioderm thin, hyaline, without integument, swimming free or col- lected out of water into a thin pulverulent stratum. Cytioplasm in the beginning homogeneous, finally granular, green, or reddish. Remarks.—I have introduced this genus as given by Professor Rabenhorst in his Flora Europa Algarum for the purpose of describing a little plant, upon which I have made some observations. As the notes were originally drawn up as a de- scription of a species, I leave them in that form. I believe it has never before been described. Protococcus, (sp. nov. ?) P. aquaticus ; cellulis globosis vel angulis, viridibus in stratum pulvereum cumulatis vel in fami- lias arcte conjunctis; cytiodermate plerumque distincto; sporis rotundatis, tegumentis duobus vel tribus protectis; tegumentis externis, crassibus; zoogonidiis ovalibus, vel subrotundatis, vel subellipticis, ciliis duobus instructis. Diam.—Max. spor. perdurant. 7755” = .00093"; microg. 7455” = -00053”. Aquatic ; cells green, globose or angular, accumulated in a green pulverulent stratum, often closely united into families; eytioderm mostly not distinct; resting spores round with two or three thick coats; zoospores oval or roundish, or somewhat elliptical, furnished with two cilia. ; Remarks.—I found this species growing in a spring near Hestonville, West Philadelphia, in the month of March, The large winter spores are round, with thick coats. Except in one instance, in which the color was a decided reddish- brown, all that I have seen have been green. How they are produced I do not know. The history of their development into the plant appears to be as follows: The first change is the rupture of their outer thick coat (fig. 4 5, pl. 7) from which the spore finally escapes still clothed with a coat of moderate thickness. The green contents next divide into a number of oval bodies (fig. 6 6, pl. 7) which FRESH-WATER ALG OF THE UNITED STATES. 87 grow, and, at the same time, separate from one another. Whilst these changes have been taking place the spore coat has been becoming gelatinous and enlarging, so that it continues to enclose its progeny. In this way a family of oval cells is formed (fig. 46, pl. 7). So far, I think, is positive. The next step I have never actually seen, but believe to be the escape of these oval bodies as zoospores (fig. 4c, pl. 7) which are of very various sizes and are elliptical, globose, or oval. They have a tolerably well-marked bright vacuole at their beak, and after swimming about actively for a time finally settle down, lose their cilia, and undergo division. They seem often to cluster together before thus becoming quiescent, so as to make little colonies (fig. 5, pl. 7). Genus CHLOROCOCCUM, Friss. Cellule spheroidex, aut singule, libere, vesicula chlorophyllosa et locello laterali pallidiori cavo ? instructz, limbo hyalino et tegumentis sepe amplissimis cincte, aut plures in stratum vel acervulos cumulate. Propagatio fit zoogonidiis cytioplasmatis divisione ortis, e cytiodermatis abavie (intellige tegu- mentum extremum) rupturis excedentibus Cells spheroidal, either single, free, furnished with a chlorophyllous vesicle and a paler lateral (hollow ?) spot, with a hyaline nimbus and surrounded by a wide coat; mostly accumulated together into strata or little heaps. Propagation by means of zoospores, which are formed by a division of cytioplasm and escape from their general tegument (the cytioderm of the original cell). Remarks.—But a few weeks after the commencement of my study of fresh- water alg, a friend, a young microscopist, asked me to look at his aquarium, as the water of it had become stagnant, opaque, and green. On examining a little of the water with the microscope it was found to be full of what I now know to have been either one of the forms already described under this genus, or else one undescribed, but still embraced within its limits, There were two sets of bodies, the one motile the other at rest. The motile forms (Fig, 5, pl. 3) were globular or pyriform, and generally contained a large, roundish, green, distinct mass. ‘They were of course provided with cilia, although at that time I was not able to demonstrate their presence. These bodies, even when moving, appeared to have a distinct wall. After a time they settled down and assumed the quiescent state. ‘The outer coat now rapidly enlarged so as to leave a considerable space between it and the green endochrome, which rapidly underwent division, forming two or more new cells which were still surrounded by the enlarged maternal coat. The num- ber of daughter-cells enclosed in the parent cell varied. A considerable quantity of the water was allowed to stand in a glass jar, exposed to the light. In a very few days all the motile forms had disappeared. ‘The contents of the vessel were allowed slowly to evaporate. The jar being tall and narrow it was some weeks now before this process was completed, before which consummation hamatococcus forms were abundantly developed. Instead of being green, and surrounded by a distant, almost sac-like wall, the cells had acquired a dark brownish-red color, were very opaque, and were protected by a thick wall, whose surface was quite rough. Unfortunately, I did not measure either the active gonidia or their progeny, the quiet cells, but I found the general 88 FRESH-WATER ALG OF THE UNITED STATES. diameter of these hematococcus cells to be one twelve-hundredth of an inch (.00083’). MM. Famnitzin and Boranetzky, in arecent paper (“Zur Entwickelungsgeschichte der Gonidien und Zoosporenbildung der Flechten,” Mem. de L’ Académie Impériale des Sciences de St. Petersbourg, 1868, Annals and Mag. Nat. History, Feb. 1869), state as the result of direct observation that this genus of alge, so called, is really a stage in the life history of the gonidia of lichens. ‘These gentlemen took thin slices of lichen thalli containing gonidia, and placed them upon pieces of fir and linden bark, which had been previously boiled to kill any plants that might be growing on them. These were then put ina glass jar inserted over a vessel con- taining water, in such way that they would be constantly exposed to a very damp atmosphere, and at the same time communication with the external air would be impossible. In another set of experiments, pieces of the lichens were allowed to lie for a long time in water, until the component filaments were decomposed into a gelatinous mass, in which the still green vigorous gonidia were imbedded. These pap-like (4reiige) masses were then washed with pure water and smeared upon pieces of linden bark. ‘The results obtained were identical in the two cases. ‘The gonidia were at first provided each with a distinct nucleus and a well-marked lateral vacuole, and resembled closely the first form of cystococcus. The next change was a division of their contents into a large number of roundish masses, with the disappearance both of the vacuole and of the central nucleus. ‘The cell- membranes were next ruptured, and the endochrome, protruding through the open- ing, formed a little ball sitting upon the parent cell. In doing this it doubled in size, so that the part without was as large as the part within, although the latter still filled the cell. The contents finally escaped, but were yet surrounded by a very thin membrane, which soon, however, ruptured, and freed the biciliated zoospores into which the endochrome had in the mean time resolved itself. These zoospores remained a long time in the motile state, but finally settled down, drop. ping their cilia, and became little round cells, which grew to three or four times their original size, Further development was not made out. Certain of the gonidia, belonging to a lichen of the genus Physcia, failed to produce zoospores, but their endochrome, divided so as to form a number of quiescent cells, which either ruptured very early the original cell-membrane and became free in the water, or else remained bound together by it into a family for a longer period. In these researches MM. Famnitzin and Boranetzky employed lichens of three genera, namely Physcia, Cladonia, and Evennia, and claim, as above stated, that their investigations prove that they developed the alge genus Cystococcus of Negeli (Chlorococeum, Fries), from the gonidia. Genus POLYEDRIUM, N acest, (1849.) Cellule singule, segregate, libere natantes, compress, 3—4—8 angulares, angulis plus minus pro- ducte, nonnunquam radiatim elongate, aut integr aut bifide, pleramque armatz, a latere oblongo- elliptic, utroque polo rotundate vel subtruncate. Cytioderma tenue, leve. Massa chlorophyl- lacea plerumque granulosa, per cellule lumen equaliter distributa, nonnunquam guttulis oleosis rubris 1-4 mixta. Propagatio adhuc ignota. (R.) Genus mihi ignoltum. FRESH-WATER ALG# OF THE UNITED STATES. 89 Cells single, segregate, swimming free, compressed, 3-4-8-angled, more or less produced as to their angles, sometimes radiately elongate, either entire or bifid; mostly armed, oblong-elliptical when viewed laterally, at each end rounded or subtruncate. Cytioderm thin, smooth; chlorophyl mostly granular, equally distributed through the cell, sometimes mixed with reddish oil-drops. Propagation unknown. Remarks.—This genus was described by Negeli in his “Gattungen Einzelliger Algen,” and, although I have never seen any specimen of it, it claims a place here, because one species has been found in this country by Prof. Bailey. P. enorme, (Razrs) Der Bary. P. tetraédiicum, angulis productis achrois profunde bilobis, nonnunquam repetito-bilobis, lobis mucronatis. (R.) Diam.—0.0011"”—0.0016". (R.) Syn.—P. enorme, (RALFs) DE Bary. Rasennorst, Flora Europ. Algarum, Sect. III. p. 62. Staurastrum enorme, RAuFs, British Desmidiex. Hab.—Florida. Bailey. “Frond irregular or quadrate, spinous; end view three or four-lobed; lobes broad, more or less emarginate or bifid, and terminated by spines, which are either simple or branched. Sometimes the front view differs but little from the end one, usually, however, there is a slight constriction or sinus at the junction of the segments, but I have never observed any difference in the endochrome at that part. The spines, which are almost confined to the angles, are irregular, some simple and some branched. The end view has three or four broad and very irregular lobes; these are spinous and more or less emarginate, and frequently one lobe is much broader and more spinous than the others. The spines on such lobe form two groups, separated by the notch; they vary much in size and are either simple and subulate, or else forked; sometimes the forked spines are again divided at the apex.”—Zalfs’ British Desmidiex, p. 141. Genus SCENEDESMUS, Meyen. Cellule polymorph, utroque polo equales vel inzequales, sepe in cornu spiniforme producte, in ztate perfecto 2-16 aut in seriem simplicem aut parenchymatice arcte conjuncte et ceenobium con- stituentes ; cytioplasmate initio homogeneo, postea granuloso, vesicula chlorophyllosa centrali vel sublaterali et seepe locello achroo laterali instructo. Propagatio fit cytioplasmatis divisione succedanea, unde gonidia oriuntur, que intra cellulam matricalem jam in cenobium planum sese conjungunt et membrane matricalis ruptura vel dissolu- tione prodeunt. Cells polymorphous, equal or unequal at the ends, often produced into a spine-like horn, in the perfect state 2-16 closely conjoined, either as a simple series or in a parenchyma-like manner so as to farm.a cenobium. Cytioplasm in the beginning homogeneous, afterwards granular, furnished with a central or sublateral chlorophyllous vesicle, and often with a lateral transparent spot. Propagation occurring as a succedaneum to the division in the cells, whence arise gonidia, which, already within the mother-cell, join themselves into a ceenobium, and are finally set free by the rupture and dissolution of the maternal cell-wall. Remarks.— According to Unger, in the genus Scenedesmus the cells never exist singly, but always in families. Two of the species here described as representatives of the genus certainly do not conform to this, for I have frequently seen them both separate and in ccenobia 12 May, 1972. 90 FRESH-WATER ALG& OF THE UNITED STATES. or families, The latter were exactly like those of the European forms, at least in one of the two species, and I do not therefore think it justifiable to indicate a new genus. Moreover, I have certainly seen single cells, belonging to a species which agrees precisely in its characters with a European form, save only in the occasional existence of these single cells. I have never studied the method of propagation, but it is said to occur by the division of the cytioplasm of a large cell into a minute caenobium composed of two or more cells, which remains for some time within the walls of the mother- cell, but is finally set free by the solution of the latter. The cells are mostly much longer than broad, cylindrical, elliptical, or oval, but in one species herein described they are habitually globular. a. Cellule inermes. a. Cells unarmed. S. obtusus, Meyen. # S. cellulis oblongis vel ovatis, utroque polo obtusis, 4-6-8 modo arcte modo laxe in seriem simplicem aut rectam aut duplicem obliquam conjunctis, diametro 3-5 plo longioribus. (R.) Diam.—Transy. max. 0.00023”—0.00028”. (R.) Syn.—S. obtusus, MeyeN. RaAseEnnorstT, Flora Europ., Algarum, Sect. III. p. 63. Hab.—Georgia: Rhode Island, Bailey. Cells oblong or ovate, obtuse at each end, 4-6-8, partly closely partly laxly conjoined into a simple series either straight or oblique and double, 3-5 times longer than broad. Remark.—I have never met with this species. S. acutus, MEyYEn. 8. cellulis fusiformibus, vel ovato-fusiformibus vel ovatis, utrinque acutis sed inermibus, inter- dum singulis sed pleramque in seriem aut simplicem rectam aut duplicem inordinate alter- nantem dispositis, arcte concretis, diametro 2-4 (67) -plo longioribus, Diam.—Trans. vag. max. ? .00016”. Syn.—S. acutus, MEYEN. RaBenuorst, Flora Europ. Algarum, Sect. III. p. 64. Hab.—Prope Philadelphia, Wood. Rhode Island, Bailey. Cells fusiform, or ovate-fusiform or ovate, acute at each end but unarmed; sometimes single but mostly conjoined into a single straight series or into an irregularly alternate double series, 2-4 times longer than broad. Remarks.—This species is common around Philadelphia. Our specimens agree very well with the descriptions and figures of the European, excepting that occa- sionally a cell is single, and that none which I have measured have attained the size given by Prof. Rabenhorst as the maximum, namely, 0.00023”. According to Rabenhorst, 8. obliquus, Ktz., is only a variety of 8. acutus, Meyen. It has been found by Prof. Bailey in South Carolina, Georgia, and Rhode Island. b. Cellule armate. b. Cells armed. FRESH-WATER ALG OF THE UNITED STATES. 91 S. polymorphus, Woop. S. cellulis fusiformibus, aut ovalibus aut ellipticis aut globosis, singulis aut 2-8 conjunctis, plerumque utroque polo aculeo unico, interdum aculeis duobus, instructis: apicibus obtusis, acutis, vel acutissimis ; aculeis gracillimis, rectis, modice elongatis, inclinatis. Diam.—z2o0"'—73'50 3 plerumque 757". Syn.—S. polymorphus, Woop, Prodromus, Proc. Am, Philos. Soc., 1869, p. 135. Hab.—In aquis quietis prope Camden, New Jersey. S. cells fusiform, or oval, or elliptic, or globose, single or 2-7 conjoined, furnished in most cases with a single spine, sometimes 2, at each end; ends obtuse, acute, or very acute; spines exceedingly slender and acute, straight, moderately long, inclined. Remarks.—This plant was found in a quiet pool, filling the water in such num- bers as to make it opaque and very green. The color of the cells, as first obtained, under the microscope, was a vivid green, but, the water containing them having been placed in a dish, during the slow desiccation which followed the color of the cells changed to a golden yellow. Fig. 1, pl. 11, represents different forms of this species magnified 450 diameters. S. quadricauda, (Turpin) Brés. 8. cellulis oblongo-cylindricis, utroque polo obtuse rotundatis, 2-4-8 arctissime conjunctis, ordine aut simplici recto aut duplice alternante, omnibus rectis, medianis inermibus vel his illisve apice uno alterove aculeo curvato instructis, extimis utroque apice sepius item dorso armatis. Diam.—0.00035"—0.00039"; long. 0.00091”. Syn.—S. quadricauda, (Turrin) Brés. Rapennorst, Flora Europ. Algar., Sect. III. p. 65. Hab.—Rhode Island, Bailey. Pennsylvania, Wood. Cells oblong-cylindrical, obtusely rounded at each end, 2—4~8 very closely conjoined either in a single straight series or a double alternating one, all straight, the median unarmed or some of them with the apex furnished with a curved spine, the external with both apices and some- times the dorsum thus armed. Remark.—Fig. 2, pl. 11, represents this species magnified 750 diameters. S. rotundatus, Woop, (sp. nov.) 8. cellulis globosis vel subglobosis, spinulis longissimis, rectis, gracillimis, acutissimis, 3-6 armatis, aut singulis aut geminis aut 3-4 arcte duplice conjunctis. Diam.—zy5y tO sas": Hab.—In aquis quietis prope Philadelphia. (Dr. Chapman.) Cells globose or subglobose, armed with three to five very long, slender, acute, straight spines, single or in pairs, or three to four closely conjoined in a twofold rank. Remarks.—The cells of this species are globular, and, when more than two, they are arranged in two rows placed at right angles one to the other. The con- tents of the cells are markedly granular, and the endochrome a bluish-green, and from the surface of the walls project outwards, very long and fine, rigid hair-like spines. It seems scarcely correct to place this plant in the genus Scenesdesmus, but I do 92 FRESH-WATER ALGE OF THE UNITED STATES, not know any other genus to which it is more closely allied, and do not feel dis- posed to indicate a new one for it. Fig. 3, pl. 11, represents a cell-family magnified 250 diameters. Genus HYDRODICTYON, Rorn. (1800.) Cellule oblongo-cylindrice, in ceenobium reticulato-saccatum connexe, omnes fertiles; alie procreant macrogonidia, que jam intra cellulam matricalem in eenobium filiale se connectunt ; alize microgonidia, que multo minora, cellule matricalis membranam perrumpunt, polo antico ciliis vibra- toriis binis et puncto rubro laterali predita sunt, brevi postea in globulos protococcoideos tranquillos transformata sporas perdurantes efficiunt. Cells oblong-cylindrical, joined into a reticulated saccate ccenobium, all fertile; some producing macrogonidia, which join themselves into a cenobium within the parent cell; the others producing microgonidia, which are furnished with two vibratile cilia and a lateral red spot, and which, escaping from the parent cell, are, after a brief period of motile life, transformed into protococcoid thick-walled spores. Remarks.—The genus Hydrodictyon comprises, as far as known, but a single species, which is common to North America and Europe. It grows in great abundance in the neighborhood of Philadelphia, especially in the ditches and stagnant brick-ponds in the low grounds below the city known as the “ Neck.” There it very frequently forms floating masses several inches in thickness and many feet in extent, so that with the aid of a rake it could be gathered by the bushel. When thus in mass the color is very generally dingy and yellowish, although the fronds, when in active vegetative life, are mostly of a bright, beauti- ful green. ‘The plant is in greatest profusion in June and July, after which time it gradually disappears, until in the autumn it is scarcely to be found, but early in the spring it reappears. The very young fronds are minute, oval, cylindrical, filmy-looking, closed nets, with the meshes not appreciable to the eye; when growth takes place, the fronds enlarge until finally they form beautiful cylindrical nets two to six inches in length, with their meshes very distinct and their ends closed. In the bright sunlight they, of course, by virtue of the life-functions of their chlo- rophyl, liberate oxygen, which being set free in the interior of the net, and its exit barred by the fine meshes, collects as a bubble in one end of the cylinder and buoys it up, so that, the heavier end sinking, the net is suspended, as it were, ver- tically in the water. I know of few things of the kind more beautiful than a jar of limpid water with masses of these little nets hanging from the surface like cur- tains of sheen in the bright sunlight. A few cells collected in the fall or early spring, if put into a preserving-jar and the water occasionally changed, will multi- ply, and in a little while become a source of frequent pleasure to the watcher. As the fronds increase in size they are always in some way or other broken up, so that, instead of being closed cylinders, they appear as simple open networks of less or greater extent. ‘The extreme length to which the frond attains is, I think, very rarely over twelve inches, with meshes of about a third of an inch in length. The construction of the frond is always the same. It is composed of cylindrical cells united end to end in such a way as to form polygonal, and mostly pentagonal FRESH-WATER ALG OF THE UNLTED STATES. 938 meshes, the size of which varies with the age of the plant. These cells, which are closely conjoined but have no passage-ways between them, are capable of independent life, so that the hydrodictyon may be looked upon as an elaborate type of a cell-family, one in which cells are conjoined in accordance with a defi- nite plan, so as to make a body of definite shape and size, yet in which each cell is an independent being, drawing nothing from its neighbors. The cells them- selves are cylindrical, with a thickish cellulose wall, and have no nuclei. ‘Their chlorophyllous protoplasm is granular, and is placed in the exterior portion of the cell, forming thus, within the outer wall, a hollow cylinder, in which are imbedded starch granules, and whose interior is occupied with watery contents. The hydro- dictyon cell, when once formed, is capable of growth, but not of going through the usual process of cell multiplication by division, so that the adult frond is com- posed of just as many and indeed the same cells, as it had in its earliest infancy. No true sexual reproduction has as yet been discovered in the water-nets. There have been described, however, two forms or methods in which the species multi- plies, both of them occurring by means of motile zoosporoid bodies. In the one case these develop immediately into the new plant, whilst in the other before doing so they pass through a resting stage. Of the life-history of the latter, the microyonidia, | have no personal knowledge. The investigation of the production and development of the macrogonidia, how- ever, has occupied considerable of the time devoted by myself to the microscope, and I have seen large numbers of specimens in almost all the stages of develop- ment. I have never been able to detect, however, any decided motion in the macrogonidia, They are formed in the protoplasmic stratum, already alluded to as occupying the outer portion of the interior of the hydrodictyon cell. ‘The first alteration in this, presaging their formation, is a disappearance of the starch granules, and a loss of the beautiful, transparent green color. Shortly after this, even before all traces of the starch-grain are gone, there appear in the protoplasm numerous bright spots placed at regular intervals; these are the centres of development around which the new bodies are to form. As the process goes on, the chlorophyl granules draw more and more closely around these points, and at the same time the mass becomes more and more opaque, dull, and yellowish-brown in color. This condensation continues until at last the little masses are resolved into dark hexa- gonal or polygonal plates, distinctly separated by light, sharply defined lines. In some, the original bright central spot is still perceptible, but in others it is entirely obscured by the dark crowded chlorophyl. The separation of these plates now becomes more and more positive, and they begin to become convex, then lenticular, and are at last converted into free, oval, or globular bodies. When these are fully formed, they are said to exhibit a peculiar trembling motion, mutually crowding and pushing one another, compared by M. Braun to the restless, uneasy movement seen in a dense crowd of people in which no one is able to leave his place. Whilst the process just described has been going on, the outer cellulose wall of the hydro- dictyon cell has been undergoing changes, becoming thicker and softer and more 94 FRESH-WATER ALG& OF THE UNITED STATES, . and more capable of solution, and by the time the gonidia are formed it is enlarged and cracked, so that room is afforded them to separate a little distance from one another within the parent cell. Now the movements are said to become more active—a trembling jerking which has been compared to the ebullition of boiling water, There is, however, with this a very slight change of space, and in a very short time the gonidia arrange themselves so as to form a little net within the parent cell, a miniature in all important particulars of the adult hydrodictyon, ‘The primary cell-wall now becomes more and more gelatinous, and soon undergoes complete solution, so that the new frond is set free in its native element. As pre- viously stated, in my investigations I have never seen the peculiar motion above described, the newly formed gonidia simply separating and arranging themselves without my being able to perceive any motion, or exactly how they fell into posi- tion. , It is evident that when the species is multiplied in the way just described, the birth of the new frond is consentaneous with the death of the old cell. But when the hydrodictyon disappear in the fall, it is months before they reappear in the spring. It is, therefore, evident there must be some other method of reproduction. This slow development of new fronds takes place, according to Pringsheim, by means of little motile bodies which he calls “* Dawerschwirmer,” which has been translated into English chronispores (statospores, Hicks). M. Braun stated already some years since that sometimes, instead of the hydrodictyon producing the ordi- nary reproductive bodies (macrogonidia), there are formed in the cells much smaller and more active bodies, the microgonidia. ‘The changes which occur in the pro- duction of these are very similar to those already described as happening when the macrogonidia are formed. When the chronispores are once formed, however, they, instead of uniting together escape in a free distinct condition into the water. They are now small ovate bodies, with a large anterior transparent space, to which are attached a pair of cilia, and their life and history, according to Pringsheim, is as follows: For a few hours they move about very actively in the water, and then, dropping their cilia, and acquiring an outer cellulase wall, pass into a quiescent stage, in which they closely resemble protococcus granules. ‘They are capable of living in this state for a long time, if kept in water. They can also endure desic- cation if the light be excluded during the process, but, if it be present, they wither and die, and cannot be revivified. After a longer or shorter period, but never shorter than three months, according to Pringsheim, they recommence their life, provided they be in water. For four or five months after this the chief change consists simply in an increase in size. The dark-green protoplasm is arranged around the exterior of the cell, within are the more fluid colorless contents, the whole body still looking like a protococcus cell. After a size of about qo Mm. is attained, the endochrome divides succes- sively into several portions. The external layers of the surrounding wall now give way in some spot and allow the inner layers to protrude and form a sort of hernial sac, into which the several endochrome masses soon pass, at the same time assuming the well-known characters of true zoospores. From two to five of these FRESH-WATER ALGZ OF THE UNITED STATES. 95 bodies are thus produced out of each original microgonidium. They are large, ovate, biciliate, and, generally, soon escaping from the hernial sac, move about actively in the water for a few minutes. Sometimes, however, they settle down within the generative utricle. In either case, after a little time, they become motionless, lose their cilia, and develop into polyhedral cells, which are structurally remarkable for having their angles prolonged into long horn-like appendages. Under favorable circumstances, at the end of a few days, the bright green endochrome of these undergoes similar changes to those described as presaging the production of the microgonidia, and is finally formed into zoospores, which, in from twenty to forty minutes, unite, within the polyhedron or large cell, into a Hydrodictyon, which is finally set free by a solution of the cellulose coat of the polyhedron. ‘The network thus formed differs in no essential way from that which arises in the better known way, except that it is composed of much fewer cells. It is generally a closed sac; but when the polyhedron, out of which it is developed, is small, it is some- times merely an open network. Its after-history appears to be identical with that of the ordinary hydrodictyon frond. Hi. utriculatum, Bors. Species unica. Syn.—H. utriculatum, Rotu. Rasenuorst, Flora Europ. Algarum, Sect. III. p. 66. Hab.—In aquis quietis. West Point, Bailey. Weehawken, (Mr. Walters.) ‘Waterholes between Van Horn’s Mills and Mueote on the Mexican boundary, Dr. Bigelow. Pennsyl- vania, Wood. Genus PEDIASTRUM, Meysy, (1829.) Cenobium planum, disciforme, libere natans, e celluarum strato unico, rarius centro entro duplicato, continuo vel perforato formatum. Cellule polygoniz, peripherice spe bilobe, lobis cuneatis et simplicibus et bidentatis, nonnunquam in cornua productis. Cenobium plain, discoid, swimming free, formed of cells in a single, rarely in the centre double stratum, which is continuous or perforate ; cells polygonal, the peripheral often bilobed, the lobes cuneate, either simple or bidentate, sometimes produced into a horn. Remarks.—The ccenobium or cell-family, or colony, in the genus Pediastrum is always discoid, and has generally a more or less truly circular outline. ‘The cells are mostly in a single stratum, but in some species there are two, more or less, complete strata superimposed one upon the other. The arrangement of the cells in this stratum, or these strata, varies greatly, as does also their number. They are mostly more or less polyhedral, and often have their margins scooped out or their angles prolonged. This may occur in such a way that the projecting point of one cell fits into the hollow in its neighbor, and the ccenobium be rendered entire, or, no such relation existing between the parts of adjacent cells, the cceno- bium may be perforated with regular or irregular openings. The outer or mar- ginal cells are often deeply notched externally, and frequently are prolonged into acute or obtuse lobe-like processes. The walls of the cells are, in adult specimens, quite thick. The contents consist of chlorophyl, protoplasm, starch granules, &e. 96 FRESH-WATER ALG& OF THE UNITED STATES. There are generally one or more hyaline spaces, besides a distinct chlorophyl vesicle, but no distinct nuclei. ; At certain periods of their existence the Pediastrums produce both macrogonidia and microgonidia. The life-history of the former is very similar to that of the same bodies in the water-nets. The ultimate fate of the microgonidia has not as yet been determined, but in all probability they go through cycles of change similar to those seen in the lives of the corresponding bodies in the Hydrodictyon utriculatum. I have not had an opportunity of watching the development of either of these reproductive forms, but, according to MM. Braun, Pringsheim, &c., their life-history, as far as known, is as follows: In most cases, all the cells of a pediastrum produce their macrogonidia simultaneously, or within a very short period of time, so that the ccenobium will be left emptied of its contents as a mere shell, the outer skeleton of its former self. When a cell is about to give birth to these reproductive bodies, the endochrome divides into two parts; each of which then undergoes a similar binary division. This is repeated once, twice, thrice, or oftener, until the endochrome is divided into 8-16-32-64 gonidial masses, the number of which, generally, but not always, corresponds to the number of cells in the colony, to which the parent-cell belongs. After the division of the endochrome is completed, a slit occurs in the outer strata of the wall of the mother-cell through which a hernial protrusion of the inmost stratum occurs. ‘The protruded part now rapidly enlarges until at last there is formed a sort of hourglass-shaped sac, one portion of which is within, the other part without, the old parent-cell. Whilst this has been going on a portion of the gonidia have escaped from the parent-cell into the outer free portion of the sac, and each end of the hourglass, therefore, contains some of them. ‘The sac with its contents now gradually withdraws itself more and more from the parent-cell until at last it lies a free globose vesicle in the water. ‘The gonidia occupy the centre, and M. Braun states, that, although he has never been able to demonstrate any cilia upon them, yet they have an active swarming motion. At first, they are irregularly heaped toge- ther in the nearly filled sac; but the latter rapidly enlarges and elongates, and the gonidia in a little while arrange themselves in a flat, tabular group within it, and cease to move. ‘Then the several individuals of this group begin to develop, becoming emarginate and assuming the form of the parent-cell, until, finally, they have all grown into the shape which is peculiar to the adult cells of the species, and after a few hours have closely cohered to form a young ccenobium. The microgonidia are formed in a very similar way by the dividing of the endo- chrome, the cracking of the outer membrane, and the protrusion and final escape of the inner. They are, however, much smaller and more numerous than the macrogonidia. When the parent vesicle first escapes into the water, they are crowded in its centre, and are nearly globose. As it enlarges, however, they elon- gate more and more, and finally become distinctly bi- or, more rarely, uni-ciliate. The cilia are much longer than the body, and are attached to the smaller end, which is prolonged into a pointed, transparent beak, about equal to the green por- tion in length. The microgonidia now become more and more restless, they, moving about very actively, and after awhile bursting the parent sac, escape into the water. FRESH-WATER ALGA OF THE UNITED STATES 97 What becomes of them after this, as has been stated, is a mere matter of conjec- ture. M. Braun' and others have described unicellular forms of several of the multicellular species of Pediastrum, and Pringsheim suggests that these are really polyhedrons developed out of these microgonidia, as is seen in the water-nets. This, of course, may or may not be the case. P. Boryanum, (Turpin) MENGH. P. ceenobio orbiculari, oblongo vel elliptico, magnitudine vario, continuo, lete viridi, e cellulis 4-8-16-32-64 (rarissime 128) composito (cellularum strato simplici, nonnunquam medio duplicato); cellulis periphericis plus minus profunde emarginatis vel bilobis, lobis cornutis, cornibus achrois hyalinis, abbreviatis vel elongatis, teretibus, obtusis vel subobtusis, interdum capitellato-incrassatis, centralibus arctissime concretis, polygonis (4-6 angularibus), in antica parte modo angulo prominulo modo plane truncatis, modo leviter repandis, omnium mem- brana decussatim punctata. (R.) Diam.—Transyv. cell 0.000795”; rarius 0.00088”—0.00094”. (R.) Syn.—P. Boryanum, (Turrrn,) Mencuint. Rapenunorst, Flora Europ. Algarum, Sect. III. p. 74. Hab.—Georgia, Florida, Rhode Island, Bailey; Pennsylvania, Wood. Cells arranged in one or more circles round one or two central cells ; the inner variable, generally concave at one side, the outer tapering into two long subulate points, the notch narrow. L. 1=2083” to 1-1633"; B. 1-27.33” to 1-2222”. (Archer.) P. Selenza, Krz. P. cenobio orbiculari, integro, e cellulis 8-16 (rarius 81 =1+5 +10+415, Krz.) formato ; cellulis periphericis‘angustis, lunatis, acute lobatis, disci cellulis leviter excisis, centrali unica 5-angulari, omnium membrana firma, subcrassa, etate provecta rubescente. (R.) Diam.—Cenobii 0.00124”—0.0035"; cell. (distantiz interlobos) 0.00026”—0.00069”. (R.) Syn.—P. Selena, Ktz. Rasennorst, Flora Europ. Algarum, Sect. III. p. 73. Hab.—Rhode Island, Bailey. Cells crescent-shaped, arranged in one or more circles round one or two central ones, connecting medium colored. (A.)? P. pertusum, K7z. P. cenobio orbiculari, lacunis pertuso, magnitudine vario, e cellulis plerumque 1+ 5+ 10415 (in formis quibusdam ad 64) composito; cellulis periphericis basi tantum laxe connexis, ad medium usque bilobis, lobis rectis, in cornua hyalina modo subacuta modo obtusa vel trun- cata plus minus productis, centralibus plus minus exacte quadrangularibus, et in antica parte et utrinque emaginatis, omnibus levibus, locellis pallioribus finis instructis. (R.) Diam.—Transyv. cell. perfecte evolut. circiter 0.00065”—0.00089". (R.) Syn.—P. pertusum, Krz. Rapennorst, Flora Europ. Algarum, Sect. III. p. 75. Cells arranged in circles round one or two central ones; inner cells quadrangular, sides concave and leaving angular vacant intervals; the outer cells with square bases, externally triangu- larly notched, the subdivisions tapering to an acute point. lL. 1-2266”; B. 1-3268”. (A.) P. constrictum, Hassat. P. cenobio orbiculari vel suborbiculari, lete viridi, continuo, levi?, e cellulis 16 (ad 1+5 + 10) vel 32 (ad 1 +6-+10-+15) formato; cellulis periphericis irregulariter bilobis, sinu 1 The best exposition of this genus is to be found in Braun’s Unicellular Algie. 2 The letter A used here signifies that the description is copied from Mr. Archer in Prichard’s Infu- soria. 13 June, 1872. 98 FRESH-WATER ALG OF THE UNITED STATES. angusto, lobis inequalibus, basi plerumque constrictis, in cornua suberassa obtusa productis, centralibus polygonis, in antica parte repandis. (R.) Syn.—P. ellipticum, Hassan. Rasenuorst, Flora Europ. Algarum, Sect. I, p. 77. Hab.—South Carolina, Georgia, Rhode Island, Bailey. Cells varying in number and arrangement; outer cells suddenly contracted into two short, cylindrical, obtuse processes. L. 1-1754” to 1-906"; B. 1-1515” to 1-1020”. B, Processes of the lobes truncately emarginate. (A.) P. Ehrenbergii, (Corpa) Braun. P. cenobio et orbiculari et oblongo, perfecte clauso, e cellulis 8 vel 16 composito et quadrato, e cellulis 4, late cuneatis, profunde lobatis, exacte cruciatim dispositis formato ; cellulis peri- phericis cuneatis a basi truncata ad apicem usque concretis, profunde bilobis sinu angusto, lobis swpe oblique truncatis, plus minus sinuato-excisis, angulis interioribus ad duplum lon- gioribus, omnibus acutis vel breviter appendiculatis; cellulis centralibus aut singulis aut pluribus (2-5-6 y. 8), omnibus flavo-viridibus, polygonis, uno latere repandis vel profunde incisis. (R.) Syn.—P. Ehrenbergii, (Corpa,) BRAuN. RABEennORST, Flora Europ. Algarum, Sect. III. p. 77. Hab.—South Carolina, Georgia, Florida, Rhode Island, Bailey. Frond minute; cells eight (seven disposed in a single series round a central one), bilobed, angular. lL. 1-2900"; B. 1-2500”. (A.) P. simplex, Mryen. P. cellulis periphericis ovato-cuspidatis, 8-10-16 basi tantum concretis, circulum simplicem ~ constituentibus, centralibus sepe nullis. (R.) ‘ Syn.—P. simplex, Meyen. Rapennorst, Flora Europ. Algarum, Sect. IIL. p. 71. Monactinus octonarius, BAILEY, Smithsonian Contributions. Hab.—South Carolina, Rhode Island, Bailey. Var.—duodenarius. Cenobio clathrato, cellulis periphericis 12, centralibus 4, regulariter cruciatum dispositis. (R.) Syn.—Monactinus duodenarius, BATLEY. Inner cells four, somewhat triangular, enclosing a central, quadrate vacant interval, and four broadly lanceolate vacant intervals between them and the outer series, to which they are united by their terminal angles, outer cells twelve, subovate, truncate below, much attenuated, acuminate. (A.) Fammry VOLVOCINE. Cenobia mobilia, globosa, subglobosa vel quadrangulo-tabulata, e cellulis viridibus cilia bina agilia gerentibus, intus vesica duplici contractibili preditis composita, membrana (tegumento, chlamyde) communi achroa hyalina plus minus ampliata involuta. Propagatio aut sexualis, monoica vel dioica (adhue in paucis tantum generibus probata) ; cellulis cenobii aut omnibus aut quibusdam genus masculinum vel feminum exhibentibus, illis in fasciculos spermatozoideorum (autheridia), has in oosporas episporio inclusas, non mobiles commutatis, aut non sexualis, gonidiis agilibus, (et macrogonidiis et microgonidiis—etiam zoogonidia yocantur). Macro- et microgonidia (cellule primordiales) cytioplasmatis divisione simultanea et multiplici orta; priora numero definita (2-48-16, &c.), majora oblonga vel rotundata, polo antico plus minus rostri- formi producta, ciliis binis per vesice membranam exsertis, puncto (ocello Ehrberg. stigma) sanguineo centrali vel parietali et locellis (vacuolis) spe binis contractibilibus instructa; ultima numero indefi- nita, multo minora, pallide vel sordide viridia vel luteola, apice ciliis instructa, plerumque jam intra cellulam matricalem vivide vacillantia, postea membrane ruptura libere erumpentia, examinantia. (R.) FRESH-WATER ALGA® OF THE UNITED STATES. 99 Cenobium mobile, globose, subglobose or in square tables, composed of green cells which have two motile cilia and a double contractile vesicle. The common tegument surrounding the cenobium hyaline, and more or less amplified. Propagation either sexual or non-sexual. The sexual monecious or dixcious; either all or some of the cells of the ceenobium exhibiting male and female characters. The male cells containing spermatozoids, the female finally converted into a quiet oospore. Non-sexual propagation taking place by means of motile gonidia (both macrogonidia and microgonidia, by some called zoogonidia). Macro- and micro-gonidia arising by the simultaneous and repeated division of the eytioplasm; the first definite in number (2—4-8-16, &e ), the larger, oblong or rounded, with the anterior end more or less rostellate, with two cilia exserted through the membrane of the vesicle, furnished with a cen- tral or parietal red spot, and often with two contractile vacuoles; the microgonidia indefinite in number, much the smaller, pale or dirty green or luteolous, furnished at the apex with cilia, mostly even within the mother-cell, moving rapidly, and finally escaping on the rupture of the membrane. Genus CHLAMYTOCOCCUS, A. Braun. Cellule globose, vel subglobose (4-8 in ceenobium fugacissimum conjuncte), cytiodermate sub- crasso firmo, cytioplasmate granuloso, fusco-rubro vel puniceo (in evolutionis gradibus quibusdam in colorem viridem mutato). Macrogonidia 2—4—8, rotundata, polo antico rostriformi producta, duo cilia longissima gerentia, nucleo centrali rubro, globulis amylaceis 4-6, non semper visibilibus instructa, tegumento amplissimo hyalino plerumque ovoideo vestita. Microgonidia multo minora, numerosa, luteola vel sordide viridia, apice rubella, ciliis binis instructa, intra tegumentum matri- cali alacriter vacillantia, denique membrane ruptura elabentia. (R.) Cells globose, or subglobose (4-8 conjoined in a very fugitive ccenobium), eytioderm thickish, firm, cytioplasm granular, brownish-red or puniceus, in certain stages of evolution changed into green, Macrogonidia 2—4~8, rounded, the frond end bearing very long cilia, furnished with a central reddish nuclei and with four to six, not always perceptible, starch granules, clothed with a yery ample, hyaline, mostly ovoidal tegument. Microgonidia much the smaller, numerous, luteolous or sordid green, the apex reddish, furnished with two cilia, moving actively within the maternal ‘egument, and at last escaping by the rupture of the membrane. Ch. nivalis (Bauer, Ac.). A. Braun. Ch. globulis, 0.004”—0.00135”. (R.) Hab.—In nive externa, Greenland. Rocky Mountains. Syn.—Ch. nivalis (Baur, AG.). A. Braun. RaAsBenuorst, Flora Europ. Algarum, Sect. III. p. 97. Globules, 0.004”—0.00135” in diameter. Remarks.—I have never seen any good specimens of this plant, merely some cells mounted in Canada balsam, and therefore ruined for scientific study, which had been collected by Dr. Kane in one of his Arctic voyages. I have also had some indications of plants in a little parcel sent me by Mr. Sereno Watson, who informs me he has seen the red snow very abundant in the higher peaks of the Rocky Mountains. It is a matter of presumption rather than determination, there- fore, that the species is identical with the European. Genus VOLVOX, Enrs. Cenobium exacte sphericum, continuo rotatum et agitatum, globum cavum quasi fingens, e cel- lulis numerossissimis wquali distantia peripherice dispositis, gelatina matricali connexis, puncto rubro laterali, locellis (vacuolis) binis contractibilibus necnon ciliis binis longe exsertis instructis, vesica communi hyalina circumcinctis compositum. 100 FRESH-WATER ALG OF THE UNITED STATES. Propagatio duplex ist, aut non sexualis aut sexualis; illa fit cellulis quibusdam certa distantia intumescentibs, multipartitis, in ccenobia filialia intra cenobium matricale evolutis, postea libere erumpentibus; hee cellulis masculis multipartitis in fasciculos spermatozoideorum mobilium, con- tractilium, pyriformium, ciliis binis instructorum, postea liberorum evolutis; cellulis femineis intu- mescentibus, non divisis, sed post feecundationem in oosporas immobiles episporio duplici cireum- datas postremo rubras evolutis. (R.) Cenobium exactly spherical, continually rotating and agitated, looking like a hollow globe, composed of very numerous cells, which are arranged on the periphery at equal distances, and are connected by the maternal jelly, and surrounded by a common hyaline bladder; they are also fur- nished with a lateral red point, with two contractile vacuoles, as well as two long exserted cilia, The propagation is both sexual and non-sexual. In the latter, certain distant cells enlarge greatly, divide into numerous parts, and evolve within the parent coenobium daughter-ccenobia, which are finally set free. In the sexual propagation certain molecular cells undergo a multipartite division into fasciculi of spermatozoids, which are motile, contractile, pyriform, and furnished with two cilia; the feminine cells are enlarged, and do not undergo division, but after fecundation develop into immovable oospores, which are finally red, and are surrounded by a double episporium or coat. V. globator, (Liyy.) Enns. V. cenobiis majoribus ad 2’, cellulis numerossissimis (ad 12,000); eccenobiis filialibus semper octo intra matricale fructificatione non sexuali evolutis; fructificatione dioica; ccenobiis masculis fasciculos spermatozoideoram numerosos rubescentes foventibus (= Spherosira volvox, Ehrb.); cenobiis femineis cellulas sexuales (oogonia) 20—40 post foecundationem in totidem oosporas globosas rubras episporio hyalino stellato circumdatas foventibus (= Vol- vox stellatus, Ehrb.). (R.) Syn.—V. globator, (LINN&,) Furs. RaBeNnHorstT, Flora Europ. Algarum, Sect. III. p. 97. Hab.—In stagnis. United States. Larger cenobium, about 2” in diameter, composed of very numerous (about 12,000) daughter- cenobia, always 8 within the maternal one, evolved without sexuality; fructification dix- cious ; male ceenobium giving origin to numerous reddish spermatozoids (= Spherosphera Volvox, Ehrb.); female coenobium, giving origin to from 20-40 sexual cells, which, after fecundation, develop mto the same number of globose red oospores surrounded by a stellate hyaline episporium, Remarks.—Some of my friends tell me they have found this species abundantly around Philadelphia. I have not been so fortunate, and have seen but a few scattered specimens, which have afforded no opportunity of studying their deve- lopment and life-history. Orvrr Zvgophycex. Alge aut uni- aut pseudomulti-cellulares, sine vegetattone terminali et ramificatione vera, Cellule singule aut geminate aut seriatim conjuncte. Multiplicatio fit cellularum divisione in unam diree- tionem. Propagatio fit zygosporis conjugatione cellularum similium binarum ortis. Alge either uni- or pseudomulti-cellular, without terminal growth or true branches. Cells segre- gate or geminate, or arranged in a single row. Multiplication taking place by a division of the cells in one direction. Propagation by zygospores, formed by the conjugation of two similar cells. Famity DESMIDIACEA. Alge unicellulares, sine ramificatione vel vegetatione terminali. Cellule forma admodum varia, plerumque in medio plus minus profunde constricte et in duas semicellulas symmetricas Civiste, FRESH-WATER ALGH OF THE UNITED STATES. 101 liber vel in fascias filifermes aut teniiformes arcte conjuncte aut in muco matricali nidulantes et in familias indefinitas consociate, Cytioderma non siliceum, plus minus firmum, leve aut varie asperatum (striatum, costatum, aculeatum, &c.). Massa chlorophyllacea in laminales axiles vel parietales, seepe e centro radiantes, distributa. Propagatio non sexualis per divisione transversa in eandem directionem repetita; sexualis per zygosporas, que per cellularum binarum conjugationem oriuntur. Unicellular alge, without branches or terminal growth. Cells of very various forms, mostly more or less profoundly constricted in the middle and divided into two symmetrical semicells, free or con- joined in filiform or teniform fascia, or involved in the maternal jelly so as to form indefinite fami- lies. Cytioderm not siliceous, more or less firm, smooth, or variously roughened (striate, costate, aculeate, &c.) Chlorophyl masses in axillary or parietal lamina, which often radiate from the centre. Non-sexual propagation by repeated transverse division in one direction; sexual by zygospores which are formed by the conjugation of two cells. Remarks.—Of all the fresh-water alge, with the exception of the diatoms, this family has attracted most attention, owing, not only to the beauty and variety of its forms, but also to their universal presence and abundance, and the ease with which their most wonderful life-histories are observed. ‘They are exclusively, as far as known, denizens of fresh-water, and preferably that which is pure and limpid. Although Mr. Ralfs states that they never grow in stagnant water, I have often found them in great abundance in such, yet never in that which was actually putrid. The same authority is also too sweeping, at least as far as this country is concerned, in stating they are never found in woods, although they are really most abundant in the open country. My experience has taught me to look for them in brick- ponds, small mountain lakes, springy fens, ditches, and, in the fall, growing among mosses and in the thick jelly composed of unicellular alge on the face of drip- ping rocks, or, to sum up in a word, they dwell in quiet, shallow waters, for I have never found them in rapidly moving or very deep water. The single cell, of which a desmid is composed, is mostly divided into two very marked similar portions, the exact counterparts one of the other, which by some have been asserted to be distinct cells. ‘Their close union and connection, and their inherent oneness are, however, so apparent that it is needless here to spend time in demonstrating that they really are halves of one individual cell. ‘They contain together all the parts found in the typical vegetable cell; an outer cellulose wall, chlorophyllous protoplasm, a nucleus, starch granules and semiliquid contents. The cell-wall, or cytioderm, as it is called in this memoir, varies in thickness and firmness. During life it is mostly, if not always, colorless; but in certain species in the dead empty frond is of a reddish-yellow. The markings upon it are various, and are not infrequently altogether absent ; they are such as fine or coarse punctations, granulations of various size, strie, furrows or elevated ribs, tuber- cles, obtuse or sharp simple or forked spines, hair-like processes, umbonations, &e. &e. These markings are within narrow limits constant in each species, and more or less peculiar, so that they afford valuable characters to the systematist. The cytioderm itself is mostly composed of cellulose free from appreciable inor- ganic matters, but in certain species contains a large amount of silex. ‘Thus, 102 FRESH-WATER ALG& OF THE UNITED STATES. according to De Barry, if Closterinm lunula be carefully burnt upon a slide, a per- fect hyaline silex cast of the cells is left. ‘The chlorophyl is variously placed in the cell, sometimes it is arranged in lamina, sometimes in spirals, sometimes in the form of radii from a central mass, These different methods afford good generic characters, and will be dwelt upon more in detail under the various genera. The color of the chlorophyl during active life is a vivid green, which, as the vital forces lessen, changes to a faded yellowish tint. Niegeli and others affirm that there is always a central nucleus in the desmid, and probably do so with truth, although in many instances I have found it impos- sible to demonstrate its presence from the size and opaqueness of the frond, crowded with endochrome, &c. In a large number of cases, however, it is very apparent. > As ordinarily viewed under the microscope the two most striking peculiarities presented by these little plants are the motion of the whole desmid in the water and the various movements exhibited within the fronds. The general movement is most apparent in the larger species, which exist free and distinct in the water, especially in the boat-shape closteria. It mostly consists of a steady, stately, slow onward movement, with sometimes backward oscillations. By virtue of it, desmids in a bottle will often congregate in such positions as are most exposed to light. ‘There have been various theories advanced as to the cause of this motion. Ehrenberg believed that he had found foot-like processes protruding from the end of the frond and giving the motile power. Others, such as Rev. Mr, Osborne and Mr. Jabez Hogg, have attributed the movements to the presence of cilia, but I think have failed so entirely to establish this that their views are more than pro- blematical, That the motion is due to vital actions, taking place especially under the action of light, is as much as can be at present affirmed with any certainty, though it is probable that the immediate agents are endosmotic currents of gas or water. The movements of the contents within the cells are chiefly of two kinds. Tak- ing Closterium lunula as an illustrative example, there will be found on ex- amination with an {th objective, a narrow, very transparent, and therefore often not very apparent layer or zone lying immediately within the cell- wall, between it and the endochrome, and dipping inward in the middle of the frond so as to communicate with the nucleus. In this zone are protoplasm, watery fluid, and scattered granules. In the ends of the fronds the different por- tions of this zone, meeting and widening, fill up the whole of the cavity, and within the space thus occupied by them, there is a globular, sharply defined, still more transparent vacuole, ‘This, some have thought to be a closed sac, with a distinct wall, but it seems really to be a vacuole lying in the midst of the inner protoplasm, which with a few granules occupies more or less completely the transparent zone already described. Sometimes the chlorophyl encroaches upon this zone at the ends so as to more or less completely surround the vacuole, within which are always found watery fluid and granules. In the protoplasmic zone and its vacuole active movements are probably always present during active life. Streams of protoplasm. appear to be constantly passing to and fro between the nucleus and the ends of FRESH-WATER ALG# OF THE UNITED STATES. 103 e the cell along the outer zone, and granules can be always seen passing backwards and forwards with an unsteady motion. When the streams of protoplasm are setting very actively from the centre to- wards one end, there will often be an accumulation of the protoplasm there, and a consequent decided lessening in the size of the vacuole, which will again expand as the retum currents arouse themselves. Within the vacuoles are seen more or less numerous smaller or larger granules in active busy motion, swarming over and about one another with an unsteady hurrying to and fro. A form of motion, similar in appearance to this, but probably of different signi- ficance, is seen in most desmids when in an unhealthy feeble condition. I have seen it most marked in Cosmariwm margaritaceum. In such fronds the endo- chrome has lost its deep green color, and become shrunken, and lying within it is a great space containing myriads of minute blackish particles swarming about actively. ‘This peculiar state and appearance is by no means confined to the desmids, for I have seen it very highly developed both in species of Spirogyra and CEdogonium. It appears to be connected with decay. Is it possible that these minute particles are foreign to the plant, vibrionic in nature ? In regard to the nature of the movements seen within a healthy desmid, some have viewed them as exceedingly mysterious, the result of the presence of cilia, &e.; but these views have been so thoroughly exploded that it is scarcely necessary even to mention them here. The movements are, in truth, precisely parallel to the so-called cyclosis of the higher plants. Protoplasmic germinal mat- ter, wherever it exists, be it in animal or vegetable, has as one of its distinguish- ing characters the power of active, spontaneous, apparently causeless movements, and it is simply the carrying out of this power or attribute which has attracted so much attention in the desmids, because it is in them so readily seen. There are, in this family, two distinct methods in which the species are multiplied one with, the other without, the intervention of anything like sexuality. ‘The non-sexual method of increase is really a modification of an ordinary vegetative process, a peculiar cell multiplication by division. In such fronds as those of the genus Cosmarium, which are composed of two evident halves connected by a longer or shorter isthmus, the first step in the process is an elongation of this neck, In a very short time there appears around the centre of this a constriction, and I believe an actual rupture of the outer coat. By this time a new wall has formed inside each half of the isthmus, and stretches also across its cavity, forming with its fellow a double partition wall, separating the two halves of the old frond, Rapid growth of the newly formed parts now takes place, the central ends become more and more bulging as they enlarge, and in a little time two miniature lobules have shaped themselves at the position of the old isthmus. These are at first small, colorless, and destitute of all markings, looking, as Mr. Ralfs says, like con- densed gelatine. ‘They, however, rapidly increase in size and firmness, their con- tents assuming a green color and their walls taking on the peculiar markings of the species. At last, the parts thus formed having assumed the shape and appear- ance of the original lobules, the two fronds, which have been developed out of one, separate, mostly before the new semicells have acquired their full size. 104 FRESH-WATER ALG OF THE UNITED STATES. What part the nucleus has in the process just described I have never actually demonstrated, but have little doubt but that it undergoes a division in the very commencement, so that the new nucleus of each secondary frond is formed out of one-half of the old one. In proportion as the form of the desmid becomes simpler, so do the peculiarities of its cell multiplication become less. In those species which are simple cylindri- cal cells, there appears to be nothing peculiar in the method of dividing, which, however, always takes place through the centre of the cell, and subsequent growth occurs, generally, only in the newly formed part. True sexual reproduction apparently does not take place as freely in this family as the former process, for whilst I have seen hundreds of cells undergoing the latter, it has not been my good fortune to meet with conjugating specimens on more than two or three occasions. The process has, however, been studied very closely by De Bary, Braun, Hof- meister, and others, and appears to consist generally in a rupture of the outer wall of two cells and the protrusion of delicate processes from an inner, often newly formed coat, with subsequent union of these, and consequently of the two cells, and afterwards a condensation of the contents in the enlarged connecting passage. The connecting passage between the fronds is really a sporangium in which the spore is perfected, the contents of the cells finally condensing it into a firm globe and secreting around themselves a thick coat. The after-history of this spore has been very successfully studied by M. Hof- meister, Whose observations were made upon Cosmarium tetraophthalmum, which he watched conjugating and forming a sort of resting spore which was perfected early in the month of July. This was composed of a thick outer coat and green endochrome lying within as a distinct ball, nowhere in contact with the invest- ing membranes. In three weeks’ time this chlorophyllous protoplasm had divided into ellipsoidal masses, or primordial cells, which soon surrounded themselves with cellulose walls and became distinct free cells in the granular fluid which filled the cavity of the original spore. In August, each of these masses was divided into two and in the month of September the process was repeated, so that out of the original endochrome eight strongly flattened primordial cells were produced. Division in some specimens ceased here, and in others took place once more, so that by the fol- lowing spring all of the living Sporangia contained eight or sixteen green daughter- cells, each of them discoid in outline with a strongly marked central notch. These daughter-cells were finally set free by the solution of the spore wall, as Cos- maria of minute size, but agreeing in all other characters with the specific form to which they belonged. According to Braun, in the larger, more or less lunate Closteria, conjugation occurs in the following method: Two fronds approach one another in such a way that they lie back to back. In the middle of each of them, there then appears an annular line or trench reaching through the cell wall, and accompanied by a dis- tinct separation of the endochrome into two halves. Whilst these changes have been progressing there has also formed a new double wall at the position of the trench, so that out of the two Closteria two pairs of separate equal cells have been FRESH-WATER ALGA OF THE UNITED STATES. 105 formed. Near to the larger or central end of each of these now appears a pouting transparent nipple-like process. The corresponding opposing processes enlarging and meeting coalesce, so that the upper half of one closterium, in the form of a daughter-cell, is finally united with the upper half of the other closterium, and the two lower halves are also joined together. ‘Thus from a single pair of fronds arise two conjugating pairs of cells, and finally two sporangia, in each of which a spore is perfected. This process does not seem, however, to be universal amidst the Closteria, for in many, if not all, of the smaller species, a pair of fronds produces a single spo- rangium, In the genus Palmogloea, in which I have had an opportunity to study the devel- opment of the spores, the process closely simulates that seen in certain of the Spirogyra. ‘The contents of the cells first became broken up and confused, and almost simultaneously the nucleus disappeared (fig. 4, pl. 11) the cells became swollen at one side and slightly bent backward so as to form jutting processes, which meeting grew together, became confluent and developed into a sporangium much larger than either of the parent cells. Into this sporangium the contents of the latter passed and soon became converted into a thick-walled spore (fig. 00, pl. 00) often completely filling the cavity, and apparently with its wall adherent to that of the latter. Genus PALMOGLGE:A, Krz. (1843). Cellule oblonge, elliptice vel cylindrice, utroque polo rotundatex, medio non constricte, plerumque in muco gelatinoso nidulantes, libere, singule vel in familias consociate, lamina chlorophyllacea axili vel excentrica, wztate provecta medio constricta, denique divisa praedite. (R.) Syn.— Mesotzenium, N®GELI. Cell oblong, elliptical or cylindrical, rounded at each end, not constricted in the middle, mostly swimming in a gelatinous mucus, free, single or associated in families, chloropbyl lamina axillary or excentric, in the early state constricted, and at length divided in the middle Remarks.—The above diagnosis of the genus is that given by Prof. Rabenhorst, and agrees essentially with that of De Bary, Negeli, &c. In the species herein described however, the axillary lamina of chlorophyl were not so pronounced, for the green coloring matter seemed often to surround the cavity of the cell, and in other specimens was broken up and diffused through it. P. clepsydra, Woop. P. saxicola et bryophila, in gelatina achroa interdum dilute viride nidulans; cellulis cylindricis, cum polis obtuse truncato-rotundatis, diametro 2-3 plo longioribus; lamina chlorophyllacea axili, plerumque indistinete, sepe nulla; plasmate dilute viride; nucleo plerumque distincto ; zygosporis subfuscis aut subglobosis aut enormiter in clepsydre forma; membrana externa enormiter excavata et sulcata. Diam.—7h35". Syn.—P. clepsydra, Woop, Prodromus, Proce. Amer. Philosophical Soc. 1869, Hab.—In rupibus et in muscis irroratis ad Chelten Hills, prope Philadelphia. 14 June, 1872, 106 FRESH-WATER ALG& OF THE UNITED STATES. P. living on rocks and mosses, swimming in a transparent, sometimes light-green jelly; cells obtusely truncated, rounded at the ends, 2-3 times longer than broad; chlorophyl lamina axillary, mostly indistinct, often wanting ; endochrome light-green; nucleus generally distinet; zygospore subfuscous, either globose or of an irregular form, somewhat resembling that of an hour-glass ; external coat irregularly excavated and sulcate. Remarks.—This species was found along the North Pennsylvania Railroad, near Chelten Hills, growing amid mosses on the rocky juttings over which the water was dripping. It occurs as a rather firm, transparent jelly, mostly of a light greenish tint, in which the cells are often placed quite thickly. They are cylin- drical, mostly straight, but sometimes slightly curved, and often completely filled with a light greenish endochrome, ‘The central lamina is irregular, and mostly not at all pronounced. In some cells the endochrome is much broken up, so that the interior is filled with little green masses with light spaces between them. In these cells the nucleus is generally not perceptible, whilst in the others it is very well marked. ‘The zygospore is often globular, sometimes it is irregularly elliptical, with a constriction in the centre, so as to give it somewhat of an hour-glass shape. The outer coat mostly fits pretty closely on the inner contents, and is very often distinctly marked with little pits, some round, some irregular in shape; in other cases, instead of being thus pitted, the spores seem to be marked with deep curved furrows. Fig. 4, pl. 11, represents this plant in different stages of growth. (See Expla- nation of Plates.) Genus PENIUM, Breés, (1848.) Cellule eylindrice vel fusiformes, rect, utroque polo rotundate vel truncato-rotundate (nee emar- ginate nec excise), medio sepius constrict. Lamina chlorophyllacea axilis, ex transverso conspecta radiatim-divergens, radii seepe fureati, granula amylacea plerumque longitudinaliter seriata includens. Individua in aqua libere natantia, singula, sparsa-vel in massa gelatinosa consociata. Cellule mem- brana levis vel granulata, achroa vel fuscescens vel rubicunda, sepius longitudinaliter striata. (R.) Syn.—Netrium, N ®GELI. Cylindrocystis, MENGH. Closterium, partim, EHRENBERG. Cells cylindrical or fusiform, straight, rounded at each end, or truncately rounded (not emarginate or excised), medianly often constricted. Chlorophyl lamina axillary, when seen transversely radi- ately divergent, arms often forked, and containing starch granules, mostly longitudinally striate. Individuals swimming free in the water, scattered and single, or associated in gelatinous masses. Cell membranes smooth or granulate, transparent or fuscous or reddish, often longitudinally striate. a. Lamina chlorophyllacea peripherice lobata vel radiatim expansa. a. Chlorophyl lamina, lobate on the periphery or radiately expanded. P. Digitus, (Enrs.) Bris. P. cellulis ovato-cylindricis, diametro 3-5 plo longioribus, utroque polo parum attenuatis, sub- truncato-rotundatis; laminis chlorophyllaceis peripherice lobatis, medio interruptis. Diam.—7hF 5" = .00173"—7225” = .0029". Syn.—P. Digitus, (Eurs.) Bris, Rawennorst, Flora Europ. Algar., Sect. III. p. 118. Cells ovately cylindrical, 3-5 times as long as broad, at each end slightly attenuate, subtrun- cately rounded; chlorophyl lamina lobate on the periphery, interrupted in the middle. FRESH-WATER ALGH OF THE UNITED STATES. 107 Remarks.—This species is probably widely diffused through the temperate por- tions of North America. I have found it abundantly near Philadelphia, as well as among the Alleghanies, and have received specimens from Dr. Lewis, collected in Saco Lake, Northern New York; Prof. Bailey also notes it as occurring in Georgia. There is one form of it which resembles somewhat in outline the modern coffin, one end being much broader and much more rapidly narrowed than the other. There is no distinct vacuole at the end, at least in any specimen I remember to have seen, although frequently large numbers of moving granules can be detected in that portion of the frond. Fig. 6, pl. 20, represents the outline of a frond of this species. P. lamellosum, Brés. P. cellulis oblongo- vel fusiformi-cylindricis, diametro 5-6 plo longioribus, medio sexpe leviter coustrictis, utroque polo magis attenuatis, obtuso rotundatis. (I.) Diam.—0.0023”—0.0029". (R.) Syn.—P. lamellosum, Bres. RaBeNnnorst, Flora Europ. Algarum, Sect. III. p. 118. Hab.—Rhode Island. (Olney) Thwaites. Cells oblong or fusiform eylindrical, 5-6 times longer than broad, often slightly constricted in the centre, more attenuate at the ends, obtusely rounded. Remarks.—I have never recognized this species, but it is one of those sent over by Mr. Olney, and identified by Prof. Thwaites. b. Lamina chlorophyllacea integerrima. o . Chlorophyl lamina entire. * Cellule in medio plus minus constricte. * Cells more or less constricted in the middle. P. margaritaceum, Enrs. P. elongato cylindricum, diametro 8-9 plo longius, medio plerumque leviter constrictum, utroque polo rotundato-truncatum ; cellule membrana nodulis seriatis quasi margaritacea; locellus in medio (circiter) utriusque cruris corpusculis mobilibus in more Closteriorum repletus. (R.) Diam.—0.00098"—0.0011". (R.) Hab.—Rhode Island. (Olney) Thwaites; Bailey. Florida. Bailey. Elongate cylindrical, 8-9 times longer than broad, in the centre generally slightly constricted, at each end roundly truncate ; membrane of the cells somewhat pearly with seriate granules; vacuole about in the centre of each crus, filled with moving granules, as in closterium. Remarks.—I have not seen this desmid, but it is in Prof. Bailey’s list; it was also among those sent by Mr. Olney to Prof. Thwaites. P. minutum, CLeve. P. eylindricum, gracile, diametro 5-7 plo longius, leve, ad polos obtusissimos (latissime rotun- datos) parum attenuatum, medio leviter constrictum. (R.) Species mihi ignota. Diam.—0.00044”—0.00063”. (R.) Syn.—Docidium minutum, Raur’s British Desmid. P. minutum, Cueve. Rasenuorst, Flora Europ. Algarum, Sect. III. p. 122, 108 FRESH-WATER ALG OF THE UNITED STATES: Hab.—South Carolina; Florida. Bailey. Rhode Island. (S. T. Olney) Thwaites. Frond slender, suture not prominent; segments four to six times longer than broad, somewhat tapering, inflation obsolete, sides straight, ends entire; e. f. without puncture. L. sts"; B. rgsx - (Archer) b. Cellule in medio non constricte. b. Cells not constricted in the middle. P. interruptum, Bree. “P. cellulis late lineari-cylindricis, diametro 5-6 plo longioribus, utroque polo subito cuneato- acutatis, apicibus obtuso-rotundatis; laminis chlorophyllaceis longitudinalibus saturate viridi- bus, ztate provecta fasciis transversis tribus pallidis interruptis.” (R.) Species mihi ignota. Diam.—0,00147’—0.00177". (B.) Syn.—P. interruptum, Brés. Rapenuorst, Flora Europ. Algarum, Sect. III. p. 119. Hab.—In fossis, South Carolina, prope Grahamsville. Prof. Bailey. Cells broadly linearly cylindrical, 5-6 times longer than broad, at each end suddenly cuneately sharpened, the apex obtusely rounded; longitudinal chlorophyl lamina deep green, in ad- vanced age interrupted by three transverse pale fascia. P. Jenneri, bares. P. ab P. Brebissonii vix discernendum, cellulis cylindricis, utroque polo rotundatis, levibus, dia- metro 23-5 plo longioribus; zygosporis plerumque globosis, membrana fuscescente subgranu- lata. (R.) Species mihi ignota. Diam.—0.00057'’—0.0006”". (R.) Syn.—P. Jenneri, Raurs. Rasennorst, Flora Europ. Algarum, Sect. III. p. 120. Hab.—In fossis, Florida. Prof. Bailey. Scarcely distinguishable from P. Brébissonii, cells cylindrical, rounded at each end, smooth, 24-5 times longer than broad; zygospores mostly globose, membrane somewhat fuscous, sub- granulate. P. Brebissonii, (Meneu.) Races. P. in massa mucosa indefinite expansa sepe cum algis alteris intermixtis; cellulis perfecte cylindricis, interdum nonnihil curvatis sed plerumque rectis, diametro 2-4 plo longioribus, utroque polo late rotundatis, in medio non constrictis; “ zygosporis angularibus vel rotundatis, membrana fuscente, subtiliter granulata.” Diam.—7e55" = .00066”". Syn.—P. Brébissonii, (Mrnau.) Raurs. Rasennorst, Flora Europ. Algarum, Sect. II. p. 120. Hab.—In fossis, South Carolina. (Prof. Ravenel.) In an indefinitely expanded mucous mass, intermixed with other alge; cells perfectly cylindri- cal, sometimes slightly curved, but generally straight, at each end broadly rounded, not con- stricted in the middle; “ zygospores angular or rounded, membrane fuscaus, finely granu- late.” Remarks.—Among the numerous desmids which I have received from Prof. Ravenel are some which, I think, must be referred to P. Brebissonii, although they do not nearly equal the size of the European form, nor even the diameter given above, which is almost the lowest limit of the mature foreign plant. I believe, however, Prof. Ravenel’s specimens are immature, FRESH-WATER ALG OF THE UNDEID STATES. 109 s Mr. Ralfs’ description of the conjugation is as follows: ‘The process of the con- jugation in this species differs from that in the rest of this genus; for, as in Hya- lotheca dissiliens, the conjugation cells enter into the formation of the containing cell and are permanently attached to the sporangium, instead of being detached, as commonly happens, in the Desmids. The sporangium is at first cruciform, then quadrate, and finally orbicular. P. closterioides, Ratrs. P. cellulis anguste lanceolatis, diametro maximo 5-6 plo longioribus, a medio in apices subtrun- cato-rotundatos sensim attenuatis ; laminis chlorophyll. saturate viridibus, medio fascia trans- versa pallida interruptis. (R.) Species mihi ignota. Diam.—0.00159"—0.00175". (R.) Syn.—P. closterioides, Ratrs. RaApennorst, Flora Europ. Algarum, Sect. III. p. 121. Hab.—Prope Grahamsville, South Carolina. Prof. Bailey. Cells narrowly lanceolate, 5-6 times longer than the greatest diameter, sensibly attenuate from the middle into the subtruncate apices; chlorophyl lamina deep green, interrupted by a median pale band. Genus CLOSTERIUM, Nirtscu. Cellule interdum cylindric sed sepius fusiformes et utroque polo attenuate, plus minus lunula- tim curvate, in medio haud constrict sed stria transversa unica vel 2-5 impresse. Cytioderma tenue, sat firmum, leve vel plus minus distincte striatum et interdum longitudinaliter costatum. Cytioplasma chlorophyllosa plerumque in laminis longitudinalibus disposita, et sub cellule polis locello achroo, plerumque globoso et corpusculis plus minus numerosis se vivide moventibus impleto instructa. Cells sometimes cylindrical, but more often fusiform and attenuate at each end, more or less lunately curved, in the centre not constricted but marked with from 1—5 transverse strie. Cytioderm thin, moderately firm, smooth or more or less distinctly striate, and sometimes longitudinally costate. Chlorophyllous cytioplasm mostly arranged in longitudinal lamina, and furnished at each end with a clear space, which is mostly globose, and contains more or less numerous actively moving corpuscles. a. Zygospore globose, rarissime angulares ; cellule crura aut non aut minus producta. a. Zygospores globose, very rarely angular ; crura of the cells not at all, or only slightly, pro- duced. 1. Cellule cylindrice, ad utrumque polum via vel paullum attenuate, recte vel leviter curvate, apicibus rotundatis vel truncatis. 1. Cells cylindrical, not at all or but slightly attenuated at the ends, straight or slightly curved, the apex rounded or truncate. C. striolatum, Enrs. C. anguste lanceolato-fusiforme, leviter arcuatum, 8-12 plo fere longius quam latum, utroque polo paulum sensimque attenuatum, apicibus truncatis seepe fuscescentibus ; membrana dis- tinctissime striata, vacuata fuscescente; vesiculis chlorophyllaceis 5-7 (in quoque crure) ; locello apices versus sito, submagno, corpuscula 12-20 includente. (R.) Diam.—7,""—,"" = 0.00152" —0.00187”. (R.) Syn.—C. striolatum, Eurs. RaAsBeNnuorst, Flora Europ. Algarum, Sect III. p. 125. Hab.—In aquis quietis, Centre County, Pennsylvania. Wood. Saco Pond, New Hampshire. (Lewis) . Narrowly lanceolately-fusiform, slightly bent, 8-12 times longer than broad, sensibly attenuated 110 FRESH-WATER ALG OF THE UNITED STATES: at the ends, which are truneate and often somewhat fuscous ; membrane very distinctly striate, when empty somewhat fuscous; chlorophyl globules 5-7 (in each limb); vacuole placed in the bent apex, moderately large, including 12—20 corpuscles. Remarks.—The measurements given are those of Prof. Rabenhorst. Our Ame- rican forms agree well with them. ¢. angustatum, Kz. GC. gracile, sublineare, diametro 16-18 plo longius, ad polos levissime attenuatum, apicibus late truncatis; costis longitudinalibus paullulum prominulis 4—5, interstitiis circiter aig latis; vesiculis chlorophyllaceis in quoque cruro 6-7; locello ab apice subremoto mediocri, corpus- culis 12-20 impleto. (R.) Diam.—y3y'—#x" = 0.00081”—0.0010". (R.) Syn.—C. angustatum, Krz. Rapennorst, Flora Europ. Algarum, Sect. IIT. p. 126. Hab.—In aquis quietis, prope Philadelphia, Pennsylvania. Wood. Rhode Island. Bailey. New Hampshire. (Lewis) C. slender, sublinear, 16-18 times longer than broad, very slightly attenuate at the ends, which are broadly truncate; with from 4-5 somewhat prominent longitudinal ribs, the interstices about z1,/” broad; chlorophyl globules in each limb 6-7; vesicle subremote from the apex, moderate, containing from 12-20 corpuscles. Cc. juncidum, Rares. C. elongatum, anguste lineare, diametro 20-35 plo longius, leviter arcuatum, utroque polo vix attenuatum; apicibus truncatis; cytiodermate luteolo, interdum longitudinaliter striato. Diam.—s2Ay5" =.0004". Syn.—C. juncidum, Rates. RaBennorst, Flora Europ. Algarum, Sect. III. p. 127. Hab.—In fossis, South Carolina. (Ravenel) In lacu Saco, New Hampshire. (Lewis) Elongate, narrowly linear, 20-35 longer than broad, slightly bent, scarcely narrowed at the ends; apices truncate; cytioderm yellowish-brown, sometimes longitudinally striate. Remarks.—I am indebted to Prof. Ravenel for specimens of this species, by” whom they were found on the slimy surface of a half dried-up ditch, associated with numerous other desmids. ‘The specimens are all smaller than the measure- ments of Rabenhorst, but much larger than those given by Mr. Ralfs. None of the plants have any chlorophyl granules—a circumstance probably simply depen- dent upon the stage of their development. The longitudinal striz are in none of the specimens very distinct, and in many cannot be demonstrated. Since writing the above I have seen specimens collected by Dr. Lewis in ‘“ Saco Pond,” near the Crawford House, New Hampshire. Mr. Archer (Pritchard’s Infus., p. 749) lays stress upon the fronds being straight in the middle, with the ends curved downwards; but I have seen numerous speci- mens in which the curve was through the whole length. Fig. 20, pl. 12, represents one of the specimens collected by Prof. Ravenel in South Carolina. 2. Cellule cylindrice, dorso plus minus convere, ventre subplane, nunquam ventricoso —inflate, , FRESH-WATER ALG# OF THE UNITED STATES. 1 2. Cells cylindrical, with the dorsum more or less convex, the belly straightish, never ventri- cosely inflated. C. Lunula, (Miter) Enrs. C. permagnum, subleve (striae subtilissime vel indistincte), semilunare, dorso alte convexum, ventre subplanum, apicibus attenuatis rotundatis ; vesiculis chlorophyllaceis numerosis spar- sis; locello distincto subapicali corpuscula numerosa includente. (R.) Diam.—37"—5" = 0.00032”—0.0045". (R.) Syn.—C.Lunula, (MULLER,) Eurp. RaBennorst, Flora Europ. Algarum, Sect. III. p. 127. Hab.—South Carolina, Georgia, Florida. Prof. Bailey. Pennsylvania. Wood. Very large, smoothish (strice very fine or indistinct), semilunar, dorsum strongly convex, belly straightish, the ends attenuate and rounded; chlorophyl globules numerous, scattered; vesi- cle distinct, subapical, including numerous corpuscles. C. acerosum, (Scurank) Enrs. (Var. nov. maximum.) C. lineare-fusiforme, sub-rectum aut leve curvatum, utroque fine sensim et paullulum atten- uatum, diametro 15-24 plo longiore; apicibus angustissime truncatis, achrois; membrana hand striata; vesiculis chlorophyllaceis 11-14 in quoque crure, in serie axilli simplici collo- catis; locello apicali parvo, corpuscula numerosa includente ; zygosporis globosis. Diam.—Transy. max. 7135” = .0017’; zygosp. 7295" =.0027”. Syn.—C. acerosum, (SCHRANK) Eurs. Rapennorst, Flora Hurop. Algarum, Sect. III. p. 128. Hab.—Pennsylvania; Wood. South Carolina, Georgia, Florida; Bailey. Linear, fusiform, straightish, or slightly curved, at each end sensibly little by little attenuate, 15-24 times longer than broad; apices narrowly truncate, transparent; membrane not striate; chlorophyl globules 11-14 in each limb placed in a simple axillary series; apical vesicle small, containing numerous corpuscles ; zygospores globose. Remarks.—The desmid, described above, was found in New Jersey, near Cam- den. It differs from the typical form of C. acerosum in its size, proportionate length to breadth, and in not being striate. The European “ forma major” (RABENH.) appears, however, to exceed it in transverse diameter, and, according to some authors, certain fronds of the species are not striate, and all authorities agree that at times the striz are exceedingly delicate. For these reasons, I think, our American form must be regarded simply as a variety. As far as can be judged from the rude figure, it is this species which Prof. Bailey identifies as C. tenue, K1z., in Silliman’s Journal for 1841. Fig. 5, and 5 a, pl. 11, represent this species magnified 250 diameters; 5 b represents the sporangium with portions of the dead fertile fronds still attached. C. areolataum, Woop, (sp. nov.) * C. fusiforme, subrectum vel normihil curvatum, lateris ventralis medio spe paullulum concavum, diametro 9-10 plo longius, utrinque modice attenuatum; apicibus truneato-rotundatis ; mem- brana crassa, et firma, rubido-brunnea, profunde distante striata, et minutissime sed distincte granulata vel areolata; suturis medianis distinctissimis 4-10. Diam.— 0.0024”. Hab.—In aquis puris quietis; Northumberland Co., Pennsylvania. 112 FRESH-WATER ALG& OF THE UNITED STATES ~~ Fusiform, straightish, or very slightly curved, the ventral side often a little concave in the middle, 9-10 times longer than broad, moderately attenuated at each end; the apices trun- cately rounded; cell-membrane reddish-brown, thick and firm, distantly profoundly striate, and very minutely but distinctly granulate or areolate; median sutures very distinct, 4-10 in number. Remarks.—I found this species growing in a quict pool of pure water, in a wild, deeply wooded ravine, near Danville, Central Pennsylvania. It was in great abundance, forming a translucent greenish jelly, one or two gills of which might have been readily gathered. Unfortunately, I had no microscope with me and cannot, therefore, determine at all as to the arrangement of the endochrome, the carbolic acid, used as preservative, having entirely disarranged this by the time I got the fronds upon a slide. ‘The empty frond is of a reddish-brown color. The membrane is quite thick and firm, and is marked with very prominent broad strie or grooves. Ina number of cases I have counted these and always found nine present upon one face of the frond. There are also upon the surface numerous minute markings not fairly visible with a lower power than a 45th objective. Under this glass they appear as minute punctations, An eighth resolves them into granules mostly of an oblong shape, arranged more or less regularly in longi- tudinal rows. Very generally, each side of the stria or groove has a close row of larger and more distinct granules forming a sort of border to it. In truth, the surface of the frond is covered with broad longitudinal bands of these granules, and the narrow smooth spaces between them constitute the stria spoken of. ‘This species is very closely allied to C. turgidum, Eurs., agreeing pretty well with it in general outline and size, I think, however, the peculiar markings upon the membrane are sufficient to separate it, and do not doubt that if fresh specimens were at hand, differences would be found to exist also in the arrangement of the cell-contents. The turning up of the ends, generally so marked in C. turgidum, is mostly entirely absent in this species, rarely there is some tendency fo it. Fig. 6, pl. 11, represents in outline a frond magnified 160 diameters; Fig. 6 a, the end of an empty cell, magnified 1375 diameters; the color of this is, perhaps, a little too dark. C. lineatum, Enrs. C. valde elongatum, gracile, quater vicies-tricies longius quam latum, distincte striatum, e medio recto cylindrico utrinque valde attenuatum, apices versus leviter incurvum, obtuso- truncatum; vesiculis chlorophyllaceis in quoque crure 20-21, in seriem unicam axilem dis- tributis ; locello parvo, ab apice remoto, corpusculis 10-12 impleto. (R.) Diam.—zp55/ = 0015". Syn.—C. lineatum, Eure. Rapennorst, Flora Europ. Algarum, Sect. III. p. 180. Hab.—Pennsylvania, Wood. Very much elongate, slender, distinctly striate, from the centre straight and cylindrical, at each end very greatly attenuate, apex bent, slightly ineurved, obtusely truncate; chlorophyl glo- bules 20-21 in each limb, placed in a simple axillary series; vacuole small, remote from the apex, containing from 10-12 corpuscles. , ee afro rs / a, . . . Remarks.—The American forms agree well with the above description; some FRESH-WATER ALG& OF THE UNITED STATES. 1B of them, however, are a little more curved in the central portion than it would imply. Fig. 1, pl. 12, is a drawing of an American plant, magnified 160 diameters. €. Cucumis, Exrs. C. oblongum, turgidum, leviter curvum, leve, diametro 4-7 plo longius, apicibus obtusis. (R.) Syn.—C. Cucumis, EuRenBERG, Verbreit. b. 28, 1V. F.28. Rasenuorst, Flora Europ. Alga- rum, Sect. ILI. Dgl 38. * Hab.—New York; Ehrenberg. Oblong, turgid, slightly curved, smooth, 4-7 times longer than broad, the apex obtuse. Remarks.—I have no knowledge of this species other than that in the above short description, which has been copied from Rabenhorst’s works. 3. Cellule semilunares, plerumque magis curvate quam in Sect. 1 et 2, ventre semper tumide, ventricoso-inflate. 3. Cells semilunar, mostly more curved than in Sect. 1 and 2, with the belly always tumid, ventricosely inflated. Cc. Ehrenbersgii, Mencu. ~ C. fusoideo-semilunare, ventre inflato, ceterum ut in C. Lunula. (R.) Diam.—ULat. 4135” =.0029". Long. z235” =.0042.” Syn.—C. Ehrenbergii, Menau. Rasennorst, Flora Europ. Sect. III. p. 131. Hab.—Prope Philadelphia. “Frond large, stout, about five or six times as long as broad, lunately curved, extremities taper- ing; upper margin very convex, lower concave with a conspicuous central inflation; ends broadly rounded; large granules, numerous, scattered ; fillets several; e. f. colorless, without strie, central suture not evident. Sporangia orbicular, smooth, placed between the but- slightly-cohnected empty conjugated fronds, the endochrome during the process of conjuga- tion emerging from the opened apex of a short conical extension from each under side of each younger segment (or shorter cone) of each pair of recently divided fronds, the conjugating fronds being produced immediately previously by the self-division of a pair of old fronds—two sporangia being thus the ultimate produce of the two original fronds. L. Jy”; B. gdp”. Archer.” Pritchard’s Infusoria, p. 748. Remark.—Fig. 2, pl. 12, represents a plant of this species magnified 160 dia- meters. C. moniliferum, (Bory) Ears. C. semilunare, plus minus curvatum, diametro maximo 6-9 plo longius, ventre inflato, utroque polo sensim attenuatum, apicibus achrois obtusis, vesiculis chlorophyllaceis in serie unica longitudinali axili dispositis, in quoque crure 7-10; locello apicali submagno, corpuscula numerosa includente (corpusculum in quoque locello unicum mobile ellipsoideum, magnitudine linez partem millesimam equans, cetera mobilia per totum corporis distributa observavit cl. Perty.) (R.) Species mihi ignota. Diam.—0.0019”—0.0022". (R.) Syn.—C. moniliferum, Rawennorst, Flora Europ. Algarum, Sect. III. p. 131. Hab.—Georgia; Rhode Island; Bailey. 15 June, 1872. 114 FRESH-WATER ALG#& OF THE UNITED STATES: “ Frond smaller than C. Ehrenbergii, stout, five or six times as long as broad, lunately curved, extremities tapering, upper margin convex, lower concave, with a central inflation, ends large granules, conspicuous, in a single longitudinal series; e. f. colorless, without rounded ; ; : ; L. 2s"—ady" B. gto’ —ate". Archer.” Pritchard’s Jnfusoria, striz, suture not evident. p. 748. Cc. Leibleinii, K1z. C. priore minus, semilunare, magis incurvum, ventre inflato, ad utrumque polum largius attenu- atum, apicibus achrois acutis; vesiculis chlorophyll. in quoqme crure 5—6, in serie simplici axillari dispositis; locello magno, apices versus sito, corpuscula numerosa includente. (R.) Diam.—7}3y"- Syn.—C. Leibleinti, KUrzina. Rapenuorst, Flora Europ. Algarum, Sect. III. p. 132. Hab.—Georgia; Rhode Island; Bailey. Pennsylvania; Wood. “Frond somewhat stout, distance between the extremities six or eight times the breadth, crescent-shaped, much curved, rapidly attenuated, upper margin very convex, lower very con- cave, often with a slight central inflation; ends subacute ; large granules, in a single series; fillets few or indistinct; e. f. somewhat straw-colored, without striw; suture evident. Spo- rangium orbicular.” Archer. Remark.—Fig. 6, pl. 12, represents this plant, magnified 260 diameters, 4. Cellule. maxime curvate, ventre non tumide. 4. Cells most curved, the belly not tumid. C. Dianz, Enzs. C. anguste fusiforme, semilunare, utroque polo valde attenuatum, apicibus subacutis; cytioder. mate achroo (vel dilutissime umbrino), striis subtilissimis medio interruptis preedito, in media parte striis transversalibus 3-5; vesiculis in quoque crure 6—7, in serie unica axili dispositis ; laminis chlorophyllaceis pluribus, sepe flexuosis; locello indistincto, corpusculis pluribus vivide mobilibus. (R.) Diam.—Lat. 7257" =.00053”. Long. 78325” = .00082”. Syn.—C. Diane, EHRENBERG. RaseEnnorst, Flora Europ. Algarum, Sect. IIE. p. 133. Y ’ ’ 1 cers , Io Hab.—Georgia; Florida; Rhode Island; Bailey. Pennsylvania; Wood. fo} ? b ? Mf y ? Frond crescent-shaped, six or eight times as long as broad, much curved, rapidly attenuated; upper margin very convex, lower very concave without a central inflation; ends subacute with a very slight emargination at the upper outer extremity; large granules in a single series ; empty frond, somewhat straw-colored, or faintly reddish, without strie, suture evident. (A.) Remarks.—Mr. Archer marks C. Venus, K1z., as a doubtful synonym of this species; not having Prof. Kiitzing’s work at hand, I do not know whether @. Venus, Kz, is really the following species or not. The two forms here known as 0. Diane, Eure. and C. Venus, Krz. are, however, I think sufficiently distinguished. Fig. 4, pl. 12, represents this species of desmid. Cc. Venus, Krz. \ . . . oe . . : 7 7 C. parvum, plus minus gracile, semicirculare, octies-duodecies longius quam Jatum, in apices subacutos wqualiter sensimque attenuatum ; cytiodermate tenui, lave; laminis chlorophylla- ceis obliteratis ; vesiculis in quoque crure 3-4; locello distineto corpusculis 4—6 repleto. (R.) Diam.—.0004”". FRESH-WATER ALG#H OF THE UNITED STATES. 115 Syn.—C. Venus, Kurzine. Rasenuorst, Flora Europ. Algarum, Sect. III. p. 134. Hab.—South Carolina. (Ravenel.) Small, more or less slender, semicircular, eight to twelve times longer than broad, equally and very perceptibly attenuate at both apices; cytioderm thin, smooth; chlorophyllous lamina obliterated ; vesicles in each crus 3-4; vacuole distinct, containing 4~6 corpuscles. Remark.—Fig. 7, pl. 11, represents in outline a frond magnified 450 diameters. ¢C. parvulum, N xc. C. parvum, semicirculare, medio non tumidum, gracile, anguste lanceolatum, sexies-octies lon- gius quam latum, apicibus acutis; cytiodermate tenui, levissimo, vacuato nonnunquam luteolo- fuscescente et subtiliter striato; vesiculis uniseriatis, in quoque crure 2—4, varius 1-7; laminis chlorophyllaceis 4-5. (R.) Diam.—Max. 0.00026"—.00062” (R.) (.0008” W.) Syn.—C. parvulum, NwGELt. Rasenuorst, Flora Europ. Algarum, Sect. III. p. 134. Hab.—Prope Philadelphia, Wood. Small, semicircular, not swollen in the middle, slender, narrowly lanceolate, six to eight times longer than broad, with the apices acute; cytioderm thin, very smooth, when empty some- what yellowish-fuscous and finely striate; vesicles uniseriate, in each crus 2-4, rarely 1-7; chlorophyllous lamina 4—5. Remarks.—I have referred to this species a desmid which I have found about Philadelphia, and which agrees in all respects with the description of Prof. Raben- horst except in attaining a larger size. Fig. 5, pl. 12, represents this plant magnified 450 diameters. Cc. Jennerii, Rares. C. cylindraceo-fusiforme, semilunare, leve, utrinque modice attenuatum, sexies-octies longius quam latum, apicibus obtuse rotundatis; vesiculis in quoque crure 5-7, in serie unica axili dispositis; laminis chlorophyllaceis 2-3; locello subaquali magno, corpusculis numerosis impleto. (R.) Diam.—0.00057”". (R.) Syn.—C. Jennerii, Raurs. Rasennorst, Flora Europ. Algarum, Sect. III. p. 134. Hab.—Rhode Island, Bailey. i Frond small, distance between the extremities six or seven times the breadth, crescent-shaped, much curved, gradually tapering (sometimes with an obscure central constriction) ; upper margin very convex, lower very concave without a central inflation; ends obtuse, rounded ; large granules in a single series; e. f. colorless without strie. L. sy’. B. y7'59”. Archer. Pritchard’s Infusoria. b. Zygospore plerumque quadrangulares, cellularum crura longe vel longissime producta, sepe setiformia. b. Zygospores mostly quadrangular, crura of the cells greatly produced, often setiform. C. rostratum, Exrs. C. corpore lanceolato-fusiformi, utrinque valde et longe attenuato, leviter curvato, striato ; cor- nibus setaceis singulis corpus vix equantibus, sepius longe brevioribus ; cytiodermate dilute vmbrino vel luteolo, dense striato ; vesiculis uniseriatis, in quoque crure 5-6; locello oblongo, sepius indistincto, corpusculis 12-15 vivide se moventibus. (R.) 116 FRESH-WATER ALGH& OF THE UNITED STATES. Diam.—0.0008"". (0.0009/’-0.0016.’" R.) Syn.—C. rostratum, EURENBERG. RABENHORST, Flora Europ. Algarum, Sect. III. p. 135. Hab.—In fossis, prope Philadelphia; Wood. Body lanceolate-fusiform, at each end greatly and for a long distance attenuated, slightly curved, striate ; crura setaceous and scarcely as long as the body and sometimes much shorter; cytio- derm light or luteolous, densely striate, vesicles uniseriate, 5—6 in each crus; vacuoles oblong, often indistinct, containing from 12-15 actively moving granules. Remark.—Fig. 3, pl. 12, is a drawing of this species, magnified 260 diameters, Cc. setaceum, Eure. C. corpore anguste lanceolato, recto vel subrecto, distincte striato, utrinque in rostrum setaceum, levissime incurvum, obtusum, longissime porrecto; singulo rostro corpore 3-4 plo longiore; et vesiculis et locello indistinctis. (R.) Diam.—Max. (plerumque) 0.0004”—0.00044.” (R.) Syn.—C. setaceum, HHRENBERG, Flora Europ. Algarum, Sect. IIL. p. 136. Hab.—Stonington. (Lewis) Pennsylvania; Wood. Georgia; Florida; Providence, Rhode Island ; Bailey. Frond very slender, from twenty to twenty-five times as long as broad, narrow-lanceolate ; upper and lower margins nearly equally and but slightly convex; each extremity tapering into a very long and slender setaceous colorless beak, longer than the body, ultimately curved downwards, ends obtuse; e. f. colorless, stria close, faint, central suture solitary. Sporangium cruciform. L. 74%”. B. ggg". Archer. Pritchard’s Infusoria. C.Amblyonema, Enrs. C. filiforme, cylindricum, leve, utroque fine parum attenuatum, apice rotundum. (R.) Syn.—C. lineatum, Eure. Battery, American Journal of Science and Arts, 1841, p. 303. C. Amblyonema, Eurs. Verbreit. p. 123. RaABenuorst, Flora Europ. Algarum, Sect. IIL. p. 138. Hab.—West Point, New York; Providence, Rhode Island, Bailey. Filiform, cylindrical, smooth, gradually attenuated at each end; the apex rounded. Remarks.—l\ have never recognized a specimen of this species, nor have I had access to the original description of Ehrenberg. Genus TETMEMORUS. Cellule cylindrice vel fusiformes, rectw, medio distincte constrict, utroque polo anguste incise cytioderma sat firmum, plerumque granulatum vel punctatum. Cells cylindrical or fusiform, straight, distinctly constricted in the middle, narrowly incised at each end. Cytioderm firm, mostly punctate or granulate. Y Brebissonii, (Meneu.) Raurs. T. diametro 4—6 plo longior; a fronte cylindricus, utroque polo non attenuatus sed rotundato- truncatus ; a latere fusiformis et a medio in apices rotundatos sensim attenuatus; cytiodermate striato-punctato. Diam.—712," = .0016”. Syn.— T. Brebissonii, Menauetnt, RaBenuorst, Flora Europ. Algarum, Sect. IIT. p. 139. FRESH-WATER ALG OF THE UNITED STATES. ir Hab.—In fossis, Atlantic States. Four to six times longer than broad; from the front cylindrical, not attenuate at the truncately rounded ends; viewed laterally fusiform, attenuated from the middle to the rounded ends; cytioderm striately punctate. Remarks.—The central constriction is more apparent in the lateral than front view. When the frond is full of endochrome the puncte on the outer wall are not apparent, but when it is empty they are seen to be small, and closely arranged in stria-like rows. ‘This species extends through all the Atlantic sea-board States. Prof. Bailey has found it in South Carolina and Florida, as well as in Rhode Island. I have collected it in Centre County, of this State. Fig. 3, pl. 21, represents an empty half frond of this species; 3 @ the outline of the frond. T. granulatus, (Brés.) Rates. T. habitu Tetm. Brébissonii, sed major et cytiodermate irregulariter granulato-punctato. (R.) Diam.—7}y" = .0013”. (.00155”. BR.) Syn.—T. granulatus, (BRéBIsSON.) RALFs. RABENHORST, Flora Europ. Algarum, Sect. III. p- 140. Hab.—Prope Philadelphia; Wood. Rhode Island; (S. T. Olney.) Thwaites. South Caro- lina; Bailey. Frond somewhat longer than T. Brebissonii, about five or six times longer than broad; in both f. v. and s. y. fusiform, the constriction a very shallow groove, ends with a hyaline lip-like projection extending beyond the notch; endochrome with a longitudinal series of large granules; e. f. punctate, the puncta scattered, except near the constriction; where they are disposed in two transverse rows. Sporangium orbicular, smooth, margin finely striated, placed between the deciduous empty fronds. L. y4,”. B. g4y”. Archer. Pritchard’s Infusoria. Remark.—Fig. 8, pl. 12, represents this species magnified 450 diameters. T. giganteus, Woop. T. maximus, oblongus, diametro 3 plo longior; apicibus haud attenuatis, late rotundatis; suturis profundis, linearibus; cytiodermate irregulariter granulato-punctato. Diam.—72%5' = .0031”. Syn.—T. giganteus, Woop. Proe. Acad. Nat. Sci., 1869. Hab.—In stagnis, Centre County, Pennsylvania. Very large, oblong. 3 times longer than broad ; with the ends not attenuate but broadly rounded ; suture profound, linear; cytioderm irregularly granulately punctate. Remarks.—I found this beautiful desmid in a stagnant pool in Bear Meadows, Centre County, in the month of August. It is very different in its outline from its nearest ally, 7. granulatus. The diameter is preserved uniform until near the end, where there is an alteration in the line of the margin, so as to cause some contraction, which is, however, wanting in some specimens. The ends are therefore broad and obtuse. ‘The size is also double that of 7. granulatus. Fig. 7, pl. 12, represents a frond of this species magnified 260 diameters. 118 FRESH-WATER ALG® OF THE UNITED STATES: T. levis, (Kurz.) Rates. T. Brébissonii formis similis sed parvior, 3-4 plo longior quam latus ; cytiodermate plerumque levissimo, interdum indistinctissime punctato. = r) arr Diam.—yzj55" = -00066". Syn.—T. levis, Kivzine. Raurs. Rasenuorst, Flora Europ. Algarum, Sect. III. p. 140. Hab.—In aquis quietis, prope Philadelphia. Similar in form to T. Brébissonii, but smaller, 3-4 times longer than broad ; cytioderm mostly very smooth, sometimes indistinetly punctate. Remarks.— Prof. Rabenhorst states that the cytioderm of this species is very smooth, and Mr. Ralfs says that he has failed to detect any punctations, but also states that “ Mr. Jenner and Mr. Ross assure me that they (puncte) are scattered as in 7. granulatus.” Ihave no doubt of their existence in certain individuals, whilst in other cases they appear to be absent. Genus PLEUROTAENIUM, Naeettr (1849). Cellule singule in aqua natantes, recta vel subrecte, cylindric vel fusiformes, valde elongate, utroque polo rotundatze vel truncate, medio leviter constricte, ex transverso circulares. Cytio- plasma chlorophyllaceum in laminis longitudinalibus pluribus dispositum, et sub utroque polo locello rotundato corpusculis se vivide moventibus impleto instructum. Cells single, swimming in water, straight or nearly so, cylindrical or fusiform, very much elongate, rounded or truncate at each end, in the end view with a circular outline. Chlorophyllous protoplasm arranged in longitudinal lamine and furnished at each end with a round vacuole containing actively moving corpuscles. Remarks.—This genus appears to include the main portion of the species, which have been described under the name of Docidium ; the remainder being represen- tatives of a number of genera. I have not had access to the original description of Docidium, and do not know in what year it was published; but, according to De Bary, Docidiwm is much the older name (‘Ueber de Conjugat.,” p. 75). M. De Bary states, however, that he prefers the name of Niegeli, because that autho- rity first defined the genus and his name expresses very clearly the character of it, as well as from the circumstance that the name Docidiwm having been made to cover a heterogeneous mass of species, its retention might cause confu- sion. I confess to thinking that this action of De Bary is not in accordance with the recognized laws of priority, but, in the absence of the original description, have thought best to follow it. P. trabecula, (Eurs.) Naxcent. P. sepe valde elongatum, octies vicies-longius quam latum, cylindraceum, utroque fine lsevissime attenuatum aut inerassatum, juxta medium constrictum sepius bigibbum (quasi biundatum), apicibus late truncatum ; cytiodermate tenui levi, achroo. (R.) Diam.—7hz" = .0013”. Syn.— Docidium Ehrenbergii. Raurs.” BAtLEY, Microscopical Observations. Smithsonian Contributions. Pleurotenium trabecula, (Eur.) Naar. Rapennorst, Flora Europ. Algarum, Se+t. III. p. 141. Hab.—South Carolina, Georgia, Florida; Bailey. Pennsylvania ; Wood. FRESH-WATER ALGA OF THE UNITED STATES. 119 (Docidium Ehrenbergit. Raurs.) Frond slender, linear; suture forming a very sharply defined rim; segments 8-12 times longer than broad, basal inflation having another smaller one above it, sides otherwise straight, parallel; ends crenate, owing to a number of emargina- tions on the edge of the truncate extremities, from three to five of the crenations being usually visible; e. f. punctate or rough with minute granules. Sporangium suborbicular or elliptic, or slightly angular, smooth, placed between the deciduous empty fronds. Ciliated zoospores formed by segmentation of the cell contents, and their emission effected through the opened apex of each one, two or three, especially, formed lateral tubes arising from be- neath the base of one of the segments. Archer. Remarks.—This species is quite common around Philadelphia; but I do not remember ever to have seen one with the cell-wall granulate. The smaller of the two umbonations near the centre is often wanting or exceedingly small, and the crenulations in the ends are very often obsolete. Fig. 9, pl. 12, represents a cell of this species magnified 160 diameters. P. Baculum, (Brés.) Dr Bary. P. priori simile, sed gracilius, angustius et plerumque longius, medio tantum semel constrictum ; cytiodermate levi. (R.) Species mihi ignota, Diam.—0.00054”—0.0009". (R.) Syn.—P. Baculum, (Brés.) De Bary. Rasennorst, Flora Europ. Algarum, Sect. III. p. 141 Hab.—Georgia ; Bailey. Frond slender, suture not prominent; segments very many times longer than broad, basal in- flation very conspicuous, solitary, sides otherwise straight, very nearly parallel, large gran- ules of the endochrome in a single series ; ends entire; e. f. without puncta. L. 74,” B. 1937 P. breve, Woop. P. robustum, diametro 4-8 plo longius, in medio distinete constrictum sed haud undulatum, utroque polo nonnihil attenuatum; apicibus truncatis et nonnihil rotundatis; cytiodermate crassissimo, dense granulato-punctato; marginibus vel rectis, vel breve undulatis. Diam.—.0038" —.00095". Syn.—P. breve, Woop. Proc. Acad. Nat. Sciences, 1869. Hab.—District of Columbia. (Billings.) Robust, 4-8 times longer than broad, distinctly constricted but not undulated in the middle, slightly attenuated towards the ends; apex truncate and somewhat rounded ; eytioderm very thick, densely minutely granulate; margins either straight or shortly undulate. Remarks.—This species was sent to me by Dr. Billings, who obtained it near Washington, D. C. The margins are sometimes straightish, but in other fronds there are three or more distinct short undulations, or rounded projections in each half margin. ‘The cell-wall is excessively thick, especially at the end—in many cases much thicker than the drawing. Fig. 2, pl. 21, represents an empty frond of this plant magnified 750 diameters. P. crenulatum, (Exrs.) Rapennorst. P. robustum, cylindraceo-subclavatum, octies-duodecies longius quam latum, medio undulato- nodulosum, stricture medie margine tumido, apicibus late truncatis, altero sepe crenulato ; cytiodermate granulato-punctato. (R.) Species mihi ignota. Diam.—0.0023". (R.) Syn.—P. crenulatum, (Eurs.) Rapennorst, Flora Europ. Algarum, Sect. IIT. p. 142. 120 FRESH-WATER ALG# OF THE UNITED STATES. ~ Docidium nodulosum, Br&s. Raters. British Desmidiz, p. 155. Closlerium trabecula, Barney. American Journal of Science, 1841. Hab.—In aquis quietis, South Carolina; Georgia ; Florida; Rhode Island; Bailey. Pennsyl- vania; New Jersey ; Wood (Docidium nodulosum.) Frond very stout, the thickened sutures forming a projecting rim; segments four to six times as long as broad, scarcely attenuated, regularly inflated at intervals so as to form an undulated margin, the basal inflation the most prominent, the others, as they approach the ends, less so, where they are indistinct or wanting; ends entire; e. f. coarsely punctate. L. dy”. B. gt,” Archer. Pritchard’s Infusoria. Remarks.—I have found this species in * Shepherd’s Mill Pond,” near Greenwich, Cumberland County, New Jersey, and also in a Spring in the Philadelphia Park, near Columbia bridge. Fig. 1, pl. 21, represents the outline of a frond of this species magnified 160 diameters. P. clavatuma, (K1z.) Dr Bary. P. subeylindraceum, multoties (16-24) longius quam latum, ad utrumque polum sensim incras- satum, subclavatum, apicibus late truncatis; cytiodermate firmo achroo, dense et irregulariter granulato-punctato. (R.) Species mihi ignota. Diam.—Max. 0.00165’—0.00147”"; min. = 0.0010’—0.00092”. (R.) Syn.—P. clavatum, (Krz.) DE Bary. RaAsennorst, Flora Hurop. Algarum, Sect. IIT. p. 141. Docidium clavatum, Kiirzine. Rawrs, British Desmidiz. ArcHER. Pritchard’s Infusoria. Hab.—South Carolina; Georgia; Bailey. Frond slender, suture scarcely prominent, segments eight to ten times as long as broad, slightly clavate near the ends, and ultimately somewhat attenuated, basal inflation sometimes solitary, sometimes having another slight one above it; ends entire; e. f. punctate. L. 24”. B. gt,” P. undulatam, (Barzey.) D. leve, gracile cylindricum, undulatum, latitudine 18—20 plo longius, medio modice constrictum; cruribus et basi et apice truncatis et crenatis. (R.) Species mihi ignota. Syn.—Docidium undulatum, BatEY. Micros. Observy. p. 36. Hab.—F lorida, Bailey. “Segments eight to ten times longer than broad, constricted six to eight times at regular intervals throughout their entire length, with the base and ends crenate, smaller than D. nodulosum, Bre&s., with more frequent and deeper constrictions. The same characters distinguish it from D. nodosum and D. constrictum.” P. nodosum, (Bartey.) D. yalidissimum, undulatum, spintlis sparsis hirsutum, medio valde constrictum, diametro 8-10 plo longius; cruribus e basi dilatata leviter attenuatis 4-undatis, apicibus quasi productis, latissime truncatis; locello apicali ratione parvo, rotundo, corpusculis paucis (ut videtur) repleto. (R.) (Species mihi ignota.) Syn.—Docidium nodosum, Battny. Micr. Obsery., pl. 1, fig. 4. Ratrs, British Desmids, p. 218. Hab.—United States; Bailey. “Frond stout; segments with four prominent nodes separated by constrictions; end view crenate. An end view shows that each node is not a simple swelling, but really formed by whorls of tubercles. ‘This species is easily recognized by the deep indentations in its out- line, corresponding to the constrictions which separate the transverse rows of knob-like pro- Jections. It is one of the largest species in the genus,’ Bailey.” Ralfs. FRESH-WATER ALG& OF THE UNITED STATES. 121 P. constrictum, (Barry) D. subvalidum, leve, latitudine 10-12 plo longius, medio valde constrictum, stricture margine non prominente ; cruribus a basi tumida in apicem late truncatum non attenuatis, 4 undu- latis. (R.) (Species mihi ignota.) Syn.—Docidium constrictum, Bainry. Ra.rs, British Desmids, p. 218. Hab.—Rhode Island, Bailey. “Frond stout, segments with moderately deep constrictions, which separate four equal, gently curving prominences; end view entire. ‘This species is at once distinguished from D. nodosum by the cross section of the nodes being a simple circle instead of an indented one,’ Bailey.” Ralfs. P. verrucosum, (Barry) D. validum, granuloso-verrucosum, latitudine 10-12 plo longius, undulatum, apicibus integris truncatis. (R.) (Species mihi cgnota.) Syn.—Cosmarium verrucosum, Battery, Amer. Journ. Sci. and Arts, 1846. Docidium verrucosum, Raurs, Brit. Desm. p. 218. Barney, Micr. Observ. p. 28. Hab.—Rhode Island; Bailey. “Segments, with numerous whorls of small prominences, which give the margins an undulated appearance, all the undulations are equal. ‘This is a very pretty species with a waved out- line, caused by the slight projections, which are arranged in numerous transverse rings,’ Bailey.” Ralfs. P. hirsutum, (Barter) D. spinuloso-hirsutum, medio valde constrictum, diametro 10-12 plo longius; cruribus et basi et apice subdilatatis, truncatis. (R.) (Species mihi ignota.) Syn.—Docidium hirsutum, Barty, Micr. Obsery. p. 36. Hab.—Florida; Bailey. “Segments many times longer than broad, slightly inflated at the base, surface hirsute. A . small species resembling D. Hhrenbergii in its form, but strongly hirsute on its outer sur- face.” Bailey. Genus TRIPLOCERAS, Batzey. Cellule singule, rect, valde elongate, processus magnorum seriebus transversis armate, utroque polo trilobatz, lobis acute bidentatis. . Syn.—Triploceras, BattEy, Microscopical Observations, p. 87, Smithsonian Contributions, 1850. Cells single, straight, very much elongate, armed with transverse series of large processes, trilo- bate at each end, lobes acutely bidentate. ae T. verticillatum, Batey. T. cellulis subeylindricis, sed utroque fine leviter angustatis et nonnihil fusiformibus, modice robustis, diametro 12-20 plo longioribus; processibus lateralibus robustis, magnis, apice emarginatis. Diam.—Cum process. 7145” = .00146”; sine process. ygiygq/ = - 00113”. Syn.—T. verticillatum, Battey. Microscopic Observations. Smithsonian Contributions, 1850. Docidium verticillatum, RAurs, British Desmids, p. 218. Pleurotenium verticillatum, RABENHOoRST, Flora Europ. Algar., Sect. III. p. 148. Hab.—Rhode Island, New Jersey, Georgia, Florida; Bailey. Saco Lake, (Dr. Lewis) Wood. Subcylindrical, but slightly narrowed at each end, and therefore somewhat fusiform, moderately robust, 12-20 times longer than broad; lateral processes large, robust, with their apices emarginate. 16 June, 1872. 2 FRESH-WATER ALGH OF THE UNITED STATES. T. gsracilie, BariLey. T. cellulis subeylindricis, utroque fine vix angustatis, gracillimis, diametro 25-30 plo longi: oribus ; processibus lateralibus brevibus, conicis. Diam.—Cum process. 7755" =.008"; sine proc. ys8oq = .0006”. Syn —T. gracille, Bartey, Smithsonian Contributions. Docidium pristide, Hopson, Magazine Natural History, v. p. 168. Pleurotenium gracile, RABENHORST, Flora Europ. Algar., Sect. III. p. 144, Hab.—In iisdem cum antecedente locis. ° Subcylindrical, scarcely narrowed at the ends, 25-30 times longer than broad ; lateral processes, short, conical. Genus SPIROTENIA, Brés. Cellule recte, cylindric vel subfusiformes, sepe in muco gelatinoso aggregate, medio haud con- strict, utroque polo rotundatie vel acuminate. Cytioplasma chlorophyllaceum in Jaminis spiralibus dispositum. Cells straight, cylindrical or subfusiform, often aggregated in a gelatinous mucus, not constricted in the middle, rounded or acuminate at each end. Chlorophyllous cytioplasm arranged in spiral lamina. Sp. bryophila, (Bris.) Rasennorst. Sp. mimina, bryophila; cellulis in gelatina matricali consociatis, oblongo-cylindricis, rectis vel subeurvatis, bis vel ter longioribus quam latis, utroque polo rotundatis ; lamina chlorophyl- lacea singula anfraectu 124. Diam.—zp5y" = . 00083” (0.00024”—. 00029". R.) Syn.—Spirotenia bryophila, (BREB.) Rasennorst, Flora Europ. Algarum, Sect. III. p. 146. Hab.—Prope Philadelphia ; Wood. “(S. muscicola (De Bary)) Frond cylindrical two to four times as long as broad, ends rounded; endochrome a single, broad, smoothly defined, widely wound spiral band, its revolutions very few (one or two).” (A.) Remarks.—I found this beautiful little desmid on the North Pennsylvania Rail- road, near Chelten Hills, growing amongst some mosses which were kept con- stantly wet by overhanging dripping rocks. It formed little transparent masses of almost colorless jelly looking much like drops of dew. It agrees well with the descriptions of the European form, except that there were generally from pp turns of the spiral, and the cells exceed somewhat the measurements of Prof. Rabenhorst. The cells are closely placed in the jelly. Fig. 10, pl. 12, represents some plants of this species. Sp. condensata, (Bréis.) Rasennorst. Sp. cellulis eylindraceis, reetis (vel leviter curvatis) octies vel decies longioribus quam Iatis, utroque polo rotundatis ; laminis chlorophyll. singulis, anfractibus subarctis (plerumque 8-12). Diam.—0.00075”. Syn.—Sp. condensata, (BREB.) Rapenuorst, Flora Europ. Algarum, Sect. III. p. 146. Hab.—Florida; Rhode Island; Bailey. Pennsylvania; Wood. Frond cylindrical, two to four times as long as broad, ends rounded; endochrome a single, broad, closely wound spiral band, its revolutions numerous. L. gg”. Br. ggg”. Archer. Pritchard’s Infusoria. | FRESH-WATER ALG# OF THE UNITED STATES. 123 Remarks.—The ouly specimens that I have seen of this species were found in a spring in the Philadelphia City Park, near Columbia bridge. Fig. 11, pl. 12, was drawn from one of these specimens. Genus SPH ROZOSMA, Corpa. Cellule compress, medio transversim profunde inciswz, itaque bilobate, in quoque lobo massa chlorophyllosa quadriradiata nugleum amylaceum involvente pradite, in filum planum teniiformem literaliter isthmis conjuncte. Zygosporx globose vel ovales, glabra. (R.) Syn.—Isthmosira, Kvz. Odontellz, spec., HARB. Isthmiz, spec., MENEG. Spondylosium, Bris. Cells compressed, transversely very deeply incised in the centre and therefore bilobate, furnished in each lobe with a quadriradiate mass of chlorophyl surrounding a starch grain, conjoined laterally by isthmuses in a teniform fascia. Remarks.—I have never found any species of this genus in America. Professor Bailey has, however, detected the following: — Sph. excavatuim, Ratrs. Sph. plerumque nudum (sine tubo mucoso) sph. vertebratum multo minus; cellulis diametro duplo-longioribus, medio excayvato-constrictis, a latere ellipticis utroque polo rotundatis; lobis brevibus truncato-rotundatis, levibus vel granulato-denticulatis ; isthmis binis parvis verru- ciformibus ; zygosporis plerumque ovatis. (R ) Latit. filor. 0.00047"”—0 00032”. (R.) Syn.—Sph. excavatum, Raurs, British Desmids, p. 67. Hab.—F lorida; Georgia; South Carolina; Rhode Island; Bailey. “ Joints longer than broad, having a deep sinus on both sides and two sessile glands at each margin at their junction, very minute, seldom more than twenty-five joints in the filament, which is fragile, and finally separates into single joints; at their junction, in the front view are two minute glands or processes, situated one near each angle, and nearly invisible before the escape of the endochrome. The joints are nearly twice as long as broad and much con- stricted in the middle; the constriction is like an excavation or broad sinus on each side, so that the margins of the filaments appear sinuated. The endochrome is pale bluish-green with minute scattered granules. The transverse view is oblong with four sessile glands, two on each side and situated near the ends.”—Ralfs’ Brit. Desm., p 67. Sph. pulchrum, Barcey. Sph. cellulis oblongo-quadrangularibus, diametro duplo-brevioribus, acute incisis, arcte con- nexis; lobis oblongis rectis, apice rotundatis; isthmis nullis, vagina mucosa ampla dis- neta. (R.) Syn.—S. pulchrum, Baitey. Ratrs, British Desmid., p. 209 (Cum icone). Hab.—West Point, New York; Princeton, New Jersey ; BAILey. “ Joints twice as broad as long, deeply incised on each side; junction margins straight, con- nected by short bands.” Remark.—* Prof. Bailey informs me that this species is twice as large as Sph. vertebratum,” RALFS. 124 FRESH-WATER ALG& OF THE UNITED STATES. Sph. serratum, Batcey. Sph. cellulis diametro duplo brevioribus, profunde et acute excisis, arcte conjunctis; lobis utrinque cuspidatis, paulum conniventibus ; isthmis nullis; vagina crassa. (R.) Syn.—Sph. serratum, BAILey, Micros. Observation. Smithsonian Contributions, 1850. Cum icone. Hab.—South Carolina; Georgia; Florida; Bailey. “ Joints broader than long, deeply notched or divided into two transverse portions with acute projecting ends, which give a serrated outline to the chain.” Bailey. Genus HYALOTHECA. Cellule breve, cylindrice, medio non profunde constrictz, a latere disciformes, in fila confervacea sine isthmis arcte conjuncte et vagina mucosa ampla achroa incluse. Massa chlorophyllosa in quaque semicellula 4-8, 5-10 radiata. Cells short, cylindrical, not profoundly constricted in the middle, disciform in the end view, closely united without intervening isthmuses into a confervoid filament, which is inclosed in an ample mucous sheath. Chlorophyl masses in each cell 4-8, 5-10 radiates. HA. disilliems, (Suirn) Brés. H. fasciis prelongis; cellulis oblongo-quadrangularibus, diametro sub-duplo brevioribus, inter- dum ante divisionem subequalibus, angulis nonnihil rotundatis, pleramque medio obsolete constrictis, sepe haud constrictis. Diam —0.00089//—0.00098/’.. (R.) Syn.—H. disilliens, (SmitH) Brés. Rapsennorst, Flora Europ. Algarum, Sect. III. p. 152. Hab.—South Carolina; Florida; Rhode Island; Battny. Rhode Island (8. T. Olney), Thwaites. Pennsylvania; Wood. Filament very long, cells oblong, quadrangular, about one-half as long as broad, sometimes before division as long as broad, angles somewhat rounded, mostly obsoletely constricted in the middle, often not constricted. Remarks.—Vhe specimens which I have identified as H. disilliens, agree with the various figures and descriptions of the European form, in every thing except that in many cases there is no constriction whatever in the centre of the cell, and when the constriction does exist, it is never so pronounced, as some of the descrip- tions indicated. The plant is very common about Philadelphia, growing in springs and ditches. Fig. 12, pl. 12, represents this part of a filament of this species. Hi. mucosa, (Merr.) Enrs. H. fasciis confervaceis, minus fragilibus; cellulis quadrangularibus, diametro squalibus vel subequalibus, medio non constrictis, ad utrumque finem (annuliformi-bicarinatis) bidentatis. (R.) Species mihi ignota. Diam.—0.00073"”—0.0008”". (R.) Syn.—Gloeoprium mucosum, Hassat, Fresh Water Algw, p. 346. H. mucosa, (Mert.) Enrs. Raxpenuorst, Flora Europ. Algarum, Sect. III. p. 132. Hab.—Rhode Island; (S. T. Olney) Thwaites, Filament scarcely fragile, mucous sheath very broad; joints about as broad as long, not con- stricted, but having at one of the ends a minute bidentate projection on each margin, the FRESH-WATER ALG& OF THE UNITED STATES. 125 adjoining ends of the next joint being similar, these projections being produced by an annular groovedrim. L. ys5”—600". B. goey—ad yr”. (Archer) Genus BAMBUSINA. Cellule oblongo-orculiformes, in filamenta articulata nodosa dense conjuncte, medio vitta trans- versa carinis duabus annuliformibus limitata instruct, itaque superne et inferne bidentate, fronte circulares, supra et infra dente unico prominente. (R.) Cells oblong-orculiform, densely united into an articulate nodose filament, surrounded by two median bands. B. Breébissonii, Kz. B. filamentis nodoso-articulatis ; cellulis diametro duplo longioribus. (R.) Diam.—0.00077’”—0.00092”. (R.) Syn.—B. Brébissonii, Kiirzinc. Rapennorst, Flora Europ., Algarum, Sect. IIT. p. 152. Hab.—South Carolina. (Ravenel) Wood. South Carolina; Georgia; Florida ; Rhode Island. Bailey. (Didymoprium Borreri, (Ralfs)) Joints inflated, barrel-shaped, longer than broad, without a thickened border at their junction; angles bicrenate, crenatures rounded; transverse view circular ; sporangium elliptic, formed within the (for some time) persistent extensions from the conjugating joints, which do not previously break up into single joints, but couple, still united in the filament, in a confused or zigzag manner, some of the joints remaining unchanged. L. 53°. B. yy's0"- ftemarks.—The specimens which I have seen agree well with the descriptions, except in regard to size; some of the cells which I measured were more than zels7 Of an inch in diameter, Genus DIDYMOPRIUM. Cellule oblongo-elliptice, modice compress, ancipites, angulis porrectis inciso-bidentatis, in fila- menta articulata biconvexa et torta sine isthmo arcte conjuncte, et in vagina mucosa incluse. Cytioplasma chlorophyllosa cellule a fronte cruciatim disposita, cujus erura e laminis duabus parie- talibus divergentibus granum amylaceum unicum involventibus formantur. Cells oblong-elliptical, moderately compressed, two-edged, with the produced angles incisely-biden- tate, closely united into a biconvex and twisted filament, which is inclosed in a mucoid sheath, cytio- plasm so placed as to be cruciate when viewed from the front (end), each erus composed of two parietal divergent lamina, each of which contains a single starch granule. D. Grevillii, Kz. D. cellulis oblongis diametro duplo brevioribus, saturate viridibus. (R.) Diam.0.0024/’—0.0031./7 (R.) Syn.—D. Grevillii, Kurzine. Rapennorst, Flora Europ. Algarum, Sect. III. p. 153. Hab.—Georgia, South Carolina; Bailey. Pennsylvania; Wood. Sheath distinct ; joints broader than long, with a thickened border at their junction; angles bidentate ; teeth angular; transverse view broadly elliptic. Sporangium orbicular, formed within one of the two conjugating joints, the endochrome passing over from one by a narrow connecting tube produced between the otherwise, but little altered, broken-up single joints. (A.) Remark.—Fig. 13, pl. 12, represents the end view of a broken filament of this species. 126 FRESH-WATER ALG#@ OF THE UNITED STATES. Genus DESMIDIUM. Cellule oblongo-tabulares, medio inciso-bilobw, lobis integris vel irregulariter dentatis, a fronte tri- vel quadrangulares, angulis obtuse rotundatis, in fila angulosa, prelonga, torta, fragiles arcte connexe, Massa chlorophyllosa (a cellule fronte visa) 3-4 radiata; quisque radius e laminis duabus lateralibus divergentibus compositus. Zygospore globose vel oblongx, glabra. Cells oblong-tabular, medianly incisely bilobate, with the lobes entire or irregularly dentate, ag seen from the front tri- or quadrangular, and having the angles obtusely rounded, closely conjoined into an angular, fragile, twisted filament. Chlorophyl (as seen from the front) 8-4 radiate; each radius composed of two lateral divergent lamina; zygospores globose or oblong, smooth. D. Swartzii, Ac. D. cellulis a fronte triangularibus, diametro 2-3 plo brevioribus. (R.) Diam.—0.00096”—0.00189". (R.) Syn.—D. Swartzii, AGARDH. RABENHORST, Flora Europ. Algarum, Sect. III. p. 154. Hab.—In aquis quietis, Atlantic States. Florida; Georgia; South Carolina; Rhode Island; Bailey. New York; Edwards. Pennsylvania; Wood. Filament triangular, equal, with a single longitudinal waved, dark line, formed by the third angle; joints in front view somewhat. quadrangular, broader than long, with two slightly angular crenatures on each lateral margin, united at the whole of their end margins by a thickened border, end view triangular; .endochrome three-rayed. Archer. Pritchard’s Infu- sorid. D. quadrangulatum, Krz. D. quadrangulare, cellulis oblongo-quadrangularibus, diametro 2-3 plo brevioribus, lobis denti- formibus obtusis, a fronte sinuato-quadrangularibus, angulis late rotundatis, lateralibus exca- vatis. (R.) Species mihi agnota. Diam.—0.0021"”—0.0029". Syn.—D. quadrangulatum, Kirzine. Rasennorst, Flora Europ. Algarum, Sect. III, p. 155. Filament quadrangular, varying in breadth from its twisting, having two longitudinal waved lines; joints in f. v. broader than long, with two somewhat rounded crenatures on each lateral margin, united by the whole of their end margins; e. vy. quadrangular; endochrome four ’ ” 1 Ar . —ql;”. (Archer) rayed. L.ys4¢’. B. eh, D. aptogonium, Brés. D. fasciis pleramque subbrevibus, nudis, perforatis; cellulis quadrangularibus, inciso-bilobis, lateralibus concavis, lobis crenatis, a fronte triangularibus (nonnunquam biangularibus), centro concavo, angulis rotundatis protensis isthmum brevissimum triplicem eflicientibus. (R.) Species mihi ignota. Diam.—0.00089"—0.00147”. (R.) Syn.—Aptogonium desnidium, Raurs, British Desmids. D. aptogonium, Brépisson. Rapennorst, Flora Europ. Algarum, Sect. III. p. 155. Hoab.—Georgia; South Carolina; Bailey. Joints in f. vy. quadrangular, broader than long, with two rounded erenatures on each Jateral margin, united at the outer portion only of each end margin by mutual projections, thus pro- ducing intervening central oval foramina. Archer. Genus APTOGONIUM, Raters. Y « . . . Ly Cellule 3-4 angulares vel compresse, non constrict ; margine laterali plane vel crenats, 1D fascias perforato-articulatas, angulares conjuncte. (R.) FRESH-WATER ALGA OF THE UNITED STATES. 127 Cells 3-4 angular or compressed, not constricted, their lateral margins plain or crenate, conjoined into angular perforately articulate fascia. A. Baileyi, Bars. “ Filament not crenated ; joints about equal in length and breadth. Syn.—Odontella? tridentata, Batney. In lit. cum icone (1846). Hab.—W orden’s Pond, Rhode Island; near Princeton, New Jersey, with sporangia,” Bailey. “ Filament triangular; joints excavated at their junction like those of Aptogonum desmidium- The joints are not bicrenate, hence the margins of the filament are entire, a character which distinguishes it from that species. The end view is triangular, with rounded angles.” RALrs, British Desmidiee, p. 208. Genus COSMARIUM, (Corba) Cellule oblong, oblongo-cylindrice, ellipticee, vel orbiculares, medio transverse plus minus con- strice, utroque polo obtuse vel rotundate et integrx, a vertice elliptice. Zygospore muricate vel Verrucose. Cells oblong cylindrical, elliptical or orbicular, more or less transversely constricted in the middle, obtuse or rounded, and entire at each end, viewed from the end elliptical. Zygospore warty or muricate. 1. Cellule sejunctz. 1. Cells separate. a. Cellule elliptice, vel subelliptice ; semicellule medio nonventricose. a. Cells elliptical or subelliptical ; semicells medianly not ventricose. * Cytiodermate granuloso vel verruculoso. * Cytioderm granular or warty. Cc. margaritiferum, (Turr.) Menau. C. paulo longius quam latum, profunde constrictum ; sinu amplo, vel modice angusto, interdum intra excavato; semicellulis semiorbicularibus, vel reniformibus vel nonnihil quadrangulis dorso plerumque late rotundatis ; eytiodermate verruculoso. Diam.—Max. 3257" = .0006” (0.00073"—0.0012”. R.) Syn.—LEuastrum margaritiferum, Ears. Baiey, Silliman’s Journal, 1841. Cosmarium margaritiferum (TURPIN), MENEGHINI. RaBENuoRST, Flora Europ. Algar., Sect. ILI. p. 157. Hab.—In aquis quietis, South Carolina; Florida; Mexico; Bailey. Pennsylvania, Wood. A little longer than broad, profoundly constricted ; sinus ample or moderately narrow, some- times widened on the inside; semicells semiorbicular, reniform or somewhat quadrangular; dorsum mostly broadly rounded ; cytioderm warty. Remarks.—I have found a form of this species growing in the vicinity of this city, which I at first was disposed to look upon as distinct, but which, in truth, grades into the typical form. In it the cells are almost quadrangular, often with their basal angles acute. The margin of the frond in C. margaritiferum, as it occurs with us, is sometimes distinctly serrate or, more correctly, crenulate from the presence of the granulations. The granules are larger than in C. botrytis, but smaller than in C. tetrophthalmum. When viewed laterally the semi-cells are roundish, or nearly so (according to Ralfs’ elliptical), and closely connected by 128 FRESH-WATER ALG# OF THE UNITED STATES. a very broad neck. I have never seen the sporangia, but, according to Mr. Ralfs, they are orbicular and inclosed in a granulated cell. Fig. 8, pl. 21, represents half of an empty frond of this species magnified 750 diameters; and fig. 21, pl. xii., a frond densely filled with living endochrome C. Botrytis, (Bory) Mrnau. ©. late ovale, profunde constrictum, diametro plerumque 1j—2 plo longius; sinu angusto, lineare; semicellulis nonnihil triangularibus, apice interdum truncatis, interdum late rotun- datis ; cytiodermate minute granulato. Diam.—z},"" = 0.0019" (0.0014”—0.0023"). (R.) Syn.—C. Botrytis, (Bory) MENEGHENI. Rasenuorst, Flora Europ. Algarum, Sect. III. p. 158. ‘ Hab.—Pennsylvania, Wood. C. broadly oval, profoundly constricted, 14-2 times longer than broad; sinus narrow, linear; semicells somewhat triangular, with the apex sometimes truncate, sometimes broadly rounded; cytioderm minutely granulate. Remarks.—In this species the semicells, as viewed transversely, are broadly elliptic in outline. ‘The end view presents a longer narrower ellipse. ‘Their out- line, when seen from the front, varies remarkably from that of a very broad semi- oval to distinctly triangular with a truncate apex. The granules are small and arranged regularly, sometimes they are very obscure. I have often seen the endo- chrome so arranged as to leave a large pyriform central vacucle in each cell, com- municating with the narrow margin between it and the cell-wall. This vacuole was apparently filled with a transparent fluid, in which were minute granules in immense numbers, in constant active motion circling among one another and pass- ing out, into and along the marginal connecting space. According to Ralfs, the sporangia of this species are large (;},’’), with branched spines. Fig. 5, pl. 21, represents an empty frond of this species; 5 a, outlines of semi- cells to show the variations, and fig. 14, pl. 12, represents a frond crowded with endochrome, magnified 460 diameters. »» OVale, Ratrs. C. magnum, ovale, compressum, profunde constrictum, diametro subduplo longius, ambitu inte- gerrimum vel crenatum, a vertice late ellipticum; semicellulis basi paulo latioribus quam longis, triangulo-rotundatis, disco punctatis, margine verrucis margaritaceis achrois hyalinis in series 4 ordinatis. (R.) Diam.—Long. 0.0053’—0.0067". Lat. plerumque 0.0041”. (R.) Syn.—C. ovale, Raxrs, British Desmidiew, p. 98. Hab.—South Carolina; Rhode Island; Bailey. Cobble Mountain, Pa. (Lewis) Wood. Frond very large, elliptic, nearly twice as long as broad, constriction very deep, linear; seg- ments somewhat broader than long, somewhat triangular, rounded at ends, rough near the margin, with a band of large pearly granules, producing a dentate appearance, the dise punc- tate; e. v. elliptic. (A.) . C. Brebissonii, Mevzan. C. paulo longius quam latum; semicellulis semicircularibus, diametro paulo longioribus, angulis inferioribus obtusis approximatis, ventre modice concavis subplanis, dorso latissime rotun- datis ; cytiodermate muricato, muricibus conicis in ordinibus regularibus collocatis. (R.) FRESH-WATER ALGA OF THE UNIDED STATES. 129 Diam.—Semicell, 0.0019”—0.0022”. (R.) Syn.—C. Brébissonit, MENEGHENI. RABENHORST, Flora Europ. Algarum, Sect. III. p. 158 Hab.—W hite Mountains, New Hampshire, (Dr. F. W. Lewis) Wood. Frond somewhat longer than broad, constriction deep, linear; segments semiorbicular, rough all over, with somewhat elongate conical scattered pearly granules; e. v. elliptic. (A ) Remark.—Fig. 6 ne 21 represents an empt frond of this Ss pecies magnified 5 2 i ? b) 5 750 diameters. Cc. suborbiculare, Woop. C. parvum, suborbiculare, paulo longius quam latum, cum margine enormiter crenato vel crenato-undulato; semicellulis a latere orbicularibus, a vertice ellipticis; sinu extrorsum angustissimo sed introrsum nonnihil execavato; cytiodermate crasso, sparse verruculoso ; granulis in semicellulis singulis subdistantibus et in seriebus elongatis, duabus (interdum unica) externis curvatis, et in seriebus duabus internis brevibus et rectis. Diam.—Lat. yy'55"” = .0012"; lat. 4855” = 0013”. Syn.—C. orbiculare, Woon, Proceed. Acad. Nat. Se. 1870. Hab.—In lacu “Saco,” New Hampshire, (Lewis.) Small, suborbicular, a very little longer than broad, with the margin irregularly crenate, or crenate undulate ; semicells from the side orbicular, from the vertex elliptical; sinus very narrow, but within somewhat excavated; cytioderm thick, sparsely coarsely granulated ; granules subdistant, in each cell arranged in one or two curved marginal series and in a cen- tral group of two or three short rows. Renarks.—The arrangement of the granules in this desmid is peculiar, one, or sometimes two rows of large obtuse pearly granules are placed at rather wide in- tervals along the whole outer margin, and then in the centre of each semicell is a group of two or three, or even more short straight rows of three or four similar but rather smaller granules. ‘The isthmus is rather broad and short; sometimes it las on it one or two granules. Fig. 9, pl. 21, represents an empty frond of this species, magnified 750 diame- ters; 9a, the outline of the end view of the same. C. tetrophthaimum, (K1z.) Bris C. tertiam partem circa longins quam latum, profunde constrictum; sinu angusto, plerumque sublineare ; ambitu obtuse crenato; semicellulis nonnihil semicircularibus, ventre subplanis, dorso rotundatis; cytiodermate verruculoso; verruculis magnis, obtusis, subordinatim dispo- sitis. Diam.— 3°55 = .0025". Syn.—C. tetrophthalmum, (K61z1ne@), BréBisson. RABENHORST, Flora Europ. Algarum, Sect. TMT p: 159: Hab.—New Jersey ; Wood. About one-third longer than broad, deeply constricted ; sinus narrow, mostly sublinear; margin obtusely crenated; semicells somewhat semicircular, belly nearly even, dorsum rounded ; cytioderm warty ; prominences large, obtuse, arranged somewhat regularly. Remari:s.—The.only specimens I have seen, and I believe the only ones hitherto 17. ‘July, 1872. 130 FRESH-WATER ALG OF THE UNITED STATES, found on the continent, were collected by myself in * Shepherd’s Mill Pond,” near Bridgeton, Cumberland County, New Jersey. Fig. 7 a, pl. 21, represents the outline of a frond magnified 460 diameters. C. ameenui, Brés. C. mediocre, oblongum eylindricum, leviter compressum, diametro duplo fere triplove longins, utroque polo rotundatum, medio profunde constrictum, sinu angusto, lineari, ambitu granulis margaritaceis achrois obsessum, a vertice ellipticum; semicellulis oblongo-rotundatis, dorso alte convexis, lateribus vero rectis parallelis, angulis inferioribus rectis et subacutis ; eytio- dermate granuloso-verrucoso, verrucis hyalinis in series regulares dispositis. (R.) Species mihi ignota. Long. 0.0017’—0.1016"; lat. 0.00087”. (R.) Syn —C. ameenum, Brésisson. Rasennorst, Flora Europ. Algar., Sect. III. p. 159. Hab.—Florida; Bailey. Rhode Island (S. T. Olney) ; Thwaites. Frond twice as long as broad, sides parallel, ends rounded, constriction deep, linear; segments rough with crowded obtuse papilla-like pearly granules ; s. v. much compressed, about thrice as long as broad; e. v. elliptic. (A.) ** Cyltiodermale glabro. ** Cytioderm smooth. C. Cucumis, Cora. ; C. ovale ellipticum, utroque polo late rotundatum, tertiam partem vel duplo longius quam latum, profunde constrictum; sinu lineari ; semicellulis angulis inferioribus rotundatis, cytiodermate glabro, haud punctato. Diam.—Max. long. 345” = 0.0026"; lat. 713,” = .0019”. Syn.—C. Cucumis, Corpa. Rasennorst, Flora Europ. Algarum, Sect. III. p. 161. Hab.—South Carolina; Georgia; Florida; Bailey. Pennsylvania ; Wood. Saco Lake, (Lewis.) Oval or elliptic, at each end broadly rounded, one-third to twice as long as broad, profoundly constricted ; sinus linear; semicells with their inferior angles rounded; eytioderm smooth, not punctate. Remarks.—TVhis species is very abundant around Philadelphia. The semicells generally each contain two large globular masses placed near the median line, which are sometimes hidden by the crowded endochrome. Figs. 15, 15 a, pl. 12, represent this species with their endochrome in different conditions ; 15 6, represents a monstrous frond, which had attempted to divide. but had not succeeded in so doing. Cc. depressum, Bairey. “Elliptical, binate, division in the plane of the longest axis. Segments entire, nearly twice as long as broad, rounded above, very much flattened at base. : Hab.—Wakes in Florida. sie IES Tad “= Al ae A This species resembles CO. bioculatum, Bris. ; but the segments are much closer together, and are angular, not rounded at the basal extremities.” BAILEY. Microscopical Observations. Smithsonian Contributions. C. pyramidatuan, Bris. C. mediocre, ovale vel subovale, utroque polo truncatum, medio profunde constrictum, duplo FRESH-WATER ALG# OF THE UNITED STATES. 131 fere longius quam Jatum; semicellulis breviter pyramidatis, angulis inferioribus rotundatis, apice (dorso) modo truncatis modo rotundatis, a vertice late ellipticis; cytiodermate pune- tato vel subtilissime granulato. (R.) Long. 0.0021”—0.0037". Lat. max. 0.0026”. Syn.— C. pyramidatum, Bresisson. Rasennorst, Flora Europ. Algarum, Sect. III. p. 162. Hab.—Georgia; Florida; Bailey. Pennsylvania; Wood. Frond scarcely twice as long as broad, suboval ; constriction deep, linear; segments pyramidal, rounded at basal angles, somewhat truncate at the ends, punctate ; e. v. broadly elliptic. (A.) Remark.—Fig. 14, pl. 13, is a drawing of this species. C. bioculatum, Brés. C. parviter, circiter tam longum quam Jatum vel paulo Jongius, profunde constrictum, sinu ex- trorsum ampliato ; semicellulis diametro duplo latioribus, elliptico-prope hexagonis angulis obtuse rotundatis, integerrimis aut levissime crenulatis; ecytiodermate levi vel subtilissime punctato. (R.) Species mihi ignota. Long. 0.00069”. Lat. 0.00066”. (R.) Syn.—C. bioculatum, Br&Bisson. RapBennorst, Flora Europ. Algarum, Sect. ITT. p. 163. Hab.—Rhode Island, (S. T. Olney) Thwaites. Frond minute, about as long as broad, constriction deep, producing a gaping notch at each side ; segments about twice as broad as long, elliptic, smooth; s. v. compressed s. v. elliptic. Sporangium orbicular with conical spines. L. ygy¢" 5 B. tye”. (A-) C. Meneghenii, Bres. C. parvum, tam longum quam latum, modo paulo-longius, modo paulo-brevius, profunde con- strictum, sinu lineari, extrorsum non ampliato; semicellulis subquadratis, leviter sinuato- hexagonis; angulis rotundatis, eytiodermate levi vel subtillissime punctato. (R. 5 7 ang , Cy Long. J7’’—-,'5"” = 0.00103’”—0.0013”; lat. ~45”’—y4y"” = 0.00081”—0.00089". (R.) Syn.—C. Meneghenii, Brézisson. Rapennorst, Flora Europ. Algar., Sect. III. p. 163. Hab.—Pennsylvania ; Wood. Frond very minute, rather longer than broad, constriction linear; segments subquadrate, bicre- nate at the sides and ends, smooth; e. v. elliptic. (A.) Remark.—Fig. 18, pl. 12, represents a frond of this species, magnified 750 diameters. C. crenatum, Ratrs. C. oblongum, tertiam partem circa longius quam latum, profunde constrictum, sinu lineari an- gusto; semicellulis e basi lata subsemicircularibus, dorso plus minus depressis vel truncatis, ambitu crenatis vel regulariter undulato erenatis, erenis 10-14; cytiodermate punctato. (R.) Species mihi ignota. Long. 0.0021”—0.0023” ; lat. 0.0015”. (R.) Syn.—C. crenatum, Raxrs, British Desmidiex, p. 96. Hab.—Rhode Island; (S. T. Olney) Thwaites. Frond slightly longer than broad, constriction linear; segments semiorbicular, ends and sides broadly rounded, crenate or minutely undulate at margin; e. v. elliptic. Sporangium orbi- F F oan ; vanes 1 cular, spinous; spines elongate, slender, swollen at the base and divided at the apex. L. 339"; 1 ” B. B71 . 132 FRESH-WATER ALG OF THE UNITED STATES. vow Cc. undulatum, Corpa. C. submediocre, oblongum, diametro subduplo longius, utroque polo late rotundatum, ambitu leviter sinuato-undulatum, profunde constrictum, sinu lineari extrorsum paullum ampliato ; semicellulis semiorbicularibus, et dorso et lateribus late rotundatis, margine undulato-crenatis, crenis 9, sublatis ; eytiodermate levi; zygosporis sphericis spinis elongatis, apice bi-tri-fidis obsitis. (R.) Species mihi ignota. Long. 0.0024”. Lat. 0.0017”. (R.) Syn.—C. undulatum, Corva. Rasenuorst, Flora Europ. Algarum, Sect. III. p. 165. Hab.—South Carolina; Rhode Island; Bailey. Frond rather larger than that of C. crenatum, slightly longer than broad, constrictions linear ; segments semiorbicular, ends and sides broadly rounded, crenate or minutely undulate at the margin; e. y. elliptic. Sporangium orbicular, spinous; spines elongate, slender, swollen at the base and divided at the apex. (A.) Semicellulz medio-ventricoso inflate. Semicells medianly ventricose. Cytiodermate levi. a Cytioderm smooth. ¢. sublobatum, (Brés.) Arcuer. C. parvum, oblongo subquadratum, diametro subduplo longius, sinu angusto lineari; semicel- lulis subquadratis, e basi dilatata ad verticem sensim angustatis, angulis et inferioribus et superioribus rotundatis, dorso late truncatis lateribusque leviter. sinuatis; cytiodermate leevissimo. (R.) Long. 0.00179”—0.00196". Lat. max. 0.0015’”—0.00157”. (R.) Syn.—C. sublobatum, (Bristsson) ArcuEr. Pritchard’s Infusoria, p. 731. Hab.—Georgia; Florida; Rhode Island; Bailey. Frond scarcely twice as long as broad, oblong; constriction linear, segments subquadrate, somewhat wider at the base, lateral and end margins slightly concave, smooth and trans- verse vein cruciform, (A.) * * Cytiodermale granulato. * * Cytioderm granulate. ¢. ornatum, Raters. C. parvum, plerumque tam longum quam Jatum; semicellulis reniformibus, diametro duplo longioribus, angulis inferioribus una cum lateribus rotundatis; dorso sub-producto late trun- catis; cytiodermate granulato-verruculoso ; zygosporis longe spinosis, spinis elongatis apice furcatis. (R.) Species mihi ignota. Long. 0.0016”—0.0015". Lat. 0.0016”. (R.) Syn.—C. ornatum, Raurs, British Desmidies, p- 104. Hab.—Rhode Island; (S. T. Olney) Thwaites. Frond in f. v. about as long as broad, constriction deep, linear; segments semiorbicular or subreniform, with a central truncate projection at the ends produced by the continuation of a central inflation, rough towards the margin and on the inflation with pearly granules; e. v. with a rounded lobe on each side. Sporangium orbicular, spinous; spines elongated, dilated at the base and slightly divided at the extremity, g13”. (Archer.) €. commissurale, (Brés.) C. minutum, fere dimidio latins quam longum, profundissime constrictum, sinu amplo basi ex-” cavato; semicellulis anguste reniformibus, diametro paene triplo longioribus, leviter ineurvis, FRESH-WATER ALG OF THE UNITED STATES. 133 t angulis rotundatis, dorso truncato-rotundatis, margine crenulato-dentatis, a dorso oblongis, medio ventricosis, utroque polo plus minus tumidis; cytiodermate granulato margaritifero. (R.) Long. 0.0010”—0.0012.” Lat. 0.0013”—0.0015”. (R.) Syn.—C. commissurale, BREBIssoN. Rasennorst, Flora Europ. Algarum, Sect. III. p. 170. Hab.—In lacu. White Mountains, New Hampshire; (Dr. F. W. Lewis) Frond small, in f. v. one-third broader than long; constriction very deep, rounded ; segments narrow-reniform, with a central, somewhat truncate projection, produced by the continuation of the central inflation, rough on the inflation and on the extremities, with somewhat large pearly granules, e. y. three times longer than broad, constricted between the central inflation and the rounded extremities. Sporangium as in C. ornatum. (A.) Remarks.—I have seen but a single specimen of this species which differed from the typical form, in having the sinus very narrow in its outer portion, and in being shorter. Fig. 16, pl. 13, represents the frond of this specimen, magnified 750 diameters. C. cxelatum, Racrs. C. suborbiculare, profunde constrictum; sinu angustissimo lineari; semicellulis inciso-crenatis, angulis rotundatis, a vertice medio nonnihil inflatis; cytiodermate granulato, granulis in series regulariter circulares positis. Diam.—Long. 4355" =.0017". Lat. pun” = .0014”. Syn.—C. cxlatum, Raurs, British Desmidiex, p. 103. Hab.—In stagnis prope Aiken, South Carolina. (Ravenel.) Suborbicular, profoundly constricted; sinus very narrow, linear; semicells inciso-crenate, angles rounded, when seen from the end somewhat inflated in the middle; cytioderm granu- late, granules placed in circular series. Remarks.—This species was collected by Prof. Ravenel in a quiet ditch near Aiken, South Carolina, sparsely scattered amidst innumerable diatoms and desmids, The number of the crenations appears to vary. In the few individuals I have seen there were six end ones besides the two very broad basal ones, if the latter can be called crenatures. Ralfs gives six as the total number, and yet every one of his figures has many more. So I think the number a character of but little import- ance. The circular arrangement of the granules is not so positive and regular in the specimens I have seen, as is represented in the figure of Ralfs, otherwise the agreement is perfect. C. Broomei, Tuwaires. C. subparvum, plerumque tam longum quam latum, nonnunquam paulo longius, obtuse quad- rangulare, profunde constrictum, sinu angustissimo lineari; semicellulis oblongo-quadrangulis, diametro duplo longioribus, angulis et inferioribus et superioribus obtuse rotundatis, ventre subplanis, dorso latissime truncatis et sepius leniter retusis vel plane convexis; cytioder- mate granulato-margaritaceo, granulis in seriebus subrectis collocatis. (R.) Long. 0.00194”—0.0022”. Lat. max. .002”, thick .0015”. Syn.—C. Broomei, Tuwattes. Rats, British Desmidiew, p. 103. Hab.—Georgia; Bailey. Prope Philadelphia ; Wood. 134 FRESH-WATER ALG& OF THE UNITED STATES. Frond in f. v. about as long as broad, constriction deep, linear; segments quadrilateral, ends straight, angles rounded, rough all over with minute granules; e. v. twice as long as broad, slizhtly inflated at the middle and rounded at the ends. Sporangium orbicular, smooth. (A.) Remarks.—The only specimens which I have seen were found in a brick-pond below the city in the month of June. They agree well with the descriptions, ex- cepting in that I should describe their central inflation as pronounced. The sinuses also are ampliate or hollowed out within. The granulations are quite large, and are arranged somewhat irregularly in rows. Fig. 15, pl. 13, is a view of the front of the frond magnified 460 diameters; fiz, 10, pl. 21, the outline of the lateral view. ce. Cellule fusiformes, cylindrice vel ovales, in medio leviter constricte. ec. Cells fusiform, cylindrical, or oval, lightly constricted in the middle. C. Thwaitesii, Kars. C. mediocre, diametro bi-triplo longius, fusiformi-cylindraceum, medio leviter constrictum, am- bitu integerrimum, utroque polo rotundatum; semicellulis e cylindraceo subconicis, e medio in apicem sensim sensimque (sed modice) attenuatum; cytiodermate levi vel indistinete pune- tato. (R.) Species mihi ignota. Long. 0.00267’—0.00287". Lat. max. 0.0012”. (R. fo) Syn.—C. Thwaitesii, RAuEs, British Desmidiex, p. 109. Hab.—Florida ; Bailey. Frond in f. v. two or three times longer than bread; constriction a very shallow groove; seg- ments subeylindrical, with younded ends ; endochrome scattered ; e. v. circular, or very slightly compressed ; e. f. not punctate, or puncta very indistinct. (A.) C. connatum, Brés. C. validum, submagnum, leviter compressum, diametro duplo circa longius, subpanduriforme, plus minus constrictum, utroque polo late rotundatum, a vertice lato ellipticum; semicellulis subhemisphericis, ambitu wquabiliter rotundatis, integerrimis ; cytiodermate punctato. (R.) Species mihi ignota. Long. 0.0035”. Lat. max. 0.00165”—0.0019". (R.) Syn.—C. connatum, BREBISSON. Ratrs, British Desmidies, p. 108. Hab.—F lorida; Bailey. Frond large, in f. v. about one-half longer than broad; constriction shallow; segments about two-thirds of a circle, coarsely punctate, and with a distinct, sometimes striated, border; e. v. circular. (A_) d. Cellule in familias connere. d. Cells united into families. C. Quimbyii, Woop. (sp. nov.) ns aye) ig © IG ~ Ly Bye, . . . . “Te . . * C. cellulis parvis, sub-ellipticis, medio profunde constrietis, in familias copulis hyalinis con- nexis ; semicellulis a fronte ellipticis et diametro subduplo longioribus, a vertice ellipticis, a latere rotundatis; sinu lato; marsis chloro-phyllaceis in quaque semicellula singulis; eytio- dermate tenue, glabro. 9 eal) Pyaton Wi Diam.—Long. 7755" = 0.001". Lat. a fronte zono = 0.00075"; a latere ,yo5q = 0.00042. Hab.—In aquis puris, New Jersey. FRESH-WATER ALG# OF THE UNITED STATES. 135 Cells small, subelliptical, profoundly constricted in the middle, joined by translucent bands into families; semicélls seen from the front elliptical, and nearly twice as long as broad, from the vertex elliptical, from the side roundish; sinus broad ; chlorophyl masses single in each cell ; cytioderm thin, smooth. Remarks.—This plant was found by my friend Mr. Quimby growing in a beau- 'tiful spring above Camden, upon whose bottom it formed a gelatinous, trans- lucent, greenish mass. ‘The cells resemble in shape those of C. cucwmis, although much smaller. ‘They are joined by bands into little families, in which the original | parent-cell is generally very distinct, it, or rather the two cells into which it first divides, remaining in the centre of the group. The bands are so hyaline that their edges can alone be distinctly seen, and hence the latter often look as though they were threads—there appearing to be two parallel threads, or two threads crossing one another, or a single thread, according as the band is flat, twisted, or on edge. It gives me great pleasure to dedicate the species to my friend Mr. Quimby, by _whom it was collected. Fig. 9, pl. 1, represents one of the family groups of this plant. Genus EUASTRUM, Enurs. Cellule vel oblong vel elliptice, medio profunde incise, symmetrice sinuate, vel lobate, tumori- bus inflatis circularibus (rare obsoletis) instructwz, utroque polo sinuato-emarginate vel inciso-bilo- bate, a vertice elliptice. Cells either oblong or elliptic, profoundly incised in the middle, symmetrically sinuate or lobed, _ provided with circular inflated protuberances (which are rarely absent), at each end sinuately emar- ' ginate or incisely-bilobate, from the vertex elliptic. A, Lobo polares in apice late sinuato-excisi. A. Polar lobe with its apex broadly sinuately excised. £. multilobatuma, Woop. E. magnum, fere duplo longius quam latum, medio profunde constrictum, et cum sinu modice amplo; a latere medio ventricosum et duplo biumbonatum, ad verticem dilatatum et emar- ginatum ; semicellulis a fronte trilobatis, lobis sinus amplissimis inter se sejunctis ; lobi basale distincte late emarginato, lobo centrale obtuso, lobo polare late leviter sinuato-emarginato ; semicellulis a vertice quinque lobulatis; cytiodermate levi. Diam.—Long. y859" = .00475". Lat. +859” = .0025”. Syn.—EL. multilobatum, Woop, Proc. A. N. 8., 1869. Hab.—In lacu “Saco;” New Hampshire; (Lewis.) E. large, about twice as long as broad, in the centre profoundly constricted, with the sinus moderately large; from the lateral view somewhat enlarged and doubly biumbonate in the middle; semicells from the front trilobate, the lobes separated by very wide sinuses, the basal lobe broadly emarginate, the central lobe obtuse, the end lobe broadly and shallowly sinuately emarginate; semicells from the vertex five-lobed; cytioderm smooth. Remarks.—The basal lobes of this beautiful desmid are distinctly five lobulate, the lateral lobules being longer and broader than the others, which, instead of | being emarginate, are obtuse. The sinuses, separating lobes and lobules, are very broad, with very obtuse angles. When the desmid is viewed from two-thirds round, so as to show the anterior and posterior lobules especially, it presents an 136 FRESH-WATER ALG# OF THE UNITED STATES. outline in which all the sinuses are of similar form, and the central and basal lobes are about equal size; whereas, when viewed from the front, the basal lobe is much the broader. When the desmid is viewed from the side it is seen to be enlarged in the centre, and provided with two distinct umbonations each side of the com- paratively narrow central sinus. Fig. 10, pl. 12, represents the front view of a frond of this plant; fig. 5, pl. 20, the outline of a two-thirds view, and fig. 5 a, the outline of a lateral view, all mag- nified 450 diameters. E. verrucosum, Eure. E. magnum, late ovatum, vix longius quam latum, medio profunde constrictum, sinu extrorsum dilatato; semicellulis trilobatis, lobis triangularibus, divergentibus, apice late et profunde sinuatis; a latere ovato-oblongum, sinuato-lobatum, lobis octo in apice rotundatis, polaribus singulis porrectis, lateralibus ternis; cytiodermate granulato-verrucoso. (R.) Species mihi ignota. Long. 0.0036”—0039". (R.) Syn.—E. verrucosum, EuRENBERG. RABENHORST, Flora Europ. Algarum, Sect. III. p. 179. Hab.—South Carolina; Georgia; Florida; Rhode Island; Bailey. Frond somewhat longer than broad, rough all over with conic granules; segments 3-lobed, somewhat divergent, all the lobes broad, cuneate, with a very broad, shallow, or external sinus. Empty frond; f. v. segments with one large circular basal inflation on surface, one smaller on each side, and two others on the end lobe; s. v. segments inflated at: the base, narrowed into a short neck, end dilated with a central sinus; e. v. oblong, with three infla- tions at each side, one at each end, end lobe having 4 divergent lobelets. (A.) E. gem:matum, Brés. E. mediocre, diametro duplo longius, profunde constrictum, sinu angusto lineari, a vertice ovato- oblongum, ambitu sinuato-lobatum, lobis 8 conformibus, rotundatis; semicellulis trilobatis, basi tumoribus 3 in seriem dispositis, lobis in apice profunde emarginatis, lobulis rotundatis, lobo polari dilatato et paulum producto; cytiodermate in tumoribus et lobulis granulato- punctato, ceterum levi. (R.) Species mihi ignota. Long. 0.00224”—0.0029". Lat. 0.00157’—0.0017”. (R.) Syn.—E. gemmatum, BreBIsson. RaBennorst, Flora Europ. Algar., Sect. III. p. 180. Hab.—Rhode Island ; Bailey. Frond scarcely twice as long as broad; segments 3-lobed, lateral lobes horizontal, deeply emar- ginate, the protuberances minutely granulate; end lobe dilated, its dilatations inclined upwards, and minutely granulate; ends with a deep rounded emargination. Empty frond slightly punctate ; f. v. segments with three granulate inflations near the base ; tr. v. broadly elliptic, with three granulate inflations at each side and one at each end; e. v. end lobe cruci- form, lobelets rounded, granulate. (A.) E. oblongum, (Grey.) Rares. E. magnum, diametro duplo triplove longius, oblongum, profunde constrictum, sinu angusto, a latere oblongo-lanceolatum, utroque polo truncato leniter retusum, ambitu undulato- sinnatum ; semicellulis (fronte) sinuato-quinquelobis, basi et in quoque lobo tumore instructis, lobis Jateralibus in apice dilatato sinuato-retusis, inferioribus latioribus quam superior., lobo polari late cuneato in apice profunde inciso, angulis omnibus rotundatis, eytiodermate levi; zy gosporis globosis verrucosis, verrucis obtuse conicis achrois hyalinis. (R.) Species mihi agnota. FRESH-WATER ALG# OF THE UNITED STATES. 137 Long. 0.0057”—0.0065". Lat. max. 0.00346”. Syn.—tL. oblongum, (GREVILLE) Raters? British Desmidiex, p. 80. Hab.—Khode Island; Bailey. Frond rather more than twice as long as broad, smooth, oblong; segments 5-lobed; lobes nearly equal, cuneate ; lateral lobes, or the basal only, with a broad, shallow, marginal concavity, all their angles rounded, terminal notch linear. : Empty frond; f. v. seg. punctate, with three large inflations, on surface near the base, two others above and two on terminal lobe; tr. v. three times as long as broad, with three sub- distant marginal inflations at each side, and one at each end, in 8, broader in proportion, more elliptic, and inflations close; e. v. end lobe notched at opposite external margins. Sporangium orbicular, beset with numerous conical tubercles. (A.) B. Lobi polares evidenter discreti et in apice anguste incisi. B. End lobes evidently separated and narrowly incised in the centre. E. crassum, (Brés.) Krz. E. oblongum, diametro subtriplo longius, profunde constrictum, sinu angusto lineari, e vertice subquadrangulare, utroque polo profunde excisum, angulis rotundatis; semicellulis ({ronte) trilobis, basi et in quoque angulo tumore instructis, lobis lateralibus latissimis unisinuatis, lobo polari paullum prominente, in apice bifido, segmentis late rotundatis; cytiodermate dis- tincte punctato, punctis in series transversas ordinatis. (R.) Long. 0.0051”—0 0073”. Lat. max. 0.0041”. (R.) Syn.—EH. erassum, (BRreBIsson) Kiirzine Rasennorst, Flora Europ. Algarum, Sect. III. p- 181. Hab.—United States. Frond about twice as long as broad, subquadrilateral, smooth; segments 3-lobed; basal lobes very broad, with a very broad, shallow marginal sinus, in which there is sometimes a slight intermediate rounded projection; end lobe creneate, rounded, terminal notch linear. Empty frond; f. vy. punctate, segments with three inflations below and two above; tr. yv. two or three times longer than broad, with three lobes or inflations at each side and one at each end; e. v. end lobe sinuate at opposite external margins. (A.) E. ormatuma, Woop. E. oblongum, diametro duplo longius, profunde constrictum, sinu angusto lineari; semicellulis a fronte trilobatis; lobis basalibus latissimis, nonnihil sinuato-emarginatis, angulis plus minus productis et rotundatis; lobo polari medio profunde lineare inciso, segmentis late rotundatis ; semicellulis a latere bilobatis, lobis basalibus profunde emarginatis et cum angulis plus minus acutis ; cytiodermate distincte ordimatim punctato. Diam.—335;5" = .00029”. Syn.—E. ornatum, Woop, Proc. A. N. 8., 1869. Hab.—Saco Lake; New Hampshire. Lewis. E. oblong, twice as long as broad, profoundly constricted; semicells from the front trilobate ; basal lobe very broad, slightly sinuately-emarginate, angles more or less produced and rounded ; polar lobe medianly profoundly linearly incised, segments broadly rounded ; semi- cells bilobate at the sides, basal lobes profoundly emarginate and with the angles more or less acute ; cytioderm distinctly regularly punctate. Remarks.—This species is close to E. crassum, from which it differs in the pro- portionate length, being only twice instead of three times as long as broad; in the size being only three-fourths as large; and especially in the peculiar lateral split- ting, as it were, of the basal lobes. 18 July, 1872. 138 FRESH-WATER ALG#& OF THE UNITED STATES. Fig. 12, pl. 21, represents the front view of an empty half frond of this species, er Bat Be A 5 7 eae magnified 450 diameters; fig. 12 4, the side view of an empty frond. E. afflime, aces. E. E. humerosum affine, paulo minus; semicellulz quinquelobe ; lobi basales quales in EH. hume- rosum sed tumores quatuor in seriem transversam simplicem dispositi, lobi intermedii valde abbreviati eorumque basi tumoribus duobus instrueti, lobus polaris magis porrectus et in apice minus dilatatus ; cytioderma subtilissime punctatum subleve. (R.) Long. 0.0038”—0.0041". (R.) Syn.—E. affine, Raues, British Desmidee, p. 82. Hab.—South Carolina; Georgia; Bailey. Frond about twice as long as broad; segments 3-lobed; basal lobes slightly emarginate, having intermediate between them and the end lobe on each side a tubercle representing middle lobes, the upper margin of which is horizontal; end lobe exserted, dilated, its notch linear. Empty frond; f. v. minutely punctate; the segments with four basal inflations, two above and two on end lobe; tr. v. elliptic, with four inflations on each side and one at each end; e. y. end lobe emarginate at opposite; e.v. end lobe emarginate at opposite external margins, producing four shallow lobulets. (A.) E. Didelta, (Turpin) Rares. E. robustum, diametro duplo longius etiam supra, in sectione transversa ellipticum, ambitu un- dulato-crenatum, in utroque latere crenis quaternis ; semicellulis pyramidalibus, quinquelobis, tumoribus 9 in series tres alternantibus ordinatis, lobis inferioribus oblique truncato-rotundatis nonnunquam leniter retusis, intermediis subadscendentibus, rotundatis, lodo polari minus dilatato, bifido, segmentis rotundato-truncatis, conniventibus, in apice tumidis ; cytiodermate distincte punctato, punctis modo irregulariter sparsis modo in seriebus rectis collocatis. (R.) Long. 0.0055”. Lat. 0.00279”. Syn.—E. Didelta, (Turpin) Ratrs, British Desmidee, p. 84. Hab.—South Carolina; Georgia; Rhode Island; Bailey. Pennsylvania; Wood. Frond rather more than twice as long as broad; segments pyramidal, inflated at the base and again at the middle, end scarcely dilated, rounded, its notch linear. Empty frond punctate; f. v. segments with several inflations in lines and two at the end; tr. v. elliptic with four inflations at each side and one at each end; e. v. end lobe entire at margin. Sporangium orbicular, with subulate spines. (A.) Remark.—F¥ig. 13, pl. 21, represents this species. E. ampullaceum, Racrs. KE. diametro duplo longius ; semicellulis trilobis, ad basin tumidis, e basi latissima subito in Jobi polaris collum attenuatis. !obis basalibus maximis integris, loco loborum intermediorum pro- cessu deutiformi, lobo polari cuneato, in apice bifido, segmentis late truncato-rotundatis; cytiodermate subtiliter punctate. (R.) Long. 0.0035”—0.0038". Lat. max. .0026”; lat. in colli (lobi polar.) 0.00085”. (R.) Syn.—E. ampullaceum, Raxrs, British Desmidee, p. 83. Hab.—South Carolina; Florida; Bailey. Frond rather more than one-half longer than broad ; segments obscurely 3-lobed, short, with broad inflated base; basal lobes not emarginate, having on each upper side a small inter- mediate tubercle between each and the end lobe; end lobe exserted and dilated, its notch FRESH-WATER ALGA OF THE UNITED STATES. 139 linear. Empty frond minutely punctate; f. v. narrow elliptic, with several inflated protube- rances, ends scarcely dilated, rounded; tr. v. with four inflations at sides and one at each end. (A.) circulare, Hassat. E. mediocre, diametro duplo longius; semicellulis trilobis (at non semper distincte), ad basin versus tumoribus quinis aut pluribus in series duas v. tres alternantes aut singulo centrali, quaternis semicirculariter ordinatis instructis, lobis basalibus sinuato-emarginatis, subito in lobum polarem apice paullum dilatatum attenuatis ; cytiodermate subtiliter punctato. (R.) Syn.—E. circulare, Hassau, Fresh-Water Alge, p. 383. Hab.—Providence, Rhode Island; Bailey. “Frond about twice as long as broad, tapering upwards into a neck, end not dilated, its notch an acute incision. Empty frond, segments with five basal inflations, four in a half circle around the fifth and two others at the extremity.” Archer. (Var. Ral fsii) Semicellula tumoribus minimis 11 in series tres alternantes ordinatis. Hab.—Saco Lake, New Hampshire; (I. W. Lewis) Wood. E. Jenneri, nobis. Frond scarcely twice as long as broad; segments 3-lobed, basal portions subquadrate, emarginate at the sides; end lobe, its notch linear. Empty frond punctate, segments with several inflations arranged in alternate lines. (Archer.) E. imsigme, Hassact. E. subgracile, diametro duplo-triplove longius, a vertice fere quadratum, lateribus coneavis, angulis rotundatis ; semicellulis basi inflatis, integris,e basi subreniformi in collum elongatum citius attenuatis, lobo polari dilatato bifido truncato; eytiodermate subtiliter punctato. (R.) Long. 0.0039’—0.0043”. Lat. max. 0.00236% (R.) Syn.—£. insigne, Hassauu, Fresh-Water Alge, p. 21. Hab.—¥ lorida; Rhode Island; Bailey. Frond rather more than twice as long as broad; sezments inflated at base, sides entire, without lateral tubercles, and tapering into a long slender neck; end lobe dilated, its notch linear. Empty frond minutely punctate ; f. v. segments with two inflations at the base ; f. v. narrower, gradually tapering to the end, which is considerably dilated; projections rounded, with a sinus between; tr. vy. subquadrate, slightly concave at sides, with a rounded lobe at the centre of each end; e. v. end lobe with a sinus at opposite external margins, angles thus protruded into four divergent rounded lobelets. (A ) E. Ralfsii, Rasenn. E. mediocre, leviter compressum, medio inflatum, diametro duplo circiter longius; semicellulis pyramidalibus, e basi ventricosa in lobum polarem rectum truncatum sinuato-attenuatis ; cytiodermate subtiliter punctato, punctis in lineas rectas ordinatis. (R.) Syn.—E. ansatum, Eur. et auctores. RABENHORST, Flora Europ. Alearum, Sect. III. p. 184. E. Ralfsvi, RaBeNuorst, Flora Europ. Algarum, Sect. III. p. 184. Hab.—South Carolina; Rhode Island; Bailey. White Mountains, New Hampshire (F. W. Lewis). “BE. ansatum, Ehrb. Frond about twice as long as broad; segments inflated at the base, taper- ing upwards without sinuations into a neck, end not dilated, rounded, its notch linear. Empty 140 FRESH-WATER ALG#& OF THE UNITED STATHS. frond punctate; f. v. segments turgid on the surface, at the middle without circular inflations ; tr. y. elliptic, with a single large inflation at each side; e. v. end lobe entire at the margin, its divisions circular. (A.) Remari:s.—I have seen only a very few specimens in a gathering made in Saco Lake, New Hampshire, by Dr. Lewis, which differ considerably from the typical form in the proportion of the breadth and length. ‘There are also certainly four, if not more, umbonations on the face of each half-cell. ‘These are nowhere dis. tinctly spoken of as existing, and Mr. Archer states there are none visible in the front view of EZ. ansatum. ‘They are, however, represented in the side view of the original figure, and are said to be very noticeable by Mr. Archer himself, when the desmid is so looked at. In the Saco Lake specimens they are always seen in the front view with great difficulty, and in some cases I failed entirely to demon: strate them, so that they do not afford a good character for the indication of a new species. Fig. 1, pl. 18, represents a front view of a Saco specimen, magnified 450 dia- meters. C. Lobi polares non-evidenter discrett. C. End lobes not evidently distinct. E. elegans, (Brés.) K rz. E. minus, oblongum, diametro duplo longius, utroque polo bifidum, segmentis introrsum rotun- datis; semicellulis sursum modice attenuatis, utroque margine laterali bi- vel tri- sinuatis, sinu superiori vel intermedio profundiori, sub polo utrinque dente acuto prominente ; cytio- dermate subtiliter punctato, punctis irregulariter sparsis; zygosporis globosis aculeatis, aculeis elongato-subulatis. (R.) Long. 0.0012"”.—0.002”. Lat. max. circiter 0.0011”. (R.) Syn.—E. elegans, (Brésisson,) Kurzina. Rapennorst, Flora Europ. Algarum, Sect. IL p. 185. Hab.—South Carolina; Georgia; Florida; Rhode Island; Bailey. White Mountains, (F. W. Lewis). Frond minute, scarcely twice as long as broad, oblong; segments with their basal portion emar- ginate at the sides, connected by a broad neck with the terminal portion; ends protuberant, rounded, acutely emarginate at the centre, pouting; s. v. with an inflation at the base of the segments, sides concave, ends rounded. Sporangium orbicular, spinous. (A.) Remarks.—According to Prof. Rabenhorst E. rostratum, Ralfs, which is noted as an American species by Bailey, is a variety of E. elegans. Its peculiarities, ac- cording to Rabenhorst, are as follows: “ Forma magis evoluta, profundior sinuata, segmentis polaribus latioribus, angulis acutis, dente paulo longiore.” Fig. 14, pl. 21, represents the outline of the frond as viewed laterally; fig. 2, pl. 13, a front view of the frond, magnified 750 diameters. E. binale, (Turpin) Rares. H. minimum, diametro paulo vel subduplo longius, in sect. transversa oblongo-cylindricum, medio tumidum, utroque polo rotundatum; semicellulis indistinete trilobis, lobis basalibus FRESH-WATER ALG#H OF THE UNITED STATES. 141 latis, rotundatis vel sinuato-bi- tricrenatis ; lobo polari abbreviato late truncato, leviter emar- | ginato, angulis acutis lateraliter plus minus porrectis; ecytiodermate subtilissime punctato. Syn.—E. binale, (Turrin) Ratrs, British Desmidex, p. 90. Hab.—Florida; Bailey. Rhode Island, (S. T. Olney,) Thwaites. Pennsylvania; Wood. Frond very minute, scarcely twice as long as broad, oblong; segments with their basal portion either entire or bicrenate at the sides, slightly contracted beneath the ends; ends dilated, not protuberant beyond the angles, its central notch acute, broad; tr. v. with two lateral infla- tions, ends truncate, angles rounded. (A.) Remark,.—Vig. 3, pl. 13, represents the front view of a frond, magnified 750 diameters. Genus MICRASTERIAS, Ac. (1827). | Cellule compress, profunde constrict, a fronte orbiculares vel late elliptic, a vertice fusiformes j cum utroque polo acuto, semicellule tri- vel quinque-lobe ; lobi basales aut integri aut pluripliciter inciso-lobulati; lobus polaris aut integer aut sinuatus aut emarginatus, et interdum angulis produc- tus et bifidus. Cytioplasma chlorophyllacea in cellule lumen subequaliter distributa, granula amylacea sparsa involvens. Cytioderma plerumque leve, nonnunquam punctatum, granulatum vel mucronatum, Zygospore globose, etate provecta aculeis simplicibus, apice bi- multi-fidis, nonnunquam repetito- multifiidis armate. Cells compressed, profoundly constricted, viewed from the front orbicular or broadly elliptic, from the vertex fusiform with acute ends. Semicells 3- or 5-lobed; basal lobes either entire or many ! times incisely-lobulate ; end lobe either entire or sinuate or emarginate, and sometimes with its angles ) produced and bifid. Chlorophyllous cytioplasm distributed nearly uniformly in the cavity of the cell, surrounding scattered starch granules. Cytioderm mostly smooth; sometimes punctate ; ! granulate or mucronate. Zygospores globose, at maturity armed with simple spines, whose ends bifid or multifid, and some- times repeatedly multifid. A. Semicellule trilobe. Lobi basales horizontales ; lobus polaris valde dilatatus, dorso plane convecus, truncatus vel leviter retusus, a lobis basalibus sinu amplissimo diseretus. A. Semicells trilobate. Basal lobes horizontal ; end lobe strongly dilated, with the back con- vex, truncate, or slightly retuse. M. arcuata, Bariey. M. mediocris, quadrangularis, paulo latior quam longa, profunde pinnatifida; lobis basalibus angustis elongatis, arcuatis, in apicem acutum attenuatis, divergentibus; lobis polaribus angustusimis, utrinque graciliter productis, in apicem acutum attenuatis, in medio dorso modice retusis. (R.) Syn.—JL. arcuata, Baitny, Microscopical Observations: Smithsonian Contributions, vol. ii. Hab.—In stagnis. Florida; Bailey. “ Quadrangular, segments three-lobed, the basal lobes long and arcuate, subtended by the trans- verse projections from the ends of the slightly notched terminal lobes.” (Bailey. ) ME. expansa, Barvey. M. mediocris, tam longa quam lata, lobis stellatim expansis; lobis basalibus angustis in apicem acutum attenuatis, divergentibus, rectis; lobis polaribus e basi angusta sensim dilatatis, in medio dorso late sinuatis, angulis acutis (sed muticis). (R.) Syn.—M. expansa, Bartry, Microscopical Observations: Smithsonian Contributions, vol. ii. Hab.—In stagnis, Florida; Bailey. 142 FRESH-WATER ALG# OF THE UNITED STATES. = Segments three-lobed, basal lobes long, subconieal, acute; termina. lobes slender, forked at the end, with the divisions much shorter than the basal lobes. (Bailey.) Mi. quadrata, Battery. M. arcuate similis, sed duplo major,-semicellularum Jobi basales minus arcuati, basi inflati, apice bidentati et cytioderma irregulariter granulatum. (R.) Diam.—0.0043"—0.0049". Syn.—WM. quadrata, BAtLEY, Microscopical Observations : Smithsonian Contributions, vol. ii. Large quadrangular, three-lobed, basal lobes elongated, slightly curved, bidentate ; terminal lobes with two slender transverse bidentate projections. Bailey. MI. disputata, Woop. M. magna, fere tam longa quam lata, subpinnatisecta, sinu acuto, lobis equalibus ; semicellulis pro- funde trilobis, lobis basalibus in apicem acute bidentatum valde attenuatis; lobo polari valde dilatato, dorso rotundato, angulis lateralibus acutissimis. Long. 7335" =.005”. Lat. 738,” =.004”. Syn.—Micrasterias incisa, Krz. BatLey, Microscopical Observations: Smithsonian Contribu- tions, 1850. Haud Micrasteria incisa, Ktrzine, Spee. Algarum, p. 171. Tetrachastrum Americanum, ARCHER, PRITCHARD’s Infusoria, 1860, p. 725. Hab.—South Carolina; Georgia; Florida; Rhode Island; Bailey. Pennsylvania; Wood. ~ M. large, about as long as broad, subpinnatisected; sinuses acute; semicells profoundly trilo- bate; basal lobes strongly attenuate into the acutely bidentate apex; distal lobes strongly dilated, rounded, with their lateral angles bidentate ; end lobe broadly dilated, lateral angles very acute. Remarks.—This desmid was first figured by the late Prof. Bailey in his Micro- scopical Observations (Smithsonian Contributions), as Jf inecisa of Ktz., and Ra- benhorst, in his Flora Europea Algarum, confirms this identification. He has probably, however, never seen the plant itself, but merely accepts the opinion of Professor Bailey. Mr. Archer (Pritchard’s Infusoria), thinks the American plant is certainly distinct from the European, and this seems to me correct. ‘The points of difference are—the American form is nearly twice the size of the European, the sinuses are much more widened outwardly, and the lobes are reduced rapidly in breadth to a mere point at the end, the dorsum of the distal lobes is also, I believe, more rounded. In his description of 7. Americanum, as he calls it, Mr. Archer states the end lobe has its angles bidentate. In the only specimen I have seen, the angles end in a very sharp, almost spine-like point. Dr. Leidy found the spe- cies abundantly at Newport, Rhode Island, and his figure agrees with mine in this respect. In regard to the name, as there is already an J. Americanum, the specific name of Archer cannot be adopted, and for a similar reason it would not do to call it M. Baileyi. Ihave then been forced to give it a new title. ° Fig. 4, pl. 13, was drawn by myself from the single specimen I have seen; fig. 4 a was drawn by Dr. Leidy from a Newport specimen. M. oscitans, Rates. M. magna, pene tam longa quam lata, subpinnatisecta, a vertice elliptico fusiformis, utroque polo bifida; lobis basalibus horizontalibus conico-productis, apice bifidis; labo polari a lobis basalibus sinu amplo ac rotundato discreto, plus minus convexo, haud raro truncato, rarius leviter retuso, utrinque producto acuminato, plerumque bidentato, (R.) FRESH-WATER ALGZ OF THE UNITED STATES. 143 Diam.—0.0047”". Long. 0.0039”. (R.) Syn.—W. oscitans, Raurs, British Desmidiee, p. 76. RABENHORST, Flora Europ. Algarum, Sect. III. p. 119. WW. pinnatifida Krz. Rapenuorst, Flora Europ. Algarum, Sect. III. p. 119. Hab.—F lorida; Rhode Island; Bailey. Frond about as broad as long, pinnatifid; lateral lobes separated from the terminal by a rounded sinus, horizontal, conical, their extremities bidentate; end lobe short, broad, its lateral pro- jections short, conical, usually bidentate, narrower and shorter than the lateral lobes; ends convex at the centre; tr. v. fusiform, e. f. punctate. (A.) Remarks.—According to Prof. Rabenhorst JL pinnatifida, Ktz., is a variety of I. oscitans, different from the typical form only in being smaller, and in having the lobes narrower. B. Semicellulx 3-vel 5-lobe, plerumque radiatim inciso-lobulate. Lobi basales assurgentes aut non aut minus a lobo polari remoti. B. Semicells 3, or 5-lobate, mostly radiately incisely lobulate. Basal lobes assurgent, either close to, or but slightly remote from the end lobes. * Semicellule trilobe. ~* Semicells trilobate. M. Americana, (Eurs.) Krz. M. magna, oblonga, subpinnatiseeta, lobis polaribus paulum remotis, pene duplo longior quam lata; cytiodermate spinuloso unde laborum margines dentato-serrati conspiciuntur ; cellula e latere conspecta oblonga, in medio leviter constricta, utroque polo bicornuta; semicellule basi tumore plus minus distincto instructe, fere quinquelobe, lobis basales latissimi iisdemque profunde bilobati, lobulis late excisis, segmentis dentato-serratis; lobis polaribus plus minus productis, in medio late excisis, segmentis profunde bifidis. (R.) Diam.—0.0041". Long. circa 0.0051”. (R.) Syn.—M. Americana, Ktrzinc. Rawennorst, Flora Europ. Algarum, Sect. III. p. 189. Hab.—In stagnis, South Carolina; Florida; Bailey. Frond angular elliptic, more or less punctate; segments 3-lobed; lateral lobes broad, cuneate, their margins concave, inciso-serrate ; and lobe broad, cuneate, end exserted, bipartite at the angles, the subdivisions narrow, and minutely dentate at the extremities ; end concave. (A.) Remark.—Fig. 17, pl. 12, represents a plate of this species. M. Baileyi, Ratrs. M. parva, oblonga, granulata; semicellulis trilobis, lobis basalibus a lobo polari sinu amplo diseretis, excisura acute triangulari in duas lacinias partitis, laciniis e basi latiori in apicem truncatum bidentatum attenuatis; lobo polari e basi angusta Jonge porrecto, sursum valde dilatato, in vertice leviter et late sinuato, angulis truncato, bidentato. (R.) Syn.—M. Baileyi, Raurs, British Desmidiew, p. 211. Hab.—New York; Rhode Island; South Carolina; Florida; Bailey. Frond granulated; segments three-lobed ; lobes bipartite, end one much exserted. (Ralfs.) M. rimgems, Batrey. M. mediocris, oblonga, margine granulata; semicellulis trilobis; lobis lateralibus bipartitis, laciniis divaricatis, apjce obtusis, truncatis vel bidentatis; lobo polari e basi angusta sursum valde dilatato, exserto, in vertice leniter sinuato, angulis truncato. (R.) 144 FRESH-WATER ALG& OF THE UNITED STATES. Syn.—M. ringens, BaruEy, Microscopical Observations, pl. 1, fig. 11: Smithsonian Contri- butions, vol. ii. Hab.—Florida; Bailey. Oblong, segments three-lobed, coarsely granulated near the edge ; basal lobes subdivided by a deep notch into two rather broad and obtuse or slightly bidentate projections ; terminal lobes exserted, emarginate ; extremities bidentate or obtuse. * * Semicellule quinque-lobate. * * Semicells 5-lobed. Mi. trumecata, (Corps) Bres. M. magna, orbicularis, aut levis aut subtiliter punctata; semicellulis quinquelobis, lobis inter se sinu obtusangulo subangusto discretis, basalibus et intermediis inciso-lobulatis, segmentis acute bidentatis; lobo polari late cuneato, in dorso truncato, modo leviter convexo, modo leviter retuso, angulis aut bidentatis aut integris. (R.) Diam.—0.003”" Long. .0036”. Syn.—WM. truncata, (Corpa,) Bresisson. Rawennorst, Flora Europ. Algarum, Sect. III. p. HONE Hab.—Georgia; Florida; Rhode Island; Bailey. Pennsylvania; Wood. Rhode Island (S. T. Olney); Thwaites. Frond orbicular, smooth; segments 5-lobed; basal and middle lobes obscurely bipartite, ex- tremities bidentate ; end lobe very broadly cuneate, bidentate at the angles, and with a slightly central concavity. (A.) Remarks.—The dimensions given above were taken from the largest specimens T have seen, but do not at all equal those given by Prof. Rabenhorst, his breadth is 0041". According to the same authority, JZ crenata, Bréb., is merely a variety of this species. Fig. 15, pl. 21, represents the outline of a frond of this plant. Mi. furcata, Ac. M. permagna paulo longior quam lata, levis; semicellulis quinque lobis (pane 7-lobis) ; lobis omnibus rectis; lobis basalibus angustioribus, bilobulatis, lobulis bifidis, sinu obtusangulo vel acutangulo, segmentis linearibus bidentatis (denticulis seepe inequilongis); lobis inter- mediis duplo latioribus, inciso-bilobis, lobulis iisdem ae loborum basalium; lobo polari non- nihil anguste cuneato, prominulo, in apice plus minus profunde sinuato-vel undulato inciso, angulis bidentatis. Diam.—7 85" = .008”. Syn.—. rotata, Raurs, British Desmidiex, p. 71. M. furcata, AGARDH. Raxsenuorst, Flora Europ. Algarum, Sect. III. p. 191. Hab.—South Carolina; Georgia; Florida; Rhode Island; Bailey. New Jersey; Wood. M. very large, a little longer than broad, smooth; semicells 5-lobed (scarcely 7-lobed) ; lobes all straight ; basal lobe narrower than the intermediate, bilobulate, lobules bifid, their sinuses acute or obtuse, segments linear, bidentate ; teeth often long and unequal ; intermediate lobes twice as wide as the basal, bilobate, their lobules of the same form as the basal lobe; end lobes narrowly cuneate, prominent, more or less profoundly sinuately or undulately cut at the apex, angles bidentate. Remarks.—According to Rabenhorst and others, there is a European form of this species in which the marginal teeth are wanting. This may exist in this FRESH-WATER ALGA OF THE UNITED STATES. 145 country, but I have never met with it. All the specimens which have come under my notice were obtained in “Shepherd’s Dam,” near Greenwich, Cumberland County, New Jersey. None of them were as large as the maximum of the European measurements of which Rabenhorst gives 0.0109” as the diameter. Fig. 5, pl. 13, represents a frond of this species, magnified 260 diameters. M. denticulata, Bree. ? M. permagna, paulo longior quam lata, levis; semicellulis quinquelobis (pene 9 lobis); lobis 5 ’ ? q iT at ? intermediis et basalibus simillimis, bilobatis, lobulis item in lobulis bifidis duobus divisis ; lobo polare angusto, cuneato, in apice plus minus inciso; margine minute denticulato. Diam.—Lat. .0092”. Long. .011.” Syn.— I. denticulata, Bresisson. Rats, British Desmidiee, p. 70, et ARCHER, PRITCHARD’S Infusoria. M. denticulata, Brépisson.? RaApBeEnnorst, Flora Europ. Algarum, Sect. III. p. 192. Hab.—Pennsylvania; Wood. Florida; Bailey. Very large, a little longer than broad, smooth; semicells with five lobes (scarcely 9); basal and intermediate lobes alike bilobate, lobules also divided into two bifid lobules; end lobe narrow, wedge-shaped, more or less incised at its apex; margin minutely denticulate. Remarks.—Prof. Rabenhorst gives M. denticulata, Bris. as merely a variety of M. furcata, AG., stating that it only differs from the latter in the marginal incisions and teeth. Not having access to the original description of Brébisson I cannot express an opinion as to whether Prof. R. is correct or not, but the specimen from which the above description was drawn up (and which is figured on plate 13) cer- tainly differs from Jf furcata very essentially in the arrangement of its lobes, and is, I feel confident, JL denticulata, Brés. of RAtrs and ARCHER. Fig. 6, pl. 13, is a drawing of this plant, as seen by myself, magnified 260 diameters. Ma. radiosa, Ac. M. maxima, orbicularis, levis, antecedenti simillima, differt inprimis segmentis ultimis tumidis in apicem bi-tri- fidum attenuatis, lobo polari vix prominulo, apice sinuato, ad utrumque angulum bi-tri- dentato. (R.) Species mihi ignota. Diam.—0.0076". (R.) Syn.—WM. radiosa, AGarpu. Rasennorst, Flora Europ. Algarum, Sect. III. p. 192. Hab.—F lorida; Bailey. Frond orbicular, smooth; segments 5-lobed; basal lobes twice, middle lobes generally thrice dichotomous, ultimate subdivisions inflated, attenuate towards the end, bidentate; end lobes emarginate, its angles dentate. (A.) uM. fimbriata, RALrs. M. magna, orbicularis, levis (nonnunquam superficie aculeis singulis sparsis) ; semicellulis quinquelobis, lobis omnibus confertis, basalibus angustioribus, repetito bilobulatis, lobis inter- mediis duplo latioribus, repetito-bilobulatis, lacinulis extremis leviter emarginatis, in angulis spinis elongatis armatis; lobo polari prominulo, in apice obtuse sinuato- vel- undulato- emar- ginato, angulis lateralibus rotundatis, ad marginem superiorem spinis singulis vel geminis obsito (rarius nudo). (R.) Species mihi ignota. 19 August, 1872. 146 FRESH-WATER ALGA OF THE UNITED STATES. - Diam.— 0051”—.0078”. (R.) Syn.—WM. jimbriata, Raurs, British Desmidiex, p. 71, et RaBennorst, Flora Europ. Algarum, Sect. III. p. 198. Hab.—South Carolina; Florida; Bailey. Frond orbicular, smooth ; segments 5-lobed, basal lobes twice, middle lobes thrice dichotomous ; ultimate subdivisions acutely bidentate; end lobe very slightly exserted, its angles very slightly produced, bidentate, ends emarginate. In transverse view is seen an inflated pro- tuberance just over the central isthmus, which may possibly exist in other species of Jicras- terias. (A.) Mi. papillifera, Bras. M. orbicularis, superficie levis, margine extremo dentato papillifera; semicellulis quinquelobis ; lobis basalibus et intermediis equilatis, bilobatis ; lobulis bifidis, laciniis linearibus bidentatis, dentibus papilliferis; lobo polari vix prominulo, in apice sinuato, angulis et margine dentato- mucronatis. (R.) Species mihi ignota. Diam.—0.0045". (R.) Syn.—JL papillifera, Brésisson. Rates, British Desmidiex, p. 72, et Kapennorst, Flora Europ. Algarum, Sect. IIT. p. 194. Hab.—F lorida; Rhode Island; Bailey. Frond orbicular, having the principal sinuses bordered by a row of minute granules, otherwise smooth; segments 5-lobed; basal and middle lobes twice dichotomous, their ultimate shal- low subdivisions terminated by two, sometimes three, gland-like teeth ; end lobe emarginate, its angles dentate. Sporangium as in I. denticulata, but considerably smaller. (A.) Mi. gramuiata, Woop (sp. nov.) M. magna, suborbicularis, arete granulata; semicellulis quinquelobis, lobis inter se sinu angusto discretis, basalibus et intermediis plerumque integris, lobo polari supra valde dilatato, in dorso medio leviter retuso; marginibus valde crenatis. Diam.—Long. 339” = .0043". Lat. qsigsg’” = -0036”. Hab.—South Carolina, (Ravenel) Large, suborbicular, closely granulate; semicells 5-lobed, lobes separated by narrow sinuses; basal and intermediate lobes mostly entire; end lobe distally broadly dilated, broadly and very shallowly emarginate; margin of frond strongly crenate. Remarks.—The only specimens of this species that I have seen were collected by Prof. Ravenel in a shallow ditch near Aiken, South Carolina, where they formed a greenish, gelatinous mass, with numerous desmids and diatoms. It is most closely aulied to M. truncata, from which it is separated by its entire lateral lobes, by its granulated surface, and its crenated margins. It also does not apparently attain as large a size as that species. The granules are very small in the central portion of the frond, but become larger as they approach the margin. Fig. 16, pl. 21, represents an empty frond of this species, magnified 460 diam- eters. Ma. Jemmeri, Rares. M. magna, oblonga, plerumque subtiliter granulata; semicellulis quinquelobis ; lobis basalibus et intermediis squilatis, confertis, cuneatis, bilobulatis; lobo polari late truneato vel late rotundato, in medio interdum leviter et obtuse emarginato, interdum nonnihil profunde emar- gvinato. Diam.—ULat. 7235’—ys'is6” = .006”—.0062”. Long. 7835,”—ys45,5” = .0062”—.0087”, FRESH-WATER ALGA OF THE UNITED STATES. 147 Syn.—WM. Jenneri, Rares, British Desmidiex, p. 76. Hab.—Prope Philadelphia; Wood. South Carolina; (Ravenel) Large, oblong, for the most part finely granulate ; semicells 5-lobed ; lobes wedge-shaped ; basal and intermediate, about equally broad; end lobe broadly truncate or broadly rounded, in the middle sometimes slightly and obtusely emarginate, sometimes rather deeply emarginate. Remarks.—I have found this species near Philadelphia, and also received it from Prof. Ravenel, by whom it was collected in South Carolina. The American plant differs from the typical form in not having the ultimate lobules emarginate, they being merely a little hollowed out in the centre, and sometimes scarcely this. The angles in some specimens are also more acute. Mr. Archer, however, speaks of a variety occurring in England, in which these lobules are not emarginate, and I do not think characters can be found separating the American from the European forms, ‘The median suture is in all the specimens very narrow and deep, a mere line, as it were,‘extending nearly to the centre. Fig. 7, pl. 13, represents a frond of this species. Mi. Torreyt, Barey. M. permagna, oblongo-orbicularis, levis, profundissime lobata; semicellulis quinquelobis, lobis basalibus profunde bifidis, laciniis inferioribus apice bidentatis, superioribus integris, lobis in- termediis profunde trifidis, laciniis superioribus bidentatis, inferioribus integris, lac. omnibus lanceolatis acuminatis, inferioribus paulum incurvis, superioribus recurvis ; lobo polari non prominente, e basi angusta seusim dilatato, in vertice acute sinuato, angulis integris acumin- atis. (R.) Species mihi ignota. Syn.—M. Torreyi, Battny. Raxes, Brit. Desmidiewx, p. 210. Hab.—Prope Princetown, New Jersey; Bailey. Frond smooth; segments 5-lobed ; basal lobes bifid, middle lobes trifid, the subdivisions nearest the opposite segments and those nearest the terminal lobe bidentate at the apex; the inter- mediate three terminating in acute points; all somewhat inflated and tapering; terminal lobe narrow, not exserted, spreading at the angles into divergent tapering points, ends slightly emarginate. (A.) M. foliacea, Battery. M. parva, subquadrata, levis; semicellulis trilobis, lobis lateralibus profunde bifidis (unde ree- tior semicell. quinquelob), lobulis ineequaliter inciso-dentatis, lobulis inferioribus rectis, superioribus recurvis; lobo polari plus minus prominente, anguste cuneato, in vertice plus minusve emarginato, angulis aut acutis integris aut productis, bidentatis. R. Species mihi agnota. Syn.—WM. foliacea, Batty. Ratrs, British Desmidiew, p. 210. Hab.—‘ Worden’s Pond, Rhode Island; Bailey.” Frond subquadrate, smooth ; segments 3-lobed; lateral lobes deeply bipartite, inciso-dentate, their margins concave, inciso-serrate; end lobe broad, cuneate, and exserted, bipartite at the angles, the subdivisions narrow, and minutely dentate at the extremities; end concave. (A.) Genus STAURASTRUM, Meyen. Cellule libere natantes, in medio plus minus profunde constrict# ; semicellule a vertice 3-6 angu- lares vel radiate. Cytioderma aut leve aut punctatum aut verrucosum aut aculeatum, nonnunquam ciliis vel pilis obsessum. Cells swimming free, more or less profoundly constricted in the middle; semicells when seen from the vertex 3 to 6 angular or radiate. Cytioderm either smooth or punctate, or verrucose or aculeate, sometimes covered with hairs or cilia. 148 FRESH-WATER ALG#® OF THE UNITED STATES. A. CyTIoDERMA LHVE VEL RARISSIME SUBTILITER PUNCTATUM. CYTIODERM SMOOTH OR VERY RARELY VERY FINELY PUNCTATE. 1. Semicellularum anguli rotundati. Angles of the semicells rounded. St. muticum, Breés. St. a fronte orbiculare, leve, profunde constrictum, nudum, vel muco plus minusve firmo inyo- lutum ; semicellulis ellipticis, a vertice conspectis 3-4 angularibus (rarius quinquangularibus) angulis rotundatis, lateribus leviter sinuato-retusis; zygosporis aculeatis, aculeis elongatis, subulatis, furecatim fissis. (R.) Species mihi ignota. Diam.—0.0013”—0 000147”. (R.) Syn.—S. muticum, Brépisson. Raxpennorst, Flora Europ. Algar., Sect. III. p. 200. Hab.—South Carolina; Rhode Island; Bailey. Seoments in f. vy. elliptic, smooth, without spines; e. v. with three or four broadly rounded 5 I , 5) vf angles, sides concave. Sporangium beset with numerous elongate somewhat stout spines, forked at the apex. (A.) St. orbiculare, (Eurs.) Rates. St. suborbiculare, lave, sepius muco matricali involutum; semicellulis divergentibus, semi_ orbicularibus, dorso nonnunquam elevatis, angulis plus minus late rotundatis, lateribus plus minus sinuato-retusis; zygosporarum aculeis elongatis, subulatis, integris. (R.) Diam. —.002’’. Syn.—St. orbiculare, (Eurs.) Raves, British Desmidiex, p. 125. RaABrnnorst, Flora Europ. Algarum, Sect. III. p. 200. Hab.—Rhode Island; Bailey. Pennsylvania; Wood. Rhode Island; (S. T. Olney) Thwaites. Segments in f v. semiorbicular, smooth, without spines ; e. v. with three broadly rounded angles, sides slightly concave. (A.) Remarks.—F ig. 17, pl. 21, represents the outline of the end view of a frond of this species. Fig. 8, pl. 13, is a drawing of the front view of a living frond. 2. Semicellularum anguli mucronati vel aristati. Angles of the semicells mucronate or bristly. St. longispinuma, (Battey) Arcuer. St. magnum triangulare, lwve, angulis in aculeos geminos validos subulatos longe productum, lateribus subplanum. (R.) Species mihi ignota. Syn.—Didymocladon longispinum, BAtLry, Microscopical Observations. Hab.—Florida; Bailey. “Large, smooth, triangular, with two long spines at each angle.” Bailey. St. dejectum, Brézisson. St. leve, parvum, sinu amplo, obtusangulo (vel acutangulo); semicellulis ellipticis (vel subtri- angularibus), dorso nonnihil convexo, utroque fine in aculeum achroum rectum vel varie cur- vatis productis; a vertice triangularibus (vel quadrangularibus), angulis sepe rotundatis aculeo interdum obsoleto imposito. Diam.—Lat. 7330’ —re800' = 0008"—.001”. Long. 42%5"—ys255” = -0008”—. 0001”. Syn.—Staurastrum dejectum, Brésisson. RABENHORST, Flora Europ. Algarum, Sect. III. p- 203. FRESH-WATER ALGA OF THE UNITED STATES. 149 Hab.—South Carolina ; (Ravenel) Wood. Smooth, small; sinus ample, obtuse angled (sometimes acute angled ?); semicells elliptic (or subtriangular ?), with the dorsum slightly convex, at the angles with a straight or curved transparent spine; from the vertex triangular (or quadrangular’), angles often rounded, with a sometimes obsolete spine superimposed. Remarks.—This species was collected near Aiken, South Carolina, by Prof. Ravenel, who found it forming with various diatoms and desmids a slimy mass in | a feebly running ditch. It agrees very well with the European form, except that it is not so large (at least the largest I ever measured did not come up to the size of their transatlantic brethren), neither does it appear to vary quite so much. In the description, I have placed in brackets those characters in which the European form varies, and the specimens I have seen do not. Fig. 18, pl. 21, represents outline of end of a semicell, magnified 750 diameters. Fig. 9, pl. 13, a front view, and 9 @ the end view, of the living frond, magnified diameters. St. aristiferum, Racers. St. leve, St. cuspidatum quodammodo simile, et eadem magnitudine sed isthmo destitutum ; semicellulis tumidis, in media parte subrotundatis, lateraliter in lobum, basi constrictum, apice aristatum productis, lobis divergentibus, a vertice tri-quadrilobo-radiatis, radiis strictis wequidistantibus cruciatim dispositis, interstitiis profunde excisis. (R.) Species mihi ignota. Diam.—Incel. arist. 0.0014”. (R ) Syn.—St.aristiferum, Raurs, British Desmidier, p. 123. RaBennorst, Flora Europ. Algarum, Sect. III. p. 204. Hab.—Georgia ; Rhode Island; Bailey. Segments smooth, in f. v. prolonged at each lateral extremity into a mamillate projection, which is terminated by a subulate, acute straight awn, the awns divergent, e. v. with three or four angles; angles inflated mamillate, terminated by an awn, sides deeply concave in the ceutre. (A) St. Lewisii, Woop. St. leve; sinu amplissimo, spinulo parvo armato et cum angulo obtuso; isthmo nullo; semi- cellulis a fronte late triangularibus, a vertice triangularibus et cum angulis nonnihil tumidis, et rotundatis; angulis spino maximo, robusto, acuto armatis. Diam.—Long. cum. spin. 45" = .0025"; lat. cum. spin. 72455". = .00225”. Sine spin. : long. gio” =.001666"; lat. y3355” .001666". Spin.: long. > 259” = -000666" Syn.—St. Lewisti, Woop, Proc. Acad. N.S. 1870. Hab.—In lacu Saco; (Lewis) Wood. Smooth, with a very ample sinus, which is armed with a small spine and has a very obtuse angle; isthmus absent; semicells from the front broadly triangular, from the vertex trian- gular, with the angles somewhat tumid and rounded; angles armed with a very large acute robust spine. Remarks.— This desmid is most closely allied to St. aristiferwm, Ralfs, but differs from it in outline as seen from the front, there being no mamellation of the ends. The spines in the sinuses are always wanting in the European species. Fig. 19, pl. 21, represents the outline of the end of a semicell, magnified 750 diameters. Fig. 11, pl. 13, represents the perfectly formed frond, magnified 750 diameters. 150 FRESH-WATER ALG# OF THE UNITED STATES. B. CyrlopDERMA GRANULATUM VEL VERRUCOSUM. CyTIODERM GRANULATE OR WARTY. Ps 1. Semicellule a vertice 3-7 angulares; anguli plus minus radiatim elongati. Semicells seen from the vertes 3-7 angled; angles more or less radiately produced. St. margaritaceuma, Eure. St. medioere, granulatum; semicellulis convergentibus, subfusiformibus, in medio tumidis, utrinque productis, truncatis, a vertice orbicularibus, 5—7 radiatis, radiis obtuse truncatis achrois, hyalinis, granulato-margaritaceis. (R.) Species mihi ignota. Diam.—0.00135"—0.0017". (R.) Syn.—St. margaritaceum, (HHRB.) MENEGHENI. RABENHORST, Flora Europ. Algarum, Sect, III. p. 206. Hab.—South Carolina; Georgia; Florida; Rhode Island; Bailey. Segments in f. v. gradually widening upwards, rough with pearly granules, outer margin eon- vex, produced at each side into a colorless, more or less attenuate, short process, having the granules in transverse lines, blunt and entire at the apex, e. v. circular, bordered by from five to seven short, narrow, obtuse, colorless, granulate marginal rays. (Archer.) St. dilatatum, Enrs. St. parvum, granulatum; semicellulis reetis, cylindrico-fusiformibus, non tumidis, utroque fine obtusis vel subtruncatis, a vertice 3-4-5 radiatis, radiis latioribus, truncatis vel rotundatis, achrois, hyalinis, granulato-margaritaceis. (R.) Species mihi ignota. Diam.—0.0008"—0.0011". (BR ) Var. alternans. Semicellulis ellipticis rectis, utroque fine rotundatis, a vertice triradiatis, radiis obtusis, alter- nantibus cum semicellule inferioris. (R.) Var. tricorme. Semicellulis fusiformibus, nonnunquam in medio subtumidis, haud raro isthmo distineto con- junctis, a vertice 3-4 angularibus, angulis truncatis vel obtusis, plus minus radiatim pro- ductis. (R.) Hab.—Georgia; Florida; Rhode Island; Bailey. Syn.—S. alternans, Brésisson. Var. alternans et tricorne. RABENHORST, Flora Europ. Algarum, Sect. III. p. 207. Remarks.—Prof. Rabenhorst considers S¢. alternans and tricorne, as simple varie- ties of S¢. dilatatum, whilst both Archer and Ralfs describe them as distinct. I have not seen either of them, and am not therefore entitled to offer an opinion. Mr. Archer describes the two species as follows :— St. alternans, Breés. Segments in front view elliptic or oblong, two or three times as broad as long, separated by a wide sinus, twisted, unequal; rough with very minute pearly granules; e. vy. with three obtuse and rounded angles, forming short, not colorless rays, alternating with those of the other segments, sides concave. Li. yos7'» Br. ype’. St. tricorne, Brés. Segments in f. v. somewhat fusiform, often twisted, rough with minute puncta-like granules, tapering at each side into a short, usually colorless process, blunt or divided at the apex; FRESH-WATER ALGA OF THE UNITED STATES. 151 ” rangium orbicular, beset with spines ultimately branched at the apex. L. yg);"—5},”. 2 leery, B. as . e. v. tri- or quadriradiate, processes short, usually colorless, sides somewhat concave. Spo- 1 2. Semicellule triangulares ; anguli non producti, obtusi vel rotundati. Semicells triangular ; the angles not produced, obtuse or rounded. St. punctulatum, Brés. St. parvum, punctulato-granulosum; semivellulis enormiter ellipticis, dorso late rotundatis, a vertice triangularibus; angulis non productis, obtuse rotundatis; lateribus modice retusis, Diam.—Lat. 1800. = .0012”. Syn.—S. punctulatum, Brésisson. Raxrs, British Desmidiew. RapBennorst, Flora Europ. Algarum, Sect. III. p. 208. Hab.—Pennsylvania ; Wood. Small, punctulate-granulate; semicells irregularly elliptic, with the dorsum broadly rounded from the vertex triangular; angles not produced, obtusely rounded ; sides somewhat retuse. Remarks.—This desmid is exceedingly common around Philadelphia, growing in the greatest abundance upon the face of wet dripping rocks. It is represented, fig. 10, pl.13. St. crenatum, Batcery. St. duplo circiter longius quam latum, in medio utrinque exsectione profunda rotundata; semicellulis e basi cuneata flabelliformibus, margine superiore crenatis, a vertice triangularibus, angulis rotundato-truncatis, crenatis, lateribus sinuatis glabris. (R.) Species mihi ignota. | Syn.—St. erenatum, Bartny. Ratrs, British Desmidiex, p. 214. RABEnnOoRsT, Flora Europ. Algarum, Sect. III. p. 220. “ Segments cuneate ; outer margins crenate; end view with three truncate and crenate angles.” 3. Semicellule vertice 3-1 radiatx ; radii in apice plerumque bi- tri- fidi vel bi- tri- spini. Semicells 3-7 radiate at the vertex ; radii bi- or tri-fid, or bi- or tri-spinous at the apex. St. polymorphum, Bréz. St. semicellulis ellipticis, subtiliter granulatis vel tenuissime spinulosis, in medio magis minusve inflatis, haud raro ventricosis, rectis, nonnunquam incurvis, utrinque processu plus minus elongato, lineari, in apice 3—4 fido vel spinulis 3-4 tenuissimis instructis, a vertice 3-4-5-6-7 radiatis, radiis achrois, aut trifidis aut rotundatis, trispinis. (R.) Species mihi ignota. Syn.—St. polymorphum, Brépisson. Ra.Fs, British Desmidiex, p. 135. RaAsenuorsr, Flora Europ. Algarum, Sect. III. p. 209. Diam.—Long. 0.001”. Lat. 0.00087”. (R.) Hab.—Florida; Bailey. Segments in f. v. broadly elliptic or almost circular, rough with minute granules (sometimes with a few minute scattered spines), processes short, stout, tipped by three or four divergent spines; e. v. with three, four, five, or six angles each produced into a short, stout process. Sporangium orbicular, beset with elongate spines, forked or branched at the apex. Archer. Var. cyrtocerum. (Sf. cyrtocerum, Bre&B.) Majus, ad gy”, longum, semicellulis introrsum ventricosis, dorso late rotundatis, utrinque pro- cessu elongato, plerumque incurvo apice bi- vel tri-cuspidato instructis, a vertice triradiatis, radiis rectis vel leniter curvatis, in apice aut bi- aut tri-cuspidatis. (It.) 152 FRESH-WATER ALG& OF THE UNITED STATES. Syn.—Var. St. cystocerum, BREBISSON. Rapennorst, Flora Europ. Algarum, Sect. IIL p. 210, Hab.—Rhode Island; (S. T. Olney) Thwaites. Seements in f. v. subcuneate, gradually widening upwards, truncate at the end margin, rough with minute granules, the lateral processes incurved, divided at the apex; e. v. triradiate, ] ] ¢ ” ar? = processes short, curved, sides slightly concave. L. g}o”. 3B. $9. (Archer) St. parodoxum, Meyey. St. semicellulis inflatis, dorso rotundatis vel rectilinearibus, angulis superioribus in radium elongatum achroum hispidum, apice trifureatim productis, seepius radio squali interposito a vertice tri- vel quadriradiatis, radiis strictis, trifureatis, longitudine corporis diam, wquan- tibus vel superantibus. (R.) Diam.—Cum rad. .0015”. Syn.—St. parodoxum, MEyEN. Raxsenuorst, Flora Europ. Algarum, Sect. III. p. 210. Hab.—In lacu Saco, New Hampshire; (Lewis) Wood. Semicells inflated, dorsum rounded or rectilinear, with superior angles produced into elongate, transparent, hispid radii with trifureate apices, often furnished also with intermediate equal radii; from the vertex three or four radiate, radii straight, trifurcate, equalling or longer than the diameter of the body. Remarks.—I am indebted to Dr. Lewis for specimens of this species, which he collected at Saco Lake. Fig. 20, pl. 21, represents the end view of an empty frond. St. arachne, Ratrs. St. parvum, gracile, granulato-asperum; semicellulis introrsum ventricoso-globosis, angulis superioribus in cornu gracile, incurvum, apice obtusum, elongatis, a vertice pentagonis, quinque-radiatis, radiis elongatis linearibus achrois, obtusis, rectis vel leniter curvatis asperis. (R.) Diam.—Sine rad. .0005”, eum rad. .00167”. Syn.—St. arachne, Ratrs, British Desmidiex, p. 136. RaBENHorst, Flora Europ. Algarum, Sect. III. p. 210. Hab.—In lacu Saco, New Hampshire, (Lewis) Wood. Segments in f. y. suborbicular, rough with minute granules, lower margin turgid, outer convex, tapering at each side into an elongate, slender, incurved process having the granules thereon in transverse lines, entire at the apex; e. y. circular, bordered by five slender, linear, colorless marginal rays. Remark.—F ig. 21, pl. 21, represents an outline of the end view of the semicell. St. gracile, Rares. St. mediocre, granulato asperum, granulis in series transversas ordinatis; semicellulis ventre valde inflatis, dorso truncatis, angulis in cornu rectum achroum gracile apice trifidum pro- ductis, a vertice triradiatis, lateribus sinuatis. (R.) Species mihi ignota. Diam.—0.0022". (R.) Syn.—St. gracile, Raurs, British Desmidiee, p. 136. Rasennorst, Flora Europ. Algarum, Sect. III. p. 211. } Hab.—South Carolina; Florida; Georgia; Rhode Island; Bailey. Rhode Island; (Olney) Thwaites. FRESH-WATER ALG# OF THE UNITED STATES. 153 Segments in f. v. triangular, ends truncate, rough with minute granules, tapering at each side into elongate, straight, slender, horizontal processes, terminated by three or four minute spines; e. v. triradiate, processes straight, sides concave. (A.) C. CyTIoDERMA PILOSUM, SPINULOSUM VEL ACULEATUM., CyTIODERM PILOSE, SPINULOSE OR THORNY, St. polytrichum, Perry. St. mediocre, tam longum quam latum, profunde constrictum, sinu acutangulo ampliato, super- ficie undique setosum ; semicellulis ellipticis vel subellipticis, divergentibus, dorso subplanis, ventre tumidis, margine setoso-ciliatis, a vertice triangularibus, angulis obtusis, lateribus subrectis. (R.) Diam.—7135" =.0017”. Syn.—St. polytrichum, Perry. Rasenuorst, Flora Europ. Algarum, Sect. III. p. 214. Hab.—Prope Philadelphia ; Wood. Moderately large, about as long as broad, profoundly constricted, with the acute angled sinus widened, surface everywhere furnished with sete ; semicells elliptical or subelliptical, diver- gent, the dorsum nearly plane, their belly swollen, the margin setose-ciliate, from the vertex triangular, the angles obtuse. Remarks.—This desmid appears to be rare in this country, as it probably is also in Europe. I have seen but a single specimen, which I found amongst other alge near Chelten Hills, north of the city. It agreed in all respects with the descrip- tion of Rabenhorst, as given above. Fig. 12, pl. 13, is a drawing of this plant, also fig. 23, pl. 21. St. Ravenelii, Woop. (sp. nov.) St. mediocre, paulo longius quam latum; semicellulis a fronte ellipticis, a vertice triangularibus cum lateribus convexis vel leniter retusis et angulis rotundatis; isthmo connexivo subnullo, lato; sinu acutangulo; cytiodermate spinis acutis, robustis numerosis armato. Diam.—Long. yxypy = 0.0014". Lat. yahq” = 0.001”. Hab.—South Carolina; (Ravenel) Wood. Mediocre, a little longer than broad; semicells from the front elliptical, from the vertex trian- gular, with the sides convex or slightly retuse, and the angles rounded ; connecting isthmus obsolete, broad sinus acute-angled; cytioderm armed with numerous acute robust spines. Remark.—Fig. 22, pl. 21, represents the front view of an empty frond of this g > | i ‘ pty plant; fig. 22 a, the side view, and fig. 22 6, the end, all magnified 750 diameters. St. hirsutum, (Eure.) Brés. St. magnum, tertiam partem circiter quam longius quam latum, plus minus dense spinulosum, sinu plus minus lineari, acutangulo; semicellulis late ellipticis vel subsemiorbicularibus, spinis tenuibus strictis hirsutis, a vertice triangularibus, angulis obtuse rotundatis, lateribus rectis vel leniter convexis. (R.) Species mihi ignota. Diam.—Sine spinis 0.0015”. Zygospor. 0.0022”. (R.) Syn.—St. hirsutum, (EHRENBERG) Brépisson. RaBennorst, Flora Europ. Algarum, Sect. III. p. 211. - Hab.—Florida; Rhode Island; Bailey. Rhode Island; (S. T. Olney) Thwaites. 20 August, 1872. 154 FRESH-WATER ALG& OF THE UNITED STATES, Segments in f. y. semiorbicular, separated by a linear constriction, covered with yery minute, very numerous close set hair-like spines; e. v. with three broadly rounded angles, the spines evenly and numerously scattered; sides slightly convex. Sporangium orbicular, beset with short spines, branched at the apex. (A.) St. Mystrix, Rarrs. St. parvum, tertiam partem longius quam latum, angulis aculeatum (cxterum leve), sinu acu- tangulo; semicellulis subquadratis, angulis late rotundatis, dorso planis, a vertice 3-4 angu- laribus, angulis late rotundatis, plus minus dense aculeatis. (R.) Species mihi ignota. Diam.—0.001”—0 00089”. CR.) Syn.—St. Hystrix, Ratrs, British Desmidiex, p. 128. RApennorsr, Flora Europ. Algarum, Sect. HI. p. 213. Hab.—Rhode Island; (S. T. Olney) Thwaites. Segments in f. vy. subquadrate, extremities somewhat rounded, end margin nearly straight, fur- nished with a few scattered, subulate, acute spines, chiefly confined to the lateral extremities; e. vy. with three or four broadly rounded angles, the spines scattered, chiefly confined to the extremities, sides concave. L. yoys"”—ydsy". Br. yg¢”—g}z". St. Cerberus, (Bainey) Arcuer. St. parvum, tam longum quam latum, sinu rotundato, superficie levi; semicellulis oblongis utroque fine sinuato-truncatis, angulis in aculeum cuspidatum productis, in medio sursum et deorsum prominentiis geminis in aculeum elongatis instructis, a vertice triangularibus, angulis in apice truncato- vel sinuato-bi-cuspidatis, sub apice aculeis geminis brevibus preditis. (R.) Species mihi ignota. Diam.—Cum. acul. 0.00114"—0.0013”. (R.) Syn.—Didymocladon Cerberus, BAiey, Microscopical Observations. St. Cereberus, (BAtLEY) ArncHER. Rapennorst, Flora Europ. Algar., Seet. III. p. 216. Hab.—Florida ; Bailey. Small, deeply constricted, segments three-lobed, lobes with four teeth, two of which project upwards and two downwards at each truncated angle. (A.) D. CYPIODERMA PROCESSIBUS NUMEROSIS, APICE PLERUMQUE TRUNCATIS ET DENTATO-FISSIS MUNITUM. CYTIODERM WITH NUMEROUS PROCESSES, WHOSE APICES ARE MOSTLY TRUNCATE AND DEN- TATELY TORN, St. furcigerum, Brés. St. validam, submagnum, cireiter tam longum quam latum, leve vel subtiliter granulatum, plerumque profundissime constrictum, sinu angusto lineari; semicellulis oblongo-ellipticis, plus minus tumidis, angulis in processus bifureum aut rectum aut divergentem longe pro- ductis, dorso processibus similibus 2, 3, 4, instructis, omnibus processibus achrois granulato- dentatis, granulis in series ttansversas ordinatis, a vertice 3-, 4-, 6-, 7-, 8-, 9-angularibus vel radiatis, angulis plus minus tumidis, in processus crassum achroum asperum in apice fissum productis. (R.) Species mihi ignota. Long. Sine process, 0.0018”—0.0019"; ¢. pr. 0.003’”—0.0032". Lat. sine proc. 0.00185”; c. pr. 0.0027”. (R.) ; : : ; Syn —Staurastrum furcigerum, Brieisson. Rasennorst, Flora Europ. Algar., Sect. Il. p- 219. ; Didymocladon furcigerus, Raurs, British Desmidies. Hab.—South Carolina; Florida; Rhode Island; Bailey. St. munitum, Woop. St. submagnum, fere 4 plo longins quam .atum, medio leviter constrictum, semicellulis a fronte FRESH-WATER ALGA OF THE UNITED STATES. 155 enormiter hexagonis, angulis in processus rectos et divergentes productis, dorso processibus similibus 4—5 instructo ; semicellulis a vertice polygonis vel suborbicularibus margine proces- sibus numerosis, plerumque 9 instructo; dorso processibus 5-8 instructis; processibus omni- bus similibus, granulato-dentatis, apice achroo simplicibus, bifureatis vel fissis. - — . * yi be . ace OFA It Qo” Diam.—A vertice cum processibus, psi5y = -00475”. Sine process. 732559’ = 002”. Syn.—St. munitum, Wood, Proceed. Ac. Nat. Sc., 1869. Hab.—In lacu Saco, New Hampshire; (Lewis) Wood. S. rather large, about one-half longer than broad, slightly constricted in the middle; semicells from the front irregularly hexagonal, the angles prolonged in straight divergent processes, and the surface furnished with four or five similar ones; semicells from the vertex polygonal or suborbicular, the margin furnished with numerous processes, mostly about nine, and also with 5-8 on the dorsum; processes all similar, granulate-dentate, their transparent apices simple, bifureate or torn. Reemarks.—This species is most closely allied to Sé. fureigerwm, Bréb., from which it is at once distinguished by the orbicular vertex. The constriction between the semicells is also very different. In Sé. munitwm it is a gradual, not very deep, hour-glass contraction ; in S#. furcigerum it is very narrow and linear. Fig. 13 a, pl. 13, is a front view of this plant magnified 260 diameters ; fig. 13 3, the end view of the same. St. eustephanum, (Eurs.) Rares. St. laterum integrorum angulis productis apice spinulosis, spinularum furcatarum corona media dorsali. (R.) Species mihi ignota. Syn.—Desmidium eustephanum. EWRENBERG, Verbreitung und Einfluss der Mikrosk. Lebens in Sud- und Nord-Amerika, t. 4, f. 23. Staurastrum eustephanum, (Eurs.) Rawrs, British Desmidiex, p. 215. Stephanoxanthium eustephanum, KUTzING. JABENHORsT, Flora Europ. Algarum, Sect. III. p. 221. Staurastrum eustephanum, RAtrs. IABENHORST (Joc. cit.) Hab.—West Point, New York; Bailey. End view triangular with six emarginate spines on the upper surface; each angle terminated by a short ray tipped with spines. (Ralfs) St. senarium, (Eure ) Rares. Antecedenti simile sed Jaterum parietibus spinulis fureatis binis (sex), corona dorsali senaria. (R.) Species mihi ignota. Syn.—Desmidium senarium, EnRENBERG, Verbreitung. T. IV. Stephanoxanthium senarium, Kurzinc. Rapennorst, Flora Europ. Algarum, Sect. ITT. p. 220. Staurastrum senarium, (Eurs.) Rares, British Desmidier. RaBentorst, (loc. cil.) Segments smooth in end view with three angles, each terminating in a short process tipped by minute spines, without lateral processes, but with six others confluent at their bases on the upper surface, divergent and forked. (Archer.) Genus XANTHIDIUM, Enurs. Cellule singule vel gemine concatenate, inflato-rotundate, profunde constricte ; semicellule compresse, oblonge, hemisphwrice vel subquadrangulares, centro in tuberculum rotundatum vel truncatum et denticulatum protuberantes, ex transverso oblongo-rotundate. Cytioderma firmum setis, aculeis vel spinis simplicibus aut bi- tri-furcato-divisis armatum. Massa chlorophyllacea radiatim expansa. Zygospore armatie. (R.) 156 FRESH-WATER ALG# OF THE UNITED STATHS, Cells single or geminately concatenate, inflated, profoundly constricted ; semicells compressed, oblong, hemispherical or subquadrangular, protruding in the centre as a rounded truncate or dep- ticulate tubercle Cytioderm firm, armed with sete, or simple, or bi- tri-fureately divided spines, Chlorophyl radiately expanded. Zygospores armed. Remark.—It has so happened that I have identified but a single species of this genus. X. aculeatum, Eure. X. parvum, singulum, sparsum, diametro ipse subsquale, ex obliquo ellipsoideum, diametro duplo longius, constrictione obtusa lineari, semicellulis oblongis subreniformibus, basi sub- planis, dorso late rotundatis, tuberculo centrali minus elevato, truncato, margine autem crenato- dentato ; cytiodermate undique aculeis subulatis obsito. (R.) Species mihi ignota. Diam.—(Sine aculeis) 0.0025”—0.0029". (R.) Syn.—Xanthidium aculeatum, HHRENBERG. RABENHORST, Flora Europ. Algarum, Sect. III. p. 222. Hab.—Prope Savannah, Georgia; Bailey. Frond inf. v. broader than long; constriction deep, linear; segments somewhat reniform; spines subulate, short, scattered, chiefly marginal ; central protuberance cylindrical, truncate, border minutely dentate. (A.) X. Arctiscon, Eure. X. semicellulis globosis, binis, aculeatis, aculeis numerosis undique sparsis crassis asperis apice trilobis. (R.) Species mihi ignota. Syn.—X. Arctiscon, EHRENBERG. RABENHORST, Flora Hurop. Algarum, Sect. III. p. 224. Hab.—America borealis ; Ehrenberg. Frond in f. v. about as long as broad; constriction forming a wide notch; segments narrowed at the base, with broadly rounded ends; spines numerous, restricted to the outer margin, scattered, elongate, stout, terminated by three or four diverging points. (Archer.) X. armatuma, (Brés.) Rares. X. maximum, validum, solitarium vel binatim conjunctum, diametro plerumque duplo longius; semicellulis subcordatis vel angulari-rotundatis tuberculo centrali subelevato, truncato, mar- gine granulato-dentato preditis; eytiodermate verruculoso et processibus swepius geminatis truncatis apice inciso-furcatis instructo. (R.) Syn.—Xanthidium armatum, (Bréstsson) Raters, British Desmidiee et RABENuorsT, Flora Europ. Algarum, Sect. III. p. 222. Hab.—South Carolina; Florida; Bailey. Saco Lake; (Lewis) Wood. Frond large, in f. v. twice as long as broad; constriction deep, linear; segments broadest at the base; efds rounded or somewhat truncate; spines in pairs, principally marginal, short, stout, terminated by three or four divergent points; central projections cylindrical truncate, the border dentate; e. f. punctate. Sporangium large, orbicular, with depressed tubercles, perhaps immature. L. 35”. B. ghy”. (A.) 270 Remark.—Fig. 17, pl. 13, is a front view of a frond, magnified 260 diameters. X. bisenarium, Eure. X. semicellulis globosis subangulosis, binis, aculeatis; aculeis fasciculatis, fasciculis in quovis globulo senis. Species mihi ignota. Syn.—X. bisenarium, BHRENBERG. Rapenuorst, Flora Europ. Algarum, Seet. III. p. 224. r Brébiesoniay a j X. Brébissonii, Raurs. Arcurr, Prircuarp’s Infusoria, p. 736. FRESH-WATER ALG OF THE UNITED STATES. 157 Tab.—America ; Ehrenberg. Frond in front view broader than long; constriction deep, acute not linear; segments subellip- tic, sometimes irregular spines subulate, geminate, marginal, central protuberance cylindrical, ’ truncate border minutely dentate. lL. (not including spines) g},”. B. g4,” to 335”. X. cristatum, Brés. X. parvum, leve; semicellulis subhemispherico-reniformibus, utroque polo aculeo unico in- eurvo, ambitu aculeis octo geminatis, a dorso ovato-ellipticis, utroque polo aculeis ternis, in medio plerumque aculeo abbreviato. (R.) Species miht ignota. Diam.—0.00196". RB. Syn.—xX. cristatum, Bréstsson. Ratrs, British Desmidier, et Rasenuorst, Flora Europ. Algarum, Sect. III. p. 224. Hab.—South Carolina; Georgia; Florida; Bailey. ' Frond rather longer than broad; constriction deep, linear; segments subreniform or truncate at ends; spines straight or curved, subulate, marginal, one at each side, at the base of the segment, solitary, the others geminate, in four pairs; central protuberance short, conical. (A.) X. coronatum, Eure. X. semicellulis subglobosis binis, aculeatis, ubique asperis, aculeis crassis apice truncatis triden- tato-coronatis quatuor utrinque dorsalibus, uno utrinque latere medio. (R.) Species mihi agnota. Syn.—X. coronatum, EuReNBeERG, Verbreitung, p. 138, et Rapennorst, Flora Europ. Alga- rum, Sect. III. p. 224. Asteroxanthium coronatum, Ktrzinc. Rapenuorst, (loc. cit.) Hab.—America; Ehrenberg. Remark.—Mr. Archer appears to think that this species is simply a form of Staurastrum fureigerum. (Breés.); see Pritchard’s Infusoria, p. 743. X. fasciculatun:, Eure. X. parvum, singulum, constrictione profunde lineari ; cytiodermate levi vel sublevi; semicel- lulis oblongo-reniformibus vel hexagonis, diametro duplo longioribus, ambitu aculeis gracilibus geminatis 4-6, a dorso ellipticis, utroque polo aculeis quatuor instructis. (R.) Species mihi agnota. Diam.—0.00228"—0.00256”. (R.) Syn.—X. fasciculatum, HuReNBeRG. RApenuorst, Flora Europ. Algarum, Sect. IIT. p. 223. Hab.—South Carolina; Georgia; Florida; Rhode Island; Bailey. Frond about as long as broad; constriction deep, linear; sezements somewhat reniform or sub- to} ? ’ ? 5 hexagonal, twice as broad as long, spines slender, subulate geminate, marginal, in four or six £ ? Fo] ’ ? 5 5) pairs ; central protuberance short, conical, somewhat truncate. (A.) Genus ARTHRODESMUS, Eurs. Cellule profunde constricts ; semicellule compresse aut oblong, utroque polo aculeo subulato firmo instruct, aut quadrangulares, angulis in aculeum rectum vel curvum productis, a dorso vel elliptice vel fusiformes. Massa chlorophyllacea in fascias quatuor radiantes disposita. (R.) Cells profoundly constricted; semicells compressed. or oblong, furnished at each end with a subu- late spine, or else quadrangular with the angles produced into straight or curved spines, the dorsal aspect, elliptic or fusiform. Chlorophyl masses disposed in four radiating fascia. 158 FRESI-WATER ALGA OF THE UNITED STATES Remarks.—I have found only a single undescribed species of this genus, but the following European forms have been detected in this country by Prof. Bailey. The genus appears to be, as Prof. Rabenhorst says, scarcely distinguishable from Xan- thidium or Staurastrum. A. octocornis, Eure. A. parvus, levis, constrictione lata excavata ; semicellulis trapezoideis, inciso-quadriradiatis, radiis in aculeum acutissimum strictum porrectis, a latere elongato-ellipticis, diametro fere triplo longioribus, utroque polo aculeum singulum gerentibus. (R.) Diam.—0.00065". (R.) Syn.—Nanthidium octocorne, RAurs. BatLey, Microscopical Observations, p. 29. Arthrodesmus octocornis, EHRENBERG. JABENHORST, Flora Europ. Algarum, Sect. III. p: 223. Hab.—Florida; Rhode Island; Bailey. Frond smooth, minute, about as long as broad; constriction a wide notch; segments much compressed, trapezoid, each angle terminated by one or two straight, subulate, acute spines, the intervals between the angles concave. (A.) a. Spine solitary at each angle. L. yy”. B. yep g”. (A.) b. Larger spines geminate at each angle. LL. ygsq”. B. gig”. (A) A. quadridenms, Woop. A. late ovalis, vel suborbicularis, paulum longior quam latus, cam margine crenato-undulato; semicellulis nonnihil reniformibus, utroque fine aculeo subulato, modice robusto, acuto, re- curyo, armatis; cytiodermate cum verruculis paucibus modice minutis in seriebus paucibus dispositis instructo; semicellulis a vertice acute ellipticis, et cum margine crenato et super- ficie sparse verruculosa. Diam.—Wat. ze55 = .00075” ;, long. gy = - 00125" Syn.—A. quadridens, Wood, Proc. A. N. 8. 1869. Hab.—In lacu Saco, (Lewis) Wood. Broadly oval or suborbicular, a little longer than broad, with the margin crenately undulate; semicells somewhat reniform, at each end armed with a subulate, moderately robust, acute, recurved large spine ; eytioderm with a few smallish tubercles arranged in three or four rows ; semicells from the vertex acutely elliptical, with the margin crenate and the surface sparsely warty. Remari:s.—Vhis species approximates A. divergens, from which it differs in the arrangement of its granules, its attaining not one-half the size, and, I believe, in the larger and more robust spines, Fig. 2, pl. 20, represents an empty frond of this species. A. Incus, (Bres.) Hassat. A. parvus tam longus quam latus, constrictione lineari obtusa vel late excisa; semicellulis oblongo-quadrangularibus, angulis externis aculeatis, internis rotundatis inermibus, aculeis longis singulis divergentibus. (R.) Diam.—Max. 0.00098”. Long. 0.00091”. Spor. (sine acul.) 0.00085”. ’ 5 Syn.—A. Incus, (Brisisson) Hassan, Fresh-Water Alge, p. 357, et RABENHORST, Flora Europ. Algarum, Sect. III. p. 226. Hab.—Georgia; Florida; South Carolina; Rhode Island; Bailey. Frond minute, smooth, as long, or longer than broad, constrictions a deep notch or sinus; Seg- ments with inner margin turgid, outer truncate ; spines subulate, acute ; sporangium orbicular, spinous ; spines subulate. (Archer) FRESH-WATER ALGA OF THE UNITED STATES. 159 A. convergens, Eure. A. levis mediocris, profunde et anguste constrictus, aculeis convergentibus armatus; semicel- lulis ellipticis vel ovato-oblongis, nonnunquam reniformibus, utroque fine aculeo longo firmo incurvo instructis. (R.) Diam.—0.00185”—0.0016". (R.) Syn.—A. convergens, EHRENBERG. JABENHOoRST, Flora Europ., Algarum, Sect. III. p. 227. | Hab.— South Carolina; Georgia; Florida; Rhode Island; Bailey. Frond smooth, broader than long; constriction deep, acute ; segments elliptic, each having its lateral spines curved towards those of the other; ends convex. L. ,Jy59’—;}," B. y357"— te ‘ 4 sta. (Archer) Fammy ZYGNEMACE. Cellule cylindric, equipolares, similes, in familias filamentosas arcte conjuncte, et cytioblasto eentrali plasmate plerumque radiante involuto, et plasmate chlorophylloso aut effuso, aut effigurato, aut (plerumque) in fascias spirales ordinato, et granulis amylaceis instruct. Filum simplex. Pro- pogatio fit zygosporis conjugatione cellularum binarum ortis. Conjugatio triplici modo, aut lateralis aut sealariformis vel genuflexa. Vegetatio fit divisione transversali repetita. Cells cylindrical, the same at both ends, closely conjoined into filamentous families, furnished with a central cytioblast wrapped up in generally radiating protoplasm, and with chlorophyllous proto- plasm effused in shapeless masses or arranged in spiral filaments, and also with scattered starcb- granules. Filament simple. Propagation takes place by means of zygospores, arising from the | conjugation of two cells. Conjugation occurring in three ways, lateral, scalariform, and genuflexuous. Growth taking place by means of transverse division of the cells. Remarks.—The family under consideration is among the commonest and most widely diffused of all the fresh-water alge. In almost every ditch or spring, or dripping moss-covered rock representatives of it are to be found, so that wherever quiet water is they may be confidently looked for. The single filaments are so minute that frequently the unaided eye cannot distinguish them, but multiplication with them is such a rapid process, that wherever found they are in great masses. These masses, when growth is active, are of a beautiful intense green, glistening and shining with the gelatinous matter which coats the threads and makes the mass so slippery. They may be found in greater or less abundance at all seasons, but as the specific characters are largely of sexual origin, non-conjugating specimens are of little value. For this reason, Zygnemas are only worth gathering when in fruit. The spores appear to be formed only in the spring and early summer, at least these are the only times in which I have found fertile filaments. In this neighborhood T have collected them in excellent condition as early as the beginning of April and as late as the latter part of June. Further south, conjugation of course commences earlier, and fine fruiting specimens received by myself from Mr. Canby were col- lected in Florida by him in February. When conjugating freely the mass of Zygnema or Spirogyra loses its beautiful bright green color and become dingy and even brownish, often very dirty looking. The collector soon learns to pass by the beautiful vivid mass, as comparatively worthless, and fasten upon the pale, wan, sickly, apparently dying specimens as prizes worthy of a place in his cabinet. In the Zygnemacee the individual plant, as ordinarily considered, is a filament 160 FRESH-WATER ALG OF THE UNITED STATES. composed of a varying number of cells placed, end to end, all alike, and each of them apparently independent of its associates. Each cell in one sense is, therefore, a perfect, complete individual, capable of living dissociated from its companions, How far the life of one of these cells is influenced by that of its neighbors is un- certain, probably to a slight extent, possibly not at all, At any rate, they are so far independent that the filament is rather a composite body than a unit of life, These cells are cylindrical, with the ordinary cellulose wall, which can commonly be stained blue by iodine and sulphuric acid, and is often distinctly composed of layers, but never has any ‘secondary markings,” each layer being precisely like that superimposed upon it. Outside of the wall is a jelly-like sheath, which is mostly not discernible from its thinness and transparency, although it no doubt exists, as is proven by the slipperiness of the general mass. The primordial utricle is always present. The chlorophyl is variously arranged, most generally in bands, either straight or spiral, sometimes in definite irregular masses, sometimes diffused through the cell. Imbedded in it are, at certain seasons, numerous minute, gene- rally shining, granules, which are either minute specks of starch, or little drops of oil, Besides these there are contained in it, especially in the bands of chlorophyl, more or less numerous comparatively large, oval or roundish bodies, with a distinct outline and a deeper color than the surrounding portions. These masses are pro- toplasm, dyed with chlorophyl-green, and are believed to be especially active in the formation of starch. At times, iodine turns them simply brown; at others it colors their inner portions blue and their outer brown, showing them to contain starch. The general cavity of the cell is occupied by fluid, in which is placed the nucleus. This is mostly single, but rarely, according to Negeli and other authori- ties, double, and De Bary states that he has seen three nuclei in a single cell. I have never seen more than one, and think that even this is not rarely absent, having certainly repeatedly failed to demonstrate its presence. It is colorless, often with a nucleolus, transparently bright, irregular in form, placed in the centre of the cell with numerous arms radiating out from it, some of them ending within the cell, others connecting it with the primordial utricle. De Bary states that this nucleus occasionally is tinged green with chlorophyl, I do not remember ever to have seen 1t so. Ihave not infrequently seen numerous minute dark granules, similar to those seen in Closterium, scattered through the inside of the cell, in active motion. Some- times they are to be found collected in vast numbers near the ends of the cells, dancing and swarming about one another, and passing off in small streams from one end to the other, coasting along close to the primordial utricle, in a word, ex- hibiting precisely the same motions as are so common among the desmids. The Zygnema filament grows in length by a process of cell multiplication by division of the simplest kind. It seems to be somewhat uncertain whether the nucleus always divides into two asa part of the process or not. These plants multiply both by the separation of cells and their subsequent growth, and, by means of resting spores, the so-called Zygospores, The first appearance of separation of the cells is an evident disposition to the rounding off of the ends of the cells. The corners are first rounded and separated FRESH-WATER ALGA OF THE UNITED STATES. 161 and this continues until only the centres of the ends are in apposition, and in a little while even these separate. This certainly, at least, is the process in certain species; but I have thought, that in other cases cells were separated by a simple splitting of the end wall, each cell retaining its half of the partition. The zygospores are produced by a process of union of two cells, to which the name of conjugation has been given, Very rarely, if ever, is there any difference between the cells before conjugation, and it has not existed in any species which has come under my notice; but, after conjugation, the receiving cell is frequently enlarged, the other remaining cylindrical. De Bary, however, states that he has found a small but constant difference between the fertile and sterile cells of Spirogyra Heeriana, The first perceptible change in a cell about to produce a resting spore, appears to be a loosening of the primordial utricle from the outer wall, and a contraction of it upon the cell contents, which thus are crowded together and more or less deformed. Simultaneously with this, or a little after or before it, the side wall of the cell is ruptured and a little pullulation or process is pushed out, which directly coats itself with cellulose and rapidly enlarges to a considerable diameter, at the same time growing in length until it meets a similar process pushing out from an opposing cell, or has attained as great a length as its laws of development will allow. When two processes meet they become fused together, the end walls are ruptured, and the contents of one cell passing over are received within those of the other, or else the contents of both cells meet within the connecting tube and there fuse together. This is the more common mode of conjugation, in which two cells of distinct filaments become joined together by a connecting tube. It is evident, that, if the filaments are fertile to their fullest extent, there will be as many of these connecting tubes as there are pairs of cells in the filaments, and a ladder-like body will be formed, the original filaments corresponding to the side pieces, the connect- ing tubes to the rounds. Hence this method of conjugation has received the name of scalariform. In the so-called “lateral conjugation,” instead of cells of different filaments join- ing, adjacent cells of one filament unite together to complete the process. The union of the two cells appears to take place in several ways. In accordance with one plan (fig. 1 a, pl. 14), connecting tubes, pushed out from near the ends of the cells, grow for a short distance nearly at right angles to the long axis of the filaments, and then bend at aright angle to themselves so as to run parallel to the filament-cells. The ends of these processes are, of course, opposed to one another, and coming in contact fuse together so as to form a continuous tube for the passage of the endochrome. Another method by which neighboring cells are sometimes connected is by the formation of coadjacent pouch-like enlargements of the opposing ends, and a subsequent fusion of these newly formed enlargements by the absorption of the end wall between them. (See fig. 2, pl. 14.) Sometimes I think the union of two neighboring cells is facilitated by a curved neck forming to one or both of them, so that they are bent at an angle to one another, and can readily be united by means of a straight tube. There is still another method of conjugation, the so-called genujlexuous, in which, 21 August, 1872. 162 FRESH-WATER ALG OF THE UNITED STATES instead of a connecting tube being formed as the medium of union, two cells of opposing filaments become sharply bent backwards, so that thei central portions are strongly thrust forward as obtuse points, which, coming in contact, adhere and allow of a passage-way between the cells being made by the absorption of their cohering walls, A curious modification of, or departure from, the ordinary method of conjugation is sometimes seen, in the union of three instead of two cells. This is, I think, very rare, but has been seen by Meyen in the genus Zygnema, as well as by Schleiden and De Bary in Spirogyra. I myself have observed it once or twice in the latter genus. One of the cells plays the part of the female, receiving the con- tents of the other two, so that in it the primordial utricles of the three, with their contracted protoplasm, are fused into a zygospore. The zygospore, however formed, varies in shape, but is mostly oval or globular, sometimes cylindrical, and when ripe is in most if not all species of a dark brown- ish color. It is described both by Pringsheim and De Bary as having three coats, but I have frequently found it impossible to demonstrate the presence of all of these, and I believe that not rarely one of them is absent. The outer coat is developed first and is the thickest and firmest. Occasionally it is double, i. e. composed of two distinct layers or parts, as in Sp. protecta, in which species the outer of these layers is the thickest, firmest, and most evident, whilst the inner layer is translucent and much less apparent. ‘The second coat contains the coloring matter, which is sometimes brown, sometimes decidedly yellowish. The inner coat is not readily seen, It is elastic, thin, and is the last of the three to be formed. The principal contents of the ripe spore are protein compounds (protoplasm), oil-drops, starch granules, and pigment. ‘The oil is generally much more abun- dant than the starch, and not rarely the minute, bright drops entirely replace the little granules. According to Prof. De Bary, the pigment frequently, but not always, reacts with sulphuric acid, as does that of the fungal family, Uredinee, striking with it a deep blue. The germination of the spore, both in the genus Spirogyra and Zygnema, is very simple. ‘The first step is an elongation and growth of the protoplasmic central mass, together with the inner transparent cellulose coat, and a consequent rupturing of the outer two coats, through which the newly forming plant protrudes and finally escapes. In this way in the genus Spirogyra an elongated club-shaped cell arises, one end of which is much larger than the other and contains all the chlorophyl. Sometimes a nucleus is perceptible in this cell, sometimes it is not. The larger end now becomes cut off by a partition wall from the smaller; if no nucleus has been previously apparent it now becomes so, and the first stage of development is completed. The filament after this grows by a simple repetition of the process of division in the larger end and the cells formed out of it. The smaller end undergoes little or no change. In the genus Zygnema, the cell that first emerges from the germinating spore is a perfect one, similar in all respects to those seen in the fully formed filament, which is developed out of it, by a simple process of cell division. FRESH-WATER ALG OF THE UNITED STATES. 163 Besides the true Zygospoves, Hassall many years since described bodies (Fresh- water Algz, vol. i. pp. 132, 156, 170), which he found in filaments of this family, and which resemble in all respects ordinary Zygospores, but are produced each in a single cell without any aid from a second cell. He affirmed that he had observed this phenomenon especially in two species, Spirogyra mirabilis and Zygnema notabilis. ‘These observations were doubted by some, whilst others, as Alexander Braun, supposed that there was a division of the cell protoplasm into two distinct portions, and then a conjugation of these within the original cell, and that Mr. Hassall had overlooked these changes. Prof. De Bary, however, states that he has seen a great many instances of this production of spores without conju- gation (all in one species), and that there can be no doubt that Hassall’s obser- vations are substantially correct, and that no division of the primordial utricle such as was imagined by Prof. Braun takes place. Spores formed in this manner, as yet have nat been seen to develop. ‘There is, therefore, no certainty that they are capable of doing so, It is possible that they are merely the results of abortive attempts at reproduction, wanting the power of development because not fertilized. Pringsheim and others have drawn from these bodies strong argument against the idea, that conjugation is to be looked upon at all as a sexual process. The arguments both for and against regarding conjugation as the simplest ex- pression of sexual life are ably elaborated by De Bary, Untersuchungen iiber die Familie der Conjugatem, p. 57, to which I must refer those desirous of following the subject further, contenting myself with expressing an agreement with the con- clusions there arrived at, namely, that in conjugation the first dawnings of sexuality are to be found. Looking at it in this light Prof. De Bary states his conviction that the spores formed in the manner last described, bear the same relation to the true Zygospore that the bud of a Phanerogam does to its seed, or the Zoospore of an Gidogonium does to its resting spore. Quite a number of bodies have been described by the older authorities as being found within the cells of plants of this family, which more recent observers have proven to be parasitic. Such are the “Spermatic spheres,” transparent spheres motile by virtue of vibratile cilia, various monads, &c¢. &e., bodies for which it has been claimed, from time to time, that they were sexual elements, spermatozoids. Genus SPIROGYRA, Linx. Cellule vegetative cylindrice, fasciis chlorophyllosis spiralibus instructs. Conjugatio aut Jater- alis aut scalariformis aut et lateralis et scalariformis. Syn.—Spirogyra et Rhynchonema, Kurzina, RABENHORST, et auctores. Salmacis, Bory. Zygnema (partim), HAssa uu. Vegetative cells eylindrical, furnished with spiral chlorophyl bands. Conjugation either lateral or scalariform or both lateral and scalariform. Remarks.—The genus Spirogyra, as defined above, has been divided by Kiitzing, Rabenhorst, and others into two genera, the characters being drawn from the method ef union of the conjugating cells; in the one case the neighboring cells of a single filament (Rhynchonema), in the other cells of distinct filaments (Spirogyra), 164 FRESH-WATER ALG OF THE UNITED STATS uniting to form the spore. This at first sight appears to be a good ground for sepa- ration, but there are certain species in which, undoubtedly, both the former and the latter method of conjugation take place indifferently. Such species make a third group so precisely between the two others as, to my mind, to fuse them toge- ther and necessitate either the acknowledgment of three genera or the denial of more than one. ‘The latter seems to me the more philosophical course, A. CONSUGATIO LATERALIS (RHYNCHONEMA). A. CONJUGATION LATERAL. Sp. elongata, Woop. Sp. articulis vegetativis diametro 7-20 plo longioribus; articulis sporiferis multo brevioribus, valde tumidis ; cytiodermate utroque fine protenso et replicato ; fascia unica, laxissime spirali; anfractibus plerumque 7; sporis ellipticis, diametro 1-24 plo longioribus. Diam.—Spor. 7857’ = .00106”. Artic. vegetat. 7455” =.0005”. Syn.—RhynchonemaelongatumW oop, Prodromus, Proc. Amer. Philos. Soe. 1869, p. 137. Hab.—In aquis limpidis, prope Philadelphia. Sterile joints 7-20 times longer than broad; fertile joints much shorter, greatly swollen; cell wall at each end produced or folded in; chlorophyl filament 1, spiral lax; turns mostly 7; spores elliptical, 2-24 times longer than broad. Remarks.—I found this species about the middle of March, fruiting in a little pool near Chelten Hills, six or eight miles north of this city. It did not form a distinct stratum by itself, but was floating, intermingled with great numbers of other filamentous alge, such as fragillarie, zygneme, &c. It seems to be most. closely allied to the European R. minimum ; it however not only attains a some- what larger size but also differs from that plant in the proportionate length of the sterile cells, in the number of the turns of the chlorophyl spiral in the cell, and in the proportionate length and breadth of the spore. Fig. 1, pl. 14, represents portions of sterile filaments magnified 450 diameters; la, a part of a fertile filament, magnified 450 diameters. Sp. pulchella, Woop. Sp. articulis sterilibus diametro 2-3 plo longioribus ; sporiferis nonnihil tumidis; fascia unica, anfractibus 3-4; sporis ellipticis, diametro fere duplo longioribus; cytiodermate utroque fine | protenso et replicato. Diam.—Artic. Steril. -455"—725y" =.00033”—.0013". Spor. 725q”—7}85" =.0012"”—.00133". | Syn.—Rhynconema pulchellum, Woop, Prodromus, Proc. Amer. Philos. Soe. 1869, p. 138. Hab.—In stagnis, prope Philadelphia. Sterile’ joints 2-3 times longer than broad; fertile joints somewhat swollen; chlorophyl band one; turns of spiral 3-4; spores elliptical, almost twice as long as broad; cell wall at each end produced or folded in. Remarks.—Vhis species was found by myself fruiting in April, 1869, im stagnant ditches below the city, and in similar localities near Camden, New Jersey. It did not occur in masses but singly, intermixed with great numbers of other fruiting spirogyras. Most of the filaments seen were about .0010’ in diameter; in but a single instance did they come much short of this. This species differs from R. clongatum, among other points, in the shortness of the tubes connecting the fertile FRESH-WATER ALGA OF THE UNITED STATES. 165 cells. I have never been able to identify an entirely sterile filament of this species; the measurements and description of the sterile cells were taken from infertile cells in filaments, which in other places had produced spores. Fig. 2, pl. 14, represents a fertile filament, magnified 260 diameters. . CONJUGATIO SCALARIFORMIS (SPIROGYRA VERA). Cytiodermate utroque fine protensum et replicatum. Cytioderm folded in at the ends. Fascia spiralis unica. * * 28 & Bw Spiral filament single. Sp. Weberi, K1z.? Sp. saturate viridis, lubrica; articulis vegetativis diametro 3-20 plo longioribus:; fructiferis ? 5 ] g ? nonnihil inflatis; fascia dentata, plerumque unica sed fasciis duabus in quavis cellula; spire anfractibus 3-8; cytiodermate plerumque utroque fine protenso et replicato ; zygosporis ellip- ticis. Diam.—Artic. steril. 7459’—725y” = .0008"—. 0012”. Syn.—S. Weberi, Kurzinc. Rapenuorst, Flora Europ. Algarum, Sect. III. p. 233. Hab.—In stagnis, prope Philadelphia. Deep green, slippery ; sterile joints 3-20 times longer than broad; fertile joints not swollen ; chlorophyl filaments mostly single, but sometimes two in certain cells, dentate; turns of the spiral 3-8; cytioderm protruded or infolded at the ends; zygospores elliptical. Remarks.—This species, which is abundant around Philadelphia in stagnant ditches, I have found fruiting in the month of April. The number of spirals fre- quently varies even in the same filament. The infolding of the walls at the end of the cells is very often wanting in the fertile cells and occasionally is absent from one end of an ordinary vegetative cell. ‘The American form agrees pretty well with the European, but is, however, larger, and also attains in its cells a greater proportionate length and has more turns of its chlorophyll spirals. The lower limits of the American form are, however, so overlapped by the upper limits of the European, that it seems to me they must be considered identical. Fig. 19, pl. 12, represents a pair of fertile filaments of this species, magnified 260 diameters; 19 a, part of a sterile filament, magnified 260 diameters; 19 6, out- line of a couple of fertile cells, magnified 260 diameters. Sp. protecta, Woop. Sp. saturate viridis, lubrica; articulis sterilibus diametro 6 plo longioribus; sporiferis vix tumidis ; cytiodermate utroque fine protenso et replicato; fascia unica; anfractibus 6; sporis oblongis vel ellipticis : membrano crassissimo. Diam.—Art. steril. z4$5" =.00146"; spor. lat. 7405”—7125” = .00133”—.0016” long. 7235” = .0033”. Syn.—Sp. protecta, Woop, Prodromus, Proce. Am. Philos. Soc. 1869, p. 131. Sp. deep green, slippery ; sterile joints 6 times longer than broad ; fertile cells scarcely swollen; cell wall folded in at the ends; chlorophyl band single ; turns 6 ; spores oblong er elliptical, spore wall very thick. Remarks.—I found this species in the latter part of April fruiting in a ditch in a meadow a little south of the mouth of Wissahicon Creek, near this city, and as late as the 25th of May in the “neck” below the city. It is remarkable 166 FRESH-WATER ALG&X® OF THE UNITED STATES, for the very great thickness of the walls of the spore. ‘There are two very appa- rent coats separated by a thin not very evident one. ‘The outer is the thickest ; it is very thick, firm, and nearly colorless. The inner coat is of a decided orange; brown. ‘The parent-cells which give origin to these spores are slightly enlarged in diameter, Sometimes the spores, instead of being elliptical, are irregular in shape. Fig. 3.a, pl. 14, represents a sterile filament, magnified 250 diameters; fig, 3, a mature spore, magnified 450 diameters. Sp. imsigmis, (Hassavv) Krz. Sp. articulis sterilibus diametro 5-14 plo longioribus ; fasciis 2 (rarius 1-8), laxe spiralibus, angustis, crenatis; articulis fructiferis nonnihil tumidis ; cytiodermate utroque fine replicato vel protenso ; zygosporis rubido-brunneis, ovato-ellipticis. Diam.—0.0015". Syn.—Zygnema insigne, Hassani, Fresh-Water Alge, p. 440. Spirogyra insignis, (HaAssauL) Kirzina. RaABENHORST, Flora Europ. Algarum, Sect. III. p. 238. Hab.—In stagnis, prope Philadelphia. Sterile joints 5-14 times longer than broad; chlorophyl filaments mostly 2 (rarely 1-3), laxly spiral, narrow, crenate; fertile joints somewhat enlarged; cytioderm at each end folded in or produced ; zygospores reddish-brown, ovate elliptical. Remark.—Fig. 6, pl. 16, represents this species. b. Cytioderma cellule fine nec protensum nec replicatum. Cytioderm not infolded in the end of the cell. * Fascize spirali unice (raro due). Chlorophyl band single (rarely two). Sp. longata, (VAucu.) Krz. Sp. dense ciespitosa, lete luteolo-viridis, valde lubrica; articulis sterilibus diametro 2-6 plo longioribus, fertilibus seepe tumidis abbreviatis ; fascia spirali lata, dentata; anfractibus sub- laxis 2-5 ; zygosporis ellipticis. Diam.—0.001". Syn.— Conjugata longata, VAucHER, Histoire des Conferves d’Eau douce, p. 71. Sp. longata, (VAucu.) Kirzina. Rapennorst, Flora Europ. Algarum, Sect. IIT. p. 238. Hab.—In stagnis, prope Philadelphia; Wood. Rhode Island; (S. T. Olney) Thwaites. Densely cxespitose, bright yellowish-green, very slippery ; sterile joints 2-6 times longer than broad ; fertile articles swollen, often abbreviate; chlorophy! filaments broad, dentate; turns of the spiral somewhat loose, 2-5 ; zygospores elliptical. Remarl:s.— According to Prof. Rabenhorst, this species attains in Europe a diameter of .0011” and the cells a length of 8 times their breadth. ‘The same authority also describes the fertile cell as being either not swollen, or moderately so (“aut non aut modice tumidis”). In all the specimens of our American forms which I have seen, the sporangial cells are very decidedly swollen. Fig. 4, pl. 14, represents portions of sterile filaments, magnified 250 diameters, and fig. 4 a, a part of a fertile pair of filaments containing immature spores enlarged” 260 diameters. FRESH-WATER ALG@& OF THE UNITED STATES. 167 Sp. quinina, (Ac.) Kurz. - Sp. saturate viridis, valde lubricata; articulis sterilibus diametro 1-6 plo longioribus ; articulis fertilibus vel haud tumidis vel nonnihil tumidis; fascia unica; spire anfractibus modo den- sioribus, modo laxioribus, nonnunquam laxissimis, plerumque 3, interdum 14—4 ; cytiodermate cellule utroque fine nee protenso nec replicato; zygosporis aut globosis aut ovalibus aut eylindricis. Diam.—Artic. steril. 7} 95" —7h35 = .0013"—.0017"; sporis 7445” =.0014”. Syn.—Sp. quinina, (Acarpu)Ktrzina. Rasennorst, Flora Europ. Algarum, Sect. HI. p. 240. Hab.—In stagnis, prope Philadelphia. Deep green, very slippery, sterile articles 1-6 times longer than broad; fertile joints scarcely or not at all tumid; chlorophyl filament single; turns of the spiral sometimes denser, some- times laxer, sometimes very lax, mostly 3 in number, sometimes 14-4; cytioderm neither infolded nor protruded at the end; zygospores polymorphous, globose, elliptical or cylindrical. Remarks.—This species is very abundant in the ditches around Philadelphia, especially in the “neck” below the city. I have found it fruiting profusely in the month of April. ‘The spores vary very much in form, some of them being globose, others elliptic, and still others cylindrical, with obtusely rounded ends. All these forms may occur in a single filament. ‘The spore cell also varies in the amount of its enlargement. In many cases it preserves its cylindrical shape completely; in ‘ther instances it is markedly swollen. Figs. 4e, 4, pl. 19, represent portions of sterile filaments of this species; figs, sa, 46, and 4d, portions of fertile filaments. tt Fascizx spirale due vel plures. tt Chlorophyl filaments two or many. Tp. decimima, (Mutter) Krz. Sp. sordide viridis, lubrica; articulis sterilibus diametro (0.00135”—0.00159") plerumque duplo-, quadruplo fere longioribus, nonnunquam subequalibus, fertilibus aut non aut modice tumidis; fasciis spiralibus plerumque 2, latis, decussatis, rarius 1 vel 3, anfractibus laxis 1-1}; zygosporis aut ovalibus ant late ellipticis vel subglobosis. (R.) Syn.—Sp. decimina, (MULLER) Kurzine. Rasrnuorst, Flora Europ. Algarum, Sect. IIT. p- 242, Hab.—Prope Philadelphia. Dirty green, slippery; sterile joints mostly from 2-4 times as long as broad (0.00135’— 0.00159"), sometimes about as long as broad; fertile joints either moderately or not at all swollen; spiral filaments mostly 2, broad, decussating, rarely 1-3, turns loose 1-14; zygo- spores either oval, broadly elliptic, or subglobose. Remarks.—I find this species marked in one of my note-books as having been found by myself near this city. I have no distinct recollection of seeing it, and, having preserved neither figure, specimen, nor description, am forced to content myself with copying the description of Prof. Rabenhorst. Sp. dubia, Kz. Sp. viridis in fructe dilute viridis; articulis sterilibus cylindricis diametro 14-23 plo longioribus ; 3 fasciis spiralibus 2-3, angustissimis, nodosis, anfractibus laxis 1-2 (= 3-6); cytiodermate utroque fine nee protenso nee replicato, nonnihil erasso; zygosporis polymorphis, aut sub- 168 FRESH-WATER ALGZ OF THE UNITED STATES: elobosis aut ovalibus, aut subcylindricis, diametro equalibus aut 2 plo longioribus; articulis fertilibus eylindricis, haud tumidis. Diam.—Art. steril. 7155” =.002; spor. 7}§5” — .002". Syn.—Sp. dubia, Kurainc. TRasennorst, Flora Europ. Algarum, Sect. III. p. 243. Hab.—In stagnis, prope Philadelphia. Green, in fruit light green; sterile joints cylindrical, 1}-2} times longer than broad; spiral filaments 2-3, very narrow, nodose, lax, turns 1-2; eytioderm neither infolded nor protruded at the end, rather thick ; zygospores polymorphous, either subglobose, oval, or subcylindrical, as broad as long to 3 times longer ; fertile articles cylindrical, not enlarged. Remarks.—I have found this species growing in the ditches below the city, fruit- ing abundantly in May. When in this condition it forms masses of a dirty, lightish, yellowish-green. ‘The spores mostly fill pretty well the fertile cells. My specimens do not agree completely with the descriptions given of the European. ‘The two forms, however, completely overlap one another, except in one character, namely, the shape of the sporangial cell. I have never seen it swollen or at all tumid in American specimens, whilst in the European it is said to be “modice tumidis.” This difference alone does not, however, seem to me sufficient to characterize a | new species. I have seen specimens of this plant collected by Dr. Lewis at Cobble Mountain. They agree well with the Philadelphia specimens, except in attaining a little larger size, .0021”, and in the sterile filaments having their walls very thick. The character of non-inflation of sporangial cells is perfectly preserved. Fig. 4, pl. 17, represents this species. Sp. rivularis, (HAssALL) RaBenu. (non Kz.) Sp. saturate viridis, lubrica; articulis sterilibus diametro 7-11 plo longioribus; fertilibus eylin- dricis aut vix tumidis; eytiodermato tenuissimo, utroque fine nee protenso nee replicato, fasciis 4, laxe spiralibus, modice angustis, nodulosis et serratis, anfractibus 24; zygosporis ellipticis, diametro 2-24 longioribus. Diam.—Art. ster. 7255"”—7 145” = .0012"”—.00146" 5 spor. 7i95’—7}25”. Syn.—Zygnema rivularis, HAssauu, Fresh-Water Algie, vol. i. p. 144. Spirogyra rivularis, (HASSALL) (non Kurzinc) RaBeNuorRS?T, Flora Europ. Algarum, Sect. III. p. 243. . Hab.—In rivulis, Florida; (CaAnsy) Woop. Deep green, slippery ; sterile articles 7-11 times longer than broad, fertile cylindrical or slightly tumid; cytioderm very thin, neither infolded nor protruded at the end; chlorophyl filaments 4, laxly spiral, moderately narrow, nodose and serrulate, turns 24; zygospores elliptical, 2-24 times longer than broad. Remarks.—This species was collected by Mr. Wm. Canby in Pine Barren Run, near Hibernia, Florida. It is rather smaller than the European forms, but does not appear to be distinct from them. Rabenhorst, indeed, states that there are only two or three chlorophyl spiral bands in a cell, but Hassall in the description of the type states distinctly that in some instances there are four bands, and alse figures the plant so. Fig. 5 a and b, pl. 17, represents sterile cells of this species, magnified 26 FRESH-WATER ALG@&H® OF THE UNITED STATES. 169 diameters. Fig. 5 ¢c is an outline of a pair of fertile cells enlarged to the same extent. Sp. parvispora, Woop. Sp. articulis sterilibus diametro 2-4 plo longioribus ; fructiferis haud tumidis, diametro 1-21 plo longioribus; fasciis spiralibus 4, angustis, nodosis, anfractibus pluribus ; zygosporis parvis- simis, ellipticis, diametro 14-2 plo longioribus; cytiodermate utroque fine nee protenso nec replicato. Diam.—Axrt. steril 7335" =.003”; spor. diam. transv. 7}55”—7}75"” = .002”—.0023”, long. 721,” ” —1500 - Syn.—S. parvispora, Woop, Prodromus, Proc. Am. Philos. Soc. 1869, p. 139. Hab.—In stagnis, Hibernia, Florida. (WM. CANBY ) Sterile joints 2-4 times longer than broad; fertile not tumid, 1-2} times longer than broad ; chlorophyl bands 4, narrow, nodose; turns many; zygospores very small, elliptical, 13-2 times longer than broad; cell wall not infolded at the end. Remarks.—I am indebted to Mr. Wm. Canby for specimens of this species, which he collected in a pond in the Pine Barrens near Hibernia, St. John’s River, Florida. It is remarkable for the comparatively small size of the spores, which do not nearly fill the perfectly cylindrical mother-cells ; indeed they are only about as long as the latter are wide. This species closely resembles S. majuscula, but is larger, does not, that I have ever seen, vary like it in the number of spores, and is especially separated from it by the very small size of the latter. Fig. 7, pl. 15, represents a fertile pair of filaments of this species magnified 125 diameters. Sp. majuscula, Kz. Sp. pallide et sordide viridis, fructus tempore fuscescens ; articulis sterilibus diametro (0.0022” —0.0025”) 244-10 plo longioribus; cytiodermate tenui homogeneo; fasciis 3-4-5 (rarius 7), modo subrectis longitudinalibus, modo laxissime spiralibus, nodosis; zygosporis globosis vel ovalibus. (R.) Syn.—S. majuscula, Kirzinc. RaABennorst, Flora Europ. Algarum, Sect. III. p. 244. Hab.—Prope Philadelphia. ? Pale and sordid green, fuscescent at the time of fruiting; sterile joints 25-410 times longer than broad (.0022”—0 0025”); eytioderm thin, homogeneous ; spiral filaments 3-4-5 (rarely 7) partly straightish and longitudinal, partly laxly spiral, nodose; zygospores globose or oval. Remarks.—Shortly after I commenced to study the fresh-water alge, I found below the city a fruiting Spirogyra, of which I preserved only a drawing, which I have since identified as apparently specifically one with the European S. majuscula, it differing only in not being quite so large; my measurement was ;},” = 0.002". Not having any specimens at hand, I have copied the description from the work of Prof. Rabenhorst. Fig. 1, pl. 15, was copied from the drawing alluded to. Sp. nitida, (Ditw.) Link. Sp. cespitibus, lubricis, saturate viridibus; articulis sterilibus post divisionem diametro sub- wqualibus, ante divisionem 2-3 plo longioribus; articulis fertilis aliis simillibus, haud tumidis ; fasciis spiralibus 4 (3-4 R.), modice latis, anfractibus 1-2; zygosporis ellipticis. 22 August, 1872. 170 FRESH-WATER ALG OF THE UNITED STATES: Diam.—0.0025". Syn.—S. nitida, (DittwyN) Link. RABENHORST, Flora Europ. Algarum, Sect. III. p, 245, Hab.—Prope Philadelphia. Occurring in lubricous turfy masses, of a deep green color; sterile joints after division about as long as broad, before division 2-3 times longer ; fertile joints similar to the others, not tumid; spiral filaments 4, moderately broad, turns 1-2; zygospores elliptic. Remarks.—This species appears to be somewhat rare, at least I have found it but once, and then only in small quantity. Rabenhorst states that there are occa- sionally only three spirals, and his maximum diameter is 0.0031”; he also speaks of the fertile joints as “ vix tumidis.” Sp. diluta, Woop. Sp. articulis sterilibus diametro subequalibus ad duplo longioribus, fructiferis haud tumidis; fasciis spiralibus 5, angustissimis, laxis, valde nodosis; anfractibus plerumque 4, interdum ~ 1; zygosporis sparsis, late ellipticis vel ovatis aut globosis; cytiodermate modice tenue, in utroque fine nee protenso nec replicato. Diam.—Artic. steril. +23,” = .003”. Syn.—S. diluta, Woop, Prodromus, Proc. Am. Philos. Soc. 1869, p. 139.' Hab.—In stagnis, prope Philadelphia. Sterile joints about as long as broad to twice longer, fertile cells not swollen ; chlorophyl bands 5, exceedingly narrow, lax, strongly nodose ; turns mostly 4, sometimes 1; zygospores few, broadly elliptical, ovate or globose; cell wall moderately thin, not infolded at the ends. Remarks.—I have found this species several successive seasons growing in the ditches in the Neck, below the city, especially in the neighborhood of the large stone barn, built by the great millionaire, and still known as “ Girard’s Barn.” The spirals are very narrow and slender, and are moderately close to one another. They are chiefly made up of a number of chlorophyl nodules, the connecting thread between which is often very faint. In all the fruiting specimens, as I have seen them, the spores have been very few in number, most of the cells of the fertile filaments appearing to have aborted, so that they are simply empty. In most cases only about every third or fourth cell contained a spore. Fig. 2, vl. 15, represents this species. Sp. setiformis, (Rorn) Krz. Sp. saturate viridis, lubrica; articulis sterilibus diametro paullum brevioribus ad 1} plo lon- gioribus; articulis fructiferis haud inflatis; fasciis 8-8, latis, dentatis, interdum nonnihil remotis, sed seepe arcte et dense conjunctis, nodosis; zygosporis globosis vel late ovalibus. Diam.—.0035". Syn.—S. setiformis, (Rotm) Kurzina. Ravennorst, Flora Europ. Algarum, Sect. III. p. 246. Hab.—In stagnis, prope Philadelphia. Deep green, slippery ; sterile joints a little shorter to one and a-half times longer than broad ; fertile joints not inflated; spiral filaments 3-8, broad, dentate, sometimes somewhat remote, sometimes closely and densely conjoined, nodose ; zygospores globose or broadly oval. Remarks.—None of the descriptions which I have seen of this species state the number of the spiral filaments, but the other characters of the American form sc agree with those of the European plant that it is probable that this one does also The plant is not uncommon in the Neck, fruiting in the spring. PRESH-WATER ALGA OF THE UNITED STATES. ie Fig. 3 a, pl. 15, represents part of a sterile filament of this species; 3b, portion of a pair of fertile filaments, both magnified 125 diameters. Sp. crassa, Krz. Sp. lete viridis, denique sordide viridis; articulis sterilibus diametro subsequalibus, post divi- sionem interdum fere } plo brevioribus, ante divisionem sepe fere 2 plo longioribus ; cytio- dermate tenui, homogeneo, utroque fine nee protenso nec replicato; fasciis spiralibus 4, dentatis vel tuberculatis, seepe aretis, subtransversis, tenuibus; anfractibus 14-4; cellulis fructiferis aliis simillimis, haud inflatis ; zygosporis globosis vel ellipticis. Diam.—Max. .0065.” Syn.—Sp. crassa, Ktz. WRaBENworstT, Flora Europ. Algarum, Sect. III. p. 246. Hab.—In stagnis, prope Philadelphia. Bright green, but finally a dirty green; sterile articles about as long as broad, sometimes after division only half as long, sometimes before division twice as long; eytioderm thin, homo- geneous, not infolded or produced at the ends; spiral filaments 4, dentate or tuberculate, often close, subtransverse, thin; turns from 14 to 4; fertile cells very like the others, not inflated ; zygospores globose or elliptical. Remarks.—This species is very common in the neighborhood of this city, occur- ring in springs, &c., but especially in the ditches in the Neck. It forms long, lubricous masses, of a bright green color, readily distinguishable by the size of the filaments, which are separated with ease by the unaided eye. I have gathered it repeatedly, in fruit, from the middle of April to the middle of June. In this state the mass has lost its bright green color, and when the filaments are closely examined, even without a glass, mmute dark points mark the positions of the spores. Fig. 4 a, pl. 15, represents part of a filament commencing reproduction ; 4 4, fila- ents which have matured the spores; 4, a pair of conjugating filaments. Genus ZYGNEMA. Cellule vegetative cylindrice. Massa chlorophyllacea initio effusa et subhomogenea, postea dis- tincte granulosa aut per cellule lumen distributa, granula amylacea duo centralia involvens, aut in corporibus duobus (in quaque cellula) plus minusve distincte stellatim radiantibus juxta nucleum centralum granum amylaceum unicum involventibus collocata. Conjugatio scalariformis vel late- ralis. Vegetative cells cylindrical. Chlorophyl masses in the beginning effused and subhomogeneous, afterwards distinctly granular, either distributed throughout the cavity of the cell, involving two central starch granules, or gathered together into two masses (in each cell), with more or less dis- tinetly stellate radii and a central starch granule placed near the nucleus, one on each side of it. Z. imsigne, (Hassaui) Krz. Z. cespitibus et plerumque natantibus vel in aqua diffusis, saturate viridibus vel swpe sordide flavo-viridibus ; articulis sterilibus diametro circiter equalibus vel duplo longioribus ; conju- gatione scalariforme (et spe simul laterali, R.); zygosporis globosis ; sporodermate levi. Diam.—Cell. ys855" = -00126"; spor. yon" —1e0550 | = 0.00093”—0.00016". Syn.— Tyndaridea insignis, Hassauu, Fresh-Water Alge, vol. i. p. 163. Zygnema insigne, (HAssauL) Kirzina. Rapenuorst, Flora Europ. Alge, Sect. IIT. p. 249. Hab.—In stagnis, prope Philadelphia ; Wood. Rhode Island; (S. T. Olney) Thwaites. 172 FRESH-WATER ALG& OF THE UNITED STATES. Cwspitose and mostly floating or diffused in the water, deep green, or a dirty yellowish-green ; sterile joints about as long as broad, or twice as long; conjugation scalariform (according to Rabenhorst sometimes at the same time lateral) ; zygospores globose; spore coat smooth, Remarks.—This species is very common around Philadelphia, forming great masses in the ditches of the “« Neck,” growing in the semistagnant water along the railroads, and forming with other algze slimy coatings on the dripping rocks of the Wissahicon and various railroad cuttings. At certain times the cells are found crowded with endochrome, at other times they are almost empty. At certain seasons this plant multiplies with great rapidity after a somewhat peculiar fashion, Constrictions first appear in the filament at the junctions of the cells, which thus look as though their ends were rounding off. ‘This goes on until the ends of the cells are greatly rounded, and are attached simply by their central parts, which soon separate. In this way (fig. 8b, pl. xv.) the filament is resolved into its com- ponent cells, or more generally into as many pairs of cells as compose it, which when once set free in the water rapidly grow into filaments by the ordinary pro- cess of cell multiplication by division. In most cases the zygospores are placed in one of the parent-cells, but I have seen instances in which some of them were formed in the connecting tubes. Fig. 8, pl. 15, represents this species. Z. cruciatum,(VaAucu.) Aa. Z. pallide viride, siccatum fuscescens vel fusco-nigrescens; articulis sterilibus brevicylindricis diametro (0.0016”’—0.00195”) eequalibus vel dimidio longioribus, rarius duplo longioribus, post divisionem factam haud raro dimidio brevioribus, fructiferis non tumidis; zygosporis plerumque globosis, maturis obscure fuscis, sporudermate subtiliter punctatis. (R.) Species mihi ignota. Syn.—Zygnema cruciatum, (VAUCHER) AGARDH. RaABENHORsT, Flora Europ. Algarum, Sect. III. p. 251. Tyndaridea cruciata, HAssaLL, Fresh-Water Alge, vol. i. p. 160.—Harvey. Battey, Microscopical Observations, p. 21. Hab.—Northern States; Virginia; Florida; Bailey. Pale green, when dried subfuscous or blackish fuscous. Sterile joints shortly cylindrical, equal or a little longer, or more rarely twice as long as broad (diam. 0.0016”—0.00195”), after division sometimes shorter than broad ; fruiting cells not tumid; zygospores mostly globose ; when mature, obscure fuscous, their coat minutely punctate. Genus SIROGONIUM, Krz. “ Cellule vegetative cylindrice, sporifere subinflate orculiforme. Fascie chlorophyllose longi- tudinales, parietales, leviter flexuose, nodose (plerumque 2-3, rarius 4 in quaque cellula), granula amylacea 7-8 involute. Copulatio genuflexa, sine tubo connexivo.” R. In specie Americana fascie chlorophyllose spirales et Spirogyre illis similes. Vegetative cells cylindrical, spore bearing cells somewhat inflated, or oreuliform. Chlorophy! fila- ment longitudinal, parietal, somewhat flexuous, nodose (mostly 2-3 rarely 4 in each cell), containing 7-8 starch granules; conjugation genuflexuous, without any connecting tubes. (Rabenhorst). In American species the chlorophy] filament spiral and like to that of Spirogyra. Remarks.—This genus was originally made by Kiitzing to contain a single species, which possesses the characters given in the diagnosis of Prof. Rabenhorst FRESH-WATER ALG OF THE UNITED STATES. we} I have met with an American plant, which has some of these characters, and at the same time others which have been supposed to belong to the genus Spirogyra. It unites the method of reproduction of Sirogoniwm and the arrangement of the chlorophyl band of Spirogyra, standing as it were midway between them. It is not midway, however, but much nearer Siregonium, for the passage from a very _ loose spiral to a longitudinal flexuous filament is a brief one, and although in some cells of S. retroversum the spiral makes a number of turns; in other long cells it scarcely gets around once, in other words the chlorophyl band is nearly straight. On the other hand, the reproduction is strictly that of S. strictum, at least in all cases which have come under my notice. ‘There is, therefore, but one of two things to be done, either to unite Sirogonium with Spirogyra, or else to give up the arrangement of the chlorophyl as an essential character of the former genus. The great variance, in the latter respect, in our American species, greatly weakens the value of any such character, and I have, therefore, preferred the latter of the two courses. S. retroversum, Woop. 8. articulis sterilibus diametro 7-15 plo longioribus; fasciis spiralibus 1, rare 2, latis, granu- latis ; anfractibus 1-9 ; articulis fertilibus valde tumidis, retroversis ; conjugatione genuflexa et sine tubo connexivo; cytiodermate nonnihil crasso, utroque fine protenso vel replicato; sporis ellipticis. Diam.—Art. steril. 7335” =.00146"; spor. lat. 7195’—7}35”=.00133"”—.0016”"; long. 7235” = 0033”. Syn.—S. retroversum, Woop, Prodromus, Proc. Am. Phil. Soe. 1869, p. 139 Hab.—In stagnis, prope Philadelphia. Sterile joints 7-15 times longer than broad; chlorophyl band 1, rarely 2, broad, granulate ; turns 1-9; fertile article very tumid, retroverted ; fertile cells scarcely swollen ; cell wall folded in at the ends; chlorophyl band single; turns 6; spores oblong or elliptical, spore wall very thick. Remarks.—I have found this species growing in stagnant ditches in the Neck below the city. In fruit the cells are almost always very markedly bent backwards, _and have a broad pouch-like dilatation in front. The spores are elliptical, and, as I have seen them, greenish and with a thin coat, but may not have been completely matured. Fig. 1, pl. 16, represents this species. Genus MESOCARPUS, HaAssatt. Cellule massa chlorophyllosa initio diffusa, postea in fasciam longitudinalem, haud raro flexnosam contracta; nucleum centralem et granum amylaceum unicum vel duo involvens. Zygospora globosa vel ovata, in tubo connexivo inter cellulas binas plus minus genuflexas formata. Chlorophyl mass in the beginning diffused in the cell, afterwards contracted into an often flexuous fascia, and involving a central nucleus and one or more starch granules. Zygospore globose or ovate, formed in the connecting tube between two more or less bent cells. Mi. scalaris, Hassaxt. M. cellulis sterilibus diametro 3-6 plo longioribus, fertilibus valde curvatis ; zygosporis ovalibus. Diam.—Max. 7335 =.0011”". 174 FRESH-WATER ALG# OF THE UNITED STATES, Syn.—UM. scalaris, Hassan, Fresh-Water Algw, vol. i. p. 166, et RABENHORS?, Flora Europ. Algarum, Sect. III. p. 257. Hab.—In fossis, prope Philadelphia. Sterile cells 3—6 times longer than their diameter, fertile strongly curved ; zygospores oval. Remarks.—This species is abundant in the stagnant ditches near Camden, It agrees well with the descriptions of the European form. I have, however, never scen it in the state in which it has “ fuscous spores.” ‘They have always been greenish, but very possibly were not fully matured, Fig. 5, pl. 15, represents a pair of cells of this species just commencing to con- jugate. M. parvulus, Hassar. M. cellulis diametro (0.00031”—0.00041”) 5-12 plo longioribus ; zygosporis globosis, plerumque 0.00062” latis, sporodermate fusco levi. (R.) Species mihi ignota. Syn.—W. parvulus, Hassat, Fresh-Water Alge, vol. i. p. 169, et RaBsnworsr, Flora Europ. Algarum, Sect. III. p. 257. Hab.—Rhode Island; (8. T. Olney) Thwaites. Cells 5-12 times longer than their diameter (0.00031”—0.00041”) ; zygospores globose, mostly 0.00062” broad, spore coat fuscous smooth. Genus PLEUROCARPUS, A. Braun (1855). Cellule exdem que in Mesocarpo; copulatio lateralis et sporifera, nonnunquam genuflexa et plerumque sterilis. (R.) Cells like those in Mesocarpus ; conjugation lateral and sporiferous, somewhat genuflexuous and mostly sterile. P. mirabilis, Braun. P. cellulis diametro (0 0011”—0.0013") 2-5 plo longioribus; zygosporis subglobosis, fuscis, levibus. (R.) Species mihi ignota. Syn.—Mougeotia genuflera, AGARDH. Batry, Silliman’s Journal. New Series, vol. iii. Pleurocarpus mirabilis, A. Braun. RAaBENHORST, Flora Europ. Algarum, Sect. II. p. 258. Hab.—West Point, New York; Providence, Rhode Island; Detroit, Michigan; Fort Winne- bago, Wisconsin; Bailey. Cells 2-5 times longer than their diameter (0.0011’—0.0013”) ; zygospores subglobose, fuscous, smooth. OrverR Siphophycex. Alge unicellulares. Cellula utriculiformis, plerumque ramulosa; ramuli vegetatione terminali prediti, sepe demum septo discreti, et alteri in oosporangia, alteri in antheridia transmutantur. Cytioplasma viride, granulosum, mucilaginosum, vesiculis chlorophyllosis et granulis amylaceis repletum. Propogatio fit aut cytiogenesi libera, aut zoogonidiis aut oosporis. Unicellular alge. Cells utriculiform, mostly branched; branches with a terminal vegetation, often finally cut off by a partition wall and transformed into antheridia or oosporangia. Cytioplasm green, granular, mucilaginous, filled with chlorophyl vesicles and starch granules. Propogation either by forming minute spores by free cell formation, or by zoospores, or by oospores. FRESH-WATER ALGA OF THE UNITED STATES. Les) Famity HY DROGASTREA. Plantule minime, terrestres, gregarix. Cellula initio globosa, postea clavato- vel pyriformi- intumescens, basi attenuata elongata et in ramulos subtillissimos hyalinos partita. Cytioplasma mucilaginosum, etate provecta gonidia divisione simultanea transformatum. Cytioderma lamellosum wtate provecta dilabens et contabescens et gonidia liberans. Plants very small, terrestrial, gregarious. Cells in the beginning globose, afterwards clavate or pyriform, with an elongated, attenuated base, divided into very fine, hyaline branches. Cytioplasm - mucilaginous, at maturity transformed by a simultaneous division into gonidia. Cytioderm lamel- late, at maturity wasting, withering away and setting free the gonidia. Remarks.—The Hydrogastree are curious little unicellular plants, which grow upon wet earth. The matured frond is swollen up at one end to form a subglobular or pyriform head, whilst at the other end it is produced into a long, much-branched, _ very fine root-like portion which enters the earth and maintains the little plant in its upright position. The green endochrome is contained almost entirely in the head, and forms generally a coat or layer in the outer portion of its cavity, the inner _ part of which appears to be occupied by a watery fluid. ‘The only specimens which I have seen of this family were found growing in the mud left by the receding water of a recently drawn mill pond, by Dr. Billings, U.S. A. When I got them they were thoroughly dried up, and consequently no opportunity of studying their development was afforded. According to Kiitzing and Braun, the species is propagated ordinarily by the breaking up of the chlorophylous layer of pro- _ toplasm lining the wall of the cell into a larger number of very small globular spores. These, although not endued with the power of motion, seem from their method of formation and history to be homologous with zoospores. In most cases they are set free by the membrane of the parent-cell becoming gelatinously softened, swelling up, collapsing, and finally dissolving away. ‘The little protococcoid cells then enlarging, develop at one end a hyaline prolongation which penetrates into the ground. Growth and development continuing the upper end of the cell swells up into the ovate or globular head, whilst the lower becomes the hyaline, branch- _ ing, root-like portion of the new frond. No indication of this method of repro- duction was discoverable in the plants which Dr. Billings sent me. The evident affinities of the family with the Vawcheriacee render it exceedingly probable that there is in it some method of sexual reproduction, as yet undiscovered, allied to that which occurs in the latter. In some of the specimens sent me, there were what appeared to be resting-spores (pl. XVI., fig. 2a), occupying the whole of the cavity of the cell, from which they appeared to be finally discharged by a decay and rupture of the outer coat or wall. How these bodies were formed, and whether they really have power to reproduce the species I cannot tell. Genus HYDROGASTRUM, Desy. Character idem ac familie. Characters that of the family. | Hi. granulatum, (Liy.) Desy. H. plerumque gregarium, sepe aggregatum, haud raro confluens; cellula e globoso-pyriformi, magnitudine seminis papaveris vel sinapios et ultra, prasino-viridi superficie pulverulenta. (R.) 176 FRESH-WATER ALG#Z OF THE UNITED STATHS. Syn.—Dotrydium argillaceum, WALLROTH. Rasennorst, Flora Europ. Algarum, Sect. III. p. 265. Hydrogastrum granulatum, (Linnzus) Desy. JABennorst, loc. cit. Hab.—Delaware ; (Dr. Billings) Wood ; West Point, New York; Providence and Newport, Rhode Island; Bailey. Mostly gregarious, often aggregate, not rarely confluent; cells pyriform, of the size of a poppy or mustard seed and larger; pea-green; surface pulverulent. Remarks.— The above description is taken from Rabenhorst’s work, and applies to the specimens collected by Dr. Billings in the State of Delaware, excepting that I did not discover any of them to be confluent, nor was their surface distinctly pulverulent. Prof. Kiitzing gives as a comparative character between this and H. Wadllrothii, the smaller size of the spores; but Prof. R. says nothing about this, There were no spores in any of the American specimens, and I think it somewhat uncertain whether or not the plant is or is not either of the European species. It is very probable that it will be discovered that the only true specific characters are sexual, and consequently have not as yet been made out in any of the forms. Certainly the descriptions of the species as at present given seem to me not to contain any reliable characters. Fig. 2 a, pl. 16, represents a very young state of our American plant; fig. 2 is the perfected frond, both magnified ninety diameters; fig. 2 a shows what is sup- posably a perfected resting spore magnified 160 diameters. Fammy VAUCHERIACEA. Algex monoiee, cespitose, unicellulares. Cellula vegetiva (thallus) vegetatione terminali, utriculi formi-elongata et ampliata, prominentiis plus minus elongatis ramosa. Propagatio aut sexualis, fit oosporis ope spermatozoidiorum fecundatis, aut non sexualis zoogonidiis. Fructificatio triplex (melius organa fructificationis tria) :— 1. Sporangium terminale, ex thalli apice plerumque globoso-clavato-tumido formatum, septo dis- cretum, cytioplasmate obscure viridi, demum in zoogonidium (zoosporam, Thur.) unicum permag- num, ciliis vibratoriis dense obsitum abeunte farctum. 2. Oogonium (ocosporangium) laterale, sessile vel prominentia, plus minus elongata vel simplici vel partita pedicellatum, cytioplasmate state provecta in oosporam singulam transmutato fetum. 3. Antheridium laterale, sessile vel e ramuli lateralis parte suprema septo discreta formatum, in quo spermatozoidea (antherozoidea, Thur.) numeros issima nascuntur, denique erumpunt. Sperma- | tozoidea oblonga, ciliis duobus inequilongis, subpolo antico ortis instructa. (R.) Monecious alge, cxspitose, unicellular. Vegetative cells (thallus) growing at the ends, elongate, utriculiform, and ampliate, more or less profusely branched. Propogation either sexual, with oospores which are fecundated by spermatozoids, or non-sexual, by means of zoospores. Organs of fructification of three kinds :— 1. Sporangia, which are terminal and mostly formed from the separation of clavately swollen, globose apex of the thallus (often of a branch) by means of a partition ; in the sporangium arises a single, very large zoospore, which is densely clothed with cilia. 2. Oogonia (oosporangia), lateral, sessile or pedicelate simple bodies, whose eytioplasm is finally converted into an oospore. 8. Antheridiwm lateral, sessile, or formed out of the end of a branch; the spermatozoids formed in them oblong, furnished with two unequal cilia, arising near the front end. FRESH-WATER ALG# OF THE UNITED STATES. W(7/ Remarks.—The Vaucheriacee are amongst our most common fresh-water alge. They occur generally in the form of vast numbers of individuals interwoven into broad mats, which have often both a felty look and feel. When growth is going on rapidly, these mats are of a beautiful vivid green; but when the process of sexual reproduction has checked the life of the individual they become dingy and dirty looking. ‘The thallus is composed of a single cell and is almost always branched. ‘The branches never have, at least in any of our species, a definite arrangement, save only in that they always arise from the side and not from the point of the thallus. In the European species, V. twberosa, however, the branches are said to arise both from the point and sides of the frond. The frond cell is generally nearly uniform in diameter and has a thick outer wall, which is composed of cellulose, as is proven by the action upon it of iodine and sulphuric acid and of the iodo-chloride of zinc solution. Within the cell are chlorophyllous protoplasm, starch granules, watery fluid, and a few scattered raphides or inorganic crystals. There is never any nucleus. The protoplasm is often very granular, and is mostly collected in a thick green layer upon the inner surface of the cell wall, leaving the centre of the cell free for the more watery contents. Growth, except in the very young fronds, consists exclusively in an increase in length, and takes place only at the ends of the thallus or in the portions near it. The branches are almost always simple, but are said in some species to give origin to secondary branchlets, and even, at times, to tertiary ones. They grow in the same manner as the main thallus, 7. e. by additions to their ends. When the thallus of a Vaucheria is ruptured by external injury, or, at times, when it is dying from some hidden cause, a number of bright green globes of various sizes are formed out of the endochrome. These appear to have the power of independent existence for some time, but whether or not they ever actually grow into new thalli I am unable to state. M. Walz asserts.that he has observed in certain species the formation of a quiet spore without the intervention of sexual organs, and that the process is as follows. The end of a long or short twig swells up, and the chlorophyl and protoplasm from the neighboring parts accumulate in the enlarged portion. A partition wall then forms at the base of the latter, which is thus changed into a closed chamber, a sporangium. The green contents then slowly gather themselves together into a denser and denser ball, becoming more and more separated, in so doing, from the wall of the sporangium, and finally secreting around themselves a distinct mem- brane. After the formation of a spore in this way, the sporangium opens at the apex and allows it to escape. The spore, after remaining quiet for some time in the water, at last germinates into a new frond, in a similar manner to an ordinary zoospore. In my earlier studies of fresh-water alge, I noticed something very similar to this in one of our species, but convinced myself that the little body was nothing but a zoospere, whose normal development had been perverted by unto- ward influences, and therefore paid no more attention to the matter. It is proba- ble that the life-history of the bodies observed by M. Walz is capable of the same explanation. 23 August, 1872. 178 FRESH-WATER ALG OF THE UNITED STATES, Although I have very frequently cultivated Vaucherias, I have never been so fortunate as to see them form their zoospores, nor indeed to see a zoospore in its motile state. The life-history of these bodies has, however, been fully and repeat- edly worked out by other observers. It is described by such as occurring in the following manner. One end of a branch first enlarges into a bulbous, often conical, point, into which the neighboring endochrome crowds itself. This point is next divided off by a partition wall from the remainder of the thallus and constitutes the zoosporangium, the contents of which rapidly condense into one or two masses, generally oval in shape, each of which eventually forms a zoospore. When the latter are matured, the apex of the zoosporangium opens, and the little bodies within slowly and gradually emerge, without any apparent cause for their motion. Sometimes, according to Cohn, instead of this steady outward passage, there are repeated forward and backward movements of the zoospores within the case. ‘The zoospore after its perfection is generally oval, and very large. Within it there are ‘one or more vacuoles, and surrounding it is a layer of colorless protoplasm. It is remarkable for having its whole surface densely covered with short cilia. Its period of motile life appears to be very brief; according to Walz, that of the zoospore of V. sericea, Lyngb., lasts only from one-half to one and a half minute, after which time the cilia are lost and a cellulose wall secreted around the mass. Germina- tion takes place by the growth of the cylindrical thread out from each end of the zoospore. True sexual reproduction takes place in this family by means of antheridia and oogonia, male and female organs. All known species are mostly if not absolutely monecious, both organs being contained in the one individual and always placed in proximity. All of the species in which the development and structure of the sexual organs have been studied, agree in the essential points, The first appearance of the antheridium is as a little pouch projecting out from the side of the thallus. This increases in size and soon assumes the peculiar shape of the species. At the same time there is a diminution, according to M. Walz, of the chlorophyl in the antheridium, so that, when the partition wall forms and shuts off the cavity of the latter from that of the thallus, there are only a very few scat- tered green granules remaining. The antheridium at the time of separation con- tains, therefore, only transparent protoplasm, which soon becomes granular, and shortly afterwards exhibits the moving spermatozoids, which appear to be formed out of the thick layer of protoplasm that lines the inner surface of the cell wall. The point of the antheridium opens so soon as the spermatozoids are perfected, and allows them to escape. The formation of the oogonia takes place very similarly to that of the antheridia. There is the same little protrusion from the side of the thallus in the commence- ment of the process, the same after-growth and increase of this pouch, and the same formation of a separating wall between it and the main body of the frond. A very marked difference, however, is to be found in the contents of the two, the oogonium from the very commencement being crowded with chlorophyl and oil globules. When the oosporangium is completed, the end of it opens, and, at the same time, the contents gather themselves into a dense protoplasmic ball, which lies in the | *« EFRESH-WATER ALG A OF THE UNITED STATES. 179 centre. The spermatozoids, which are at this time already free in the water, are very minute, longish, ellipsoidal or ovate masses, provided with two unequal cilia. ‘These commonly both arise together from one end of the body, and are directed in oppo- site directions—one backwards, the other forwards. According to M. Walz, how- ever, in V. sericea the cilia arise from the opposite ends. According to De Bary, | the spermatozoids of V. aversa, Hassall, contain reddish pigment-granules. M. Walz states that he has twice seen the process of impregnation in V. sericea, Lyngb., and describes it essentially as follows: After the bursting of the antheridium and the formation of the opening in the oogonium, the spermatozoid clustered around the little orifice in the latter, but were apparently debarred entrance by the presence of a glutinous jelly. After a time, however, one, and then another, forced a passage through this obstacle until finally a number gained access to the protoplasmic ball within. Over this they swarmed, pushing it and retiring and butting against it until some of them actually forced their way into it and were absorbed by it. Im- pregnation being now completed, the oospore acquired a very sharp definite outline, and secreted in a very short time a membrane around itself. The changes which followed during its maturing consisted of the acquiring of a thick coat and the replacing of the chlorophyl within by a reddish-brown coloring matter. The ripened resting spore of almost all the Vaucheria is provided with three coats, of which the middle is the thickest. The contents consist of protoplasm, reddish-brown pigment, and numerous oil globules. Genus VAUCHERIA. Genus unicum, character idem ae familiz. The only genus of the family, having the same characters. V. sessilis, (Vaucu.) De Canporie. V. laxe intricata, pallide et subsordide viridis; thallo ecapillari, parce ramoso; oogoniis 2-3 approximatis, rarias singulis, ovatis vel ovali-oblongis, plus minusve obliquis, rostratis; anthe- ridio intermedio, ramuli modo brevi hamato, modo recto subulato, subclavato, modo elongato et incurvato, haud raro circinato sustentato; oosporis maturis fusco-punctatis, membrana triplici involutis. (R.) Syn.—V. sessilis, (VAucH.) De CanpoLLe. Raxsennorst, Flora Europ. Algar., Sect. III. p. 267. V. cespitosa, (Vaucu.) AGarpu. RapBennorst, loc. cit. Hab.—Salem, North Carolina; Schweinitz. Common at West Point, New York; Waterville, Maine; Culpepper Co., Va.; Bailey. Laxly intricate, pale and subsordid green; thallus capillary, sparsely branched; oogonia 2-3, ! approximate, rarely single, ovate or oval-oblong, more or less oblique, rostrate; antheridia intermediate, sustained upon branches partly shortly hamate, partly straight subulate, sub- clavate, partly elongate and incurved, and not rarely circinnate ; oospores at maturity, fus- cous-punctate, surrounded by a three-fold membrane. Remark.—I think I found this species near Philadelphia in my eatliest re- searches, but cannot speak certainly, having preserved neither notes nor specimens, 180 FRESH-WATER ALG& OF THE UNITED STATES, WV. velutima, Ac. V. thallo repente, ramulis erectis, numerosis, fastigiatis, in cespitem velutinum lete viridem intricatis ; oogoniis lateralibus singulis, globosis, sessilibus, antheridio paulo longiore unico subulato leviter incurvato consociatis (R. ie Species mihi ignota. Diam.—Oogonii 0.0023”—0.0027". (R.) Syn.—V. velutina, AGARDH. RABENHORST, Flora Europ. Algarum, Sect. III. p. 274. Hab.—Salem, North Carolina; Schweinitz. Common at West Point, New York; Waterville, Maine; Culpepper Co., Va.; Bailey. “ Filaments exceedingly tough, interwoven into a dense, velvety, green stratum, pellucid below and creeping over the mud; branches near the extremity erect, fastigiate, and more or less crooked; vesicles solitary, globular, on short lateral peduncles.” Carmichael. V. geminata, (VAucu. ) DE CANDOLLE. V. obscure vel sordide viridis, in ceespites dense intrientas thallo capillari, tenaci, dichotomo ; oogoniis duobus (rarius 1 vel 3), ovatis vel obovatis, oppositis, distincte pedunculatis, angheridio intermedio subulato, plus minus recurvo; oosporis maturis fusco-maculatis, sporodermate achroo e stratis tribus composito involutis; sporangiis in eodem vel proprio thallo, eyathiformi- ampliatis truncatis et angulato-cornutis. (R.) Syn.—V. geminata, (V aucun.) De CanpotiE. RABENHORST, Flora Europ. Algarum, Sect. III. p- 269. Hab.—In stagnis, prope Philadelphia; Wood. Obscure or sordid green, densely interwoven into a turfy mass; thallus capillary, tenacious, dichotomous ; oogonia two (rarely 1-3), ovate or obovate, opposite distinctly pedunculate, antheridia intermediate, subulate, more or less recurved; oospores at maturity spotted with fuscous, their coat transparent and composed of three strata; sporangia in the same or a sepa- rate thallus swollen cup-shaped, truncate and horned at the angles. Remarks.—I have found this species in fruit but once, then it grew in a ditch below the city. Not having mounted any of it, nor having written a description of it at the time, I have been forced to simply copy that of Prof. Rabenhorst. V. polymorpha, Woop. V. in cxspites dense intricata; thallo capillari, tenui; antheridiis corniculatis ex ramuli lateralis apice formatis; ramulis fertilibus interdum et oogoniis et antheridiis instructis, interdum antheridiis solum ; oogoniis plerumque geminis, interdum singulis, globosis vel ovatis; sepe breve rostratis, plerumque distinete pedunculatis sed rarius sessilibus; oosporis enormiter subglobosis vel ovatis; sporodermate achroo, e stratis duobus composito. Syn.—V. polymorpha. Woop, Prodromus, Proceedings Amer. Philos. Society, 1869, p. 140. Hab.—In aquis, prope “ Buffalo Bayou,” Texas; (Ravenel.) Czespitose ; thallus hair-like, thin; antheridia corniculate, formed of the apex of lateral branches ; fertile branches sometimes furnished both with oogonia and antheridia, sometimes with antheridia alone ; oogonia sometimes single but mostly in pairs, occasionally shortly rostrate, generally distinctly pedunculate but sometimes sessile ; oospores irregularly subglobose or ovate, surrounded by a transparent double spore coat. Remarks.—This species was collected by Prof. Ravenel near the city of Houston, Texas. As I received the mass, it was labelled as being obtained from “a shallow slimy pool formed by drippings from the side of a ravine near Buffalo Bayou.” The species probably grows in the water, evidently forming turfy mats. It is FRESH-WATER ALG OF THE UNITED STATES. 181 remarkable from the fact that, whilst in many cases the little branches which pro- duce the antheridia give origin to the spores also, in others they do not; so that there are numerous antheridia, which are unconnected with any female organs. When a branch does produce both of the reproductive organs it usually forks into three short branchlets, thus giving origin to a pair of sporangia and a single curved, hooked antheridia. Sometimes, however, there is but a single female branchlet, and I have even seen a sporangium, immediately sessile upon a branch, which at its apex gave origin to a male organ. In the coat of the perfected spore, I have not been able to find more than two distinct strata, Figs. 3 and 3a, pl. 20, represent sporangia and antheridia of this species; 3 6, a simple, young and only partly formed antheridia, magnified 160 diameters; 3 ¢, a perfected spore magnified 260 diameters. V. sericea, Lynapye. V. aquatica vel terrestris, cespitosa, vel sordide vel lete vel luteolo-viridis; thallis tenuibus, dense intricatis, laxe et vage ramosis, ramisque spe adscendentibus vel erectis ; oogoniis ses- silibus vel brevissime pedicellatis, 1-6 seriatis, unilateralibus, oblique et enormiter ovalibus, ore laterali producto rostellatis ; antheridiis in thallo ipso juxta oogoniis sessilibus, cylin- draceo-subclavatis, deflexis; spermatozoideis oblongis, puncto rubro notatis (teste de Bary), in utroque polo cilio unico preditis. Syn.— V. aversa, HAssauu, Fresh-Water Algae, p. 54. V. sericea, Lynesyr, RABENHORST, Flora Europ. Algarum, Sect. ITI. p. 271. Hab.—Prope Philadelphia; Wood. Aquatic or terrestrial, occurring in turfy mats of a yellowish, dirty, or bright green color; fronds thin, densely intricate, laxly and vaguely branched, often together with the branches ascending or erect; oogonia sessile or very shortly pedicellate, 1-6 seriate, unilateral, obliquely irregu- larly oval, their lateral mouths produced into a rostellum or beak; antheridia sessile upon the thallus itself near the oogonium, somewhat cylindrical, subclavate, deflexed especially in age ; spermatozoids (according to De Bary) oblong, marked with a red point and furnished with a single cilia at each end. Remarks.—I can perceive no constant differences between V. sericea, Lyng. and V. aversa, Hass. The extreme forms differ somewhat, but both are very common about Philadelphia, and everywhere grade into one another. Prof. Rabenhorst thinks that the two forms are scarcely distinct, and states that the most character- istic differences are, that in V. aversa, the thallus is much thicker, and the oogonia larger and more erect, whilst the oospores are smaller and consequently do not fill the cavity of their case. These differences are, except the last, simply differences in size, and seem to me to depend simply upon circumstances of growth. The rela- tively smaller size of the spore is a very frail hook indeed to hang a species upon. The plant grows in springs and actively running water abundantly in this neigh- borhood ; also on very wet ground, especially on that which is habitually overflowed, such as the face of dams, neighborhood of springs, &c. In the water, it is frequently on the ground, but also often clothes such objects as stones, largish sticks, &c. OrverR Nematophycex. Algee multicelullares, chlorophyllose, membranacee vel filamentose, ramificatione aut instruct aut destitute. Propogatio fit aut oosporis aut zoogonidiis, sed nunquam conjugatione. 182 FRESH-WATER ALGA OF THE UNITED STATES, Multicellular, chlorophyllous algw, membranaceous or filamentous, furnished with or destitute of branches. Propagated by oospores or zoospores, never by conjugation. Famity ULVACEA. Thallus membranaceus vel foliaceus, vel filiformis (Schizomeris ?) rarius crustaceus, e cellularum strato unico formatus, aut expansus aut tubuloso- vel vesiculoso-concretus. Propogatio fit zoogonidiis, eytioplasmatis divisione repetita ortis. Zoogonidia oblonga, polo antico ciliis vel binis vel ternis vel quarternis instructa. Thallus membranous or foliaceous, rarely crustaceous, composed of a single stratum of cells, either expanded or tubularly or vesicularly concreted. Propagation by means of zoogonidia, formed by the repeated division of the cytioplasm. Zoogoni- dia oblong, furnished with two, three, or four cilia at the anterior end. Genus PROTODERMA, Kz. Thallus crustaceus, indeterminatus, substrato arcte adherens, e cellulis anguloso-rotundatis, irregulariter ordinatis, arete connexis compositus. Propagatio ignota. Thallus crustaceous, indeterminate, closely adherent to the substratum, composed of closely con- joined irregularly arranged angularly rounded cells. Propagation unknown. P. viride, Krz. P. viride, lubricum. Syn.—P. viride, Kurzina. Rapennorst, Flora Europ. Algarum, Sect. III. p. 307. Hab.—In aquario; Wood. Green; slippery. Remark.—I have seen a plant, which I take to be this species, growing on the glass and on pebbles in the aquarium of my friend, Dr. Frické. Genus ULVA, Linn. Thallus membranaceus, plane expansus, angustus vel latus, nonnunquam latissimus, magis minusye undulato- crispatus, swpe laciniatus, haud raro perforatus, e cellularum strato unico formatus, callo disciformi parvo affixus, wtate provecta spe libere natans. Cellule anguloso-rotundate, cceloplas matice, parenchymatice connexe. Vegetatio cellularum divisione in duas directiones repetitia. Propogatio fit zoogonidiis, in cel- lulis quibusdam cytioplasmatis divisione 4, 8-16 ortis, ciliis vibratoriis quaternis longitudine cor- poris longitudinem vix superantibus instructis. Thallus membranous, expanded, narrow or broad, sometimes very broad, more or less undulately curled or ecrisped, often laciniate, not rarely perforate, formed of a single stratum of cells, fastened by a small discoid thickened portion, in advanced age often swimming free. Cells angularly glo- bose, joined into a sort of parenchyma. Growth occurring by the repeated division of the cells in two directions. Propagation by zoospores, 48-16 of which are formed at once by a division of the endochrome of certain cells, and are furnished with four vibratile cilia scarcely longer than the body, U. merismopedioides, Woon. U. ampla, membranacea, late expansa, dilute viridis, tenuis, radiatim et enormiter plicata, ambitu sepe subrotundata; margine undulato, interdum suberenato ; cellulis enormiter ovalibus vel angularibus, nucleo destitutis, quarternariis et in familias Merismopediarum modo obseure associatis. Diam.—Cell. max. rae50. i .00041', plerumque SO SO” = 00016 — .00025. FRESH-WATER ALGA OF THE UNITED STATES. 183 Syn.—U. merismopedioides, Woop, Botanical Report of the United States Geological Ex- ploration of the Fortieth Parallel, p. 413. Hab.—In torrentibus, Diamond Range (alt. 6000 ft.), Rocky Mountains; (Sereno Watson) Wood. Thallus ample, broadly expanded, membranaceous, dilute green, thin, radiately and irregularly plicate with its outline often somewhat rounded; its margin undulate or at times almost crenate; the cells irregularly oval or angular, destitute of nucleus, quarternary and obscurely arranged in families after the manner of a merismopedia. Remarks.—The largest fronds of this species that have come under my notice are about three inches long by two broad, thin, easily torn, and not all gelatinous. The portion by which they have been attached is very evident, near one of the margins, and from it broad undulations or folds radiate. Sometimes the frond is split up into palmate, lobe-like parts. The cells are not closely approximate, but are placed in a homogencous translucent membrane, in such a way as to remind one of a Merismopedia. I do not feel certain that this plant is distinct from U. orliculata of Rabenhorst, though for the present I have preferred so to consider it. His description is very brief and incomplete, as is also the original one of Thuret, which I have con- sulted. Prof. R., however, gives U. latissima of authors as a synonym of JU. or- biculata, and certainly this plant is distinct from U. latissima, Harvey, of our coast. Again it seems impossible that a plant growing near the summit of the Rocky Mountains should be identical with one found on the coast of France. Prof. Sereno Watson found this plant growing on rocks in a mountain stream of the Diamond Range, at an altitude of 6000 feet. Genus ENTEROMORPHA, Link. Thallus membranaceus, tubulosus vel utriculiformis, basj affixus (saltem initio, postea ssepe libere natans), e cellularum strato unico compositus, spe ramosus, haud raro ramosissimus. Propogatio ‘fit zoogonidiis. Hee zoogonidia proceantur in cellulis quibusdam 8-16 cytioplasmatis divisione repetita, in polo antico rostriformi ciliis duobus corpus duplo superantibus priedita. (R.) Thallus membranaceous, tubular or bladder-shaped, affixed by the base (at least in the beginning, often afterwards floating freely), composed of a single stratum of cells, often branched, not rarely very much branched. Propagation by means of zoospores, 8-16 of which are formed by the repeated division of the protoplasm of a cell. Their anterior beak-like portion provided with two cilia whose length is not less than twice that of the body. E. imtestimalis, (Liny.) Linn. E. teres, forma et magnitudine admodum varia, sepe pedalis etiam supra, leptoderma, saturate vel pallide viridis, filiformis vel intestiniformis, plana vel bullosa; cellulis 3-5-6 angularibus. (R.) Species mihi ignota. Diam.—0.00048” — 0.0008”. (R.) Syn.—EL. intestinalis, (Linn&uUs) BAILEY, Silliman’s Journal, N. 8., Vol. III., et Rasenuorst, Fiora Europ. Algarum, Sect. IIT. p. 312. Hab —Hudson River, from Newburgh to New York City; Narragansett Bay, Rhode ! Island; Bailey. Terete, very various in size and shape, often a foot or more in length, smooth, deep or pale green, filiform or intestiniform, plain or bullose ; cells 3-5-6 angular; their diameter 0.00048" — 0.0008". 184 FRESH-WATER ALG OF THE UNITED STATES, Genus SCHIZOMERIS, Kz. ? Thallus filliformis, cylindricus, hie illic valde contractus, basi attenuata affixus. Vegetatio fit cel- lularum divisione initio in duas postea in tres (?) directionem. Propogatio fit zoogonidiis. Zoogo- nidia in thalli juvenis cellulis orta, ovata, polo antico ciliis tribus instructa. Thallus filiform, cylindrical, here or there strongly contracted, adnate by the strongly contracted base. Growth in the beginning by the division of the cells in two directions, afterwards in three directions. Zoogonidia formed in the cells of the young thallus, ovate, their anterior end furnished with three cilia. Remarks.—The plant from which the above generic description has been drawn up grows abundantly in our ditches below the city. Whether it really belongs to the genus Schizomeris or is the representative of a new group is somewhat uncer. . tain. I have never seen the European plant, but, if I understand the descriptions of it, the cells in it are all arranged in a single plane. This certainly is not the case in the old plants of our North American form, for in them the cells are so placed as to make a thick opaque filament, the outside of which everywhere pre- sents the outer walls of cells. ‘The life history of the European species has not been at all worked out, and I have refrained from actually indicating a new genus, | in the absence of absolute knowledge upon the subject, because the specific cha- racters of the two plants are so much alike. I have had some opportunities for studying the life history of our American plant. The zoospore (Fig. 1 ¢. pl. X VIL.) is of the ordinary conical or ovate form, with a very decided transparent anterior end, from which arise three cilia. As the number three is a rare one for cilia to exhibit, I have examined several zoospores | with care, and am very certain that they had no more or less. It is, therefore, probable that the number is fixed for the species, although just possible that my finding several individuals in agreement was accidental. The zoospore after a period of free life, during which its motion is very active, becomes quiescent, and, its cilia withering away, attaches itself by its smaller end to some twig, stone, or other support. At the same time it appears to change its shape somewhat, grow- ing longer and narrower, and the smaller end spreading out to form a little foot. Simultaneously with these changes the young plant acquires a cellulose coat, and so becomes a perfect cell, in which I have never been able to detect any nucleus. After a while the cell thus formed divides transversely into two, which, of course, lay end to end. Each of these cells then grows until it attains a certain size, and then the transverse division is repeated. In this way the process goes on until finally a long filament is produced, which is composed of but a single series of cells. These cells are much broader than long, and are placed end to end, so that the cylindrical frond is made up as it were of disks laid one upon the other. When the filament has in this way reached a certain stage of development, one of two things occurs, either the cells begin to divide at right angles to the plane o! their previous division, or else the production of zoospores takes place. In the first instance each cell divides into two, four, or more cells. This division, I believe occurs in three if not all directions, so that each original cell is represented by ¢ number of cells, and a sort of compound filament arises, out of which the maturet FRESH-WATER ALGA OF THE UNITED STATES. 185 large trichoma is formed by a continuation of growth, and, perhaps, by a repetition of the division. I have never been able to discover that any reproductive process whatever takes place in this compound filament, and am very confident it never produces zoospores. It is very possible, however, that it may in some way give origin to resting spores, although, as above stated, no indication of this has ever come under my notice. The zoospores are formed in the young fronds as follows: The endochrome in the cell concerned gradually separates in the ordinary manner into several distinct masses, which soon assume a more or less irregularly globular or pyriform shape. Whether the number of these masses is fixed for the single cell or not Iam unable to state. These changes occur almost simultaneously in a number of consecutive cells, commencing with the most distal and rapidly spread- ing towards the base of the filament. When they are pretty well advanced, the walls of the cells undergo some alteration, probably a gummy degeneration, whereby they become soluble in the water. As the division of the endochrome occurs first in the most distal cells of the filament, so does also this change in the cellulose coat. When the endochrome masses are well shapen and distinct, they begin to exhibit motion, becoming uneasy, restless, changing their position, rolling on themselves, and pushihg against one another. At the same time solution of the cell walls com- mences, the partitions between the cells disappearing, and the outer walls spread- ing. These changes go rapidly forward, and in a little while the zoospores stream out from the fading end of the frond, jostling and crowding as though eager to enter upon their new life. Fig. 1 a, pl. 17, represents the basal portion of an old filament which has failed to form zoospores, magnified 125 diameters. Fig 1 was drawn from a young filament during the process of forming zoospores; owing to their rapid motion, the cilia of the latter could not be seen. This figure is enlarged 250 diameters. Fig. le represents a zoospore which has just become quiescent, and still retains its cilia, although they have lost their motile power. Fig. 1 d, e, c, represent the very young plant in different stages of growth. They are all magnified 450 diameters. S. Leibleinii, Kz. ? 8. late viridis vel saturate nigro-viridis. Diam.—Max. 73,”.=.08". Syn.—S. Leibleinii, Kivzinec. Rapenuorst, Flora Europ. Algarum, Sect. III. p. 311. Hab.—In fossis, prope Philadelphia. Bright green to deep blackish-green; largest diameter of the frond +35”. Remarks.—Owing to the profusion of zoospores produced by a single filament at one time, it is very usual to find large numbers of the younger plants attached so closely to some central body as to form dense masses of a beautiful green color. The support of these small masses is often entirely concealed, and I have frequently seen them moving freely about the jar, without any apparent cause, until the mystery was solved by finding that some unfortunate snail carried the forest on his back. The oldest filaments are perfectly opaque, showing, under the microscope, by transmitted light, no trace of their structure. 24 August, 1872, 186 FRESH-WATER ALG& OF THE UNITED STATES, The species is exceedingly common in the later summer and early fall months in the ditches and sluggish streams around the city, especially in the Neck. Famity CONFERV ACE. Fila articulata aut simplicia aut ramosa, vegetatione terminali non limitata instructa. Articuli plerumque plus minusye elongati, sed nonnunquam diametro breviores, cylindrici, rarius tumidi, Cytioderma plerumque manifesto lamellosum. Massa chlorophyllosa granulata, vesiculas amylaceas involvens, parietalis vel in «tate provecta sepe in celluls centro contracta. Vegetatio fit utriculi primordialis divisione semper in unam eandemque (transversam) directionem repetitia. Propagatio fit zoogonidiis. Filaments articulate, simple or branched, growth terminal, unlimited. Joints mostly more or less elongated, but sometimes shorter than long, cylindrical rarely tumid. Cytioderm mostly plainly lamellate, chlorophyl masses granular, surrounding fine starch granules, parietal or often in the centre of the cell. Growth taking place by division of the primordial utricle always in one direction, namely trans- versely. Propagation by means of zoospores. Genus CONFERVA, (Linn.) Linx. “Fila articulata simplicia. Articuli cylindrici. Massa chlorophyllosa homogenea vel granu- lata, vesiculas amylaceas involvens. Propogatio ignota.” (R.) : Threads articulate simple. Articles cylindrical. Chlorophyl mass homogeneous or granulate, including amylaceous vesicles. Remarks.—A large number of forms of the genus Con/ferva have been described as distinct species by Kiitzing and other authors. The characters assigned to these species, however, do not seem to me in any way distinctive. I cannot believe it possible at present to recognize, define, and describe species in this genus, and believe that further studies must be made in their life-history, and other characters discovered before the different forms can be separated. Probably, as was the case with the Gidogoniacew, when their sexual life is made out, in it will be found the vital differences. No doubt there are many species common to Europe and Ame- rica, but I have been entirely unable to determine them. Among the very earliest of my observations upon the fresh-water alge, before experience had taught how and what to observe, was one made upon what I suppose was a species of this genus. I have never met with the plant since, but as the observation has direct bearing upon the method of propagation, I mention it here, imperfect as it un- fortunately is. The plant was found growing on the mud along the Schuylkill River, near Gray’s Ferry Bridge, below the city. The filaments were simple, of great length, and uniform in diameter; fig. 7 a, pl. 18, represents a portion of one magnified 500 diameters. The cells varied from about as long as broad to three times as long. The amount of endochrome in the cells also varied very much. In most of them, it was not nearly sufficient to fill the cavity, and was arranged as a central superficial band. Many of the cells were seen engaged in the production of zoospores. (Fig. 7b, pl. 18.) Such were well filled with endochrome, which gradually condensed itself into a globular or pyriform mass in the centre of the cell. This, after a short time, began to exhibit activity, rolling upon itself and finally pushing about as much as its confined quarters would allow, until at last it FRESH-WATER ALGHZ OF THE UNITED STATES. 187 escaped into the water, through the cell wall. Each cell in this way gave origin to a single zoospore. ‘The walls did not melt away in the water, and, as a number of consecutive cells underwent these changes at the same time, the filament or a portion of it was left as an empty shell. ‘The zoospores were of the usual shape, with a bright anterior spot or beak. The number of cilia was not noted. After a time they settled down generally in clusters, attaching themselves to some foreign particle, dropping their cilia and acquiring a cellulose wall. (Fig. 7e, pl. 18.) They then elongated, underwent the ordinary cell division in a transverse direction, and, by the repetition of this, gradually grew into filaments similar to that from which they sprang. Fig. 7d, pl 18, represents a young filament just formed in this manner, magni- fied 500 diameters, Genus CLADOPHORA, Krz. (1843.) Fila cellularum serie simplici formata, varie ramosa. Rami filo centrali similes. Cytioderma plerumque crassum, lamellosam. Cytioplasma parietale. Filaments composed of a simple series of cells and variously branched. Cytioderm mostly thick and lamellate. Cytioplasm parietal. Remarks.—The Cladophora are branched plants of rather rigid habits, which grow both in salt and fresh water. They are readily recognizable by their comparatively still appearance, the absence of gelatinous matter about them, and by the want of regularity in their branching. A large number of species have been described, most of which are marine. They are exceedingly difficult to define, and it is very possible that their hitherto undiscovered sexual reproduction may be finally found to afford the only true characters. I have identified two European forms as growing near this city, and a third has been recognized by Prof. Harvey, as found in our northern States. I have never seen the production of zoospores in this family, but they are said to be formed by the simultaneous division of the layer of chlorophyllous proto- plasm, which fills the outer part of the cell cavity. ‘They exhibit the power of very active motion even before their exit from the cell, which occurs through a papilloid orifice, mostly at the end of the cell, sometimes in its side. Their cilia are sometimes two, sometimes four in number, and their life-history appears to be precisely similar to that of other zoospores. CL. slomerata, (Liny.) Ramuli fili primarii in parte superiore atque ramorum ordinis seeundi et tertii plerumque fasciculato- vel penicilliformi-aggregati. Cellule maximi vegete cytioplasmate cellularum parieti retiformi- vel subspiraliter applicato. Cellule fructiferee semper terminales, inferiores semper steriles videntur. (R.) Syn.—Cl. glomerata, (Kttzina) RapBennorst, Flora Europ. Algarum, Sect. IIT. p. 337. Hab.—Wake Ontario; Pickering. Falls of Niagara; Lakes Erie, Huron, and Michigan ; Fourth Lake, near Madison, Wisconsin; Bailey. “Filaments tufted, bushy, somewhat rigid, much branched, bright grass-green; branches crowded, irregular, erecto-patent, repeatedly divided ; ultimate ramuli secund, subfasciculate ; articulations 4—8 times as long as broad.” 188 FRESH-WATER ALGH OF THE UNITED STATES. Remarks.—Prof. Harvey says (Smithsonian Contributions): “T have received ; ; . » . ‘ oe e 1 North American specimens from Milton, Saratoga County, N. Y., and from Lake : 2 a Erie; also from the Mexican Boundary Surveying Expedition, CL. fracta, Ditiw. Clad. prima juventute affixa sed postea libere natans et cespites formans; ramis ramulisque sparsis, divaricatis, honnunquam refractis ; ramulorum eytioplasmate non spiraliter ordinato; cytiodermate szepe crassissimo ; cellulis fertilibus haud terminalibus, plerumque in ramulorum medio, aut eorum basi. Syn.—Cl. fracta, (DILLW.) RABENHORST, Flora Europ. Algarum, Sect. III. p. 334, Hab.—In flumine Schuylkill, prope Philadelphia; Wood. West Point, New York; Provi- dence, Rhode Island; Bailey. In the young state fixed, but afterwards floating free and forming matted masses ; branches and branchlets scattered, divaricate, somewhat refracted ; cytioplasm of the branches not spirally arranged ; cytioderm often very thick; fertile cells not terminal, mostly in the middle of the branches, sometimes in their base. Cl. brachystelecha, Rasennorst. C. per totam vitam innata, obscure viridis, sicca pallida, pygmea, 2-4, rarius 6 linea longa, ramosissima, intricata, plerumque culmigena; ramis primariis 31y’”—5'” = 0.00295” —0.0022" erassis, ramulis ultimis y/”— 7)” = 0.00147” — 0.00128” crassis; articulis diametro 4-12 plo longioribus ; cytiodermate subcrasso, hyalino, subtiliter plicato-striato ; eytioplasmate imprimis cellularum superiarum laxe spiraliter ordinato. (R.) Syn.—Cl. brachystelecha, RABENHORST, Flora Europ. Algarum, Sect. III. p. 343. Hab.—Prope Philadelphia; Wood. Fixed through the whole life, obscure green, pale when dried, dwarfish, 2—4, rarely 6 lines long, very much branched, intricate, mostly attached to culms; primary branches 0.00295” — 0.0022” thick, ultimate ramuli 0.00147” — 0.00128” thick; articles 4-12 times longer than thick ; eytioderm thickish, hyaline, subtilely plicately striate; cytioplasm, especially of the upper cells, laxly spirally arranged. Remarks.—I have notes of having identified this species at some time, but, having kept neither specimens nor detailed memoranda, have simply copied the description of Prof. Rabenhorst. Famity OKDOGONTACE. Alge monoice vel dioice. Fila articulata aut simplicia aut ramosa, cellula basali obovato-clavata, basi plerumque lobato-partitia vel scutata innata. Propagatio fit tum zoogonidiis tum oosporis fecundatione sexuali ortis. Zoogonidia formantur singula in quavis cellula, forma late ovali vel globosa, polo antico achroo corona ciliorum vibratoriorum predita. Oogonia singula vel plura (2-5) continua, plus minusve tumida, in quoque oospora singula, matura rubro- aut flavo-fusco-colorata, ante germinationem in zoosporas plerumque quatuor dilabens se format. Antheridia brevi-filiformia, 1-2-3-10-articulata, plerumque singula aut oogonio aut filo vegeto in- sidentia aut in individuis variis sepe cellula obovato-clavata subtentata. Monecious or dizcious alge. Filaments articulate, either simple or branched, fixed by the basal cell which is obovate-clavate, mostly with its base lobately parted or shield shaped. Propagation sometimes by zoospores, sometimes by resting spores, the result of sexual impregna- tion. Zoospores formed simply in certain cells, broadly oval or globose, their anterior end trans- parent, and furnished with a crown of vibratile cilia. Resting spores single or in series of from FRESH-WATER ALG OF THE UNITED STATES. 189 two to five, more or less tumid, single in each sporangium, at maturity reddish or yellowish fuscous, before germination dividing themselves into (mostly four) zoospores. Antheridia shortly filiform, 1-2-3-10 articulate, mostly single, either upon the sporangium or vegetation cell. Remarks.—The Cidogoniacee have been by previous writers simply divided into two genera, Gidogonium and Bulbochete. The plants represented by these two divisions have certainly many characters in common, as in the production of their zoospores and spermatozoids as well as in their peculiar method of cell division. Yet they are so very diverse in some particulars in regard to the latter, as well as in their habit of growth and in the formation of their sporangia, that it has seemed to me that the differences between them were more than sufficient to cha- racterize merely genera, and that to each of these groups should be awarded the rank of a sub-family. Again, in the old genus of Gidogonium, we have very distinct groups, separated by differences in the most important of all the characteristic portions of the plant— the sexual apparatus. These groups are the so-called Monecious, Gynandrous, and Diecious Gidogonia; the monecious division comprising those plants in which one individual gives origin both to the female and male germs; the gynaundrous, those species in which the plant that produces the female germ gives origin also to a peculiar zoospore, the so-called androspore, which, after a period of motile life, settles down and develops a dwarf plant, the andrecium, in which the spermato- zoids are developed ; and the diecious group containing species in which the male and female plants are distinct individuals. Dr. Pringsheim states (Morphologie der CEdogon., p. 43) that these groups pass into one another, but in my opinion, by his own showing, they are sharply distinct. The nearest approach to such passage is between the first and second groups, and consists simply in the fact that in certain species the androspore when it settles down develops into a one-celled instead of a two or three-celled antheridium. ‘This to me does not seem to indicate a union of the groups, for the essential difference is not in the form or complexity of the an- theridium, but in the circumstance that in the one case the female filament develops a spermatozoid capable of fertilizing the germ, whilst in the other it gives rise to a body which does not possess that power at all, but does have the capability of giving origin to a second plant, in which the spermatozoid is developed. The groups, therefore, appear to be sharply and distinctly definable. In the Bulbochetiea but a single genus has as yet been discovered, and this is distinctly gynandrous, but it seems probable that hereafter other plants of this subfamily will be found which are monecious or dizcious, so that we will have in the two subfamilies two parallel groups of genera. For the reasons above indicated I have ventured to divide the family into two subfamilies, the one comprising three, the other a single genus. The peculiarities of growth, production of zoospores, and sexual development will be found described under the particular subfamilies, 190 FRESH-WATER ALG#& OF THE UNITED STATES, SubraAmMinty GADOGONIE/. Filamenta stricta, haud ramosa, sine setis veris, sed swpe apice setiforma, elongata, hyalina, Filaments simple, not ramose, without true seta, but often with their apex seta-like, elongate, hyaline. Remarks.—The CGidogoniacee are small filamentous plants, whose size is sufficient to render them visible to the unaided eye, and yet not sufficient to make each indi- | vidual distinctly apparent. ‘They grow mostly in quiet water, attached to almost any and eyery thing that can afford a foothold, frmging with apparent indifference stones, twigs, sticks, dead leaves, bits of glass, boards, ete. I have seen such | masses of them crowding the Se surface of a physa as to entirely conceal the animal and its shell, and present the curious spectacle of a perambulating, waving — forest of bright green. ‘The individual filament is composed of cylindrical cells, which are always without a nucleus, and have their chlorophyl diffused instead of being collected into bands or stripes. ‘The walls are mostly quite thick and marked near the distal end with circular striae, whose numbers bear relation to the edge of the cell, for these strie are the results of the peculiar method of cell multiplication by division, each one marking one such division. When an cedo- gonium cell has attained sufficient maturity and is about to divide, the first per- ceptible change is the appearance of a little circular line or streak near its distal end. About the same time and in the same place a fine partition is formed by an outgrowth from the primordial utricle, a probably double delicate wall of con- densed protoplasm separating the upper end of the parent cell from the lower or main portion. ‘The upper end now begins to develop into a new cell. This de- velopment takes place by the formation of an entirely new layer of cellulose inside the little cell, 7. e. between the new primordial utricle and the old cell wall, and afterwards by the lengthening of this layer by interstitial deposit in the usual way; the thick wall of the parent cell in no way directly participates in the — growth (fig. 2 6, pl. 17). It is evident that as the new wall grows the old cell wall must be as it were raised up upon it, borne away as a little capping from the basal portion of the parent cell. Consequently when a young cell is watched during this process the little line-like incisure of the parent cell is seen to widen until it becomes an evident trench, and this trench grows wider and wider, until at last it is so broad as to be no longer a trench, and the little end of the parent cell simply caps its offspring. When the latter has fulfilled its allotted period ot growth, the process is repeated, the line of separation appearing this time just below the edge of the first cap. It is plain that the second new cell when formed must have a double cap crowning its extremity. At each repetition a new layer is added to the thickening cap, until at last it may be composed of six distinct layers, each projecting just beyond the next older one. Under the microscope the increased thickening of the distal end of a cell bearing such a crown-piece is not sufficiently evident to at first attract attention, whilst each edge of a layer appears asa stria, It is plain that the number of these strie represents the number of FRESH-WATER ALGA OF THE UNITED STATES. 191 times division has occurred; if there be four striw, four times; six striw, six times, &c. Besides this method of development, in many species new cells are formed by a sort of pullulation, occuring in the end cell of the filament. ‘The primordial utricle appears to rupture the wall of the distal extremity of the latter and grow out into a little pullulation, or teat, which very soon becomes separated from the , parent cell, by the reformation, as it were, of the end wall of the latter. ‘The _ new little cell thus formed coats itself with cellulose, and rapidly grows, especially in length, always, however, or’at least for a length of time, remaining of a smaller _ diameter than the cell from which it sprang. By a repetition of this process a _ succession of cells is formed, each one of which, like the successive joints of the _ field telescope, is a little smaller than its proximal neighbor and contains less chlorophyl, until finally the cells are reduced to exceedingly fine, perfectly trans- _ parent, colorless cylinders, which together form a seta or hair. Reproduction takes place among the Gidogoniacee, both by means of zoospores and sexual organs. ‘The former of these are quite peculiar, and, therefore, require especial notice. Only a single one is ever produced in a cell, and there is consequently no divi- sion of the chlorophyllous protoplasm preceding their formation. ‘The first change noticeable is a sort of confusion of the cell contents, the protoplasmic portion of which loosens itself, as it were, from the walls, and collects in a mass at the distal end of the cell. This mass after a short time assumes a more or less irregu- larly globose shape, and simultaneously the parent cell begins to separate from its distal neighbor. This separation appears to take place commonly by a solution of an exceedingly fine ring of the wall of the parent-cell, just at the origin of the transverse partition separating the two cells, and it is therefore brought about not by a splitting of the end partition wall, but by a circumcision of the side walls of the cell, and consequently the cavity of the latter is thrown open, the end wall remaining with and closing the distal cell, whose contents have not undergone change. On the other hand, observation leads me to think that sometimes there is a splitting of the end wall. According to my observation, sometimes the filament is completely broken in two, but very commonly the two cells remain attached by one corner, opening from one another as it were on a hinge-joint (fig. 2 /, pl. 17). The gathering of the protoplasm, already spoken of, into a ball, is a slow process, and the escape of this ball, through the opening formed in the manner described, takes place even more slowly. ‘The motion is not at all perceptible, with a power of a thousand or twelve hundred diameters. During the passage the ball becomes more or less twisted and deformed, but as it emerges the uncompressed portion shortens and swells out, and when the mass of protoplasm is at last free in the water, it soon assumes a globular or regularly ovate shape. The mother-cell, thus bereft of its contents, is left dead and void. The primordial utricle indeed still re- mains within, but it has lost all its wonderful powers, and is nothing but a shrunken, twisted, or folded dead membrane. What is the cause of the motion of the zoospore within the cell it is very difficult to determine. It certainly is not vibrating cilia, When the zoospore first escapes, it is, as already stated, an irregular lump 192 FRESH-WATER ALG& OF THE UNITED STATES, of strongly chlorophyllous protoplasm, homogeneous or with one or more roundish masses of darker green within it. As it assumes its shape, however, a very dis. tinct transparent spot appears at its smaller end. Whether this is an absolute vacuole or not, I have never been able to satisfy myself, but I am rather inclined to believe that it contains highly refractive transparent protoplasm, As this spot is perfected the cilia make their appearance. Whether they are actually first formed there, or whether, as is more probable, they are formed inside the cell, and are so folded against the general mass as to be invisible, I have never determined, Dr. Pringsheim, however, figures them within the cell. I have seen them in their early development long before motion commenced in them, but they were always perfectly formed as soon as apparent. ‘They are present in great numbers, making a crown or ring around the edge of the transparent beak-like end. When ’ they commence to vibrate, their action is at first very slow, and the waves of motion run’ through them deliberately from one cilium to the other, but soon, however, the motile impulses succeed one another more and more rapidly, until the general mass of the zoospore begins to tremble, then to rock, and finally dart- © ing off the little body hastens hither and thither through the water. The zoospore of an Gidogonium is always readily distinguished from most other similar bodies by its large size and peculiar motion, which is a forward movement combined with a distinct rolling on its long axis. After a time the zoospore, coming in contact with some speck of matter to which it can attach itself, ceases its movements, the cilia rapidly wither away, and the end to which they have been attached swells out or elongates into a broad, or narrow, simple, bifid, or trifid process, placed at an angle to the main axis of the cell, so as to form the so-called foot, the holdfast that anchors and fixes the new plant. Whilst this is taking place, the general form of the zoospore alters into that of a cylinder, a cellulose wall is secreted all about it, and the first cell of the new plant is complete. As soon as this cell is sufficiently matured, it begins to undergo division in the manner already described, and to develop into the new filament. In regard to the time when these zoospores are given off most abundantly, and the circumstances that influence the process, I can only state that it occurs when there is least tendency to the production of resting spores, probably in youngish plants, and I have thought was favored by a full supply of light, with a moderate temperature, Sexual reproduction occurs among the Gidogoniacee in accordance with three distinct types, to which the name of monecious, diacious, and gynandrous has been severally applied. The characteristic differences are to be looked for in the production of the antheridiz or male plant, the female germ being always pre- pared in essentially the same way. In most instances two cells are requisite for the production of the latter. At first there is nothing by which cells set apart for the formation of the female germ can be distinguished from ordinary cells. ‘The proximal one of the pair finally, however, undergoes changes similar to those seen when a zoospore is to be formed, namely, a sort of confusion of the endo- chrome, and finally a gathering of it into a mass at the distal end of the cell. In- stead of there being a solution of the side wall of the cell, however, the end wall FRESH-WATER ALG OF THE UNITED STATES. 193 undergoes absorption, so that the cavities of the two cells are more or less com- pletely thrown into one. All or nearly all of the contents of the proximal cell now slowly pass into the distal one, which thus becomes crowded with chloro- phyllous protoplasm. At or before this period, the distal receiving cell undergoes a change in form, widening out greatly, and sometimes appearing actually to shorten, so that it is in most instances resolved into a more or less regular globose or oval cell. As the sporangium or spore-case thus formed perfects itself the endo- chromes of the two cells become completely fused into one mass, which gradually condenses and assumes a regular shape, until, in the form of the perfected female or receptive germ, it is a dark, opaque ball more or less completely filling the spo- rangial cell, At the same time, in order to afford passage for the male germ, an opening is formed through the walls of the sporangium. ‘This happens in two ways. The simplest of these is by the formation of one or more circular openings or pores in the wall. ‘This pore is sometimes below, sometimes above the equatorial line. Its position, numbers, and form afford good specific characters. The second method is by the development of a little trap-door entrance at the distal end of the spore- case. ‘This method is unknown in our American flora, and, never having seen it, I must refer to the papers of Pringsheim for details. The above-described mode of origin of the sporangium is the common one. In O. mirabile, Woop, however, but one cell is concerned. This cell grows to an enormous size, far beyond that of its fellows, and its endochrome collects into the upper half of it, to be at last shut off from the lower half of the cell by the forma- tion of a new cellulose partition or end wall; or, in other words, the parent cell divides by a modified process of cell division, different from that common in the family. The distal daughter-cell contains all the endochrome. After the changes are completed, the appearance is the same as ordinarily presented, namely, an empty cell surmounted by the sporangium. Sometimes, even in plants in which the ordi- nary process occurs elsewhere, a single cell appears at times to have sufficient vitality to develop into a sporangium without aid from its neighbor, so that the latter will preserve its integrity, and the resting spore finally lie in proximity to a cell full of endochrome. In the monecious Gidogoniacee, a single filament produces both the male and female germs. Certain cells appear to be set apart to develop into sporangia, whilst others give origin to the spermatozoids. No such plants have as yet been detected in North America, and I, therefore, pass on without speaking more in detail. The second method in which the spermatozoids are produced is the most com- mon in our flora; it is the so-called gynandrous plan. In this the single filament produces the female germs directly and the male germs indirectly. The former arise in the way previously described, whilst the latter are the resultant of a complex series of life actions, as follows: One of the main cells of the originating filament, differing in no perceptible way from its fellows, instead of like them developing new cells, divides up by a simple process of cell division into two or more cells, each one of which contains very largely of chlorophyllous protoplasm. The protoplasm within each of these secondary or daughter-cells soon condenses into an irregularly ovate or conical mass, which often, even within the cell, may be seen to 25 September, 1872. 194 FRESH-WATER ALG& OF THE UNITED STATES, have the transparent beak of the zoospore (pl. 18, fig. 2d). Inside of the cell the androspore, as it is called, shows no cilia, but when it is set free by a more or less complete solution of the cell wall, it assumes the form of the ordinary CElogo- nium zoospore, With a crown of cilia, whose vibrations soon cause it to dart through the water. ‘Lhese androspores are of course much smaller than an ordinary zoospore, and after a period of active motion, they attach themselves to the parent filament, generally either on or near the sporangial cell, ‘Their first life-actions, after settling, are precisely like those of the zoospore, namely, dropping of the cilia, enlargement of the smaller end into the so-called “ foot,” an elongation of the general mass, and the secretion of an outer coating of cellulose. In this way a peculiar-shaped, somewhat ovate cell is formed, which contains a great quantity of rich protoplasm with mostly a small amount of chlorophyl. From such cells are developed the mostly two- or three-celled, perfect antheridia, which in gynandrous Cilogonia are generally to be seen, during the period of fructification, in numbers attached to the filament, mostly in the neighborhood of the sporangium, Their pro- toplasmic contents are remarkable for the activity of their movements, and I have seldom seen more beautiful and rapid cyclosis than they display—currents setting in all directions—particles actually brushing against one another (pl. 17, fig. 2). The spermatozoids are formed in the distal cell, sometimes one, sometimes more. In the species O. mirabile, Woop, (pl. 18, fig. 2 g, 2 6) in which I have most carefully studied their origin, two are produced in the single cell. This cell is in the commencement of the process, although comparatively poor in chlorophyl, crowded with a rich solid protoplasm, which divides into two distinct masses, some- what in the manner seen in the commencement of ordinary cell division, As there is no distinct nucleus, of course there are no precedent nuclear changes. The masses thus formed gradually assume a more or less perfectly globular shape inside the cell, although I have never been able to see that they there develop cilia, and finally are set free by the lifting up of the end of the mother-cell, like a little trap-door. ‘Their mode of escape through the exit thus offered is similar to that of the ordinary zoospore, which they resemble, except that they are much smaller, are much less rich in chlorophyl, and have the anterior clear space less defined. They are said to be furnished with a crown of cilia similar to that of the zoospore, I myself have never seen these, but do not doubt their existence. In the diacious Gidogonia there are distinct filaments, male and female, one of which produces the oosporangium with its contained germ, whilst the other gives rise directly to the spermatozoids, The resting spore which develops after impregnation is variously shaped, but in most instances is round or oval. It is often, if not always, furnished with two coats, the outer of which is thick, firm, and frequently provided with surface appendages, such as tubercles, ridges, spines, ete. Besides these there is also, probably, a very delicate inner coat. ‘The spore appears to be set free from its case by the decay of the latter, there being never, at least that I have seen, any regular dehiscence. Although [have made several attempts, it has never been my good fortune to observe anything ike germination of these resting spores. Prof. Chr. Vaupell, however, has published an account of the manner as observed by himself. Some water containing fruitful FRESH-WATER ALG OF THE UNITED STATES. 195 CGdogonia was allowed to dry in a glass, towards the close of September, and the greenish residue was placed in water in the following January. By March the resting spores were everywhere in active germination. The first change was a rupture of the two outer coats and the escape, through the slit, of the contents, still surrounded by a very delicate hyaline membrane. By this time the proto- plasm had divided into usually four (sometimes only two or three) greenish masses, each of which was oval in shape and had its own extremely thin, hyaline coat, and was therefore a perfect cell, The old outer shell of the spore laid discarded in the water and soon decayed, and in a little while the hyaline sac surrounding the four daughter-cells itself disappeared, leaving them exposed and naked. After awhile each of these cells opened at one end by means of an annular split, cutting off the apex of the wall and allowing it to lift off like a little lid. Through the circular opening thus made, the contents now emerged. ‘The point of the inner mass was colorless and directed towards the orifice, and the whole moved vigorously backwards and forwards until it finally escaped, as a perfected zoospore. This little body simulated very closely the ordinary zoospore, both in appearance and life-history, growing, after a brief period of activity, into an ordinary filament, in precisely the same manner as the zoospores. Genus GADOGONIUM. Antheridia et oogonidia in individuo unico. Antheridia and oogonidia in the same individual. Remark.—No species of the genus Gidogonium, as here defined, has as yet been discovered in this country. Genus PRINGSHEIMIA. Dioica. Antheridia et oogonidia in individuis distinctis orta. Diecious Antheridia and oogonidia arising in distinct individuals. P. inequalis, Woop. P dioica; cellula basali biloba; plantis femineis quam plantis masculis permulto majoribus; oogoniis enormiter globosis vel subovoideis, poro laterale supra medium posito instructis; oosporis forma eadem, sed paulo minoribus. Syn.—CGidogonium inequale, Woop, Proc. Amer. Philos. Soc., 1869, p. 141 Hab.—In stagnis, prope Philadelphia. O. diccious, basa] cell bilobate; female plant very much larger than the male plant; oospo- rangium irregularly globose or subovoidal, opening by a lateral pore above the middle; resting spores of the same form as the sporangium, but a little smaller. Remarks.—This plant seems to be more closely allied to O. gemelliparum, Pringsheim, than to any other species. It agrees with it in the inequality of the male and female plants, in the shape of the sporangium, and the position of the lateral pore. The diameter of the female plant is often nearly four times that of the male, and the difference in length is much more apparent. ‘Che mother-plant is composed of from 3-6 cells in the most distal of which the spermatozoids are formed. I am not able to state how many of these bodies are formed in a single cell, having 196 FRESH-WATER ALG# OF THE UNITED SPADES: only seen the latter when more or less completely emptied, but, judging from the relative sizes, there must be several. In a cell containing a single spermatozoid, that body moved about freely, and at last escaped, apparently through an orifice in the end wall of the cell. It made two attempts before getting out, and during its passage was distinctly constricted in the middle. It resembled in appearance an ordinary zoospore, but was of course much smaller, and was nearly devoid of color, having but a slight greenish tint. I found this species growing abundantly in the stagnant ditches of the Neck, below the city. Fig. 1 a, pl. 18, represents a young female plant; 1 6, a fertile plant with imma- ture spores. 1c was taken from the supposed male plant alluded to in the text. The latter figure is magnified 450 diameters, the others 250. Genus ANDROGYNIA. Gynandra. Androspore in plantis femineis orte ; postea hance affixe et in antheridiis se formant. Gynandrous. Androspores arising in the female plant; after affixing themselves to this and developing into antheridia. A. multispora, Woop. A. oogoniis singulis, vel binis vel ternis continuis, globosis instrueta; poro laterale distale; oosporis globosis, oogonii lumen replentibus ; antheridiis plerumque pluribus, planta feminea insidentibus, cellula inferiore multo majoribus. Syn.—Oecedogonium multispora, Woop, Proc. Amer. Philos. Soe., 1869, p. 141. Hab.—In stagnis, prope Philadelphia. Oosporangia single or bi- or triseriate, globose with a distal lateral pore; oospore globose, about the same size as the sporangial cavity; antheridia bi- or tricellular, curved, with the lower cell much the largest, generally adhering in considerable numbers to all parts of the female plant. Remarks.—This species differs from its nearest European congeners, Gilogon. Rothii and Gi. depressum, very markedly in the bicellular antheridia. I have never seen the spermatozoids actually emerging from their mother-cell, but have seen in the terminal antheridial cell a pair of oval bodies, which I took to be those bodies. Fig. 3, pl. 17, was taken from a filament of this species magnified 500 diameters. It shows spores in different stages of maturity, with an empty basal cell in one case, and in the others without. Also male plants, one of them containing partially formed spermatozoids. The small arrows indicate the direction of cyelotic currents. A. mirabilis, Woop. A. rare setigera; articulis diametro 2-8 plo longioribus; oogoniis plerumque singulis, rare geminis, nonnihil ovatis, infra latis sed supra contractis et medio tumidis ; poris lateralibus duobus supra medium positis ; oosporis aut late ovalibus aut subglobosis ; sporodermate haud signato; antheridiis plerumque bicellularibus, interdum tricellularibus, plerumque in filo vegetativo infra oogonium aut in oogonio insidentibus ; spermatozoideis singulis et geminis. 7. tic. vere Silty, 3" 5 C ”" Diam.—Artic. veget 795y"—7} 35" =.0004”—0017". Spor. 7385”—7285".0024”—0027". Syn.—CEdogonium mirabile, Woop, Proc. Amer, Philos. Soc., 18 . 142. ? ’ ? Pp Hab.—In rivulis quietis, prope Philadelphia. FRESH-WATER ALG# OF THE UNITED STATES. 197 A. rarely setigerous; articles 2-8 times longer than broad ; oosporangia mostly single, rarely geminate, subovate, in the lower portion broad, in the middle swollen, in the upper part con- tracted; the 2 lateral pores situated above the middle; oospore subglobose or broadly ovate, its coats without markings; antheridia generally bicellular, sometimes tricellular, numerous, placed generally upon the female filament either upon or below the oosporangia. Remarks.—This species was found growing in a rather stagnant brook in the meadow by “Robinson’s Knoll,” at the junction of the Schuylkill River and Wissa- hickon Creek, near Philadelphia. The filaments, which vary very greatly in size, are in their early history attached to dead leaves and sticks, but finally, I think, float free in the water. The larger, fruit-bearing filaments are remarkable for their crookedness. None of the threads that I have seen ended in a seta-like portion. The fruit is produced in abundance, but very rarely is there more than a single spore in any one place. ‘The method of the formation of the sporangia differs from that of all the other Qidogonia which have come under my notice. Instead of two cells being concerned but one cell is employed. The cell (fig. 2 a, pl. 18) that is to be used for such a purpose grows much beyond the ordinary size, until it is nearly or quite twice as large as its neighbors. All the time it is well filled with chlorophyllous protoplasm. ‘This now contracts and finally is all packed into the upper half of the cell. At or even before this time the lateral apenings become appa- rent. There are two of them, situated just in the angle where the cell at its upper end commences to contract to the size of its fellow. At this time I think fertiliza- tion takes place, although I have never actually seen the spermatozoids enter the orifices. The cell (fig. 2 4, pl. 17) now divides into two by forming a wall separa- ting the lower empty half from the upper full one, which is to be the sporangium, The contents of the latter now condense into a ball, and it itself becomes more tumid in the middle. Finally a reddish-brown broadly globular spore (fig. 2 ¢, pl. 18) is formed. I have not been able to make out more than one distinct thick coat. The surface of the spore is smooth. The androspores are formed in a cell (fig. 2d, pl. 18) which has grown beyond the normal size and then divided into four or five short cells, each of which gives origin, I believe, to a single androspore inits interior. ‘The antheridia are numerous, from 2 to 6 being commonly attached to the lower portion of the sporangium, or to the cells just beneath it. They (fig. 2 e, pl. 18) have a rather large foot, and are generally curved at the base. ‘The distal of the two cells composing them is crowned with a little cap, and produces one or sometimes two spermatozoids. These (figs. 2 6 and 2 g, pl. 18) during their escape are always very much squeezed out of shape, but when free become globular or slightly pear-shaped. They are highly transparent and contain a few green granules. Their motion is at first slow, but soon becomes very active. ‘The mode of egress from the cell is obtained by the cutting off of the upper end of it, the little cap opening like a trap-door. After this cell has been emptied, sometimes a second similar one is formed, which bears it aloft. I have never seen spermato- zoids produced by this second cell. A. HWuntii, Woon. Filuma plerumque in setam longam, terminalem coloris expertam productum ; oogoniis pler- umque singulis, globosis, interdum nonnihil hexagoniis, medio nonnihil tumidis, poro laterale 198 FRESH-WATER ALG# OF THE UNITED STATRS, infra medium posito; oosporis globosis, oogonii lumen haud replentibus, superficie lineis eleyatis spiralibus quatuor instructa; antheridiis bicellularibus (interdum tricellularibus ) Diam.—Spor. s}5” =.002". Syn.—CGidogonium Huntii, Woon, American Naturalist, 1868. Hab.—In aquario meo. Filaments mostly produced into a long apical seta; oogonia mostly single, globose, sometimes somewhat hexagonal, somewhat tumid in the middle, the lateral pore placed below the — middle; oospore globose, not filling the eavity of the spore case, its surface with four spiral ; elevated lines or ridges; antheridia bicellular (sometimes tricellular ?). Remarks.—This little plant appeared in my aquarium some years since, forming - a delicate fringe upon the various aquatic plants growing therein. Its color is a bright yellowish green, deepening to a very dark green in cells which are crowded } with granular protoplasm. The filaments vary very greatly in size, the largest I have seen were 54> of an inch in diameter. ‘They are provided with long, termi- nal seta, which are much more universally present than in any of the other species , I have met with. ‘The first step in the formation of a spore is the emptying of a_ cell into its distal neighbor, so that each spore case is placed at the end of an empty cell. These sporangia may be single or they may be in series of two or more, separated only from one another by the eruptive cells just spoken of. The color of the mature spore is a very dark reddish-brown. The antheridia is bicellular, slightly curved, somewhat stipate, with a distinct foot. Its most common position is on the vacated cell just below the spore case. ‘The zoospores, as I have seen them, are always globose. I have named this species after my friend, Dr. J. Gibbons Hunt, a well-known microscopist of this city, to whom I am greatly indebted for aid in my earlier microscopic studies. Fig. 2, pl. 17, represents different forms and parts of this plant. 2 a shows. the end of a filament and the long seta-like lip. 2 was taken from two’ cells,, one of which had just undergone division, and shows very plainly the method of procedure ; lying as it were between the cells, and bearing the end of the lower one upon it, is the new little cell. Fig. 2.¢ represents a fertile filament with two mature spores and one not fully grown. Fig. 2d was drawn from a filament just forming a spore, and shows the male plant in situ. Fig. 2 e represents a male plant (magnified some 1300 diameters) with the outer terminal cell scarcely more than a primordial utricle. The contents of the lower cell were in a state of in- tense motion; and the arrows are meant to indicate the directions of the currents. Fig. 2 / represents a portion of a filament with a zoospore just escaped and still quiescent. A. echimata, Woop (sp. nov.) A. valde elongata ; articulis diametro 6-14 plo longioribus; oogoniis globosis, plerumque de: pressis, ad .0014” crassis ; oosporis oogonii forma et ejus lumen replentibus, valde aculeatis poro laterale supra medium posito; antheridiis bicellularibus ? Diam.—Spor. ys\fyq"=.001". Cell. gqgq”— zoho" = .00033” —.0005”. Hab.—In stagnis, Alleghany Mountains, FRESH-WATER ALG# OF THE UNITED STATES. 199 O. gynandrous, very elongate ; joints 6-14 times longer than broad; sporangia globose, mostly depressed, about .0014” in diameter; oospores of the same form as sporangia, whose cavity they almost fill; covered with sharp spines; the lateral pore placed above the middle; an- theridia bicellular ? Remarks.—1 found this distinct species in a little stagnant pool in the wilderness, known as Bear Meadows, in Centre County, of this State. ‘The fila- ments are very long, and were matted together into a sort of fibrous mass. The male plants were few in number, and were attached to the female plant in the neighborhood of the sporangia. I have not seen any composed of more than two cells. ‘They are furnished with a well-marked foot, above which there is a short neck. As I have seen them they are nearly straight. I have not been able to make out more than one coat to the spores. This coat is very thick, and is furnished with numerous thorn-like spines. These are very sharp at the points, but at their bases are mostly very robust. Fig. 3, pl. 18, represents a spore of this plant magnified 750 diameters, SupramMiLty BULBOCH ZTE. Filuma ramosum, setis strictis hyalinis achrois e basi bulbosa et plus minus elongatis instructum., Filaments branching, furnished with straight, hyaline, more or less elongated seta, arising from a bulbous base. Remarks.—The Bulbochetee are at once separated from their allies the Gidogo- niee by their bushy, branched habit of growth. The shape of the individual cell is also entirely different, for instead of being regularly cylindrical they are almost always markedly dilated at their distal end, so as to be somewhat clavate, nor is the filament or its branches ever ended by a long seta-like series of narrow colorless cells. Many or all of the cells are, however, furnished with a single very long unicellular unbranched hair. These hairs are colorless, hyaline, and provided with a markedly and abruptly bulbous base. The Bulbochetee grow in similar positions to their allies, but are not nearly so common, nor when present do they grow in such abundance, very rarely, if ever, forming the dense forest-like fringes or the matted masses that some species of the Gidogoniee do. They are reproduced both by zoospores and resting spores. The manner of the development of and growth of the plant from the zoospore is very peculiar. I have never myself studied it, but Prof. Pringsheim gives the following account: When the zoospore first settles down it produces a cell closely resembling that of an Gidogonium. The first change which occurs in this cell is the formation of a small, conical, transparent, colorless space at the apex, which space in a little while becomes separated from the mother-cell by a distinct par- tition-wall, and at the same time the apex itself is ruptured, and the point of the little growing cone pushed through the opening. This rupture does not take place irregularly, but by a sort of circumscribed dertiscence, similar to that of the Cidogonium, the top of the mother-cell being lifted up like a little trap-door, and finally pushed aside as the new conical cell grows elongate and becomes converted into a hair. After the formation of this apical hair, the mother-cell undergoes division in a manner similar to that of an Gidogonium. Near its distal end a 90 FRESH-WATER ALGA OF THE UNITED STATHS, circular slit appears, and at the same time a partition forms, so that from the mother-cell are developed a small apical and a large basal daughter-cell. ‘The his- tory of the former of these is simply one of growth as regards the main axis, It inereases in size but does not give origin to new cells. All such cells are formed out of the basal daughter-cell, which, as already described, divides into a new apical and basal cell—the apical only to grow in the main filament—the basal to divide anew, It is always the basal cell that undergoes division, throughout the whole life-history of the plant, one cell alone contributing to the growth of the main filament. The filament thus formed bears upon its distal end the hair which grew upon the original spore-cell, and this hair is, save only the basal cell, the oldest part of the filament. The cell upon which it rests is the next oldest, the next to it in position, the next in age, and so on (from older to younger) down to the basal cell, the oldest of all, lying next to the latest born. Although the cells of the main filaments do not contribute to its development, yet it is from them that the lateral branches are formed. The production of a branch begins by the appearance of a clear space near the apex of the cell, but this clear space is placed, not exactly at the apex, but a little to one side. It soon becomes distinctly conical, enlarges, bursts through the old cell-wall, is cut off by a cellulose partition from its parent, and develops into a hair similar to that first formed, but placed at an angle to the long axis. It is remarkable that the opening for the exit of the growing hair occurs, not by a circular transverse slit, but by a longi- tudinal one, the two halves of the old cell-walls separating as the little cone pushes its way between them and persisting as a sort of sheath to its base. When the hair is perfected the cell from which it grew undergoes division in the usual way, save only that the cutting off of the old wall is done obliquely instead of trans- versely, so that the partition is oblique instead of horizontal, and the new cell grows at an angle to the old, instead of in the line of its axis. The new cell, consequently, is the starting point to a branch at an angle to the main filament. This branch, like the main filament, grows only by the repeated divisions of its primal basal cell, and bears aloft its seta. Secondary branches may arise from it precisely in the way that it arose from the parent stem, and thus at last is formed the bushy plant of the Bulbochetee. The zoospores closely resemble those of the Gidogoniee, and are oval or glo- bose masses of chlorophyllous protoplasm, with a transparent space at the smaller end, surmounted by a crown of cilia. Their mode of formation and whole life- history are also similar to that of the (Hdogonie zoospores, up to the time when in their germination they begin to produce new cells. Sexual reproduction amongst all the known Bullochetee is similar in its general aspect to that seen among the gynandrous Gidogoniee, but differs considerably in detail. The oogonia are mostly formed in lateral branches, Their position in these branches varies in the various species. Since any cell from the next to basal to the most distal of all crowned with the terminal seta may be converted into a oogonium, according to Pringsheim, the cell which is to form the oogonium arises in the usual way, by the division of a ccll into two daughter-cells. The new daughter-cell, which is to develop into the FRESH-WATER ALG OF THE UNITED STATES. 201 sexual part, does not, however, rupture the old wall of the mother-cell, but grows out beyond it, and there dilates. ‘The new cell is therefore divisible into two parts, /a proximal cylindrical portion, contained within the walls of the mother-cell, and a distal more or less globular piece beyond the latter. ‘The chlorophyllous proto- _ plasm now collects in this dilated portion, leaving the basal cylindrical part bare and empty. The oogonium is not, however, formed directly from this upper portion | (the primitive oogonium, as it may be called), but a new wall forms within the latter ' and then it undergoes division much as did the primary cell. In this way it is that the upper and lower portions of the old wall, 7. e. that of the primitive oogonium, remain as a sort of basal sheath and cap to the fully-formed sporangium. ‘The little hole by which the spermatozoids find entrance to the contents. of the oogo- _ nium is always formed in the upper half of the wall of the latter. As stated, all the species of Bulbochwtew as yet known are gynandrous. The ' antheridia resemble those of similar Gidogoniee, and their life-history is very similar. _ The development of the resting spores is said to take place as follows: The first change is in the color of the spore, the bright red becoming green, especially near the margins of the cavity. ‘The outer wall is then ruptured and the spore grows into a long oval body, whose contents are chiefly green with a sprinkling of the original red. The protoplasm of this oval body gradually divides into four masses, which become more and more distinct, until they are at last well formed zoospores, similar to those produced in the more ordinary method, except, perhaps, that they are redder. They are finally set free in the water by a solution of the cell wall surrounding them, and enter upon a brief free existence, to settle down after a - little and grow into a fully-formed plant. Genus BULBOCHLETE., Androspore in planta feminea orte, postea hance affix et.in antheridiis se formantes. Androspore arising in the female plant, afterwards affixed to it and developing into the antheridia. B. igmota, Woop. B. sparse ramosa, elongata; articulis diametro max. (qs49” = -00077") 15-3 plo longioribus; oogoniis long. hy” = .0025”, lat. g445” =.0018”, interdum lateralibus et sessilibus, inter- dum inter ramulorum cellulas vegetativas positis, dissepimento nullo; oosporis ovalibus, longi- tudinaliter nonnihil oblique et distante costatis, in wtate provecta aurantiaco-brunneis, sporo- dermate crasso; antheridiis 3-4 cellularibus, stipitatis. Syn.—B. ignota, Woop, Prodromus, Proc. Amer. Philos. Soe., 1869. Hab.—In aquis quietis, prope Philadelphia. B. sparsely branched, elongate with the joints 13-3 times longer than broad (y3455” = .00077”"); oosporangia .0025” long by .0018” broad, sometimes lateral and sessile, sometimes placed upon the apex of a branch, sometimes situated in the length of the branches between their cells; the empty cell which supports the sporangium without dissepiment; oospores, oval, filling rather closely the cavity of the spore-case, longitudinally somewhat obliquely and dis- tantly costate, when mature orange brown; spore-coat rather thick; antheridia 3-4 celled, searcely stipitate. Remarks.— When I described and figured this species I had never seen the mature fruit, but very recently Mr. Quimby has communicated specimens to me. 26 September, 1872. 9()2 FRESH-WATER ALGA OF THE UNITED STATES: The color of the spore is orange brown, and the thick coat is slightly tinged with yellowish. The mature oosporangium is somewhat flattened at the sides, not so elliptical as the young spore, which I have figured. Fig. 5a, pl. 18, represents a fragment of a filament showing young sporangial cells magnified 260 diameters ; 5 b, represents a branch with a youngish spore in it, magnified 460 diameters; fig. 5c, was taken from a male plant. B. dumosa, Woop. B. articulis diametro 14-2 plo longioribus; oogoniis plerumque in ramorum brevissimorum apicibus positis sed interdum lateralibus, plerumque setam terminalem gerentibus; oosporis evormiter ovalibus aut ovatis, nonnihil indistinete longitudinaliter oblique snbarcte. striatis ; antheridiis bicellularibus, stipite instructis, cellula basale medio tumida, supra swpe contracta, Syn.—B. dumosa, Woon, Prodromus, Proc. Amer. Philos. Soc., 1869, p. 142. Hab.—In aquario meo. Joints 14-2 times longer than broad; oosporangia generally placed upon the ends of short branches but sometimes lateral, mostly carrying a terminal seta; resting spores irregularly oval or ovate, somewhat indistinctly obliquely longitudinally and rather closely striate; an- theridia bicellular, furnished with alittle stipe, their basal cell tumid in the middle, frequently contracted above. Remarks.—This species appeared spontaneously during the latter part of the winter upon some large fresh-water algze which I was cultivating. It branches irregularly and sometimes somewhat profusely, so as to have quite a bushy habit. The antheridia appear to produce a single spermatozoon in the terminal cell; at least as far as my observation has gone this is true. I think I have always found the distal cells of fertile plants emptied of their contents, as though they had fur- nished the androspores which had grown into the antheridia. This species is closely allied to B. gracilis, of Pringsheim, from which it differs in the position of the oogonia, in the relative breadth and length of the cell, and the number of cells composing the antheridia. Vig. 6 a, pl. 18, represents a filament of this species magnified 260 diameters; 6 6, a male plant magnified 750 diameters. B. Canbyii, Woop. B. permagna ad .035” longa, sparse ramosa; articulis sterilibus diametro 2-8 plo longioribus: oogoniis lateralibus vel in ramulorum apicem positis, transverse enormiter ovalibus ; oosporis, transverse enormiter ovalibus, plerumque nonnihil triangularibus, oogonii lumen replentibus; sporodermate crasso, haud costato, enormiter punctato; antheridiis bicellularibus. ' Y 7 a t An , Diam.—Cell. steril. 7355"—gFo0 = .00066—001. Spor. transv. 7355” = -00226. Syn.—B. Canbyti, Woon, Proe. Amer. Philos. Society, 1869, p. 142. Hab.—In aquis quietis, prope Hibernia, Florida; (Wllliam Canby). B. very large, attaining a length of more than one-third an inch, sparsely branched ; sterile Joints 2 to 8 times longer than broad ; oosporangia lateral or placed upon the ends of branches, irregularly transversely oval; oospores of a similar shape, often a little triangular, filling the cavity of the sporangium; spore coat thick, not costate but irregularly punctate. Remarks.—It affords me great pleasure to dedicate this very handsome species to Mr. William Canby, by whom it was collected in Florida, as an acknowledg- ment of favors received, and as a testimony of respect and high regard for him FRESH-WATER ALGA OF THE UNITED STATES. 203 personally, and as being among the foremost students of American phanerogamic botany. _ This species is more nearly allied to S. minor than to any other of the European forms, but differs from it very essentially in size and habit. It is always, as I have seen it, except in very young plants, sparsely and mostly dichotomously branched, and attains a very great length, at times probably exceeding the third of an inch. The spore is mostly sessile upon the distal ends of the cells of the filament; in all such cases I have noticed that the cell upon which it was borne was divided in its ‘middle by a partition into two cells. Not unfrequently the spore is raised upon a short branch. ‘The male plants are attached to the female filaments generally in the neighborhood of the sporangium, to which they sometimes fasten themselves immediately. They are shortly stipitate, and composed of two cells. The mature spore is transversely oval, now and then slightly triangular, and is nearly of the color of burnt sienna. Its.coat is thick, often slightly yellowish, and has on its outer surface irregular punctations, looking like corrosions. These are not detach- able, except when the ruptured spore is more or less completely emptied of its contents. ‘The sporangium closely invests the spore, and when the latter is matured undergoes a circular division, so that the top falls off and allows the spore to escape. Fig. 6 ¢, pl. 16, represents a portion of a filament, magnified 260 diameters, with a young sporangium and young male plants attached; 6 6, represents a very young plant, magnified 260 diameters. Fig. 6 a, was taken from a mature plant, and shows the mature spore. Fig. 6 e, shows in outline a sporangium and male plants attached; whilst 6 d, was drawn from a sporangium which had perfected its spore and undergone the natural dehiscence. Famty CHROOLEPIDEZ. Algw aeree, aureo-, aurantiaco- vel rubro-fusco-colorate, siccatw spe cane. Fila varie ramosa, cytiodermate crasso vel subcrasso, firmo, subcartilagineo predita, in pulvinulos minutos vel in stra- tum tenue aut incrassato-tomentosum densissime aggregata vel implicata. Cytioplasma oleosum vel ‘granulosum, aut rubellum, aureum, aut flavo-fuscum, interdum viride tinctum, post mortem plerum- que expallescens. Propagatio fit zoogonidiis. Arial alge. Golden orange, or reddish fuscous, often grayish when dried. Filaments variously branched, furnished with a thick, or thickish, subcartilaginous cytioderm, densely aggregated into minute cushions, or a thin or tomentosely thickened stratum. Cytioplasm granular or containing oily particles, reddish-golden, or yellowish-fuscous, sometimes tinged with green; after death often colorless or nearly so. Propagation by zoospores. Remarks.—The plants of this family are so different from the others of the order, that it is a matter of considerable doubt whether or not they should be classified with them. They rarely possess distinct, well-pronounced chlorophy], and form mats or strata of some shade of reddish, grayish, or brownish, so that they are very different in appearance from the other Confervacee. I do not think their position can be certainly fixed until their life-history has been more fully developed. In assigning them this place I have simply followed Prof. Rabenhorst. a4 FRESH-WATER ALG OF THE UNITED STATES) The only specimens that have come to my notice are in a dried condition, and consequently no possible opportunity has been afforded of studying the manner of reproduction, No one has as yet, at least to my knowledge, discovered any sexual reproduction in the family, but the method in which the zoospores are produced has been carefully studied, especially by Drs. Caspary (Kegensburg Flora, 1858) and Hildebrand (Botanische Zeitung). ‘The little motile bodies are not produced in the cells indiscriminately, but in certain ones set apart for the purpose, to which the name of zoosporangia is very applicable. ‘These are large, globular, thick walled cells, which are generally provided with a protuberance at the top and marked by transverse wrinkles. They are most frequently situated upon the end of the filament or one of its branches, but are rarely placed in the middle of the thread, and still ‘more rarely the cell next below the zoosporangium elongates itself sideways and up- wards into a thread, so that the reproductive cell is left as a lateral one-celled branch or process. When the zoosporangium is sufficiently matured the endochrome breaks up into a number of minute masses, the future zoospores. Finally the crowning papilla of the mother-cell ruptures and allows the contents to escape as a well-formed vesicle, containing the perfected zoospores. It is said, however, that sometimes the vesicle is wanting, and the zoospores are discharged into the water. In the ordinary course, after a little while the vesicle lying in the water bursts and sets its motile contents free. The zoospores themselves are very small, accord- ing to Hildebrand, ;$,-;¢,;mm. in length, by ;4,-,2;mm. in breadth. In accord- ance with the same authority they are, when first discharged, cylindrical, but in a little while become flattened, and shaped like a flaxseed. ‘They are biciliate and contain a large number of small, orange-colored particles. From thirty-two to sixty-four of them are formed in one zoosporangium, and neither light nor time of day appear to have any influence upon their birth. Hildebrand states that their motile life lasts from eighteen to thirty-six hours, but according to Caspary, after continuing in motion for about an hour, they grow sluggish, sink, become globular, then elongate themselves and shortly undergoing transverse division, actively com- mence to form the new filament. Genus CHROOLEPUS, Ag. Fila distinete articulata, intricata, enormiter ramosa. Filaments distinctly articulate, intricate, irregularly branched. Cc. aureum, (Liyné.) Krz. C. filis ramossimis, in stratum aureo-brunneum, ad duas tres lineas crassum, cespitosum ct molle intricatis vel in cxespitulos aggrewatis ; articulis enormibus, diametro sesqui-, duplo triplove longioribus. Diam.—Max = .001”. Syn.—C. aureum, (Linné.) Ktrzine. Rapennorst, Flora Europ. Algarum, Sect. II. p. sil. Hab.—Little Falls, New York; Godwinsville, New Jersey ; (Austin). Texas; (Ravenel). Filaments very much branched, interwoven to form a yellowish-brown softish mat, two or three lines in thickness; joints irregular, 14-3 times longer than broad. Remarks.—I am indebted to Mr. Austin for specimens which are labelled “Forms dense yellow-brown cushions on rocks, at Little Falls, New York and FRESH-WATER ALGA OF THE UNITED STATES. 205 Godwinsville, New Jersey.” As dried, the plant is in extended, gray, felt-like masses. ‘The walls of the articles as seen with the microscope are thick and uregular, and the joints themselves are also very irregular, the end ones being often ‘swollen and rounded so as to give the branches a sort of bulbous termination. Among the Alge collected in Texas by Prof. Ravenel, is a dried specimen (No. 100), labelled “On Bark, Houston, Texas,” which I cannot separate from this ‘species. It occurs in small tufts, which, as dried, are of a very decided orange, ‘and, no doubt, were still brighter during life. The articles are not so uregular as in Mr. Austin’s specimens, but excepting in this and color when dried they agree very well. Besides these I have several specimens from the same source, which are in extended mats and agree in all respects with their northern brethren. Our American form appears to attain a greater diameter in its individual fila- ‘ments than does the European variety, but I know of no other character separating ‘it from the latter; and consequently must consider them identical. ‘The measure- ment given is an extreme one, .009” being commonly the limit. Genus BULBOTRICHIA, Kotz. Fila indistinete articulata, achroa, firma, ramosa; rami in apice intumescentes, sporangia con- stituentes. Filaments indistinctly articulate, translucent, firm, branched; the ends of the branches swollen so as to form sporangia. B. albida, Woop (sp. nov.). B. strato albido, coriaceo vel crustaceo ; filis arcte intertextis, enormiter ramossissimis, coloris expertibus; sporangiis viridibus. Hab.—In muscis, Northern New Jersey ; (Austin.) Forming a white leathery or crustaceous stratum; thread closely interwoven, irregularly and plentifully branched, colorless; sporangia greenish. Remarks.—This curious little plant, which was sent me by Prof. Austin, occurs in minute white patches growing on mosses at the base of stumps in woods. Some- times these are encrusted abundantly with the carbonate of lime, when they are hard and crustaceous. ‘The sporangia appear to vary greatly in size; sometimes they resemble very closely a single spore (probably their commencing stage). The bases of the branches are rarely, if ever, furnished with the bulbous swelling, given by Rabenhorst as a generic distinction, but such enlargements do occasionally occur in the course of the filaments and branches, The filaments are composed _of a series of cells, which are in places long, and have their end walls thin and not readily seen. Fig. 5, pl. 16, represents a part of a plant magnified 460 diameters. Famty CHATOPHORACEA. Alge aquatice vel palustres, rarius terrestres, plerumque monoice vel dioice. Fila varia, spe dichotome ramosa, haud raro fasciculatim ramulosa, plerumque in cespites vel pulvinulos eumulata, in muco gelatinoso subliquido vel firmo nidulantia. Propagatio fit tum oosporis, tum zoogonidiis, Zoogonidia oriuntur aut singula aut geminis aut eytioplasmatis divisione 8-16 in quoque sporangio. 206 FRESH-WATER ALG# OF THE UNITED STATES. Aquatie, paludal, or rarely terrestrial alge, mostly monecious or diecious. Filaments various, often dichotomously, but not rarely fasciculately branched, mostly aggregated into turfy masses or little cushions, and generally surrounded by a firm or subliquid gelatinous mucus. Propagation both by zoospores and resting spores. Zoospores arising either singly or by the division of the cytioplasm into 8-16 in each sporangium. Genus STIGEOCLONIUM. Fila articulata, simpliciter ramosa; rami ramulique sparsi, rarius faseiculatim approximati, in apicem acutum, seepe piliferum achroum attenuati et plerumque longe protensi, sepius ramellig brevibus subulatis instructi. (R.) Filaments articulate, simply branched ; branches and branchlets sparse, rarely fasciculately ap- proximated, with their ends acute and frequently prolonged into an attenuate transparent seta or hair, and very often furnished with short subulate branches. Remarks.—Plants which are certainly referable to this genus are abundant in every place in which I have ever looked for fresh-water alge. I confess, how- ever, that although very much time has been given to their study, I have not been able to make out any distinct specific characters, nor any identifications from the diagnoses of M. Rabenhorst. In a certain spring northeast of the city, there grows one of these forms, which I have closely watched for several seasons. In the earlier state it appears at times to possess the characters of a young Chetophora (pl. 19, fig. 1), forming a small gelatinous base out of which the threads soon escape as they lengthen. It constituted a sort of mucoid layer adhering to the boards lining the stones with waving masses of projecting filaments six or even eight inches in length. The filaments were mostly about z5!;5’ in diameter and much interlaced. The cells varied greatly in length, some being scarcely as long as broad, whilst others were eight or ten times longer. ‘The short cells were generally densely filled with endochrome, whilst the long ones were nearly empty. ‘The branches often ended abruptly, but were more frequently tipped with a long seta-like point. The method of branching is as varied as can be imagined, as is shown by fig. 4, pl. 16, and fig. 1, pl. 20, all taken from different plants of this species. I have frequently seen the production of zoospores, but no other method of repro- duction. In all cases a single motile body (fig. 4, pl. 16) was formed in each cell. These minute bodies are globular or pyriform, and within the cell exhibit no motion whatever. ‘Their escape takes place very slowiy through a lateral slit in the wall. No cause of the motion is visible, and during the passage the zoospore is often very much squeezed out of shape. According to Braun (Ver- jungung), these zoospores possess a red eye-spot. I had not read his description at the time my observations were made, but did not notice any. The zoospores germinated in the usual way, elongating and growing into a cell with a transparent seta-like end, and finally undergoing repeated divisions to form the plant. M. Braun states that he has observed another process, in which the contents of a single cell undergoes a perpendicular division, so as to form four small zoospores, which escape from the cell in the same way as the larger one, and further says that he has never known these microgonidia to germinate. FRESH-WATER ALG OF THE UNITED STATES. 207 Genus DRAPARNALDIA, Ag. Fila articulata ramosa, e cellulis magnis, maxime hyalinis, fascia chlorophyllosa latiuscula ornatis, semper sterilibus formata, fasciculis penicillato-ramulosissimis, e cellulis minoribus fertilibus com- positis, plus minus dense obsessa. Articuli terminales omnium ramulorum inanes achroi steriles, in pilum byalinum plus minus elongati. Filaments articulated, branched, formed of large cells which are chiefly hyaline, but furnished with a transverse chlorophyllous fascia, more or less densely clothed with penicillately ramulose fasciculi, formed of smaller fertile cells. Terminal articles of all the joints empty, trausparent, sterile, and elongate, in a more or less hyaline hair. BD. slomerata, (Vaucu.) Aa. D. filis ramisque primariis achrois vel subachrois, ad 0.00147” crassis, articulis inferioribus diametro xqualibus vel paulo brevioribus, geniculis manifesto constrictis, fasciis chlorophyl- losis angustis dilute viridibus; ramis primariis subrectangulo-patentibus, swepe oppositis ; ramulorum fasciculis confertis, patentibus, alternantibus vel oppositis, dense ramellosis, sub- ovalibus, obtusis. (R.) Syn.—D. glomerata, (VAucnER) AGARDH. Rapgnnorst, Flora Europ. Algarum, Sect. IIL. p. 38. Hab.—Rhode Island; (S. T. Olney) Thwaites. Filament and primary branches colorless or subcolorless, and reaching 0.00147” in diameter, lower articles about as long or a little shorter than broad, manifestly constricted at the joints, chlorophyl fascia narrow, light green; primary branches subrectangularly patent, often opposite; fasciculi of branches crowded, patent, alternating, or opposite, densely ramellose, suboval, obtuse. Remarks.— According to M. Thwaites the true Dr. glomerata grows in Rhode Island, as he so identified specimens sent to him by Mr. Olney. ‘These specimens were, however, in all probability dried, and if this was so, I confess not to attach- ing much weight to the identification. The Draparnaldia, common near Philadelphia, is at once so like and yet so different from the description of D. glomerata, that I am unable to fully satisfy myself whether it be a variety of the European species or distinct from it. It differs very greatly in the thickness of the stem and _pri- ‘mary branches. I have given above Prof. Rabenhorst’s description of the Euro- pean variety, and now append one of th> plant growing in this neighborhood, Vor. maxima. Dr. filis achrois, ad 0 004” crassis, articnlis plerumque diametro duplo longioribus, in medio sepe valde tumidis; ramis primariis achrois vel subachrois, oppositis vel alternantibus vel ternatis, elongatis, dense ramellgsis, cum ramulis lanceolatis; ramulorum extremorum fasciculis dense ramelosis, ovatis vel late lanceolatis, plerumque confertis; ramulorum articulis inferioribus plerumque diametro (ad ;,!,,”) subequalibus, articulis superioribus diametro duplo aut triplo longioribus, plerumque piliferis. Hab.—Prope Philadelphia ; Wood. Filament transparent, attaining a diameter of 0.004”, its articles mostly twice as long as broad, strongly swollen in the middle ; primary branches colorless or subcolorless, opposite, alternate or ternate, elongate, densely ramellose with the ramuli lanceolate ; fasciculi of extreme branches densely ramellose, ovate, or broadly lanceolate, mostly crowded, inferior articles of the branches mostly about as long as broad (,,',,”), superior articles two to three times as long, mostly piliferous. 908 FRESH-WATER ALG OF THE UNITED STATES. Remarks.—In this form there are almost always numerous little clusters of branch- lets, growing immediately from the main stem or large branches; such clusters are more rigid, more open, more broadly ovate, and less markedly piliferous than the others. D. plumosa, (Vaucuer) AGaArpuH. D. filis ramisque primariis hyalinis, pleramque 5” = 0.00179" crassis; articutis diametro eequalibus vel dimidio brevioribus, rarius paulo longioribus, geniculis vix aut modice con- strictis, fasciis chlorophyllosis angustis lete viridibus ; articulis inferioribus ramulorum dia- metro ( 73,/"—s}3'”) equalibus vel subduplo longioribus, pene torulosis, superioribus eylin- dricis ad 53,/” attenuatis, diametro duplo triplo-quintuplo longioribus, plerumque non pili- feris; ramulorum fasciculis dense ramellosis, elongatis, acute lanceolatis, erecto-subap- pressis. (R.) Syn.—Dr. plumosa, (VAucuER) AGARDH. Raxpennorst, Flora Europ, Algarum, Seet. IIL. p- 382. Hab.—In rivulis et aquis quietis Filament and primary branches hyaline, mostly gy’” = 0.00179” in diameter; articles as long as broad or one-half shorter, rarely a little longer, scarcely or slightly constricted at the joints, chlorophyl fascia bright green, narrow; lower articles of the branches about as long as broad (,3,/"—-s5},3’”) or nearly twice as Jong, somewhat torulose, the upper ones cylindrical, as small as 5},’", two to five times longer than broad, mostly not piliferous ; fascicles of branches densely branched, elongate, acutely lanceolate, actually subappressed. _ Remarks.—1 have found a Draparnaldia frequently, which I believe to represent the European D. plumosa. As I have preserved, however, no specimens or descriptions, I have simply copied the description of Prof. Rabenhorst. D. Billingsii, Woop. D. valde gelatinosa; filis et ramis primariis achrois ad 729,” crassis, sparsissime ramosis, articulis diametro 2-6 plo longioribus, seepe medio valde tumidis ; fasciis chlorophyllis dilute viridibus, sepe nullis aut subnullis; ramulorum fasciculis distantibus, late ovalibus vel late triangularibus, alternantibus vel oppositis vel triplice verticellatis, sparse ramosis, patentissi- mis; ramulis pilis longissimis robustis terminalibus instructis; oosporis globosis, moniliforme conjunctis ; sporodermate crasso. Syn.—D. Billingsii, Woop, Proe. Am. Philos. Soc., 1869, p. 1438. Hab.—In aquis quietis, prope Philadelphia. Frond very gelatinous, filament and primary branches attaining a diameter of 3}9”, very sparsely branched, their articles 2-6 times longer than broad, often very much swollen in the middle; chlorophyl band light green, frequently almost or entirely wanting ; fascicles of branches distant, broadly oval or triangular, alternate, opposite, or in whorls of three, very open; ultimate branchlets terminating in a long, robust, hyaline hair; resting spores globose, with thick walls, arranged in long moniliform sometimes branched filaments. Remarks.—1 found this plant about the middle of March, 1869, floating on the surface of a little pool in the woods near Chelten Hills, a few miles north of Philadelphia. To the naked eye it appears as a gelatinous mass, resembling a Tetraspora, but when closely examined this translucent jelly is seen to be filled with rather distant greenish points, which are the little clusters of branches, The largest specimens I have seen had attained a length of nearly two inches. The filaments are very transparent and have the branches placed at long intervals. FRESH-WATER ALGA OF THE UNITED STATES. 209 The ultimate branch groups are ovate or oval, and are remarkable for their open- ness, the branchlets being few in number and widely separated. Most of the ulti- mate branchlets are prolonged into a remarkably strong long hair. ' The cells of the main filaments are beautifully transparent, and are sometimes cylindrical but more generally are barrel-shaped. Both secondary and primary branches are often arranged singly, sometimes in pairs, not unfrequently in threes. When placed between two plates of glass and examined closely by the unaided eye, this species is readily distinguishable from our other Draparnaldia, by its fas- ciculi of branches being so widely separated as to be not at all confused with one another. I have a single specimen which I believe to be in fruit. The resting spores (fig. 6, pl. 14) are in long branched chains. They are more or less globose, with a very thick outer transparent wall, and an inner green endochrome, which very probably becomes brownish at maturity. Except when they are branched, these series of spores remind one very strongly of the filaments of some nostocs. I dedicate this very beautiful species to Dr. J. S. Billings, U.S. A., to whom I am under the greatest obligations for aid in the prosecution of this research, and whom I have ever found to unite the greatest scientific liberality with a strong en- thusiasm for and able prosecution of the study of these lower vegetable forms. Since describing this species I have received the Microscopical Journal for 1869, containing Dr. Hicks’s paper upon D. cruciata. The original description in the Linnean Transactions had escaped my notice. D. cruciata and D. Billingsii are exceedingly closely related, yet if Dr. Hicks’s description and figures be accurate they are probably distinct. Thus in the last species the ramuli are not placed at right angles to the main filament, nor are they ever in fours, both of which are given as characters of D. cruciata. ‘They are, on the contrary, in D. Billingsti at various angles, and commonly arise singly, but not unfrequently in pairs, and very rarely in threes. It is worthy of remark, on the other hand, that the figures of Dr. H. do not entirely agree with his description, as in no case are there more than two and frequently but a single branch at one place. ‘The cells of the main fila- ment are also more barrel-shaped in our species than one would infer to be the case with D. cruciata. After all, however, I think it very possible that both forms belong to the one species. Fig. 6, pl. 14, represents a small portion of the frond with fertile branches mag- nified 460 diameters, Genus CHAETOPHORA, Scuranx. Fila articulata ramique primarii radiatim dispositi, e cellulis vegetativis elongatis, fascia chloro- phyllosa in morem Draparnaldie et Stigeoclonii ornatis compositi, sursum in ramulos numero- sissimos, brevius articulatos, articulis extremis attenuatis sepe inanibus non aut vix piliferis in- structos, fasciculatos plus minus dense congestos divisi, massa gelatinosa firma, coriacea vel dura involuti, thallum globosum vel subglobosum aut plane expansum varie lobatum et fissum con- stituentes. (R.) Filaments articulated, with the primary branches radiately disposed, composed of elongated vege- tative cells, ornamented with a chlorophyllous fascia like a Draparnaldia or Stigeoclonium, distally 27 September, 1872. 210 FRESH-WATER ALG OF THE UNITED STATES. resolved into very numerous fasciculate, more or less densely congested branches, with shorter joints, their end joints alternate, often empty, either not or scarcely piliferous; surrounded by a firm coriaceous or hard jelly, so as to form a globose, subglobose, or expanded thallus. Remaries.—I have never seen the production of the zoospores in this genus, but they are said to arise one in a cell, and to escape by a sort of lateral splitting of the wall. Cc. elegans, (Roru) AGARDH. Ch. thallo globoso vel subgloboso, pisi vel cerasi magnitudine, dilute vel saturate viridi, nitido, superficie levi vel quasi tuberculata, elastice molli, nonnunquam indurato; fasciculorum ramulis laxis vel confertis, articulis extremis brevi-cuspidatis, seepe piliferis. Syn.—C. elegans, (Rotu) AGARDH. Rapennorst, Flora Europ. Algarum, Sect. III. p. 384. Hab.—United States. Thallus globose or subglobose, of the sizeof a pea or cherry, light green, with the surface smooth or quasituberculate, elastic but soft, sometimes indurated ; branches of the fasciculi lax or crowded ; end articles shortly cuspidate, often piliferous. Remarks.—One of the commonest of our fresh-water alge is a plant belonging to this genus, which I think is probably the C. elegans of Roth. I am, how- ever, unable to discover any characters separating C. pisiformis, C. elegans, and perhaps C. tuberculosa, and hardly know by which of the three names our Ameri- can form should be known, Our plant grows generally in shaded pools, springs, and ditches in great abundance, adhering as little translucent balls to grasses, leaves, twigs, or anything that may be in the water. The size of the frond varies from the young one, not so large as a pin’s head, to the old matured one, which may be nearly an inch in diameter. The color also varies greatly. It is always some shade of a pure green. The surface is mostly smooth, but sometimes it is so puckered up as to be a mass of large flat tubercles. It is these forms that I suppose to represent C. tuberculosa. ‘The thallus is generally elastic, but at the same time soft, so that although readily compressed and pushed out of shape, it is entirely mashed with some difficulty, especially as, owing to its slipperiness, it constantly escapes from the grasp. In regard to the individual filaments, the method of their branching and the proportionate length and breadth of the cells vary very much in different in- dividuals and probably at different ages of the same individual. Fig. 5, pl. 6, represents rather indifferently well a young individual of this species. C. endivizfolia, (Rorn) Ag. Ch. thallo lineari, subplano, semipollicari vel pollicari, nonnunquam valde elongato, Ite vel obscure viridi, dichotomo-subretiewatum-laciniato (nonnunquam habitu Riccie fluitantis) ; filis ramisque primariis plerumque achrois, passim viridi-zonatis, parallelis; ramulorum fasci- culis lateralibus, plus minus densis, divaricato-patentibus; articulis plus minus tumidis, diametro qualibus vel subsqualibus; geniculis constrictis; eytioplasmate granuloso effuso. (R.) Species mihi ignota. Syn.—C. endiviefolia, (RorH) AGArpH. Rasennorst, Flora Europ. Algarum, Sect. III. p- 383. FRESH-WATER ALGA OF THE UNITED STATES. DALI Hab.—South Carolina; (Ravenel) Wood. Rhode Island; (S. T. Olney) Thwaites. Thallus linear, flattish, of half to a whole thumb’s breadth, sometimes greatly elongate, bright or obscure green, dichotomously subreticulately laciniate (sometimes with the habit of Riccia fluitans) ; filament and primary branches mostly colorless, sometimes zoned with green, parallel; lateral fasciculi of branches more or less dense, divaricately patent ; joints more or less tumid, diameter equal or subequal; joints constricted; cytioplasm eflused granulate. Remarks.—I have never seen a living or well-preserved specimen of this species, and have, therefore, here simply copied the description of Prof. Rabenhorst. Prof. Ravenel has sent to me dried alge labelled, and I think correctly, as belonging to this species, but their condition did not allow any scientific study of them, Genus PILINIA, K1z. Fila articulata, erecta, simplicia vel dichotome ramosa, basi affixa, in stratum crustaceum sub- spongiosum, fragile aggregata. Propagatio adhue ignota. Filaments articulate, erect, dichotomously branched, fixed by the base, aggregated into a some- what spongy fragile crustaceous stratum. Method of propagation unknown. P. diluta, Woon, (sp. nov.) P. rupicola, in strato cano-viridi disposita; filis ramisque fascitulatis, apice obtusis; articulis diametro 14 plo—34 plo longioribus. Diam.—Max. 0.0004”. Hab.—In fontibus maximis, prope Bellefonte, Centre County, Pennsylvania; Wood. Growing on stones and rocks, forming a grayish-green stratum ; filaments and branches fasci- culate, with the apices obtuse ; joints 15-3} times longer than broad. Remarks.—Near Bellefonte, Centre County, Pennsylvania, there issues from the limestone rocks the largest spring I have ever seen, giving rise to a ercek-like tor- rent, which supplies the city with water, and passes on scarcely diminished in volume. In this spring grows the curious alge under consideration, forming a somewhat lubricous crustaceous and stony stratum on the stones and rocks in the basin. This stratum is of a grayish-green color, and is quite friable, breaking in the direction of the filaments with the greatest possible readiness. When placed under the microscope it is seen to be composed of filaments whose course is a direct one from the under to the upper surface. They are apparently rigid, pre- serving their courses, and not being intermatted. They are composed of cylindri- val, confervoid cells, and are dichotomously branched, and yet when viewed as a whole the filament and its branches form a sort of fasciculus. The basal cell or cells appear to be globular. When I collected this plant I was forced by cir- cumstances to put the specimens in carbolic-acid water for future study, and, therefore, I have had no opportunity of studying their method of reproduction. I am not altogether satisfied in referring this plant to the Pilinia, and yet all the most important of the characters given by Rabenhorst are preserved by it. It certainly, however, differs very greatly from P. rimosa, Ktz. Genus APHANOCH TE, Braun. Fila distincte articulata, prostrata, repentia, interdum in stratum irregulare plus minusve concreta } ramulis repentibus vel adscendentibus 3 cellulis chlorophyllaceis, apice vel dorso setigeris. Propa- gatio zoogonidiis. DD FRESH-WATER ALG OF THE UNITED STATHS. Threads distinctly articulate, prostrate, creeping, sometimes more or less concreted into an irregular stratum; branches creeping or ascending; chlorophyllous cells with the dorsum or apex setigerous, > “ - = 45 : rac Propagation by zoospores. Remarks.—Sexual reproduction has not as yet been discovered in this genus, According to Dr. Braun (Verjiing., Translation of the Ray Society, p. 184, &e.) two zoospores are generally formed in a cell by a division of its contents parallel to the septa, but occasionally this division not taking place, the cell contents are resolved into a single zoospore. ‘The zoospores themselves are nearly globular, biciliate, and unprovided with any reddish eye-spot. A. repens, Braun. A. filis procumbentibus plerum ue simplicibus ; articulis cylindricis aut tumidis, diametro sub- equalibus ad 1-2 plo longioribus ; setis e cellularum dorso egressis, plerumque singulis sed in- terdum geminis, interdum nullis. Dianii=—ATti Cs soo soo = 0002o 00047. Syn.—A. repens, Braun. RAsBeEnnorst, Flora. Europ Algarum, Sect. III p. 391. Hab.—In Gidogoniis, prope Philadelphia ; Wood. Filaments procumbent, mostly simple; articles cylindrical or tumid, from as long as broad to twice as long; seta arising from the back of the cells, generally single, sometimes geminate, sometimes wanting. Remark:s.—The specimens from which the above description was drawn up, were found growing on the filaments of Cidogonium mirabile, Woop. They were re- markable for the rarity with which they were branched, for in but two or three cases out of a great number, were any branches detected. The articles were fre- quently twice as long as broad. In both these particulars the plant differs from the typical European A. repens, but the descriptions of that form are so short and imperfect that I have preferred retaining the name for the American plant. Fig. 5, pl. 14, represents an ordinarily formed specimen magnified 460 diameters. It had been kept for some time in weak carbolic-acid solution, and although the green of the chlorophyll was perfectly preserved, the stumps only of the sete were visible. How long the perfect sete are I canndt at present say, not having made any notes on the fresh specimens. Genus COLEOCH ATE, Brés. (1844). Fila articulata ramosa aut in pulvinulum conjuncta aut in thallum planum subdisciformem parenchymaticum concreta; articuli oblongi, antice plus minus dilatati, angulo superiori vel dorso sepe in setam basi vaginatam producti. Propagatio fit tum oosporis foecundatione sexuali ortis, tum zoogonidiis. Zoogonidia in quaque cellula fructifera unica, forma subglobosa vel late ovalia, polo antico ciliis vibratoriis binis instructa. (R.) Filaments articulated, branched, either conjoined into a little cumulated mass or parenchematously concreted into a plain subdisciform thallus; articles oblong anteriorly, more or less dilated, often furnished with a long seta on their dorsum or superior angle. Propagation occurring by means of oospores, formed by sexual organs or by zoospores. Zoospores subglobose or broadly oval, formed singly in the fertile cell, furnished at their anterior pole with vibratile cilia. Remarks.—I have seen a large number of specimens of, as I believe, two distinct species of this genus, but never having found any fruiting fronds, have not been FRESH-WATER ALG# OF THE UNITED STATES. 213 able to identify them. One of the forms grows in this immediate locality, and is very probably C. scutata, Bréb. ‘The other was collected in Northern Michigan. It is characterized by its frond never being disciform, although composed of a single plane of cells parenchematously united. Crass RHODOPHYCEA. Alge multicellulares, vegetatione terminalis non limitata preedite plerumque trioice. Thallus e cellularum seriebus vel stratis singulis vel pluribus compo- situs, aut nudus aut e cellularum strato corticatus, forma quam maxime varius; membranaceus (Porphyridium), crustaceus (Hildenbrandtia), filamentosus et verticillatim ramosus (Batrachospermum, Thorea), fascii- formis (Bangia), foliaceus, ete. Cytioplasma plerumque rhodophyllo (Cohn), rarius phycho-chromate coloratum, granula amyloidea vel amylacea et sepe guttulas oleosas includens. Propagationis organa triplicis indolis, saepissime in plantas distinctas disposita. 1. Organa mascula vel antherida e fasciculis cellurarum plerumque mo- niliformibus ramosis, denique in spermatozoidea vel spermatia foecun- dantia (Sporidia I. Ag.) oblonga vel ovalia, achrod, immobilia dissolutis formata. 2. Organa feminea vel cystocarpia Ktz. e soris nonnunquam monilifor- mibus formata, qui e placenta saepissime corticali evolvuntur, nudi vel cuticula mucilaginosa vel involucro inclusi, denique sporas (polysporas) numerosas immobiles mox germinantes emittunt. Foecundatur cysto- carpium statu primordiali ope organi piliformis (¢richogyne Thuret et Bornet) quorum spermatia copulantur. 3. Tetrasporangia e cellula corticali unica valde intumescente formata, divisione utriculi primordialis cruciata quadrilocularia; in quoque loculo (cellulis secundariis, sororiis) spora unica (fetraspora) se format, quae sine foecundatione germinat. (R.) Multicellular alge, mostly triecious, furnished with unlimited not ter- minal vegetation. Thallus composed of cells in rows or in a simple or multiple stratum, either bare or provided with cortical strata of cells, exceedingly various in form; membranaceous (Pophyridium), crustaceous (Hildenbrandtia), filamentous and verticillately branched (Batrachospermum, Thorea), fasciate (Bangia), foliaceous, &e. Cytioplasm mostly rhodophyltlous, rarely phycochromatously colored, including amyloid granules or starch and frequently oil drops. Propagation by means of three immotile organs, generally placed upon distinct plants. 1. Antheridia composed of mostly moniliformly branched fascicles of 214 FRESH-WATER ALG OF THE UNITED STATES. cells, which dissolve into oblong, oval, transparent immotile spermato- zoids (Sporidia Ag.). 2. Cystocarpia Ktz., or Pistillidia, formed of somewhat moniliform sori, which are evolved from a generally cortical placenta, and are naked or surrounded by a mucilaginous cuticle or involucre, and finally emit numerous immotile spores (polyspores), which quickly germinate. The fecundation of the cystocarpia occurs in their primordial state by con- tact of the spermatia with a piliform organ known as trichogonia. 3. Tetrasporangia formed of single, greatly swollen cortical cells, be- coming cruciately quadrilocular by division of the primordial utricle; in each loculus (secundary or sister cells) a single spore (fetraspore) forms, which germinates without fecundation. Famity PORPHYRACEA. Thallus mucoso-membranaceus, foliaceus vel filamentosus, e cellularum seriebus vel strato unico formatus, pleruamque purpurascens, valde lubricus. Vegetatio fit cellularum divisione in duas vel omnes directiones repetita. Propagatio fit tetrasporis. Cystocarpia nondum observata. Thallus mucous-membranous, foliaceous or filamentous, formed of cells in series or in a single stra- tum, mostly purplish, very slippery. Growth taking place by repeated division of the cells in two or all directions. Propagation by means of tetraspores. Cystocarps not yet observed. Remarks.—The only species of this family as yet observed in North America can hardly be said to have a definite thallus. They are rather multitudes of cells heaped together and closely attached to one another into a shapeless expanded mass, Genus PORPHRYDIUM, Nase. (1849). Thallus mucoso-membranaceus, suberustaceus, longe lateque expansus, e cellulis globosis vel . polyedricis compositus. Propagatio adhue ignota. Thallus mucous-membranous, suberustaceous, long and widely expanded, composed of globose or polyhedral cells. Propagation-unknown. P. cruentum, (Ac.) NAza. P. thallo saturate purpuro-sanguineo, lubrico; cellulis anguloso-rotundatis. (R.) Diam.—0.00027”—0.00035". (R.) - Hab.—New York. Syn.—P. cruentum, (AGarp.) Nagcet. Rapennorst, Flora Europ. Algarum, Sect, TIT. p- 397. Thallus deep crimson purple, slippery ; cells angled and rounded. Remarks.—The only specimen I have seen of this species was a little speck, adherent to a bone picked up on Governor’s Island, in New York Harbor. It is very probable that it was a recent arrival, brought over, perchance, by some emi- grant. For it I am indebted to Dr. Billings, U.S. A. The description and FRESH-WATER ALGA OF THE UNITED STATES. Oy measurements given above are copied from Prof. Rabenhorst’s work. My specimen agrees well with it. P. magnificum, Woop. P. cellulis globosis vel subglobosis, sepe nonnihil polygonis et in massam indefinite expansam confluentibus ; cytioplasmate purpureo, granulato; cytiodermate crasso, haud lamelloso. Diam.—Cell cum. tegum. 538 55—agy50-. Tegum. 35459—16h0- Syn.—P. magnificum, Woon, Proc. Am. Philos. Soe., 1869, p. 144. Hab.—In terra humida, Texas; Prof. Ravenel. Cells globose or subglobose, often somewhat polygonal and conjoined into an indefinite mass; endochrome purple, granulate; cell wall thick, not laminate. Remarks.—This species, which was collected in Texas by Prof. Ravenel, growing, I believe, on wet sand, is very distinct from the European plant, differing essenti- ally in size and form. In some instances the cells have a greenish tint, but this is possibly owing to immaturity, as such cells seem smaller than others. The whole mass to the eye has a very rosy purple tint, and although under the microscope it appears much darker and more purple, yet it often retains some of the roseate hue. At the edges of the masses the dark-reddish color often gives way to a very decided greenish tint, presenting an appearance which is very well represented in the drawing of the preceding species, in M. MENGEHINI’S Monographia Nostochinearwm ttalicarum, &e., Memoire della Reale Academia delle Scienze di Torrino. ‘The cells are often closely united by their thick coats into a very coherent mass. With the ordinary cells I have occasionally seen other larger ones, of an orange color, with very thick walls. Are these resting spores ? Fig. , pl. 19, represents single cells of this plant magnified 750 diameters. Fammy CHANTRANSIACEA. Thallus filamentosus. Fila articulata, e cellularum serie unica formata, ramosa, stricta, nuda, raro passim corticata, rami superne fasciculatim ramellosi; articuli cylindrici. Cytioderma, homo- geneum, maxime hyalinum. Cytioplasma homogeneum, plerumque purpurascens. Propagatio fit polysporis immobilibus, ovalibus, in ramellorum apice vel lateraliter formatis, corymboso aggregatis. Antheridia subglobosa, terminalia. Tetraspora raro observate. Thallus filamentous. Threads articulate, formed of a single series of cells, branched, straight, hare, rarely here and there articulate; branches above fasciculately branched ; joints cylindrical Cytioderm homogeneous, mostly hyaline, cytioplasm homogeneous, mostly purplish. Propagation by immovable oval polyspores formed on the ends of the branches or laterally and corymbosely aggregate. Antheridia subglobose terminal. Tetraspores rarely observed. Genus CHANTRANSIA, FRIEs. Familie genus unicum. The only genus of the family. ¢C. expansa, Woop. C. cespitosa, in lapide stratum saturate violaceo-purpureum lubricum, indefinite expansum, formans; filis purpureis, modice ramosis, fere 2 lineas longis et ramis plerumque strictis et rectis, sepe elongatis; ramulis fertilibus brevibus, ascendentivus; articulis diametro 3-8 plo 216 FRESH-WATER ALGA OF THE UNITED STATES; longioribus, extremis obtusis; polysporis in ramellis lateralibus racemosim et confertim cumulatis, ovalibus vel nonnihil obovatis. Diam.—Fil. 3257" =-0004". Spor. transv. 3050 =-00027 long. 5355 = .0004”. Syn.—C. expansa, Woon, Prodomus, Proc. Amer. Philos. Soc., 1869. Hab.—In rivulis, prope Philadelphia. Cxspitose, forming a dark purple, slippery, indefinite stratum on stones; filaments purple, moderately branched, almost 2 lines long, together with the branches strict and straight, often elongate; infertile branches sometimes very few, sometimes very numerous ; fertile branches short, ascending ; joints 3-8 times as long as their diameter, the final articles ob- tusely rounded: polyspores racemose, crowded on the fertile branches, oval or somewhat ovate. Remarks. —This species was found growing in a running stream, forming a felty slimy coating upon large stones, looking so much like a stratum of Oscillatoria, that when I gathered it I thought it probably was a representative of that genus. The stratum, however, when carefully examined, is seen to be made up of an in- definite number of minute, very closely approximate tufts. The color was a dark dull purple. ‘The plant may possibly be the Chantransia violacea, of Kwtr1z1ne, which it resembles in many particulars, but it is nearly twice as long and the fila- ments are considerably thicker. Its habit of growth also seems to be essentially different from that of the European plant, so that I have finally decided to con- sider it a distinct species. ‘The exact locality of its growth is in a thickly-shaded portion of the stream that runs along the North Pennsylvania Railroad, just this side of Chelten Hills. Fig. 2, pl. 19, represents a filament magnified 125 diameters; fig. 2a, a part of a fertile branch magnified 460 diameters, C. macrospora, Woo (sp. nov.). C. cespitosa, subpollicaris, olivaceo-grisea vel saturate violaceo-purpurea; filis ramosis et ramis plerumque strictis et rectis, et elongatis; articulis diametro 3-8 plo longioribus; ramu- lis fertilibus brevissimis ; polysporis singulis vel geminis, sparsis, spe distantibus, globosis, interdum nonnihil ovalibus. Diam.—Fil. plerumque .0008—max. .001. Polysp. .0009. Hab.—South Carolina; (Ravenel). Cespitose, about an inch long, olive-gray to deep-violet purple ; filaments a good deal branched, with the branches mostly straight and elongated ; fertile branches very short; articles 3-8 times longer than broad ; spores single or geminate, few, often distant, globose, or sometimes slightly oval. Remarks.—I am indebted to Prof. Ravenel for specimens of this species pre- served in carbolic-acid water. They are labelled, “ Dull olive green, growing against wooden boards in spring, Nov. 5, 1869. Aiken, South Carolina.” The most of the mass is of the color noted, or at least approaches it, but a portion is almost blackish purple. The species is a very distinct one, characterized by the larger diameter of its articles and spores, by the paucity and shape of the latter, as well as by its variance in coloration. In some old specimens the cell wall is distinctly lamellate. I have only seen fruit on the purple filaments. The FRESH-WATER ALGA OF THE UNITED STATES Diy spores, apparently not mature, have a greenish-brownish tint. I have also received from Prof. Ravenel dried alge, which, apparently, are the same species as those from which this description has been written, but which, not being in fruit, cannot be absolutely identified. They are, as dried, of a bright bluish-green, and attain the length of an inch and a half or more. Fig. 3, pl. 19, represents a part of a branch of this plant magnified 460 diameters. Famty BATRACHOSPERMACEA. Alge dioice. - Thallus filamentosus, articulatus, ramosus, aut violaceus, violaceo-purpureus vel ceruleo-viridis, muco matricali involutus; filis primariis ramisque e cellularum serie unica centrali primaria et seriebus numerosis secundariis parallelis continuis vel interruptis externis com- positis, aut ramulorum fasciculis verticillatis globoso vel subgloboso dense conglobatis equali distantia obsitis, aut ramulis simplicibus vel dichotomis dense ubique vestitis. Vegetatio terminalis. Diecious alge. Thallus filamentous, articulate, branched, violet or violet-purple or bluish-green, covered with mucous; primary filament and branches composed of a single central series of cells, and numerous external, parallel, continuous, or interrupted secondary series; either furnished with globosely or subglobosely densely conglobate, equally distant verticillate fasciculi of branches, or everywhere densely covered with simple or dichotomous branches. Vegetation terminal. Genus BATRACHOSPERMUM, Rots, 1800. Thallus filamentosus, moniliformis, e cellularum serie unica medullari, accessoriis parallelis corti- cata compositis, ramulorum fasciculis subgloboso-conglobatis obsessus. Thallus moniliform, composed of a simple series of medullary cells and cortical accessory parallel series, clothed with subglobosely conglobate fasciculi of branches. Remarks.—The Batrachosperms are amongst the very largest of the fresh-water alge, forming gelatinous branched masses from a few inches to even more than a foot in length. ‘The fronds are very freely and very irregularly branched, and are evidently composed throughout, 7. e., both in regard to the main filaments and the branches, of two portions, a central axis and much more slender short transverse branchlets, which often end in a long hair, and are arranged more or less exclu- sively in groups, so as to form, to the naked eye, at regular intervals, little balls or knots, the whole plant thus presenting a sort of moniliform aspect. Sometimes, however, these glomeruli are placed so closely together, and grow so large that they become confluent, and the branch to which they are attached appears as a uniform thick and very gelatinous cylindrical cord. The axis both of the stem and the branches of a Batrachosperm consist ori- ginally of but a single series of cells. The development of new cells takes place in two ways, the one of which results simply in an increase in the length of the axis, the other in the production of branches. The first of these is the ordinary process of cell multiplication by division, and occurs only in the end cells, so that no new cells are ever formed in the central portions of the axis, which increases in length solely by the addition of new cells at the end, and by longitudinal growth of the old ones. The first step towards the formation of a branch is the produc- tion of a little pouch-like protrusion near the upper end of a cell. This increases 28 September, 1872. 218 FRESH-WATER ALG OF THE UNITED STADES in size and soon being cut off from the parent-cell by a partition, forms a complete cell, the starting point of a new branch. If this cell has been formed alone, with- out companions, it is the beginning of a main branch, and divides after a very brief period transversely, the new cell thus arising in a little while itself divides, and so the process goes on until the axis of a large branch, similar to the parent axis is developed, and which, like the parent axis, increases only by a division of the end ecll and longitudinal growth of the central ones. When a glomerulus is to be formed instead of a single pouch, a number appear around the upper end of a cell, and become cut off as new cells. Each of these is the starting point of a new row of cells, which not only grows, at least up to a certain point, by the division of the end cells, but which also gives rise to a large number of branches in a way precisely similar to that in which it itself was developed, i. e., by the formation of little lateral protrusions, &c. These secondary branches have a life-history similar to that of the branch whose offspring they are. They continually give origin to new branchlets in the way just described, which branchlets themselves produce fresh offshoots, and so it goes on until at last the forest of branchlets making up the dense glomerulus is evolved. It has been just stated that the original axis of the main filament or any branch is composed of a single simple series of large cells; when an old Batrachosperm is placed under the microscope, however, it is at once evident that the axis is in reality formed of such a series lying in the centre and covered over and often hidden by numerous longitudinal series of smaller cells. These latter do not belong to the original axis, but are secondary additions to it, and arise in this way. Whilst a glomerulus is being developed certain of the basal cells of its constituent branches give origin in the usual manner to branchlets, which, instead of growing outward to form a part of the glomerulus, grow upwards or downwards, closely hugging and finally enveloping the original axis, and at last forming a distinct cortical layer to it. Very frequently in well-advanced Batrachosperms there will be seen scattered among the glomerulus large, round, firm, dense balls composed of a great number of small closely-attached cells. These are the reproductive bodies. According to H. Graf zu Solms-Laubach (Botanische Zeitung, 1867, p. 161), they are the result of sexual reproduction, and are developed from antheridia and trichogonia (female organs) in the following manner :— The antheridia are small roundish cells full of a colorless protoplasm, which is remarkable for the very numerous bright granules which it contains. ‘They occur either scattered or in groups, and are placed upon the upper ends of peculiar ovate cells, also filled with a colorless protoplasm. Most frequently there is a single antheridium to the basal cell, sometimes two; the latter number appears never to be exceeded. When matured, the antheridia open and allow their contents to escape in the form of roundish or flattened bodies, which never, as far as known, acquire cilia, and have, therefore, no power of spontaneous motion. ‘These bodies, which are believed to be spermatozoids, are unprovided with anything like an external membrane, and are composed of protoplasm identical with that in the antheridium. Whilst these changes are occurring, certain cells in other localities are being trans formed into female organs, to which our author applies the name of Trichogonia. FRESH-WATER ALGA OF THE UNITED STATES. 219 These are borne upon cells similar to those supporting the antheridia. At first they are not markedly different from the other cells, but soon undergo a very rapid growth. ‘This is not, however, regular, and is not partaken of by a band of tissue about one-third way from the basal end, so that at last a long somewhat flask- shaped cell is produced, with a very marked contraction at the point indicated, separating it into two portions. ‘The wall of this cell is thin but very distinct, and the cavity is filled with a homogeneous or very sparsely granular protoplasm, which is continuous through the narrow neck-like portion. After a time there appear one or more large irregular vacuoles, with actively moving corpuscles in them, and at the same time the neck appears to be stopped with a slimy substance. Careful examination with reagents shows that this is cellulose, and that it does not com- pletely block the passage-way through the isthmus. At this time there appear lying upon the free end of the trichogonia globular or flattened bodies, without external membrane, corresponding in all respects with those already described as being pro- duced in the antheridia. The end of the trichogonium generally enlarges at this period into a sort of roundish knob, and by and by the end wall between this and one of these globules becomes absorbed, so that there is a free communication between the two. Whilst this is going on the globule acquires a thin, delicate coat, and there appears in it a vacuole similar to those preexisting in the tricho- gonium. The first result of this impregnation of the trichogonium is the deposit of new cellulose, and the complete blocking up of the passage-way through the isthmus or narrowed portion. Already before the fecundation, the upper cells of the branches supporting the trichogonia have produced numerous branchlets, which growing upwards more or less completely cover that organ. After impregnation the cells near to the trichogonium become much larger and broader, their vacuoles disappear, and are replaced by a dense granular dark greenish-brown protoplasm. These cells now show a great activity in the production of numerous branches in the usual way, but it is the upper two alone which, with the trichogonium that they support, are concerned in the formation of the fruit glomerulus. These put out all over their surface an immense number of protrusions, which soon in the ordinary way become the parents of as many twigs or branchlets, which growing and branching, precisely as do the vegetative branches, soon become excessively crowded. ‘The base of the trichogonium participates also in this production of branches, and at last a dense ball is formed of pseudoparenchymatous tissue by the forced adhesion of the crowded twigs. The central cells of the glomerulus thus formed are very large and bladder-like. The outer part of the ball is com- posed of innumerable radiating rows of small cells, the end cell of each branch being roundish so as to present a convex external face. At maturity these cells open and allow their contents to escape as round masses, which appear to have no membrane, but begin at once to grow and secrete cellulose. Their after- history has not been made out with absolute certainty, but they are believed to directly develop the new plant. 220 FRESH-WATER ALG& OF THE UNITED STATES, BB. moniliforme, (Rorn.) B. pollicare, bi- tripollicare, raro pedale, muco gelatinoso plus minus firmo involutum, viola- ceum, fuscum, rufo-brunneum, purpureum vel ceruleo-viridiscens, vage ramossissimum ; ramu- lorum articulis omnibus conformibus, oblongo-subclavatis, extremis nonnunquam setigeris; internodiis nudis vel ramulis accessoriis singulis sparsis instructis. Diam.—Tetrasp. globulus 4235 = .006. Syn.—B. moniliforme, Roru. RaBENuoRstT, Flora Europ. Algarum, Sect. III. p. 405. Hab.—In aquis puris, Michigan; Gray. New York; Bailey. Virginia; Jackson, Alabama; Tuomey. South Carolina; (Ravenel) Pennsylvania; New Jersey; Wood. One inch to a foot in length, clothed with a more or less firm gelatinous mucus, violet, fuscous, reddish-brown, purple, or bluish-green, vaguely and -profusely branched; joints of the branches similar, oblong-subclavate, the outer ones sometimes setigerous; internodes naked or furnished with a few scattered accessory branchlets. Remarks.—This specics is very abundant in fresh, cool rivulets, in springs, in limestone waters, in pine-barren streams, and even occasionally in ditches, wherever I have botanized in Pennsylvania and New Jersey. It varies greatly in size, in color, and other particulars. The branchlets, as I have observed them, are most generally not setigerous, but at times they are provided with seta of moderate length. I have found numerous fruiting fronds, but in none of them was the fruit in great abundance, not nearly so much so as in the Rocky Mountain species. B. vagum, (Rorn) Acarpu. B. vage ramossissimum, uni- vel tripollicare, fuseum vel wrugineum ; internodiis inferioribus ramellis numerosis obessis, superioribus nudis vel subnudis; ramulorum articulis extremis setis longissimis instructis. Diam.—Tetrasp. globulus 3435 = .00333. Syn.—B. vagum, (Roru) AGarpe, RasBennorst, Flora Europ. Algarum, Sect. III. p. 406. Hab.—In aquis quietis, Uintah Mountains, Nevada; (S. Watson). Vaguely branched, one to three inches long, brownish or eruginous; internodes—the inferior covered with a dense mass of branchlets—the superior naked, or nearly so; last articles ot the branchlets provided with an extremely long seta. Remarks.—I have received from Mr, Sereno Watson some half a dozen dried alge, which I have referred to B. vagum, with some doubt. They are labelled as having grown in shallow water, ina beaver pond, in Pack’s Caiion, Unitas, Uintah Mountains, Nevada, at an altitude of 7000 feet. All the descriptions of B. vagum which I have seen are singularly imperfect; in none is it stated how large the spore masses grow, and how plentifully the branchlets are provided with seta. As far as the descriptions go, however, my specimens agree with them, and I have, there- fore, refrained from indicating a new species. The plants are remarkable for the profusion and extreme length of the seta, and for the quantity of fruit which they produce. The fruit masses are small but very compact, scarcely more than half the size of those of the preceding species. The verticles of branchlets are often completely joined, and as it were almost swallowed up by the mass of inter- vening scattered branchlets which arise directly from the main axis. In the distal FRESH-WATER ALG# OF THE UNITED STATES. 29] portions of the fronds, however, the glomeruli are more fasciculate and more distinct, for although sometimes so close as to be almost confluent at their spread- ing edges, at their bases they are distinct. This species very probably attains a much larger size than indicated by my specimens, and possibly varies as much in color as B. moniliforme. Genus TUOMEYA, Harvey. “Frond cartilaginous, continuous, solid, at first transversely banded, afterwards annularly con- stricted ; composed of a longitudinal axis, and two strata of peripheric cells. Axis columnar, consisting of several longitudinal cohering filaments, beset with closely placed whorls of moniliform ramelli, whose branches anastomose horizontally and vertically into a cellular peripherie membrane, which is coated externally with moniliform filaments, gradually developed. Fructification probably in the superficial filaments. T. fluviatilis, Harvey. Hab.—On stones, in rivers and streams. River in Alabama; Prof. Tuomey. Near Fred- ericksburg, Virginia; Prof. Bailey. Fronds tufted, an inch or two in height, scarcely as thick as a hog’s bristle, much and irregu- larly branched, bushy; the branches alternate or secund, scattered or crowded, twice or thrice divided, and set with scattered patent ramuli which are slightly constricted at the in- terstices, and taper to an obtuse point. When young the branches and ramuli are perfectly cylindrical, and when examined under a low power of the microscope show a surface com- posed of minute, dotlike cells, placed close together, and marked at short intervals with dark- colored transverse bands. These bands disappear under a higher magnifying power. They are indications of the nodes of the axis of the frond seen through the peripheric stratum. In old, fully developed specimens the branches and ramuli are annularly constricted at short intervals, the nodes becoming swollen, whilst the internodes remain unchanged. When a young branch is bruised between two pieces of glass the axis may be readily extracted. It consists of several parallel longitudinal jointed threads combined together at closely-placed nodes, from which issue horizontal dichotomous filaments, composed of roundish or angular cells. These excurrent filaments spread both horizontally and vertically, and their branches anastomose into a cellular mass or fleshy membrane, which forms the inner peripheric stratum. In young plants a portion of the frond, between the axis and periphery, is hollow, but in older ones the cavity is quite filled up with cells. The external surface of the cellular peri- phery is clothed with a coat of moniliform filaments gradually developed, and forms what is above called the second peripheric stratum. These are found only in fully-grown specimens; they consist of much smaller cells than those of the inner stratum; they are more strongly colored, and I consider them to be connected with fructification. The color is a dark olive. The substance is brittle, rigid when dry, and the plant scarcely adheres to the paper. The generic name is in memory of the late Prof. Tuomey, of Tuscaloosa.” Remarks.—I have no knowledge of this plant, and have simply copied the de- scription of Prof. Harvey; Smithsonian Contributions, 1846. Famtty LEMANEACEA. Alge rivulares vel fluviatiles. Thallus e prembryone confervacea enascens, setaceus, subsimplice vel fasciculatim ramosus, cavus, nodosus, e cellularum stratis internis et corticatis formatus. Noduli plerumque papillarum corona instructi. Polyspore numerose, in seriebus ramosis moniliformibus fasciculatim aggregate, sine fecundatione germinantes. 999 FRESH-WATER ALG OF THE UNITED STATES. “ Alge growing in streams and rivers. Thallus developing from a confervoid prothalloid filament, setaceous, almost simple or fasciculately branched, hollow, nodose, composed of internal and corti- cal strata of cells. Nodules generally provided with a corona or papilla. Polyspores numerous, fasciculately aggregated in branched moniliform series, germinating without fecundation. Genus LEMANEA, Bory. Genus unicum. The only genus. Remarks.—The plants belonging to the genus Lemanea are quite peculiar in aspect and habit. ‘They grow exclusively in fresh water, especially frequenting streams whose current is rapid, and whose waters are chilled by the mountain air, Their frail, tubular, scarcely-branched fronds offer but little resistance to the water, whilst their lower end is swollen into a sort of discoid root, which adheres firmly to the stones. ‘The frond is mostly blackish or brownish, and is formed of two distinct portions or layers, of which the outer or cortical is composed of small closely cohering, colored cells; the inner of much larger cells, which have thick colorless walls, and are placed so as to leave more or less numerous interspaces, In the immature frond there is also a longitudinal central column, besides some slender many-jointed filaments, passing obliquely through the cavity, but as final development takes place these seem to disappear. ‘The mature frond is alternately contracted and expanded throughout its length. In the narrow portions the inner tissue often blocks up the tube entirely, whilst the dilated parts are loosely filled with the spores, which are produced within the frond. The spores themselves are oval, thickish-walled cells, whose endochrome changes from greenish to a very decided yellow during the process of maturing. They are joined together to form rows or series, which are not simple, but are very much branched, so that from a central basal row arises a complex bush-like mass (pl. 20, fig. 4). These spore- clusters are always distinct, a number of them existing in each sporangial node of the frond. Dr. B. Wartmann described, nearly twenty years ago, very fully the way in which the spores germinate and develop into the frond. The first step, according to this authority, consists in the elongation of the spore and the projection of one end, which is soon cut off by the formation of a transverse partition, and consti- tutes a new cell. This multiplying in no strikingly peculiar way soon develops into a branched confervoid filament. A large number of these filaments are gene- rally produced in one place at one time and form a very apparent greenish layer. Finally certain cells in branches of these filaments swell up and become very much broader than their fellows, undergoing, at the same time, division so rapidly that they become very short. By and by they divide also in the direction of their breadth, so that instead of a simple series of cells there arises a compound mass. This is the beginning of the new frond. At first it is dependent upon the parent filament, but soon acquires a root-like process at the base and develops rapidly into the complex cartilaginous plant. FRESH-WATER ALG& OF THE UNITED STATES 293 L. torulosa, (Roru) Aa. L. subsimplex, plerumque arcuata, cartilaginea et nonnihil rigida, 1-2 pollices longa; nodulis approximatis, papillis applanatis, plerumque 4-6 enormiter verticellatis, vel nonnihil sparsis, interdum nonnihil confluentibus ; sporis ovalibus. Diam.—Sporis. transv. max. yx85q"—yshhh0"- Syn.—L. torulosa (RorH) Ac. Raxpennorst, Flora: Europ. Algarum, Sect. IIT. p. 411. Hab.—In flumine, Kentucky; (Short) Harvey. Pennsylvania; Virginia; New York; New Jersey ; Wood. Subsimple, mostly arcuate, cartilaginous and somewhat rigid, 1-2 inches long; nodules ap- proximate, with their papules applanate, mostly 4-6, irregularly verticillate or somewhat scattered, sometimes slightJy confluent; spores oval. Remarks.—This plant attains a length of about two inches, and grows in masses attached to rocks, often forming a sort of turfy covering to them, in rapidly run- ning water. In mass it has a grayish or blackish appearance. The filament has a grayish groundwork, with a dark band at the position of the nodes, which are enlarged and inclose the spores. The transverse outline of the filament is a very irregular circle. I have found this species very abundant in the rapid water of the Schuylkill, just above Flat Rock Tunnel, on the Reading Railroad, eight or nine miles above Philadelphia. Prof. E. D. Cope has sent me specimens collected by himself in swift streams in Western Virginia, and Mr. Austin has obtained it in similar situations in Northern New Jersey. Mr. Austin has also sent me specimens collected in Canada West. L. fluviatilis, Ac. L. simplex vel parce ramosa, quatuor uncias longa (interdum spithamea?), recta vel subrecta ; nodulis subremotis, papillis verticillatis magnis obssesis; sporis globosis vel subellipticis. Diam.—Spor. zs'950—r12000 - Syn.—L. fluviatilis, AGARDH. RABENHORST, Flora Europ. Algarum, Sect. III. p. 411. Hab.—In rivulis, Alabama; T. M. Peters. Simple or sparsely branched, 4 inches long (sometimes growing of a span length ?), straight or nearly so; nodules rather distant, papille verticillate, large, prominent. Remarks.—The only specimens I have seen of this species were sent me by Prof. Ravenel. This plafit is larger and heavier than L. torulosa, from which it is also readily distinguished by its very large prominent papille. These are in slightly irregular whorls of three or more. The spores vary in shape from that of a globe to that of a somewhat four-sided ellipse; in the latter case being some- times nearly twice as long as broad. Prof. Rabenhorst speaks of the plant attain- ing the length of a span. I have never seen it over four inches, L. catenata, Krz. L. ad uncias 5 longa, regulariter constricta, simplex, compressa, arcuata, in massa obscure violacea; papillis nullis; sporis enormiter ovalibus vel subglobosis. Diam —Spor. transv. max y_xe25q/ = .001”". Syn.—L. catenata, Kiirzine. Rasenuorst, Flora Europ. Algarem, See. IIT. p. 412. 224 FRESH-WATER ALG OF THE UNITED STATES. Hab.—In rivulis frigidis montanis Diamond Range, Rocky Mountains ; (Sereno Watson). About 5 inches long, regularly constricted, simple, compressed, arcuate, in mass obscure violet; papules wanting ; spores irregularly oval or subglobose. Remaris.—I have received specimens of the plant from which the above diag- nosis was drawn, from Mr. Sereno Watson, labelled ‘“ Mountain stream, Diamond Range, altitude 6500 feet.” In the dried state they are closely interwoven into a dark purple, rigid thin mass. When soaked out they preserve the same color in mass, but each individual stem has a general light yellowish, neutral ground tint, with dark-purplish or greenish-black bands at regular intervals. At the position of these bands the filament is nearly round and contracted, whilst between them it is compressed and enlarged. The spores are placed, not at the swelling, but at the constrictions, corresponding to the dark rings in position. They are quite irregular in shape, and of a faint yellow tint. ‘The filaments between the little knots of spores appear to be hollow. Their walls are everywhere very thin when compared with ZL. torulosa, hence they are more flaccid. ‘The species agrees in every respect with Prof. Rabenhorst’s diagnosis of L. catenata, K1z., a native of cold mountain streams of Germany and Switzerland. I regret, however, very greatly that I have had no opportunity of comparison with European specimens, or 0 a fuller description. SUPPLEMENT. Tue following species, of which the author has not seen specimens, were inad- vertently omitted from their proper places in the monograph. ‘They are all con- tained in the Nereis Boreali-Americana of Prof. Harvey. The following descrip- tions and remarks are simply copied from the work mentioned. Tetraspora lacunosa, Cuavy. Frond at first tubular, then flat, or irregularly lobed, membranaceo-gelatinous, pale-green, every- where pierced with roundish holes of various sizes. Chauv. Alg. Norm. Breb. Alg. Fal. p. W1,¢.1. Kitz. Sp. Alg. p. 227. 2. Godeyi, De Breb. Kitz. Tab. Phyc. t. 30, f. 3. T. perforata, Bailey, M.S. Hab.—In fresh-water streams. Abundant near Westpoint, Prof. Bailey ; Providence, Rhode Island, Mr. Olney. (v. 8s. in Herb. T.C.D.) Frond at first funnel-shaped, afterwards splitting open, and then flat, expanding upwards and irregularly lobed, everywhere pierced with roundish holes of various sizes, large and small intermixed. These holes increase in size and numbers with age, and thus at last the frond becomes an open network. The substance is very gelatinous, but rather firmer than in some other species of the genus. The color is a pale green; and the hyaline gelatinous membrane is filled with roundish granules set in fours. Kiitzing’s figure of 7. Godeyi answers well to our plant. I have not seen any authentic specimens of TJ. /acwnosa, which is referred by Kiitzing to his 7. lubrica, var, 3., but the description given of it applies to the American plant. When care- fully dried, it forms a very pretty object for the herbarium. (Chlorospermee, p. 61.) (Harvey, p. 61.) Nostoc (Hormosiphon) arcticum, Berk. Fronds foliaceous, variously plaited, green or brownish; filaments at length (their gelatinous envelope being dissolved) free. Berk. in Proc. Lin. Soc. fide An. Nat. Hist. 2d Ser. vol. 10, p. 302. Hab.—On the naked soil, in boggy ground. Assistance Bay, lat. 75°40’ N. Dr. Sutherland. (v.s.) “ Fronds foliaceous, variously plicate, sometimes contracted into a little ball. Gelatinous envelope at length effused ; connecting cells at first solitary, then three together; threads, which are nearly twice as thick as in N. commune, breaking up at the connecting cells, so as to form new threads, each terminated with a single large cell, the central cell becoming free.” Berk. l. e. “Tt grows,” says Dr. Sutherland, “upon the soft and almost boggy slopes around Assistance Bay; and when these slopes become frozen at the close of the season, 29 October, 1872. ( 225 ) 226 SUPPLEMENT. the plant lying upon the surface in irregularly plicated masses becomes loosened, and if it is not at once covered with snow, which is not always the case, the wind carries it about in all directions. Sometimes it is blown out to sea, where one can pick it up on the surface of the ice, over a depth of probably one hundred fathoms, It has been found at a distance of two miles from the land, where the wind had carried it. At this distance from the land it was infested with Podure, and ] accounted for this fact by presuming that the insects of the previous year had de- posited their ova in the plant upon the land, where also the same species could be seen in myriads upon the little purling rivulets, at the side of which the Nostoc was very abundant.” At p. 205 of his Journal, Dr. Sutherland further mentions having tried it as an article of food, and found it preferable to the Tripe de Roche of the arctic hunters. Its nutritive qualities are probably equal to those of the jelly derived from other Alge. (Chlorospermee, p. 113.) Nostoc flagelliforme, Berk. and Curt. Terrestrial; frond cartilaginous, linear, very narrow, compressed and often channelled, much branched, irregularly dichotomous; branches solid, densely filled with moniliform curved threads. Berk. and Curt. No. 3809. Hab.—On naked aluminous soil, at San Pedro, Texas, Mr. Charles Wright. (v-.s.) Fronds several inches in length, half a line in diameter, lying prostrate on the surface of the soil, much branched in an irregularly dichotomous manner; branches exactly linear, com- pressed, often channelled on one or both sides, thinned in the middle and incrassated to the edge. Substance firm and elastic, cartilaginous, solid, densely filled with moniliform, curved or curled, interlaced threads, which are set longitudinally in the frond, and lie nearly parallel to each other. Color dark olive. A very curious and most distinctly marked species, differing from others of this genus, much in the same manner that Chetophora endiviefolia does from the ordi- nary globose forms of Chetophora. (Chlorospermew, p. 115.) Nostoc microscopicum, Carn. Fronds densely aggregated, very minute, giobose or oblong, immersed in a blackish crust; fila- ments few. Carm. in Hook. Brit. Fl. 2, p. 399. Harv. Man. Ed. 1, p. 184. N. muscorum, Hass. Br. Fr. Wat. Alg. p. 292, t. 14, fig. 4. Hab.—‘ Stones in a small stream, Baffin’s Bay,” Dr. Sutherland, fide Prof. Dickie. I have not seen American specimens. In Britain this species grows among mosses on exposed calcareous rocks, but not in water. The above specific charac- ter is taken from the British plant. The fronds are rarely more than the tenth of an inch in diameter, and contain two or three beaded filaments lying na copious transparent jelly. (Chlorospermew, p. 115.) Genus Hyprurus, AG. Frond fixed at base, cylindrical or compressed, elongated, branched, gelatinous. Structure: Seriated, but separate, cellules, filled with bright-green endochrome, inclosed in gelatinous parallel tubes, ranged longitudinally in the frond, and surrounded by a common gelatinous envelope. SUPPLEMENT. 997 Of this genus several species have been described by authors, all having a close resemblance to each other, and all very variable in ramification. Indeed it is almost impossible to fix characters by which they can be permanently kept apart ; and instead of adding another specific name to the already too numerous list, I prefer to consider the American specimens received as constituting a luxuriant variety of the best known of the established species. All previously recorded species or varieties of these plants are natives of rapid rivers and streams in various parts of Europe. (Chlorospermee, p. 118.) Hydrurus penicillatus, var. occidentalis, Harv. Frond very long (1-2 feet or more), much branched; branches very irregular, scattered or crowded, wormlike, tapering to a fine point, naked or clothed with feathery villous ramuli; cells ellipsoidal or pear-shaped, twice as long as their diameter. Hab.—On the rocky bottom of rivers and streams, in a strong current. Santa Fe, New Mexico, Mr. Fendler, February to April, 1847. (vs. in Herb. T.C.D.) Fronds attached at base, one or two feet long, from one to four lines in diameter, very much and irregularly branched; branches scattered or crowded, simple or divided, a foot or more in length, attenuated to a fine point, sometimes smooth and naked, but generally densely clothed with slender, villous ramenta, spreading to all sides. The gelatinous tubes or sheaths in which the cells are seriated are very obvious, and lie close together in longitudinal, paral- lel strata. The cells are of large size, bright-green color, and variable shape; some are twice as long as others. This I had at first supposed to be a new species, but now regard it as a very gigantic state of H. penicillatus, Ag., which under various forms and of various sizes is common in alpine streams in Europe. I fear characters derived from the shape and size of the cellules are not more to be depended upon than are those taken from the ramification. (Chlorospermec, p. 118.) Draparnaldia opposita, Aa. Frond vaguely much branched ; joints of the main filament as long as broad, or shorter; pencils of ramuli mostly opposite, densely set, lanceolate-acuminate in outline, plumose, bi-tripinnate, the apices much attenuated. Ag. Syst. p.59. Kiitz. Sp. Alg. 851. Lyngb. Hyd. Dan. tab. 65, fig. A. Batrachospermum Americanum, Schweinitz. Hab.—In clear streams. New York, Professor Bailey. New Jersey, Mr. Jackson. (y.s.) Frond 2-3 inches long, gelatinous, capillary, irregularly much branched; the branches patent, lateral, more or less divided, and set with lesser ramuli. Main filaments with short articula- tions, as long as their breadth, or shorter, transversely banded. At every two or three nodes and sometimes at every node a pair of opposite penicillato-multifid ramuli are thrown off. These are bright green, ovato-lanceolate in outline, much acuminated and twice or thrice pin- nate, their pinnules somewhat constricted at the nodes, and tapering at the apex into long, needle-like, hyaline points. Their cells are commonly nucleated and filled with endochrome. Whether this be permanently distinguishable from D. g/omerata is doubtful. _It has externally the aspect of that species, but its microscopic characters are nearer those of D. pluwmosa. ! ne > 2 ta ao, (Girt eee Ae - { 40) Ad» ae. a Sree ae ii\fig ip im 9 fei Ge ge i i (has f 1 T age \ } : te face Ts ees 1} en tts sire ait y ie " — - sat ee ; i it a — ion , GEOGRAPHICAL LIST OF SPECIES. Crass PHYCOCHROMOPHYCEZ. Orper CYSTIPHOR. Family CHROOCOCCACE@. Chroococcus refractus, Wood. Hab. near Philadelphia. Hab. near Philadelphia. Hab. Benton Springs, Owen Co., California. multicoloratus, Wood thermophilus, Wood. Gloeocapsa sparsa, Wood. Cezlospherium dubium, Griin. Hab. near Philadelphia. Merismopedia Hab. near Philadelphia. ITab. Spring Mills, Montgomery Co., Pa. nova, Wood. convoluta, Bréb. Orper NEMATOGENE.®. Family OSCILLARIACE. Oscillaria chlorina, Atz. Hab. near Philadelphia. Hab. New York. Hab. Northern U. States. Hab. Schuylkill River, near Philadelphia. Hab. near Philadelphia. Hab. near Camden, New Jersey. Hab. West Point, New York. Hab. near Philadelphia. Hab. New York; Philadelphia. Rhode Island; New York; Virginia. Hal. Warm Springs of Washita. corium, Ag. decorticans, Gener. Frohlichii, Avz. imperator, Wood. limosa, Ag. muscorum, Ag. neglecta, Wood. nigra, Vauch. tenuis, Ag. Hab. tenuissima, Ag. Chthonoblastus repens, Kiz. Hab. New York; Massachusetts ; Rhode Island. Lyngbya bicolor, Wood. Hab. Schuylkill River, near Phila- delphia. Hab. Whale Fish Islands, Davis Straits, British Aierica. muralis, Ag. Family NOSTOCHACE. Sub-Family Nostrocex. Nostoc Austinii, Wood. alpinum, Kz. ITab. New Jersey. Hab. Alleghany Mountains; Clover Mts., Nevada; Baffin’s Bay, British America. calcicola, Ag. calidarium, Wood. ceruleum, Lyn. Cesatii, Bals. comminutum, Atz. commune, Vauch. depressum, Wood. punctatum, Wood. pruniforme, Agh. verrucosum, Vauch sphericum, Vauch. Hab. Catoosa Springs, Georgia. Hab. Benton Springs, Owen Co., California. Hab. New Jersey. Hab. Kansas. Hab. near Philadelphia. Hab. New Jersey ; Rio Bravo. Hab. New Jersey. Hlab. New Jersey. Hab. New Jersey. 5 Hab. Maine. Hab, Centre Co., Pennsylvania. Sub-Family SpeRMostrex. Anabena gelatinosa, Wood. gigantea, Wood. flos-aque, Kitz. Cylindrospermum comatum, Wood. flexuosum, Rab. macrospermum, Ate. minutum, Wood. Dolichospermum polyspermum, Ktz. subrigidum, Wood. Hab. near Philadelphia. Hab. near Philadelphia. Hab. Round Pond, West Point, New York. Hab, Niagara, Canada. Hab. near Philadelphia. Hab. South Carolina. Hab. near Philadelphia. Hab. near Philadelphia. Hab. New Jersey. Family RIVULARIACE/. Nostochopsis lobatus, Wood. Gloiotrichia angulosa, Roth. Hab. Schuylkill River, near Phila- delphia. Hab. Hudson River, near West Point. ( 229 ) 230 Glolotrichia incrustata, Wood. Hab. Schuylkill River near | Philadelphia. Rivularia cartilaginea, Wood. Hab. Northern Michigan. Zonotrichia minutula, Wood. Hab. Clear Pond, Adirondack Mountains. Hab, Cave of the Winds, Niagara. Hab. Cave of the Winds, Niagara. mollis, Wood. parcezonata, Wood. Dasyactis mollis, Wood. Hab. Cass River, Northern Michigan. Mastigonema Hab. Philadelphia. Hab. Alleghany Mountains, Centre Co., Pennsylvania. elongatum, Wood. fertile, Wood. halos, Wood. sejunctum, Wood. Hab. Stonington, Connecticut. Hab. Northern Michigan. Cass River, Mastigothrix fibrosa, Wood. Hab. near Philadelphia. Family SCYTONEMACEZ. Scytonema Austinii, Wood. Hab. Little Falls, New Jersey. Hab. South Carolina. Hab. Niagara River, Niagara. Hab. South Carolina. Hab. Cumberland Co., New Jersey. Hab. Cumberland Co., New calotrichoides, Ktz. cataracta, Wood. cortex, Wood. dubium, Wood. immersum, Wood. Jersey. Myochrous, Ag. Hab. West of Crow’s Neck, West Point. Negelii, tz. Hab. near Bellefonte, Centre Co., Pennsylvania. Hab. South Carolina. Hab. Aiken, South Carolina. Hab. South Carolina. Ravenellii, Wood. simplice, Wood. thermale. Tolypothrix distorta, Mul. Hab. near Philadelphia; West Point, N. Y.; Rhode Island; Madison, Wisconsin. Family SIROSIPHONACEA. Sirosiphon Hab. South Carolina. Hab. South Carolina. Hab. Salem, Massachusetts; New Jersey. Hab. Mount Tahawus, Adirondack acervatus, Wood. argillaceus, Wood. compactus, Ag. Crameri, Br. GEOGRAPHICAL LIST OF SPECIES: Sirosiphon guttula, Wood. Hab. South Carolina, Hab. South Carolina. Hab. New Jersey. lignicola, Wood. neglectus, Wood. pellucidulus, Wood. Hab. near Hibernia, Florida. Hab. Northern New Jersey. Hab. South Carolina. pulvinatus. scytenematoides, Wood. Stigonema Ravenellii, Berkeley. Hab. Lookout Mountains, Georgia. Ciass CHLOROPHYLLACE. OrpER COCCOPHYCEH Family PALMELLACE. Pleurococcus Hab. Boiling Springs, near Belle- fonte, Centre Co., Pennsylvania. pulvereus, Wood. seriatus, Wood. Hab. New Jersey. Palmella dura, Wood. hyalina, Lyn. Hab. near Philadelphia. Hab. From Rhode Island to Wisconsin. Jesenii, Wood. Hab. near Philadelphia. Pagerogalla stellio, Wood. Hab. Bear Meadows, Alleghany Mountains, Centre Co., Penn- sylvania. Tetraspora bullosa, Roth. gelatinosa, Roth. Hab. Salem, North Carolina. Hab. Salem, North Carolina; Newburgh, New York. lubrica, Roth. ITab. Northern Atlantic States. Dictyospherium pulchellum, Wood. Hab. near Philadelphia. Rhaphidium faleatum. Hab. near Philadelphia. polymorphum, Fr. Hab. near Philadelphia. Family PROTOCOCCACE. Polyedrium enorme, Ral/s. Hab. Florida. Scenedesmus acutus, Meyen. Hab. Rhode Island; near Phila- delphia. Hab. Georgia; Rhode Island. Hab. near Philadelphia. Hab. Rhode Island; Penn- sylvania. obtusus, Meyen. polymorphus, Wood. quadricauda, Turp. Mountains. rotundatus, Wood. Hab, near Vhiladelphia. GEOGRAPHICAL LIST OF SPECIES. Hydrodictyon utriculatum, Roth. Hab. West Point and Wee- hawken, New York; Mexi- can Boundary; Pennsyl- vania ; New Jersey. Pediastrum Boryanum, Tur. Hab. Rhode Island; Pennsyl- vania; Georgia; Florida. Hab. South Carolina; Georgia; Rhode Island. Hab. South Carolina ; Rhode Island. Hab. Rhode Island; South Carolina; Georgia; Florida. Hab. Rhode Island, Hab. Rhode Island. constrictum, Hassall. duodenarius. Ehrenbergii, Corda. pertusum, Atz. Selenza, Atz. Family VOLVOCINE. Chlamydococcus nivalis. Tab. Greenland ; Rocky Mountains. Volvox globator, Linn. Hab. United States. Orpver ZYGOPHYCEZ. Family DESMIDIACEZA. Palmoglcea clepsydra, Wood. Hab, near Philadelphia. Penium Hab. South Carolina. Hab. South Carolina. Hab. Pennsylvania; New York; Georgia. Brébissonii, Men. closterioides, Ralfs. Digitus, Ekrb. Hab. near Grahamsville, South Carolina. Hab. Florida. Hab. Rhode Island. margaritaceum, Ehrb. Hab, Rhode Island. TTab. Rhode Island; South Caro- lina; Georgia. interruptum, Bréb. Jenneri, Ralfs. lamellosum, Bréb. minutum, Cleve. Closterium acerosum, Schr. Hab, South Carolina; Georgia ; Florida. Hab. West Point, New York; Providence, Rhode Island. Hab. Rhode Island ; New Hamp- shire; Pennsylvania. Hab. Northumberland Co., Pennsylvania. Hab. New York. Hab. Georgia; Florida; Pennsyl- vania; Rhode Island. Hab. Philadelphia. Hab. Rhode Island. Hab. Saco Lake, New Hampshire ; South Carolina, Amblyonema, Ehrb. angustatum, Atz. areolatum, Wood. Cucumis, Ehrb. Diane, Virb. Ehrenbergii, Men. Jennerii, Ralfs. juncidum, Ralfs. 231 Closterium Leibleinii, A¢z. Hab. Georgia; South Carolina ; Pennsylvania. lineatum, Fhrb. Lunula, Miiller. Hab. Pennsylvania. Hab. South Carolina; Florida; Georgia; Pennsylvania. maximum, var. Hab. Pennsylvania. Hab. Georgia; Rhode Island. Hab. near Philadelphia. Hab. near Philadelphia. Hab. Stonington, Connecticut ; Providence, Rhode Island ; Pennsylvania; Georgia; Florida. Hab. Centre Co., Pennsylvania. Hab, South Carolina. moniliferum, Bory. parvulum, Neg. rostratum, Ehrb. setaceum, Lhrb. striolatum, Hhrb. Venus, Kitz. Tetmemorus Hab, Atlantic States. Hab. Centre Co., Pennsylvania. #lab. Rhode Island; Pennsyl- vania; South Carolina. Hab. near Philadelphia. Brébissonii, Men. giganteus, Wood. granulatus, Bred. levis, Ktz. Pleurotenium Baculum, Bréb. breve, Wood. Hab. Georgia. Hab. District of Columbia. Hab. South Carolina; Georgia. Hab. Rhode Island. Hab. Rhode Island; New Jer- sey; Pennsylvania; South Carolina ; Georgia ; Florida. Hab, Florida. Hab. United States. Hab. South Carolina; Georgia ; Florida ; Pennsylvania. clavatum, A?z. constrictum, Bailey. crenulatum, Ehrb. gracile, Rab. hirsutum, Bailey. nodosum, Bailey. Trabecula, Ehrb. Hab. Pennsylvania; New Jer- sey ; South Carolina; Georgia: Florida. Hab. Florida. Hab. Rhode Island. undulatum, Bailey. verrucosum, Bailey. Triploceras Hab. Rhode Island; New Jersey ; New Hampshire ; Florida; Georgia. Hab. with the last. gracile, Bailey. verticillatum, Bailey. Spirotzenia bryophila, Bréb. condensata, Bréb. Hab. near Philadelphia. Hab. Pennsylvania; Rhode Island; Florida. Bar busina Hab. Florida; Georgia; South Carolina; Rhode Island. e Brébissonii, Atz. Didymoprium Grevillii, Atz. Hab, Pennsylvania; South Caro- lina ; Georgia, Sphezrozosma excavatum, Lalfs. pulchrem, Bailey. serratum, Bailey. Hyalotheca disilliens, Smith. mucosa, Mert. Desmidium aptogonium, bréb. GEOGRAPHICAL LIST OF SPECIES. Hab. Rhode Island; South Carolina ; Georgia ;, Florida. Hab. New York; New Jersey. Hab. South Carolina; Georgia ; Florida. Hab. Rhode Island; Pennsyl- yania; South Carolina; Florida. Hab. Rhode Island. Hab. South Carolina; Georgia. quadrangulatum, A?z. Hab. South Carolina. Swartzii, Ag. Aptogonium Hab. Atlantic States. Baileyi, Ralfs. Hab. Rhode Island ; New Jersey. Cosmarium amenum, Bréb. bioculatum, Lréb. Botrytis, Bory. Brébissonii, Men. Broomei, Thw. exlatum, Ral/fs. commissurale, Bréb. connatum, Bréb. crenatum, Ra/fs. cucumis, Corda. depressum, Bailey. Hab. Florida; Rhode Island. Hab. Rhode Island. Hab. Pennsylvania. Hab. White Mountains, New Hampshire. Hab. Pennsylvania; Georgia. Hab. near Albany, New York : South Carolina. Hab. White Mountains, New Hampshire. Hab. Florida. Hab. Rhode Island. ITab. New Hampshire ; Pennsyl- vania; South Carolina; Georgia; Florida. Hab. Florida. margaritiferum, Turp. Hab. Pennsylvania ; South Meneghenii, Bréb. ornatum, Ralfs. ovale, Ralfs. pyramidatum, Bréb Quimbyii, Wood. sublobatum, Bréb. suborbiculare, Wood. tetropthalmum, Kiz. Thwatesii, Ralfs. undulatum, Corda. Buastrum affine, Ralfs. ampullaceum, Ra/fs. Carolina; Florida; Mexico. Hab, Pennsylvania. Hab. Rhode Island. Hab. Penusylvania. Hab. Peunsylvania ; Georgia; Florida. Hab. near Philadelphia. Hab. Rhode Island ; Georgia ; Florida. Hab. Lake Saco, New Hamp- shire. Hab. New Jersey. Hab. Florida. ITab. Rhode Island; South Carolina? Hab. South Carolina; Georgia. Hab. South Carolina; Florida. Euastrum. binale, Turp. Hab. Florida; Pennsylvania; Rhode Island. circulare, Hassal. Hab. Rhode Island. crassum, Bréb. Hab. United States. Didelta, Turp. Hab. South Carolina; Georgia; Pennsylvania; Rhode Island. elegans, Bréb. Hab. United States. gemmatum, bréb. Hab. Rhode Island. insigne, Ralfs, Hab. Florida; Rhode Island. multilobatum, Wood. Hab. Saco Lake, New Hamp- shire. oblongum, Greville. Hab. Rhode Island. ornatum, Wood. Hab. Saco Lake, New Hampshire. Ralfsii, Rabenh. Hab. South Carclina ; New Hamp- shire; Rhode Island. verrucosum, Ehr. Hab. Rhode Island; South Carolina ; Georgia; Florida. Micrasterias Americana, Virb. Hab. Florida; South Carolina. arcuata, Bailey. Hab. Florida. Baileyi, Ral/s. Hab. New York; Rhode Island; South Carolina; Florida. denticulata, Bréb. Hab. Pennsylvania; Florida. disputata, Wood. Hab. Atlantic States. expansa, Bailey. Hab. Florida. fimbriata, Ralfs. Hab. South Carolina; Florida. foliacea, Bailey. Hab.Worden’s Pond, Rhode Island. furcata, Ag. Hab. Atlantic States.” granulata, Wood. Hab. South Carolina. Jenneri, Ral/s. Hab. near Philadelphia. oscitans, Ralfs. Hab. Florida; Rhode Island. papillifera, Bréb. Hab. Flovida ; Rhode Island. pinnatifida, A/z. Hab. quadrata, Bailey. Hab. Florida. radiosa, Ag. Hiab. Florida. ringens, Bailey. Hab, Florida. Torreyi, Bailey. Hab. near Princeton, New Jersey. truncata, Corda. Hab. Atlantic States. Staurastrum alternans, Bréb. Hab. Georgia; Florida; Rhode Island. arachne, Ralfs. Hab. Saco Lake, New Hampshire. aristiferum, Ralfs. Hab. Georgia ; Rhode Island. Cerberus, Bailey. Hab, Florida. crenatum, Bailey. Hab. Rhode Island. eyrtocerum, var., Bréb. Hab. Florida. dejectum, Bréb. Hab. New York; South Caro- lina. dilatatum, Eh7b. Hab. Southern Atlantic States. eustephanum, Ro/fs. Hab. West Point, New York. furcigerum, Bréb. Hab. South Carolina; Florida; Rhode Island. Staurastrum gracile, Rul/s. hirsutum, Lhrb. Hystrix, Ral/s. Lewisii, Wood. longispinum, Arch. margaritaceum, Ehrb. munitum, Wood. muticum, Bréb. orbiculare, Lhrb. paradoxum, Mey. polymorphum. polytrichum, Per. punctulatum, Gréb. Ravenellii, Wood. senarium, Lhrb. tricorne, Men. Xanthidium aculeatum, Ehrb. Arctiscon, Ehrb. armatum, Bréb. bisenarium, Lhrb. cristatum, Bréb. coronatum, Ehrb. fasciculatum, Ehrb. Arthrodesmus convergens, Ehrb. Incus, Bréb. octocornis, Ehrb. quadridens, Wood. GEOGRAPHICAL List OF Hab. South Carolina; Georgia; Florida; New York; Rhode Island. Hab. Florida; Rhode Island. Hab. Rhode Island. Hab. Saco Lake, New Hampshire. Hab. Florida. Hab. Saco Lake, New Hampshire. Hab. Saco Lake, New Hampshire. Hab. South Carolina; Rhode Island. Hab. Rhode Island; Pennsylvania. Hab. Saco Lake, New Hampshire. Hab. Florida. Hab. near Philadelphia. Hub. Pennsylvania. Hab. South Carolina. Hab. America. Hb. Georgia; Florida; Rhode Island. & Hab. near Savannah, Georgia. Hab. North America. Hab. South Carolina; Florida; New Hampshire. Hab, America. Hab. Southern Atlantic States. Hab, Amarica. Hab. South Carolina; Georgia; Florida; Rhode Island. Hab. South Carolina; Georgia; Florida; Rhode Island. Hab. Georgia; Florida; South Carolina; Rhode Island. Fas. Florida; Rhode Island. Hab. Lake Saco, New Hampshire. Family ZYGNEMACEZ. Spirogyra crassa, Kiz. decimina, J/i/. diluta, Wood. dubia, Aiz. elongata, Berk. insignis, /7as. longata, Vauch. Hab. near Philadelphia. Hab. near Philadelphia. Hab. near Philadelphia. Hab. near Philadelphia. Hab. near Philadelphia. Hab. near Philadelphia. Hab. Rhode Island; near Phila- delphia. 30 October, 1872. Spirogyra majuscula, Atz. vitida, Dill. protecta, Wood. parvispora, Wood. pulchella, Wood. quinina, Ag. rivularis, Hassall. setiformis, Roth. Weberi, Atz. Zygnema insigne, Hassal. cruciatum, Vauch. Sirogonium retroversum, Wood. Mesocarpus sealaris, Hassall. parvulus, /assall. Pleurocarpus mirabilis, Braun. SPECIES. 933 Hab. near Philadelphia. Hab. near Philadelphia. Hab. near Philadelphia. Hab. Hibernia, Florida. Hab. near Philadelphia. Hab. near Philadelphia. Hab. Florida. Hab. near Philadelphia. Hab. near Philadelphia. Hab. Rhode Island; near Phila- delphia. Hub. Virginia; Florida; Northern States. Hab. near Philadelphia. Hab. near Philadelphia. Hab, Rhode Island. Hab. New York; Rhode Island; Michigan; Wisconsin. Orper SIPHOPHYCE#. Family HYDROGASTRE A. Hydrogastrum granulatum, Linn. TTab. Delaware. Family VAUCHERIACE 2. Vaucheria aversa, Hassall. geminata, Vauch. polymorpha, Wood. sessilis, Vauch. velutina, Ag. Hab. near Philadelphia. Hab. near Philadelphia. Hab. Texas. Hab. New York; Maine; Vir- ginia; North Carolina. Hab. New York; Maine; Virginia ; North Carolina. Orper NEMATOPHYCE. Family ULVACEZ. ~ Protoderma viride, Atz. Ulva merismopedioides, Wood. Enteromorpha intestinalis, Linn. Schizomeris Leibleinii, Ktz. Hab. Philadelphia. Hab. Diamond Range, Rocky Mountains. Hab. Hudson River; Narra- gansett Bay. Hab. near Philadelphia. 234 Family CONFERVACEZ. Conferva. Hab. United States. Cladophora Hab. near Philadelphia. Hab. Pennsylvania; New York ; Rhode Island. Hab, Lakes Ontario, Erie, Huron, and Michigan. brachystelecha, Rab. fracta, Dill. glomerata, Linn. Family G2DOGONIACE As. Androgynia Hab. Florida. near Philadelphia. echinata, Wood. Huntii, Wood. mirabilis, Wood. Hab. Hab. Hab. near Philadelphia. multispora, Wood. near Philadelphia. Pringsheimia ineequalis, Wood. Hab. near Philadelphia. Bulbochete Canbyii, Wood. dumosa, Wood. Hab. Gibernia, Florida. Hab. near Philadelphia. ignota, Wood. Hab. near Philadelphia. Family CHROOLEPIDE A. Chroolepus Hub. New York; New Jersey : Texas. aureum, A?z. Bulbotrichia albida, Wood. Hab. Northern New Jersey. Family CHAZTOPHORACEZ. Stigeoclonium. Hab. Hastern United States. 2 Draparnaldia Billingsii, Wood. glomerata, Vauch. Hab. near Philadelphia. Hab. Rhode Island. maxima, var., Wood. Hab. near Philadelphia. GEOGRAPHICAL LIST OF SPECIES. Drapernaldia plumosa, Vauch. Hab. near Philadelp) Chetophora elegans, oth. Hab. Eastern United St endiviefolia, Roth. Pilinia diluta, Wood. Hab. Centre County, Pennsylva nia. Coleochete. Hab. Eastern United States. Aphanochete repens, Braun. Hab, near Philadel Cuass RHODOPHYCEZ:. Family PORPHYRACEA. Porphrydium cruentum, Ag. - Hab. New) magnificum, Wood. Hab. Te Family CHANTRANSIACEA. Chantransia expansa, Wood. Hab. near Philad macrospora, Wood. Hab, South Cai Family BATRACHOSPERMACE@, Batrachospermum 5 Hab. Eastern United St Hab. Uiutah Mountains, Nev: moniliforme, Poth. vagum, Roth. Tuomeya fluviatilis, Harv. Hab. Aiabama; : ; Family LEMANEACEZ. Lemanea catenata, Kitz. Hab. Diamond Range, | fluviatilis, Ag. torulosa, Roth. BIBLIOGRAPHY. Agardh (Karl Adolf). Dispositio Algarum Sueciz. Lunda, 1810-12. Anmiarkningar om sligtet Lemania, samt beskrifning om tvenne nya arter deraf. Kongl. Svenska Ve- tenskaps Akademiens Handlingar, xxxv., 1814, p. 33. Beskrifning af en ny art Conferva. Ibid., 1814, pp. 195-200. Synopsis Algarum Scandinavie, adjecta dispositione universali Algarum. Lunde, 1817. De metamorphosi Algarum. Isis von Oken., 1820. Regensburg Flora, vi., 1823. Species Algarum rite cognite cum synonymis, differ- entiis specificis et descriptionibus succinctis. Gri- phiz, 1823-28. Systema Algarum. Lunde, 1824. Aufzihlung einiger in den Ostreichischen Lindern gefundenen neuen Gattungen und Arten von Algen, nebst ihrer Diagnostik und beigefiigten Bemerkungen. Regensburg Flora, x., 1827. Einige Bemerkungen iiber Hrn. Dr. Meyen’s kritische Beitriige zum Studium der Siisswasser Algen. Regensburg Flora, 1829. Icones Algarum Europearum. Leipzig, 1828-35. Agardh (Jakob Georg). Observationer pa Sporidiernes rérelse hos de gréna Algerne. Stockh. Acad. Handl., 1836. Beobachtungen iiber die Bewegung der Sporidien in den griinen Algen. (Translation.) Regensburg Flora, 1840. Bidrag till en noggrannare kinnedom af propagations— organerne hos Algerne. Stockh. Acad. Handl., 1836. Observations sur la propagation des Algues. (Trans- lation. ) Annal. de Scien. Natur., vol. vi., 1836. Anadema, ett nytt sligte bland Algerne (Anadema, genus novum confervearum familie). Stockh. Acad. Handl., 1846. Algologiska Bidrag. Ofversigt af Konigl. Vetenskaps Akademiens Férhandlingar, Sjefte Argangen, 1549. Nya Algformer. Ibid., Arg. xi., 1854. Species, genera etordines Algarum. Lunde, 1848-63. Allman (George James). On a new Genus of Alge belonging to the family Nos- tochinee. Annals and Magazine of Nat. History, 1843, vol. xi. p. 161. On an undescribed Alga allied to Coleochete scutata. British Association Report, 1847. On an apparently undescribed genus of Fresh-Water Alge. Ibid., 1847. On two undescribed Algz. Acad., 1847. On microscopic Algz as a cause of the phenomenon of the coloration of large masses of water. Phytolo- gist, vol iv., 1852. Proceedings Royal Irish Alquen (F. d’). Notes on the Structure of Oscillatori#, with a descrip- tion of a new species, possessing a most remarkable locomotive power not Cilia. Journal Microscopical Soc., iv., 1856. Amici (Giovanni Battista). Descrizione di un’ Oscillaria. Firenze, 1833. Andrejewsky (Erastes). Ueber die Vegetation in den Biadern von Abano. Graefe und Walther’s Journal fiir Chirurgie und Augen Krankheiten, 1831. Annalen der Chemie und Pharmacie, 1832. Note sur les végétaux qui croissent autour et dans les eaux thermales d’Abano. Annals Scien. Nat., Bot.) 1835. (Translation.) Edinburgh New Vhilos. Journal, xix., 1835. Archer (William). Description of two new species of Staurastrum. lin Nat. History Soc. Proceedings, 1856-9. History Review (Dublin), vi., 1859. Society, 1860. Notice on some cases of abnormal growth in the Des- midiacee. Ibid., 1256-9. Ibid., 1859. _—Ibid., 1860. Observations on the genera Cylindrocystis, Mesotznia, and Spirotenia. Microscop. Journal, N. 8., vol. viii. On the conjugation of Spiroteenia condensata. Ibid., N.5., vol. viii. Catalogue of Desmidiacee (Dublin) Nat. Hist. Rev., vol. iv. Microscopical Journal, vol. vi. Supplementary Catalogue of Desmidiacee (Dublin), with descriptions and figures of a new genus and four new species. Natural Histor. Rev., vol. v., 1858. Notice of the occurrence near Dublin of a Unicellular Alga, believed to be allied to that alluded to by Hofmeister (Ann. Nat. Hist. 3ser. vol. i., Jan. 1858). Nat. History Review, v., 1858. Dublin Zool. Bot. Assoc. Proceedings, i., 1859. On a new species and genus of the Desmidiacez, with some remarks on the genera Micrasterias and Euas- trum. Nat. Hist. Rev., vol. vi. On the occurrence of Zoospores in the family Desmidi- acer. Dub. Nat. Hist. Soc. Proce. iii. Journ. Micros. Society, 1860. Nat. Hist. Review, vol. vii. On Asteridium occurring in Penium digitata. Micro- scopical Journ., vol, vil. Record of the occurrence new to Ireland, with a note of a peculiar condition of the Volvocinaceous Alge Stephanosphera pluvialis, &c. Ibid., vol. vi. On two new species of Saprolegniex. Ibid., vol. vii. Description of a new species of Cosmarium and of a new species of Xanthidium. Dubl. Nat. Hist. Soc. Proe. iii., 1859-62. Nat. Hist. Review, vol. vii., 1860. ( 235 ) Dub- Nat. Journ. Micros. 936 BIBLIOGRA PIL ¥. ao Archer (William).— Continued. Description of a new species of Micrasterias, with re- marks on the distinction between M. rotata (Ralfs) and M. denticulata (Bréb.). Dublin Nat. Hist. Soc. Proce., vol. iii. Micros. Soc. Journ. ii., 1862. Description of a new species of Cosmarium (Corda), of Staurastrum (Meyen) ; of two new species of Closte- rium (Nitzsch), and of Spirotenia (Bréb.) Dub- lin Nat. Hist. Soc. Proc. iii. | Microscop. Soc. Journ., vol. ii., 1862. On a new species of Ankistrodesmus (Corda), with re- marks as regards Closterium Griflithii (Berk), and C. subtile (Bréb.). On Closterium aciculare (West). Microscopical Soe. Journal, ii., 1862. An endeavor to identify Palmoglwea macrococea (Ktz.), with description of the plant believed to be meant and of a new species, both referrible to the genus Arthrodesmus (Ehrb.). Dubl. Nat. History Soc. Proc. iv., 1862. Hedwigia, 1864. Description of a new species of Cosmarium (Corda), and of Arthrodesmus (Ehrb.). Duo. Nat. Hist. Soc. Proc., iv., 1862-3. Observation on Micrasterias Mahabuleshwarensis (IHob- son), and on Docidium pristide (Hobson). Dubl. Nat. Hist. Soc. Prue., iv., 1862-63. Microscopical Journ., N. §., vol. vi. Description of a new species of Docidium from Hong Kong. Dubl. Nat. Hist. Soc. Proc., iv. Micro- scop. Journ., N.S., vol. vi. Description of two new species of Cosmarium, of Pe- nium, and of Arthrodesmus. Microscop. Journ. N.5., vol. iv., Hedwigia, 1864. Description of two new species of Staurastrum, Mi- croscopic Journ., N.8,, vol. vi., 1839. On a new genus and species of Desmidiacee. Ibid., N.5., vol. vi. On some eases of abnormal growth in Desmidiaceer. Ibid., N.5., voi. vi. Ardissone (Francesco). Enumerazione delle Alghe della marca di Ancona. Fano, 1866. Areschoug (Johann Erhart). De Hydrodictyo utriculato. Lunde, 1839. Ueber die Vermehrungsart des Wassernetzes (Hydro- aes utriculatum Roth). Linnea, vol. xvi., Om Achlya prolifera, vixanda pa lefvande fisk. Stock- holm, Ofversigt Kon.Vetensk. Akad. Férhand. 1844. Algarum minus rite cognitarum pugillus primus, tab. 12. Linnea, 1842. Die Arten der Gattung Ulva. Nova Acta R. Soc. Scient. Upsal. Ser. III., vol. i. Copulationem hos Zygnemacex. Stockholm, Ofver- sigt K. Ventensk. Akad. Forhand. 1853. Regens- burg Flora, 1855. Arrondeau (Theodore). Essai sur les Conferves des Environs de Toulouse. Act. Soe. Linnwa de Bordeaux, 3 Series, tom. iv. Observations sur l’Organization du Zygnema orbiculare, Hassal. Sessions des Congrés Scientifique de France, xix., 1852. Bailey (John Whitman). A sketch of the Infusoria of the family Baccilaria, with some account of the most interesting species which have been found in a recent tour in the U. States. American Journal of Sciences, First Series, vol. xli. p- 184. Vol. xlii. p. 88. Vol. xliii. p. 321. Bailey (John Whitman).— Continued. On some new species of American Desmidiacem from the Catskill Mountains. American Geologist and ‘Nat. Assoc. Reports, 1843. Amer. Journ., N. §, vol. i., 1846. y Notes on the Algw# of the United States. Thid., New Series, vol. ili. pp. 80, p. 329 Voi. vi. p. 37, Microscopical Observations made in South Carolina, Georgia, and Florida. Smithsonian Contributions, vol. ii. Notes on new species and localities of Microscopical Organisms. Ibid., 1853, Balfour (John Hutton). Observations on the Spores of Cryptogams, and on the reproductive processes in some Alge@ and Fungi, Proceedings of the Royal Suciety of Ediuburgh, Ses- sion of 1867-8. Bary (Heinrich Anton de). Bericht iiber die Fortschritte der Algenkunde in den Jahren 55, 56, 57. Botanishe Zeitung, 1858. Sur la génération sexuelle des Algues. Annal. Scien. Natur. (Bot.), tome v., 1856. : Beitrag zur Kenutniss der Acilya prolifera. Bota- nische Zeitung, 1852. Uber die Copulation der Desmidiacien. Berlin, Bo- tanische Zeitung, 1557. Regensb. Flora, 1857, p. 693. Ueber das vortweltliche kleinste Siisswasser Leben in dEgypten. Monatsberichtder Kon. Akad., 1853. Beitrag zur Kenntniss der niedersten Algenformen nebst Versuchen ihre Entstehung betreffend, Sitzungs- bericht d_ k. Akad. zu Wien, Bd. xi., 1853. Beitrag zur Naturgeschichte der Schwiirmsporen. Ver- handlungen des naturhist. Vereins der preus- sisch. Rheinlande und Westphalens. Bonn, 1855. Untersuchungen iiber die Familie der Conjugaten, Leipzig, 1858. Uber die Algengattungen (idogonium und Bulbochete. Verhandlungen der Senckenberg naturwissen. Ges- selschaft zu Frankfurt, 1854. Beitrag zur Kenntniss der Nostocaceen, insbesondere der Rivularieen. Regensburg Flora, 1863. Linige neue Saprolegnieen. Pringsheim’s Jahrbuch fiir wissens. Botanik, bd. 2, 1860, p. 169. Bary (Heinrich Anton de) et M. Woronin. Beitrag zur Kenntniss der Chytridieen. Freiburg, 1863, also Berichte der naturforschenden Gesell- schaft in Freiburg, Band. iii., Heft. ii. Bauer (Francis). Microscopical Observations on Red Snow. The Jour- nal of Science and Arts (Royal Inst. Gr. Br.), vol. vii., 1819. Some experiments on the Fungi which constitute the coloring matter of the Red Snow discovered in Baf- fin’s Bay. Philosophical Transactions, London, 1820. 5 Microscopical Observations on the Suspension of Mus- cular Movements in Vibrio tritica. Philosophical Transactions, 1823. Ann. Sci., vol. ii., 1824. Begglato (Francesco Secondo). Delle terme Euganee memoria, Padova, 1838. Berkeley (Miles Joseph). Gleanings of the British Algw, being an Appendix to the Supplement to English Botany. London, 1833. BEB OXG eR PAyP HOY, Berkeley (Miles Joseph).— Continued. Deseription of Closterium Griffithii, n. sp. Aunals and Magazine of Natural History, 1854. Ona new vegetable Parasite on fishes. Gardener’s Chro- nicle, 1864. Annalde Scien. Natur., vol. ii., v. Ser. Note on the recent discoveries in relation to the Micro- gonidia of Fresh-water Algz. Journal of Linnean Society i., 1857. Biasoletto (Bartolommeo). Dialeune Alghe microscopiche Soggio. _‘ Trieste, 1832. Ueber die Metamorphose der Algen. Regensburg Flora, | 1838. Ueber microscopische in chemischen Solutionen entste- hende Algen. Isis von Oken, 1833. Bory de St. Vincent (Jean Baptiste M. A G.). Memoire sur les genres Confervaet Byssus. Bordeaux, 1797. Essai monographie sur les Oscillaries. Paris, 1827. Braun (Alexander). 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Ibid., Beitriige zur Physiologie der Phyeochromaceen und Florideen. Schultzes Archive fiir Mikrose. Ana- tomie, 1867. Ueber Chlamydomonas marina, Cuhn. Hedwigia, 1865. Comelli (Francesco). Intornuo alle alghe microscopiche del Dr. B. Biasoletto Relazione. Udine, 1833. Corda (August Karl Joseph). Observations microscopiques sur les Animalenles des eaux thermales de Carlsbad. Almanach de Carlsbad, 10e année, 1840. Die Algen. Sturm’s Deutschland Flora, 2e Abthei- lung 1829-32. Corti (Bonaventura). Osservazioni microseopische sulla Tremella. 1774. nega; BP Bua OG ReASe HOYe: Crouan (P. L., and H. M., brothers). Observations microscopiques sur le genre Mesogloia Agardh. Aunal. Scien. Natur. (Bot.), vol. iii., 1835. Observations microscopiques sur la dissémination et la germination des Kctocarpes et sur le: Conferva scutulata. Ibid. vol. xii. Note sur le genre Spirulina. Mewoires de la Société lmper. des Scien. Nat. de Cherbourg, ii., 1854. Currey (Frederick). On Stephanosphera pluvialis. vol. vi., 1858, p. 131. On some British Fresh-Water Alge. Microscopic Journ., Ibid., p. 207. Dareste (Camille). Mémoire sur la Coloration de la mer de Chine. Annal. Scien. Natur., vol. i., 1854. Mémoire sur les Animaleules et autres corps organisés qui donnent a la mer une couleur rouge. Annal. Scienc. Nat. (Zool.), 1855. Davaine (C.). Conterve parasite sur le Cyprinus Carpio. Mémoires de la Soc. de Biologie, 1851. Recherches physiologiques et pathologiques sur les Bactéries. Comptes Rendus, Paris, 1868, tome Ixvi. Recherches sur les infusoires du sang dans la maladie connue sous le nom de sang de rate. Tbid., lvii., 1863. Journ. de Pharmacie, xliv., 1863. De Candolle (Augustin Pyramus P.). Flore Frangaise. Tom. vi. Rapport sur les Conferves. Journal de Physique, de Chimie et de Histoire Naturelle, liv., 1802. Notice sur la Matiére qui a coloré le lac de Morat en Rouge au Printemps de 1825. Mémoires de la Société Nat. de Genéve, tome iii., 1826. | Edinburgh Journ. Scien., 1827. Derbes (Alphonse) et Solier. Sur les organs reproducteurs des Algues. Scien. Natur. (Bot.), xiv., 1850-1551. Mémoire sur quelques points de la physiologie des algues. Paris, 1856. Antoine Joseph Jean Aunal. Dillenius (John Jacob). Historia Muscorum. A General History of Water Mosses and Corals, containing all the known species, their names, places of growth, and seasons. 85 tab. London, 1763. Dillwyn (Lewis Weston). Synopsis of the British Conferve. 1810. London, 1800- Dippel (Leopold). Beitriige zur Lésung der Frage “Kommt der Zellmem- bran blos ein Wachsthum von Aussen nach Innen zu, oder besitzt dieselbe zugleich ein solches von Innen nach Ausen. Botanische Zeitung, ix., 1851. : Zur Primordialschlauchfrage. Regensburg Flora, xxxix., 1856. Ueber die Fortpflanzung der Vaucheria sessilis. Regensb. Flora, xxxix., 1856. Zelltheilung der Ulothrix zonata. 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Ueber eine contagiédse Confervenbildung auf dem Was- sersalamander. Archiv fiir Anatomie, Physiologie, und wissensch. Medecin. (J. Miller), 1839. Fernere Erlauterung der contagiésen Confervenbildung auf l'roschen und Wassersalamandern. —Ibid., 1840. Hartig (Theodore). Ueber die Funktionen des Zellkerns. Zeitung, xii., 1854. Ueber das Verhalten des Zellkerns bei der Zellen- theilung. Ibid., 1854. Beitrige zur Entwickelungsgeschichte der Pflanzenzelle. Botanische Zeitung, 1855. Botanische Ueber die Conjugation der Spirogyren. Regensburg, Flora. Harvey (William Henry). Manual of the British Alga. London, 1844. BIBLIOGRAPHY. Harvey (William Henry).— Continued. Index Generum’ Algarum. A Systematic Catalogue of all the Genera, Marine and Fresh-water. London 1860. Phycologica Britannica, or a History of the British Sea- weeds. London, 1846-1851. Nereis Borealis Americane. Part III. Chlorospermez. Smithsonian Contributions. , Hassal (Arthur Hill). Observations on the genera Zygnema, Tyndaridex, and Mougeotia, with descriptions of new species. Annals and Mag. Natur. History, &., 1842. Observations on a new group (Vesiculasperme), genus (Vesiculifera), and subgenus (Vesiculifera com- posita) of Fresh-water Conferve, with descriptions of species mostly New. Ibid., 1842. Observations on the growth, reproduction, and species of the branched Fresh-water Conferve, mostly new, with observations on some of the genera. —_Ibid., 1843. Observations on some points in the anatomy and phy- siology of the Fresh-water Alge. Ibid., 1848, Observations on the genus Mougeotia, or two new genera of Fresh-water Alge, and on Tyndaridea, with de- scriptions of species. Ibid., 1843. Observations on the growth and reproduction of Entero- morpha intestinalis. Proceedings of the Linnean Society of London, 1849. Essay on the distribution, vitality, structure, modes of growth and reproduction, and uses of the Fresh- water Conferve. Ibid., 1849. 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Die Algenflora des mittleren Theiles von Franken. Diagnosen von ein und fiinfzig entdeckten neuen Ar- ten und drei neuen Gattungen. Mit 13 Tafeln. Niirnberg, 1867. Retschy. Ueber Tremella nostoc. Regensburg Flora, 1°47. Richter (P.). Description of Pleurotenium nobile n. sp. 1865 Hedwigia, 245 Ripart. Recherches sur organization du genre Isomeria Ktz. Ann. Sci. Nat., 5 ser. (Bot.), tom. vii., 1867. Notice sur les Algues recoltes, &c., dans les Pyrénés. Bulletin de la Soc, Botanique de France, vol. xv. Avril, 1868. Observations sur le Mougeotia genuflexa, Ag. et sur la formation de ses spores. Annal, Scien. Natur. (Bot.), vol. ix., 1868. Robin (Charles). Note sur la presence de la Sarcine dans l’ail. Mé- moires de la Soc. de Biologie, 1854. Rose (August). Ueber Bulbochete setigera. Tledwigia, 1852. His most important papers omitted. Rossman (Julius). Ueber Urin-Sarcina. Regensburg Flora, 1857. Bemerkung iiber Sarcina Welckeri. Virchow Archiv., xiv., 1858. 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[A conferva described on p. 9-10.] Williamson (W. C.). On structure of Volvox globator. Trausactions, vol. i. Microscopical Wood (Horatio C.). Notes on some Alge# from a Californian Hot-Spring. American Journ. Science, vol. xly., N. Series. Prodromus of a Study of North American Fresh-Water Alge. Transactious of the American Philosophi- cal Society, June, 1869. Wyman (Jeffries). Observations and experiments upon living organisms in heated Water. American Journ. Science, N.S., vol. xliv. Zanardinlii (Giovanni). Synopsis Algarum in Mari Adriatico, cui accedunt Mono- graphia Siphonearum. Memoire della Realc Ac- cademia delle Scienze de orino. Series 2, tome iv., 1852, Ueber Cladophora Heufleri. Verhandlungen des zoolog.-botan. Vereins in Wien, Bd. ii., 1853. ANABAENA, 37 flos-aque, Atz., 38 gelatinosa, Wood, 38 gigantea, Wood, 38 Androgynia, 198 echinata, Wood, 198 Huntii, Wood, 197 mirabilis, Wood, 196 multispora, Wood, 196 INDEX. Chroolepus, 204 aureum, Ktz., 204 Chthonoblastus, 21 repens, Atz., 21 Cladonia, 88 Cladophora, 187 brachystelecha, Rabenhorst, 188 fracta, Dillwyn, 188 glomerata (Linn.), 187 Ankistrodesmus falcatus (Corda), 85) Closterium, 109 Aphanochete, 211 repens, Braun, 212 Aptogonium, 126 Aptogonium, Baileyi, Ral/s, 127 Arthrodesmus, 157 convergens, KhArb., 159 incus (Bréb.), 158 octocornis, Hhrb., 158 quadridens, Wood, 158 Bameusina, 125 Brébissonii, 125 Batrachospermacee, 217 Batrachospermum, 219 moniliforme, Roth, 220 vagum, Roth, 220 Bichatia, 12 Botrydium, 175 argillaceum, 176 Bulbochetezx, 199 Bulbochete, 201 Canbyii, Wood, 202 dumosa, Wood, 202 ignota, Wood, 201 Bulbotrichia, 205 albida, Wood, 205 Czx.Losprzriom, 13 dubium, Grun., 13 Chetophora, 209 elegans (Roth), 210 endiviefolia (Roth), 210 pisiformis (Roth), 210 tuberculosa (/oth), 210 Chetophoraceer, 205 Chantransia, 215 expansa, Wood, 215 macrospora, Wood, 216 violacea, Avz., 216 Chantransiacex, 215 Chlamydococcus, 99 nivalis, 99 Chlorococcum, 87, 88 Chlorophyllacez, 77 Chroococeacezx, 10 Chroococeus, 11 multicoloratus, Woud, 11 refractus, Wood, 11 thermophilus, Wood, 12 Chroolepidex, 2/3 32 October, 1872 acerosum, (Schr.), 111 areolatum, Wood, 111 amblyonema, Ehrb., 116 angustatum, A?vz., 110 Cucumis, Fhrb., 113 Diane, Ehrb., 114 Ehrenbergii, Mengh., 113 Jenneri, Ra/fs, 115 juncidum, Ralfs, 110 Leibleinii, Atz., 114 lineatum, Fhrb , 112 Lunula (Miller), 111 maximum (var.), 111 moniliferum (Bory), 113 parvulum, Negel., 115 rostratum, Lhrb., 115 setaceum, Hhrb., 116 striolatum, Ehrb., 109 trabecula, Bailey, 120 Venus, Kéz., 114 Coecophyecee, 78 Coleochete, 212 scutata, Bréb., 213 Collema bulbosum, 25 Conferva, 186 muralis, Dillw., 22 Confervacez, 186 Conjugata longata, Vaucher, 166 Conjugation, 161 Connecting cells, Thwaites, 23 Cosmarium, 127 amenum, Bréb., 130 bioculatum, Bréb., 131 Botrytis, Bory, 128 Brébissonii, Menegh, 128 Broomei, Thw., 133 celatum, Ralfs, 133 commissurale, Bréb., 132 connatum, Bréb., 134 crenatum, Ralfs, 131 Cucumis, Corda, 130 depressum, Bailey, 130 margaritiferum (Zurp.), 127 Meneghenii, bréb., 131 ornatum, Ralfs, 132 ovale, Ralfs, 128 pyramidatum, Bréb., 130 Quimbyii, Wood, 134 sublobatum, Bréb., 132 suborbiculare, Wood, 129 tetropthalmum, Atz., 129 Cosmarium. Thwatesii, Ra/fs, 134 undulatum, Corda, 132 verrucosum, Bailey, 121 Cylindrospermum, 39 comatum, Wood, 41 flexuosum, Pabenh., 40 macrospermum, A?fz., 40 minutum, Wood, 39 Cystiphore, 10 Cystococcus, 88 Dasyactis, 50 mollis, Wood, 50 Desmidium, 126 aptogonium, Br¢b., 126 eustephanum, Ehrb., 155 quadrangulatum, Avtz ,126 senarium, Khrb., 155 Swartzii, Agardh., 126 Dictyospherium, 84 pulchellum, Wood, 84 Didymocladon cerberus, Bailey, 154 furcigerum, Bréb., 154 Didymoprium, 125 Brébissonii, A?¢z., 126 Borreri, Ral/s, 125 Grevillii, Atz., 125 Docidium clavatum, A?tz., 120 constrictum, Bailey, 121 Ehrenbergii, Ra//s, 118 hirsutum, Bailey, 121 minutum, Ral/s, 107 nodosum, Bailey, 120 nodulosum, Bréb., 120 pristide, Hobson, 122 undulatum, 120 verrucosum, Bailey, 121 verticillatum, Ra/fs, 121 Dolichospermum, 41 polyspermum, 42 snbrigiduin, Wood, 43 Draparnaldia, 207 Billingsii, Wood, 208 eruciata, Hicks, 209 glomerata ( Vauch.), 207 maxima, Wood, 207 opposita, Marv., 227 plumosa ( Vauch.), 208 ENTEROMORPHA intestinalis (Linneus), 183. Euastrum, 135 affine, Ralfs, 138 ampullaceum, Ralfs, 138 ansatum, Hhrb., 139 binale ( Turpin), 140 circulare, Hassal, 139 crassum (Bréb.), 137 ( 249 ) 250 Euastrum. didelta (Turpin), 138 elegans (Bréb.), 140 gemmatum (Bréb ), 136 insigne, Ralfs, 138 Jenneri, Archer, 139 margaritiferum (7urp.), 127 multilobatum, Wood, 135 oblongum (Greville), 136 ornatum, Wood, 137 Ralfsii, Rabenh., 13). verrucosum, Lhrb., 136 Evennia, $8 GroorapuicalL List, 229 Globulina, Turpin, 12 Gloeoeapsa, 12 sparsa, Wood, 13 Gloeoprium mucosum, Hassal, 124 Gloiotricha, 45 angulosa (Roth.), 47 incrustata, Wood, 45 Hererocysts, 23 Hyalotheca, 124 disilliens (Smith), 124 mucosa (Mert.), 124 Hydrodictyon, 92 utriculatum, Roth, 95 Hydrogastree, 175 Hydrogastrum, 175 granulatum (Linn.), 175 Hydrurus, 226 penicillatus, Ag., 22 occidentalis, M/arv., 227 Istamta, 123 Isthmosira, 123 LemaneEA, 223 catenata, Atz., 223 fluviatilis, Ag., 223 tornlosa (Roth), 223 Lemaneacee, 221 Lyngbya, 22 bicolor, Wood, 22 copulata, Harvey, 22 muralis, Ag., 22 MasticonemA, 51 elongatum, Wood, 53 fertile, Wood, 51 halos, Wood, 52 sejunctum, Wood, 53 Mastigothrix, 55 fibrosa, Wood, 55 Merismopediu, 14 convoluta, Bréb., 15 glauca, 15 Mediterranea, Nay., 15 nova, Wood, 14 Mesocarpus, 173 parvulus, Hassal, 174 sealaris, Hassal, 173 Mesotenium, 105 Micrasterias, 141 Americana (Ehrb.), 143 arcuata, Bailey, 141 Baileyi, Ral/s, 143 denticulata (Bréb.),145 disputata, Wood, 142 expansa, Bailey, 141 fimbriata, Ralfs, 145 foliacea, Bailey, 147 furcata, Agurdh, 144 INDEX. Micrasterias. granulata, Wood, 146 incisa, 142 Jenneri, Ra/fs, 146 oscitans, Ralfs, 142 papillifera, Bréb., 146 pinnatifida, Atz., 143 quadrata, Bailey, 142 radiosa, Agardh, 145 rotata, Ra/fs, 144 ringens, Bailey, 143 Torreyi, Batley, 147 truncata, Corda, 144 Microcystis, Menegh., 12 Monactinus duodenarius, Bailey, 98 octenarius, Bailey, 98 NeMAToGENEX, 15 Nostoc, 27 arcticum, Berkeley, 225 Austin i, Wood, 27 alpinum, Atz., 29 ceruleum, Lyngb., 31 calcicola, Ag., 33 calidarium, Wood, 34 Cesatii, Bals., 32 comminutum, Atz., 36 commune, Vaucher, 37 cristatum, Batley, 29 depressum, Wood, 30 flagelliforme, Berk. and Curtis, 226 lichenoides, 30 microscopicum, Carm , 226 punctatum, Wood, 32 pruniforme, Agh., 28 sphericum, Vauch., 30 Sutherlandii, Dickie, 29 verrucosum, Vauch., 28 Nostochacee, 23 Nostochopsis, 44 lobatus, Wood, 45 Nematophycezx, 181 OscILbaRta, 17 chlorina, Avitzing, 18 corium, Agardh,17 decorticans, Gener., 17 Froblichii, Atz., 18 imperator, Wood, 20 limosa, Agardh, 19 muscorum, Agardh, 17 neglecta, Wood, 20 nigra, Vauch., 17,19 tenuis, Ag., 17 tenuissima, 4g., 17 princeps, 21 Oscillariacer, 16 Odontella, 123 tridentata, Bailey, 127 (Edogoniacex, 188 (Edogoniex, 190 (Edogonium, 195 Huntii, Wood, 197 inequalis, Wood, 195 mirabilis, Wood, 196 multispora, Wood, 196 PAGEROGALLA, 81 stellio, Wood, 82 Palmella, 79 dura, Wood, 80 hyalina, Lyngb., 81 Jesenii, Wood, 79 Palmellaceex, 78 Palmogleea, 105 Palmoglea. clepsydra, Wood, 105 Pediastrum, 95 Boryanum (Turpin), 97 constrictum (Hussal), 97 duodenarius, 98 | Ehrenbergii (Corda), 98 pertusum, Aiitzing, 97 selenea, Aitzing, 97 Penium, 106 Brébissonii (Menegh.), 108 closteroides, Ralfs, 109 digitus (Hhrb.), 106 interruptum, Bréb., 108 Jenneri, Ralfs, 108 lamellosum, Brébisson, 107 margaritaceum, Hhrb., 107 minutum, Cleve, 107 Petalonema, 66 alatum, Berkely, 66 Phychochrom, 16 Phycochromophycee, 9 Phycocyan, 16 Phykokyan. 16 Physcia, 88 Pilinia, 211 diluta, Wood, 211 Pleurocarpus, 174 mirabilis, Braun, 174 Pleurococeus, 78 pulvereus, Wood, 79 seriatus, Wood, 78 Pleurotenium, 118 baculum (Bréb.), 119 breve, Wood, 119 clavatum (Kiz.), 120 constrictum (Bailey), 121 crenulatum, Bhrb., 119 gracile, Rabenh., 122 hirsutum (Bailey), 121 nodosum (Bailey), 120 trabecula (Ehrb.), 118 undulatum (Bailey), 120 verrucosum (Bailey), 121 verticillatum, Raben., 121 Polyedrium, 88 enorme, Ralfs, 89 Porphyracee, 214 Porphrydium, 214 cruentum (Aq.), 214 magnificum, Wood, 215 Pringsheimia, 195 inequalis, Wood, 195 Protococcacee, $5 Protococeus, 86 Protoderma, 182 viride, Avitzing, 182 RaaApuipium, $5 faleatum, 85 polymorphum, /resen, 85 Rhynchonema, 163 elongata, Wood, 164 pulchella, Wood, 164 Rhodophycez, 213 Rivularia, 47 calcarea, Sm., 50 cartilaginea, Wood, 47 Rivulariacee, 43 Satmacis, 163 Scenedesmus, 89 acutus, Meyen, 90 obtusus, Meyen, 90 polymorphus, Wood, 91 quadricauda (Turpin), 91 rotundatus, Wood, 91 Schizomeris, 184 Leibleinii, 185 Scytonema, 57 Austinii, Wood, 58 calotrichoides, Aiitzing, 61 eataracta, Wood, 62 cortex, Wood, 64 dubium, Wood, 63 immersum, Wood, 59 Myochrous, Ag., 61 Negelii, Ktz., 59 ocellatum, Harvey, 63 Ravenellii, Wood, 64 simplice, Wood, 57 thermale, 60 Scytonemacee, 55 Siphophycee, 174 Sirosiphon, 73 acervatus, Wood, 74 argillaceus, Wood, 73 compactus (Ag.), 69 coralloides, 75 Crameri, Briigg, 70 guttula, Wood, 73 lignicola, Wood, 72 neglectus, Wood, 71 pellucidulus, Wood, 69 pulvinatus, 75 scytenematoides, Wood, 68 Sirogonium, 178 retroversum, Wood, 173 Sirosiphonacex, 67 Spermatia, 23 Spermosirex, 37 Sphzrozosma, 123 excavatum, Ralfs, 123 pulehrum, Bailey, 123 serratum, Bailey, 124 Spherozyga, 43 Carmichelii, Harvey, 43 polysperma (Ktz.), 43 Spirogyra, 163 crassa, Kiifzing, 171 decimina, Mi ler, 167 diluta, Wood, 170 dubia, Kiitzing, 167 elongata (Berk.), 164 insignis, Hassal, 166 longata ( Vauch.), 166 majuscula, Aiitz.), 169 nitida (Diller.), 169 protecta, Wood, 165 parvispora, Wood, 169 INDEX. Spirogyra. pulchella, Wood, 164 quinina, Ag., 167 rivularis (//assal), 168 setiformis ( Roth), 170 Weberi, Aiz., 165 Spirotenia, 122 bryophila (Bréb.), 122 condensata (Gréb.), 122 Spondylosium, 123 Staurastrum, 147 alternans, Bréb., 150 arachne, Ral/s, 152 aristiferum, Ra/fs, 149 Cerberus, Bailey, 154 erenatum, Bailey, 151 cyrtocerum, Bréb., 151 dejectum, Bréb., 148 dilatatum, Fhrb., 150 enorme, Ra/fs, 89 eustephanum, Ralfs, 155 fureigerum, Bréb., 154 gracile, Ralfs, 152 hirsutum, Hirb., 153 Hystrix, Ral/s, 164 Lewisii, Wood, 149 longispinum, Archer, 148 margaritaceum, Hirb., 150 munitum, Wood, 154 muticum, bréb., 148 orbiculare, Ekrb., 148 paradoxum, Meyen, 152 polymorphum, 151 polytrichum, Perty, 153 punctulatum, B)é., 151 Raveunellii, Wood, 153 senarium, /hrb., 155 tricorne, Menegh., 150 Stephanoxanthium eustephanum, Kiitzing, 155 senarium, Avifzing, 155 Stigonema mammellosum, 77 Ravenelii, Berkeley, 76 Stigeoclonium, 206 Termemorvs, 116 Brébissonii (Menegh.), 116 giganteus, Wood, 117 granulatus (&réb_), 117 lavis (Kiitz.), 118 Tetraspora, 82 bullosa (Roth), 84 251 | Tetraspora. gelatinosa (Roth), 83 lubrica (Roth), 82 perforata, Harvey, 82 Tolypothrix, 65 distorta (Miller), 65 Triploceras, 121 gracile, Bailey, 122 verticillatum, Bailey, 121 Tuomeya, 221 fluviatilis, Harvey, 221 Tyndaridea cruciata, Hassal, 172 insignis, Hassal, 171 Unva, 182 latissima, Harvey, 183 merismopedioides, Wood, 182 orbiculata, Rubenhorst, 183 Ulvacee, 182 VaoucueriA, 179 aversa, Hassa/l, 181 cespitosa (Vaucher), 179 geminata ( Vaucher), 180 polymorpha, Wood, 150 sessilis ( Vaucher), 179 sericea, Lyngbya, 181 velutina, Agardh, 180 Vaucheriacee, 176 Volvocinex, 99 Volvex, 100 globator, 100 XanrTurpiom, 156 aculeatum, Ehrb , 156 Arctiscon, Hhrb., 156 armatum (5réb.), 156 bisenarium, Khrb., 156 Brébissonii, Ra/fs, 156 coronatum, Lhrb., 157 cristatum, Bréb., 157 fasciculatum, Lhrb., 157 Zonorricara, 48 minutula, Wood, 50 mollis, Wood, 48 pareezonata, Wood, 49 Zygnema, 171 insigné ( Hassal), 171 eruciatim, Vauch., 172 Zygnemacee, 159 Zygophycez, 100 Line CORRECTIONS OF THE PLATES. Plate IV., for Zonotricha read Zonotrichia. Plate VIII., for wm in the terminations of the specific name of figures 2, 3, and 4, read us. Plate IX., fig. 3, for argillacea read argillaceus. Plate X., fig. 4, for Botryococcus pulchellus read Dictyospherium pulchellum. Plate XI., fig. 5, 6, and 7, for Cesmarium read Closterium. : 7 Plate XIL., fig. 1 and 20, for Cosmarium read Closterium. ; 2 Plate XV., fig. 8, for insignis read insigne. i Plate XVI., fig. 4, for Bulbotricha read Bulbotrichia. Plate XVIIL, fig. 1, for Pringsheimii read Prinasheimia. Plate XXI., fig. 7, for tetraopthalmum read tetropthalmum. EXPLANATION OF PLATES. Pe lpAS Dh abarele Fig. 1. Fig. 2. The end of a filament of an Oscillatoria supposed to be identical with O. Fréhlichii Kiitzing. A single filament of Oscillatoria chlorina, Kiitzing, magnified 750 diameters. ag Fig. 3. O. nigra, Vaucher. Fig. 3 a. Represents a portion of a mat or mass of Oscillatoria nigra, Vaucher; there is too much green in the color. Fig. 3 6. Represents several filaments separated from the edge of the mass and slightly magni- fied. Fig. 3c. ° OP vey 0 0” Pee.eos”* Srp oe Oo a Poses AFTER NATURE BY OR M. CG. WOOL y. SINCLAIR & SON, PHILADELPHIA Fig. 1. MASTIGONEMA FERTILE. Fig 3. ZONOTRICHA MOLLIS Fig. 6. DASYACTIS MOLLIS. “2. MASTIGONEMA SEJUNCTUM * 4. ZONOTRICHA PARCEZONATA “ 6. CELOSPHARIUM DUBIUM. © 9¢e* FRESH WATER ALGAZ. P PLATE V. aS T. SINCLAIK & SON, PHILADELPHIA [ossteoenes = AFTER NATURE SY DR. H.C. wooD Fig. 1. MASTIGONEMA ELONGATUM | Fig. 3. MASTIGOTHRIX FIBROSA, Fig. 6. CHROOCOCCUS MULTICOLOR- “ 2. MASTIGONEMA HALOS. “4. SCYTONEMA RAVENELII. ATUS., 5. CHROOCOCCUS REFRACTUS, FRESH WATER ALGA. PLATE VI. AFTER NATURE BY DR.H.C.WOOD *. SINCLAIR & SON, PHILADELPHIA. Fig.1. SCYTONEMA THERMALE. Fig. 4 S. CORTEX. « 9, S. CALLITRICHOIDES. | “ §, CHAZTOPHORA ELEGANS. “ 3. S. DUBIUM. FRESH WATER ALGA. Suede Wh — AFTER NATURE BY OR. H. OC. WOOD : T. SINGLAIR & SON, PHILADELPHIA. ’ Fig. 1. SCYTONEMA CATARACTUM. Fig. 3. RHAPHIDIUM POLYMORPHUM. « % SCYTONEMA IMMERSUM. “4. PROTOCOCCUS. i FRESH WATER ALGAZ. ———— AFTER NATURE BY DR.H C. WOOD. PLATE Vill. 8 00 Bs Ss CA | >| C4 | 2 (| a | | \ @\ | =| |e | | a \ \ fe | | =2\ | | S| | is |£ || | \ED \ | =| 260 \ 2 \ low! \ es ' /oe\ \ JSEe- Fig. 1. TOLYPOTHRIX DISTORTA. “2. SIROSIPHON PELLUCIDULUM. * 3. SIROSIPHON COMPACTUM. SINCLAIR & SON, PHILADELPHIA, Fig.4. SIROSIPHON NEGLECTUM. Fig. 7. GLOEOCAPSA SPARSA, “*°§ SIROSIPHON GUTTULA. “ 8. MERISMOPEDIA NOVA. “6. SCYTONEMA NAGELLI!. FRESH WATER ALGA. PLATE IX. — | | AFTER NATURE BY DR 4 C.woOoD. r TSINCLAIR & SON PHILADELPHIA Fig. 1. SIROSIPHON SCYTENEMATOIDES. Fig. 2, SIROSIPHON LIGNICOLA. | Fig. 3. S. ARGILLACEA. Fig 4. STIGONEMA RAVENELLI!. FRESH WATER ALGA. PLATE X. w% 4 ae ave AFTER NAT B URE By DR. H.C, WOOD. T. SINCLAIR & SON, PHILADELPHIA Fig. 1. SIROSIPHON PULVINATUS | Fig “8. PALMELLA JESSENI, ) Fig. 4. BOTRYOCOCCUS PU} CHELLUS. 2. PLEUROCOCCUS SERIATUS 5 } “ 4. PALMELLA DURA. / FRESH WATER ALGA SS L PLATE Il. ATRL tas a 4 fez 160 IES IVES DEG RD Netiscen AFTER NATURE BY DR. HC woCcD Fig. 1. SCENEDESMUS POLYMORPHUS. 2. SCENEDESMUS QUADRICAUDA. Fig. 3. SCENEDESMUS ROTUNDATUS 4. PALMOGLOEA CLEPSYDRA. 5. COSMARIUM ACEROSUM. T. SINCLAIR & 50K PHILADELPHIA Fig. 6: COSMARIUM AREOLATUM. ~ COSMARIUM YENUS. He APYER QATAR BY OR.H.C WooD Fig. 1, COSMARIUM LINEATUM * 2. ©. EHRENBERGII, “ 3. C. ROSTRATUM “ 4. C. DIAN, " 5 ©. PARVULUM. ; ~ ig. 6 6. 9 “ 40 Cc. LEIBLEINI! TETMEMORUS GIGANTEUS., TETMEMORUS GRANULATUS PLEUROT-ENIUM TRABEGULA SPIROTANIA SRYOPHILA Fig. 11. ye Bee 13 14. 6 SPIROT-ENIA CONDENSATA HYALOTHECA DOISSILIENS DIDYMOPRIUM GREVILLI! COSMARIUM BOTRYTIS., ©. CUCUMIS PLATE X\ es FRESH WATER ALGA. PLATE XII sto AD 2," nS AFTER NATURE BY OF.H.C-WOOD. Sin Ain & Se H Fig. 1. EUASTRUM RALFSI!! | Fig. 6. IBID VARIETY DENTICULATA Fig. 10. STAURASTRUM PUNCTULA F 13. STAURASTRUM MUNITUM “ 2 EUASTRUM ELEGANS | “7. M. JENNERII TUM l4 OSMARIUM PYRAMIDATUM 8. EUASTRUM BINALE } © §, STAURASTRUM ORBICULARE “ 41. STAURASTRUM LEWISIi 15 ARIUM BROOMET “ 4 MICRAST. DISPUTA } “ 9. STAURASTRUM DEJECTUM “12. STAURASTRUM POLYTR Lb OSMARIUM COMMISSURALE “ 5. MICRAST. FURCATA | CHUM 17, XANTHIDIUM ARMATUM FRESH WATER ALGA. PLATE XIV. fe | AFTER NATURE BY OR. HC. WOOD TSINCLAIR ASOK PHILADE Fig. 1. RHYNCHONEMA ELONGATUM. Fig. 3. SPIROGYRA PROTECTA. Fig. 5. APHANOCHAETE REPENS, "2. RHYNCHONEMA PULCHELLUM. * 4. SPIROGYRA LONGATA. “6. DRAPARNALDIA BILLINGSII. ay i FRESH WATER ALGA. AFTER NATURE GY DR H.C wOOD Fig. 1. SPIROGYRA MAJUSCULA. 2 SPIROGYR* DILUTA. Fig. 3. SPIROGYRA SETIFORMIS « 4. SPIROGYRA CRASSA. “ 5. MESOCARPUS SCALARIS PLATE XY. 7. SINCLAIR & SON, PHILADELPHIA Fig. 6. SPiIROGYRA NNSIGNIS “7. SPIROGYRA PARVISPORA 8&8. ZYGNEMA INSIGNIS. FRESH WATER ALGA. PLATE XVI. — AFTER NATURE BY OR.H.C WOOD T. SINOLAIR & SON, PHILADELPHIA Fig. 1. SIROGONIUM RETROVERSUM. Fig. 4. BULBOTRICHA ALBIDA. “ 2. HYDROGASTRUM GRANULATUM. “ § BULBOCHATE CANBY}. « 3. STIGEOCLONIUM. “« 6, STIGEOCLONIUM, FRESH WATER ALGA. PLATE XVII. am = 1 f 5c %, oN a hs a a: Ss 4 pa NA Ja \ =n 250 he ~*~ ; i *e) | % | Nea aX % = | | ey t a ; Ao! ae 2e0 | é | a cM } \ ‘ | { | | | } } i } | i r—|" | j 1] | | i\\ | | / { ee Oe ee 26 7 ss : : hc 2 AFTER NATURE BY DA. H.C.WOOD. T. SINCLAIR & SON. PHILADELPHIA. Fig.1. SCHIZOMERIS LEIBLEINII. Fig. 4. SPIROGYRA DUBIA. “ 2. OEDOGONIUM HUNTII « §. S. RIVULAIS. « 3. O. MULTISPORA. FRESH WATER ALGA. PLATE XVIII. r - ee = pe 6606 . © Pind @0o0 J == AFTER NATURE BY DR. HC. wooo T. SINCLAIR & SON, PHILADELPHIA Fig. 1. PRINGSHEIMI! INAEQUALE. Fig. 5 BULBOCHATE IGNOTA. “ 2. OEDOGONIUM MIRABILE. “6 BULBOCHATE DUMOSA. “ 3. OEDOGONIUM ECHINATUM. “7 CONFERVA. “4 OQEDOGONIUM. PLATE XIX. FRESH WATER ALGZ. T. SINCLAIR & SON, PHILADELPHIA. AFTER NATURE BY DR.H.C. WOOD. Fig. 3. CHANTRANSIA MACROSPORA. Fig. 1, STIGEOCLONIUM. «4. SPIROGYRA QUININA. “2. CHANTRANSIA EXPANSA. PLATE XX T. SINCLAIR & SON, PHILADELPHIA Fig 3. VAUCHERIA POLYMORPHA TIGEOCLONIUM, “2, ARTHRODESMUS QUADRIDENS. Fig. 6. EUASTRUM MULTILOBATUM. “6, PENIUM DIGITUS. | * 4. LEMANEA TORULOSA. | es = . ied twa ee | ee ee 7 i a - i : i i - 5 ft 1 . . a q ’ S . i , ’ Fi . — i , . ¥ ‘ \ j = t ‘ / ‘ ' + t * ‘ ’ ; " AFTER NATURE GY OR, H.C. WOOD Fig. 1. PLEUROTCENIUM CRENULA-| Fig, 6. COSMARIUM BREBISSONI!. 7. COSMARIUM TETRAQPTHAL « “ “ PLATE XXI. 2 PLEUROTCENIUM BREVE 3. TETMEMORUS BREBISSONII, 4. CLOSTERIUM LINEATUM. 5. COSMARIUM BOTRYTIS. i 8. COSMARIUM MARGARITIFE- 9. COSMARIUM SUBORBICU to. Ji. 12 13. 14. 15. 16. COSMARIUM BROOME!. MICRASTERIAS JENNERII. EUASTRUM ORNATUM E. DIDELTA. —. ELEGANS. MICRASTERIAS TRUNCATA. MICRASTERIAS GRANULATA. 18. 19. 20. 23 T. SINGLAIR & SON, PHILADELPHIA. -. STAURASTRUM ORBICU- LARE. ST. ST. ST 51 ST ST. DEJECTUM. LEWSI!, PARADOXICUM. ARACHNE RAVENELLI/. POLYTRICHUM. ae SMITHSONIAN CONTRIBUTIONS TO KNOWLEDGE. — —— 262 - - AN INVESTIGATION OF THE ORBIT OF URANUS, WITH GENERAL TABLES OF ITS MOTION. BY SIMON NEWCOMB, PROFESSOR OF MATHEMATICS, UNITED STATES NAVY. [ACCEPTED FOR PUBLICATION, FEBRUARY, 1873. ] ADVERTISEMENT. In the investigation of the Orbit of Uranus which forms the subject of the accompanying memoir, as well as in that of the Orbit of Neptune previously published in the Smithsonian Contributions, a large amount of arithmetical computation has been required, especially in the reduction and com- parison of observations. The cost of this, in accordance with the spirit of the Institution in advancing science, has been defrayed from the income of the Smithson fund. As required by the rules of the Institution, the accompanying memoir was referred to competent authority for examination, and the persons selected for this purpose were Professor J. H. C. Coffin, of the Nautical Almanac Office, and Professor Asaph Hall, of the Naval Observatory. JOSEPH HENRY, Secretary S. I. Wasuineton, 1873. PHILADELPHIA: COLLINS, PRINTER, 705 Jayne Street, PREFACE. Tue present work was undertaken as far back as the year 1859. But the labor devoted to it at first amounted to little more than tentative efforts to obtain numerical data of sufficient accuracy, and to decide upon a satisfactory method of computing the general perturbations of the planet. The elements of Neptune employed in the earlier computations were found to deviate too widely from the truth to be used in computing the perturbations of Uranus with the first order of accuracy, and it became necessary to correct them. ‘This was done during the years 1864 and 1865, and the investigation was’ printed by the Smithsonian Institution in the latter year. It was then found that the adopted elements of Uranus also differed too widely from the truth to serve as the basis of the work, and they were provisionally corrected by a series of heliocentric longitudes derived from observa- tions extending from 1781 to 1861. Finally it was found that the adopted method of computing the perturbations, that of the “ variation of elements,” though not deserving of the disfavor into which it has fallen of late years, was practically inapplicable to the computation of the most difficult terms, namely, those of the second order with respect to the disturbing forces. Indeed, it appeared to the author that the only method of computing those terms which was at the same time general, practicable, and fully developed, was that of Hansen, But, were this method adopted, all that had previously been done would have been. useless, even for the purpose of comparison and verification, owing to the expression of the co- ordinates in terms of a disturbed mean anomaly. It appeared to the author that, although this form of theory led to expressions having fewer terms than the other, it was not without its relative disadvantages. Other considerations being equal, he conceived that astronomers generally would greatly prefer to see the perturbations expressed directly in terms of the time, owing to the ease with which the results of different investigators could then be compared, and with which corrections to the theory may be introduced. Under these circumstances the method described in the first chapter of the present paper was worked out. ‘The question how much it contains that is essen- tially new is one that the author has never closely examined: it is, however, certain ( ili ) iv IPIRVE REALE. that the mode of considering the subject is well known, being that employed by La Place, Herschel, De Pontécoulant, Encke, and perhaps others. ‘The method of forming the required derivations of the perturbative function from the analytical development of that quantity, he has not seen elsewhere. With these improved elements and methods the work was recommenced in 1868. The earlier investigations being merely provisional, it has not been deemed neces- sary to present them in the present work. Some of the results, corrected for errors of the older elements, are, however, given for the purpose of comparison, ~ Although this investigation has absorbed the greater part of the author’s leisure for more than five years, it is only through the aid of the Smithsonian Institution and Nautical Almanac that he has been enabled to bring it to a conclusion within that time. At an early stage of the work Professor Henry responded favorably to a. request for aid by the employment of computers; it was, however, not found practicable to use such aid until the perturbations had been completed, and the provisional theory concluded. Then, the comparison of theory and observation, ~ and the construction of the tables, involved a large amount of mechanical compu- tation, and on this part of the work a number of persons have been employed by the Institution at various times, among whom may be mentioned Professor F. W. Bardwell, of the University of Kansas, and Dr. C. L. F. Kampf, late of the Ob- servatory of Leiden. Every part of the work has, however, been done under the author’s immediate direction, and, as nearly as possible, in the same way as if he had done it himself, a result which, in one or two cases, has been attained only by the expenditure of an amount of labor approximating that saved by the employ- ment of the computer. In presenting the steps of the investigation, the end has been kept constantly in view to render as easy as possible the detection and correction of any error, or the introduction of any alteration in the elements or other data. It is, of course, impossible to present the steps of the computation with any approach to fulness without far transcending the limits of the printed work: The results given are, therefore, those which it was supposed would be most useful to the future investi- gator of the same subject. There is reason to believe that the original computa- tions will ultimately become the property of the National Academy of Sciences, so that they may always be referred to for the clearing up of any difficulty in the printed text. The author’s acknowledgments are due to Professor J. H. C. Coffin, Superin- tendent of the Nautical Almanac, and Mr. E. J. Loomis, of the Nautical Almanac Office, for reading the proof sheets of the last twelve tables during the absence of the former abroad. WASHINGTON, July 31, 1873. TABLE OF CONTENTS. » INTRODUCTION . . 9 : : : ¢ 5 : : CHAPTER TI. METHOD OF DETERMINING THE PERTURBATIONS OF THE LONGITUDE, RADIUS VECTOR, AND LATITUDE OF A PLANET BY DIRECT INTEGRATION. Notation and general differential formule 5 : Formation of the required derivatives of the nertarbanive Snetion : Correction of these derivatives for terms of the second order . 3 ° Integration formule for perturbations of radius vector : : 6 c Development of functions of rectangular co-ordinates . : : : 5 Integration of perturbations of radius vector . : : Formule for perturbations of longitude to terms of the acennd order Motion of the orbital planes. : Perturbations of the second order dependiaete on the motion of the erbial mane Reduction of the longitude to the ecliptic : é . : Expressions for the latitude CHAPTER II. APPLICATION OF THE PRECEDING METHOD TO THE COMPUTATION OF THE PERTURBATIONS OF URANUS BY SATURN. Data of computation. 5 : c Numerical expressions for R onl its denvatires - ; ° Perturbations of radius vector ; c : : . Perturbations of longitude ; 0 F : c , 3 : : Perturbations of latitude : : : : : j 3 : : CHAPTER III. PERTURBATIONS OF URANUS PRODUCED BY NEPTUNE AND JUPITER. Adopted elements of Neptune . F Development of & and its derivatives for the ection ef Neptune The term of long period between Neptune and Uranus : : Perturbations of the longitude produced by Neptune . - c : Perturbations of the radius vector produced by Neptune : 6 : . Perturbations of the latitude produced by Neptune. : : : Perturbations produced by Jupiter. 4 . . . 2 PAGE 6 10 12 13 14 17 22 24 25 27 29 9 vo 2° vo 44 49 51 53 54 55 58 60 61 62 vi TABLE OF CONTENTS. CHAPTER IV. TERMS OF THE SECOND ORDER DUE TO THE ACTION OF SATURN. Preliminary investigation of the orbit of Saturn : : : . 0 5 Perturbations of Saturn and Uranus. 5 : ; 5 ; Formation of the expressions for the terms of the sseota order A ‘; : 5 Perturbations depending on the square of the mass of Saturn Z A : Perturbations depending on the product of the masses of Jupiter Saal Saturn : CHAPTER. VY. COLLECTION AND TRANSFORMATION OF THE PRECEDING PERTURBATIONS OF URANUS. Terms independent of the position of the disturbing planet. . 5 : : Secular variations 3 z S S 3 : Auxiliary expressions on which the penned nore depend ¢ ° : 5 P Reduced expressions for the latitude of Uranus : 5 ; - . 5 Positions of Uranus resulting from the preceding theory 3 0 5 ‘ 5 Elements III of Uranus : é : : rs : c 2 CHAPTER VI. REDUCTION OF THE OBSERVATIONS OF URANUS, AND THEIR COMPARISON WITH THE PRECEDING THEORY. Reduction of the ancient observations . : A . . : : 5 Their comparison with the provisional theory . 0 : . : c 0 Discussion of the modern observations 6 . , : ¢ - c Reduction of the results to a uniform system . é : . : Adopted positions of fundamental stars o : : Discussion of eorrections to reduce the different ppeenationen to a homogeneous system Table of these corrections c 5 Results of the observations from 1781 to 1830 Observations from 1830 to L872 : : 5 6 Table to convert errors of right ascension and declination of race inte errors of fenetade and latitude Tabular summary of results of eoremrations| 1830 to 1872 Corrections to be applied to the positions of Uranus in the Berlin Talenged ana the Nautica! Almanac to reduce them to positions from the provisional theory ¢ CHAPTER. VII. FORMATION AND SOLUTION OF THE EQUATIONS OF CONDITION RESULTING FROM THE PRECEDING COMPARISONS. Expressions of the observed corrections to the longitudes of the provisional ee in terms of the corrections to the heliocentric co-ordinates Expressions of the same quantities in terms of the corrections to the clement of he ranus and the mass of Neptune : Table to express errors of Hekecenticn co-ordinates as errors of clones Discussions and solutions of the equations thus formed Concluded corrections to the elements of longitude Corrections to the inclination and node of Uranus . . . . . . PAGE 65 68 69 76 17 79 80 81 93 98 99 127 131 151 158 161 162 165 173 173 TABLE OF CONTENTS. Vil CHAPTER VIII. COMPLETION AND ARRANGEMENT OF THE THEORY TO FIT IT FOR PERMANENT USE. PAGE Correction of the coefficients of the long inequality between Uranus and Neptune for the terms of the second order . 0 ° : ¢ : : - 178 Concluded elements, or elements IV of Grane , ‘ 5 2 6 5 alist Long-period and secular perturbations of the elements “ . 5 . 182 Table of these perturbations from A.D. 1000 until A.D. 2200 : 5 ° ss Mean elements of Uranus 3 : : : . : - 184 Expressions for the concluded theory of Uae : : : ° ° - 185 CHAPTER IX. . GENERAL TABLES OF URANUS. Enumeration of the quantities contained in the several tables 5 5 AUK) Precepts for the use of the tables : : : : ° : ° co UG Examples of the use of the tables : : o c . . . eS Tables of Uranus . : 3 ; ° . : ° . - 206 Subsidiary tables . : : . ° : : ° : 5 | AY) ERRATA. Pages 100 to 105. In computing the latitude from the provisional theory the values of the secular terms of 8 and 5% on page 97 have been interchanged. The provisional latitude, therefore, requires the correction —0".53 Tsin v + 07.53 T cos v where v=g + 12° 45’ + 2e sin g. This correction is not applied in the subsequent investigation. Its effect would have been to change the value of 6 deduced on p. 176 by probably 0’.2 or 0.3. The effect on the other elements of latitude would have been much smaller, and therefore unimportant. Page 122, line 15. Add: the corrections in the sixth column being omitted. Page 151. Add foot-note: In forming these comparisons the corrections to the heliocentric longitude in the sixth column of the provisional ephemeris, pages 100 to 105, are not applied. Page 159. Equation 7. In this equation the coefficients of 5, and $p have been multiplied by 4, instead of 2, the factor of 67. The effect of this error enters into all the subsequent results, but in the comparisons of theory and observation it is corrected. Page 184. The element here represented by x (kappa) is the same which, in the preceding chapters, has by mistake been represented by &, and which is defined on p. 24. The & of Chapter VII! is, therefore, not the same with that of preceding chapters. ON THE ORBIT OF URANUS. [EN DRODUC LEON. THE connection of the planet Uranus with the most brilliant astronomical achievement of the century lends a peculiar interest to its theory. ‘The researches of Adams and Le Verrier showed that the observed motions of that planet were represented, at least approximately, by the action of a theoretical planet having the longitude of Neptune. Peirce showed that the action of Neptune itself accounted for these motions within the limits of possible error of the observations used by Le Verrier. It remains to be seen whether the agreement between theory and observation still subsists when the comparatively few observations used by those investigators are reduced with the more refined data now at our disposal, and when the great mass of additional observations made both before and since the date of Le Verrier’s researches are included. The circumstances connected with the discovery of Neptune have been so exhaustively recounted by a number of authors that it would be difficult to add anything not already familiar to astronomers without transcending our present limits. I shall therefore confine myself to such an account of previous researches on the theory of Uranus as may give an idea of their nature and extent, and facili- tate their comparison with the methods and results of the present investigation. The perturbations used by Bouvard in his tables are those of the Mécanique Céleste. Although not affected with any striking error, the numerical methods adopted in their computation are necessarily too rough to allow of much interest attaching to their comparison with the results of the more recent researches, It is essential to a clear understanding of subsequent researches that we classify the methods which have been or may be adopted in the computation of the general perturbations of the planets. This computation comprises two distinct operations: (1) the development of the disturbing forces, or some quantities of which these forces are functions; (2) the integration of the equations of motion under the influence of these forces. In each of these operations three methods have been employed. In developing the perturbative function, we have first the purely analytic method used by the great geometers of the last century. In this method this function is developed in powers of the eccentricities and mutual inclination of the orbits of the two planets, and the numerical coefficients are found by substituting the values of the elements in these expressions. It is only applicable when the eccentricities 1 March, 1873, Gt» YD THE ORBIT OF URANUS and mutual inclination are small, and has for that reason fallen, of late, into a certain disrepute. ‘The extended tables published by Le Verrier’ have, however, added so much to its facility for use that it is not wholly unworthy of attention. At the other extreme stands the purely mechanical method, in which special values of the disturbing force are computed for many combinations of the mean anomalies of the two planets, and the values of the coefficients in the general expression for the force thence deduced. Between these two stands what I conceive we may designate as the Cauchy- Hansen method, in which the development is made mechanically with respect to the one planet, but the eccentric anomaly of the other is retained as an undeter- mined quantity. The germ of this method is found in several papers, by Cauchy, in the earlier volumes of the Comptes Rendus of the French Academy, which have since been combined into a homogeneous memoir by Puiseux.” The object had in view by these authors is only the computation of inequalities of long period. But Hansen has taken up the essential principle of the method, first, in his prize memoir on the perturbations of comets, crowned by the French Academy of Sciences, about 1848, and afterwards in his ‘‘ Auseinandersetzung einer zweckmdssigen Methode cur Berechnung der Stérungen der kleinen Planeten,”* and applied it to the general development of perturbations. Among the three methods of integration, the first in point of analytical elegance and generality, but the last in order of convenience in use, is that of the variation of elements, a method with which the name of La Grange is inseparably associated. In the second the direct integration of the differential equations which express the perturbations of longitude, latitude, and radius vector is effected by special devices. In the first of these methods the problem is presented in this form: The equations of motion being completely integrated for the action of the principal forces only, how must the arbitrary constants of integration vary in order that the same expres- sions may represent the motion of the planet under the influence of the disturbing forces? In the second method, the same thing being presupposed, the question is, what expressions must be added to the integrals of undisturbed motion in order that the sum may represent the integrals of the disturbed motion ? The third is Hansen’s method, in which the co-ordinates are partly expressed in terms of a certain function of the time known as the disturbed mean anomaly, determined by the condition that the true longitude in the disturbed orbit shall be the same function of the disturbed time that the longitude in the elliptic orbit is of the simple time. : Although the last two methods have a great advantage over the first in the com- putation of the periodic perturbations, I conceive the first to be best adapted to the computation of the secular variations, and perhaps, of terms of very long period in the mean longitude and the elements of the orbit. 1 Annales de l’ Observatoire Impérial de Paris. Tome I. 2 Annales de VObservatoire Impérial de Paris. Tome VII. 3 Abhandlungen der Kéniglich Sichsischen Gesellschaft der Wissenschaften. Band VY. VI, VII. THE ORBIT OF URANUS. 3 In his researches on the motion of Uranus, the first thing done by Le Verrier was to recompute the perturbations by Jupiter and Saturn. It will snfficiently describe his method of doing this to say that in the developments he used the purely mechanical method for the action of Saturn, and the algebraic development of the perturbative function for the action of Jupiter, while in the integration he used the method of the variation of elements. After completing the perturbations of the first order he made the earliest attempt at a complete determination of those of the second order. Beginning with the terms of this order which arise from the secular variations of the elements, he determines them by recomputing the terms of the first order for the epoch 2300, and assuming that the general term will then be given by interpolating between the two terms thus found, supposing them to increase uniformly with the time. This proceeding has the sanction of such high authority that it is worth while to call attention to its want of rigor. The dif ferential coefficient of each element being given in the form ue —k cos bt, dt k being a function of the elements, the perturbation of the first order will be Bie éa == sin bt. b When we take into account the variation of /, and suppose it of the form &, + kt, the process is equivalent to supposing that in this case ca = oe sin bé, whereas it really contains the additional term, ‘4? ze 68 bt, which appears to be neglected in the process in question. It will be seen that the neglected coefficient is equal to the secular variation of the term during the time that its argument requires to increase by an amount equal to the unit radius. It is therefore the more important the longer the period of the inequality. To obtain the periodic terms of the second order Le Verrier begins by determin- ing the ten principal terms of the perturbations of the elements of Saturn produced by Jupiter. Next he takes up the terms in the mean longitude of Uranus which depend on the square of the mass of Saturn. The only sensible terms he finds are — 1.17 sin (f*— 387) —0".35 cos (¢’ — 36) + 0’.43 sin (¢” — 42’ + 47) — 0.21 cos (¢” — 4¢’ + 42), ¢, ¢’, and ¢” being the mean anomalies of Uranus, Saturn, and Jupiter, respectively. The terms depending on the product of the masses of Jupiter and Saturn are then taken up. Fifteen arguments are found the coefficients of which vary from a small fraction of a second to one or two seconds, while a single one of long period amounts to 32”. When the method of variation of elements is used, it is necessary not only to determine these variations to quantities of the second order, but, in the transforma- 4 TH hy OR BL LO UR AN UES: tion of the perturbations of the elements into perturbations of the co-ordinates, to carry this transformation to terms of the second order also. This Le Verrier avoids by showing that the terms of the lowest order with respect to the eccentricities thus introduced are destroyed by certain terms in the perturbations of the elements, so that it is only necessary to omit both classes of terms. These terms are of that fictitious class which disappear of themselves by a simple change of elements. When, instead of the eccentricity and longitude of the perihelion, we take h and k, which represent the products of the eccentricity into the sine and cosine of this longitude respectively, these terms disappear of themselves both from the perturba- tions of the elements and of the co-ordinates. It is not likely that any of the neglected terms of this class exceed 0".1, As soon as the elements of Neptune were known, the nature of its general action on Uranus became of interest. ‘This subject was taken up by Prof. Peirce, whose results are found in the Proceedings of the American Academy of Arts and Sciences, Vol. I, pp. 334-337. This paper is accompanied with a comparison of his theory of Uranus with observations, to which similar comparisons of the theories of Adams and Le Verrier are added. ‘This comparative exhibit is of sufficient interest to be given here. ‘The numbers given are probably excesses of computed over observed longitudes. ReEstDUAL DIFFERENCES BETWEEN THE THEORETICAL AND OBSERVED LONGITUDES OF URANUS, FROM THE THEORIES OF PEIRCE, LE VERRIER, AND ADAMS. From Le Ver- rier’s best or- bit of Uranus from the mo- dern observa- tions without any external planet. From Le Ver- rier’s original theory with his best orbit of hypotheti- cal planet, of which the mass is 1 93 22° From Adams’s original ory with second hypo- thetical planet of which mass is the- | his | From Peirce’s theory of Neptune adopting for its mass That of Struve from his own | observations of the satellite Tiise- ATANIDawounrke7AwWo ows =e oe PORTA WR WO DAIS ~ x — SNYNSHNSSSSHKSHNHRUNS WNWONrPrFRUOUDMDDoOWNwWAC US seal hl Fee Oe ROSS Sore ies wpowraAnpnooarteHwoonm cr lbarse ll ‘/ 99.6 = oo a) RO It 09 BOS BO 0 OD mMmwwoerInwwwnmwmwaot That deduced by Peirce from | Bond’s & Las- sel’s observa- tions combined 1 Is780° ~ x eS a ee PRED HE EHOM OH OOH MAW so 7 DwWRWOTHRANAMNAOYSOS That deduced by Peirce from Bond’s obser- vations of Las- sel’s satellite Tosi ~ ~ 1} ++]4+1 ]+++] ] |+++ SCH PSNESSSSOS SOM ARMS SK OWMDOOCWrFRMOWWNUOCOOC-T18M In this paper Professor Peirce presents the results of a complete computation of the general perturbations of Uranus by Neptune in longitude and radius vector, THE ORBIT OF URANUS: 5 but without any details whatever of the investigation, or any statement of the methods employed. ‘The minuteness of the residuals in the last column of the preceding table shows that employing these perturbations by Neptune, and those of Le Verrier by Jupiter and Saturn, we had a theory of Uranus from which quite accurate tables might have begn constructed. But this never seems to have been done. ‘The ephemeris of Uranus in the American Nautical Almanac was intended to be founded on this theory, but the proper definitive elements do not seem to have been adopted in the computations, as the ephemeris does not correspond with the theory. Although twenty-five years have elapsed since the epoch of these researches, I am not aware of any published work of importance on the theory of Uranus during the interval. Mr. T. H. Safford has, however, made a very extended investigation of the subject, but has published nothing more than a brief general description of his work, which may be found in the Monthly Notices of the Royal Astronomical Society, Vol. 22. Like Professor Peirce, he took Le Verrier’s perturbations by Jupiter and Saturn, but, instead of using general perturbations by Neptune, he computed the effect of the action of this planet by mechanical quadratures for the whole period of the observations of Uranus, and thus corrected the elements and the mass of Neptune from modern observations alone. The mass in question deduced was 1 20039 Mr. Safford does not give the representation of the modern observations, but pre- sents the following comparison of the ancient ones, alongside which we place for comparison the corresponding numbers of Peirce’s theory and those of the present investigation. EXCESS OF OBSERVATION OVER THEORY. Date. No. of obs. Safford. Peirce. Newcomb. 1690 1 HE BH) == 83 = iit” 1715 3 — 4.2 — 8.7, — 8 1750 2 == PP 2 eeciee ) 1753 1 = (6) heer ) ene 1756 1 — 0.9 — 4.0 1764 1 + 0.4 1769 8 4+ 4.5 + 6.0 —1.4 6 THE ORBIT OF URANUS. CHALRAE RA: METHOD OF DETERMINING THE PERTURBATIONS OF LONGITUDE, RADIUS VECTOR, AND LATITUDE OF A PLANET BY DIRECT INTEGRATION. Ler us conceive a plane determined by the condition that it shall pass through the sun and contain the tangent to the orbit of a planet at any moment. If the planet were acted on by the sun alone, the position of this plane would be invariable, but, under the influence of the disturbing forces of the other planets, it is subject, at each instant, to a motion of rotation around the radius vector of the planet. We may regard this as the instantaneous plane of the planet’s orbit. The disturbing and the disturbed planet will each have its own instantaneous plane. Let us now put :— v, the longitude of a planet counted from a determinate point in the instantaneous plane of its orbit. v, its distance from the node of intersection of its own orbit with that of another planet. y, the mutual inclination of the two orbits. o, sind y. y, the radius vector of the planet. p, its logarithm, u, the attractive force of the sun upon unit of matter at unit distance. a, the mean distance corresponding to the observed mean motion of the planet, determined by the condition “ify : m and n being as usual the mass and mean motion, ay, the value of a corrected for the constants introduced by the perturbations, so that, as in the elliptic motion, we have p=loga + f (I, e, 3), we shall have in the disturbed motion p = log a, +f (, e, a) + periodic terms only. a,, the mean distance of an outer planet, whether it be a disturbing or disturbed planet. », the logarithm of a. a, the ratio of two mean distances, taken less than unity. R, the perturbative function. THE ORBIT OF URANUS. 7 a h, the coefficient of any term of — R, so that we have Vet : >; TM os N a m’ being here the mass of the disturbing planet. a, the mean distance of the planet from the node, or the mean value of v. w, the distance of the perihelion from the node. g, the mean anomaly. i, the mean longitude, or the mean value of v. a, the angle of eccentricity so that e = sin y. r,, the radius of the planet in the undisturbed ellipse. r;, the quotient of 7, divided by the mean distance, which is a function of the eccentricity and mean anomaly only. T, the time after the epoch 1850, Jan. 0, Greenwich mean noon, counted in Julian centuries, : : athe Be iC y, the integrating factors of the periodic terms, or the ratio y’ N being the change l of the angle in unit of time. u, the eccentric anomaly, and, in the tables, the argument of latitude. We have for the value of I m’ m'r : os Se a _ Se oe ee a ee cos Vv cos V-+sin ysin v’ cos 2 V -—2rr’(cosvcosv-sin vsinv'cosy)+r? 7 v D U or, if we suppose 7 replaced by its value in p, namely ce we shall have TUN Vis Oxo) With this value of 2 it is well known that the differential equations for the longi- tude and radius vector of a planet are CR Og sa) dt? dt? iP op Cv dr dv oR ENG = : dt? sig ap dt dt" dv sf (1) zs If we multiply the first of these equations by 2 sy and the second by 2 ee and ¢ add them together, putting, for brevity, OR do , oR dv ee 2 Op di ' Ov dt ae @) and then integrate, we shall have dr dv ue A+m) _ , Sete ge = (C+ f D. Rat) r 8 THE ORBIT OF URANUS. C being the arbitrary constant added to the integral. Adding this equation to the first of equations (1) we have 1 d(r") a ate e -+}- m) $4 Y ; ok ‘ je : = "(20+ 2f Di, Rat + ay (3) Let us now represent by 7, that elliptic value of + which satisfies the equation 1%) #A+™) _o¢ "o (y 7 a 7 Subtracting this equation from the last we have ia as) —nuwi-+ m) (= — eS ) ate ( 2f Dd Rdt + a ny Sie in which no constant is to be added to the integral, and both sides of the equation are of the order of the disturbing forces. As there is a decided advantage in taking the logarithm of the radius vector as the variable instead of r itself, we substitute for the latter its value T—=Cl, T =Ch and put op = p = (05 Then : ° A We Bee oye 2 poe ie eae Sl .) 6) tog te Cte (1+ 2% + +5 dp + ete B (7° — 7") = 79 dp + 79°dp? + ete. 1 1 é dp” ree ee Substituting these values in the above equation, carrying the development only to terms of the second order, and transposing those terms to the right hand side of the equation, and putting w’=u(1 +m), we find 2 (4 2 U cy) 2 fine Ne 2 Ee) + = Ge dp) = ( 2 Dat -f = ee d ey ) =e ~ 5 (4) an equation which gives the perturbations of radius vector. The general mode of solving this equation by successive approximation is familiar. The principles on which the successive approximations are made being the same, we shall begin by assuming that we have obtained first approximations to the values of dv, dv, dp, dp’, dy, and that from these we wish to pass to a second approximation. We must first carry this approximation into the functions of R in the second mem- ber of (4). To effect this we must show how, from the development of R in terms of the elements and the time, we may form its successive derivatives with respect to the quantities which enter into it. &, while originally a function of v, Vv, Ps Ps and y, is, in its developed form, a function of A, 2’, a, w, e, &, 2, , and y, the development being effected by substituting for the first set of quantities their values in terms of the second. ‘The substitution is as follows: v=A aE Fq, v=W4+ Fy, (5) Pe ae ae Ome ie THE ORBIT OF URANUS. 9 Fy being the equation of the centre, and gg the part of p depending on the eccen- tricity in the elliptic motion. It follows that if we express the developed expression for Rasa function of a, 2’, g, J, 2, 2, which we may do by putting we shall have by successive differentiation OR = OR ov Le oR OAT ie LOW ton Ove Ol OR Ov OR. “O42 (OV? «OA COV? OR a Ok op sek (6) On 6p 00 ~~ Op SO ee ORS out dp? dn op etc. ete. etc. and in general omtztm tw R onmtntm i Pp 6A" Ov" O2" 50" — Sv" dp" Ov" O™ Thus, by expressing the developed # in the above form, we may find the derivative of any order with respect to v, Vv, p and p’, by taking the corresponding derivative with respect to 2, 2’, » and w. The developed # is usually expressed in the form (Pesos oe cos (en! + in +t j'a! + jo) Q a, being the mean distance of the outer planet, whether disturbing or disturbed, and A a function of e, e, a, and y. Substituting for o its value in g, this equation will become mh Sate ODS He ap i R= Ecos (+5) 4 +(6-49)4—J'9 — Jy). Putting for brevity N=tv +2+ 70 + Jo, oR mh the formule (6) give == D = (+7) sin N Se E EG iy 008 (1) = oe oes = > m’— 4 cos N Op on and in general h n+n Outwtntn Pp 0 ra cos Ov" ov’ dp” op” i Se m (é +i) Cay) 2 March, 1873. On" Ov™ sin 10 THE ORBIT OF URANUS. The formation of the derivatives in the second member of this equation demands attention. In the analytic development of the perturbative function each value of h is composed of a series of terms each of the form EX A, E being a function of the eccentricities and mutual inclination, and A a function of a of the form (0) a BP $ Dartt OF 4 att! OOF 4 ete, pa 28+2n—1 O*6® 2) panes 8 Oa” (8) (0), (1), ete., being numerical coefficients connected with the coefficients V“ tabu- lated by Le Verrier, in Tome I of his Annales de Observatoire, by the relation ) vo ny) ( LO 8T en: and 6") being, as usual, the coefficient of cos i@ in the development of s 3? “) (1 — 2a cos @ + a”) in multiples of cos @, and x — 1 the sum of the exponents of the eccentricities in Z. It would have been much more convenient if in effecting this development the derivatives of b had been taken with respect to » instead of a. In fact the n},(%) derivative oe when expressed in terms of the derivatives with respect to » is of a the form 7b) YA Boag) Oa" =m On +n 2 a aan ete. == Ln re On" Therefore, when expressed in terms of the derivatives with ae to », A will be of the form 60 e oD =i aKO) paces Bee oe (coy? + Cy G+ (2 Se + ete.) oA OA from which the derivatives Ae a c tc., may be found with great facility. As in the actual developments of R which we possess, the values of A are given in the form (8), we must find the expression for the first two derivatives of its several terms with respect to », which we easily do by the application of the sym- ’ bolic formule D.z=aD. =a(Dz +a) ). Beginning with the case of s = 3, we have ob!) obo Ov 7 on OO9 be 2 Ue Be eq tat THE ORBLE Ok OU RAN UIS: ll 8 (a" cone n npr) n+1 (i) ce TO ee, Bf on 0a" Caw. 0 ( O7b® @ ) o” De n+1 2h n a B® Or F240 Oa re 42 ge t Ont la eRe pare | eaeatan ole wet), olen Sa rope) On consequently we have for the derivatives of A from formule es oA ve ob) 2 te aD —— (0) a g =e (1) (a5 3) JL (2) ( Qa? = Oras 3 © aa OPA 0b) se Ho) AG) 52h (0) Ol + a? ) 4 ale a + 8a? a lage oo) ete. The derivatives of A being formed in this way, those of are immediately deduced from the equations oh OA ee Ov = 2 dn” oh OA Sy pe “On 2H On" When s is equal to 3, A is of the form bo a {Ou + aya S 14 Qo oI fete. | The quantity within parentheses is of the same form with A, in the case of s If we represent it by A’ we shall have i tim DA OA’ 7 =a( Op +A) =a" — + A 07A OA’ CA ge At oy $8 er A’ being the same form with A, the derivatives ae and — will be of the form 7) (9), substituting 3 for the index $, and (0), (1), ete., for (0), (1), ete. In the case of s = § the derivatives are obtained in the same way, which is too simple to need elucidation. We have now to pass from the derivatives of h to those of onl the coefficients ay of the perturbative function. The form of these derivatives will depend not on whether the planet is disturbing or disturbed, but on whether it is an outer or 12 THE ORBIT OF URANUS. inner one. Let us then suppose for the present, that a and » refer to the inner planet, and put », for the logarithm of the mean distance of the outer one. We then have for the derivatives relatively to » h o” panied a, rT 1 Oh OD Phila on: and for the first derivative relatively to », using the symbolic notation, The symbols in the second member being distributive, we have by successive differentiation oo ke ; Gy sk as Foe a (Pn — DM The quantity A is a function of a, the ratio of the mean distances or of C?—%, C being the neperian base. Hence IDasle = ID. which substituted in the last equation gives . h es (10) Gaps > a= q, (De FIN h This formula gives for the first two derivatives h Ole Oy 1 h oh Ov, ——( a =) es “ 1 Os i aa ye) OL u or ae abt 25 ae) Substituting in the general formule (7) these expressions for the derivatives relatively to » and », we have expressions for the derivatives of # relatively to v, V, p, 9’, it being understood, however, that all the quantities are expressed in functions of the elements of elliptic motion. In order to compute the perturbations of the second order we must carry R and such of its derivatives as enter into the differential equations (1) to quantities of the first order with respect to the perturbations. Let us then represent by y,, Vv, Por P'or Yo, the elliptic values of v, Vv, p, p’, and y, which we have assumed in the first approximation to the perturbations, and by dv, dv’, etc., the quantities to be THE ORBIF OF URANUS. 13 added to v,, v)’, etc., to make the true values of v, v, etc., whether perturbations or corrections of the elements. We shall then have OB, OR, ate One ee LOR: sk = & dv + e dv + ——* §p + ap dp’ + a oy OR OR, 8 Cas = oat 2 not +3 ¢ ee 1: 2 oA i i OV, sono pie Fes oy (11) OR oh, oe OR OR, OR se v+———*_ $v’ —q~ 6p 4 = 6p Opdv. TF Op.0v', 1 6% ue 6p OP's eg Sate! The value of D’,R may be found either by equation (2), or by differentiating with respect to the time as introduced by the co-ordinates of the disturbed planet. When quantities of the first order only are considered the latter operation is very simple, but it is different when terms of the second order come in, because the true longitude of the planet is then expressed in terms not only of its own mean longi- tude, but also of the mean longitude of all the disturbing planets. The result can still be obtained in the same way by separating all the mean longitudes introduced by the co-ordinates of the disturbed planet from those introduced by the co-ordinates of the other until after the differentiation relatively to ¢. Let us now resume the equation (4), representing its second member by wQ, so that it becomes a a oe, where IR 1a? (7.7dp") =2)DRat + oe a as Opyaaiemenat: 4 = By the operations already given @ has become a known function of the time. It is well known that the integration of (12) may be effected by finding two values of 7,"dp which satisfy this equation when the second member is neglected, or, in other words, by finding two variables « and y which satisfy the equations ae , wl+m) a — 0 dt’ is Fo z : Py ud amy dt? a mae: when the required integral is 09 = — ae as : y xQdt — x fy Qdt } ; iG GE The above differential equations are satisfied by the rectangular co-ordinates of the planet in its assumed elliptic orbit. The position of the axes of co-ordinates being arbitrary we shall take the line of apsides for the axis of X, the perihelion eae on the positive side. If we put €) = sin, we have w (1 + m) cos p an 1 = = nay : = Vaud + m) cosy = 14 TEE ORB Oe UE eASN UES: Let us, for convenience, replace 2 and y by two other variables & and y connected with them by the equations . i= Ue y =ancosy. £ and y are then functions of the eccentricity and mean anomaly only, and may be developed according to the multiples of the latter. Substituting the last three expressions in the preceding value of 7,*dp it becomes an oD Ne ae eens &Qdt — N at. ry Op Tansee Efn@ If we put 7? for the value of 7, when the mean distance of the planet is put equal to unity, so that 7,, like & and y contains only the eccentricity and mean anomaly, we shall have fo | nJ EaQdt —Ef naQadt } (13) 1+m We must now express & and y in terms_of the time, or of the mean anomaly. Putting for the present w for the eccentric and v for the true anomaly, we have, by the theory of the elliptic motion, x2 =7 COS Vv = a (cos u — ee), y =rsinv =a cosy sin u, from which follow & = cos u—e, y= sin U. As & and y are to be expressed in the form == >, Coseg, n= 4 >q;sin ig, the finite integrals extending to all values of i from —cc to +cc, we shall deduce general expressions from p, and qg, arranged according to the power of the eccen- tricity. Since u=g+tesinu, we have by Lagrange’s theorem : e Osin’g é& 0’ sin* COS u = COs g —e sin’ g — nites I _ ete.; : : 2! og 3! Og? or Aree 2 ney Rae Vaeirr using the notation Cl res Meo) 855 oc n= Tia —— 1). We then have OM THE ORBIT OF URANUS. 15 Substituting in the general term of the above series for sin g its value in imaginary exponential functions 2sing=V—1 (ec 9¥—1— ¢9V=}) we find by the binomial theorem, using the notation of combinations, oc” Gi Dia 8) n! no PEAS ~ s!(n—s)! ~S) co} DH Ont) gine t! g — V A Yea —(n-1). 9 V1 C (ul) 7 Va C —~(n—3) 9 V—1 smi g=(Vv — c eric + Ce — eee n41 n+ ui —1) Cet)gV=1 —1)yF ems V1 \ (= DCE +1) Differentiating n — 1 times with respect to g, and putting together the first and last terms, the one after the first, and that before the last, and so on, we find qt Omastl cg Cgum: == (n 4. 1 \re (Gee! V1 a: cog a) 1 —— Fim =a C(n p= iD ee elt)! -—1 (om ame Ou —1 etc. io ( Zin Vai Substituting for the exponentials their values in circular functions, and dividing by 2”t? we have Om sing, ag —= — . | (n + 1)" cos (rn + 1) 9g — Cm —1)"" cos (n— 1) 2 + C(n — 3)" cos (n — 3) g — ete. \ n+l the series terminating at the last positive coefficient of g. Substituting this last value in the general term of the series which gives cos wu, we have n =oc cosu= > n=0 ! De e” . \ (n+ 1)" cos (n+ 1) g — C(n — 1)" cos (n — 1) g- ete. \ n+1 Let us now substitute for m another variable 7, putting in the first term of the last factor i= n- 1, in the second i =n —1, in the third i=n— 3, etc. The limits of finite integration with respect to 7 will then be in the first term, +1 to +oc, in the second term, —1 to +cc, in the third term, —3 to +o, etc. ete, But all the coefficients of g will then be 7, and the formula supposes the factor of cos 7g to vanish whenever 7 is zero or negative; whence, those elements of the finite integral in which i is negative must be omitted, and all the terms must be taken between the limits + 1 and + oc. Making the proposed substitution we have 16 THE ORBIT OF URANUS. i=e yi—2 “7 1 jit . —»> pee ae eo = ire mis cos U = \ (i—1)! 2°? oe gi Cee + apy Qs etc. \ COS 1g i= yi—2 ,i—1 Sag bees & We jet fs i=1 (—1) 127 {!- 2 EDL. SA 2iG+1) (+2) ay l. — €€ } cos ig We have, therefore, for all values of i different from zero a6 , a é —— — 1 —_— OPES tc. Eales eye | aa GLI) TD $05 ¥GL)G4+2043) } cS To obtain the value of p, we remark that the only constant term in cos wu arises from the term —esin?g; its value is therefore —$e. The constant term in & cos u — e¢ is therefore 3 e, whence Po = —e. (15) si—2 i—1 The values of g; may be obtained in a similar way by developing sin w by LaGrange’s theorem. But the development is rather more complex, and it is easier to derive them from p;. Let us take up the equations & =—cosu—e y = sin wu u—esinu=g Considering u, like £ and y, as a function of the independent variables e and g, we have by differentiation Ou O(esinw) ——a() 0e 0e _ Ou O(eé) ___—_ sina ioe a toe Yr eicose (a) Ou 1 6g 1—ecosu : cee ey le un eiey an ae Oe og Ou og og (0) Comparing (a) and (4) 0§ _ _ O(n) dg te 1 tee Putting in this equation for £ and y their developed values this equation becomes Sp, sintg —> aoe sin ig which gives by equating the coefficients of sin ig ='Sp, de. (16) THE ORBIT OF URANUS 17 The following are special values of p, and q,, developed to the sixth power of the eccentricities, as derived from the preceding formule : py = — de oa 7 Pig go 9016 Ee, eS GIG Burd an ted SORi'y, Peo 9g" FIDO" Th ccmege i 125, 4875, Ps 384° — 9216" 7 eel p tO ~ 16807 4 Pi = 46080 © (16y ie: 1 4 6 tong: aS) 9216. cP Dee See ae Bis) 2 are aero eS ei ae ae Vig 198 120" 1 4 OL oe a IQS) BOE ds — 384° — 9916" ie de = 80 ° __ 16807 |, t: = 46080 Having the developed £ and x in terms of time, let us resume the equation (13). As only purely linear operations are performed on @ in this equation, it follows that if we represent its several parts by Q,, @., etc., and by Opis dpa, etc., the values dp obtained by putting Y= Y,, @ = @,, etc., we shall have dp = dp: + dp2 + ete. We have, therefore, only to find the separate values of 7;°Sp corresponding to the different terms of Q, and to take their sum. Let us then represent, as before, by ™ hos (ta + ia + Jo’ + jo) ay any one term of &. 3 April, 1873. 18 THE OR BLL OR TURAN UES We then have, considering only terms of the first order with respect to the dis- turbing forces, ; mihn _. a mee sin N, ay JDR= __ mihy COeINE a ye oF =m being extended so as to include all values of j and 7’ which correspond to the given values of i and 7’, we shall have for the general terms of Q, mJ kecos (i+ ig) + h,sin (70-4 ig) a, or, when we represent the angle 7’ + ig by N, m™ Yk, cos N, + k, sin N, Os ales cos N, + k, sin This we are to combine with the values of £ and y 1S p, cos ig,* Esq singg, in the general integral formula (13). If we substitute them mm this formula, and represent by u the coefficient of ¢ in the value of N we shall have to integrate differentials of the form qa Ss i] — ae (m as ig) in which the coefficient of the time ¢ in the angle is 7+ in. Let us represent by y, the integrating factor n ena The formula (13) will become by these substitutions, which, though a little com- plex, offer no difficulty, ECT: | ska op = * eS 2 16 a, (1m ae! Pid X 4 Ykecos LN, (i+) )g]+h, sin IM YD ma ry —v_y| fhecos EN, +-(i—J)g]+hsin IN +i —J)gli + {r4;—0_1} [k, cos [N,—(i—j )g]+4, sin [M—(i—J) 9] | tivi—r} (h.cos[M—C+ 3914 sin [M—C 91} The sign > of finite integration here includes the separate combination of every value of ¢ with every value of j, except those combinations which make the Nese * The indices 7 and j, in these equations, are not to be confounded with the coefficients of 2 and in the general terms of & and @. We need not use the latter at present. 20 THE ORBIT OF URANUS. coefficient of the time under the sign sin or cos vanish, and so render the corre- sponding value of » infinite. These cases have to be treated separately. To find, from the expression, the coefficient of the sine or cosine of cosine N,+ ug in 7,°dp, we put, in the four lines of this equation, as follows: In the first, ttjsuos.g7= u—t; “ seconds t—jf=u.if= t=—uU; “« third, —i+tjg=u..g= uti; “ fourth, — i—yoeu .. g=—u—t. In the above expressions 7 and j being independent, and including all values from —oc to -+oc, i and w will also be independent, and include the same range of values. Substituting for 7 its value in w the coefficient of L many | [%. cos (N, + ug) + &, sin (N, + ug)] 16a, (1+ m) becomes Pi Tu—i) Ye) V; ) > + p; Vi—w : =, Vu) —_ + Pi Nuri) Yap Vi ) + Pi Iw Ca = Venti). Since q; = — q_, this expression reduces immediately to 25 | PiQe—y(% = —Va-o) + PiGi+w Maso— Pa )s or, substituting 7 — wu for ¢ in the second line a t 2> (P.:4G—w + Pu—wi) (1% oo Vie): Hence, writing NW instead of ,, A uareens : p=3 a(L pm) eh Peo Peo) (Vay) [h.cos( N--ug)-+-k,sin( N--ug)] (19) This expression fails for the particular case N= ug, where the value of »_, will be infinite. If we take each term of Q of the form , m : — (Kk cos ug + hk sin ug), ay and substitute in the general expression (13) it will be found that the terms in 7,*dp which have the infinite values of » as a factor are to be omitted, and replaced by ee me ny in Zp koo—E Sq, ky} (20) The two parts of 7,°sp thus found include all the terms of the first order with respect to the disturbing forces. But when terms of the second order are taken into account, we shall find terms in @ proceeding from secular variation in which the time appears as a factor, outside the signs sin and cos. Let us represent such of these terms as depend on any angle N by q=™ (k, cos N+ k, sin VN) DEH OR BIT OR UR A NUS: 21 and use the symbol »;, as before, to represent the ratio of the mean motion of the planet to the coefficient of ¢ in the angle N-+ig, so that if «’ represents the coefficients of ¢ in N we have v1) v= 7? n 1 Vv, SO 1 win ae — v n we find the expression Weed L a ' (7h, —vknt) cos (N + ig) + (v2,k,—v_knt) cos (N — ig) (2k, + v,knt) sin (N+ ig) + (v2, hk, + v_jhnt) sin (N — ig) } fx Qdt — : = a | ik (v7k, — v,knt) sin (N + jg) — (v2,k, — v_jkint) sin (N — jg) — (nfl ayant) 00s (N+ J) + (os + 9st) c08 (N— ji) If we now put for brevity vik, + vdegnt =e, v7k, — vt = 8, the general value of 7,°3p becomes 1 mar? Saale aden) Zs ( © —«@ )cos(N+C¢+7)9) + —s8 )sm(V+C¢+4+ 7) 9) ee +(e, —¢,)cos(N-+ @ —j) 9) + (65— 4 )sin(N+ G—JZ) 9) ) + 24) c05(N— (J) 9) + i= 4 )sin(V—G—J) 9) L + (¢..— c,) cos (N— (@ +-3) 9) + (8, — 8)sin (N— G+ 7) 9) If, as before, we transform this expression by putting in the first line g=u—v7; in the second “ Jj=t—uw4; in the third « g=t-+u; in the fourth “ jg=i—w: the value of 7,*sp reduces to 1 ma 8 alm) * ‘ fe Gu-) Cu —% )¢08 (N+ ug) + PG» (i— 8x1) Sin (N+ ug) } (21) = Peri (Cu 49 — C1) 608 (N + Ug) + Pur Si— Sug) 811 (NW xg) or, putting 7 — w for 7 in the last line, sl mon as a,(1+m) D2, PQe—yp — Pes (Cup — &x) COS (N+ ug) + (8; — $,_:) sin (N-++ ug)} 29 DE EVO RR Ble Oi SUeRrAGN) Uns: to which expression is to be added, in lieu of the terms which will have ixfinite values of » as a factor. ee © and i? being the factors of Ea nt cos ug and ie nt sin ug in the expression for Q. “ - The formule 19, 20, 21, and 22 give the complete expressions for the perturba- tions of the logarithm of radius vector by successively substituting in it all the terms of Q. Perturbations of Longitude. We now pass to the perturbations of longitude. In the Mécanique Céleste (Premiére Partie, Liv. ii. Chap. vi.), Laplace gives an equation (Y) by which the perturbations of longitude, which are of the first order, may be derived from those of the radius vector without the formation of any other derivatives of R than those which enter into Y. But the formula does not seem easily adapted to the case in which the perturbations of the second order are taken into account, we shall therefore derive all the perturbations of longitude from the second of equations (1). By integration this equation gives lv su 1 See “eo Te Sie ov wes \ C being the arbitrary constant of the integral. Representing, as before, by sub- script zeros the values of the co-ordinates corresponding to the ellipse to which the orbit is supposed to reduce itself when the disturbing forces vanish, we have dy _ancos) uC al, Tileae nk because the constant to which the integral must reduce itself in the elliptic motion 2 jg CeO lu v—v, = ov, we find Subtracting the last equation from the preceding, and putting div _ uw (OR 1 DV ay = Ge ae dt + (> a= =) a’n cos . Developing BL to terms of the second order with respect to the disturbing force = 1 1 fo (1 — 23p + 25p? — ete.), which, being substituted in the last equation by putting _ ent Cel +m’ N° — Oninas gives , adv an? OR nae = Lm 1 — 20) Sd —2n cosh (Sp— Sp), (23) which is rigorous to quantities of the second order. THE ORBIT OF URANUS. 23 The most convenient mode of making the numerical computation of the second order terms by means of this equation will depend upon circumstances. If the perturbations of longitude and radius vector of both planets are already known with a sufficient degree of approximation for the computation of formula (11), it will be more convenient to form at once the complete values of all the quantities which enter into the equations (12), (13), (19) to 22), and (23), so that no steps of the process shall have to be repeated. If such perturbations are not known, they must first be computed, and it will then be necessary to begin with the perturba- tions of the first order, and afterward add those of the second. There is, how- ever, one class of terms of the second order which it will be most convenient to take account of from the beginning, namely, those arising from the constant term in dp and dp’. This is effected by correcting the mean distances for an approximate value of these constants at the beginning of the computation, and then proceed- ing in the usual way. ‘This is in fact what we have supposed to be done in the preceding investigation. ‘The values of dy, év, dp, dp’ in formula (11) will then contain only periodic terms. In computing the terms of the first order we determine the value of dp from the equations (19) and (20), using the value of @, im (18). Then those of év are obtained by integrating the equation div an*rz”? (OR, a ; = = 9) 24 ema =e ~ dt — Qn Cos (24) Having found the values of dv and dp for both ae they are to be substituted in (11), to obtain SR, §© “and 5 oe But, rigorously, dv and év are not the op same with dv and dv’, owing to the movement of the orbits of the planets, and the corrections for dy are also to be added. Considering, for the present, only the perturbations of the second order, which depend on ¢v, dv’, dp, and dp’, we may use the following equation for ¢#, and similar ones for its derivatives: oR ae (i soi ay 2 by ee (25) Having thus found df, and hence D'sR by nee na and then § , we form the quantity p S OR 1d (7006p) , 1 69” ; 8Q=2S Di Radt +9 % Sera eo (26) which is the difference between the value of Q, in (18) and that of @ in (12). The terms in ép arising from 6@ are then to be computed by the formule (19), (20), (21), and (22), when we shall have dp accurate to quantities of the second order. Let us represent these additional terms by dp. Subtracting (24) multi- plied by 7,7 from (23), recollecting that the Jp which appears in the second term of the former is really de=—é’p, we find, neglecting quantities of the third order, rp” a 4 {se s© oT = O5p os a} — 2n cost (5p — 5p”) 24 THE ORBIT OF URANUS. from which the terms of dv of the second order are obtained by multiplying by ~ ry, and integrating. Motion of the Orbital Planes. The general theory of the motion of the planes of reference, especially of the motion of the instantaneous orbit, has been so often treated that I can scarcely hope to add anything essentially new to it. I shall, however, endeavor to pre- sent the differential equations of the motion in a simple and general form, and one in which the geometrical conceptions of the problem shall be made as clear as possible. The orbital plane of each planet being at each moment osculatory to that part of the orbit which the planet is actually describing, its only motion is one of rota- tion around the radius vector of the planet as an instantaneous axis. ‘This rota- tion may be resolved into two others around any pair of rectangular axes fixed in the moving plane. But the rotation produced by any one planet is most simply expressed when referred to axes, one of which coincides with the common node of the two orbits. The rotation produced by each separate planet must, therefore, be first referred to its node on the moving orbit, and then the combined rotations must be resolved into two around axes assumed at pleasure. To effect this, let us suppose positive rotation around an axis to be such that an observer looking from the origin along the positive direction of the axis sees the right hand side of the plane move downwards, and the left hand side upwards. Let us also denote the first axis in the order of longitude the principal axis, or that of X, and that 90° farther advanced the secondary axis, or that of Y. Let us now put dq, the instantaneous rotation around the axis of Y; dp, the instantaneous rotation around the axis of Y. Let us also put, relatively to any disturbing planet, dy, the instantaneous rotation around the ascending node of the disturbing planet on the orbit of the disturbed one. dk, that around the corresponding secondary axis. Then, from the known equations for the perturbations of the inclination and node of an orbit, we find, that, if any term of the perturbative function be repre- sented, as before, by mh —— cos (12+ 00 +-9'70'+ Jo), 1 the differential rotations 7 and / will be given by the equations dy mh an Neer: eee es : T= a, cong {Ct Deoty +44) cose } sin W dk man oh —- = — — - -cos NV. dt a, COS») Oy a As R is actually developed, the mutual inclination y docs not explicitly appear, but is replaced by o= sin by THE ORBIT OF URANUS. 25 Making this substitution, and putting also ifity+is— these equations become dn man th : : ; eee : j) oh + sin N di a, cosy cos 3 ger +3) oh | : : (27) dh __ man cos dy Oh Aes dt 2a,cosy Oa To pass to the general rotations dp and dq, let us represent by 6,, 6,, etc., the lon- gitudes of the ascending nodes of the several orbits of the disturbing planets on that of the disturbed planet. We shall then have dq. dy; Ssi dk; | eos 0 sin 6; rae (28) DDS cat ore y sin 0, ane he 9 : dt’ These equations completely define the instantaneous motion of the orbital plane. They cannot, however, be rigorously integrated in their present form because p and q as integrals have no completely defined signification. ‘To do this it is neces- sary to express the differential rotations dp, dq, etc., in terms of the differentials of any elements we may select to define the position of the orbital plane, and then to integrate the equations thus formed. But, for the purpose of constructing tables of the planets we may consider p, g, etc., to represent small rotations of the planes of which the powers and products may be neglected, and the integration is then quite simple. Perturbations of the second order depending on the motion of the orbital planes. R being a function of the five quantities of 7,7’, v, Vv, and y, the motion of the orbital planes introduces terms of the second order by changing the values of v, Vv, and y. ‘These terms we have hitherto neglected. ‘Io investigate them let us refer the rotations of both planes as given by (28) to the node of the disturbing on the disturbed planet as the principal axis. If we represent by dy, dk, dy’, and dk’ the rotations corresponding to this axis, and designate by the subscript 1, the quantities which refer to the disturbing planet whose action we are considering, and by 2, 3, etc., the other planets, the equations (28) will be replaced by these 1, : Tk dn =" 4S c0s(0,—0) & —S sin (0,— 0) 9,» . dt dt dk dh, ap dy; 3 s = ae; + ¥ cos (6; — 1) — 7 re sin (0; — 0,) - dé’ the summation commencing with i = 2. ° By formule of the same kind we are to find the differential rotations dz’ and dk of the orbit of the disturbing planet, produced by the action of all the planets. 4 April, 1873. 26 THE ORBIT OF URANUS. These rotations will be around the same principal axis with the rotations dy and dike, but around a secondary axis in the plane of the disturbing orbit, and therefore making an angle y with the secondary axis of the disturbed orbit. A geometrical construction will now show quite simply that the infinitesimal rotations dy, dh, dy, and dk’ will produce the following changes in v, V, and y. dv = cot ydk — cosec ydk’ dv’ = cosec ybk — cot ydbk’ (29) dy = bx'— bn If we substitute these values in the general formule (11) the terms of the second order added to d# will be OR oR ok — ( By Coby + 5,7 cosec y ok (30) OR oR a ( jy COsee ¥ -- a cot y) sid OR + (5, (Ox! — nr). Oy ( The first two terms of this expression may be put into the form Ook , OR OR OR \ 1 ( oe ob sr) (cosec y + cot vy) — 3 (s a eer ) (cosec y — cot y) \ ok (See OR oR = 1B ( oy + gy) Ceoseey + coty) + (5, —“yr) (Coser y — coty) } Bh But, cosec y + cot y = cot hy = © bY o -osec y — cot y = té 1 a oy = coseey — coby — tan i and in the general term of R, by (7) oR Mila” Pee a we ary oR mh Ok mh |. ") gin N. By ag TS) sin. Making these substitutions, and putting, as before, ifjtitj=— the above value of SR reduces to mh SR= Qa, {ccot 3 y (64 — 0k) + @+7—1 —7) tan 3 y (dk + d2’)} sin (31). + a aS (dy' — dn) cos N THE ORBIT OF URANUS: 27 The corresponding terms of § ou oR ; : and o-, , and may be obtained in the same way ov Z dy eG EKOU ER Ont ne, een . by substituting —— and —— for & in (30) and continuing the corresponding sub- OV 9) stitutions of the general terms of the derivatives of 2 as given on page 9. The equation (31), besides being of the second order with respect to the disturb- ing forces, is also of the second order with respect to the mutual inclinations. For ok, dk’, dy, and dy’ are of the first order with respect to both quantities, and, when- ever ¢ is not zero, / is a quantity of the second order, containing o? as a factor. It is, therefore, only in exceptional cases that the terms of the second order depend- ing on the motion of the orbital planes can become sensible. Reduction of the longitude in the orbit to lo gitude on the ecliptic. The integration of (23) gives a value of Sv, which, added to the longitude in orbit corresponding to the pure elliptic motion gives the longitude in the disturbed orbit, counted from a fixed point in the moving plane of that orbit. The position of this fixed point is completely determined by the condition that the instanta- neous rotation of the plane in question around the axis perpendicular to itself is always zero, so that the motion of the point of reference is always perpendicular to the direction of the plane. But, although this instantaneous rotation is zero, the integrated rotation is not rigorously zero when we consider the terms of the second order. It follows that the value of v, the longitude in orbit, and the position of the plane of the orbit do not rigorously determine the position of the planet: we must also know how the fixed point of reference has changed its position in con- sequence of the motions which the plane has undergone. Let us consider the relative positions of this plane at two epochs. If the fixed point were equally distant from the common node of the two planes, the integrated rotation of the plane around its own axis would be zero. But, these distances not being equal, their difference is a correction to be applied to the longitude of the planet in its orbit. Suppose, now, that at the end of any time the inclination of the actual orbit to the primitive orbit is @, and the distance of its ascending node from the present position of the moving axis of x is @. A rotation around the line of nodes will not change the quantity sought. But, if we represent the infinitesimal rota- tion around an axis perpendicular to it by dr we shall have cos § dp — sin 0 dg =dr, dq and dk being the instantaneous rotations around the respective axes of # and y. By this rotation it is easy to see that the relative distance of any two fixed points, one on each plane, from the node, will be altered by the quantity, dr (cosee @ — cot ) = dr tan § 9, the relative longitude of the fixed point on the moving plane being increased by this amount. The correction to the longitude in orbit from this cause is, therefore, di =dr tan 4 ¢ = tan $ ¢ (cos 6 dp — sin 6 dg). 28 THEO} RIB LO TU ERZACNGURSE Counting the integrated values of p and q in a direction perpendicular to the moving plane we have : tan p sing tan p tan cosy. —— q tan p which, being substituted in the expression for dl, gives i eer (tan gdp — tan pdq). The approximate value of the integrated correction is therefore 1 ol = 9 fat — pdq). (82) For every pair of periodic terms in p and q, such as g=s sin pl, p==s cos ut, é7 will contain the secular term —— § s? wf, which will be confounded with the mean motion, and, if it were not so confounded, would in few or none of the larger planets amount to a second in a thousand years. If the secular terms in p and q be d= stp — sb 67 will vanish. We hence conclude that these terms are entirely unimportant in the present state of astronomy, and that, if we consider the positions of the plane of the orbit at two epochs, we may consider the points of departure in them to be equally distant from their common node. We have therefore only to consider the motion of the inclination and node due to the change of the position of the orbit and of the ecliptic. If we put $, the inclination of the orbit of the planet to the ecliptic, 6, the longitude of its node counted on the ecliptic, z, the longitude of the same node counted from the same fixed point in the moving plane of the orbit from which v is counted, Then, the longitude of the planet on the ecliptic, or Z, will be given by the equation tan (L — 0) = cos ¢ tan (v — 7), or, when developed in powers of @, L=v+4—r4 D, (33) where D is the reduction to the ecliptic, the value of which is D = — tan’? @ sin 2 (v—z) + $ tan’ § ¢ sin 4 (v — 7) — ete. Let us refer the instantaneous rotations of the orbit and of the ecliptic to the fixed points of reference in the two planes; g being the rotation around an axis passing through the sun and the fixed point, and p that around an axis in 90° greater longitude, and the accented quantities referring to the ecliptic. We then have THE ORBIT OF URANUS. 29 oo — cost a sin t = i dy ap cos a —sin @ dt S a = cosec (—sine“ TF 7 + cost - :) (34) + cot o( sin“ ——— ya F) ee = cot hee ag ++ cos “i + cosec@ ( (cos t ép — sinc 6g) the terms dependent on dp’ and éq’ being omitted because, being purely secular, they may be included in the mean values of @ andz. Substituting in the expres- sion for 63 cos 353 = cos @ {sin v ég — cos v ep}. (37) In the case of all the larger planets both cos 3 and cos @ may here be put equal to unity, when the expression for §3 will become Sy, q bY 03 = sin v dg — cos v ep. (38) To develop this expression in purely periodic terms we must substitute for v its value in terms of the mean longitude or mean anomaly, namely, v=1+ desing + 7e sin Qy + ete. ; suppose the terms of dp and ég depending on any argument, WN to be dp = —a,sin N—a,cos N (39) dg= a,sinN-+a’,cos N and put a for the longitude of the perihelion, so that l=xa+g then, to terms of the first order with respect to the eccentricities, we have 62 = —e(a, cosa + a’, sin z) sin N—e(a, cosa + a’, sin 2) cos N f(a, +a’ : cosa + (a, —a,) sium} sin(N-+ gq) }(a, —a’,) cosa + (a’, + a,) sna} cos(N-+ gq) ( a’.) cosa + (a’, + a,) sina} sn(.V— gq) i (a, 4 a’,) cos + (a, — a,) sinz} cos(N— gq) } (a, + a’,) cosa + (a’, —a,) sina} sin (N+ 29) U( ) ( ) ) ( ) ) ( (40) , a,— a, 2 Wane ar aeer se WI WI WI Wi I= I= l= cos 2 + (a’, + a,) sinz} cos (N + 29) 3 €)(a, —a@’,) cosa + (a’, + a,) sin 2} sin (N — 29) He, + a’,) cosa + (a', —a,) sinz} cos (N — 29) The point of the orbit from which a and v are counted is entirely arbitrary, and, in considering the action of but a single planet, it will be most convenient to count them from the common node, in which case 1 must be replaced by o, and dp and dg by d& and dy. Thus, deducing the perturbations of the latitude imme- diately from the formule (27), we shall have ce 63 = sin v dy — cos v Ok. THE ORBIT OF URANUS. 51 ChHPAVE an K: APPLICATION OF THE PRECEDING METHOD TO THE COMPUTATION OF THE PERTURBATIONS OF URANUS BY SATURN. Data of Computation. THE elements of Uranus, adopted in this computation, were deduced from the comparison of nine normal heliocentric longitudes at intervals of 3697 days extend- ing from 1781, December 26, to 1862, December 18, with corresponding provisional places derived from the elements given in the “ Investigation of the Orbit of Nep- tune,” with perturbations produced by Jupiter, Saturn, and Neptune. As the perturbations are to be entirely re-computed, and the elements to be re-corrected from more extended series of observations, all the details of this first approxima- tion will be omitted. The resulting elements of Uranus are given in the follow- ing table, together with the adopted elements of Saturn, which are nearly the same as those employed in the theory of Neptune, except that the inclination and lon- gitude of the node have been corrected to agree with observations :— Elements IT. of Uranus. Elements I. of Saturn. m4 GSS Ge ole 90° 4’ 0” E Dass sy HBO) 14 48 45.0 6 13 1 58 1220 0 co) 0 46 20 A 48) BY 4 e 0469276 0560050 e in seconds, 9679.5 11551.9 n 15426.10 43996.13 m i == = 1 21000 3901.6 In computing the perturbations of the radius vector, one of the largest terms will be a constant. To avoid the necessity of computing separately the perturba- tions of the second order, which depend on this constant, we shall include an approximate value of it in the mean distance. This approximate value is, in the action of an outer or an inner planet, 5 log a = — 4 m' MoD, 6%). In the action of an inner or an outer planet, § log a’ = +1 mM (6) +aD.6"). M being the modulus of the system of logarithms. Using the values of 6? anda D, 6), which are found in different works relating to Celestial Mechanics, we find that the different planets produce the following changes in 6 log a, the units being those of the seventh place of decimals: — 32 The uncorrected mean distance is deduced from the mean motion by the rela- tion VEER OMB Action of Venus, oe Earth, oe Mars, x Jupiter, o Saturn, 33 Uranus, ae Neptune, ce Sum, 6 log a We thus have The following functions of the elements are derived from the preceding ele- Uncorrected mean dist. (log) Action of the planets Corrected log a On Saturn. 4 22 + 24 J 8 -+-10865 Js 235 =e 59 -L 0010870 + 0001812 o—_tad+m™) ar nr ‘ OF URANUS. On Uranus. 22 + 25 + 3 +8780 43081 + 0011792 + .0001965 Saturn. Uranus. 0.979496 1.282901 Teel a 7 0.979677 1.283098 ments by well known formule :— The following functions of a, necessary in computing the coefficients A, are derived from Runkle’s Tables, published by the Smithsonian Institution:— y (mutual inclination) eee eee log sin § y =o 1 log cos § y pena ae 8.232373 9.9999367 ¢ (long. of ascending node of Saturn on Uranus) 126° 44’ 51” o (’ wo! —- @ = (0) log sin (@) log cos (a) sin 2(@) cos 2(a) a 1 — a 41 31 40 323 18 21 281 46 41 —9.990759 +9.309888 —0.39966 — 0.91667 0.497249 4.04438 s. OMAHA TKH WWrH OS THE ORBIT OF URANUS. ; 3 n Jnr pli) Vulues of a" D® b?, be aDab? a DY a Dh) as Ds bi ol i Bo 2.14447 0.33969 0.5878 1.081 3.44 13.6 0.55207 68314 .4990 id 3.40 13.8 0.20836 47198 .7396 1.152 3.99 13.9 0.08687 .28491 -7123 1.463 3.68 14.5 0.03793 .16270 9632 1.596 4.30 15.1 0.01702 O90 L0 3998 1.485 4.87 16.9 0.00777 04896 £2653 1.231 4.98 19.1 0.00359 02624 .1682 0.940 4.60 20.5 0.00168 .01392 1022 0.679 3.91 20.4 0.00079 0.00733 0.0615 0.463 3.11 18.8 Derivutives with respect to (loga =) of a® D® b?. i — nO 1 2 3 4 Dob® Do(aDab) Dela? D2?) Do(a®D3,%) Dela D459 0.33969 0.9275 2.257 6.68 27.4 .68314 1.1821 2.175 6.93 27.4 47198 1.2116 2.631 7.05 28.3 -28491 0.9972 2.888 8.07 29.2 16270 0.7259 2.722 9.09 32.3 09010 0.4599 2.285 9.35 36.4 04896 0.3143 1.762 8.67 39.0 02624 0.1944 1.276 7.41 08.9 .01392 0.1161 0.879 5.93 36.0 0.00733 0.0688 0.586 4.50 31.2 Second derivatives, i D2be Di(aDob?) D3(a2D2b) D(a? D3?) 0 0.9275 3.184 eS) 47.4 1 1.1821 3.357 11.28 48.2 2 1.2116 3.843 12.31 49.4 3 0.9972 3.885 13.85 53.4 4 0.7259 3.448 14.53 59.6 5 0.4899 2.775 13.90 64.4 6 0.3143 2.076 P19 65.0 Values of a Db) i ab? aD? a DIY at Dive 0 1.865 2.674 8.104 30.8 1 1.267 2.844 Teal 30.8 2 0.761 2.412 7.63 29:9 3 0.433 1.790 6.92 28.7 4 0.240 1.224 5.73 26.8 5 0.130 0.792 4.41 23.5 6 0.070 0.493 3.20 19.6 April, 1873. 33 34 THE ORBIT OF URANUS. Derivatives with respect to (log a = ».) i Dx(ab’?) Dv(a*Dab'}) — De(a* Dab’) i IY =— JOB) —— ey oh 0 4.539 13.452 By). ] 4A 111 13.46 o4.1 2 3.173 12.45 02.8 3 D2 10.50 49.5 4 1.464 8.18 44.0 5 0.922 5.99 36.7 6 0.563 4.19 29.2 Second derivutives. i a,DoB” a, Do BY 20, Do BY 0 4.539 7S) 82.0 1 4.111 Via7 81.0 2 3.173 15.62 Titel 3 2,223 WE TPZ 70.5 4 1.464 9.64 60.4 5) 0.922 6.91 48.7 6 0.563 4.75 37.6 a,E° a, E® 2a,E9 6a,E2 aDoE® a,DoE® a,DoE® if) 1.267 4.111 15.46 54.1 4.111 WI) 7 81.0 1 1.313 3.856 12.95 54.0 3.856 16.80 719.8 2 0.850 3.167 11.98 51.8 3.167 15.15 75.8 3 0.500 2.318 10.31 48.4 2.318 12.63 69.0 4 0.281 1.573 8,24 43.1 1.573 9.82 59.6 5) 0.155 1.014 6.18 36.6 1.014 7.20 49.0 The notation B® and EF? is that of Le Verrier in his development in the first volume of “Annales de ( Observitoire Imperial de Paris.” Numericil expression of R and its derivutives. We next proceed to the computation of the coefficients A and their derivatives. As an example of the most convenient form of computation we present in full that of the coefficient of ”“ cos (i2/—(i—1)A—o) in the expression of R for the action a of Saturn on Uranus. In this computation I use the tables given by Le Verrier in his “Annales de 0 Observatoire,” tome i, pages 358-383, comparing the develop- ment with that of Professor Peirce in the Astronomical Journal, vol. i, as a control. i u (@) (1) x=Dab) Aa, (50)(2) a, (50)() (00) (1) XaDa (2) Xa? Dt (3) Xa3D8z Aa,(51) a,(51) (0) x5 (1) XaDa (2) Xa2D%a (3) X30 Aa,(52) a,(52) (0) xaE,@ QQ) xq, 4,0 Aa, (60) a,(60) ze X(50) 18x (51) 4 ee? (52 1 e€?X (60) THE ORBIT OF URANUS. 35 Oe — le | 5} =o =I 0 +1 2 3 4 5 6 +4 3 2 1 0 =I =3 =3 af 5 +0.5212, 0.8334 + 1.1041, 0 —1.10414 —0.83344 —0.5212 —0.3034 —0.1702 —0.0929 —0.2849 —0.4720 — 0.6831 —0.3397 —0.68314 —0.47198|—0.2849 —0.1627 —0.0901/—0.0490) | | Bee eRe GSS) net ae ses Ae ght | fn eal eee Pat sae i} f lis apa | | +0.2363 40.3614 —15.7565 —0.3397 —0.78728 —1.30542|—0.8061 —0.4661 —0.2603|—0.1419 —13.55] —11.25] — 5.52 0.00} 0.00 | + 2.92 |4 5.73 |4 6.83 | + 6.46 + 8.54] + 7.78] + 4.78] +0.51 0.00 | + 1.18 |} 2.56 |+ 3.25 | 4+ 3.06 | + 1.43] + 0.74) — 0.00; —0.59 | — 1.00 | — 2.92 |— 2.85 |— 2.82 | — 2.40 — 0.73] — 0.58| — 0:59| —0.54 | — 0.59 | — 0.58 |— 0.73 |— 0.80 | — 0.74 a ieee DA ec ene ee eal eee || Maree — 4.31] — 3.31| + 47.20! —0.62 | — 1.59 | + 1.30 |4 4.71 |4 6.46 | + 6.38 —18.76| —13.33] — 4.42) 0.00 | + 4.42 | +13.33 |+-18.76 |4-19.41 | +-17.01 | | +13.10| +10.38] + 4.10) —0.68 | — 1.37 | + 2.83 |4 6.27 |+ 7.48 | + 7.02 + 1.43 0.00) — 1.00 —2.35 | — 3.00 ffs 592 |= ye) |——nGaiion |i e60 e465) 1-18 1-08) == 1.18) |) — 11d) |= 1-461 1-60)))— Te4e ! | eta | + 32.36 ees ete dP aviiee a aka) | = — — | — 5.69| — 4.10| + 29.86] —4.11 | — 1.13 | + 9.09 |+-16.44 |4-18.53 | +-16.95| | —3.00 | — 3.40 — 2.63 0.00 | + 263 | + 340 |4 3.00 |4 2.25 | 4 1.55 eer Sa Set 3:86) 400 | 3.86 |e Sly = 232) (4 1.57 |) 4 101 eeonlll eee + 16.18... Poe EE all ees on ieee | 3 as : —0.68 | — 0.23) 4 17.43) +411-| + 6:49 | + 6:57 |e 5.32 |4 3.82'| + 2:56 | +6.61'| 410.12] —441.22] —9.51 | —50.049) —36.556 —29.573|—13.05 7.28] —3.97 | —0.09 | — 0.07] + 1.04, —0.01 | — 0.035] + 0.028'+ 0.103/4 0.14 | + 0.14| | —0.08 | — 0.06] + 0.45, —0.06 | — 0.017| 4+ 0.138\4 0.248|4+ 0.28 | + 0.26] | —0.05 | — 0.02, + 0.14 40.03 | + 0.051) + 0.0524 0.043|4 0.03 | + 0.02 | | | | a = +6.39 | + 9.97| —439.59 —9.55 | —50.050, —36.338 —22 179|12.60 | — 6.86] —3.60 | | The derivatives of / with respect to (log. a =») are computed in precisely the same way by simply substituting for 6), aDab’?, to » as given in the above table of constants. The quantities Aq, (50), ete., which appear in the third series of terms above express that part of the perturbations of Uranus caused by the action of Saturn etc., their derivatives with respect + In units of the third place of decimals. 36 THE ORBIT OF URANUS. on the sun, They are each of the form NV x a, N being a numerical coefficient given by Le Verrier under the coefficient for each term, The derivative of this expression with respect to » is — 2N x a~*, so that for the corresponding terms in D,h and Dzh we have ADzh = — 2Ah AD:h = + 44h The values of A and its derivatives, corresponding to any one argument 7 and 2, are to be combined into two terms depending the one on the cosine, the other on the sine of the argument. Let us represent by g the mean anomaly of Uranus, and let us put / for the mean longitude of Saturn counted from the perihelion of Uranus, or, more exactly, for the arc 2’—o. Put also N=ig+i, Geo Sei =tls P=jot+jo, P=jot+ yo. Then, for each value of N there will be several values of P corresponding to dif- ferent powers and products of the eccentricities and inclinations in h. Distin- guishing these values and the corresponding values of A by subscript numerals, we shall have a series of terms of & of the following form-— h,cos(N-+ P") m + h,cos(N-+ P’,) PGA |) Sey nconres 725) | + ete. etc. | and by putting h,= h,cos P+ h, cos P’, +h, cos P, + ete. h,=—h, sn P| — fh, sin P’; — ha sin PP’, — ete. (41) The above terms may be condensed into m he te fk =—h, cos N+ —h, sin N, a, ay which are of the form supposed in the preceding theory. In order that the derivative of R, with respect to the true longitude of Uranus, may be expressed in the form ) o = = v, sin VN + = v, cos N we must, by (7), put %, = — (t+ 7) h, cos P’, — (6 +4.) h, cos P’, — ete. (42) v= — ((+ 4) h, sin P’, — (6+ 72) A, sin Py — ete. Jas Joy Tepresenting the several values of j in the different terms which correspond to one and the same set of values of 7 and 7, TEE ORB IME Os UERVAUN UES oT ' To obtain the derivative with respect to y we notice that all the appreciable terms in the different values of h, which depend upon the mutual inclination, are of the form On Ay mee ; ee eee where og =sin}y. ‘These equations give Ooh ohodc : ae ae a = Ao cosy = 2 Asiny. Oy Oady Consequently — = ,dsiny cos(N-+ P). and the various terms depending on the same argument (i’,7) may be condensed into two, exactly as in the case of F itself, The different co-efficients 4 and D.h, computed in the way already described, are given in extenso in the following table. At the top of each individual column is given the value of P, or of jo +-7’0’, corresponding to the values of 4 below, and imme- diately under P is given its modified value, or P’, to be used in condensing the terms, putting for brevity (o) =o —w. Pand F’ are therefore regarded as constant angles the numerical values of the sines and cosines of which may be obtained from the values of @ and w’ already given. The condensed h, and hf, are given in the two right hand columns. All the numbers are given in units of the third place of decimals. VALUES OF h. tO 497% a Lee ans —3481. == 348i ae as + 205. 9. + 204. — 10.29 84.9 . se aRa ENG 5 =e 36. + 0.90 15.8 AE Ors h, = iL == 0551 — 93.30 +285.20 + 48.15 — 47.138 + 30.63 + 17.96 + 9.98 + 5.38 crea | Eex03 — (Nv! i—.03 |—.016 3\—.007 0 0 0 co bo bo piel ecocooessosoo™ PeeINS, ee SSO So) fete ete ll [el = 38 THE ORBIT OF VALUES OF h. URANUS. ! h —— Od) — (») 0.83 Held) 5.290 5.458 2.74 0 Qw f+ 27 65 .03 16 h +0.11 “LOS (oD =E018 +0.111 40.061 +0.04 10.02 SOHO , —wa 2a , =O in || | a Q Mi 5 30—wa] Qu h — alte == (Ni ——246 7160 —. 847 eT} Oi TD > bo ett \lIrDowwrrmorocos || # BO SD bO bo CO S| h maT .02 08 lily .122 107 | 086 |) —.06 h ea | — 02 }4b.0il (0) |b. _—.016 |++.029 (DL) JL OPAL —.008 |+.014 —.005 |+.009 OR eexO) h +.02 |_ —©) ' 9 9./ (4) 2) | = 4S 2) el! ¢ 9 74( 0) |) <= OY) 3(w) ) ——> == 2(@)) ie AD =) w h —— 4: —, (0s) ==, 11(0}5) let —.106 —.090 h a2 on (ee —— (NY =—048 —.044 —.036 h +.04 +.08 4.106 4.112 +.1038 +.086 VALUES OF Doh. h 0 +.01 4.008 4.008 +.007 +4..006 ! (0) Bi hyalac3 +8749.59 + 467.72 4+ 277.00 + 153.87 4+ 82.12 Dyh 3.18 3.24 159 40 075 2.69 .83 5 aR .62 90 171.51 8749.12 472.46 278.25 155.46 83.54 + Lf -+ + + -+- THE ORBIT OF UR AN US: VALUES OF Dh. 39 9 +(e) ' o — sw | Doh — 9.28 — 9.74 — 4.63 —956.40 + 30.16 oll Doh | Stesiige4 + 18.5 | 4907.62 id Orally 71.56 86.47 74.60 54.9 36.9 23.2 Doh 06 08 .09 alts) 26 .63 29 .68 90 69.86 84.30 73.49 53.9 36.7 23.4 ' — (> ra —(+) Doh =—0866; —64.55 2.70 — 8.78 —15.870 —18.93 ——3. 07 50 115 444444444 7.232 7.85 6.99 5.59 Doh 40.47 10.60 43.02 10.60 40.463 40.32 40.21 40.13 10.09 H+ | + | 9 t 20 o | — (») Doh | oy | —o 30 —oa'| + Qa Dah eDah 22 | +. 08 | 09 09 08 07 .05 Doh, == —0.08 —(0.59 —0.84 —1.38 ce —l. The values of D:h, needed in computing the perturbations of the second order with respect to the masses being obtained in the same way, by the simple substitu- tion of the second derivatives of the functions 6,aDab, ete., for those functions themselves in the expressions for /, it is not necessary to present the details of the computation. After obtaining h and its derivatives, it will be found convenient to change the arrangement of the terms. the sum of the indices are a constant. Hitherto we have kept in one series those in which Now, we shall put together all those in which 40 THE ORBIT OF URANUS. the index of the disturbing planet has the same value, arranging the individual terms of each series according to the index of the disturbed planet. ‘Thus, the index of the product of any term, as 2 cos N, by any multiple of the mean anomaly of the disturbed planet, as jy, will be found in the same series with that of N itself, and j lines above and below. The next process will be the formations of the required functions of the mean . dv dp @ eae ; ; anomaly of Uranus, de de® log ry. ‘Their values are as follows :— pe fat doy ass dv ay et sng ndt ila 1.001103 +.0005507 +.093933 cos g —.0468889 cos g -+.0468889 sin g +.0938294 cos g -+.005507 cos 2g —.0016494 cos 2g = +-.0032988 sin 2g -+-.0055012 cos 2g +.000336 cos 3g —.0000732 cos 387 +.0002196 sin 8g -++.0003357 cos 3g +.000020 cos 4g —.0000035 cos 4g -+.0000142 sin 4g -+.0000206 cos 49 Considering only those terms which are of the first order, the value of D’,.R may be found in two ways, the agreement of which will afford a check upon the entire development of the perturbative function, and upon the computations of & and oO OV tained in the mean anomaly of a single planet, whereby each term in & of the form These are (1) by direct differentiation, with respect to the time as con, R=" hceosN a will produce in DA the term R= ™ inh sin N- ay and (2) by forming the expression OR dv, Fe Ne SG : ov dt is dp dt As several “mechanical multiplications,” like those indicated in this last 2 ’ : expression, are to be performed, the following example of the form of com- putation is presented. It exhibits the formation of the product of those terms IR. . aes Iv ° of 2" in which 7’ = — 1 by SLO OV dt i eo 4 0 a 2 3 4 5 a & (sin) -$.02 + 1.68 —287.49 +8481.14 4.54.49 4 6.988 +1119) eeeoeme m Uv ents 0 0 0 + 08 <= 13848 16832 = 21558 4 0;3ameenmme ye ta08 1948 2-169:39 35 Welsh) eeolss) Pe uosn hone 0 qnaren, eae 0 0 0.00 — 0.79 + 9.576 -FO.I5 stem A CZ AON Set NOMS) temo? 0 0 0 0 x .000168 +.58 + 0.01 0 0 0 030 = 1048) 2 ig a a, OR dv 1 = ip Cle — 296 — 123-87 =-3470123 -Eo17-28 E1925 ono dons mn OV G DEER ORBIT OF URANUS: 41 The multipliers on the left are each one-half the coefficient cos jy in the ex. 6 aU : : pression for dt? and each *product is placed in the two columns corresponding respectively to V+ yy and N — jg. All the derivations of J, necessary in the computation of the perturbations of the first order are given in the following tables. First we have the values of Df obtained by direct differentiation, as indicated in the preceding formule. 5 cay Next we have ae and = , obtained by the formule (7) and (42). ‘The products ov Oo j Iv, 3} 7) ; ; 5 ; ‘ a by = and of o by f, being formed in the simple way just pointed out, and with the values of the component factors just given, their sum is next shown. This sum should agree accurately with DR. ‘The discrepancies are shown in the next two columns. The only apparently large discrepancy is found in the argu- ment 5g—5/. It probably arises from the incompleteness of the computation of R 5 OR ; : and a far as they depend on this argument. As the entire term does not Vv amount to 0’.01, I have not sought to correct it. The great value of this check arises from the fact that it gives a complete con- trol of the correctness of the development of the perturbative function, ab initio, since the two valves of DJ? are derived from different terms of that development. It also controls all the computations except that of oi This quantity being Op multiplied by quantities of the order of the eccentricities in the second value of IR, an error in its value will produce a discrepancy of only j1, its own amount in DR, and may therefore be overlooked. The derivative in question must there- fore be checked by a complete duplicate computation. In the column next following are given the integrating factors », for which the expression is n 1 vn in an 7 nN, vi For each value of 7 the values of » are therefore the reciprocals of a series of num- bers in arithmetical progression, the common difference being unity, 6 April, 1873. THE ORBIT OF URANUS: , m! OR mm’ Ook m! Dear Oy = rae FG, Gy sin cos sin cos. cos sin Onno 0 0 a ayene + 0.487} —1244.31 tiie 1, — 48.15 4+ 48.96 | + 10.20 | + 49.07 | — 63.52 | —118.99 2, ——- O64 + 8.28 -L 4.48 + 3.20 + 3.42 — 10,14 3, 4+ 0.45 AY 0,64 |) = 105507 | —=) (0:089| 10sec 4, + 0.16 0 + 0.06 | — 0.06 | + £0.10 = 0.01 ll 0 + 0.68 + 0.02 |— 0.04} + 0.08 | — 0:05 sel Se) 003 1 -o.05. || SEA aves] Se isa |p Seon Somes 0, 0 0 —— 287.42 Onno + 276.62 + 34.79 1 +3481.42 — 3: +3481.14 = ie —5267.70 + 4.35 O46 + 94.28 + 71.06 + 54.42 + 71.23 — 200.44 ORS 3, a: 5.74 | 42 15/65 - | 26 6.988) =e 9i4gor = 2800 ean 4, A 300) se 66 ise Res Pee oreo et sie. ties ieee Be a= 015 + 0.15 + 0.10 0 =P Onin —— nn OSOIR se) Se “Ont #006 | O81 te Oe SE soit Pe oem 0, 0 0 S658" Fe soeon = 6 Se aneMesibos iE + 93.30 —430.35 + 96.81 —430.37 — 87.14 —458.30 2, = 403:238 — 20.58 — 410.42 == MOL — 676.60 — 1.76 3; — 91.89 + 64.32 — 57.01 — 64.98 — 98.97 — 94.93 4, =) “5:46 18:30) je das: 1 2 o9G ==) Gone, =alores 5, a 0788) |) Se ord) BE Sop ee coe salon eens B, 4+ 0,27 St OT Se 0L20 y= t0x02 |) 2) ore | eon pees == 38729 |) 2108 | == 8867) Sy o0 |e 240502" = aia 2, ue 102) 3) ais'so |i brs || Seo araallee tere ed cual meso omnes 34, = Wy teail + 2.37 — 254.15 — 0.02 — 363.02 — 0.43 4, S74 |) BOLO | = pode Oe n= 2990) 5 trosee) wlan Gets a — 9.65 + 16.95 —— 0223 + 13.0 — 1.82 ——= LSA0M 6, + 0.87 + 2.76 + 1.29 + 1.5 = EAS ener: 1G + 0.34 + 0.24 + 0.27 0 JE Ose as O07 4. + 3.03 + 18.78 Se alee! + 18.78 | + 6.383 | — 25.296 4, — 144.96 = 3.60 == Wels + 1.5 —— LOO eh eoi De — 49.90 + 35.00 — 38.64 + 34.58 — 52.17 — 43.7 6, i850 |) ME seG aN BONO) alee |e Otel ema ih 4° 0075. | 42 2265: ieee ae | Seeger Sarg eeemons 8, 2 0:35 |) 4 0726" ee 408s) | Seon @ 1S corso acne — —— 79205 + 4.05 —= 73,9) + 92.4 ——| 9993 =——) ean 6, = 88.8 + 22.7 = 3,8 ++ 22.3 == 8Boe — 27.2 1, zeae or 4+ 10.4 me Pe Peel oe nc ees fee Ii! 8, O07 == 12°3 + 0.9 ae ales) Je lel let 9, a 083 + 0.3 S083 + 0.1 + 0.3 = (Orn THE ORBIT OF URANUS. 43 ee ok dv, OR dp, oe igs mM cena: Discrepancy. v k. k, — feet J sin cos sin cos 0 ORO 0 —.01 sents —1244.3 — 48.05 + 48.99 +.10 —.04 +1.0 + 32.78 — 20.90 — ONG S JE RO |] Spill! —.01 SLR + 4.06 | — 1.86 =—— 0:45 | -- 0.65 0 +-01 # + 0.44 | + 0.08 + 0.11 | — 0.03 | —.05 | —.03 +0.25 == 0103) 12) 0:02) | —.03' || —.06' |) =0:206098] == 0108 | — 0108 — 0.05 + 2.16 AN) +.01 —0.259601 | + 23 == 108 — 0.06 | + 0.09 | —.06 | +.09 | —0.350693 | + 276.62 | + 34.79 +3481.41 a= IES —.01 0 —0.539942 | —1508.18 ae bits See 942 90) + 71.06 | —.08 0 —1.173630 | + 20.85 —286.59 - 5.78 | + 15.63 | +-04 | —02 | +6.75940 | — v9.56 | 194.17 eee Lo | = a6: |) —.03 0 -+0.871126 | — 1.10 | + 1.79 sees Ould ee ORI INO | 048 -POde5G Eee 0503 (Neen Oats + 0.16 | * 0.10 | —01 +.04 | —0.149162} + 0.19 |— 0.13 BEN LOLOW || =) 0L0T +) e02 +.07 | —0.175312 | + 6.34 | + 31.39 + 93.24 4305310 |iaen00 —.02 —0.212580 | — 47.47 — 275.3: — 408.28 — 20.60 0 —.02 —0.269971 | — 897.05 + 3.35 — 91.90 + 64.33 | —.01 +.01 —0.369806 | — 166.93 —142.50 — 45.7 = 18.17 | —.01 | —.13 —0.586813 | — 17.44 — 41.13 = 0.91 aE OEE +.03 0 —1.42022 + 4.05 — 8.75 + 0.29 + 0.15 +.02 —.02 +3.379T — 1.53 + 1.09 OSS ln O Sma ——s061 9 0 —0.132342 | + 29.78 | — 12.65 SE SL I Se TELS 0 12 —0.152528 | + 8.79 | — 82.56 — 254.29 + 2.37 +.02 | 0 —0.i79981 | — 454.55 — 1.28 — 71.88 + 50.13 | —.04 —.07 —().219482 | — 107.20 — 88.50 OO + 17.01 | —.04 +.06 —0.281202 | — 5.00 — 27.61 + 0:92 | + 9.73 | +.04 | —.03 |-—0.391910; + 9.15 | — 4.30 + 0.35 + 0.19 +.01 | —.05 —0.6426 + 0.79 — 0.38 JET Bel) || Sans | Sey 0 +0.11893 | + 7.05 | — 29.73 — 145.0 + 3.51 —.O04 ==.09 —0.13500 — 230.83 — 3.28 — 49.9 + 34.85 0 —.15 —0.15605 — 67.74 — 54.60 — 4.7 + 13.95 | —.20 +.09 —0.18490 — 3.63 — 19.80 + 0.8 + 2.62 +.05 —.03 —).22685 “hb 1.65 — 3.36 + 0.3 + 0.2 —.05 —.06 —0.2934 + 0.57 — 0.27 — 179.3 + 2.7 —.25 —1.35 | —0.1080 — 116.4 — 3.5 — 32.4 + 22.7 —.10 0 —0.1211 — 41.3 — 32.7 — 3.5 + 10.5 a0) +.10 —0.13TT — 2.6 — 13.9 a OES fe OF 0 | © —0.1597 apes — 2.6 + 0.3 + 0.2 0 —.10 —0.1901 + 0.5 — 0.2 4h THE ORBIT OF URANUS. ay ay 3 eae The values of *f DRelt are formed from at DR by simple multiplication by ) U y, and proper changes of sign. ‘The values of &, and &, are then formed by adding ay, oR % ' the terms of 2 —f D'.Rydt to the corresponding terms of =a “Op. Perturbations of radius vector. Let us now resume equation (19), and put for brevity ma imal (1+ m) ee If we give to w the successive values 0, +1, —1, +2, —2, +38, —3, we have 13, pig: —v -:) [k.cos N+-k, sin N} } + 3 i (Pie) es (%—1a-») (k,cos(N-+ g) +h, sin(N+9)} rp = MX +43, (piGery +P Wem—?_-wyy ){4-cos(N— g)--k,sin(N—g)§ +43; (PQi-» — Pent) P—"%e-») (&.cos (N-+-29)-+-&, sin( N29) -f- ete: etc! ete. etc. the finite integral being taken with respect to all values of 7 from — oc to + o, and the terms in which the angles N--wg vanish being omitted, Proceeding far- ther to expand with respect to 7, if we collect similar terms we shall find the indi- vidual terms in 7*)p to be as follows: 7.25 1 { Ah — v4) ea Poe (¥2 — v2) Poh (y Lema e10 P:iq2 + Poh) (v2 — v1) P24 + psqz) (v3 — V_2) ete. ete. :,cos N+ k,sin N} +++ 4 Poh (% — v_) (P.g2 + po) (41 — v_2) (2s + Psq2) (Ye — v_ 3) etc. ete. ii: {k,cos (N+ g) +h,sin(V+ g)} k igs cos (N — g) + &, sin (N—gq)i ae i a + (ings + Psi) (3 — va) {, cos (N+ 2g) +k, sin (N + 29)} + etc. etc. Pog2 (Yo — V2) =F (2 P93 + ps4) = Fife ak aL ae ae etc. {i:, cos (N — 2g) +k, sin (VW — 2g9)} Pods (¥3 — %) PGs + PH) (% — Va) Piz — Poh) (r% — ») ete. ete. is {k, cos (N+ 3g) +h, sin (N + 89) } | f | r ye) | ae | THE ORBIT OF URANUS. 45 | Pos (% aes a) + (P.%+ Pin) (v2 — vs) 14 MN Sih we cae Ie ff CYA?) == 4 ya + (Pi — Poi) GS a. v_») py’, COS (NV 39) + k, sin (N 3g) i f + ete. etc. J A law of the factors of &, cos(N-+ uy) + k, sin (N+ ug) which will be noticed in the above expression, is this: Representing this factor by A’,, we have Ki =K), the index 7 representing the coefficient of g in NV, so that only half the values of K need be separately computed. As the computation of 7,%sp from these formule can be arranged in such a way as to be very simple, the computation of the terms in which the index 7’ is —1 is here presented quite fully. The logarithms only are omitted, being used only in the cases in which they are more convenient than a table of products. In prac- tice I find it convenient to write them in red ink immediately under the numbers which they represent. ma a (1+ my’ numerator represents the mean distance of the disturbed planet, as deduced from the observed mean motion by the equation a’n? = u (1-++-m) while a, represents the mean motion of the outer planet. When the outer planet is the disturbed one, the ratio First, to find Jf it will be noticed that in the expression the ain the a . . — would be unity, but that, to avoid a large class of second order terms, a, has 1 been corrected for perturbations in the beginning (p. 32). In the case of Uranus disturbed by Saturn, we have in consequence log ~ = 9.999803. ay M = 285.44 Whence in units of the sixth place of decimals. Computing the values of p, and q; from (16) we find, for Uranus, LM pn = + 142.56 LM pa =—-+ 0.0784 4 Uf Poh — — 10.044 4 M(p.qz+ pon) = + 3.8433 4 (pds + Pst) = + 9.0028 In units of the sixth 4M pode = — 0.2358 place of decimals. 4 MUpqs+psm)=+ 90.118 4M pq, — — 0.008 {Mpa tpg) = + 90.005 4 W(pge—Pm)— + 9.008 J In the computation the first three lines are copied from previous pages. 46 THE ORBIT OF URANUS. +=—] ay 3 29, = 0 eT 2 3 4 5 —0.1459 —0.1709 |—0.20610 |—90.25960 —0.35062 |—0.53994 |—1.17363 + 6.75940 +0.87113 | +.0.4656 ke +0.08 +1.23 276.62 |—1508.18| + 20.85 | — 79.56 | —1.10 +0.03 ke —0.08 | —1.28 |+4 84.79] +5.79 | 286.59 | +194.17| +1.79 | 40.18 4142.56 ee — .0887 | — .14452 | — .28034 | — .82301 | +7.29934 | +2.04476 | —6.2938 | —0.5535 =) 10:0784 || v,—v2 — .2047 | — .3690 | — .9675 | +-7.0190 |41.2218 |+1.0055 | +1.4913 | —6.5186 — 10.044 y—¥, — .0535 | — .0910 |— .1893 | — .63368 | +-7.9330 | —5.8883 | —0.4056 + 3.3433 a — .1797 | — .3338 | — .9140 | 47.1100 |+1.4110 | +1.6392 | —6.4417 + 0.0028 Ss — 23 —1.00 +6.96 +1.13 40.82 +0.86 +1.41 — 0.2358 %»—V% — .144 | — .280) |— .823 | +7299 |--2:045 || —6.294 + 0.118 ¥y—V4 — .369 |— .968 {17.019 | 11.222 |+1.005 | +-1.490 — 0.008 %3—% = 88° l= 1.41 41.64 + 0.005 %y—v_| —1.00 10291 Amel | llreaceosicss +0.82 +0.56 + 0.0003 bop SF SH) | eeccascee +7.93 — i 142.56 X(—»,) | —12.35 | —20.60 |—39.965 | —117.327 | +-1040.59 | 4+-291.49 | —897.2 —78.8 0.0784 X(,—v5) | — 0.02 | — 0.03 | — 0.075 -4-0.550 +0.10] 0.08 0.1 —05 Ky —12.387 | —20.63 | —40.0£0 —116.777 |+ 1040.69) +291.57 | —897.1 —19.3 — 10.044 x(,—») +0.532 | 40.914 | 41.901 | + 6.364 | —79.68 | 459.14 4.1 ==) 313433 <@5—9=,)) | 08601) |) e126) ||) —s!056) | =2238770) | f= 472) 05-48) ole + 0.0028 X(,—»,) | —0.001 | —0.003 | +-0.020 | + 0.003 0 0 0 UG cepeonaas —0.20 —1.135 | +30.137|—74.96 | +64.62 | —17.4 Hi@a 9 || essed —0.07 —0.205 | — 1.1385|+380.14 | —74.96 | + 64.6 — .2358 xXG,—y) | -20:034 || 0.066 | 0.194 || —1-7211 || —0148 | == 1-48 + 118 X(—,) | —0.043 | —0.114 | +0.828 | --0:144 | +012 | 4+ 0.18 I |i sesecneas’ |} soscnaced +1.022 | —1.577 | —0.386 | + 1.66 We || esseocees || asonmecon —0.009 | —0.048 | 41.02 | + 1.58 = 1003 Gea) (|) os 0 | SE any —.011 — ig} + .005 XO) — .005 + .035 +.004 +.004 iG —.007 | —.009 ies MT cpncsncen |] cen cone —.002 +.042 KAM it — 26 |—11076 |+ 176117 | +21698 | —23197 +991 =) K, ke 0 0 0 —=314 | 45452 | 1563 | —5i42 +19 K_, ke 0 — 56 41712 +628 | +5964 =i 0 0 Ke ke 0 0 0 0| ++ 282 | -12378 —8 sy) Kane es -- 42 21 +126 0 0 0 0 K, ke 0 0 0 0 0 0 +8 0 K_,ke + 3 JL il 0 0 0 0 0 0 7,2 3p (cos) —1 — 9 —9343 |+176557 |\—17508 |—22453 —4151 —115 oaegie 0 0 0 —438 | 48283 —=821 | — 1053 —195 ai { 0 —438 +8283 Ssor|| 1058 —195 =—5 ae 0 0 0 0 —26 +485 —48 =o SAA | —26 +485 —48 =62 ea 0 0 0 x .000168 +30 a} —4 wl 0 —2 +30 —-3 cos +3 (cos)| + 8 4+ 34 | —1112 |4175235 |—10815 |—22986 | —5227 | —875 K, ke aL il ++ 26 —1393 —676 | —298254 | +56614 | —1606 ==il() K,, ks 0 0 0 — 39 +175 | +21483 | 412548 =30 K_, ks 0 — 7 —6 | —8637 | —14556] 116 3 0 Ky ke 0 0 0 0 +36 =i +104 +322 K_,ks 0 0 —293 —306 — 1 0 0 0 K+, ks 0 = i 0 0 0 0 0 —1 r,? dp (sin) +1 + 8 —1692 —9658 |—312600 |+78204 /4+11044 +281 046015 0 0 0 aio —453 | —14666 | +3669 +518 AGILE: { 0 = 46 —453 | —14666 +3669 | +518 +13 0 002751 |§ 9 0 0 0 —5 —27 —860 +215 MEME Nh) fi == Oh] —860 +215 +30 0 0 0 ~-000168 —5 — 53 + 13 +2 0 0 —2 —3 cos ¢ 3 (sin)| — 9 —151 | —2992 |—24186 |—309359 464029 |413864 | +961 THEH ORBIT OF URANUS. 47 In forming the next ten lines, it will be noticed that the value of »,, correspond- ing to any vertical column is found w columns to the right. It is therefore necessary to extend the line y two columns at each end. The extension on the right is, however, omitted for want of space. In performing the subtractions it will be convenient to copy the v’s again on the lower edge of a horizontal strip of paper, and, in forming the differences »,—y,/ to lay the strip above the line of »’s, and w—w' columns to the right. On the left of each line of differences is written the factor by which that line is to be multiplied. The mode of formation of the A’s is evident from the formula. It will be seen that the same computation which gives K, gives also K_,, only the latter belongs w columns to the right. Each /, and &, is multiplied in succession by all the A’s which lie below it in the same column, but the product by A’, is to be written w columns to the right, and that by A_, w columns to the left. The sum of the products in any one column gives the coefficient of © (ig +71) in the development of 7,7¢p. a’ cos This quantity being multiplied by - oo = ae ia uae have cos {ép, which only needs to be multiplied by see Y = vere to give dp. The units of 7,%%p and ép correspond to the ninth place of decimals. All the periodic terms are to be treated in this manner, all the series of values of k, and /,, including the constant term, being subjected to the same process. But, when 7 and i are both zero, vy will be infinite. Here we simply omit the v, treating it as if it were zero. We thus obtain the complete value of the terms with constant coefficients in 7,°Sp and ¢p which are given in the following table. The terms multiplied by the time are still to be computed. They are derived from (20), which may be put in the form Top — = iM} Q n=puk'? Oa ED@. hee) } nt. This expression is computed ae aL Pu AS) Qn ko p, kee —g, ke () AO NS3 ——IAArsi 0 0 +-175.18 0 Meeneoooiy) | Bae, 99cm) 2010) 2 32:75, 90.89 peeeen2sd | 0G 0284, = 186 6 09 E004 We have now > pk = + 208.02; 2 MS pk? = + 29689 —Sg¢,k = + 20.98; LMS gk = + 2988 7 0p = — 210 nt + 2986 ntcos g + 29681 nt sin 9 + Tntcos2g + 697 nt sin 29 +. 2Zutcos3g + 24 1¢ sin 3¢q, in units of the ninth place of decimals. The value of cos {dp is obtained from them by multiplying by 77°, exactly as in the case of the constant terms. 48 THE ORBIT OF URANUS. cos pdp +2986 nt +70 né + 2nt |\— 355045 +14875 —283 —20 =a = —9343 +176557 —17508 —29453 —4151 —115 Sasi —56 +710 4.20180 +4TT9T +1534 42071 —653 151 —52 +4348 +1564 -1133 —210 né - |-+29681 né +697 nt +24 nt —1503 +49 1 ae Fi —1692 —9658 —312600 +78204 +11044 +281 +1 —276 +3723 —13 +6512 +6202 —d954 +1755 +64 +221 +15 +1280 +685 +252 +87 +121 SLING +389 +201 +53 + 7 +12 +138 tan +20 + 3 cos sin —70 nt +2978 nt |+29633 né +209 né +2084 nt +14 nt +139 nt — 354347 —18411 —1496 —1536 —21 —112 — 2 . 4.3 9 +34 sisi = 1112 |) —=2999 +175235 | —24186 —10315 |—309359 —22986 | +64029 —5227 | +13864 —373 +961 — 3 —100 +3728 +484 +6796 +6230 —5610 +494 +45 +229 +88 +1314 +756 +272 100 +122 441 +400 42921 +63 19 sas +142 +84 +24 ah 1.8698 2.00035 2.18786 2.52504 8.28542 2.39559 2.1235 1.6292 1.6993 1.78303 1.88683 2.02348 2.22401 2.60786 2.98438 1.5772 1.63886 1.71074 1.79691 1.9045 2.0479 2.2634 1.5408 .9385 .5946 6589 +139n't rs wb Sb eH ) —329 +5874 —31660 —6019 +943 +527 —190 —1464 +231 —13050 —3295 18 +144 +50 aE TSi —5578 == 1720 —50 +70 —2432 —908 +14206 +4454 = 49 —99 +10 +160 —23860 +18310 +104 0 +7 —1551 +26114 +926 —6860 UT —421 —20 +653 —833 sii —3126 —1943 163 = — 638 —58 —1540 —590 —100 —i4 THE ORBIT OF URANUS. 49 Perturbations of Longitude. The perturbations of the longitude are now to be computed by formulz (24). To do this in the most simple way we remark that the numbers given on page 42, 6 under the heading a are those represented in formula (42) by v, and», If v we put ite equation (24) may be put into the form ee ear gee = \ Z ey a\eon Vp } ; but we have from (42) 5 ! OR my. my afk —dt => (ee v,sin N— — v, cos y). OV a, a, If now we represent the numerical values of cos 1p, already found, by > (p, sin N + p, cos NV), and if we substitute these expressions in the above value of oe , the latter will ¢ become a2 =, "> {(v. — 2p,) sin N-+ (Vv, — 2p.) cos N}, ¢ where we put for brevity Vv, => Mvv,, Ve = — Mv,. The numerical expression for 7,~* is given on page 40, and by multiplying the quantities within brackets by this expression, after the manner explained on pages 40 and 41, we form the terms of Be Multiplying each of these terms by its corresponding value of », changing cos to sin and sin to cos, we have the coefficients in the expressions for dv given on page 50. As previously mentioned, before commencing the above computation, I had computed all the perturbations of Uranus by the method of ‘perturbations of the elements,” using the formule developed in my Investigation of the Orbit of Nep- tune. The two results are here placed side by side, for the purpose of comparison. The discrepancies in the various coefficients, expressed in thousandths of a second, are shown in the sixth and seventh columns. It will be seen that the largest discrepancies, and indeed the only ones (with a single exception) exceeding one-tenth of a second, occur in the coefficients of the terms 2,'—1 aug 3g'—/. Here the errors are almost certainly in the computation from perturbations of the elements. Owing to the long period of the term 3¢—l they would not become sensible in the course of any one century, 7 April, 1873, 50 THE ORBIT OF URANUS. PERTURBATIONS OF THE LONGITUDE OF URANUS PRODUCED BY THE ACTION OF SATURN, AND DEPENDING ON THE FIRST POWER OF THE DISTURBING FORCES, From comp. preceding 6U From pert. of elements 5U Discrepancy. 0.434294 dp prs Hero & (—) _ PONE Or w bo SRS OU CS. Ope AO w oo ~ J++4++ Ou 09) ST So (OU > 09 —— 25006 —— ONO iret — 0.004 nt + 0.028 + 0.036 + 1.282 2.079 0.648 1.163 0.503 0.0384 0.037 0.824 0.355 0.047 0.026 0.053 0.006 0.228 0.103 0.013 0.005 0.003 0.074 0.038 0.005 0.002 + J + +- + + + + + + he + + + -{- | | J}+++ ” + 10.9690¢ + 12.231 nt + 0.717 nt + 0°048 nt 4.735 0.169 0.005 0.039 0.718 8.522 +143.463 +115.86 == 5.616 + 0.329 0.017 0.814 0.009 1.607 1.830 2.956 0.378 0.012 0.041 0.019 0.267 0.165 0.063 0.032 0.022 0.008 0.077 0.044 0.013 0.002 0.008 0.027 0.016 0.004 —1.228 nt —0.072 né —0.004 nt l++ +44 p}+t+44t4+ 4444444 4441 SoorFSoN ROS OWE [+++] cos ” +10.9645¢ 12.271 né + 0.720 né + 0.045 nt 0.166 0.014 0.005 0.719 8.595 +143.465 116.08 + 5.621 + 0.331 0.033 0.818 0.012 1 676 1.902 2.991 0.445 +++ +++ 0.015 ° 0.050 0.017 0.263 0.191 0.066 0.018 0.023 0.002 0.075 0.057 0.015 0.002 0 0.026 0.025 0.003 002 net. ” 00456 -040 nt 0 .003 nt 0 0 — bo or bo ew po ww o wor ovate CN CO _ a pp Oo +13 nt +1 nt 0 —1541 p+++++ +) ++++ i (Jo) + +++ EEE + SSS iPS SS) Oh ou OS Seo = oO eo corocoeccrFNOrF COrFWwNese S&S eS & WwW THE ORBIT OF URANUS. 51 Perturbations of Latitude. These are computed from the formule (27) and (40), no reductions being made from d& and dy to dp and dq, but the perturbations of the latitude being computed directly from the former by (40). We have only to represent the expressions for dk and dy by ok = — Sa, cos N— Ya, sin N 6, = a’,cosN-+ Sa’,sin N and substitute o for 2 in the equations (40) from which §3 is computed. The principal steps of the computation are shown quite fully in the following table. ‘The values of 5 h 1 it ele COs Oy 2 Oo are first formed from those terms of f, on pages 37 and 38, which contain o as a coefficient. ‘Then, having for each original term of 2 oR m oh ee eee See COS| NV oy a, Oy all the terms which have the same coefficients of 2 and 4’ in N are combined into two depending on g and / as shown in the case of / on page 36. The coefficients of these terms, in units of the third place of decimals, are given in the columns headed oe oy The value of ee sin N being formed for each term of £, all the terms depending o on the same multiples of % and %’ are combined into two, of which the coefficients are given under the proper heading. The terms of (¢ +7) cd sin N being formed in like manner, we have, by adding the last two expressions, all the quantities which enter into the formule (27). .To integrate these equations thus forming the numerical values of d&% and dy we have only to multiply each term in the second, third, eighth, and ninth columns of the table by the corresponding values , may . My . ore , for which we may use the value of ———_ already given. a, COS sin 1 The quantities given in the four columns under é& and — dy show the values of —a,, —a,, —a’., —a’,, corresponding to each argument. From these the terms of §3 are formed by equation (40) with the modification mentioned above. 52 THE ORBIT OF URANUS. PERTURBATIONS OF THE LATITUDE PRODUCED BY SATURN. th =h dy 2 an ish = (i+j)shsin N| qx — 4m Xx bk io sin sin cos sin cos sin “W 0 0 0 +0.008 0.00|+- .008 — 2.92|—1.94 |— 0.11/+ 0.17/+0.84 |— 1.77|4 0.73/+-0.203|—0.172)/—0.104|—0.043/+-0.246 —11.10)+1.30 |410.73/4 0.07/+-0.05 |+ 1.37|4+-10.78)|—0.042|—0.326|+-0.042 —0,317|\—0.049 — 1.71\41.31 + 1.71/4+ 0.01) 0.00 |4 1.32/4+ 1.71)—0.026|—0.033/+4-0.026 —0.033/—0.008 + 0.1 |-1.0 |— 0.1] 0.00] 0.00 |— 1.0 |— 0.1 |4-0.012/—0.001|+-0.012,—0.001] 0.000 —52.6 |4+6.4 |—52.8 |4+ 0.03-+0.02 |4+ 6.4 |_52.8 |—0.098|-++0.804/—0.098|—0.804|—0.004 + 7.06|+-1.53 |+ 5.14|— 4.91] 0.00 |— 3.38/4 5.14|—0.144|—0.145]+-0.0704-0.106|—0.661 +46.67 0.00 | 0.00/4+-59.45, —0.02 |4-59.45|— 0.02|4+1.482} 0 |—1.887|—0.001|4-0.084 — 1.84/— 2.86/—0.99 |— 0.20/4+ 0.93/+-1.22 |— 0.06/4+- 1.02]/—0.127|+-0.197|-+0.004'-+-0.070 + 0.90 — 6.98|+0.78 + 6.44/4+ 0.1240.16 |4 0.90\4 6.60|—0.358|—2 781|40.358'—2.630|4-0.091 + 1.09 -38}+1.09 + 1.38/4+ 0.02 40.01 |4+ 1.11)4+ 1.39|—0.056|—0.071|+ 0.057 —0.071|—0.048 + 7.6 25 |—7.6 |= 1-5 0.00 0.00 7.6 |+ 1.5 |4+-0.067|+-0.013|+-0.067 +-0.013 +0.003 S618} —1.3 |+-10.7 11 40.53 1.4 |4-11.2 |4+-0.613 —0.112)+-0.014 +0.116/-++0.008 ==0:27 +1.70 + 0.91 -65.—7. 3.35|— 6.44|—0.003/—0.112)—0.042 —0.080/+-0.040 —14.51 0.00 0.00 } ; 7.01/— 0.21/—0.230} 0 |+-0.112,—0.003] 0 —0.42 |— 0.19 97) 41.11 1.39 + 0.92 —0.062/+-0.050|+-0.030 +-0.020|—0. 063 0.44 |4+ 3.67 .02 +-0.22 0.46 + 3.89) +-0.015|-+-0.146/—0.016 +0.134|—0.080 ;+0.80 + 0.99 -02 +0.02 0.82 4+ 1.01)4+0.067|+-0.083)—0.068 +0.084/+-0.008 +1.22 |+ 0.97 .09 +0.33 |4+ 1.31/4+ 1.30/—0.010|—0.008/—0.011 40.012 +0.050 0.00 | 0.00 ; @ 4.34) 0.00/—0.090} 0 |40.046/ 0 |+4-0.004 5|—0.13) |— 0.15 40.85 .70|—0.026/+-0.022)+-0.013 +-0.009|—0.012 +0.24 + 2.00 +0.22 ; .22 4-0.002 +-0.041/—0.004 +-0.037|/—0.010 36 +0.55 |-+ 0.66 40.02 : .68) +-0.013 +-0.015|—0.013 +-0.016|+-0.004 +0.82 |+ 0.78 +0.32 ae —0.005 —0.003)—0.006/+-0.008|+-0.022 0.00 0.00 —0.02 48) -02/—0.038) 0 +0.020) 0 +0.002 —0.01 +0.59 6 —0.013 +-0.010|-+-0.006)/+-0.004|—0.004 +0.12 +0.19 + 1.2 |+0.001 +0.014)—0.001)+-0.013)+-0.004 5 10.04 .35|4+- 0.04|—0.017] +0.008 dk = -+- 4.77 T Secular terms { w= ate Taos a. THE ORBIT OF URANUS: 53 CAEICAGE Rives or PERTURBATIONS PRODUCED BY NEPTUNE AND JUPITER. Tue perturbations of Uranus by Neptune were originally computed with ele- ments of both planets quite different from those finally adopted. But the last computations, on which the concluded values of the perturbations depend, were made with the concluded elements of Neptune found in my investigation of the orbit of that planet." They are as follows: ° ' ” Nt, 43 17 30 6, 130 tin 83 E, 339 5 39 >, Ti eG n, 7864.935 e, 0.0084962 log a, 1.478141 Mass, a7s00 Hence follow the following functions of the elements of Neptune and Uranus: a = 0.638195 Oe 48s wo = 247 45 20 my ple oO 29:6 o=sin}y= 0.013161 M= 37.522 (in units of 6th place of decimals). From these values of the elements are obtained the following values of the various terms in the development of the perturbative function, and of ». As the developments have been formed on the same principle as in the case of Saturn, it is deemed unnecessary to give the details of the process. It is only necessary to remark that the indices 7’ and 7 are the coefficients of /’ and g respectively, the mean longitude of Neptune, or /’, being counted from the perihelion of Uranus. 1 Smithsonian Contributions to-Knowledge, Vol. XV. 54 THE ORBIT OF URANUS. ACTION OF NEPTUNE. oO R m =-— xX Op a, +1135.65 18.07 0.41 0 0.21 5.21 134.19 25.45 1.12 0.05 0.85 13.53 375.92 60.090 Sol 0.09 0.83 12.09 200.37 : + 601.13 §2 33 9.8: —150.02 5.05 -56 + 13.8 0.24 | — 0. — 0.58 0.75 : JL BED 9.47 By i + 38.3 111.16 pile + 444.7 38.58 + —149.10 5.205 0.36 0.65 6.97 63.03 27.43 4.84 0.45 0.03 4.93 36.16 19.04 4.14 0.490 | 271 0.036 -026 3.41 28 20.90 16 12.99 2.54 3 40 oil 0.487 -276 0.044 -034 2.35 20 12.20 L118} 8.8 1.71 2 69 1.05 0 45 0.26 0.049 0.039 7.0 0.1 59 1.13 2.07 0.81 0.39 0.25 0.050 0.042 4.1 0.1 3.9 0.8 1.6 0.6 0.3 0.2 0.05 | + 0.04 +358.26 +368.26 — 64.80 4+ 2.98 0 | S= PWS OSES oto Veao of = + 1.6 — 295 +509.94 — 8457 §.83 0.24 25 4.0 ke OO i SRW S SOSHoSoS OR 1S bo RR OoOrPNwW WNHROS sw. ++1+1+ +1 oS ou bo H 32 ~I Ci) ire) ey -+50.7820 + 0.98069 + 0.49512 — 0.28814 — 0.40478 | — 0.68006 — 2.19559 + 1.88844 + 0.65378 — 0.25249 — 0.33777 — 0.51004 — 1.04100 +25.3910 + 0.96210 — 0.22468 — 0.28984 — 0.40804 — 0.68929 — 2.21843 + 1.82073 + 0.6455 — 0.2538 — 0.3400 — 0.5152 — 1.0628 +16.9273 de ie) WHat wWwoRoe bose Wresa Sorbo -1 @ SHENIS She EHS ComMS Ror ~pup. Iti+/+ +1444 +] CHeH DHE Tae Seuss Deo oO So _ HO bo bt tg | +++ | Z w ++ [|+) +++ +4+/ 4/4 see wNagnw Sis! Lar ll | Ue S oO cs NRONS SHRORS SHU SORS — Se el el? a a 2 wo) eb bo Or oy bo SrAOWS WAND He Ss Si Sw * SrPwewea mb we larllsr || ao rary i=) > cS ae pwwts 1 rs NH DONSMN Hwan [++ +/4/+++1)41+ +14141 Hs 1 bo bo aD eo Naw Ow S SContrPowo ror bo POWs ons rt OV OD He OO oO ot i | H om-t oc eae is bo Lien J i) , OA WEWUE AE WHO OTh ¢ TS eek ll SS OS od etre flere elder sissies Ie bo or gro bo | ivy) Ll or) Gee soso ehe bo ORS HP hipe oon Pike abha e wo on COOH RA — @M~Ib ar lar il I | 1 le 6) terete tl | crrse ph bo (o 6) for oO ~) [t1++ [4144+ [4144+ PM HANS Rt ae a SH w-3SLn bo © bo bo res So Nn 2 000 Hip _ — cea eo So BOO Not S ee I+) ++ +/+) 4+ 414+ Swowan-t © ae oo ot Io Co ow eo ce io} ra 4 ob ao + + -f + f- $ + oo 6 | + ee 4} 4 @) |} — 2 | a ee 6 | + 54 ab | aes Py | 1} + 7/4 6}+ on 4) 4 53} 2/4 8} 4 7 | + 6a 5 | Ay | = 3} + -f + + 4 ae + + + |+ ae || +Lt+ +4++ Lito} ce aalllat= | ~ > bo nw Sow 1 1 — me op Sm NHOUDH So =r ODDO W OS D> } ~1b9 Ww bo =) —_ mest bo BR bon DB SWEAANS oo DE RW Hott he= Wo bo Biwnoe > WOR eaH Lah) vllarilay aelsel ae (rep ae M100 = to oS a So — or rae S pee oveoS Cm, ORIN Bog a10 53 B ouS> gH Lt bo > “I= Go COW aAtMHWw 7) SOU WOOD o by bo alle Sesh oan SAS as Tle sleet saise ley | eo + iB” oo oO SP IDS lor) w io R, cos ig. Put also i VG yom Noe ey, Pe OR, Saal ne = ate IR é€ Express any set of corresponding terms of the preceding perturbations in the form él =L, sin N+ L,cos N; etl — edn = F, sin N+ F, cos N; ce = LH, cos N+ E, sin N; 60 = A,cos N-++ A, sin NV. We shall then have 280 = 5(V".F, + VE.) sin(N-+ ig) -- 5(V", F,— V’,E,) sin(N— ig) +3(V", F.— VE,)cos(N+ ig) + 5(V", F.+ V,E,) cos (N — ig) + 28 239 => (Fk, E,— RF’, F,) cos (N+ tg) + > (Kh, E, + FR’, F,) cos(N — tg) +3 (RE, + BF.) sin (N+ ig) +3 (Ry Ey — RF.) sin(W— iy) + 25n 1e rical values of V v’, and f” are as follows: The numerical values of V’, V’, &’, and R” are as foll V,= 1.99835 V.= 1.99945 Tees Onis V,= 0.11722 Vi= 0.00714 V’,= 0.00714 V,— 0.00044 V',— 0.00044 R,= + 0.02348 R= — 0.99753 R= 0.99917 RP, = — 0.07020 R’, — + 0.07030 RP, = — 0.00466 R’, = + 0.00467 The final results of the entire computations are given in the following table: In the columns év,, we have the complete perturbations of the longitude computed cok OV we have, under the caption év,, the perturbations of the true longitude deduced from the long period perturbations of the elements, as set forth in the last para- graph, omitting the constants added to the perturbations. Under dv; we have the 8 April, 1873. by the direct method from the values o : — Q, etc., already given. Next 58 TEE OUR TB et OR sup ReAgN TUES: perturbations of the longitude deduced from all the remaining terms of the per- turbations of the elements. ‘The sum of the columns $v, and dv; shows the entire perturbations computed by the method of variation of elements, ‘Thus, in $y, and Sv, + dv, we have two complete sets of perturbations computed by methods entirely independent. ‘The differences of the results, expressed in thousandths of a second, are given in the last two columns of the table. ‘This comparison gives rise to remarks similar to those suggested by the per- turbations of Saturn computed by the same methods. ‘The only terms in which the difference of results amounts to as much as one-tenth of a second are those of very long period, and those very nearly the period of Uranus, where a more accu- rate value is not at present of great importance, because the error will be com- pensated by the corrections of the element during several centuries. RB RS EE ET PERTURBATIONS OF THE LONGITUDE OF URANUS PRODUCED BY NEPTUNE. bu, bv, 3U, Discrepancy. Ug sin cos sin cos sin cos sin cos dé ” OO —0.4262 ¢ —Il | —.065 nt —1.181 nt —2 | —.004 ni —0.069 né 0, 0 See JOG |S One BSG EI 101046) == 204005 ==3)]) 4 101003 1—3 | + 0.147 | — _ 0.001 rere HOOS + 0.146 | —0.002 1 1 —2) + 2.509 | — 0.019 sda 5106 + 2.509 | —0.010 0 4) —l1/+ 39.658 | — _ 0.080 Eee 5.01510 +39.673 | —0.081 15 1 Oe) Op sare nee + 4249 | —0.478 8 | 383 —l;+ 0.280 | — 0.032 oaac 5 ONC + 0.275 | —0.032 5) 0 =9 | 0017 == 020028 9—4 | + 9978 | =e. 10,097 |/4> 12)89)! 22 ors 0r09s: |= 02002 4) mato 1 —3}-+ 49.015 | + 0.413 | + 47.22) + 0.44 |} + 1.797 | —0.021 2 6 —2 | + $40.93 | + 17.388 | 4. 805.64) + 7.43 | 435.355 | 01098 || Gham —lI | —3475.4 + 180.36 —3474.32 | +180.10 | — 0.700 | +0.095 | 380 165 0 | — 162.07 | == “8101 |== 161.96) 4. (8.01 | = 01067 =|—-201015" |e —l|— 9.447 | 4+ 0.468 |— 9.50) + 0.47 3,—5 | — 0.076 0.000 pac0 doo — 0.077 | —0.003 1 3 —4/— 1.162 0.000 ae Sera — 1.153 | —0.011 9 11 Seg) |= W7.9860 11102998 ane ie —17.285 | 40.229 1 1 —2 |— 22.085 | + 4.037 S506 S400 —22.07T | +4.020 8 17 —l|}— 0.673 | + 0.082 eons ones — 0.682 | +0.079 9 3 0) ==) "OL05i 1-1-1 0°006 4,—6 | — 0.036 | + 0.002 |— 0.015} + 0.002 | — 0.027 | —0.003 6 3 —) |— 0.558 | + 0.037 |— 0.25 + 0.04 — 0.315 | —0.007 T 4 —4/— 7.968 | + 0.750 |— 4.08 + 0.68 — 3.908 | +0.059 20 ll —3 | — 75.00 + 12.832 | — 69.55 +11.67 — 5.733 | +1.067 | 283 95 —2 | + 146.78 — 54.218 | +146.72 —54.10 + 0.126 | —0.079 66 39 AA | $46 19605 9 22579: || 25 46.81") ones 59 THE ORBIT OF URANUS. PERTURBATIONS OF THE LONGITUDE— Continued. uv, bu, 5U, Discrepancy. Ug sin cos sin cos sin cos sin cos AA AA pa —0.009 0.000 ——OnOS —0.002 6 2 ——6 —=())-1K(0B} +0.006 ——Oenulis. —0).004 10 10 5 —0.986 —0.042 —0.994 —0.042 8 0 4: +3.366 —0.670 +3.370 —0.662 4 8 a 3210 =2 151169 SEBO | SYS 4) Bly 17 = 2 40.077 —0.017 +0.075 | —0.002 2 15 ant +0.005 —0.001 6— i —0.050 —().004 0.00 0.00 054i 00,02; 4 2 —= (5 —0.387 —0.020 +0.02 O01 —0.425 ——Os0NS 16 3 — 8 Saal —0.320 40.40 —0.14 +0.855 | —0.169 6 11 —4 +7.781 —2.825 +6.80 —2.54 -+9.857 | —0.318 | 124 33 Sie 0020 +5.073 een 15.03 +0.017 | —0.022 72 65 eo) 0.43 10.246 —0.42 +0.26 pees — 0.027 —0.002 == 7 —0.186 —=O2003 —0.189 —0.009 3 6 = ff +0.272 0.046 +0.260 —0.047 12 1 ee) (neni: +0.217 —=OEoS +0.202 33 15 4) 0.459 +0.250 —0.419 | 10.255 | 40 5 ne ORO -L0.005 gag) 0.013 0.000 as —().084 —0.002 —0(0.093 —0.007 9 5 — i -+0.125 —0.022 ats pictaig +0.115 —0.024 10 2 — 6| —0.194 +0.082 O08 1) SEOLOne | ata Bs) SOs) al 10 = 5; —0.839 +0.463 —0.66 +0.30 | —0.110 | +0.067 | 69 96 — 4 0.647 O20 OL +0.64 (eos 979) 02040 —0.001 —0.048 | —0.005 8 4 = 8 +0.063 (Ona: +0.002 (On Oia 61 0 ay —0.055 +0.020 (02059 + 0.025 4 5 6 -+0.080 —().049 +0.083 —0.054 3 5 —a) +0.059 —0.050 10—10| —0.020 0.000 —0.025 | —0.005 5 5 ==i9) =£0.035 —0.005 — 8 —0.027 —0.010 ae +0.026 == (NON — 6 +0.092 —0.079 The perturbations of the logarithms of the radius vector are given in a form similar to those of the longitude. Under §p, we have the complete perturbation. dp: the effect of the perturbations of long period. Under But under dp, we have only the difference between dp, and dp, it being deemed unnecessary to present in full the perturbations of the radius vector as compaed by the other method. dp, being employed in computing dv, may, in fact, be regarded as completely checked by its affording a correct value of the latter. In the last two columns dp, is reduced to common logarithms by multiplying the coefficients by the modulus 0.434294. sin ++ ++] 1 +I ano oo POF S mewWwWwWoe conwnococo owooco +++ mpPpowoc lets ewe oS | | (> a) 60 THE ORBIT OF URANUS. PERTURBATIONS OF THE LOGARITHM OF THE RADIUS VECTOR OF URANUS PRODUCED BY NEPTUNE. U g 5p, Sp. 8ps Msp cos sin cos sin cos sin cos 0, 0 +138 + 60 Il ——o 8} + 4 0 + 2 =—o + 68 0 + 30 sh +523 + 1 +227 0 — 68 iG a) +] rs —ael —o 24 + 94 ae ae elt 4+ 9 0 + 1 = +1416 =) | Segia A118 + 42 0 seas —o + 20025 —— eid) +19490 —180 +535 +1 +232 =a +1663 +116 | -- 1726 SEs | 3 Gy) AS 27 0 +3912 +194 + 3927 +194 == 15 0 7 Ba) — 3 0 el a — 38 0 = I =} Sil = & ——229 — —284 ee ——124 =i + 16 + 2 + $7 4,—6 — il 0 —_ 5 —— wg) == jl — aS ease == Ill —1 == fy | 254 — OD, —— 11153 —— 0) 36 — 9% == i) —3} —1759 —300 —1681 — 289, == 7S! ifs} — 34 aS) —143 Gl |) 58 = i0 ES Lok HL § | Nore "62 ==> Rs Ge nO + 2 el D6 — 4 0 == 09) a; = AD oe ly aay +103 +20 + 45 eS5} + 39 +15 Pre 6,—T — 2 0 — 0 =5 il =36 a hi +1 shee sesseye mela + 1 nit 285 + 42 +10 sEh10 ney 51390), EG apie ——4. +180 +65 +164 + 61 + 16 + 4 + 4 3 + 13 + 8 + 14 + 5 eels + 3 0 iG — 8 0 = & 36 Si aieng 5 = —l17 =— en ——4. — 5 => 3 — 2 8,—8 — 4 0 —— 9, = aE 5 1 po —6 — 7 —= 3 — 3 —) —19 IU) = 8 9,—9 oy 0 — | 3 Sens 0 aia tee ar == 2B — il ae 6 + 2 + 2 + 1 i ] THE ORBIT OF URANUS. 61 PERTURBATIONS OF THE LATITUDE PRODUCED BY NEPTUNE, = (tj) dy x h sin N an m!anX sin sin cos i cos 4/ 4/ 4/ 4/ —21.65 0 0 0 0 0 + 1.08)+-2.73)— 4.00/\—1.66)-+ .01 — 3.99 —1.67/— 0.008 | +0.021 | +0.030 | —0.012 |-++.080/+.020 +19.32 .39|--19.32/+-9.39|— .01| +19.31/+-9.39/— 0.073 | —0.036 | —0.073 | 0.036 |+-.016'|—.002 +14.58 -0 |—14.7 |+-7.0 0 —147 |+7.0 | +0.045 | —0.022 | —0.045 | —0.022 |+-.007|—.002 — 1.16 6 —2.40/—0.07 + 3.97 —2.4 | —0.006 | +0.014 | +0.020 | +0.012 |+-.201/—.046 — 34.26 0 0 |+41.76 + 1.76) 0...} —0.540 0 + 0.028 0 —.008)/—.037 + 3.74\4+-2.32) \—1.48 —0.24 — 3.04 —1.48) —0.057 | +0.035 | +0.046 | —0.022 |4-.364|+.084 +15.77|—7.8 +7.75) 0 +16.25 +7.75; —0.081 | —0.040 | —0.083 | +-0.040 |-++.060 +.007 +10.52 +5.06) -6 |+5.0 Oo | 10.6 l4-5.0 +0.027 | —0.013 | —0.027 | —0.013 |+-.002\—.001 — 2.60/—2.38\4+ 3.65|—2.28|4-0.36 4.0.04 -01, —2.24) —0.010 | 40.009 | +0.016 | +0.009 |-++.062/—.014 0 U 0 |+9.88 —0.01 .£8 —0.01| —0.264 0 +0.078 0 +.008)—.004 +2.04 —1.04'—1.53 +-0.04| .65 —1.00 [—2.283] [0.802] [+1.431] [—0.393]/+.326/+4-.075 —9.52/+19.56/+-9.32/4-0.08) 0 —0.143 | —0.072 | —0.049 | +0.071 |+.002) 0 +3.6 A }43.5 5|—2.00 .12/—2.0 0 0 +1.73 -20'—0.54 2 -68 +-7.00 |-+-0.22, -51|—0.22) 0 E +0.017 | —0.008 | —0.017 | —0.008 |+.002] 0 0.05 .60,—2.0 | —0.010 | +0.006 | 40.011 | +0.006 |++.023!—.004 0 0 | 8 —01128 | 0 +0.041 —.004/ 4.041 ‘+0.26 -28] 10.108 | —0 029 | —0.053 | +0.004 |+.303\-+-.074 —0.04 | —0.199 | —0.106 | —0.217 | +0.101 |—.090'—.026 0 3 : +0.008 } +0.001 | +0.008 | —0.001 |—.013) 0 H~1H aPee Ho So SSF=1590 ow —1.63 -53)\—1.7 | hes 0 -0 |-—1.7 | —0.008 | +0.004 | +0.008 | +0.004 |4+-.022'—.004 0 0 0 13 | 0) || 01070 0 +0.023 0 —.008)-+.006 +1.34 .59|—0.22 +0.30 2.56 +-0.08) +0.051 | —0.011 | —0.020 | —0.001 |+.048)4+-.010 —5.37/4+10.57/+-5.04 + 0.25 —0.07 82 +4.97 [—1.852] [—1.056] [—2.127]|[40.977]/—.047| 0 +0.32) .72|/— 0.32 —0.02|-+-0.02 -74—0.30) +-0.013 | +0.002 | +0.013 | —0.002 |—.001; 0 4 | —0.005 | +0.003 | +0.005 | +0.003 +-.012|—.003 0 La. | 0 —0.038 +0.013 0 ——10 12 .97 +0.24) +0.030 | —0.005 | —0.010 | —0.001 |—.109'—.028 73 +3.45) +0.107 | +0.067 | +0.131 | —0.059 |—.038/4.001 -69 —0.32) +0.024 | +0.005 | +0.025 | —0.004 |+.018) 0 om in ee wt —1.40 -8 |—1.4 |+0.5 0 0 0 |+4.0 0 +1.01 -22, —0.04'—1.75)+-0.28 —3.89) .43/+ 3.54/+.0.30 —0.09) +0.33/— 1.66,—0.33 —0.03 +0.01) 0 0 |+42.8 0 2.80) 0 —0.022 0 -+0.007 0 —.010)-+.002 3 +0.76 .03/-+0.04 —1.46)4-0.24— 1.49 +-0.28) +-0.019 | —0.003 | —0.006 | —0.001 |—.028|—.008 3 —2.79} : 2.45 +0.31 —0.10)4 5.46 42.35) +0.033 | +0.022 | +0.045 | —0.019 |—.010) 0 5 —0.31 —0.04 +0.02 49, —0.29 [+-0.190] [40.041] [40.195] [—0.038]/+.015) 0 ram ri bo bo arar lil ShaAOpR AE aT iB bo [eter Pata jee) °$ 0 —0.012 0 +0.004 0 —.006 +.001 Re 0 |—1.1'7/4-0:20 17 +0.20) +0.012 | —0.002 | —0 004 | —0.001 |—.008|—,004 9 —2.00) .52 +1.68/+-0.30/—0.10 -82-+1.58} +0.013 | +0.011 | +0.020 | —0.008 |+.014) 0 +0.27| .19, —0.27 —0.04,+-0.02) .23;—0.25) —0.021 | —0.005 | —0.022 | +0.004 |+.002) 0 atest | el a 0 0 |41.3 0 : 0 —0.008 0 +0.003 0 —.003 +.002 5 +0.50 0 |—0.91 +0.16) -91/+0.16} +0.007 | —0.002 | +0.002 0 —.003/—.002 —1.41 2.38'+1.14'+-0.27/—0.09 -65 +1.05| +-0.006 | +0.006 | +0.011 | —0.004 |4+.004) 0 |-+0.22,— 0.97, —0.22; 0 |+-0.02) -97,—0.20} —0.008 | —0.002 | —0.008 | +0.002 |+.002} 0 | Gteatel ene Secular term, %—=+1/.25 T %=— 17.25 Tcos v The terms dS and Sy, which are inclosed in brackets, are of very long period, and are therefore omitted in forming the values of §2 in the last two columns 62 THE ORBIT OF URANUS. Perturbations produced by Jupiter. The series in which these perturbations are expressed converge so rapidly that I deem it unnecessary to present the details of the computation. They have been computed by both methods, and the separate and independent results are given in the following table, where dv, represents the perturbations computed by the method developed in Chapter I, and ¢v, those computed by the method of variation of elements. The apparently large discrepancy between the coefficients multiplied by the time arises from the circumstances that in the form of development the mean motion, and hence the mean anomaly, appears affected by the perturbation 31”.2t, Accord- ingly when we enter the table which gives the true longitude in terms of the mean anomaly in the form vl sin (J — 2) -+ ete., we may consider this quantity 31’.2¢ as a secular variation of 7 — 7 producing in» the term dv == 62" 4et cos (J — x). In én, this term is left in its primitive form, while in dv, the value of 7 is supposed to include this term, and the secular terms are only those which arise from the secular variation of the eccentricity and perihelion. It is also to be remarked that the terms which are independent of the mean longitude of Jupiter, or those in which 7’ = 0, are not comparable, as they corre- spond to slightly different elliptic elements in the two theories. THE ORBIT OF URANUS. 63 PERTURBATIONS OF URANUS BY JUPITER. 6v, Diff. COS dp | cos sin |cos cos < ”" ” | -sace 31. | soce +31.1982¢ — 0.160nt} +37.585né— 0.1622né\— 1.5406n¢| — 0.010n¢; 2.2 0.0095né — 0.0899 eres 5 0.0006né— 0.0054nt "346 1.361 030 0.086 oo WNrF Cor WNrFES WNreS 0.017 |+ 0.011 1.232 0.001 53.084 0.002 3.565 0.082 0. 0. ily _ — HD bo to 164 0.050 031 0.014 176 0.515 0.263 0.037 0.083 0.008 0.014 0.003 0.025 0.025 0.005 0.015 0.037 0.017 | bo +++] +442) oSrss =woss osso t Te co -1 er) ST Ww bo OV H CWRrHOW Wormer bo ne OD (i=) bo | os moor 0 1 2 3 4 1 2 3 4 5 PERTURBATIONS OF THE LATITUDE. on t sr or Ad 40.071 —0.041 40.030 +0.036 +.060 —0.012 —0.075 —0.012 40.075 —.010 —0.003 —0.012 —0.003 40.012 —.002 —0.006 +0.042 10.006 40.042 +4.005 —0.297 +1.949 40.297 41.949 +.001 —0.161 —0.191 —0.189 —0.190 —.494 49.611 0 42.628 0 —.024 +0.070 +0.021 +0.066 —0.002 +.002 40.055 —0.088 —0.055 —0.088 +.010 +0.002 —0.011 —0.002 —0.011 = 017 — (HKG) —0.050 —0.109 -+0.046 —.009 —0.014 0 —0.011 0 —.002 (sk — Secular terms - 3 ss on. — é cos V 64 THE ORBIT OF URANUS. CEVA PALE Rave TERMS OF THE SECOND ORDER PRODUCED BY THE ACTION OF SATURN. Preliminary Investigation of the Orbit of Saturn. For the accurate determination of the perturbations of a planet it is essential that the functions of the time which are substituted for the co-ordinates of each planet in the expression of the disturbing forces should approximately represent the true places of the planet. The difference between the true place and that implicitly assumed in the investigation should be so small and of such a character that, when multiplied by the mass of the disturbing planet, and by the factors introduced by the process of integration, the result shall be insensible. If one of these factors is so large as to make a perturbation of an order of magnitude approxi- mating that of the inequality which gives rise to it, if will represent an inequality of very long period in the elements, which, though apparently sensible, may be neglected for a great length of time. The perturbations hitherto found have been computed on the hypothesis that the disturbing action of Saturn on Uranus is the same as if both planets moved in the elliptic orbits corresponding to the adopted elements. We have given formule for the computation of the corrected perturbations when, to the co-ordinates of the two planets corresponding to the adopted ellipse, we add corrections represented by dv, dv’, dp, etc. These corrections are now to be taken of such magnitude that when thus added they shall very nearly represent the actual motions of the planets. Generally, it is considered sufficient to take for these corrections the perturba- tions of the first order. But this presupposes that the elliptic elements are nearly correct, which does not hold true in the case of the old elements of the outer planets. Bouvard’s Tables of Saturn, the elements of which have been adopted, are subject to recurring errors amounting to 30” or more. Moreover, when we substitute the new and more accurate perturbations for the old and imperfect ones adopted in the tables, the chances are that the errors will be increased. Desiring that the theory shall be as far as possible free from doubt, we begin with a pre- liminary investigation of the orbit of Saturn, the design of which will be to give the co-ordinates of that body in terms of the time with sufficient certainty and accuracy to serve for computing the perturbations both of Jupiter and Uranus. As usual, the first step in this investigation will be the determinations of the per- turbations of the planet. THE ORBIT OF URANUS. 65 General Perturbations of Saturn. The perturbations produced by Jupiter will be taken from the exhaustive prize memoir of Hansen.’ As the perturbations required are those of the co-ordinates, it will be necessary to transform those of Hansen into the usual form. Hansen gives the true anomaly v in the form ©=9 + ntiz +e, sin (g + néz) + e, sin 2(g + ndz) + etc., C1, , ete., being the coefficients of the multiples of the mean anomaly in the usual development of the elliptic true anomaly. Whence, neglecting the second power of n°z, dv = niz(1 +e, cosg + 2¢, cos 29 + etc.). To make the development sufficiently rigorous it is only necessary to increase gy by + néz in this expression. In the same way, we have for the perturbations of log r, 2 dp = dpa + néz (e® sing 4+ e® sin 2g + ete.) 8po being Hansen’s perturbation, and e the negative coefficient of cos 7g in the development of the elliptic log x. Hansen having adopted ,;45.5 as the mass of Jupiter, it will be necessary to multiply his perturbations by 1.0216 to reduce them to Bessel’s mass. Thus the perturbations by Jupiter hereafter given have been obtained. The perturbations by Uranus and Neptune have been computed by the preceding general method, and are given in the following table. Iu the table d’ is the mean longitude of the disturbing planet, Uranus or Neptune, counted from the perihelion of Saturn. dp is the perturbation of the Naperian logarithm, in units of the seventh place of decimals. GENERAL PERTURBATIONS OF THE LONGITUDE IN ORBIT AND THE LOGARITHM OF THE RADIUS VECTOR OF SATURN. Action of Uranus. Action of Neptune. bu év 5p ° (s) a sin cos Ket B 4/ 4} 40.23 | —0.05 41.95 0.00 | +0.02 —1.11| +0.02 =|) 60100 40.02 | —0.03 +0.10 | —0.04 0.00 | sF a -+ I ae | =x Sto Onhws KONIoc wr was as NG) woe —_e — See an a = FSS bo mwo Dew wet DRA De OOM api Meo pone u— = He oo 09 OAS S&S > loo be [a cee ae || ows ocooco bo oO ao 1 Untersuchungen iiber die gegenseitigen Storungen des Jupiters und Saturns. Von P. A. Han- sen. Berlin, 1831. 9 April, 1873. 66 THE ORBIT OF URANUS. I have submitted these perturbations to such duplicate computations and other checks as lead me to believe that none of the terms can be in error by more than a small fraction of a second, but, as they are not intended to form the basis of a definitive theory of Saturn, I do not vouch for their absolute precision, In this provisional correction of the orbit of Saturn only heliocentric longitudes have been employed. ‘These were derived for a series of dates from Airy’s redue- tion of the Greenwich observations, the modern Greenwich observations, and the Washington observations. For these dates the value of n‘z for Saturn was computed from the formule found on pages 189 and 190 of the work of Hansen, already quoted, omitting all terms less than 1”, and including only tenths of seconds in the results. The dates, the resulting values of ndz, of the factor e, cos (g + § ncz) + 2¢, cos 2(g + 4 nbs), and of the concluded év are as follows. ‘The formule for év is év = 1.0216 n°z {1 +e, cos(g + §n‘z) + 2, cos 2(g + §ntz)}. Gr. Sosa Rion. BeZ We ee 1751 May 31 — 1947.7 —.0990 —— Dail 1757 Aug. -7 ——Q134-2 —.0652 — 2038.2 1758 Aug. 27 = PINS) —.0474 P53) Gil Oct aa ae A OED, + .0244 — 2664.5 1763 Nov. 1 —2880.3 + .0729 Gil) 7, 1h 1765 Nov. 23 —— si 08), I +.1082 —3004.0 iii Heba 26 == 333}! (0) +.0419 B95) (1? 1780 May 24 So, _ —.0956 — 2640.7 1794 Nov. 16 = SVL Ih +.1017 3337/33 7/9) 1802 Feb. 23 = Bi LSL 7 +.0529 —— 845) 1823 Nov. 13 F753) + .0944 —3036.8 1831 Feb. 18 —— 343) [tshd) +.0659 =e Olden 1838 May 19 IAN 7(5}, 7 —.0S66 77/719) 1845 Aug. 17 3) | — (7A = 222029 1852 Nov. 15 — 2847.0 + .0863 SN GY9).3 1860 Feb. 14 3161.4 +-.0740 — 3468.3 1867 May 15 = 9397/35, I a) Sey, IPT {0 The perturbations by Uranus and Neptune were computed from the values of their terms just given. The principal terms, the sum of which make up the helio- centric longitude resulting from the adopted elements, are shown in the first of the following tables. In the next table we have after the date the heliocentric longitude from Bouvard’s Tables, as deduced from the longitudes given in Airy’s reductions of the Green- wich Observations, from the Astronomisches Jahrbuch for 1831, and from the Nautical Almanac. Then follow the corrections, roughly deduced from observa- tions made near the opposition. Adding these columns, we have the longitude from observation. THE correction of the elements. ORBIT OF URANUS 67 To the right of these are the equations of condition for the Perturbations by Long. of Mean Equation of _____| Red. to | Nuta- True Perihelion. anomaly. centre. Ecliptic. tion. longitude. Jupiter. | Uranus.} 5 EPS une. o r ww or " © 0 ' " yy, ” G00 ” ° row 88 41 27.5 | 159 55 50.1) +2 3 56.6 | —29 52.7 | 45.7 |—2.7 |4-1 36.9 | 115.8 | 250 12 25.8 88 46 41.2) 23613 7.1|—5 7 38.1 | —33 58.2 |—36.6 |+1.0 |—1 20.5 | —11.8 | 31916 4.1 88 47 31.4-| 248 25 53.1 | —5 48 30.4 | —35 53.2 |—35.2 |+1.7 |—1 35.8 | —15.0 | 330 46 36.6 8850 7.6 | 286 26 29.8|—6 16 1.3 | —44 24.6 |— 3.3 |42.9 |—0 43.5 | —14.1 8 15 13.5 88 51 51 6 | 311 44 13.6 | —d 0 58.2 | —52 37.1 |+11.9 |+2.4 |+0 43.3 | — 4.9] 34 43 22.6 88 53 35.1 | 336 55 56.1 ; —2 41 10.6 | —58 24.0 |419.2 |41.9 |4+1 36.6) + 6.7; 6212 1.0 Sen OO ORGalGDI40e 2:3 |e=1=6) 0035-9159) 17.2) |—3.29) | 058) |= 16 Sire | + 4.8) 159 38 54.6 89 543.4) 154 8 4.8) +2 37 57.7 | —44 0.7 |—53.8 |—1.7 |41 37.5 | —14.5 | 245 8 13.2 89 17 50.7 | 331 610.4|—318 5.2 | —62 17.9 9.6 |-+3.5 |+1 30.9 | —14.1|} 56 5 17.9 89 23 55.7 | 5956 17.9 | +5 44 46.6 | —57 5.5 |—18.6 |42.5 |—1 37.3) + 1.4] 154 6 2.7 89 42 7.0 | 325 22 26.0 | —3 51 58.0 | —50 36.8 |—27.9 |—3.0 |+1 21.6 | 414.7 | 5023 3.6 89 4811.9) 54 1033.1} +525 2.6 | —61 11.7 /+13.3 +2.5 |—1 33.2 | — 7.4 | 148 21 11.1 89 54 16.1 | 142 44 37.0) +3 40 38.1 | —46 17.9 |— 3.4 |+2.1 |+1 29.7 | — 4.3 | 235 34 37.4 90 0 20.4 | 231 18 40.9 | —4 47 36.6 | —37 0.9 |—35.5 |+0.9 |—1 11.8 | Papal 8 | 3115 52 5202 90 6 24.6 | 319 52 44.8 | —4 21 53.0 | —52 39.3 |4- 4.7 |—0.9 |41 8.8] —18.4] 44 45 31.3 COMBE T 48126 4887 F-51464) 57-4853) 423058 |= 3.4 |= 95.0) | Sila | 149949 314 90 18 32.8 | 137 0 52.6 | +4 931.2 sine i 8.2 |+-1.3 |+1 21.6 | — 4.1 | 230 52 59.7 eae i feteiec | 20tR tane foe Equations oF Coxpittox. ul ” 1751 May 31 | | 250 13 38.0 | — 8.5 | 250 13 29.5] 63. 7=+0.905e —445n +0.605e 41.82e50 1757 Aug. 27 | 319 1T 8.5 | —18.0 | 319 16 50.5] 46.4 0.94 —39 —1.53 +1.14 1758 Aug. 27 | 330 47 37.9 | —14.3 | 330 47 23.6} 47.0 O23 =A) Saye ALO.E0 1761 Oct. 6 8 15 13.8] + 0.2 8 15 14.0 0.5 1.03 —39 —1.99 —0.50 1763 Nov. 1] 384 43 42.3) +15.3| 34 43 57.6] 35.0 LO 8) Gey a1 1765 Nov. 23) 62 12 20.7 | +20.4/ 6212 41.1 § 40.1 1.11 —38 —0.88 —1.88 1773 Feb. 26 | 159 40 3.8 | 418.8 | 159 40 22.6] 88.0 1.04 —28 -+1.93 —0.79 1780 May 24/245 9.17.6] + 7.8] 245 9 25.4] 72.2 0.90 —18 -+0.76 +1.74 1794 Nov. 16) 56 5 38.8 + 14.7 | 56 5 53.5 | 45.6 1.10 —6 —1.08 —1.78 1802 Feb. 23/154 7 2.7| +10. 8 | 154 % 13.5] 70.8 1.05 + 2 +1.85 —0.98 1823 Nov. 13 | 50 23 34.5 | +21.3| 50 23 55.8] 59.2 0G) 4293-3 1831 Feb. 18 | 148 22 38.5] + 3.0 | 148 22 41.5 | 90.4 1.07 +383 +41.75 —1.16 1838 May 19 | 235 36 11.2) + 3.4 | 235 36 14.6] 97.2 0.91 +35 41.08 +1.56 1845 Aug. 17 | 315 53 42.2 | +15.7 | 315 53 57.9 | 65.7 0.938 +42 —1.43 +1.28 1852 Nov. 15} 44 46 15.4| + 5.9 | 44 46 21.3] 50.0 1.09 +57 —1.42 —1.53 1860 Feb. 14 | 142 44 16.2 | —13.7 | 142 44 2.5] 91.1 1.08 +65 +1.68 —1.35 1867 May 15 | 230 54 47.8 | + 5.1 | 230 54 52.9]113.2 +0.92 +62 +1.28 -+1.45 A normal equation for de is obtained by taking the sum of all the equations. That for én is formed by subtracting the sum of the first seven from the sum of the last seven, and those for bY ce and e\o by taking the sum of the equations in which the coefficients of ¢e or eo are greater than unity, after changing the signs of the equations in which they are negative, The normals thus obtained are 63 THE ORBIT 17.198 + 81dn — 2.168e — 3.15edo = +1069.3 4 939.3 + 208.1 He 5358 0.04 +587 — 24 2207 = Oia =n0 These equations give 1256183. Y==s0169 421.58 + 2.19 —- 3161 ae19l66 + 64.8 (Epoch, 1800.) a én =-+ 0.268 ée = + 12.6 eo =—+ 8.2 —Substituting these values in the seventeen equations of condition we have the following residuals, or excesses of theoretical over observed longitudes: Ui 1) Se yt eee 3 02 ey 5 = 190) 6 yal (ties Se we Be es Re These residuals are much larger than they should be, and I scarcely know to what cause to attribute their magnitude. ‘The results are however amply reliable for the purposes of the investigation, and lead to the following elements of Saturn: 1, 10 11 12 13 14 15 16 iz 1°) Ul 90 6 26 14 50 3.2 £12207 30 2 29 39.2 43996.395 .0560660 log (a+ da), 0.979676 Epoch, 1850, Jan. 0, Greenwich mean noon. It will be seen that the adopted position of the plane of Saturn’s orbit is retained. It was corrected from observations before the perturbations were finally computed. Of the above corrections, those of the epoch and mean motion need not be taken account of in the corrections of the co-ordinates, since the mean longitude remains in the formule as an arbitrary quantity to the end. The effect of the correction The corrections of eccentricity aud perihelion They are allowed for by adding to dv and dg of the mean distance is insensible. are therefore alone to be retained. the terms OF URANUS. =e TOeT | ++ 6.4 i 0.4 0.0 3.7 4.1 eh THE ORBIT OF URANUS. 69 dvu= 8§e sing —edw cosg = + 25".2 sin g — 16’.4 cosg; dp = — de cosy — eda sing = — 12.6 cosg — 8".2 cos g. Perturbations of Saturn and Uranus. The following expressions include, with these corrections, all the perturbations of Saturn and Uranus which can produce any appreciable perturbations of the second order in their mutual action. In these expressions the initial letter of each planet is put for its mean longitude counted from the perihelion of Uranus. PERTURBATIONS OF SATURN. Argument. to . 2. =} o> bo _ = OO ~~ J df J 2 2S 27 5 — 2S) > rs oe Mork. mor) 9 eb OD Or bo bO em Oo bo we ARRBOWODWNRAN REDE oO Ot —— aS 2U —28 Sia) 3U —28 sul tee eee) | eee 0 6 8 4 4 3 —_ PERTURBATIONS OF URANUS. Argument. ° cS) rn 2.2) ma or o~ U U— § 2U— § 3U— S 29'S: 3U —2S 4U —28 5U —28 oo coor R RN DO aa mRoOLbTOSOS +44 | Wee oSNwWAaS SHMODAND Fede) sinalt| Let us now resume the equation ; : sQ=2f Dsra+ 5°” _! pp wry 4) Op ll 2 To 70 TUESE OM Bale LOR WiaeAGNa Uae Beginning with the last two terms of this expression, it may be shown at the outset that they are quite insensible. The effect of the constant terms in dp and dp’ has already been included by correcting the logarithm of the mean distance by their amount; they are therefore omitted. The largest remaining term is 64’, the square of which is only 0’.02. In the product 7;°dp’ the largest terms are + 0.014 — 0.013 sin g — 0.011 sin (8g — 27’) — 0.011 cos (4g — 27) which may be entirely neglected. We shall therefore only consider in §@ the terms , IR 2 DsRdt + ee As already remarked, & is rigorously a function only of V, p, and 9’, V being the angle made by the radii vectores of the two planets. But, in the analytical development of R, the quantity V is considered as a function of v, v’, and y, so that we have R —= 1K; 5 V,V; y). In the previous computation of the perturbations of Uranus, we have supposed R to be a function of Pros p, etc. The corrections to & and its derivatives with respect to v and p are now given by the equations (11), with the modifications shown on pages 24 to 27. The derivatives of R, which enter into these equations are formed as follows: If, in the value of 2 produced by the action of Saturn on Uranus, we consider any term of the form mh as Ne in ae wat + ja + 7" the accented quantities always referring to Saturn, but a, being the corrected mean distance of Uranus, then we shall have the following terms in che derivatives of 2. cos NV where OR mh Tag ae —@ i+ 7)sin N on et 28 ah +7’) sin N OR m Oh OR m oh ag” = a a cos V 9 Ok nt mh “i +i) ON, O Vv: THE ORBIT OF URANUS. wal OR mh ,. , eee ig eres, °| OVvOv ss (+ 7) +7’) cos N OR om oh : : } Ovdp cot er (x Si =) ((+ 7) sin .V __ ok _@R = OV OVOP OR mn’ oh ; fr Ovo’ mo a, On ( tae ) sll N OR m Oe aes Opov’ a (7 =i Ov )¢ +7) sin N Ok m Oh oh = x 5) —~ N Op” a, (n+ I SSSA ) Ss ee OR Op Ops OR moh . Oh ARG) aa Eas ae s N Opdog’ Al On + a) co All the numerical data necessary for the computation of these derivatives have been given in Chapter II]. Combining the terms having the same argument, we find the following values, omitting those given in Chapter IH, and those which ; ie PR : are derived from the others by mere addition, The terms of — are also omit- =a 9 : 3 oO” : ted, because they are sensibly the same with those of _ ,, changing the alge- OVOV braic sign. a, OR m! Ovov sin cos +0.2874 0 0 +0.2872 +0.2691 0 40.0066 —0.0310 +0.0310 0.0067 -+0.0060 —0.0322 —0.0102 —0.0491 +0.0490 —0.0101 +.0.0249 +0.1194 = 3 AST) —0.0010 —0.0021 | —83.4810 —5.2684 0 —0.0968 +0.4304 —0.4304 | —0.1039 —0.0792 +0.4583 +0.0388 +0.0173 —0.0179 +0.0389 +0.0398 -+0.0163 —0.0045 —0.0035 40.0021 —0.0080 | +0.0155 +0.0066 —0.0544 —0.0712 +0.1426 —0.0689 "| —0.1760 +0.2403 +0.4104 +0.0120 oor al 40.8233 | —1.3515 40.0014 —0.0053 —0.0191 +0.0390 —0.0226 | +0.0288 +0.0547 —0.0070 —0.0093 40.0171 —0.0210 +0.0197 +0.0304 +0.0570 —().0650 +0.1957 +0.1013 —O TOL .2840 +0.2542 0 —0.0065 10.7624 | —1.0884 0071 —0.0014 —0.0130 +0.0368 —0.0193 +0.0213 | 0547 +0.0515 —0.0499 +0.1987 +0.1450 | —0.2144 2647 +0.1448 —0.0015 —0.0016 +0.5789 —0.7644 —0.0005 {DD THE ORBIT OF URANUS. a, OR m Ove cos i cos ete 40.172 “aye +0.002 — 0.026 —0.565 +0.556 0 +0.070 0.015 +0.015 +0.070 —0.003 +8.749 —8.750 —0.001 +0.889 0.180 —0.176 +0.889 +0.084 | +0.248 —0.230 +0.169 +0.018 +0.472 —0.941 | +0.013 +0.012 +0.004 +0.013 +0.021 +0.073 +0.068 —0.122 | +40.219 0 +0.278 —0.834 —0.007 +0.015 —0.001 -+0.020 +0.042 +0.001 +0.155 —0.619 0 The derivations with respect to y and the node have been omitted because they are quite insensible. The terms of )# depending on these derivatives are given by equation (31). In the case of Uranus disturbed by Saturn the largest values of the coefficients 1 1 Zihcothy; sh tanhy,} are only about .05, while the largest coefficients in df, dX’, and dy are less than 10’. Hence the largest terms in (31) will be of the order of magnitude 0”.5 multiplied by the mass of Saturn, and may therefore be omitted entirely. Omitting them oR the values of dR, Ja — and 5 become p oR oR oR oR = A a \ , x 2 , 6sk= a év + ay oe ss a dp + ag! 6 oR OR OR oR OR : ‘Ov OV? SPatse OvOv te ees + aap? ok Oo OR 5 = orp” + sept + aye 8 + Sap All the separate factors from which the second members of these equations are formed have already been given. Forming their products in the way described in Chapter II, we have the result given in the following tables. The expressions for §R are arranged so that the value of DSR can be obtained from them by direct differentiation. This is done by distinguishing the time introduced into R by the co-ordinates of Uranus from that introduced by the co-ordinates of Saturn, THE ORBIT OF URANUS. ee = 9 \ Dp 5 ae 5 on bu on rae cee § oF bu’ ok 5p. ) m (OV Op i) m (ov Op 5 Cams: cos sin Ci Sted ” ” ” mn? ji = 1e40r == 183t Ply 39 +375 +7336 a, 0 + 0.06¢ — 0.28: | +2, —990 —558 ane = 73 —261 o—1 — 0.24¢ 4+ 0.206 | +4, + 23 — 42 Hip + 0.03¢ 4+ 0.04 a == 105¢ — 0.95¢ 0, 2 —64% —319 3,—1 + 0.02¢ — 0.30¢ L. +773 +5616 2, +153 —103 0,—2 + 0.93¢ — 0.83¢ a + 24 — 26 2 —12.69¢ 411.60¢ 4, 0 0 je) — 0.24¢ + 0.308 = — 0.63¢ — 0.59§ J—2, 5—2 + 90 + 60 = —625 +73: is — 1.60¢ = IL 0, Se a0 — 76 — + 0.25¢ — 0.29¢ Ue + 61 4719 3,—3 + 0.0S8¢ =EeOSO2E 2, — 20 + 6 s—3 — 0.36¢ — 0.34¢ i 497.5 == B09 3—4 + 0.19¢ — 0.17% J—3, +94.1 at aed 4,—4 4+ 0.15¢ 0.0 J—2 + 66 +243 Pie —8823 —6114 UU S: 0, +699 +522 i 0 9 = 38! eae p= J. ©) + 6 —4, 72 + 70 == 9 09 =I) —153 3}. +299 +147 1, 0—3 + 14 ls —2, +937 +706 oll + 56 = 2 —l, —1978 +1117 jes rea a 2 0 eee J. 8 4+ 43 —241 — 1 + 19 9a, (OR OR_) 1,+2—2 ey JLB = tan oot on © 1—242 aos 0 ys 1-34 ==798 Fess ool + 30 +100 U Ss’ Ss cos sin Fact. né 1,+3—2 +140 +112 ” ”" ” — 9 — cen i) tort zn STi) S120 spo | 01as2 US J 3; + 59 +801 +0.148 A t OF 0 1-4 4.946 ; 4, —139 +353 +1.448 ib —2469 =5 fi ; 2) mens a oA 3,—1—2 —122 + 26 +0.148 4 Tee ioe 4, — 60 saO9 SS 7 &, + 94 60 +2.148 —=9 om g x ae et tee Wo | Ses ela 950 0 ©| —2.852 = +1638 188 1 / a i880 0 wy: Gy | lass % ae == 1-882 iL 89 89 2. +197 —450 07852 ry ra aya 3, ili + 38 +0.148 1 a: —3.852 seo! + 6 — 8 i oa Fe ome Bea 7% 3p ae tee it — 80 3 —1.852 ae 4478 —280 : pe aa %, 110 — 90 L416) —1.852 2 a6 +158 —0.852 —3, 4— ¢ 28 : ‘ 36 = ! as ape 3, 4 12 = 39 40.148 i; +297 —286 10 May, 1873. 74 THE ORBIT OF URANUS. +] 1+ +441 THE ORBIT OF URANUS. 75 ” ” 0 — 84 +2469 —2947 + 44 —1i6 ==0.9 — 17 aad) +1231 ear —196 =i + 84 + 42 +1638 8 +1642 0 mained + 89 - =-- $3 Aad = il Hae tek Je ap + 95 — 42 +132 +143 +4478 SEAT 0 — dT m 7 ene ee Lee ell apse secre || te [o/) + 27 + 71 —442 —449 297 286 +310 —375 3: —— 598} +1980 5 +1987 +219 Se + 51 — 92 i —104 0 —T13T —306 —— (2 a +180 —625 0 ==> 61 + 40 +110.0 +282.3 +132 —8893 R 312 73 —9020 0 0 93 +406 + 48 +897 —9ITT —2195 +904 — il 75 TEE OR BET On SU RAGN Gis soe the terms of SR introduced by the perturbations of Saturn, namely, oR ob ‘ie ap ees dv and dp’ as constant, although they are expressed as a function of the mean longitude of Uranus, as well as of Saturn. The mean longitude of Uranus thus introduced is therefore represented by U’, which is regarded as constant in! taking D’,R, and a ee eae to vary. 3 OR = os - Uranus, their complete a ees with respect to the time, are to be taken. But their expressions contain the mean longitude of Saturn as well as Uranus. The mean longitude of Saturn thus introduced is represented by S’, and is to be con- sidered variable in obtaining D'6k, while Sis considered constant. The ratio of the coefficient of ¢ to m in the various terms of this part of §f is given to the right of each corresponding term. The value of DSR being once obtained, there is no longer any distinction necessary between U, U’, or between Sand S’. The similar terms are therefore combined by putting S’ = S; U'= U. From the above values of 25D’,R and 26 ee op % dp, the differentiation aaegoaiel by D', should be performed by con- Again, in the terms © ép,Since ¢v and dp represent perturbations of we form the following value of a ok Sas 1 f 3D Rdt + 55 and of the other quantities which enter the ee - the co-ordinates. We shall begin with those terms which depend only on the mutual action of Satum and Uranus, because they are few and small, and the only terms which are sensible are those in which the coefficient of the mean longitude of Satum is —l. We shall therefore confine ourselves to these. And, instead of employing the con- densed formule, we shall make the computation in full by (13). a > 5) & 39=2 (sp Rat OF uy “1 93Q nv m 5 mv m OP sin ” ” 47 + 78 — 10 +148 + +153 1440.53¢ |+307—0.51¢ |4+ 1340.02¢/4+ 49+40.05¢|4 36—0.06¢/— 840.011 24-+0.02¢ |4+ 62—0.08¢ |— 414-2.37t | 4-138+-1.62t Re 29—1.85t |—164+2.11¢ 604-4.23¢ |— 28-+43.84¢ |—270—0.13¢ |+157—3. 05¢ | +250—0.13¢ |+ 85—2.70t —525+-0.31¢ |4+240—5.49¢ |4 7-+-2.09¢ |— 31-+42.30¢ |— 3042.11¢ |4 14—1.926 962 40.15¢ | +120—2.74¢ |—262-+0.15¢ |_120-42, Tt | ” ” a Ooi oe +: 024+.0003 333-4.00437¢ 14200002 —.071 sin ar yr — .023+-.00002¢ —.036-+.00033¢ —.406-+.00432t +.079—.00169¢ +.019 sin ” ” 40.002 +0.00008 +0.043 +0.0006¢ +0.620-+0.0098¢ +2.69 —0.00) 1¢ +0.132 OE €os 0.021 40.0000 0.05840. 00004 +0.803—0, 0097 +0.830—0.0153¢ +0.040-+0.00004 THE ORBIT OF URANUS. ball The computation of these terms being extremely complex, a check upon their accuracy is desirable. In the case of the secular variations of the coefficients, the coefficients of the time are easily obtained by substituting in the integrated per- turbations the variations of the eccentricity and perihelion of Saturn. ‘Thus I have found év = +0.0103 ¢ sin (2g — 1) — 0.0094 ¢ cos (2g — 7) -+-0.0027 ¢ sin (dy — 1’) — 0.0138 ¢ cos (8g — 1’) The greatest discrepancy is found in the coefficient of sin (8g —U), and it amounts to 0".0038¢, or about 0”.4 ina century. But, owing to the great period of this term, nearly 600 years, this difference, during any one century, will be nearly eliminated through the mean longitude and mean motion. It may also be remarked that in this case the terms derived from the pertur- bations of the elements are undoubtedly the correct ones, and will therefore be employed. The terms which the preceding integration fails to give, owing to the constant terms introduced into £)@ and 43 Q, are found by (22). We thus have ee = + 0".36 Uy Guiee — a 0.27 Ue aie 36 sin g — y".27 cos g} dv=41] ss }0".36 cos g + . 27 sin g} = & $0" .0000038 cos g + 0".0000029 sin g}. The greatest effect of these terms amounts to less than one-twentieth of a second in a century. They may therefore be neglected in the present theory. The other terms containing the square of the time are yet smaller. Applying the terms of the second order thus found to the terms of the first order depending on the corresponding arguments, the perturbations of Uranus by Saturn become cos pdp T here represents the time counted in centuries from 1850.0. The other terms remain the same as given on page 50. Perturbations ie on the product of the masses of Jupiter and Siturn. The values of §D’,R, 56 fe and 3 oe depending on the products of the masses p 78 THE OR BL OY Uy RAy NaUES: of Jupiter and Saturn, are given on page 74. The computation from these data being conducted in the same way as in the case of the terms of the first order, it is not necessary to give much more than the results. ‘These are shown in the fol- lowing table. ‘Ihe indices to the left represent the coefficients of the mean longi- tudes of Uranus, Saturn, and Jupiter, all counted from the perihelion of Uranus, Column y gives the ratio of the mean motion of Uranus to the coefficient of the time in each argument. ‘The perturbations of the common logarithm of the radius vector are expressed in units of the seventh place of decimals. sin tr —0.2364 +0.002 —0.3095 —0.020 —0.4480 +0.004 —0.003 —0.8127 +0.016 —().024 —0.2960 —0.007 —0.001 —0.4204 —0.108 —0.007 —0.7254 —0.014 —0.012 —2.6420 +0.164 —0.267 +1.6090 0.005 —0.605 —0.6551 40.012 —0.002 —1.8997 40.175 —0.015 42.1115 +0.078 +0.512 0.6786 40.007 +0.024 +0.754 —0.030 0.081 +0.430 043 +0.005 +0.301 40.001 —0.003 —0.2170 —0.002 —0.032 —0.2771 051 +0.035 —0.3833 O10 +0.050 —0.6215 032 40.052 —0.362T 010 —(). 004 —0.5692 OTS —(0- 154 les OO .349 OT +4.1150 —0.510 —0.453 —0.5250 —0.25: —0.034 —1.1051 —4, 43: Soeir 410.5152 —0.546 —0.032 +0.9132 40.206 —3.254 +0.4773 +0.015 —0.192 —0.9497 41.824 —0.519 —18.9250 440.650 -10.500 +1.0558 46.237 21.866 + 0.5136 +0.467 —0.539 +0.3393 +0.007 —0.017 +0.5558 —0.050 — 9.003 +0.3573 —0.046 +0.032 +0.2632 —0.045 -+0.032 +0.2084 +0.006 +0.007 +0.1724 0 0 THE ORBIT OF URANUS. 79 CEHVAP TER: V. COLLECTION AND TRANSFORMATION OF THE PRECEDING PERTURBATIONS OF URANUS. Tue terms of the perturbations which neither contain the elements of the disturbing planets, nor depend on the secular variations of the eccentricity and perihelion, admit of being greatly simplified by a slight change in the arbitrary elements. ‘These terms are as follows: (1) In the longitude of Uranus 4/ 4/ 4 4) 4/ Action of Jupiter, -+31.2116¢ +25.657sing -+1.397sin2g —1.859cosg —0.087 cos 2g Action of Saturn, -+10.96906 -+ 8.545sing +0.461sin2g —4.735cosg —0.169 cos 2g Action of Neptune, — 0.4262 + 0.697sing -+0.046sin2g —0.088cosg —0.005 cos 2g Total, +41.7544¢ +34.899sing -+1.904sin2g —6.682cosg —0.261 cos 2g (2) In the value of cos dp, units of 7th place of decimals. Action of Jupiter, —10089 —492 cos g —33 cos 2g — 2sing +1 sin 2g Action of Saturn, — 3543 —184 cos g —15 cos 2g —l5sing Action of Neptune, + 138 — leosg — lcos 2g + Ising Total, —13494 —677T cos g —49 cos 2g —16 sing +-1 sin 2g Let us first consider the first or constant term in the perturbation of each co-ordinate. If we suppose a change of $m in the mean motion of a planet, the corresponding change in dp will be 23n ip = —5— p on If, then, we increase the mean motion of Uranus by 41”.754, the corresponding change in dp will be —18045, and in cos dp, —18025. Subtracting these from the above perturbations, the secular term in the mean motion will disappear, and we shall have for the constant term of cos de +4531 This same change in the mean motion will produce a secular term in the equa- tion of the centre of the same nature with that produced by the secular variation of the perihelion. The differences of the values of the secular terms, found by the two methods employed in Chapters II. and IIT., proceeds from the fact that in the one case the effect of the above term in the mean motion is included, and in the other excluded. 80 THE ORBIT OF URANUS If we subduct the effect in question when necessary, the remainder will be the effect of the secular variation of the longitude of the perihelion of Uranus, to which we shall revert presently. Let us next introduce such a change in the eccentricity of Uranus as shall pro- duce the term 34”.899 sin g, and ascertain its effect on the other terms, For this purpose we must determine ¢e by the condition C= *é) Se = 34”.899 which gives ce = 17’.464 = .0000847. A change of this amount in ée will introduce the following terms in dv and dp dv = 34’.899 sin g + 2".048 sin 29 cos dp = 20 — 844 cos g — 59 cos 2g. Subtracting these terms from the expressions previously found we have dv = — 0’.144 sin 2g — 6".682 cos g — 0".261 cos 2g. cos dp = + 4511 + 167 cos g + 10 cos 27 — 16 sin g + 1 sin 29. Again, let us put edn = 3’.342 = .0000162, we shall have the elliptic terms dv = — 6".682 cos g — 0".391 cos 2g cos Yép = — 162 sin g — 11 sin 2g. Subtracting these expressions the constant terms, independent of the mean longi- tude of the disturbing planets, are reduced to dv = — 0’.144 sin 27 + 0.130 cos 2g. cos Yop = 4511 + 167 cos g + 10 cos 2g + 146 sin g + 12sin 2g. 0.43429 dp = 1969 + 73 cosg + 4cos 2g + 63 sing + 9 sin 2g. In the last equation we have introduced the constant +-.0000008 produced in dp by the combined action of Venus, the Earth, and Mars. ‘The effect of each planet is computed by the approximate formula So = : m'(b® + aDab”). Secular Variations. The following inequalities result from the secular variations of the eccentricity and longitude of perihelion produced by each of the disturbing planets, 7 being the time expressed in centuries. From the variation of the eccentricity 4/7 a? 4/ Action of Jupiter, dv = —1.2167 sing —0.072T7 sin 2g —0.005 T sin 3g Action of Saturn, —9.1827' sin g —0.538 7 sin 2g —0.032 7 sin 3g Action of Neptune, —0.5027' sin g —0.0307' sin 2g —0.002 7 sin 3g THE ORBIT OF URANUS. 81 4 af Action of Jupiter, Msp = +13T cos g +17 cos 2g Action of Saturn, +98 T' cos g +7T cos 29 Action of Neptune, + 6T cos g The secular variation of the longitude of the perihelion is yy Action of Jupiter, +122.17 Action of Saturn, 4118.47 Action of Neptune, 4+ 51.17 Total, éxn=—-+291.6T The effect of this secular variation on the longitude and radius vector is wv " “ Action of Jupiter, év = —11.46T cos g —0.671T cos 2g —0.047 T cos 3g Action of Saturn, —11.11T7 cos g —0.651T cos 2g —0.039 T cos 3g Action of Neptune, — 4.80T cos g —0.2817' cos 2g —0.016 7 cos 3g Total, —27.37T cos g —1.603 7 cos 2g —0.1027 cos 3g ” n” Action of Jupiter, Msp =—120T sin g —8T7'sin 2g Action of Saturn, —117 7 sing —8T sin 2g Action of Neptune, — 507 sin g —3T sin 2g For the purpose of conveniently tabulating the perturbations, we shall express them in a form similar to that adopted in the theory of Neptune. Let us select, from the terms of the periodic perturbations produced by any planet, all those in which the difference between the indices ¢ and 7?’ is the same. For example, in the perturbations of the longitude produced by Jupiter, let us consider the terms dv =+1.269sin¢ — J) — 3.495 sin (2g— 2) + 1.182sin( g — 2/) + 0.074 sin (8g — 2/) — 0.005 sin (2g — 32) + 0.011 sin (47 — 31) +0.002cos( — 2) — 0.092 cos (2g — J) + 0.515 cos ( g — 22) — 0.005 cos (8g — 2/) — 0.001 cos (4g — 32) These terms may be expressed in the form + 0.094sin (g —?) év=sing X + 0.520 sin 2(g — 1) — 2.226 sin (g —/) + 1,256 sin 2(g — l) + 0.006 sin 8(g — 2) + cosy X ——_——“—SS OO wo 11 May, 187 —4.764cos (g—/2) — 1.108 cos 2(g — 1) + 0.016 cos 3(g — 1) — 0.090 cos (y —?) + 0.510 cos 2¢g — 2) 82 THE ORBIT OF URANUS. In general, a series of terms of the form Ya, sin (2A + sg) + 2b, cos (iA + 89) + Sa’, sin ((A — 8g) + 2 0’, cos (A — 89), may be put in the form {= (a,—a’,) cosiA — ¥ (6, — &,) sini A} sin sg + {XS (a,+a’) sintA + 3 (6; + 8) sinzA} cos sg. All the periodic terms containing only g and / in the arguments may be put into | this form by taking A=g—l, so that the coefficients of sin sy and cos sy may all be expressed as a function of the single variable argument A. The “pestnniei ams of the elements may be “eine to perturbations of the co-ordinates expressed as the sum of several products of slowly varying functions into the sines and cosines of the multiples of gy. We have, in fact, 6v = el +(2- ve *) Se x sing (oie °Vedg X cos g ee 8) fe < sin 2g (G3 a °) eby X cos 29. + ete. + ete. It appears, therefore, that all the perturbations in which the arguments contain | the mean longitudes of only two planets may be put in the form dv = (v.c.0) + (v.c.1) cos g + (v.c.2) cos 2g + ete. + (v.s.1) sing + (v.s.2) sin 2g + ete. Mdp = (p.c.0) + (9.6.1) cos g + (p.c.2) cos 2g + ete. + (.s.1) sing + (.s.2) sin 27 + ete. We have next to reduce to the same form those terms which contain the mean longitudes of both Jupiter and Saturn, and which are given on page 78. We have here twenty-four terms, each greater than 0’.04. As most of these terms depend on three independent arguments, they cannot be included in a double entry table, while, if we include them as perturbations of the longitude in tables of single entry, we shall have to enter twenty-two tables with as many different arguments. But, by taking, for the argument A, the middle one in each series of arguments which jlgreail on the same multiples of Jupiter and Saturn, and expressing the terms above and below it in each series as coefficients of sin g, cos g, sin 29, and THE ORBIT OF URANUS. 83 cos 2g, we may reduce the number of arguments to eight, and the number of tables to seventeen. Consider, for instance, the terms of the second series, — 0.108 sin(— g + 2S—J) — 0.007 cos (— g + 28 — J) — 0.014 sin ( 2S— J) — 0.012 cos ( 2S — J) +0.164sin¢ g+2S—J) —0.267cos( g+2S—J). These terms may be allowed for by adding to (v.c.0), (v.s.1), (v.c.1), the terms (v.c.0) = — 0.014 sin (2S — J) — 0.012 cos (28 — J) (v.8.1) = + 0.260 sin (2 — J) + 0.272 cos (28—J) (v.c.1) = + 0.056 sin (2S — J) — 0.274 cos (2S —J). From the perturbations of longitude and radius vector already given, we readily find the following values of (v.c¢.0), (v.s.1), ete. Acticn of Jupiter. A,=I'—g (v.c.0)=+53.064 sin A, —0.004cos A, — 0.277sin 2A, +0.036 cos 2.4, — 0.025 sin 3A, (w.c.1)=+ 2.226 sin A, —0.090cos A, (v.s.1)=—0.094sin A, —4.764 cos A, — 1.256 sin 2A, +0.510 cos 2.4, —0.520 sin 2A, —1.108 cos 2.4, — 0.006 sin 34, +0.016 cos 3.4, —11".467 —1".227 (v.e.2)=-+ 0.121 sin A, —0.038 cos A, (v.s.2)=—0.056 sin A,—0.175 cos IO OO re bO so OO He OO rR OO OAT © bo ys) Deo BPR} .89 PU ete el al We CoOCOCOrFRNWHROS ae ae TA 12 May,1873. 90 THE ORBIT OF URANUS: VALUES OF (v.¢.1) Q) @) (@) (4+) (6) Jupiter Saturn Neptune Neptune Jupiter & Satarn Sum. (sec.) (sec.) (sec. ) (long per.) (long per.) 4/ /f Wd ” A ” 1700 +17.19 +16.67 +17.20 +37.63 —1.98 +76.71 1750 +11.46 +11.11 +4.80 +28.67 —2.28 +53.76 1760 10.31 10.00 4.32 26.46 —2.36 438.73 1770 Os LLYS 8.89 3.84 24.10 —2.45 43.55 1780 8.02 1.78 3.36 21.60 —2.53 38.23 1790 6.88 6.67 2.88 18.95 —2.61 32.77 1800 9.73 5.56 2.40 16.16 —2.69 27.16 1810 4.58 4.44 1.92 13223 —2.16 21.41 1820 3.44 33583 1.44 10.15 —2.83 15.53 1830 2.29 2.22 0.96 6.92 —2.90 9.49 1840 + 1.15 So Heli +0.48 + 3.53 —2.97 + 3.30 1850 0.00 0.00 0.00 0.00 —3.03 — 3.03 1860 — 1.15 — 1.1 —0.48 — 3.68 —3.08 — 9.50 1870 — 2.29 —- 2.22 —0.96 — 7.51 —3.13 —16.11 1880 — 3.44 — 3.33 —1.44 —11.49 —3.16 —22.86 (v.s.2) consts— 044s () (2) (3) (4) (5) Sum. A uM di yr u "Nr 1700 +0.11 +0.81 +0.04 —12.46 —0.72 —12.36 1750 +0.07 +0.54 +0.03 —8.3l —0.70 —8.51 1760 0.06 0.49 0.03 —T.48 —0.69 —.73 1770 0.06 0.43 0.02 —6.64 —0.68 —6.95 1780 0.05 0.38 0.02 —).81 —0).66 —6.16 1790 0.04 0.32 - 0.02 —4.98 —0.65 — 5539 1800 0.04 0.27 0.02 —4,15 —(.64 —4.60 1810 0.03 0.22 0.01 —3.32 —0.63 —3.83 1820 0.02 0.16 0.01 —2.49 —0.61 —3.05 1830 0.01 0.13 +0.01 —1.66 —0.60 —2.27 1840 +0.01 +0.05 0.00 —0.83 —0.58 —1.49 1850 0.00 0.00 0.00 0.00 —0.57 —0.71 1860 —0.01 ——O205 0.00 +0.82 —0.56 +0.06 1870 ——Os0) — O11 —0.01 1.65 —0.54 0.84 1880 —0.02 —0.16 —0.01 +2.48 —0.53 +1.62 (v.¢.2) const. = + 0/7.13. (1) (2) (3) (4) (5) Sum. A ” ” y W " 1700 +1.00 +0.98 +0.42 +2.20 ——() aval +4.62 1750 +0.67 0.65 +0.28 +1.68 —0.12 +3.29 1760 +0.60 0.59 0.25 1.55 —0.12 3.00 1770 | +0.54 0.52 0.22 1.41 —0.12 2.70 1780 +0.47 0.46 0.20 1.26 —0.13 2.39 1790 +0.40 0.39 0.17 1.11 —0.13 2.07 1800 +0.34 0.3: 0.14 0.94 —0.13 15 1810 +0.27 0.26 0.11 0.77 —0.14 1.40 1820 +0.20 0.20 0.08 0.59 —0.14 1.06 1830 +0.13 0.13 0.06 0.39 —0.14 0.70 1840 +0.07 +0.07 + 0.03 +0.20 —0.15 +0.35 1850 0.00 0.00 0.00 0.00 —0.15 —0.02 1860 —0.07 —0.07 —0).03 —0.21 —0.15 —0.40 1870 —0.13 —0.13 —0.06 —0.43 —0.16 —0.78 —0.20 —0.68 —0.16 == i15)'5) 1880 —0.20 —0.08 THE ORBIT OF URANUS 91 (v.s.3) Q) (2) (3) (4) (5) Jupiter Saturn Neptune Neptune Jupiter & Saturn Sum. (sec. ) (sec. ) (sec. ) (long per.) (long per.) 7 " ”f 4? 47 47 1709 +0.01 +0.05 0.00 —0.76 0 —0.70 1750 0.00 +0.03 0 —0.51 0 —0.48 1760 0 0.03 0 —0.46 0 —0.43 1770 0 0.03 0 —0.40 0 —0.37 1780 0 0.02 0 —0.35 0 —0.33 1790 0 0.02 0 —0:30 0 —0.28 1800 0 0.02 0 —0.25 0 —0.23 1810 0 0.01 0 —0'20 0 —0.19 1820 0 0.01 0 ——(Oe15 0 —0.14 1830 0 -+-+0.01 0 --0.10 0 —0.09 1840 0 0.00 0 —0.05 0 —0.05 1850 0 0.00 0 0.00 0 0.00 1860 0 0.00 0 +0.05 0 +0.05 1870 0 —0.01 0 +£0.10 0 0.09 1880 0 —0.01 0 +0.15 0 +-0.14 (v.¢.3) q) (2) (3) (4) Sum. 4, 47 " uw " 1700 410.06 +0.06 40.02 40.13 40.27 1750 -+0.04 +0.04 +0.02 +0.10 +0.20 1760 0.04 0.04 0.01 +0.09 +0.18 1770 0.03 0.03 0.01 +0.08 +0.15 1780 0.03 0.03 0.01 -+-0.07 +0.14 1790 0.02 0.02 0.01 +0.06 +0.12 1800 0.02 0.02 0.01 +0.06 +0.11 1810 0.02 0.02 +0.01 ++0.05 +0.09 1820 0.01 0.01 0.00 +0.04 +0.06 1830 +0.01 +0.01 0.00 +0.02 +0.04 1840 0.00 0.00 0.00 +0.01 -++0.02 1850 0.00 0.00 0.00 0.00 0.00 1860 0.00 0.00 0.00 —0.01 —0.02 1870 —0.01 —0.01 0.00 —0.02 —0.04 1880 —0.01 —0.01 0.00 —0.04 | —=(0506 For THE RADIUS VECTOR. VALUES OF (p.c.(0) Q) (2) (3) (4) (5) Sum. ur " " ” wt " 1700 +0.4 +3.5 —0.2 +164 +13 +181 1750 +0.3 +2.3 —0.1 +110 7 +120 1760 0.3 2.1 0 99 6 107 1770 0.2 19 0 88 5 95 1780 0.2 1.6 0 rare 4 83 1790 0.2 1.4 0 67 3 72 1800 0.2 1.2 0 56 2 60 1810 0.1 0.9 0 45 2 48 1820 0.1 0.7 0 34 | + 1 36 1830 +0.1 0.5 0 22 0 93 1840 0 +0.2 0 + 11 0 + 11 1850 0 0.0 0 0 — | — Fl 1860 0 —0.2 0 — ll — 2 — 13 1870 = —0.5 0 29 aang OG 1880 pil —0.7 0 — 33 — 3 — $i 92 THE ORBIT OF URANUS: (p.s.1) const. = + 63. qd) (2) (3) | (4) (5) Jupiter Saturn Neptune Neptune Jupiter & Saturn (sec. ) (sec. ) | (sec.) (long per.) (long per.) i WwW | ” | ” +180 +176 +15 +396 4120 +115 +50 +302 108 105 45 280 96 94 40 255 84 82 35 228 72 70 30 200 60 59 5 170 48 47 2 139 36 35 106 24 23 72 + 12 + 12 oh BY 0 0 0 Sh) = Ag 739 24 23 80 == (36 85 —122 (p.e.1) const. = + 73. (2) (4) 4‘ i —147 42941 ==198 +1496 Eeyss 2s 1346 nis aes, 1197 69 —4 1047 — 59 4 897 — 49 = 747 9 598 — 39 Loe 448 = 90 en | 299 2 4+ 149 0 0 0 aa] — 149 20 1 298 29 Ca PS aay He Om OR OO He OI OO const. = + 5. (3) (4) ” 428 421 19 17 15 =~ jay - = = { > _ NOD K SOF NW WROD AATH bo a> > a + _ ie) + | NDF OFM NNW ROAD T bo et OS em bo bo Oo He He Or Or eceooeocececocoeoces oc sg} — Pre TN rR te THE ORBIT OF URANUS. 93 (p.€.2) const. = + 4 (1) (2) (3) (4) () Jupiter Saturn Neptune Neptune Jupiter & Saturn (sec. ) (sec. ) (sec. ) Cong per.) (long per.) A uy —1 —10 il T —l 6 —l 6 —1 5 —ll 4 0 2 +157 +105 95 84 74 63 52 42 31 21 10 0 10 21 — 3l oooocoocoocococecoc & oooooococoocqcooco oo oooococooco DRE oe = bw Reduced Expressions for the Latitude of Uranus. If we represent by V,, V2, V; the distances of Uranus from its descending nodes on the respective orbits of Jupiter, Saturn, and Uranus, we find the following perturbations of the latitude, which are independent of the mean longitude of the disturbing planets. 63 = —0.0114¢t cos V, — 0.0477t cos V, — 0.0125¢ cos V; th + 0.245 " + 0.386 sin g + 0.266 cos g — 0.043 sin 2g + 0.006 cos 29. To find how far the last five terms may be represented by simple corrections to the elliptic elements, we first represent the effect of minute corrections to the inclination and node of Uranus as a function of its mean anomaly. Putting w for the argument of latitude of Uranus, we have to a sufficient degree of approxima- tion 63 = sin udp — sin @ cos wd u =g+o-+ sing sinu = —esina + cososin g+ sinwcos g + e cos @ sin 29 + e sin cos 2g 94 THE ORBIT OF URANUS. cos u—= — e Cosa + coswcos g— sinwsin g + € cos w cos 2g —esinga sin 2g. Substituting these values of sin w and cos w in the expression for 53, and putting sin 9d) = 00, we have 68 = ecos od) — esina dp + ( cos ad? + sino’) sin g+ ( sinodd — cos ws) cos g + (e cos add + e€ sin wS'9) sin 27 + (e sin od — e€ Cos @5'0) cos 29. To represent the numerical coefficients of sin gy and cos g in §3 we must put cos add + sin oS) = 0".386 sin add — cos wd’) = 0 .266. Since wo = 95° 3/, this gives ” op = 0.231; 66= £é0.409; §2 — — 0.013 4, + 0.386 sin g + 0.266 cos g + 0.018 sin 2g + 0.013 cos 29 Subtracting this expression from the corresponding terms of 43, we have left 63 = + 0".258 — 0".061 sin 2g — 0’.007 cos 2g. The first term of this expression shows that the mean orbit of Uranus at the present time is a small circle of the sphere one-quarter of a second north of its parallel great circle. If we put v = longitude of Uranus in its orbit, referred to the equinox and ecliptic of 1850, we have V, = 0 — 127° 37 V,==v — 126 45 V;=v— 155 32 Substituting these values in the first three terms of 53, and multiplying the last term by the factor (1 + «) by which the adopted mass of Neptune, ;,;}55, must be multiplied to obtain the true mass, we find 63 = (4.69 + 1”.14u) Tos v — (5.24 + 0".52u) 7'sin v. To these terms must be added those which arise from the motion of the ecliptic. . In the absence of any exhaustive investigation of the obliquity and motion of the ecliptic, I adopt the elements of Hansen, employed in his “ Tubles du Soleil,” because they are a mean between the results of others, and are very accordant with recent observations. The secular motion of the obliquity there employed is — 46".78. THE ORBIT OF URANUS. 95 Hansen mentions — 5.39 as the corresponding motion at the equinox of 1850, found by Olufsen, but I cannot reproduce this result from the secular diminution with any masses of Mercury, Venus, and Mars, which seem to me probable. The expressions in terms of the masses given by Le Verrier are (Annules de [ Observa- toire Imperial de Puris, tome ii, p. 101), Secular change = — 47.59 — 0.52 — 28.90»' — 0.83” Mot. at equinox = + 46.89 + 0.62» + 7.57)’ + 0.732”. In this expression the masses of Mercury, Venus, and Mars are represented by inl” foe and i apes 3,000,000? 401,847’ 2,680,337 changes in the masses of the other plants is insensible. From the researches of Le Verrier on the motions of the four inner planets I conclude that the following are about the most probable distribution of the correc- tions of the masses necessary to produce the motion of the obliquity given by Hansen, namely, respectively. The influence of admissible a 2 nie mae 2 ONS Ff ge == enls a 10 These values give for the motion at the equinox of 1850 457.43 Introducing the secular variation of these motions we have, for the change in the latitude of any celestial body near the ecliptic, arising from motion of the ecliptic, 63 = (5.437 + 0".19 7”) cos v + (46”.78T — 0”.06 T°) sin v. Combining this with the change arising from the motion of the orbit of Uranus, we find 63 = {(10".12 4+ 1.140) 7+ 0°.19 T?} cosv + §(41".54 — 0".52u) 7 — 0.06 7} sin v. We may represent these expressions in the usual way by secular variations of the inclination and node of Uranus. But, owing to the small inclination, and consequent rapid motion of the node, it will be necessary to include the coefficients of the second power of the time. On the other hand, no distinction between ¢ and @ is necessary. Putting @ for the inclination of the orbit, @ for the longitude of the node referred to the equinox of 1850, and p =sin ¢ sin 6, q = sin ¢ cos 0; we have sin 8 =— pcosv+ qsinv cos 333 = — dp cos v + dg sin v. 96 THE ORBIT OF URANUS. From the expressions for p and q we obtain cos ¢ Dp = sin 0 Dp + cos 0 Dg; sin @ Df = cos § Dp — sin 0 Dg. And, neglecting (D,)” X sin ¢, we have farther, cos 9 D?.g = sin p (D0)? + sin 6D°,p + cos 0D*q; sin pD?,6 = — 2 cos oDD + cos OL", p — sin 017g. Since @ is only 46’ we may put cos @ and cos 3 both equal to unity in these expressions, while we have, for 1850, sin’ j= 0.9573 cos # = 0.2890 Dip = — 10"12—1".14u Dq =+ 41.54 —0 .52u LP p=— 0.38 Dg-—= — 10-12 log sin ¢ = 8.129606. The above formule then give Doe O31 1704, D§ = — 3167".5 + 12’. 6u Dig = + 07.26 Dip = + 5".6 Oe Ce ON) WLS f= 9; — Gilets — 1246p) rears or, adding Struve’s precession, we have when @ is counted from the mean equinox of date, G= 62 (8577 218462) P3297 Using the valnes of @ and @ given by these expressions, the latitude, secular variation included, will be given by the expression sin 3 = sin ¢ sin (v — 6). If we take from a table, as the principal term of the latitude, the value of sin $y Sin (v — 0), the secular term to be added will be {(2’.31 — 1794,) F-07183) 7") sino 6) If we represent, as before, by o the variable distance of the perihelion from the node, this term will be allowed for by adding to (6.s.1), (.¢.1), ete., the terms THE OMB TD O) he UR AN Uist 97 (b.c.0) = — e sin wd, (O:3 1) COS add, Gc) — sin wd9, (5.8.2) = —€ cos wd9, (6.c.2) = esinwdd; where dp = (27.381 — 1.244) 7+ 0".187? Putting in the above expressions © = 95°3’ + 3459’T, cos @ = — .0880 — .0167T, sin o = + .9961 — .00157, we find (b.c.0) = — (0".11 — 0°.06.2) 7 (6.8.1) = — (0 .20 — 0 11u)T— 0".057T? (6.c1)= (2.30—1 .24u)7T+ 0.127 (6.8.2) = —0.017 (6.c.2) = (0.11 —0.06.)T. We have, finally, to consider the terms cf long period in dy and ¢/ which have been omitted from the periodic perturbations produced by Neptune, in computing the terms of $3 on page 61, and which are as follows: dy= 1'.48cos(2l— g) —0".39 sin(2//— |g) — 2.12 cos (47 — 2g) + 1.00 sin (4/'— 29) -+ 0 .20 cos (6/’ — 3g) — 0 .04 sin (6/'— 39) + constant = 0’.00 sk&—= 0".80cos(2l— g) — 2’.28sin(2’— g) — 1.06 cos (4/' — 2g) — 1 .85 sin (47 — 29) ++ 0 .04 cos (67 — 3g) + 0 .19 sin (67 — 3y) + constant = 0".564. For the period during which Uranus has been observed, these values of dz and ¢k may be replaced by the following: én = — 0’.80T which are to be multiplied by the factor 1-+-u. The corresponding perturbation of the latitude will be 68 =sin vin — cos vik. Putting for v its approximate value v=g+o-+ sing and developing to quantities of the first order with respect to the eccentricities, we have sin v = sin (g +) + esin (2g + 0) —esinao cos V = cos (g +) +e cos (2g + w) —e cosa. 13. May, 1873. 98 THE ORBIT OF URANUS. Substituting for @ its value, 12°45’, and for éy and d/% their above values in the expression for 63, we find that the terms of $3 in question will add the following terms to (b.c.0), (6.8.1), ete. (b.c.0) = — .010 dy + 046 é& = + 0".027 (1 4+ w) (6.8.1) = + .979 dy + .221 dk = — 0.727 (1 + p) (6.c.1) = + .221 én — 9715 6k = — 0 44701 + lL) (6.8.2) = + .046 dy 4+- O11 dk = — 0.047 (1 + wu) (b.c.2) = — (b.c.0) = — 0.02714 pn) These values will be employed in the construction of the provisional ephemeris, but not in the tables. Collecting all three classes of terms discussed in this section, we have the following constant and secular terms in (4.c.0), (0.8.1), ete. (b.c.0) = + 0'.26 + (—0".09 + 0.082) 7 (6.8.1) = (— 0.92 — 0".61u) 7 — 0".05T? (b.c.1) = (+ 1.86 — 1 .682)7+4 0.127? (6.8.2) = — 0.06 — 0.057 (6.c.2) = —0.01 + (0.09 — .084)7 Positions of Uranus resulting from the preceding theory. The next step in order is the preparation of an ephemeris of the planet for comparison with observations. As this provisional theory is, for future use, super- seded by the tables appended to the present work, it seems unnecessary to enter very fully into the details of the computation of the ephemeris. ‘The perturba- tions of the longitude, logarithm of radius vector, and latitude, were first com- puted by the formule already given. év = (v.c.0) + (v.c.1) cos g + (v.c.2) cos 2g + ete., + (v.s.1) sin g + (v.s.2) sin 29 + ete., Mdp = (p.c.0) + (p.c.1) cos g + (p.¢.2) cos 2g + ete., + (p.s.1) sin g + (9.8.2) sin 2g + ete., 63 = (b.c.0) + (6.6.1) cos g + (6.8.1) sin g. Each coefficient (v.c.0), (v.c.1), ete., is composed at most of the following quan- tities: 1. The five classes of secular, long period, or constant terms, the separate values of which, with the sum of all, are given on pages 89 to 93. 2. Periodic terms due to the action of Jupiter, Saturn, and Neptune, given on pages 83 to 87. 3. Terms depending on the product of the masses of Jupiter and Saturn, given on page 88, omitting those depending on N, and N,, because they are given in column 5 of the terms of the first class. The sum of the perturbations thus computed is given in the third column of the following ephemeris. An approximate value of the perturbations produced by Neptune alone is inde- pendently computed for every fourth date, and the result is given in the fourth DHE ORBLE OR URANUS: 99 column. ‘The secular and long period terms are here taken from columns (3) and (4) of the tables on pages 89 to 93. The elliptic co-ordinates were then derived from the following elements, which are a little different from those employed in the computation of the perturbations. Elements HI. of Uranus. ty LOS? 1270 é, 28° 25 29.5 GUS ih 5G De 0) AEN 201.0 e, 0469436 e, (insec.) 9682".81 n, 15426.196 log a, 1.2828989 Red. to Ecliptic, — 9".37 sin 2 (v — 6) The longitudes thus found are corrected for lunar, but not for solar nutation, and the results are given in the fifth column. The column “correction” arises in this way: after the comparison of the ephe- meris with observations was nearly completed, it was found that some errors had crept into the former, the most important of which was the employment of a mean anomaly, g, corrected for secular variation of the perihelion in the computation of the perturbations from the preceding formule. As a large portion of the com- putations on the provisional ephemeris had been made by assistants furnished by the Smithsonian Institution and Nautical Almanac, I deemed it prudent to make a careful recomputation of the perturbations for every sixth date during the entire period of the modern observations. ‘The longitudes actually printed in the fifth column are the results of the original incorrect computation, while the numbers in the next column show the several corrections to be applied to obtain the results of my final revised computation. During the period of the modern observations the ephemeris is computed for intervals of 120 days, and the selected dates are all exact multiples of that interval before or after the fundamental epoch, 1850, Jan. 0, Greenwich mean noon. For convenience of reference the dates are numbered from an epoch earlier by 212 intervals, and the number is given in the second column. Between 1796 and 1801 no observations worth using were made on Uranus, the ephemeris has, therefore, not been extended over this interval. THE ORBIT OF URA NUS. Herniocentric EePHEMERIS OF URANUS FROM THE PRECEDING PROVISIONAL THEORY, [The longitudes are corrected for lunar but not for solar nutation.] Approximate Date. Sum of A ¢ y , Gans No. perturba- eae Longitude. Correction.| Latitude. mean noon. tions. Neptune. / ” : 4 ° / AA ” / ” YR OF — 8 1690, Dee. 23 +4 28.5 +236.1 59 40 35.2 OMe Dec. 24 +4 28.5 59 41 16.5 —10 7.3 1712, April 2 +8 37.9 155 24 Bah ie oe April 3 +8 37.9 155° 25) 295 > 58. 1715, Tie 4 +6 58.3 +136.2 169 10 21.4 +46 2.1 Mar. 5 +6 58.2 169 I1 se a Mar. 10 Sl GMOs 169 15 0.0 ‘ April 29 +6 51.1 +134.8 169 53 42.2 ane oe April 30 +6 51.1 169 54 28.8 oP : 1748, Oct. 21 —0 28.9 — 65.0 316 13 19.3 =A ae Oct. 22 —(0 28.8 316 13 58.8 —_ ; 0, S 3 Q —43 47.2 1750, Sept. 1: 0 6.3 — 51.3 823 44 14.9 43 y Ben 14 Hi 6.3 323 44 54.0 —43 47.3 Oct. 14 +0 6.6 — 50.8 324 4 22.1 ie ae Oct. 15 +0 6.6 324 5 1.0 —43 92. Dec. 3 +0 8.0 — 49.8 324 36 49.5 —44 ie Dec. 4 +0 8.0 324 37 28.5 —44 0. 1753, Dee. 3 —0 5.6 — 25.6 336 25 Be re ie iDechums —0 5.6 336 25 39.3 —46 1.2 1756, Sept. 25 —0 49.5 — 0.4 347 25 ote ay: os Sept. 26 —0 49.5 347 26 13.6 == : 1764, i 15 +1 24.0 16 13 35.0 —38 ore Jan. 16 +1 24.0 16 14 14.2 —38 1768, Dee. 27 +1 33.1 86 3 35.2 —27 42.4 Dec. 29 +1 33.1 +106.9 36 4 44.1 Page re S Dec. 31 +1 33.1 36 6 5.0 1769, Jan. 15 +1 33.5 36 i ie ae a 5 Jan. 18 +1 33.5 36.18 4.5 — Jan. 21 +1 33.5 +106 36 20 4.5 age oa 2 Jan. 24 +1 33.5 36 22 4.4 i 1781, Jan. 1 1 | +3 57.46 | +178.9 86 36 13.23 | 40.16 |+11 3.63 May 1 2 | +3 53.41 88 2 34.70 +12 11.14 Aug. 29 3 | +3 48.58 89 29 7.41 +13 ae Decwmoa 4 | +3 43.09 90 55 51.40 14 2 1782, April 26 5 | +3 37.18] +180.4 92 22 46.89 +15 31.53 Aug. 24 6 | +3 30.97 93 49 53.89 +16 37.47 Dec. 22 T | +3 24.52 95 17 ae +0.18 ate a ee 1783, April 21 8 | +3 18.03 96 44 42.73 8 47. i Aug. 19 9 | +3 11.87] +180.1 98 12 24.96 sc aor Ibe, Wr |} 1) |} es} BO 99 40 19.00 +20 55. (SS eA rileton ee lle Se Oas 101 8 25.13 tee ore Aug. 13} 12 2 56.24 102 36 43.29 +23 0.92 Nee i 3 i 52. 82 +178.6 104 5 13.79 | 40.23 |+24 2.46 g1785, April 10) 14 | +2 50.58 105 33 56.61 +25 3.14 Aug. 8 15 | +2 49.99 LOT 2eolesG tees aa Dees |) IG |) SER} ZG) ys 108 31 58.18 = : 1786, April 5] 17 | +2 50.53] 4175.5 | 110 1 16.88 = 97 59297 Aug. 3) 18 |) +2 53.29 111 30 47.68 +28 57.02 Deca 19 | +2 57.31 113 0 30.11 | 40.31 |+429 o3:08 ,1787, Mar. 81) 90 | +3 9:38 114 30 23.81 +30 47.92 July 29) 21) +3 8:51 |) 171.0 116 0 28.62 +31 41.72 Nov. 26} 22 | +3 15.48 117 30 44.11 +32 34.33 Logarithm Radius vector. 1.2876828 1.2876789 1.2626650 1.2626640 1.2623890 1.2623893 1.2623911 1.2624066 1,2624109 1.3001855 1.3001875 1.3013714 1.3013731 1.3014165 1.3014181 1.3014856 1.3014872 1.3025789 1.3025796 1.3029363 1.3029363 1.3003508 1.3003486 1.2955331 1.2955274 1.2955216 1.2954753 1,2954659 1.2954566 1.2954476 1.2785972 1.2780883 1.2775834 1.2770827 1.2765870 1.2760968 1.2756124 1.2751345 1.2746633 1.2742012 1.2737412 1.2732907 1.2728469 1.2724105 1.2719806 1.2715564 1.2711405 1.2707299 1.2703265 1.2699294 1.2695394 | 1.2691551 THE ORBIT OF URANUS. 101 HELIOcENTRIC ErHEMERIS OF UrANUs.—Contlinued. Date. Sum of [EEL Logarithm Greenwich Ko. perturba- PEE Longitude. Peter Latitude. Radius aan EG Aen produced by eae eee Neptune. | 7 uM u ° / " | / ” 1788, Mar. 25/ 23 | +3 23.02 | 119 1 9.79 | +33 25.72 | 1.2687775 July 23] 94 3 31.36 120 31 45.79 | 34 15.81 | 1.2684065 Nov. 20] 25 3 39.82) 1165.1 | 122 2 31.13 | 40.28 | 35 4.58 | 1.2680423 1789, Mar. 20) 26; 8 38.67 | 198} BS PEER | 85 51.99 | 1.2676852 July 18) 27 8 57.40 | 125 4 29.49 | 36 38.01 | 1.2673353 Nov. 15] 98 4 5.97 126 35 41.50 | 37 22.50 | 1.2669932 1790, Mar. 15| 99 | 414.35| 4157.8 | 198 7 1.70 | 38 5.50 | 1.2666592 July 13} 30 4 92.48 | 129 38 29.88 | 38 47.00 | 1.2663336 Nov. 10| 31} 4 30.00 131 10 5.37 | +0.25 | 39 26.85 | 1.2660167 1791, Mar. 10} 32 4 36.95 | 132 41 47.99 | 40 5.04 | 1.2657087 diel; S|) 88 4 49195 | -+149:1 || 184 13 37-15 | 40 41.58 | 1.2654082 Nov. 5| 34 4 48.11 | 135 45 32.62 41 16.44 | 1.2651215 1792, Mar. 4) 35 4 52.33 137 17 34.16 41 49.53 | 1.2648434 July 2| 36 4 56.03 | 138 49 41.78 42 20.89 | 1.2645770 Oct. 30) 37) 4 58.32) +139.3 | 140 21 54.47 | 40.46 | 42 50.35 | 1.2643210 1793, Feb. 27| 38] 4 59.50 141 54 12.11 | 3 18.00 | 1.2640766 June 27/ 39] 4 59.85] 143 26 34.79 43 43.81 | 1.2638448 Oct. 25| 40 4 58.84 144 59 1.64 44 1.75 | 1.2636267 1794, Feb. 22} 41] 4 56.80| +128.6 | 146 31 32.77 44 99.74 | 1.2634226 June 22| 42| 4 53.48 | 148 4 7.47 | 44 49.85 | 1.2632327 Oct. 20 3 4 49.13] +122.9 | 149 36 45.84 | 40.47 | 45 17.99 | 1.2630577 1795, Feb. 17] 44] 4 44.20 151 9 27.73 45 24.94 June 17 45 | 4 38.00 152 42 112.55 45 38.44 Oct. 15 | 46 | 4 31.04 154 15 0:15 45 50.64 1801, Jan. 17) 62 | +3 22.92 179 8 50.23 +44 34,51 | 1.2627599 Mar. 17] 63 3 26.59) + 57.3 | 180 36 54.94 44 12.86 | 1.2628920 Sept. 14] 64 | 3 31.06 182 9 57.40 3 49.24 | 1.2630362 1802, Jan. 12] 65]! 8 36.26 183 42 57.23 43 23.66 | 1.2631922 Mar. 12| 66] 3 41.88 185 15 54.08 42 56.34 | 1.2633599 Sept. 9| 67 3 47.78| + 43.7 | 186 48 46.54 | 40.35 | 42 26.97 | 1.2635387 1803, Jan. 7] 68 3 54.01 LSSeole shale 41 55.86 | 1.2637281 Mar. 7] 69 4 0.29 189 54 18.55 41 22.99 | 1.2639280 Sept. 4) 70 | 4 6.438 191 26 56.80 40 48.28 | 1.2641386 1804, Jan. 2/°71 | 4 12.95} + 30.7 | 192°59 30.15 40 11.79 | 1.2643591 Mar. 1| 72 4 18.74 194 31 56.83 39 33.63 | 1.2645891 Aug. 29) 73 4 94,38 196 417.13 | +0.14 | 3853.72 | 1.2648287 Dee. 27) 74 4 29.34 197 36 30.24 38 12.21 | 1.2650777 1805, April 26| 75 A BEGO || 2b IO | 1ey eeep.ey 37 29.07 | 1.2653360 Aug. 24] 76 4 37.42 200 40 33.91 36 44.40 | 1.2656037 Dec. 22] 77 4 40.11 202 12 23.39 35 58.15 | 1.2658808 1806, April 21] 78 4 41.83 203 44 4.25 | 35 10.48 | 1.2661674 Aug. 19] 79 4 49.55| + 5.9 | 205 15 36.11 | 40.21 | 34 21.34 | 1.2664631 Dec. 17} 80 4 49.19 206 46 58.67 33 30.88 | 1.2667689 1807, April 16) 81 4 40.52 208 18 11.39 32 39.05 | 1.2670847 Aug. 14} 82 4 37.43 209 49 13.98 | 31 45.96 | 1.2674103 Mecee U2 188) | 4 Serle 50) P21 20516296 30 51.63 | 1.2677459 1808, April 10) 84 4 28.30 212 50 49.02 | 29 56.12 | 1.2680922 Aug. 8] 85 4 21.63 214 21 20.46 | +0.25 | 28 59.51 | 1.2684491 Dec. 6] 86| 4 14.32 215 51 41.72 28 1.80 | 1.2688166 1809, April 5| 87 | 4 5.84/— 15.4 | 217 21 51.97 27 3.03 | 1.2691945 Aug. 3| 88| 3 56.63 218 51 51.53 26 3.28 | 1.2695829 Dec. 1] 89 3 47.07 290 21 40.62 25 2.66 | 1.2699816 1810, Mar. 31] 90 3 37.16 221 51 19.10 24 1.10 | 1.2703905 July 29} 91 3 27.15 | — 24.3 | 223 20 47.07 | +0.23 22 58.68 | 1.2708094 Nov. 26] 92 | +3 17.52 224 50 4.83 191 55.38 | 1.2712377 102 THE ORBIT OF URANUS. SS TE AE EE eee ar HEI IOCENT E aa —— . 4 NTRIC EPHEMERIS 5 / = = S OF UR ANUS. Continued Date. c A pproxim: “Ips - Sum of Approximate | i tee No. Sete pentane ens Peer | L f on. Sear WRarelnvaeyal. ongitude. >|Correeti A ogari ; fone: pictareay itude Correction. Latitude. fe z ee EA vector. 1811, Mar. 26] 93 | +: y oe a a ’ . 9° 3 99 y i ip ily 24 |code ane lock 226 19 12.04 es lecs | Boe 227 48 9.06 +20'51:52 Rio Mee (anil oe hs aso) amma ee 18 4169 | one July 18) 97 2 37.26 ar Ui 25.80 1 13088 Nov. 15| 98| 2 31.98 932 13 59.15 30.80) 1813, Mar, 15| 99 | ee | 933 42 ee | TON | Te oaeel eae July 13 | “li 2 27.86 | —39.1 | 235 10 21.88 15 22.35 | 1.2739838 < 00, 2 25:05 936 38 18 96 | 14 14.92 | 1.27446 Nov. 10)! 101 | 9 99.98 256 38 18.26 1 ‘ omit 51 a 2 424.00 92 5 aot ) July 8} 103 9 99.5 239 33 39.61 | ate 8.79 | 1.2754409 Tastes 22.54 | —44.6 2 . 0 50.21 | 1.2759: Nov. 5] 104 99 241 1 4.63 0.29 59367 1815, Mar. 5|105| 9 24.24 eeprom ab |) asca: re” Tal 9 | 105 2 26.58 943 55 Ae 8 32.19 | 1.2769379 uly 3/106| 23 ee are 36 Revel as |) oats 245 92 17.17 7 22.86 | 1.277403 re eed mee oes) fogs sts 3 eo |e June 27 | 109 9 44.51 248 15 31.47 3.80) Oct. 2510 ae 51 249 41 52.08 3.54.09 | 1.2789600 Ha Satta 0 | 2 50.43 | Sa ee UG —0.01 2 44.46 | 1.2794688 Sie eb. 22 | 111 9 56.15 P 20) 8) 142 1 34.89 Dy) Fae oeimen| wien: 54.4 | 952 33 59.49 34.82 | 1.2799795 On 301 dia! 9010s 253 59 46.00 + 025.25) 1818, Feb. 17| 114| ee 255 25 21.18 — 04 ee June 17/113. | 3 32.96 956 50 44.48 138.44 ee ne TE 8 oe ane ae see | corto. eee 128 1819, Feb. 12| 117] 3 ae 959 40 55.14 sage a June 12] 118 e ee 961 5 49.95 yee 2 128270 Sane Colbie Rewrsins 262 30 17.04 6 27.83 | beat ee mar Meret hace oar 1 85.53 Ee Fe ae 20 3 45.03 Se a OO 8 49.77 | 1.2844143 au i es | 3 46.37 266 49 AES 9 49.52 | 1.2848927 sie nana 3 45.07| —56.4 | 269 30 2.43 12 1.40/7e Soa | TGR 3 42.35 970 53 21. 2 13 6.44 | 1.2863087 Sept. 29 | 125 2 92 92 270 53 21.55 TR elie oone 272 16 27.96 14 10.83) ee M: i a 26 | 3 32.86 972 20 ae 5 15 14.57 | 1.2872402 May oT 127 8 98.18 nee 973) 39 22-99 16 17.5 QRhr 2 Sept. 24 | 128 Beate —56.2 275 2 4.25 0.1 58 ee en at OA 3 18.32 ar ouere ial Aad 0 | 17 19.84 | 1.2881632 823, Jan. 22) 129 P , 276 24 34.06 2 aes 3 9.68 Seaton 18 21.32 | 1.2886214 aa 92 | 130 2 0.94 ar athe De ers 19 22.01 | 1.2890772 Sept. 19 | 131 9 50: 279 8 59.05 20 21.8 1894, Jan. 17| 132] 2 50.36 | —56.1 | 280 30 54.82 0 2.89 ee ae a ioe 2 40.10 | 981 59 56) Til an 20.91 | 1.2899826 er 133| 2 29.74 Bees hate ; 22 19.10 | 1.2904315 aon SePt 18) 13 2 19.49 oe Te ele | 07 | 23 16.40 | 1.290877 1825, Jan. 11) 185 | 2 9.9 : BS a Seee 60 2419 ie May 11) 136 = 9.25 —)5.5 985 56 52.92 Hees 129i 1896, mept : ee | 1 50.21 288 38 53.43 = 2.71 | 1.2921967 May 6 ie | Aa A 939 59 Ay 26 56.22 | 1.2926290 Mey eee eee ue oat oui | 97 48.71 | 1.2930570 1897 | 140 | 1 26.44 Sch an a ee —0.09 | 28 40.26 | 1.2934804 827, Jan. 1/141 | 1 20.91 Ber eae 29 30.76 | C28 May 1| 142 | apie 294 1 8.83 30 20. Fe. Aug. 29 143 bee 295 21 99.15 30 20.16 | 12a Dec 27) 144 | 1 10.52| —54.0 | 296 41 97.79 ae Lone Prt a 1 7.92 Be aren ons 31 55.92 | 1.2951196 328, April 25| 145 | 1 P 298 1 25.72 39 49 : 5 | 479 EAE 32 42.14 | 1.2955136 Aug. 23| 146 | 1 999 21 16.45 | —0.07 | 33/27.: Ane. 23) | 3 5] Saar 33 27.94 | 1.2959002 . 21) 147 | +1 3.11} —52.9 302 0 36.75 34 11.25 | 1.2962790 2 36.75 —34 54.11 | 1.2966496 ES RECEP EE THE ORBIT OF URANUS. 103 HeELIoceNTRIC EPHEMERIS OF URANUS.— Continued. Approximate | | Date. Sum of f Sime | Logarithm Greenwich No. perturba- ala yg Longitude. |Correction.| Latitude. Radius mean noon. tions. Ixus ucediby | vector. Neptune. / uy WV ° / uW " / i 1829, April 20} 148 | 41 3.68 303 20 6.73 3) BST | 1.2970111 Aug. 18 149 IL fa} 315) 304 39 30.08 36 16.26 | 1.2973630 Dec. 16) 150 eo | 305 58 46.96 36 55.60 | 1.2977052 1830, April 15) 151 1 10.14 —2.3 307 17 57.34 +0.08 37 33.68 | 1.2980368 Aug. 13 | 152 Nee Te! 308 37 1.72 38 10.56 | 1.2983571 Dee. 11 | 153 1 2.63 309 55) 59.75 38 46.13 | 1.2986661 1831, April 10 | 154 1 21.69 B11 14 51.68 39 20.49 | 1.2989638 Aug. 8) 155 1 26.06 —52.2 SLITS Silat 89 53.54 | 1.2992501 Dee. 6) 156 1 30.26 313 52 17.74 40 25.29 | 1.2995247 1832, April 4) 157 1 34.13 SoM LORoIEGS +0.05 40 55.74 1.2997873 Aug. 2] 158 1 37.50 | 316 29 19.67 | 41 24.90 | 1.3000379 Nov. 30] 159 1 40.39 —)2.2 317 47 41.94 41 52.65 | 1.3002768 1833, Mar. 3 160 1 42.02 319 5 57.95 42 19.10 1.3005047 July 28) 161 1 42.86 320 24 8.3 42 44.17 | 1.3007216 Nov. 25] 162 1 42.45 321 42° 12.89 43 17.88 | 1.3009280 1834, Mar. 25} 163 1 40.92 — 2.9 323 0 11.92 +0.15 43 30.24 | 1.3011231 July 23) 164 1 38.01 324 18 5.47 43 51.18 | 1.3013090 Nov. 20] 165 1 34.01 325) 39) 93299 44 10.74 | 1.3014851 1835, Mar. 20} 166 1 28.76 | 326 53 37.59 44 28.92 | 1.3016515 July 18/} 167 Th 22, IS) —2.9 328 11 16.27 44 45.69 | 1.3018084 Noy. 15 | 168 1 14.89 | 329 28 50.92 45 1.14 | 1.3019567 1836, Mar. 14] 169 1 6.50 | 330 46 21.41 | +0.19 45 15.12 | 1.3020960 July 12) 170 0 57.31 332 3 48.20 45 27.71 | 1.3022265 Noe sob iT: () ZE(D1L |) BN) 333 21 11.76 45 38.98 ; 1.3023482 SSieevlare sO 0 37.27 834 38 32.35 45 48.84 | 1.3024612 July 7) 173 0 26.69 315 55 50.40 45 57.28 | 1.3025656 Nov. 4] 174 0 15.92 | 337 13 6.16 | 46 4.45 | 1.3026622 1838, Mar. 4/175 | -+0 5.23) —55.1 338 30 20.17 | +0.17 46 10.04 | 1.3027503 July 2) 176 }—0O 5.21 339 47 32.76 46 14.36 | 1.3028297 Ocizs 177 0 15.76 341 4 43.81 46 17.29 | 1.3029008 1839, Feb. 27 | 178 0 25.58 342) 20 54235 46 18.83 | 1.3029634 June 27) 179 0 34.82] —56.9 | 243 39 4.45 | 46 19.00 | 1.3030176 Oct. 25)) 180 0 43.68 344 56 14.08 | 46 17.84 | 1.3030630 1840, Feb. 22] 181 0 51.42 | 346 13 24.23 | +0.17 46 15.28 | 1.3030996 June 21 | 182 0 58.36 347 30 34.70 | 46 11.29 | 1.3031269 Oct. 19 | 183 1 4.34 —58.4 | 348 47 46.09 46 6.01 | 1.30381447 1841, Feb. 16 | 184 RAN | 8090 4 58.23 | 45 59.35 | 1.3031528 June 16 |° 185 NT Se36 351 22 11.79 | 45 51.33 | 1.3031509 Oct! 14) 186 iE GI 352 39 26.97 45 41.91 | 1.3031392 1842, Feb. 11 | 187 Vi (a4 fe} 0.5 |) Sis) AS 2eeon +0.15 | 45 31.07 | 1.3031164 June 11 | 188 1 18.13 355 14 2.89 | 45 18.90 | 1.3030817 Oct. 9 |) 189 Teles 256 31 24.13 | 45 5.3 1.3030351 1843, Feb. 6] 190 5 S50 357 48 47.74 | 44 50.50 | 1.3029767 June 6) 191 1 12.83 —66.7 359 6 13.65 44 34.22 | 1.3029060 Oct. 4] 192 1 9227 0 23 42.19 44 16.64 | 1.3028228 1844, Feb. 1) 193 eh 1 41 13.59 | +0.07 43 57.59 | 1.3027267 May 31) 194 OF 5929 2 58 47.55 43 37.23 | 1.3026173 Sept. 28 | 195 0 54.61 — 64.9 4 16 24.34 | 43 15.53 | 1.3024947 1845, Jan. 26) 196 0 49.19 5 34 3.83 42 52.43 | 1.3023588 May 26 | 197 0 43.90 6 51 46.06 42 27.99 | 1.8022095 Sept. 23 | 198 0 38.92 8 9 31.09 42 2.22 | 1.3020472 1846, Jan. 21) 199 0 34.37) —67.7 9 27 19.04 | 40.15 41 35.11 | 1.3018722 May 21 200 0 30.55 10) 45 9.82 | | 41 6.76 | 1.3016851 Sept. 18 | 201 | —0 27.21 2 3 3.39 |\—40 37.01 | 1.3014861 104 THE ORBIT OF URANUS: HELIOCENTRIC EPHEMERIS OF URANUS.— Continued. Date. Sum of Approximate Logari Greenwich No. perturba- Re aneatee Longitude. Correction | Latitude. Feat mean noon. tions. Neptune. vector. / “ Wt ° 4 ” / WW 1847, Jan. 16| 202 | —0 25.64 13 21 0.27 —40 6.03 | 1.3019758 May 16! 203 0 24.73) —70.2 14 39 0.21 39 33.75 | 1.3010544 Sept. 13 | 204 0 24.95 15 57 3.61 39 0.23 | 1.3008219 1848, Jan. 11 | 205 0 26.43 17 15 10.46 | +0.06 | 38 25.40 | 1.3005790 May 10 | 206 0 28.95 18 33 21.30 37 49.42 | 1.3003263 Sept. 7 | 207 0 32.75 | —72.8 19 51 36.15 37 12.18 | 1.3000640 1849, Jan. 5| 208 0 37.59 21 9 55.34 36 33.73 | 1.2997993 May 5| 209 0 43.47 22 28 19.02 35 54.16 | 1.2998115 Sept. 2/ 210 0 50.27 23 46 48.13 35 13.36 | 1.2999994 Dee. 31] 211 0 Bye | 75.8 25 5 22.56 | —0.08 | 34 31.47 | 1.2989959 1850, April 30 | 212 i Geis 26 24 2.38 33 48.40 | 1.2986203 Aug. 28 | 213 1 15.11 27 42 48.39 33 4.37 | 1.29838078 Dec. 26 | 214 1 24.38 29 1 40.87 32 19.06 | 1.2979881 1851, April 25 | 215 1 34.833] —77.9 30 20 39.78 81 32.79 | 1.2976615 Aug. 23] 216 1 44.28 31 39 45.91 30 45.46 | 1.2973287 Dee. 21} 217 1 54.21 32 58 59.55 | —0.09 | 2957.09 | 1.2969894 1852, April 19 | 218 2 4.12 34 18 20.89 29 7.71 | 1.2966436 Aug. 17] 219 2 13.61] —80.5 85 37 50.43 28 17.33 | 1.2962916 Dee. 15 | 220 2 23.00 36 57 28.19 27 25.95 | 1.2959338 1853, April 14 | 291 2 31.54 88 17 15.06 26 33.62 | 1.2955701 Aug. 12} 222 2 39.58 39 37 10.77 25 40.35 | 1.2952015 Dec. 10 | 223 2 46.62 | —89.6 AO UGe ip 0s0'7 24 46.17 | 1.2948970 1854, April 9 | 224 2 52.64 42 17 31.26 23 51.08 | 1.2944458 Aug. 7 | 225 2 57.77 43 87 56.94 22 55.08 | 1.2940583 Dec. 5] 226 3 1.60 44 58 31.59 21 58.25 | 1.2936645 1855, April 4| 227 3 4.03] —84.6 46 19 17.49 21 0.51 | 1.2932645 Aug. 2]| 228 3 5.35 47 40 13.98 20 1.97 | 1.2928582 Nov. 30] 229 3 5.82 49) 226") On ill 19 2.66 | 1.2924449 1856, Mar. 29 | 230 3 3.95 50 22 39.57 18 9.55 | 1.2920239 July 27 | 231 Te) SSRI 51 44 9.08 17 1.67 | 1.2915954 Nov. 24/| 239 2 57.62 53 5 49.32 16 0.03 | 1.2911592 1857, Mar. 24 | 233 2 52.95 54 27 40.66 14 57.72] 1.2907155 July 22} 234 2 47.58 55 49 42.98 13 54.71 | 1.2902648 Nov. 19 | 235 9 41.65] 87.9 57 11 56.10 | —0.10 | 12 51.08 | 1.2898057 1858, Mar. 19 | 236 2 35.18 58 34 20.16 11 46.89 | 1.2893395 July 17) 237 2 28.82 59 56 54.59 10 41.96 | 1.2888659 Nov. 14] 238 2 22.46 61 19 39.71 9 36.91 | 1.2883853 1859, Mar. 14 | 239 DNN6735))) ==88k6 62 42 35.14 8 30.87 | 1.287898] July 12) 240 2 10.33 64 5 41.38 7 24.61 | 1.2874049 Nov. 9]| 241 D Bil 65 28 57.59 | —0.13 6 17.90 | 1.2869061 1860, Mar. 8 | 242 2 0.67 66 52 24.15 5 10.84 | 1.2864023 July 6} 243 1 57.10 89.1 68 16 0.90 4 3.42 | 1.285894] Nov. 3] 244 1 54.37 69 39 47.78 2 55.74 | 1.2853816 1861, Mar. 3] 245 1 52.61 "1 3 44.98 1 47.83 | 1.2848658 July 1] 246 1 51.92 72 27 52.38 — 0 39.69 | 1.2848468 Oct. 29 | 247 1 BOG} || —399),83 73 52 10.31 | —0.19 |+ 0 28.60 | 1.2838257 1862, Feb. 26) 248 1 53.51 15 16 38.24 1 37.00 | 1.2833029 June 26 | 249 1 55.99 76 41 16.65 2 45.46 | 1.2827788 Oct. 24) 250 1 59.54 78 6 5.46 3 53.95 | 1.9822542 1863, Feb. 21| 951] 2 3.98] —88.5 79 31 4.97 5 2.40 | 1.2817290 June 21| 252 | 2 9.43 80 56 15.07 6 10.80 | 1.2812047 Oct. 19] 253 2 15.63 82 21 36.01 | —0.04 7 19.09 | 1.2806819 1864, Feb. 16| 254 | 92 29.59 83 47 7.95 8 27.25 | 1.2801609 June 15 | 255 2 30.20 | —S87.2 85 12 50.83 9 35.22 | 1.27964 21 Oct. 13| 256 | —2 38.39 86 38 44.88 + § 42.95 | 1.2791263 THE ORBIT OF URANUWS. 105 HELIOcENTRIC EPHEMERIS OF UrRANUS.—Concluded. Date. Sum of EISEN Logarithm Greenwich No. perturba- Eee Crs Longitude. Correction.) Latitude. Radius mean noon. tions. produced by vector. eptune. 7 "f Us ° f tf ”v 7 u 1865, Feb. 10) 257 | —2 47.06 88 4 50.26 +11 52.20 | 1.2786142 June 10] 258 2 55.93 89 3 Celun 12 57.54 | 1.2781063 Oct. 8) 259 3) 5201 —§5.2 $0 57 35.61 —0.09 14 4.34 | 1.2776029 1866, Feb. 5] 260 8) deen | 92 24 16.00 15 10.74 | 1.2771044 June 5] 261 3 22.61 Wes BL ts}sh) 16 16.70 | 1.2766112 Oct. 3) 262 3 30579 95 18 13.06 17 22.18 | 1.2761237 1867, Jan. 3 263 3 38.49 —82.5 96 45 29.91 18 27.15 | 1.2%56419 May 31 | 264 3 45.25 98 12 59.44 19 31.54 | 1.2751656 Sept. 28 | 265 3 50.96 99 40 41.71 —0.08 20 35.37 | 1.2746948 1868, Jan. 26 | 266 3 55.67% 101 8 36.54 } 21 38.51 | 1.2742299 May 25 | 267 3 59.06 | —9.1 102 36 44.31 22 40.96 | 1.2737709 Sept. 22 | 268 4 1.02 104 5 5.00 23 42.75 | 1.2733175 1869, Jan. 20) 269 4S a 105 33 38.41 24 43.69 | 1.2728693 May 20] 270 4 0.81 107 2 24.67 25 43.81 1.2724264 Sept. 17 | 271 3 58.59 | —T5.1 108 31 23.55 | —0.11 26 43.07 | 1.2719888 1870, Jan. 15] 272] 3 54.96 110 0 34.97 27 41.44 | 1.2715563 May 15 | 273 3 50.23 J11 29 58.50 28 38.77 | 1.2711290 Sept. 12 | 274 3 44.34 112 59 33.91 H 29 35.116 | 1.2707070 1871, Jan. 19) 2% 3 37.90 —69.8 14 29520579 30 30.41 | 1.2702903 May 10) 276 3 30.56 ns tae) TR | 31 24.57 | 1.2698791 Sept. 7 | 277 3 22.61 | 117 29 28.39 | —0.18 32 17.57 | 1.2694735 1872, Jan. 5 | 278 3 14.50 118 59 48.05 | 33 9.31 | 1.2690788 May 4) 279 3 6.19 —C4.2 120 380 17.95 33 59.79 | 1.2686804 Sept. 1) 280 | —2 57.65 122 0 57.87 +34 48.97 | 1.2682934 The next operation would be to interpolate these co-ordinates to intervals of time suitable for the computation of a geocentric ephemeris, to correct the longi- tudes for solar nutation, and then to compute the geocentric right ascension and declimation. ‘This operation has not, however, been completely carried out except for most of the observations before 1830, and for three of the oppositions observed since, the latter being computed only as a check upon the accuracy of the com- parisons. As a general rule, it may be said that wherever a complete geocentric ephemeris, with the heliocentric ephemeris from which it was computed, were available, these ephemerides were made use of in a manner which will be more fully described hereafter, while, in all other cases, the geocentric places were computed directly. It may also be stated here that Hansen’s Tables du Soleil have been adopted as giving the places of the sun to be used in computing the geocentric places, 14 May, 1873, 106 THE ORBIT OF URANUS. CiHPACPAI ER, Walt: REDUCTION OF THE OBSERVATIONS OF URANUS, AND THEIR COMPARISON WITH THE PRECEDING THEORY. Tue observations of Uranus naturally divide themselves into two distinct classes. (1) The purely accidental ones, made previous to the recognition of the planet by Herschel in 1781, and therefore without any suspicion on the part of the observers that the object was not a fixed star, and (2) the systematic observations made since. The first class are nearly all so uncertain in comparison with the second that I have hesitated over the question of employing them at all. If nothing buta determination of the elements of Uranus were called for, they would certainly not be worth using, since these elements may be determined with entire certainty from the observations which have been made during the entire revolution of the planet since 1781. But the mass of Neptune is also to be determined, and it is at least possible that these observations, uncertain though they are, may add materially to the weight of this determination. I have, therefore, determined to include them all, re-reducing them when there seemed to be good reason so to do, The earliest observations are those of Flamstead, published in the Historie Celestis. The observations themselves, as printed, together with the principal elements for reduction, are given in the following tables. The first column of the table gives the name of the star. The second gives the clock time of transit over the wire of the quadrant as given by Flamstead. The time, it will be seen, is only given to entire seconds. We must, therefore, expect to find a probable error, of which the mathematical minimum is 0°29, and of which the minimum we can reasonably expect is much greater. Next we have the apparent right ascensions of the stars as computed. For these data I am indebted to Prof. Coffin, Superintendent of the American Ephemeris. The mean places are mostly derived from the “Star Tables of the American Ephemeris,” and from the two Greenwich Seven Year Catalogues, while the reduction to apparent place is made with the modern constants. The fourth column gives the apparent clock correction for sidereal time, in which is included the effect of deviation of the instrument from the meridian. The clock keeping mean time, the errors are in the next column reduced to those of sidereal time at the moment of the transit of Uranus. The next two columns give the corrections for clock rate, and for deviation of the instrument from the meridian, as inferred from the observations themselves, both being referred to the time and position of the transit of Uranus. TEE OR BN Ome UR AUN Wis: 107 In the last column we have the seconds of concluded correction for clock and instrument to be applied to the observed time of transit of Uranus. 1690, December 23. Right Ascension. Star. Time of Tr. R. A. of star. Clock. Cr R. Deve, C72 h. m. s. fo NG Eh 0 tre © Gs mss Ft Ss. b a Arietis, 7 48 40 1 49 52.1 a8 08 47.9 98 29.3 =n =aieo 31.6 gq Ametis, 8 3054 232 8.4 —5 58 45.6 58 33.9 —0.7 +1.7 32.9 o Arietis, 8 33 17 2 34 31.0 —5 58 46.0 58 34.7 —0.7 42.9 32.5 ¢ Arietis, 8 40 21 2 41 38.0 —5 58 43.0 58 33.0 —06 —0.2 33.8 6 Arietis, 8 5244 254 3.0 —5 58 41.0 58 33.0 —05 40.5 33.0 y Tauri, eet eo Lol at —=o On oo4 OS loll) Ol P= 13) 1325 Uranus, 9 41 49 AwTauri, 9 45 3 3 46 31.1 —5 58 31.9 58 32.4 0.0 —0.8 33.2 a Virginis,19 415 13 8 58.2 —5 55 16.8 56 49.3 a Bootis, 19 58 314 14 1 34.0 —5 56 57.5 58 39.0 Hourly rate of clock, —0°.6 Deviation of instrument for each degree of Z. D., —0.5 hs ms is: Transit of Uranus, 9 41 49.0 Correction for clock and instrument (mean), —5 58 32.8 Observed R. A. of the planet, 3 43 16.2 1690, December 23. Declination. Z. D. observed. Refraction. Declination. Eq. point. a Arietis, Pas)? Dat Oi +0’ 33” 21° 58’ 53” 51° 28’ 36” az Arietis, 39 18 55 +0 41 IG YY 4€ 51 28 40 o Arietis, 37 41 O +0 44 13 46 58 51 28 42 e Arietis, 31 23 15 +0 35 20 4 37 ol 28 221 6 Arietis, 32: 55 55 +0 37 18 31 40 51 28 12 y Tauri, 28 20 55 +0 31 23 6 53 OL 289 Uranus, 31 52 35 +0 36 A Tauri, 30 15 55 +0 34 2112. 3 51 28 32 Circle reading for Uranus, corrected for refraction, BO bey IU Equatorial point on circle, 51 28 30 Declination of Uranus, from observation, +19 35 19 1712, April 2. Light Ascension. Ne Tbs 8 Uranus, 9 30 19 ik tm m. &: e Virginis, 12 0 19 12 47 52.4 +47 33.4 e Virginis, 12 14 51 13 2 31.5 +47 40.5 ¢ Virginis, 12 32 11 13 20 5.7 +47 54.7 The discordance of clock errors, and the time which intervened between the transit of the planet and that of the first star, seem to render an accurate reduction impossible. 108 TEE ORB TT sO 2 UR esGN UES: 1715, March 4. Right Ascension. Ake aeAS Cc. Ces hem. aes. h. m. s. h. m. &. s. d Leonis, 1 50) 19 S10F 45 22a —— 1A 6r5 20.5 Uranus, LEP PAT 6-Virginis, 12 49 41 11 45 23.6 —I1 4 17.4 21.1 hy “ms 8. Clock time of transit of Uranus, 12 2a ak Correction for clock and instrument, —] 4 20:8 Right ascension of Uranus from observation, ll 22 40.2 1715, March 4. Declination. Vt 10). R. Dee. Eq. point. d Leonis, 46° 19’ 40” +1 0” +5° 8 6’ 51° 28) 467 Uranus, 46 33 10 +1 1 6 Virginis, 46 13 20 +1 0 5 14 17 51 28 37 Circle reading for Uranus, 46° 34 11” Equatorial point, 51 28 42 Observed declination of Uranus, +4 54 31 1715, Murch 5. Right Ascension. T. R. A. C. Cc’ np oftts Ge Ing ah Eb 1s. elo s. ad Leonis, 11 46 24 10 45 52.5 —1 0 31.5 25.5 Uranus, 122299 b Virginis, 12 45 49 11 45 23.6 —1 0 25.4 29.1 he mse Transit of Uranus, Ny 2 Correction for clock and instrument, —1 0 27.7 Observed right ascension of Uranus, 11 2273s The large apparent clock rate, and the colons after the time of transit, both throw doubt on this observation. Declination. The circle readings for the stars are the same as on the day preceding, while that for Uranus is 50” less. The declination is therefore 50” greater, or +4° 55’ 21”. 1715, March 10. Right Ascension. T. R.A: C. C’. Ns Tb lw sth Eb h. m. Ss. 8. d Leonis, Wil Qs) ays} 10 45 52.5 == 40 oro 59.5 p® Leonis, 11 32 28 10 52 25.1 —0 40 2.9 58.1 Uranus, HOF 12), b Virginis, 12 25 18 11 45 23.6 —0 39 54.4 58.2 é h m. 8. Clock time of transit of Uranus, 12° 1 42 Correction for clock and instrument, —0 39 58.6 Observed right ascension of Uranus, 11 21 43.4 THE ORBIT OF URANUS. 109 Declination. 76.10). R. Dee. Kq. Pt. d Leonis, 46° 19’ 35” +1’ 1” Oe toy (0) il Dish a p> Leonis, 47 58 35 +1 3 3 29 25 ol 28 63 Uranus, 46 27 0 +1 1 b Virginis, 46 13 25 +1 0 5 14 17 51 28 42 Circle reading for Uranus, AG 28 1 Equatorial point on circle, ol 28 49 Observed declination of Uranus, +5 0 48 1715. April 29. Right Ascension. 4th RarAy C. Cr: hom. oS: h. ints Gy inh fey GE s. o Leonis, 8 42 11 TI 62883 +2 24 17.3 18.7 Uranus, 8 50 44 y Virginis, o) (oy 11 31 14.1 +2 24 19.1 16.4 17 Virginis, wo) Gis} Giese WY fs} (545) +2 24 28.5 19.8 x Virginis, 11 32 48 13 57 47.6 +2 24 59.6 33.1 The discordance of the clock corrections makes a satisfactory determination of the right ascension very difficult. I deem it best to reject the doubtful observa- tion of 17 Virginis, and the discordant one of x Virginis. The result will then be Iie Ti Es Observed transit of Uranus, 8 50 44 Correction for clock and instrument, 2 24 17.6 Observed right ascension of Uranus, Peto FG Declination. Z. D. R Dee. Kiq. Pt. o Leonis, 43° 52’ 40” -+0! 55” 7° 34 dl” ol? 28) 26; Uranus, 45 45 30 +0 5 y Virginis, 43 20 20 +0 54 ie gut 51 28 25 17 Virginis, 44 34 10 +0 56 6 53 33 51 28 39 x Virginis, 60 23 5 +1 41 —8 55 42 51 28 64 Circle reading for Uranus, 45° 46’ 29” Mean equatorial point, 51 28 38 Observed declination of Uranus, +5 42 9 The next observations in the order of time are two by Bradley, discovered by Mr. Hugh Breen, but still unpublished. The following are the results as given by Mr. Breen in the Astronomische Nachrichten, No. 1463. Mean Time. Reals Newb: 3 Honmee Ss h. m. Ss. oar ye " 1748, October 21, 76 18.4 21 4 37.93 107 29 1750, September 13, 10 8 57.8 21 40 0.23 104 42 33.9 110 THE ORBIT OF URANUS: Mr. Breen remarks: “The right ascensions are very accurate. It has been assumed that the N. P. D., on 1750, September 13, is identical with « Capricorni, with which it was compared. ‘The first observation was by the transit instrument, and the second by the quadrant.” No ground is given for the above assumption respecting the N. P. D, for the second observation ; it may, therefore, be omitted as valueless. In the year 1750 we have also two observations by Le Monnier at Paris. For these, and all the other observations by the same observer, 1 shall adopt the results given by Bouvard in the Connaissince dis Temps, for 1821, p. 341, with the cor- rections indicated by Le Verrier, in Connaissance des Temps, for 1849, pp. 125 and 126. ‘The necessary uncertainty of the observations is such that, considering that Bouvard reduced them with the star positions of the “ Fundamenta,” scarcely anything will be gained by a new reduction. 1753, December 3, we have another observation of right ascension by Bradley, I adopt the result kindly communicated by my distinguished friend, Dr. Auwers, Ing is h. om: s. ) 1753, December 3, 0 33. R. A. = 22 23 21.59 1756, September 25, Observation by Mayer, at Gottingen. I adopt the result given by Bessel, in Fundamenta Astronomia, p. 284. hy am: fe} ’ " 1756, September 25, 10 12. R. A. = 348 0 54.5 Dec. =—6 1492 The following is a tabular summary of the preceding results, with their com- parison with the provisional theory. In the computation of the geocentric place the places of the sun were derived from Hansen’s Tables. I am indebted to Pro- fessor Coffin for a duplicate computation of the geocentric places from the pro- visional ephemeris, which was executed by Mr. Joseph A. Rogers. Right Ascension. Declination. Correction to theory. Observation. C Observation. nA t Long. hm ~°s ‘ de 28/3) 43 16: 35 16 424 4|11 22 40.2 |38. 54 3 +28 . 5/11 22 81.5 |29. 55 2 +44 ar.10/11 21 43.4 |41. 8 | 56 8 | 236 p29) | ileal 5 = 016 5 4s ae +2 . 21/21 4 87.93 a wise | 489.3 1750, Sept.13 |21 40 0.23 ° / ” Oct. 14 |324 15 24.6 i—15 1 40.4} 47.0 +35.9 THE ORBIT OF URANUS. HL Right Ascension. | Declination. Correction to theory. Observation. | | ays c Long. ot on | | Date. | | fo , 7 8 o24 34 53.5 hyenas 22 23 21.60 ° , ” ~ = | 51/348 0 54.5 1764, Jan. 15| 12 37 39.0] 1768, Dee. 31 52.0 s i 4 aah ee 1 SS) Dbo co or NAR TR OOS Rr © me OL ee Dec. : 31 ; 1769, Jan. 15) 31 Jan. 16) 31 Jan. 31 Jan. 2 Jan. Jan. ~ Ww TOR chr es eo Or no bo or) ro Hm CO OF woo 6 DOH He He Or i oS ise) DC hm = on ono a 0.0 Wh WL NWWWNWr GS or) 1 ¢ ore | 5 pa pp pS bo Where no declination has been observed the observed corrections in right ascen- sion have been changed to corrections in longitude on the hypothesis that the theoretical latitude is correct. The approximate formula is da cos 5 sin i” cos £ = sin « cos a, € being the obliquity. d= where DISCUSSION OF THE MODERN OBSERVATIONS. Reduction of the Published Results of Observations to a Uniform System. We have now to discuss the great mass of observations made at the principal observatories of the world since the discovery of the planet by Herschel, in 1781. To make all the data of reduction rigorously homogeneous and uniform, it would be necessary to completely re-reduce the greater part of the observations made before 1850, using the modern values of the constants of reduction, and to com- pare each observation separately with the geocentric place deduced from the pro- visional theory. Such a reduction and comparison would be extremely desirable. Their execution would, however, involve an amount of labor far greater than it is now possible for the author to bestow upon the problem. We must, therefore, adopt the reductions which have been already made, applying such systematic corrections for reduction to a uniform system of star places as we have the means readily to determine. No reduced places are employed unless we can find data for some more or less accurate determination of these corrections, a rule which necessitates the rejection of a great mass of observations made at the minor obser- vatories of the European continent, and published in the Astronomischen Nach- richten. We still have the following rich collection of materials at our disposal: 112 THO OR BIT OF VU RASNTURSE 1. Observations at Greenwich, 1781 to 1872. 2, Paris, 1802 to 1827, and 1837 to 1869. 3. Konigsberg, 1813 to 1835. Vienna, 1822, and 1827 to 1839. PS Y ~ net 2. «| Speier, 182729) 62-94 5 == “Cambridge 1828 coslst2: i 2 + = © Edinburgh, 1836tolere 8 2. . £ .”.) Berlngt838 to ls42: 99. 2 8 592) Pulkowa, 11841 andas22: 10: . &£ i & +) Washington, 1861 steasi2: Ii es . «-. % ) Meiden, 1863 toda 2s Santiago, 1854 and 1855. As to ie Serer distribution of these observations in time, we may remark that during the first three or four years the planet was zealously observed at Greenwich, Observations then began gradually to fall off until 1798, in which year we find but one. From this time until 1814 only one or two observations were made at each opposition. ‘They become a little more numerous, until 1829, when there is a sudden increase. Few interruptions have occurred since. With regard to the other observatories it may be said that from 1802 until 1830 there is a gradual increase in the number of observations, and that since the latter year the number of observations is entirely satisfactory. A great number of the observations were reduced with the star places of the Tabule Regiomontane, and the entire Paris series are reduced with the star positions of Le Verrier, given in his “Annales de 0 Observatoire Imperial de Paris,” Tome II. As a preliminary to the discussion of the systematic corrections to the principal published reductions, I have prepared the following table, showing the corrections which must be applied to the places of the equatorial fundamental stars in the above catalogues to reduce them to the adopted standard, namely, Dr. Gould’s coast survey list in right ascension, and Auwers’ standard in declination. In the table of right ascensions the first column after name of the star gives the annual variation of that co-ordinate for the epoch 1860.0, as derived from Le Verrier’s tables of right ascensions just cited. Next we have the correction to this annual variation, expressed in units of the fourth place of decimals, to reduce it to that given in the “ Star Tubles of the American Ephemeris,” the positions in which are founded on Dr. Gould’s Catalogue. The fourth column gives the correction to the right ascensions of Le Verrier for 1860, in hundredths of a second of time. Subtracting from this column sixth-tenths of the preceding, we have the corre- sponding corrections for 1800. The last four columns give the corresponding num- bers for the right ascensions of the Tabule Regiomontane. The table of declinations shows, for different epochs, the corrections necessary to reduce the tabular positions to those given by Auwers in his paper on the declina- tions of the fundamental stars THE ORBIT OF URANUS. 113 I. Riagur Ascensions. Corrections to Corrections to Le Verrier’s : Ann. var. of Date. amn. var. Tab. Reg. 1860. | Ann. var.| R. A. Rapa. 1860. Ann. var. R. A. R. A. 1860. 1800. 1860. 1800. 3 s ; 8 a Andromede, | +35.0844 | 4+ 14 | + 2 = (? 3.0840 +18 + 8 — 3 y Pegasi, 3.0801 | 4 15 + 3 — 6 3.0824 —8 —%9 |—4 a Arietis, 3.3644 | + 12 | Oo seg se3636) wih u=20 te =E Gt 26 a Ceti, SUG ee On ih ees le ee Sok 26T ang Tee eg a Tauri, oe EY eee oie ee | 0 | 3.4335 +10 Set hese 6 Orionis, 2.8797 |} + 6 + 2 =e 2.8800 + 3 + 5 + 3 B Tauri, Sell j= 2! + 1 + 3 3.7888 —21 — 4 + 9 a Orionis, 3.2460 | + 5 aig I CEC gte464 aoe a eee a Geminorum, 3.8409) | — 2 0 == 3.8386 +21 +18 + 5 a Canis Min. 3.1462 | + 5 + 9 + 6 3.1455 +12 +7 0 Bp Geminorum, 3.6828 | + 5 SL B 0 3.6807 +26 +13 |— 3 a Hydre, 2.9485 | 4+ 12 | +6 | —1 | 2.9469 428 +16 | aan a Leonis, aoso) eis i eet = 3) a ore anon |) piss ee 6 Leonis, 3.0654 | +4 11 + 4 = 3) 3.0640 495 +12 | — 3 a Virginis, 3.1495 | +20 | +5 | —T | 8.1497 418 SE) a) a Bootis, 2.7325 | + 16 ES} a; 2.7327 414 Sep ees a? Libre, Spee Peeese = be | Score rou | =6) “eins a Corone, ONS 780 elon ecien le —iG) Ih e2-5373 7 |) + 5 | —5 a Serpentis, 2.9488 | + 8 + 4 —1 2.9513 —I7 — i +3 a Scorpii, s6658 | 2= 16 | 42-4 1 =— 6) | Msr66T2 ee | ees ies} a Herculis, 2.7322 | + T aD — 2 2.7319 ak, ty) 0 |—5 a Ophiuchi, 2.7808 | + 10 0 — 6 2.7783 +34 415 |—5 a Lyre, 2.0312 | + 2 — I — 2 2.0305 +9 +1 —4 y Aquila, 2.8520 | + 9 t. ® es 2.8546 il —8 |+2 a Aquile, 2.9281 | 4+ 4 + 1 —1 2.9281 +4 —3 |—5 6 Aquile, 9.9466 | + 5 aEE8 0 | 2.9496 B95 a 12" 1 Se3 a’ Capriconii, 3.3338 | + 4 sts & 0 3.3349 <4 —6 |—2 a Aquarii, 3.0829 | + 13 Te Ve || Sesinepe 420 TY Wea eee |) seen | Be ath =a legac [2-7 || Sie |e 1a Pegasi, 2.9898 | + 11 a 2.9830 1g '| —1/—6 Sum, 125 +81 —i2 | +210 +71 | —55 Mean, +8.5 +.027| —.024 +7.0 4.024 \—.018 15 May, 1873. 114 THE ORBIT OF URANUS. Il. DECLINATIONS. Corrections to Tabule Regiomontane. Corrections to Le Verrier. 1800. 1820. 1840. 20. 1840. ” 4 ” Andromede, +0.2 +0.1 ; ; 0.0 Pegasi, +0.1 +0. : +0.3 Arietis, 0.0 +0.: . +0.3 Ceti, =H 20) A eon Tauri, ; +0.1 +0. PP 0.0 Orionis, DEE +0.1 +0.6 : : +0.7 Tauri, a0 0.0 c : +0.5 Orionis, i 0.0 : ‘ +0.3 Geminorum, ; Pt 5 F +1.0 Canis Min. F 0.3 : 2 +0.5 Geminorum, Nee ; ; 2. +0.5 Hydre, 2 3 a 0.7 Leonis, ue : : +0.5 Leonis, 2 : Re +0.3 Virginis, 1). 0.6 ; i -+0.8 a Bootis, i id us ; +0.4 a Libre, 5 : l +0.6 a Corone, ; F we +0.4 a Serpentis, i); 6 : ‘ +0.9 a Scorpii, 5 : : +0.7 a Herculis, ; < +0.7 Ophiuchi, BE ; 0.6 Lyre, ; BC E +0.4 Aquile, : b AL(I\; 0.4 A quilie, ; +0.4 Aquile, f +0.8 * Capriconii, : a +1.1 Aquarii, : +1.0 Piscis Aust. 2.3 a Pegasi, : +0.2 Mean +0.60 THE ORBIT OF UKANUS. 115 The correction to the reductions to apparent place given in the Tabule Regio- montana on account of the correction to the constant of Nutation is :— In right ascension: —0".46 sin Q —0".18 sin Q sin a tan § — 0’.24 sin Q cos « tan 8. In declination: —0’18 sin Q cos a + 0.24 cos Q sina. The terms which contain tan 6 as a factor may be entirely neglected, as they are small, periodic, and contain tan § as a factor which is sometimes positive and sometimes negative. I shall also neglect the corrections in declination, as their sum is sensibly 0”.21 sin (a — Q) the effect of which will generally be confounded with the accidental errors of observation. The only correction we shall apply on account of nutation is, therefore, da = —0*.030 sin Q. The values of this expression at the dates when it is zero, a maximum, or a minimum, are as follows :— y- s. 8. I7718'5) 08 1920.3 00 1783.1 .00 1825.0 +.03 1787.7 +.08 1829.6 .00 1792.4 .00 1834.3 —.03 190 203 1838.9 — .00 1801.7 00 1843.6 +.03 1806.3 +.03 1848.2 .00 1811.0 00 1852.9 —.03 1815.6 —.03 1857.5 00 1820.3. .00 Having adopted this system of standard positions, we may adopt two ways of reducing the observations to it. One is to compare the positions of the stars adopted in the published reductions with the standard, and apply the mean differ- ence to the reduced place of the planet. Another is to make a similar com- parison of the standard catalogue with the positions of the fundamental stars which have been deduced from the observations by a system of reduction uniform with that employed in reducing the observations of the planet, and to regard the mean difference as a correction applicable to all the positions of the planet. If the standard catalogue and the observations are both free from systematic error, the results obtained in these two ways should be substantially identical. These are, however, conditions which we cannot expect to find fulfilled. In the follow- ing discussions I have sometimes used one, sometimes the other, and sometimes 116 THE ORBIT OF URANUS. combined both, the choice being determined by circumstances. We shall con- sider the different series of observations in succession, Greenwich Observations from 1781 to 1830. These observations are completely reduced by Airy and compared with Bou- vard’s Tables, in the work Meduction of the Observations of Planets made at the Royal Observatory, Greenwich, from 1750 to 1830. London, 1845. The con- cluded positions given in this work depend mainly on the star places of the Tabule Regiomontane, both in right ascension and declination. If we consider the first four oppositions—1781-1785—as forming a single group of which the mean epoch is 1783, we find that the general correction to the Tabule Regio- montane for this epoch is In right ascension, —0*.030; In declination, +0".08. If, on the other hand, we consider only the particular stars compared with Uranus, the result will be a little different. ‘The number of times each of the fundamental stars has been compared with Uranus, and the correction in right ascension cor- responding to each star, are nearly as follows:— s. 8. a Arietis, Ni— Com——0109 NS es a Tauri, 2 —.01 — .02 y Pegasi, 2 —.03 — .06 2 Tauri, 19 +.13 +2.47 a Orionis, a3 = e02e +0.66 a Canis Minoris, 33 —.02 — 0.66 2 Geminorum, 34 =i —2.38 a Leonis, 7 —.11 — .17 2 Leonis, 2 —.07 — 14 ‘The mean correction from these data comes out —0°.008, differing by 0°.022 from the general mean correction. Our choice between the two corrections depends on whether we are to consider the relative positions of the Tabule Regiomontane, or those of the standard catalogue, as nearest the truth at the epoch 1788, and particularly upon whether we are to consider the large correction to the proper motion of 3 Tauri as real. In the absence of exact data for settling this ques- tion, the mean of the two results, or —0°.020, has been adopted. A similar anomaly is exhibited by the declinations. It is probable that the declinations of Uranus during this period mainly depend on stars in the first twelve hours in right ascension, for which the mean correction is about —0".30 instead of -+-0".08. I have adopted —0’.16. Changing these corrections to lon- gitude and latitude, we have, during the period 1781-1786 :— Correction to observed longitude, = —0".30 ; Correction to observed latitude, —(.19. THE ROUBLE ORR TURAN UsSe 117 During the years 1788-1798 the above systematic difference in right ascension does not appear. ‘The most probable correction seems to be Aa = — 0°.025; Ad = 0".00. Whence A long. = — 0". 34; A lat. = — 0".10. Between the years 1800 and 1823 the stars used for comparison are so widely scattered that I consider it safe to apply only the gencral mean correction for the epoch 1813, which is aS — OOo" Ab = +0".66. Whence A long. = 0".00; A lat. = +0 .66. From 1825 to 1830 more than half the weight of the right ascension comes upon the stars a, 3, and y Aquile, the mean correction to which, during this interval, is —0°.035. The general mean correction at this epoch is +0°.002. I think the right ascensions of these three stars in the Tabule Regiomontane are really too great at this epoch by the entire difference of these results. We may, in fact, hereafter regard the positions of the standard catalogue as sufficiently accurate. ‘The mean corrections to be applied will then be No MONG IN) = +0'.83. Whence A long. = —0".05 ; A lat. = +0".86. From the year 1831 until the present time the Greenwich observations are regu- larly reduced in the several annual volumes of observations. But a reduction of the observations from 1831 to 1835, executed by Mr. Hugh Breen, is given in an appendix to the volume for the year 1864. ‘The results here given differ from those published by Pond in the several annual volumes for the same interval. The right ascensions are altered only by applying the constant correction —0*.030, which is found necessary to reduce Pond’s right ascensions to those of the Tabule Regiomontane. This correction I have verified. ‘The mean correction to reduce the right ascensions of the Tabula Regiomontane to our standard is at this time +0°.005. On the other hand, when we compare the concluded right ascensions of stars within six hours of Uranus, as given by Pond in the Greenwich observa- tions for 1834, with our standard, we find a mean correction of —*.034 to reduce his positions to the standard, which implies a correction —*.004 to Breen’s reduc- tion. The two results being +*.005 and —*.004, I have applied no correction whatever. In the paper in question the declinations are completely re-reduced, using im- proved data of reduction, but, so far as I see, making no changes in Pond’s method. The results differ strikingly from those of Pond, and suggest the desirableness of a complete re-examination of all Pond’s determinations of decli- nation. Having no catalogue of observed declinations of standard stars reduced in this same way, we cannot directly determine the systematic correction to the declinations. I therefore proceed as follows: A comparison of Pond’s observed declinations of standard stars with Auwers’ normal catalogue show that the former require the following corrections near the parallel of Uranus: 118 EGE OUR B 1) (OLE SUEReAGN Uns? Joa, S333} = 1s! 1834 — 2.10. Then comparing Airy’s reduced declinations of Uranus with Pond’s, we find the following mean differences: In 1831, Airy — Pond = — 3”.18 1834, — 3.50. To reduce Airy to Auwers we must there apply to the declinations In 1831 + 1”.76 1834 + 1.40. I have regarded the correction + 1’.60 as applicable throughout the period in question. During this interval the corrections in right ascension have been derived by the following two sets of comparisons: (1) A comparison of the several collected six and seven year catalogues with Gould’s standard, from which it appears that they require the following general corrections in right ascension : Six year catalogue of 1840 + 0°.047 Six year catalogue of 1845 + 0.002 Seven year catalogue of 1860 + 0.008 Seven year catalogue of 1864 + 0.022 (2) A comparison of the corrections applied to the right ascensions of the indi- vidual years to reduce them to the standard of the catalogue, as given in the introduction to each catalogue. ‘The sum of these two numbers gives the correc- tions for each year. A slightly different method is to regard the above correction for each catalogue as applicable to all right ascensions which depend fundamentally upon that cata- logue. I have sometimes combined both methods so as to derive what seemed to be the most probable result, and sometimes used but one. The corrections to the declinations during the interval in question have been derived from Auwers’ “ Tafeln zur Reduction der Declinationen verschiedener Sternverzeichnisse auf ein Fundamentulsystem,” Astronomische Nachrichten, No. 1536. These tables include the Greenwich seven year catalogue for 1860, when the correction corresponding to the declination of Uranus is about +0’.45. The corrections for the previous catalogues vary between 0’.35 and 0”.68, The cor- rection corresponding to the interval 1861-67 has been derived by a direct com- parison with Auwers’ declinations, and the result is + 0".44, agreeing with the two preceding catalogues. But, on making a similar comparison with the annual catalogue for 1569, a considerable change was found, the correction being — a a change of more than half a second. I shall use this correction for and after the beginning of 1868, as the change is probably due to the introduction of a new constant of refraction in the reduction of the observations for 1868 and subse- quent years, THE ORBIT OF URANUS. 119 Cambridge. An extended series of planetary observations was commenced here by Professor Airy, in 1827. The series was continued by him and Professor Challis, his sue- cessor, until 1842. During the first three or four years the combined right ascen- sions depend on a few special stars, and mainly on a? Capricorni. ‘Taking the mean correction to the adopted right ascensions of the stars actually compared as they are given in the introduction to each annual volume, giving to each star a weight proportional to the number of comparisons, the following corrections are deduced: 1828 — 0°.10 1829-31 —0.16 1832-37 —0.19. - In the introduction to the volume for 1838 it is stated that the adopted right ascensions are diminished by the average amount of 0°.083, which would still leave a correction of —0*°.107. Actual comparisons in two subsequent years give 1840, Aa = — 08.087 1842, — 0.069. Although the positions deduced from each year’s work were adopted for clock correction the year following, without any change of equinox, it seems that there was, effectively, a progressive change of about 0°.01 annually in the equinox as adopted. No declinations were observed until 1830. On comparing the declinations deduced from several years’ work with Auwers, it was evident that the correction increased with the polar distance of the star. The law of increase could be well enough represented by supposing the correction proportional to N. P.D. ‘Thus the following corrections were deduced in three different years. IRV eae Ws i8 x NSLS D: in degrees 100° 1”.00 « N. P. D. in degrees SA = pies erciasiaea te 1840, 100 : 1”.03 « N.P.D. in degrees 842 — ; 2 : Hees 100 From which the correction for other years was deduced by interpolation. But, on applying these corrections, the results were found systematically different from those of other observatories, and on referring to Auwers’ corrections to Airy’s Cambridge Catalogue, it appeared that the mural circle required a large correction near the declination of Uranus during this period. ‘The above results were there- fore altered so as to conform as nearly as practicable to Auwers’ law. Evlinburgh. In reducing the observations of 1836 Henderson uses the right ascensions of the Tabule Regiomontane, to which the general correction is at this epoch +-*.007, 120 THE ORBIT OF URANUS. But, if we take only the stars near Uranus, with which the latter was necessarily most frequently compared, the corrections will be negative. Comparing the con- cluded positions of the stars from a Serpentis through 0" to 3 Orionis, we find the following mean corrections : In right ascension, — 0°.012; in declination, — 0’.09. In subsequent years it is stated that the adopted positions of clock stars used each year are derived from the right ascensions observed at Greenwich, Cam- bridge, and Edinburgh, during the year or the two years preceding, without any statement whether corrections were applied for difference of equinoxes. In some subsequent years the following corrections are deduced, sometimes from the adopted and sometimes from the concluded positions : 183a, Aq 103.000); 1840, Aa=+0.015; ADec.= 0’.00; 1844, Ac=-+0.970; A Dec. =+0.49. Paris. All the positions of planets given by Le Verrier, in his ‘Annales de U Observa- toire Imperial de Paris: Observations’ depend both in right ascension and N, P. D. on his adopted positions of fundamental stars, the corrections to which have already been given. As the corrections to the individual star places used by Le Verrier are not generally of a systematic character, the general mean corree- tion is employed, which is :— In right ascension — 0°.024 + 0°.085 7, In declination + 0".12 + 1.207, T being the fraction of a century after 1800. In 1854 a new and larger catalogue was introduced, and for this and the follow- ing years the correction in declination is derived from Auwers’ tables. A summary of the adopted corrections after 1830, as deduced from the pre- ceding comparisons and discussions, is given in the following table:— TABLE OF ADOPTED SYSTEMATIC CORRECTIONS. Greenwich. Paris. Konigsberg. Berlin. Cambridge. Edinburgh. Aa Ad Aa Ad Aa Ad Aa Ad Aa 46 8 s " s ” 8 " 5 ” ecco tl brecs tage 4] aero nl eee ANO I Sepa | cece |) tse) |) 5.3 |) ily .00 4 .00 'Ds| 02] 0.91) 5...) 4-029) —— i les 9) Iso —l.4 —1.3 oes : Sint .. | —1.2 | +1.0 ; KO Wodes || LW 4 4 40.9 alee a9 1 Os | Sp eile x 295i ape ae —0.2 page r ( THE ORBIT OF URANUS. 121 TABLE or ADOPTED SysTeMAtic CorrEecTIONS.— Continued. Greenwich. Paris. Konigsberg. Cambridge. Edinburgh. b 2 Aa Ad By) ” = ”, n 1842 1843 1844 1845 1846 1847 1848 1849-53 1854-55 1856-60 1861 1862-65 1866 1867 1868 00 | 1869 | . Bee Mt cee \(Leteeed | eieeel ieee dl heer eat 1870 : ; Serica merce | eens a FS Tact | rere mais ate 00 1871 : BOR | Bare An Miser pore Me i ercde aig Homer tlhe anne, |b aransge || Se Soe 00 1872 5 : Edecent, Mptrsrstcasell ie ereesreietl | rstsictat | |) keeper ste eee ee I Soak 00 —0.3 Santiago .00 | +0.6 LW eee eee Washington | 00 00 00 oo2Toocoococqcoqcocqcoeo so WWwWWwWWwWWwWwh dD DWN Ree Pitter ener bo bo bo bo tO KS COME eo Pim ot een +4444 | as & Applying the preceding corrections to the positions of the planet as originally reduced and published, we have a series of observed positions as nearly homo- geneous as it 1s possible to make them with the means now at our command. The next step in order will be the computation of the geocentric place of the planet from the provisional theory for the moment of every observation, to be compared with the results of the latter. ‘The complete execution of this labor, ab initio, is, however, at present impracticable, and it is proposed to diminish it by making use of the published comparisons with the older tables. This can be done without danger of serious error, and with all the more ease that owing to the great distance of Uranus the errors of the solar tables are, for the most part, without appreciable effect upon the computed geocentric place of the planet. The method of making the comparison is different with different series of observations, and each series must therefore be described and discussed separately. ‘The general plan, how- ever, has been to replace observed and computed absolute positions by observed and computed corrections to the geocentric positions deduced from Bouvard’s Tables. To carry out this plan it is necessary to have at our disposal an ephemeris both of the heliocentric and geocentric positions derived from these tables. The corrections to the latter given by the observations are then given by direct com- parison. To obtain the corrections given by the provisional theory, the heliocentric longitudes, latitudes, and radii vectores given by that theory are interpolated to the dates of the heliocentric ephemeris from Bouvard’s Tables, and compared with that ephemeris. The differences are then changed to differences of geocentric place by the usual differential formule, and thus the corrections given by theory are derived. The difference between the two sets of corrections is the difference 16 May, 1873. 192 THE ORBIT OF URANUS. aa between the provisional theory and observation. A condensed summary of the results for each of the principal series of observations is here presented. Greenwich, 1781-1830. In Airy’s reductions, already referred to, we have given for the moment of each individual observation a heliocentric place computed from Bouvard’s Tables, and the geocentric longitudes and latitudes thence deduced. The observed right ascensions and declinations are then changed to longitudes and latitudes, and the apparent error of the tables thence deduced. The means of these errors are taken for groups of observations, and expressed in terms of the errors of heliocentric longitude, radius vector, and latitude. The mode in which these means have been treated is fully shown in the following table. ‘The first column gives the mean date of each individual group of observations. The next three give the mean excesses of the co-ordinates interpolated from the heliocentric ephemeris, p. 100, and corrected for solar nutation, over those printed in the “Computations of tabular place, etc.,” in the Greenwich reductions. In the fifth column these corrections are changed to corrections of geocentric longitude. In the next two columns we have the mean corrections to Bouvard’s geocentric places given by observation. It is the negative of the mean error of tabular place printed in the “ Reductions,” corrected by the numbers already given to reduce the star places to a uniform system, ‘Then we have the difference between these two sets of corrections, or, the mean correction to the geocentric place of the provisional theory as given by observation. Lastly, we have the differential coefficients for expressing the errors of geocentric in terms of the errors of heliocentric co-ordinates taken without change from the Greenwich volume. From Provisional Theory. | From Observations. Correction to Correction to tabular position in Proy. Theory. a - Correction to Greenwich Reductions. Mean Date. = Log. | Geoc. Geoc. Hel. Hel. Tiong: Ree Ven Lat. long. long. | lat. lat. ’” iL " + bo Co 0D He bow occ No aT OO TT OO CER ROD ORWW ROUEN ; } > 1781, Oct. 10 Noy. 13 as anaes Dec. 27 533 [1 1782, Jan. 31 Mar. 4 Oct. 10 Nov. 26 | 1783, Jan. 11 Feb. 24 | Oct. 10 Nov. 1 Dec. 15 1784, Jan. 29 Mar. 12 | Oct. 30 | Dec. 14 > Foran i | Doe oe hm bo G2 bo DOr SA!L AOWe| NO wWaTH SOs) SSeS bo Ot ee — ie Ole Sea el FO ONWOS WWM OMeDe ro oo oe ISN RUS ++ |4+/+4+444 to = S | to 99 bo bo Pima) Oe F — Co Hm bO RCO ++4+++ Ss Ee aoomnssd teal elisa tow nes Ll ce ee oe ee ey SiS SS) SS} Om Roop oo | eeriese=ts TAR WRWOWS BAC O OH bor wowwhnDp wwork PROP eS Sms tH -T oe FNwWWwWNPe — e2 no co bo ++ THE ORBIT OF URANUS. 123 From Provisional Theery. From Observations. Correction to Correction to tabular position in 2 Prov. Theory. Mean Date. Greenwich observations. Corecsionito Log. Long. ae Tete Geoe. Geoc. Hel. |No.of long. | long. lat. Obs. ” ” —l. 3 i =) by : =I = + 9¢ —o. . || =e | —3.% —3.8 —4, —4, 954] 4. 1785, Jan. 16 Feb. 13 Mar. 20 Nov. 8 1788, Mar. 13 Oct. 26 1789, Jan. 18 Apr. 8 Oct. 31 | 1790, Jan. 24 Nov. 5 1791, Jan. 29 Apr. 14 Noy. 10 1792, Feb. 5 Noy. 15 1793, Feb. 8 | Nov. 14 1794, Feb. 15 Nov. 19 1795, Feb. 20 Nov. 29 1796, Feb. 24 1797, Feb. 1800, Mar. 1814, May 2: 1815, May 1818, June 1819, June 1820, June 1823, July 1825, July 11 1826, July 16 1827, July 20 1828, July 23 1829, Aug. 7 Octs 4 1830, July 30 Aug. 29 Sept. 20 Oct. 14 Noy. 13 la Vel el = = o!| »& © eo bo Py -T bo | i _ ) — aids |) 3 RSwl oom De] ie Co He bo bye Lene at fs VI Hm He CO a Hm He HD pr me bo Or on oe oon bo S © PI CUCN eS) ot wm bo on ee or rs or 09] — = eo > — — for) on ~ He OO to enon noe bo (>) or) me t+ ]++]+++ oro — CS i=) or) Lh) oO S o hoe |+|/++ Ho} toe ae er fora) Suen Wage gone or oie oo Bae ay a Le GE en OL ae of ou Co bo bo ++ bo oe CS ory org oo wo | a ee ee ee ee if + + + + =| + —+- + gas + 9 + + + = + + + + | + + mm ee He CO OO oo oo — Co > im oO < oT ~~ to 09 2 Ww | Va ales He bo |_| Wr on (ool oro. =r) Pose] | eae Se se BH EE PE ee HE EE Pe 2 or wl al $44t4t4t+t4+4+444 [+ mPocoooorFre — Oe OR em OL bo Se aos CO CORD SO OH OAT OMT SO Ob tn +++++4+4+4+4++ 4444+ 4+ + pelllbac ee eas PATO HR OD eRe Re POD Pw SS OS Mie Hou oie — Si — NDP WHE WNUOaAD CS Siscis oe S.ce io i> SLI CCE [fe Te EI as i COVCOSKrVSOAPOHANRW FERRO ODO ee RO OO WOH DF HOO DWwWooeo-s10% Cr Corr WUC r-ATSO DHE HODOmMoNDanmrwoaurn fr) SOSH EE PH NSHIasts A Hot Soimoe 3 © 124 THE ORBIT OF URANUS. Paris, 1801-1827. A complete reduction of this series is found in Le Verrier’s Anwales de 0 Obser- vatoire Imperial de Paris, Observutions, tome I. No comparison with any ephemeris is given here, nor is there any complete ephemeris to compare them with. A complete geocentric ephemeris was therefore computed from the provisional theory for the principal groups of the Paris observations. The individual observations being compared with it, the resulting mean corrections are given in the following table: Mean date. Aa max.) N. Mean date. Aa Ab N. 1801, March 24, —*.02 -+1".2 2|1813, May 20, +519 =S1¢6eia 1802, April 1, + .08 +06 13/1814, May 27, + .21 408 4 1805, April 22, +.10 +2.2 18/1815, May 24, —.02 1422 % 1806, April17, —.01 —1.6 5|1816, Junel, —.01 +0.8 7 1807, April 28, + .17 +0.4 16/1817, June5, —.08 416 5 1808, April 28, + .02 +1.4 6/1818, June7, -+.12 42.2 9 1809, May 5, + .20 +0.1 9|1819, June 18, —.07 —+1.8 59 1810, April 30, + .22 42.6 16/1820, June 20, —.20 —24 85 1811, Febr’y 18, + .21 +2.2 3/1821, June 22, + .05 41.0 5 1811, May 17, +.14 42.6 8| 1823, July 18, + .02 41.8 95 1812, Febr’y 16, - + .28 2/1824, July 13, + .04 0:0" a 1812, May 10, +.16 +83.0 6 1827, July 25, —.05 +06 5 1813, Febr’y 25, + .44 3 Total number of observations in right ascension, 175, The observations in this series exhibit numbers of discordances of that class which leave the astronomer in doubt whether the observation should be retained or rejected. ‘This remark applies more especially to the declinations. If we de- termine the probable error of an observation in declination by the condition that it is that amount which the error falls short of as often as it exceeds, it is found to be about 2”.. Then, if the errors followed the commonly assumed law of proba- bility, only about one in six of the errors should exceed 4”, and one in twenty- three 6”. But errors of these magnitudes are much more numerous, the deviations often amounting to six or eight seconds. I have rejected only a few in which the discordances approached 10”. Bessel’s Kénigsberg Observations, 1814-1835. I have made a complete re-reduction of the right ascensions of this important series, and of most of the declinations. In order to avoid the necessity of apply- ing systematic corrections, Dr. Gould’s right ascensions and Dr. Auwers’ declina- tions were used throughout in these reductions. In this work a selection of the fundamental stars observed by Bessel was made for each observation of the planet, to be used for clock error. These were chosen so that the mean of their right ascensions and declinations should be as near as practicable to those of Uranus, a condition, however, which could not generally be fulfilled for the decli- nations, owing to the southern position of the planet. Bessel’s instrumental cor- THEE OER B Leh OF Ui RAN) US 125 rections were applied to his observed times of transit over the mean wire, and the resulting time was employed as that of transit. Kach time, compared with the computed right ascension of the star gave a value of the clock correction, which was reduced to the time of transit of the planet by the known daily rate. If the instrumental errors were always accurately determined, the mean of these clock corrections would be used to obtain the right ascension of Uranus. But it was frequently found that the clock error varied systematically with the declination of the star, so that it was deemed advisable to add to the clock correction a term varying as the simple declination, which was deduced from all the stars, and used to reduce the correction to the parallel of Uranus. It was intended to give the results of this reduction for each observation, but on comparing the results with those of Fleming in the Astronomische Nachrichten, Band 30, it appeared that the results were not materially better than his. It does not, therefore, seem necessary to give more than the mean results for each opposition. From Bessel’s declinations, with the old Cary circle, I was unable to obtain any satisfactory results, owing, apparently, to a want of knowledge of some peculiarity of the instrument. Fleming’s reductions were therefore adopted. They are designated by the letter Fin the following list. Mean Corrections to the Provisional Ephemeris given by Bessel’s Observations at Kénigsberg, 1814-1829, Mean date. Aa Ad N. Mean date. Aa Ad N. 1814, May 22, -+'11 +2”5F 9 / 1822, June 24, +°10 +178 qf PSioy May 25, -— 13° --1-8F 11 | 1823, July 4, — 05 Shor 8 1816, May 27, + .06 +1. .2F 11) 1824, July 6, + .01 —1.0F 5 1817, June 6, S213 -§2.3F 8/1825, July 16, + .01 —24F 5 feteswnme 6, 802 24377 13/1826, July 18, = 01 =310K 7 1820, June 21, + .02 +4.1 Z| Ties), divi; Bo, ells BO = 7 1821, June 23, +.12 41.5 5 | 1829, Aug. 1, —10 —10Fr 9 Total numbers of observations, 103. Results of Observations at various Ol.servatories, from 1827 to 1829 inclusive. During these three years we have, besides the observations already quoted, the following :— 1. Observations by Schwerd, at Speier, of which the originals are given in Astronomische Beobachtungen angestellt auf der Sternwarte des Kénigl. Lyzeums in Speyer von F. M. Schwerd, Speyer, 1829-30, and of which the reduced results are found in the Astronomische Nachrichten, Band 8, S. 264. 2. The series by Airy, at Cambridge, commenced in 1828, and found in the Cambridge Observations. 3. Littrow’s Vienna Observations, found in the first series of Annalen der NK. Sternwarte ii Wien. The mean corrections to the provisional.ephemeris given by these series are shown in the following table. ‘The observations have been divided in the usual 126 THER Os Biles OPE UPR ANG UE Se way into groups of about a month each, and the mean date and mean correction found for each group. ‘The Paris and Konigsberg results are repeated for the sake of clearness. ‘The small figures show, as usual, the number of observations employed in forming the mean, Aa Ad Date. Observatory. Original. Corrected. 1827, July 22, Speier, —(*.16; —0*.14 July 25, Paris, —() .03, —0 .05 -+0".5, September 15, Vienna, —().11,, —0 .10 0.0, October 14, Vienna, —().18, —0.17 — 2.2, 1828, July 25, Kkonigsberg, —0.15, —0.15 —3 9, July 29, Vienna, —() .24, —0 .20 —1 4, August 14, Vienna, —( .13,, —0 .09 +1 .15 August 27, Speier, —) WO, —O. 09 September 18, Vienna, —0 .08, +0 .01 +1 .0, September 25, Cambridge, —0.05, —0 .16 October 17, Vienna, —(0 13), —0 09 0.0% October 17, Cambridge, —0 .02,, —0.12 1829, August 1, Kkonigsberg, —0.10, —0 .10 —I1.0 August 6, Cambridge, —+0.i1, —0 .08 —1.1, August 28, Speier, =m (a ()) omnes (I (EE September 23, Cambridge, -++0.21,, +0 .05 November 6, Cambridge, -+0.25; +0 .09 Observations from 1830 to 1872. Since the year 1830 heliocentric and geocentric ephemerides of Uranus com- puted from Bouvard’s ‘Tables are at our disposal. We make use of those in the Berlin Astronomisches Jahrbuch for the years 1830 to 1833, and of those in the Nautical Almanac from 1834 forward. ‘The system of comparison is the same as that already explained, That is to say, we deduce separately: (1) Mean corrections to the geocentric longitude and latitude of Uranus in the ephemeris as derived from observation. (2) Mean corrections to the same, given by the provisional theory, as derived from a comparison of the heliocentric positions of that theory with the heliocentric positions in the ephemeris. Then (1) — (2) is the correction to the provisional theory given by observation. The process of forming (1) and (2) is shown quite fully in the following pages. Kach individual printed observation was first compared with the printed ephemeris, and a correction to the latter was thence deduced. When this correction was given with the observations themselves, it was of course not recomputed, unless in some doubtful cases, The observations were then divided into groups, usually of about a month each, and coinciding in time with the grouping of the Greenwich results, The mean of the dates and the mean of the corrections were then taken separately for each group and each observatory. ‘The separate results are shown THE ORBIT OF URANUS. ’ WY in the proper columns of the following table, under the head “Mean dates,” Mean cor. in R. A., and Mean cor. in Dec. These means are those given by the observations as printed, without the application of the systematic corrections on pages 120 and 121. In the columns “Corrected mean” these corrections are applied ; this column would therefore exhibit no systematic differences between the results of the different observatories, unless the observations of Uranus were affected by errors different from those which affect the positions of the fundamental stars. A careful comparison of the differences in various parts of the table shows that this is unfortunately the case. A weight is next assigned to each individual result depending on the number of observations, the general sufficiency of the data of reduction, the mean discordance of the individual observations, and the quality of the instruments. ‘The critical reader will notice a lack of homogeneity among the weights assigned, of which I shall speak presently. ‘The mean of the separate group-results is then taken with regard to these weights, and also the mean of the mean dates, using for the latter the relative weights adopted for the several right ascensions. ‘Thus, we have a mean result derived from all the observations for each month, or other group-period, which is written under the horizontal lines. These corrections to right ascension and declination are next changed to correc- tions of longitude and latitude, using for this purpose the following table, which is computed from the formule of Gauss: cos H = sine cosa sec 6 = sine cos/ sec) Aj sin Ecos y) cos EB as cos b cos Ab = — cos E cos § Aa + sin E Ad. The differential coefficients in this table are expressed as a function of the right ascension of Uranus only, which may be done because, owing to the small inclina- tion and great distance of the planet, its geocentric position on the celestial sphere is never more than about 2’ from some point of the projection of its heliocentric orbit. ‘The coefficients of Aa are multiplied by 15, that the right ascension may be expressed in time. To Convert Errors or Riagut ASCENSION AND DECLINATION OF URANUS INTO ERRORS OF LonGirupE AND LATITUDE. When the Right Ascension exceeds 12", enter with R. A —12", and change the signs of the quantities Ob ‘) ae and ol. Ou on) Logarithms of a | & | & ob o Oa | tol) ran) ral) —0.7761 +9.6000 9.9626 5S | | + 0.924 —0.7757 +9.5996 : 9.9626 —0.7743 Be +9.598° a 9.9628 —0.7720 Bt ‘ if = 9.9632 —).7687 a 9.593 x 9.9637 — 0.7643 9.5896 eG 9.9645 —0.7588 9.5849 9.9653 —0.7525 9.5794 ~ 9.9662 —0.7451 ve 9.57% 9.9673 —0.7365 : 9.5656 ss 9.9685 —0.7270 Pes 9.557: 5 9.9698 —0.7162 : 9.9711 128 THE ORBIT OF URANUS Th \ o Convert Er JRRORS LIG SC oF Rigut ASCENSION AND DECLINATION.—C' oN. — Continued Logarithms of eoAL at 0b él y : is 5 ob ob & ‘ = 2k od Od aa ob ot ob Qo (yn 1.1395 — 0.7044 | = g On 0d 0) 10 1.1395 | —0.6915 229 o 5375 a5 | 9.9725 | $13.84 = 20 | 1.1395 | —0.6773 ~ 222 9.5260 9.9741 Be 5.06 30 | 1.1395 Thane ieee 49.5134 —128 | pe 4 a Barr 0.934 40 | 1.1395 Sheva 768 | 72-8095 aie biga2 4.75 nes 50 1.1394 Bo Sean +0.4843 ip end 4.58 104 } t —0.6266 Saas 49.4676 Saye! 9.9788 441 0.31 | a) See 7 i981 | 9-9804 4.23 oe 10 | 1.1394 | Eee 915 | 19-4495 9.9821 : Ge 20 | 1.1393 | —os6x1 —233 | 9.4081 B.O8oT | aore | aca oa lee Sor iedelso3 ip pags mors +9.4081 —216 nee | 3.86 a +0.96-4- 40 | 1.1392 | 915094 Sy +9.3844 meee en 3.66 a 50 | 1.1391 | 0.4795 999 +9.3586 —_| 258 9.9 8.45 9 “4795 397 | +9.3302 —og4 | 9-9883 3.24 a a 0 1.1390 is 2 = 3319) 9.9898 3.02 oa 10 | 1.1390 0.4109 229 + 9.2990 9.9912 | i 20 1.1389 | —o 3710 —399 + 9.2644 —346 9 S rl =-13/8--— | —2's04= | 012 go | 11388 Deane eee aliceece So 2938 Zoe O18) | sl 1.1387 | —0.2769 —498 +9.182 2.36 50 | 1.1386 | _o.2198 Spee honiaai aces eae 2.14 bie 2198 345 | +9 0781 — ary aaeee 1.91 = 0 | 1.1385 | —0.1539 ee 1.67 0 2 10 | 1.1384 | =oi0753'— 28 +9.0130 + | 9.9977 50° | aciaes | 2ptere igen aeceeces 277 2S ie ae 30 | 11382 | —9.e54 —125| Terrisy —956 | 55990 1.19 0:03 | eel Bo} aeaseoi | Bese Ome eee Zi7oi| eeeeee 0.71 He —9.378 48.240 399 | 9-9997 aa 0.05 ee eo: oe as 9.9999 Ae 0.03 Jo) | 1.1378 Fs es Ge 0 413.74 a 137 “ ROY Ygy.. | Opec | 0 0.0 ue 11376 tose t175 —8.540 Beas aaeee xvas Bie. +1.004- ECs edie en coer en es 757) 999° 0.4 ‘0: 50. | aiasyanl eer i 96 a hey SE) O71 ae +0.0743 F733 | —8.9353 +956 | 9-9990 Gigs 0.05 0 | 1.1373 } +-0.1526 4-777 | 9-9984 1.18 ne 1o | 1.1373 | 0.2184 +858 ee 9.9977 . 20 | 1.1372 | 10.275 SprteHe oa sa gee Be 9.8977). EIST | | fazt | eo aesoi tot ereaa ie. 1.65 0.12 40 | 1.1371 250 Fy oieaa aaa oeees 1.8 13 HSE ol eek ai zu | ots ; 559 | —9-2644 9.992: 2.3 0 | 1.1370 | +0.4448 oe 4346 | 9-9925 256 ae 0 1.1370 +0.4773 +325 —9.2990 | 9.9912 j 20 | 1.1370 | +0.5071 +29 — 9.3302 +312 | 9 9898 413.74 | 4-2.78— |o——0.20 30 | 1.1370 | 0.9344 +273 | 9/3644 +284 | 9883 al) on | a 40 | 1.1370 | 40.5597 1253 —9.3844 13% soe 3.22 0.23 50 1.1370 40.5830 933 | 92-4081 ee ete 3.43 0.24 -08 4914 —9.4297 216 | 99 Be 3.63 0.26 0 | 1.1370 +198 | 9 9837 3.8% 27 < 0.6044 83 0.27 10 | 1.1370 ste —9.449 : 0.6241 +197 5 9.985 7 20 1.1370 uf oes +184 -—9.4676 +181 aoa =F15.7-- | -4.02— | 0198 30 11371 Tipesaa +169 —9.4843 eee 9 Baan 4.20 aoa +0.96+4+ 0 | 1.1371 ee 155 | —9-4995 ioe 4.38 : 50 | 1.1372 Bireecs Bee | —9.5134 Lee a oeae 4.55 He “6592 13) | —9-526 26 | 9 97 2 "35 0 | 1.1373 | 40.7023 ze ist BY Sig || ee 498 Hes 10 | 1.1374 Ay ol —9.53 ; : . 19 5375 7 20 | 1.1375 alinroee ti09 Seek allay 9725") ‘18.7 | El eee 93-4 1376 gg | —9:5573 : 5.18 : an | 1376 | torsar P| Castas 1 $2) sans sat | 03s 50 | 1.1378} Lo.7510 + 76 | ily concen Uae renee: 5.43 0.37 10 Sites Tease te 5.54 Rs o | 11379 | 40.7575 + 55 9662 5.64 oy a 1.1380 +0.7632 + 57 9.5849 4 g7 | 9:9653 | +18 2 .138 , | —9.5896 one 7 30 | 1.1382 to. ure + 35 Bree oo ase ih ea ees +e 40 | 1.1383 | 0. GE SE oe |) Soe eee aitboe 5.85 0.39 5 1.1385 | 10.7754 1 17 a eae EE = 9.9628 6.90 0.39 of aang + 7 196 T “| 9.9626 Us 0.39 40.7761 sonra 98 O40 senees 0 ..., | 9:9626 13. +13.8+ } +5.97— | —0.404 | +0.92+ THE ORBIT OF URANUS. 129 We thus have, for the interval occupied by each group of observations, a mean correction to the geocentric longitude and latitude of the planet given by obser- vations, which are found in the ninth and tenth columns of the table, on the same horizontal line with the mean corrections in right ascension and declination from which they are derived. ‘The next step is to obtain the corresponding corrections given by the provisional ephemeris. This correction has been first obtained for every twentieth day of each of the forty-two oppositions included in the table. The heliocentric longitude, latitude, and radius vector were interpolated to the most convenient twenty-day intervals, and compared with the corresponding co-ordinates in the heliocentric ephemeris. This ephemeris was of course the one corresponding to that with which the observations were compared, namely, the Berliner Jahrbuch for the years 1830-33, and the Nautical Almanac for subsequent years. These comparisons are fully given at the end of this chapter, and the resulting corrections to the printed ephemeris are given in the proper columns of the table. These corrections to the heliocentric co-ordinates were then changed to corrections of geocentric longitude and latitude by the following formule, Put a’, the projection of the planet’s radius vector on the ecliptic ; p, the projection of the planet’s distance from the earth on the same plane ; p, this distance itself; 4,3, the planet’s heliocentric longitude and latitude ; I, the sun’s geocentric longitude ; RF, its radius vector; M, the modulus of the common logarithms ; él, 66, the corrections to the geocentric longitude and latitude ; dp, the correction to the common logarithm of the radius vector. Then We [LG cos (Le — 2) f 3 me. bp oa sin (ZL — a) Msin 1” Sel inh ao “9 / r 6= = \ 15 me tan?3 cos(L—a) } 63 pP p J 2 = fe tan 3 sin(L — 2) da iy 2 rR? 7 : Sp ee [It c=) § 80 Bape The last term in SJ and the last two terms of 6b have been omitted in the com- putation, as they scarcely ever exceed a few hundredths of a second. 17. May,1873. 130 THEE (OUR BIT OR UPR PAGN UES: The values of &7 and 60 are printed in the last two columns of the table. The formula for 66 might have contained the additional term 66 = sin lia do being the correction to the obliquity of the ecliptic adopted in the ephemeris to reduce it to that employed in the provisional theory. ‘This correction is, how- ever, deferred until we come to form the equations of condition. From the values of é¢ and $6 thus obtained we are to find the mean values during each group of observations. If these quantities varied uniformly, the proper value would be that corresponding to the mean date of each group. But the second differences are so large that this value would generally be in error by one- or two-tenths of a second. Owing to the minuteness of this difference, it has been considered that when the mean date was near the middle of a twenty-day interval, the correction §¢ interpolated to that date without regard to second differences would furnish a sufficient approximation to the required mean value of éd during an interval of about 30 days. In other case the value of ¢2 was inter- polated to 5-day intervals through the period of each group of observations, and the mean value taken. During the years 1850-1863 the sun’s longitude employed in the ephemeris required a gradually increasing correction, amounting at the latter date to about — 3”. A-small correction of which the maximum value is about 0’.15 was applied to <1 to reduce it to the value it would have had if Hansen’s tables had been employed. The corrected mean values of (7 and ¢6 thus obtained are given in the last two columns of the following table, being inclosed in brackets and printed immediately above the values of Ad and Aé derived from observation. I deem it proper to mention that the mechanical labor of constructing these tables of comparisons, in the manner just described, was in great part performed by Dr. C. L. F. Kampf, who was employed by the Smithsonian Institution to assist me in the work. Before using it I subjected the whole of the work to a careful revision, altering especially the relative weights of the corrected means in many cases. As the assigned weights now stand, each set of results which are combined into a single mean has its own unit of weight, which does not necessarily coincide with that of any other set. The use of a uniform scale of weights through this series of observations, and the assignment to every final mean of a weight equal to the sum of the weights of the quantities whose mean was taken, would have led to weights in many cases quite fictitious, owing to the obvious presence of systematic errors in the results. For this reason I have made no further use of the weights found in this table, and their lack of homogeneousness therefore does no harm. TGR ONeill e OPE TURR AGN TU; Siam 131 MEAN CORRECTIONS 10 THE EPHEMERIS OF URANUS IN THE BERLINER JAHRBUCH AND THE NauvtTicaL ALMANAC. a ete. - eg | Observed corrections in R.A. | Observed corrections in Dec. | Corr. to Geocentrie | Observatory. ~ [R. A. of Mean dates. ] fee = Uranus. ] Mean. | No. of | Corrected | Mean. No. of | Corrected Longitude.) Latitude. obs. mean. obs. mean. Konigsberg, Cambridge, [20" 40] Konigsberg, Cambridge, [20" 36") Cambridge, [20° 37™] Cambridge, [.20" Be] i Cambridge, | Noy. 14 [.20" 37™] 1831 Greenwich, | Aug. 3 Cambridge, | Aug. 8 : : pos : 3.8] [410.0] PAU? S8=]] | wes Gs be ea Bee : Bae Sa 3. —+11.1 Greenwich, Sept. | Cambridge, | Sept. 15 nG 3 -76 [110.0] f20" 52"] | Sept. rene ieee TOME pes” ees 22.2 | 411.8 Greenwich, | Nov. | Cambridge, eel -T)[4 9.8] [20° 507] is 5008 aD 65 sob oe .0 | +10.8 Greenwich, Konigsberg, Cambridge, Vienna, pai 17") Cambridge, pa1* 12") Cambridge, Vienna, Pe oy Cambridge, Vienna, [21 10°] 132 THE ORBIT OF URANUS. MEAN CoRRECTIONS TO THE HPHEMERIS OF URANUS.—Continued Observatory. [R. A. of Urauns. } Greenwich, K6nigsberg, Cambridge, [21" 3 gm] Greenwich, Cambridge, Vienna, [21> 287] Greenwich, Cambridge, Vieuna, [2i" 26") Cambridge, Vienna, [21> 26") Cambridge, Vienna, [21" 49] Greenwich, Cambridge, [21> 45™] Greenwich, Cambridge, Vienna, [21> 41™] Cambridge, Vienna, foi" 4174 Cambridge, Vienna, [2b e2y Greenwich, KGnigsberg, Cambridge, Vienna, [22 4™] | Mean dates. 1833 Aug. 22 Aug. Aug. 15 Aug. 15 Sept. 18 Sept. 19 Sept. 11 Sept. 18 Oct. 11 Oct. 12 Oct. 14 Oct. 12 Nov. 18 Nov. 14 Nov. 16 18384 15 13 14 10 16 a) vo Observed corrections in R.A. Observed corrections in Dec. Corr. to Geocentric | No. of | Corrected obs. mean. Mean. No. of obs. Corrected mean. Longitude. | Latitude. |+ —— Or bo o eo oO for: wo - toe oo 2 9 — b oO =¥ aes & ee eee) —0.1 —0.1 2 O. 298 —5.9 [—32.2] |[-+10.8] —33.4 | 119.3 [—81.2] |[-+10.6] —33.0 | 411.9 [—30.3] |[+10.3] —32.0 | 410.4 +11.) te [411.1] 411.9 [-+10.9] +11.6 [-+10.7] 411.8 [410.5] A115 — 4 hil 11.3] in pats THE ORBIT OF URANUS. 133 MEAN CorrecrIons TO THE EPHEMERIS OF UrRANuUS.— Continued. | | Observed corrections in R.A. Observed corrections in Dec. Corr. to Geocentric Observatory. | ES | [R. A. of Mean dates. | | | Uranus. ] | Mean. | No.of Corrected | Mean. | No. of) Corrected Longitude.| Latitude. obs. | mean. obs. lean. i 1885 s ” ” ” / | Cambridge, | Sept. 15 | —3.17 9 —— 2G i) —5.8, | Vienna, Sept. 14 | —3.3 9 — 6.7 9 —).85 alae [—45.8] |[-+11.2] [228 1™] | Sept. 15 —5.8 | —48.1 | +11.6 | Greenwich, | Oct. 10 | —3.27 4 | —3.27,|}— 6.53) 4 , Cambridge Octaae li e——Selel 8 | —3.30,;— 4.1 | 8 83 0 | See eae S_ [44.4] [411.1] el ra Oct. 15 oobi 506 |) oes) o606 --- | —0.3 —46.9 | +11.4 Greenwich, | Nov. 27 | —3.21 6 —3.21, | — 5.6 7 —4.4, ; Cambridge Nov. 26 | —3.00 | 10 | —3.19,} — 4.5 9 —).7, | : ie |e | Saba 2 |[—43.0] [+10.7] [21" 57™] | Nov. 26 ere 5o0. || Bee) Eroyete soo || ==5,2 9) || qalil4 1836 Greenwich, | July 22 | —3.80 7 | 3584, | —1:0:5 8 —9.5, } Cambridge, | July 25 | —3.60 3 | —3.89, | — 8.9 2 —9.9, [55.3] 411.5] eons ooul | 2...) || secaatsGile 2.054 | ceeizore: oli—56-5ui-edles Greenwich, | Aug. 24 | —3.78 7 —3.82,| — 9.8 8 |— 8.8, KGnigsberg, | Aug. 30 | —3.63 5 || —3.63) Cambridge. Aus: 16) |) — 3278) |) 12) |) 35972 | —=— 953" |) 12) || 10.30) | Edinburgh, | Aug. 19 | —4.09 | 9 | —4.10,|— 9.3 t i= 9.8 Vienna, Aug. 20 | —3:77 1 | —.81, | —12.5 | ITH, [—54.6] |[-++11.6]] paces AS bt LAT Ee YT 04.6 .6 [22" 207] | Aug. 22 Siseais oo6 || Shy acos || ooo || —=9.6 —56.6 | 411.5 Greenwich, | Sept. 13 | —3.77 CG |) sole || — Gh |] —.8, Cambridge, | Sept. 16 | —3.70 | 10 | —3.89,|— 8.4 | 10 —9.4,, Edinburgh, | Sept. 16 | —4.01 8 | —4.02,|— 8.6 8 —8.6, Vienna, Sept. 15 | —3.59 9 —3.63, | —10.4 9 0), 4 bead aoe “*s_|(—53.4] |[+11.6] 22 Ge) °|Septmeloa| sss) | ea |—se8Te |) < 556 |) 22. | esis!) |—56.9) [nerd Greenwich, Oct alesse 7 —3.71,/— 8:6 | 8 —1.6, Gambrideens | Oct. 16) ——se5 lel) p38 708 = Br 5n | LIN 9852 Edinburgh, | Oct. 15 | —4.05 8 —4.06, | — 7.9 4 —T.9, Vienna, Oct. 11 | —3.57 | 10 | —8.61,|—10.2 | 10 9.25 [—51.8) |[-+11.3] Peta Oct. 15 600K -.. | —3.7T 566. || con || ==) —54.8 | +11. Greenwich, | Nov. 13 | —3.59 | 11 | —3.63,/— 9.1 | 11 = sale Cambridge, | Nov. 13 | —3.37 8 | —3.56,| — 7.9 7 —8.9, Edinburgh, | Nov. 17 | —3.78 8 | —3.79, Vienna, Nov. 9 | —3.56 3 | —3.60, | —10.2 3 —— 9) 2 [—50.6) [11.1] fe2eeea | Novae) 22— Il aq. eazereBe| 226 | eee) | 85 S52: 8) |e Iaco Cambridge, | Dec. 12 | —3.40 7 | —3.59, | — 8.0 7 —9.0, Edinburgh, | Dee. 12 | —3.29 5 —3.30) | — 8:8 3 —8.8, [50.0] [-++10.9] fe es (el = ees | 50: of [22" 137] Dec. 12 Dido ... | —3.52 S060 ... | —8.9 —§1.5 | 410.1 1837 [—62.9] [11.5] Greenwich, July 22 | —4.23 4 —4.26, | —13.4 4 |—12.5 | —63.2 | +12.0 [22" 407] 134 THE ORBIT OF URANUS. Mean Correcrions To THE EPHEMERIS OF URANUS.— Continued. | Observed corrections in R.A. | Observed corrections in Dee. | Corr. to Geocentrie § Observatory. | . | [R. A. of Mean dates. | | Uranus. ] Mean. No. of Corrected No. of Corrected Longitude.| Latitude. | obs. mean. Mean. | obs. mean. : | 1837 8 s i ee " " Greenwich, | Aug. 18 | —4.30 | 10 | —4.33,|—13.4 | 10 | —12.5,, Cambridge, | Aug. 18 | —4.09 | 14 —4.28, | —12.6 ll | —13.4, Edinburgh, | Aug. 22 | —4.40 6 —4.40, | —12.7 6 | —12.7, Paris, Aig. 4 432 9 —4.33, | —12.0 10 | —11.4,, Vienna, Aug. 25 | —4.29 A) 3 |] S113 4 | —11.6, [02.9] [411.7] raa® 36] | Aug 18 | .... |... |—4383 | .... |... | =12%5))eecaeieemnee Greenwich, | Sept. 17 | —4.23 | 14 | —4.26,)—13.4 | 14 | —12.5,, Koénigsberg, | Sept. 11 | —4.10 8 —— alo Cambridge, | Sept. 17 | —4.06 | 14 | —4.25,|/—11.9 | 15 | —12.7,, Edinburgh, | Sept. 10 | —4.39 4 | —4.39, | —11.5 3 | —11.5, Paris, Sept. 18 | —4.20 | 12 —4.19,}—12.3 | 13 | —11.7,, Vienna Sept. 13 | —4.12 5 115. |) 18.9 5 | —12.9, y a es — + [61.7] |[+ 11.6] 22h 32] Sept. 16 eee ... | —4.91 ers ..- | —12.3 —62.3 | +11.5 | Greenwich, | Oct. 16 | —4.13 | 11 —4.16,|—12.9 | 11 | —12.0, Cambridge, | Oct. 17 | —4.00 | 10 | —4.19,|—11.6 | 11 |—12.4,, Paris, Oct. 17 | —4.05 4 | —4.04, | —11.2 4 —10.6, Jienné Oct. Sais 2 —4.47. | —15.5 2 —14.5: Vienna, ch ts 4.44 me 15.5 | oF [59.8] [411.4] [22' 28™] Octal aeier: Bevel = —— 4G Sond ... | —12.0 —61.5 | +11.3 F Greenwich, NOVAS 4 lO) 5 —4.13, | —12.4 5 | —11.5, Cambridge, | Nov. 4 | —3.96 4 —4.15, | —12.3 3 | —13.1, F Paris, INjOWAN OM 4-400 3 SNS, || ILL 83 3 | —10.7, Vienna Nov. 2 | —4.09 2 || ath 1A | <1) 2 | —11.9: pede EBS! (esses TS |f 59.0] eee 22" 27] | Nov. 6 Jaan] cos. lon | Sy eee Greenwich, | Dee. 2 | —4.05 2 | —-4.08, | —14.0 | lsh Cambridge, | Nov. 30 | —3.87 6 | —4.06, | —11.6 T | —12.4, Paris, Dee. 8 | —4.03 T | —4.02, | —10.9 7 —10.3, Vienna, Dee. 8 | —4.28 2 | —4.31: | —13.0 2 | —12.0: sane [2 tae Set 5 Ee faa 93") | Dec. 5 | .... |... | =405 | .... |... | Set) eoiea ia 1838 Greenwich, | Aug. 20 | —4.72 | 11 iG || 9) ‘ 1 | ee Konigsberg, | Aug. 25 | —4.65 5 —4.65, | —18.6 By | S118, Cambridge, | Aug. 19 | —4.67 | 11 | —4.78,|—15.8 | 12 | —16.3,, Edinburgh, | Aug. 25 | —4.96 3 —4.95, | —15.0 4 | —15.0, Paris, Aug. 20 | —4.80 9 | —4.79, | —16.6 9° | —16.0, [—70.9] [411 7] 2h Si Atos eee ee —4.76 eevee ... |—15.8 |= a aie Greenwich, Sept. 12 | —4.62 8 | —4.66,} —16.2 8 Ab Kénigsberg, | Sept. 16 | —4.65 | 14 | —4.67,,|—18.5 | 14 | —17.5, Cambridge, | Sept. 16 | —4.61 | 11 | —4.72,]—14.9 | 12 | —15.4,, Edinburgh, | Sept. 15 | —4.74 9 | —4.73,| —15.8 7 | —15.8, Paris, Sept. 12 | —4.71 6 —4.70, | —16.0 6 | —15.4, Vienna, Soom Wl | el |) i) zee) ali) ae) ato Berlin, Sept. 6 | —4.72 if —4.74, | —17.5 7 | —16.5, [—70.4] (4a fea® 47=] |-Sept. 15 | 2... | ..../=28000] 2.2 | 22. | GRO Cee THE ORBIT OF URANUS, 135 MEAN CorREcTIONS TO THE EPHEMERIS OF URANUS.— Continued. H Observed corrections in R.A.| Observed corrections in Dec.| Corr. to Geocentric Observatory. [R. A. of Mean dates. | Uranus. } Mean. | No. of | Corrected No. of | Corrected |Longitude.| Latitude. obs. mean. Mean. obs. mean. 1888 s s " " " wt Greenwich, | Oct. 15 | —4.67 T | —4.71, | —15.5 0 | Slane | Cambridge, | Oct. 17 | —4.49 7 | —4.60, | —15.3 6 | —15.8, (Edinburgh, | Oct. 16 | —4.56 | 12 | —4.55, | —15.0 8 | —15.0, | Paris, Oct. 16 | —4.66 | 7 | —4.65,;—15.9 | 7 |—15.3, ; Vienna Oct. 14 | 4.66 9 | —4.68, | —17.3 9 | —16.3, a a ‘ Heracles Se ES ees [22h 44m] | Oct. 16 —4.63 —16.2 | —69.7 | +10.8 Greenwich, | Nov. 9 | —4.52 6 | —4.56, | —16.0 6 | —15.2, | Cambridge, | Nov. 15 | —4.37 | 10 | —4.48, | 15.1 9) 1556; Edinburgh, Nov. 16 || —4.70 8 —4.69, | —17.4 2 —17.4, j Vienne Nov. 7 | —4.93 | 4 | —4.95;|—18.3 | 4 | —17.3, WEDS ea ; ! \[—66.9] |[+11.3] p22 49™] Noy. 14 2 —4.58 Bessie ... | —15.8 | —68.8 | +10.9 Greenwich, | Dec. 9 | —4.60 WL GL | 113-6) |) al) le Cambridge, | Dee. 15 | —4.96 | 5 | —4.37, 15.1 7 |—15.6, Edinburgh, | Dee. 17 | —4.18 3 | —4.17, | —14.7 1 |—14.7, i Deca ole 4a4. eS Ohmi al —14.5 ; Paris, D o! 7 yh ee es eee eet [228 43™] | Dee. 10 —4.38 —15.0 | —65.7 | +10.4 1839 Greenwich, | Aug. 22 | —5.28 | 5 | —5.32,|—21.3 | 5 |—20.6, Cambridge, | Aug. 24 | —5.11 | 8 | —5.22, | —20.7 1 ||—=215 02 Paris, Aug. 23 | —5.20 3 ==) Qe | —— 2087 3 | —20.1, i Aug. 20 | —5.2) 2 | —5.20; Edinburgh, | Aug. 29° 5) =: 20% 9.99 |p 49.79 [23 6™] | Aug. 23 —5.22 so || SHO ee SEU Greenwich, | Sept. 10 | —5.14 | 12 | —5.18,|—21.0 | 12 | —20.3,, Konigsberg, | Sept. 12 | —5.11 | 12 | —5.13,|—21.7 | 12 | —20.7 Berlin, Sept. 10) — 5:10) | 14 | — 5.10, | 21-1 | 14 | —20:1, Cambridge, | Sept. 17 | —5.12 | 11 | —5.23,|—20.0 | 10 | —20.3,, Paris, Sept. 14 | —5.93 | 10 | —5.22,|—20.7 | 10 | —20.1,, Edinburgh, | Sept. 16 | —5.16 | 17 —5.15, | —20.5 5 | —20.5 ; Sepia ilmee5n3o — Se || — xi. 19). Vienna, Se ee ae Waals el eee “2 |[—79.0] |[ + 11.6] fas" 37] Sept. 14 i —20.3 | —78.8 | +11.0 | | Greenwich, Ocil27 == sa8 5 —5.22, | —20.2 | ail). Cambridge, (Oye, GY |) — 8 —5.18, | —19.2 5 —19.5, Edinburgh, | Oct. 15 | —5.14 | 10 | —5.13, | —19.9 || 0.6) i Octa plON 5 : =f), || —=PX0). 13) | 1925 Vienna, I LO} Seu |e Ls fae ; 0.5 Pe nt 4) |E-411.5] ees os Oct. 14 nee soe || — 1s —19.6 | —78.7 | +11.8 Greenwich, | Nov. 12 | —4.98 6 | —5.02, | —20.2 6 | —19.5, Cambridge, | Nov. 19 | —4.85 G. | — bya. || it |) —p).2b Paris, Nov. 9) || —=4799 2 | —4.98, | —20.4 2 |—19.8, Edinburgh, Nov. ea —4.90 3 —4.89, sec 1 —18.3, [—75.3] (EI) [22" 57™] | Nov. 14 —4.97 —19.4 | —15.7 | +10.4 Greenwich, | Dec. 6 | —4.84 e489 50 2 | —18.3 Cambridge, | Dec. 15 | —4.83 Se 4294 190 2 || aif). Ze Paris, Dee. 3) 4.90 3 —4.89, | —17.8 3 1-25 | Edinburgh, Dec. 28 | —4.96 | 4 | —4.95, | —18.5 3 18.55, [—13.9] c-+11.0] (22h or] | Dee. 12 so |e} ugiee soo |S ac) Pee a 136 THE ORBIT OF URANUS. MEAN CorRECTIONS 10 THE EPHEMERIS OF URANUS.— Continued. Observed corrections in R. A. | Observed corrections in Dee. | Corr. to Geocentric Observatory. | = | [R. A. of Mean dates. | | Uranus. ] Mean. No. of | Corrected Mean. | No. of | Corrected | Longitude.| Latitude. obs. mean. obs. mean. ~ ~ s Greenwich, =— eal Cambridge, , —5.64 Edinburgh, . i —5.65 [2 gh 207] ee) -~T ho 3c S| bo bo bo wo On bo eo | sa Greenwich, Konigsberg, Cambridge, Edinburgh, Paris, Berlin, Vienna, [esses bo 19 He Co — a o Cy bo bo to 1 CO }) pw Son - es & — oO e cae | ko aw | [—87.6] |[-+11.5] — 89.5 | tee Greenwich, Cambridge, Edinburgh, Paris, Berlin, Vienna, [23 147] o ov ee = OU Bm bo Orc LA bo bo bo to bo tO rt = bo Co OD 00 Se SS >) OM =T bo -1 09 > ~ | | ho we oo Greenwich, Cambridge, Jdinburgh, Paris, Vienna, : ‘ : arr 7) [411.3] [23" 12™] sietoke ane Roa se see +10.3 Greenwich, Cambridge, Edinburgh, Vienna, D. [—82.2] [+11.0] Greenwich, Paris, ° t ° ad. : [—96.9] [+11.3] [23" 377] 2 fe) oa eee pea” Ape 28. —96.2 | +10.6 Greenwich, Konigsberg, Berlin, Edinburgh, Paris, Vienna, Co bo bo bo G2 bo SOHOSS won aro : Mees f—96.5] |[+11.4] eae see Sainte lca ee (3 —96.3 | +10.4 THE ORBIT OF UR AN US: 137 MEAN CorRRECTIONS TO THE EPHEMERIS OF URANUS.—Continued. Observed corrections in R.A. | Observed corrections in Dee. | Corr. to Gedcentric | Observatory. [R. A. of Mean dates. | 4 Uranus. | Mean. | No. of) Corrected Mean. No. of Corrected |Longitude.| Latitude. § obs. mean. obs. mean. | ' = || ~ s Greenwich, —6.13, Berlin, ! —).87, Edinburgh, at. é 3.0 —6.04, Paris, 5 D. —6.09, Vienna, b [23" 28") > =I -f -T > © bo -T eer et ee) bo bo bo bo bo bo at lh w= | CO & io bo bw to Lele] 7} |[+-11.2] +10.5 | -T } Greenwich, Berlin, Edinburgh, Vienna, [23* 267] ow -r CO on - bo bo bo to Neier) —) Greenwich, Berlin, Edinburgh, Paris, [23 26") Greenwich, Cambridge, [23" 51™] Greenwich, K6nigsberg, Berlin, Paris, Cambridge, Edinburgh, Pulkowa, Vienna, —_ _ [org eA —6. —6.69, =—GaGse —6.67, —6.59) — 6.64, —6.80: Nees atte | a Pee Seo [23" 47™] ; beh a ll caconl A el kaka oe me ees 32.0 | —104.3 | +10.3 Greenwich, ‘ ds : —6.68, Silas é Lf i Berlin, te ‘ 36 —6.53, D. —29.7, Paris, ? 2 —6.63, 31.5 —30.9, Cambridge, : Of —6.67. Sil, 0 —31.7,, Edinburgh, Wall! 5. y —6.59, 31. —30.8, Vienna, f 9 —6.731 34.6 —33.6, [103.6] [23" 44™] Sal) coos | ceo EGOS) Sesame se | Sere hese Greenwich, , & : —6.38, 31, —30.6, Berlin, cl 8 ——6. 30. —29.2) Cambridge, , Be bad: 30. —31.0, Edinburgh, : eos —6.3% Vienna, i B 88) ON) See E1008) -LOrs) [23" 42™] 5 ik Bee tal (pore 5. sece |) con |= SLO NOOR | SELOC 18 May, 1873. 138 THE OR BAT OF URANUS. MEAN CorREcTIONS TO THE EpHEMERIS OF URANUS.— Continued. l | Observed corrections in R.A. Observed corrections in Dee. Corr. to Geocentric j Observatory. = [R. A. of | | Mean dates. | Uranus. J Mean. | No. of Corrected Mean. | No. of | Corrected Longitude.| Latitude. obs. | mean. obs. mean. 1842 s s dA 7 uw Greenwich, | Dec. 16 | —6.3 8 | —6.33, | —30.2 8 | —29.8, i Berlin, Dec. 10 | —6.16 4 —6.15, | —31.4 4 —30.4, Paris, Dee.’ (14 || {6.98 |) 25) 2 :699"==309 1) 5 esoeas Cambridge, | Dec. 16 | —6.22 9 622.9) 3 ON 8 | —30.4, Edinburgh, | Dec. 13 | —6.25 2 ——6. 183 Pulkowa, Dee. 13 | —6.28 33 —6.28, 92h m era 9) . 20 6 a Be Boe al 23" 417] | Dee. 15 —6.27 ® —30.2 | — 98.2eeny ‘eae 1843 (— 97.5))\[+10.3] | Edinburgh, | Jan 9 | —6.32 7 | —6.25 |—31.4 4 |—31.0 | — 98.2] + 8.6 (Greenwich, || Aug. 2051) Tells) (\)==1204 54 35e Tse Paris, Mage) 21) 1-20 |) 2354 oe sis Oeil us 2 eae Ca (ee ets |114 37) (ana iO? GE] Aug. 20 —7.10 —35.7 | —111.9| + 9.6 | Greenwich, Sept. 17 | —7.23 9 | —1.16, | —35.4 9 | —35.0, i Paris, Sept. 15 | —7.17 ! 11 —1.16; | —36.8 || 4 36.2), | Edinburgh, | Sept. 18 | —7.26 T || 7.19), | —36.8 5 | —36.4, ; Pulkows Sept. 22 | —7.18 | 10 | —7.18 Bia igs Wi eel eae ean. pais [—114.1]|[-+10.6] (ORFS) Sept. 19 —T.17 —35.8 | —112.9| +10.0 Greenwich, Octane 1Sih—ellO avo —jf_03, || 35.8} 10) | 3429") | Paris, Oo, ii |) ey 6) || —7.063 | =3ir 1 5 | —36.5, Edinburg! p || oO be —— (Re. || 8s ly dinburgh, | Oct. 17 7.05 Beebe 30.1 5 34.7, [—112.6][+10.5] [23" 59") | Oct iy —1.02 —35.2 |—110.6| + 9.6 | Greenwich, Nov. 20 | —7.01 9 —6.94, | —34.3 9) | —=3359) | Konigsberg, | Novy. 11 | —6.84 5 | —6.83, | —34.4 5 | —33.4, | Paris, Nov. 15°} —7.03 | 3 | —7.02,}—36.3 | 3 | —35.7, | Hdinbur } Ry aac UO fas Bm ease: Kdinburgh, Nov. | 15 6.93 8 —6.86 BD) 7h 6 ; 35.3, [—110.1] [-+10.4] [23" 567] | Nov. 15 —6.90 —34.5 | —108.6| + 9.5 1844 ! Greenwich, A 3 | 6.70 3. | —6:64) | —=34. I 83 || 83.7, f Edinburgh, Jan i —6.67 6 | —6.60, aria an. 5/2 17709) OM lesen ase € ¢ 2 t Paris, ee hae 6.72 2 | —6.71, | —35.5 2; | —34:9) (—106.1]|[+ 9-9] [23° 56™] Jan. 7 6.64 —— Ame —105.0 + 8.2 | Greenwich, es si Sats |) 10) 767396) | LOlp ees onae | Paris, ABB ET ST | 8 | 1.88, 40.00 | 8 ee ee ro® 22) Aug. 18 —7.68 39.9 | —121.2| + 9.6 | Greenwich, | Sept. 25 | —7.74 | 11 | —7.68,]—39.9 | 11 | —39.5, | Edinburgh, | Sept. 19 | —7.67 | 11 | —17.60,|—40.3 | 10 | —39.8,, Kénigsberg, | Sept. 17 | —7.65 | 10 | —7.64,|—40.7 | 10 | —39.1, Damia o ps 4 5 ¢ : f Paris, Sept. 10 | —-7.638 i0 62, —— 3 O64 15 —38.8,, [—122.6] [+10.3] [0* 18™] | Sept. 17 —7.63 —39.3 | —120.6| + 9-3 TECH OPA IB Mee OlR EE UpRE AGN Ulises MEAN Corrections TO THE EPHEMERIS OF URANUS.— Continued. 139 Observatory. [R. A. of Uranus. J Greenwich, Edinburgh, [o® 15™] Greenwich, Edinburgh, fo® 127] | Edinburgh, Paris, (0 10™] Greenwich, Edinburgh, OE aI} Greenwich, Greenwich, K6nigsberg, Paris, NO? Sey Greenwich, Paris, (0® 297] Greenwich, [o* 27°] Greenwich, [o® 25") Greenwich, Paris, [o" 50] Greenwich, K6nigsberg, Paris, [o> 46™] Greenwich, Paris, [o" 4] | Mean dates. 1844 Oct 1%, Oct 13 “Oct. 15. Nov. 26 Nov. 19 Nov. 23° Dee Dee. 22 Dec. 20 1845 Jan. 13 Jan. 14 | Jan. 14 Sept. 18 Sept. 30 Sept. 14 Oet. 17 Oct. 20 Oct. 18 Nov. 9 Dee. 14 1846 Sept. 8 Sept. 12 Sept. 11 Oct. 8 Oct. 4 Oct. 14 “Oct. 9 | Nov. 10 Nov. 16 Nov. 14 | Observed corrections in R.A. |Observed corrections in Dec. Corr. to Geocentric | No. of Corrected | . No.of, Corrected | Latitude. Mean. Longitude. obs. mean. Mean. obs. | mean. s s ” " ” x ” —7.70 9 | —7.64,/—39.2 | 9 | —38.8, —7.58 | 10 7.51, | —40.5 14 ORO! \¢—191.5) |¢-+10.2) | Sa —39.0 | —120.0 |.+ 9.4 | 7.44) 9) | = 7/38, | —38.7 2) |) BRB} Se | eat ama ‘|[—118.2] [+ 9.9] 3 o839/3) 116268 == 8.8 —T.24 5 = eee ; —7.10 | 2 | —7.09, | —39.3 3 |, Sse 115.7] /C-+ 9.6] even Sain eels ar eeauial —7.18 | 1 | —7.10,]—38.4 | 1 | —38.0 Saas |i a elias! h |f—114.07 f+ 9.5] = FOS — 35.0) | I | 8 — OF | By |) ee | 6 | —42.3 |[—131.0] |[-+ 9.6] | | ¥ =I Se OH =A |) 1) |) OS, |B ||) |) 2, —8.16 | 5 | —8.15, | —43.4 5 | —42.4, —8.24:| 5 | —8.23, | —43.9 3 | 8 ey [181 6] [+ 9.99 Suit = 7195 || OB) |e Ol — Op) || TO |) SI! || to |] aD) |) Ze nit {| @ |) — O05 | — Gule—=43 518 (—130.2]/[4 9.8] —8.08 9.3 SILO |b G9 —8.06 6 | —7.98 |—43.3 | 7 | —42.9 |[—128.4]|[-+ 9.6] : —126.9 |[+ 7.9 ES || Bo) acct | i 8 | —39.8 |[[—124.8]|[4 9.3] ae | SSNOMET || eG | —8.80 | 8 | —8.76, | —46.2 8 | —-45.8) —8.73 | 19 | —8.72,,| —46.9 | 14 | —46.3,, [139.9] [+ 9.2] Sie | 2G. | Sale) | Sb BES! } | | | : — Se. |) |] EO, | GS || 2634. —8.61 | 11 | —8.61,)—44.8 | 11 | —43:8, —8.69 | 13 | —8.68,|—46.7 | 13 | —46.1, [139.7] [+ 9.2] —8.65 | ABS atoll | Ae 8) | = OUG 2 CSD Sei 4 048 6 | —46.4, | 8948) || (9) |) —=8: 47, || —4'6r4 9 | —45.8, [136.8] |[+ 9.0] i500) Ca eeetOm eeatSbelelpeten ea | 140 THE ORBIT OF URANUS. MEAN CoRRECTIONS TO THE EpHEMERIS OF URANUs.— Continued. Observatory. [R. A. of Urauns. ] Greenwich, Paris, [o> 397] Greenwich, { Paris, fo" 40™] b Greenwich, eS Gy | Greenwich, # Paris, Be 2] # Greenwich, Paris, 0-5" Greenwich, Paris, o> 5471 | Greenwich, ) Paris, [o® 54”) } Greenwich, 4 Paris, fi™ 19") Greenwich, Paris, [1 15™] Greenwich, Paris, pe sues Greenwich, Paris, ple ge] | Greenwich, | Paris, le By Observed corrections in R.A. Observed corrections in Dee. Corr. to Geocentric Mean dates. Mean. |No. of Corrected Mean. |No.of) Corrected | Longitude. | Latitude. obs. mean. obs. mean. 1846 s s ” ” ” 7 Dee. 15| —8.3i, |) 9) \)—=_8!9%1| ==46%0 1) TON toeom Dees A161) a0ste5. || ie) =aesode eden ie ae — 45.55 |, 134.548 Dec. 16 8.26 —45.6 | —131.7 | 46.5 1847 Jan.) 13") 8:25 | 4 | 28:90.) "43.0 Yaa oe Jan) dd |/==8.14 ||| 6 |S ToS etree aes [130.9] | [48.6] Jan. 12 —8.15 —44.0 | —199.6 | oy Sept. 3 | —9.21 | 8 | —9.16 | —49.4 | 8 | —49.0 |[—1484q)ieeumam eree: —144.9 | 668 Oct. 1) 2905: II) 6 d= Zore0n| s==49401 6 nle eee Oct, 12"|| 9:93 ||) 1) 2 ougae| —=4979) | aioe ee 148.0] [48.7] Oct. 10 —9.21 —48.6 | 145 50a Nov, 31) 9.10) }]/ 99) | |, —-Os0 sey =492o" I/ignlle = —dgage Nov. 12 9.08 | a0 | ==9:06, | 249.3 | “9 | Z24asige ey Gaae S — "19 | 146: 97) fieeeeam Nov. 8 9.06 —48.8 | 14/6) ieee Dee. 3) W896. lieti| Sele = 40M ean eco “Dec. 12) es Ses Il Oy le = sreas | Seaekh all Toraleecetiaos r—143.0)| [48.3] Dee. 9 seeegies 48.2 | —14016n manne 1848 i Jan.) 1109) 8283) deo |b —6- ON =a | oe gare Fans LL ||) 22825904) Vos ena, | aa eee OF (—130.4]/[48.1 Jan. 11 —8.57 —46.9 | —136.20 seu Sept. 8/9684 1) yp i 29u Oe ret tl rales One Sept. 22 | —9.82 | 11 | —9.80,,! —51.5 | 9 | —50.9 sci ee — —_———|f__ 156.4] |[8.0 Sept. 17 USS —90.8 | 15219 | eae Oct. 19) 29889 |) aE ONT 5989 I aralleeeeee Oct. 16 9.79 |) Tae LON Ty 59.9) || 04 ==> eee ate — ———— —___—|r—156.4 ; Oct. 17 a 57 |Sneees ae Nov. 13 | 9:68) 7 9.68.) 259.34) “ya |= seoe Nov. 13) | 29155 10 4) e953" 20D | aig) eee at a = —|f—154.6 Dee. 14 || 29:48 || %5 |) 89.96. | 59 | oe ene Dec. 9 | —9.43 | 5 9.41,| —51.3 | 6 | —50.%, Dee 19 | laagisen —50.7 [—150.2 1849 Sri: ; Jan. 6. |, —=9:3% 0 20.372) 512 ater oes Jan, 18 | = 910 |) 1 | 29/09) P=—=S0L0u ees cage] | E08] Jan. 12 —9.93 —50.1 | a 1aein eee THE ORBIT OF URANUS. 141 MEAN CorRRECTIONS TO THE EPHEMERIS OF URANUS.— Continued. Observed corrections in R.A. Observed corrections in Dee.| Corr. to Geocentric Observatory. [R. A. of Mean dates. Uranus. ] Mean. | No. of) Corrected | Mean. (No.of Corrected | Longitude. | Latitude. obs. mean. obs. | mean. " " 8 Greenwich, ppt. Bo, =i BE | —52.9 |[-+165. [+7.3] [lh 34] ES i linac Ata | eer ee eae eT G Ossi ene Greenwich, Cian 28 ——I1(0) 4 5: Hose Paris, ti, 10.2 : of : : 2 |[_164. [41.4] pir 307] 16 aaa .-. | —10.2 seis es 53.8 | —I161. +5.8 Greenwich, 3 3 — ONT Paris, Nov. | 2 —— OED, = : pe ed fi eo [1® 267] lov. eS! ae eee ees |eok 53. Bites Greenwich, Paris, Scie ceeecets sat plb 247] rcs sobs aes SOB | oaee Bi 2 : eee Greenwich, Paris, [46 — — — —s dee fim 24") an. Se Sloe eee arte || e286. Greenwich, Sept. 53.5 3-0 |[—I71- [+6.! piso] tet BSG a9 5O0C nie moo | —168.1 +5.9 Greenwich, : = ON92s Paris, ot. ——=il(0).315) [113 r-46 —= —— ~ —- —1iv.a | VF a 47>] CeO ean alleen TOredalh > a-eim her 5.3 | —169.6 | +5.3 Greenwich, vov. f =O aio} Paris, Voy. = OnSite : 55.5 pees eas EET TOCA Tl eed] [> 437] Nov. oo0d yer) |i ——LOs82 Bee-3 Dio 55. —169. +5.6 Greenwich, ; 5 —10.56, Konigsberg, Xe, —10.48, : wnat aaa ; __|-=16 9:09) | f--6.3) pe Ate) : sepe ti| cee |= Ube | See cel lio .6 | —165.0 | 5.2 Greenwich, ; Osa 54.5 4 |[—163.5]| [+6.0] Ete coe so0¢ ane Sppe fepe be .... | —160.4 | +3.5 Greenwich, 11 : eas Konigsberg, | Sept. 19 “11. |) Samo) cone oe cecpins F ot Say i Paris, Sept. 14 LY, 9; | -180.0]| [45.7] eee iSepteei4)| 22a: ||. --a=UIed6ll) “lace Vos. Soars — 176.9") ae’ KGnigsberg, | Oct. 2% 5| 3 | —11.75, O° Paris, Oct. 2: BI eS ll LARS Sear Neel | ie ects ciate ae [45.8] IN| ROCROSS0| 0. ||... | ause 142 MECH ORRbue DORR URANUS. MEAN CorRECTIONS TO THE EPHEMERIS OF URANUS.— Continued. Observatory. [R. A. of Uranus. ] Mean dates. | | | | Observed corrections in R.A. Observed corrections in Dee. Corr. to Geocentrio Mean. No. of obs. | Corrected mean. No. of obs. Corrected | mean. Longitude. Latitude, Greenwich, Paris, 2" 0") Greenwich, Paris, le ais Greenwich, Paris, [it 547] Greenwich, i Paris, fl® 54™] Greenwich, (2) 225) Greenwich, j Paris, zEaS= Greenwich, 1 Paris, fa" 14") Greenwich, KGnigsberg, Paris, (2* 10") ' Greenwich, {| Paris, [2" oF Greenwich, Paris, [2 397] Greenwich, Paris, 345] 19 16 ‘Dee. 19 1853 Jan. Jan. Jan. 12 15) 13 17 16 Sept. 17 Sept. Sept. Oct. 15 Oct. 27 Oct. 22 s —11.50 —11.48 —12.69 —12.66 s ==11e50! —11.49 —l1. = ess: 37, = 49 =n: = "10598" —11.06 —10.85,] 0s —10.86 —12.06, —12.23, —12.04, Say —12.00,| —12.06, —12.03 eee! —11.75,, Sie —11.42, S43) = —58.2 —59.0, —d8.4 =p. SE 5 Gee 58.3 FS —178.1 [—179.9] —l1T7.5 [—175.7] —172.3 [—187.4] ie —184.5 [—189.6] —ieir2 —185.6 [—1s4.8] = 1eies [—181.0] —176.8 [—194.6] —191.6 [—196.7] —192.7 THE ORBIT OF URANUS. 143 MEAN CORRECTIONS TO THE EPHEMERIS OF URANUS.—Continued. Observatory. [R. A. of Uranus. | Mean dates. Observed corrections in R.A. Observed corrections in Dec. Corr. to Geocentrie Mean. No. of | Corrected | obs. mean. Mean. No.of obs. Corrected mean. Longitude. | Latitude. Greenwich, Paris, 22 312] Paris, [at 27] Greenwich, Paris, Paris, fi2® 26™] f Greenwich, [2" 53") Greenwich, Paris, [2® 507] Greenwich, Paris, ON Santiago, [2 487] Greenwich Paris, Santiago, [oe 447] Greenwich, f2® 42™] Greenwich, Paris, [3 9] Greenwich, Paris, ana Santiago, [38 5] Greenwich, Paris, Santiago, [3* 07] Ss 9 GA —12.65,, =" | [[—206.2)] —202.1 144 THE ORBIT OF URANUS. MEAN CorRRECTIONS TO THE EPHEMERIS OF URANUS.—OContinued. Observatory. Mean dates. | Observed corrections in R.A. Observed corrections in Dec. Corr. to Geocentric [R. A. of | Uranus. } Mean. | No. of Corrected Mean. |No.of Corrected | Longitude. | Latitude. obs. mean. obs. | mean. 1856 s 8 ss 4 t Greenwich, | Jan. 22 | —12.97| 6 |—12.97, | —56.1 6 | —55.6, Paris, Feb. 10 | —12.87 2 |—12.85, | —54.8 2 | —54.6, [—199.5] [41 4] [a 58") | Jan. 27 —12.94 55.4 | 194.7) eee Greenwich, | Oct. 16 |—14.36] 4 |—14.36, | —53.8 4 —53.3 Paris, Oct. 16 |—14.38| 2 |—14.36, [—213.8}| [-40.7] [3* 277] Oct. 16 —14.36 | 53.3) (210205 sie Greenwich, | Nov. 18 | —14.23| 8 |—14.23,| —54.8 8 | —54.3, Paris, Nove Lit |) 14532 6 |—14.30, | —54.0 7 | —53.8, [—214.7)| [40.7] [3* 22™] | Nov. 18. —14.26 | —54.1 | 210.2 | 209g Greenwich, | Dec. 18 | —13.92| 5 |—138.93, | —55.9 5 | —55.4, Paris, Dec. 19 | —14.11| 7 |—14.09, | —55.0 | 6 | —54.8, [—211.6]| [40.6] —_—-= 07) . eS eS] Dee. 19 —14.03 sal |i 577 —l2 1857 Greenwich, Jan. 21 |—13.59| 10 |—13.60,,) —53.4 8 | —52.9, Paris, Jan. 12 |—13.87] 3 |—138.85, | —65.3 3 | —59.1, [—206.4]| [0.5] [3° 15") | Jan. 19 —13.66 —53.5 | —202.3 | —04 Greenwich, | Oct. § | —=114-68) |) 47 |= 1469) |) 4901 4 | —48.6 |[—216.8]| [0-3] (anes ee 213.0) | sie Greenwich, | Nov. 6 |—14.80| 4 |—14.81,| —51.3 | 5 | —50.8, Paris, Nov, 10/2243 (209 oMeteiy ets |) saloons Konigsberg, | Nov. 19 |—14.77) 3 |—14.75,| —652.7 3 | —51.5, [—218.9]| [0.4] (stay Noy. 11 —14.80 — 5129) | 22911525 ees Greenwich, | Dec. 11 |—14.55; 6 |—J4.56, | —53.2 5. | —52)7, Paris, Dec. 16 |—14.48| 7 |—14.46, | —52.8 6 | — 52:65 Konigsberg, | Dee. 8 | 14.56) 1 |—14.54,) 638.1 | 1) 62.51, 914 19] 0.59 [3 36™] | Dec. 13. —14.51 52.6 | —219.9) |e 1858 Greenwich, | Jan. 17 |—14.13| 13 |—14.14,,| —52.8 | 13 | —52.3,, Paris, Jan. 16 |—14.19| 8 |—14.1%,| —52.8 | 9 | —52-60/- 91 97/ ¢0.6] [3 327.2] | Jan. 17° SVs —52.4 | —207.5 | —28 Greenwich, Paris, [3* 317.9] Greenwich, [a 323i Greenwich, Paris, [3® 58".5] Feb. 1] Feb. 12 Feb. 11 Nov. 14 Nov. 20 Nov. 17. —13.81 | —13°91 —15.16 | 15919 —52.1 —52.7 —48.2 —47.8 10) = ieee 1 | = 53%5, = 57 11 10 THE ORBIT OF URANUS. 145 MEAN CoRRECTIONS TO THE EPHEMERIS OF URaNnus.— Continued. Observed corrections in R.A. Observatory. [R. A. of Mean dates. Uranus. ] Mean. No.of Corrected obs. mean. 1858 s 8 Greenwich, | Dec. 16 | —14.96) 6 |—14.97, Paris, Dee. 16 |—14.73| 5 |—14.76, [3" 58".6] | Dec. 16 Bie; 1859 Greenwich, | Jan. 19 | —14.56} 8 |—14.57, Paris, Jan. 10 | —14.72) 4 |—14.70, [3" 50.0] | Jan. 16 —14.61 Greenwich, Feb. 18 |—14.18] 5 |—14.19. Paris, ad, Wh || SS ei 6 |= 14.35 [BAS Heb: 1 |—14.28 Greenwich, { Oct. 25 |—15.43| 6 |—15.44 Posy |... Greenwich, | Nov. 18 | —15.56} 6 |—15.57, Paris, Nowe iy | 15.61/93)" 8) | =S1'5 7595 [42 17.2] | Nov. 17 | —15,58 Greenwich, | Dec. 16 |—15.43) 9 |—15.44, Paris, Dee. 11 |—15.46} 5 |—15.44, 4" 12™5] ] Dec. 14 —15.44 1860 Greenwich, Jan. 16 |—15.08} 8 |—15.09, Paris, Jan. 15 | —15.08/ 8 |—15.06, (4* 8™0] | Jan. 16 —15.08 Greenwich, | Feb. 17 | —14.64) 12) |_14.65,, Paris, Feb. 6 | —14.79 2 —14.77, fa" 7™.0] | Feb. 15 — 14.67 Greenwich, c4* 40.9] Greenwich, Paris, [4® 35".8] Greenwich, Paris, [4 317.6] Greenwich, Paris, [42 267.8] 19 Nov. Nov. Dec. Dee. Dee. 1861 Jan. 13 Jan. 13 13 Jan. May, 1873. |Observed corrections in Dec. Corr. to Geocentric | | Mean. No.of Corrected wv —19. —4). —48. —48.2 —4s. —48.: —45.5 —44. orb So Oreo —45.8 —A4.: Latitude. Longitude. jobs. | mean. ale 7, hi = Mt ; 6 | —48.7, i Roma ta [—220.7] | [—1.6] — MO OHI || TEI Sa——aguee o) emse 2h | onie agit eeen | AGS || Oil fy |) —_ Sil DuleAtrebe ee 2209 ROTO OT] [ETCH HES. | ANGI || Be Gi) Opi Pee oy| =o.) =P IG.B" |) —B6 5 || 2.8. oP ee EON Cy a ere JOS | OPO) || AG 3) || ye eee f= 92397 (2s 4 = 220 ee ——429 oe OM deenasdes| (eo g’6])) (==. 9H eV SOG |) hes 12) || 44.5 | 183,9)]]| [2.837 —44,.5 | —210.4 | —5.0 |[—229. 4] —219.3 THE ORBIT OF URANUS. Mean Correcrions To THE ErneMeRis oF URANUS.— Continued. Observed corrections in R. A. Observed corrections in Dec.| Corr. to Geocentric Observatory. = ——— [R. A. of Mean dates. | | | Uranus. | Mean. No. of Corrected No.of Corrected | Longitade. | Latitude, obs. | mean. Mean. obs. | mean. 1861 2 | s ” | ad ” Greenwich, Feb. 10 )—15.12} 2 |—15.13, —40.7 | 3 | —40.3 Paris, Feb 5 heme 5.23 1 —15.20, | [—217.9]| [—3 1] f4" 210] | Feb: 8 j—=15-15 | —40.3 | —215.0 | —6.0 | Greenwich, | Nov. 10 |—15.99} 9 |—16.00,/ —30.9 | 8 | —30.5 |[—226.2]| [—4.0] Fe Seta Be SS gee ex —223.4 | —6.1 | Greenwich, | Dee. 9 |—16.01| 4 |—16.02,| —83.4 | 4 | —33.0, Paris, Dec. 10 | —16.04| 11 |—16.01,,] —32.8 | 13 | —32.6,, Washington,| Dec. 18 | —16.10| 6 GCs | [227.3] [—4.0] [4 50".8] | Dee. 12 —16.03 —32.1 | 904,47 \iiam 1862 Greenwich, | Jan. 19 |—15.67} 8 |—15.67, | —35.1 9 | —34.7, Paris, Jan. 20 |—15.72| 5 |—15.69;| —34.1 5 | —33.9, [—223.2]| [<0 f4> 447.9] | Jan. 19 —15.68 34.4 | —220.2 | —6.3 Greenwich, | Feb. 22 |—15.18} 7 |—15.18,| —34.3 if || 883.9. Washington, | Feb. 19 | —15.31} 3 |—15.31, Paris, Feb. 15 |—15.47) 5 |—15.44,| —34.6 | 6 | —34-4,)- oo, 69] Caaigg [4* 43".1] | Feb. 19° —15.30 | —34.1 | —215.10 eee Greenwich, | Nov. 9|—16.13| 9 |16.13 | —24.5 9 | —24.1 |[—226.2]| [—4 7 Pa Caoene —223.6 | —7T1 Greenwich, | Dec. 12) |—16.27 | % |=16297| —2722) | 7 |) —2eese Paris, Dec. 11 | — 16.97 5 |—16.24. | —26.5 5 | —26.3, Leyden, Dee. 8} —16.16) 4 [—16.13,] —25.2 | 4 | —25.2, [—297.6]| [—4.7] [5" 10".7] | Dec. 19° —16.23 —26.3 | —295.4 | 1.3 1863 Greenwich, Jan. 7 | —15.96 | 10 |—15.96,,) —98.3 || 10! ;,—2i7-9)5 Paris, Jan. 20 | —15.94| 2 |—15.91, 97.4 |. 1 | —27.2, Washington,| Jan. 16 | —15.92| 4 |—15.92, Leyden, Jan. 16 | —16.03| 6 |—16.00, | —27.9 6 | —27.9, [—224.8]| [—£ 8) Bf =. 6] |idan Ta —15.96 —27.9 | 222.9 | 6.8 T Greenwich, Feb. 15 | —15.55| 10 |—15.55,,] —29.2 | 10 | —28.8,, Paris, Feb, 15) 2215.47 | in (ME istaa Bogie || vn eee Washington,| Feb. 9 564 2 |—15.64, Leyden, Feb. 15) ==15 5% | 6. |= .545 O85 7 | —28. 5, [—219.3]] [—4.8] [5" 2.92] | Feb. 15. —15.50 —98.6) | —-2160Nleam Greenwich, Mar. 3 |—15.36} 2 |—15.36, | —29.0 3 | —28.6, Paris, Mar. 5 |—15.30| 4 |——15.27, | —28:0 4 | —27.8, [—215.9]| [—4 8] [5% 2") | Mar, 4 Sian —98.2 | —213.3 | —1.0 THE ORBIT OF URANUS. 147 MEAN CORRECTIONS TO THE EPHEMERIS OF UrnaANUS.—Continued. Observatory. [R. A. of Uranus. ] Observed corrections in R.A. Observed corrections in Dec.| Corr. to Geocentric Mean dates. | No. of | Corrected | Mean. |No.of Corrected Latitude. obs. mean. obs. Inean. Mean. Longitude. Paris, Leyden, [5* 347] Greenwich, Leyden, [5" 29") Washington, Leyden, [5" 25") Washington, 1 ney | Greenwich, [5" 21") Paris, Washington, Leyden, [5* 49") Greenwich, Paris, Washington Leyden, [bt 45™] ’ Washington, Leyden, [b* 417] Greenwich, Washington, OS wees Greenwich, Leyden, Washington, [e™ 11] Nov. 21 1863 s Nov. 29 —16.30; Nov. 14 —16.20, —16.24 Dee. 19 Dee. 14 Dec. 18 —16.32, —16.48, BAGG 1864 Jan. 15 Jan. 10 —IBMils NR —16.02 Se 5ui [—219.6] | [—5.7] 17.8 y) —— —15.57 g, |[—216.1] | [—5.6] —-215.6 —T.3 —16.29, 10 : (62028 18 : —=l6216% 17 [—223.4] —220.0 [—218.9] —217.4 [215.3] —214.0 148 THE ORBIT OF URANUS. MEAN Correcrions TO THE EPHEMERIS OF URANUS.— Continued. Observed corrections in R.A. Observed corrections in Dee.| Corr. to Geocentric Observatory. | — — [R. A. of Mean dates. Uranus. ] Mean. | No.of) Corrected | Mean. |No.of Corrected | Longitude. | Latitude. obs. mnean. obs. mean. 1866 s s iM ” " " Greenwich, | Jan. 10 | —16.20| 8 |—16.20,) — 7.2 9) == 628) Paris, Jan. 212 | —16.04 16,05] <5 6.83 1 —siGale Washington,| Jan. 17 | —16.06 QJ UlGsO95 || —s (8 OF soso, Leyden, Jan. 17 | —16.02} 6 |—15.99,| — 7.1 6 | — 7.1, [—291.6] [ 1.3] — $$ ee | |e en — [05 [6" ye Jan. 14 —1'6.09 —— 9020) 9/221) cnt Greenwichy Neb; W4al—=15. Si SON 1 oeeTes Oban olen ei Washington,| Feb. 14 | —15.81 6 |—15.81,| — 8.4 8 | — 7.3, Leyden, Feb. 14 | —15.75} 8 |—15.72,| — 8.6 8 | — 8.6, [—217.3] |[— 1.3] [6 o™] | Feb. 14 —15.79 — 8.2 || 2916.0 ine Greenwich, Wiles WE | aii Alf |) Ol yd) = BT 2) — 7.7, Washington, | Mar. S) | ail.) 6 |—15.49,| — 9.7 6 | — 8.6, § Leyden, Mar. 3 |—15.59| 2 |—15.56,) — 9.0 2) — 9.0, ([—212.9] [— 7.1] [6° 0m] | Mar. 9 15.50 — 8.5 | 919 9 |) aaieie Washington, | Oct. 13 |—15.36] 5 |—15.36 | 4 4.4 5 | + 5.5 |[—209.6] |[— 7.5] ROSS CS Mie catata te rele 2s —211.2 | — 7.9 Greenwich, Nov. 9 |—15:65) 92 ||—Il5.65,| —- 3.5 2) + 329, | Washington,| Nov. 15 | —15.72| 10 |—15.72,| + 3.8 | 10 | + 4.9, (215.1) (fe ae [6" 35] | Nov. 14 Sia 4 46 | 229556 9s Greenwich, | Dee. 11 | —16.01] 10 |—16.01,,| + 2.8 | 10 | + 3.2, Washington,} Dee. 12 |—16.00) 7% |—16.00,| + 2.0 | sp Soll, Leyden, Dec. 10 |—15.87] 2 |—15.84,| + 1.6 | ae Le: [—218.2] [— 8.0] [6" 317] Dec. ll TSR j + 2 ——J1OeT = oe ; 1867 Creme, | dei, Wil | — tH) || & | 1589,|] — 4 5 | — 1.0; Washington) 7sans 21015293) om | n9 Seen Oh eh OMe Leyden, Jan. 9 | —15.82)| 4 | — 15.793 | — 0:3 3 0.3, [—217.8] |[— 8.0] 6" 2477) |) Tanyas 15.90 —= 04" || aie 9.3 Paris, Feb. 13) | —15.72)| 3 |—15.69,| — 2.4 4 | — 2.0, f Washineton,| Feb, 15 | —15.65 9 |—15.65,| — 2.0 | 10 | — 0.9, Leyden, | Feb. 18 | 15.54 5,516 — 0.8 8 | == Wt [214.5] | 1.9] [6® 20"] | Feb. 15 ae eT ee 8.6 Greenwich, Mar. 2 | —15.45 te al 5 || Ss BG 8 | — 2.5, Washington,| Mar. 11 | —15.32] 4 —15.32,| — 2.5 2) | —— esas (—211.1] |E 7.8] [6 19") | Mar. 5 —15.40_ — 2.1 | 2 see Greenwich, | Dee. 11 | —15.61 2 15.601, Se 85 5 | + 8.9; Washington,| Dec. 18 | —15.70| 3 —15.70,| + 6.9 3] + 8.0, [—213.1]|[— 8 6] [6 517] | Dee. 13 Be 4 8.5 | —215.3 | — 1004 THE ORBIT OF URANUS. MEAN CorRECTIONS TO THE EPHEMERIS OF URANUS.— Continued. 149 Observatory. [R. A. of Uranus. ] Mean dates. Observed corrections in R.A. Observed corrections in Dee. Corr. to Geoventric Mean. No. of obs. Corrected mean. Mean. No.of obs. | Corrected mean. Longitude. Latitude. Greenwich, [6" 45™] Greenwich, Leyden, Washington, Paris, [6® 41"] Leyden, Washington, [6" 39™] Washington, ii 16™] Washington, io We Greenwich, Washington, Rit 10™] Greenwich, Washington, Lie" S| Greenwich, Washington, Paris, qh 1™] Greenwich, Washington, [6" 59") Greenwich, ee 327] Greenwich, live 25™] Greenwich, Washington,’ [7® 217] s Ss |; —15.67, 28 ao} 31, 5.31 2215.04. i516 —15.06, 15.08 —14.81, —14.87, Syeevi| —14.50, Sey —14.57; 1 16. [—210.: —212.: [—205. 205. [—198.: = 301.5 [—202.0 —205.6 [206.5 — lee 2 |[—201. —204. [—197. —202. [—199.3 —204. f=107e —901. ” (— 8.7] —10.1 150 THE ORBIT OF URANUS. MEAN CoRRECTIONS TO THE EPHEMERIS OF URANUS.— Continued. Observed corrections in R.A. |Observed corrections in Dec.| Corr. to Geocentric Observatory. 2 eis =— [R. A. of Mean dates. | Uranus. ] Mean. | No. of Corrected | Mean. |No.of; Corrected | Longitude. | Latitude. obs. mean. obs. mean. ” 1870 s 8 Greenwich, | Mar. 12 | —14.29] 5 |—14.28, 16. +16.0 Washington, Mar. 11 | —14.30] 6 |—14.30, Be +16.3 193.6] \C— oy] | [7 1g") | Mar. 12/ .... | .../—14#29) .... | .. | 41699) loess 1871 Greenwich, 9 —13.98 ; +24.6 |[—190.0] [—10.2]] | [7 47") cece [eee | sees | tees ae | aes | rr Greenwich, Jy, is 130i és .6 |[—188.8]|[—10.1]] | (78 41°] BRO wee | sees |e | oe. || = Greenwich, | Mar. [toast ; 5 |[—184.9] [—10.0]] | [7* 38°] was wilteceal) wena eke aS ie) eee Greenwich, : : | 1831155 : [—177.9] [10.4] cs" 11") eM 5 o> | 1865 ani Greenwich, Washington, Ese Gat Greenwich, Washington, [s* 17.0] [—179.8] [—10.6] 188, 3y)|=miee bo w seer eee eee a bo to ry [—179.3] |[—10.5] | 186.6 | =e bo Cr) ie) <) or .) Greenwich, Mar. 15 Washington, | Mar. 15 Sci anal iemlaL e '[—1176.4] |[—10.4] [7 57™.9] | Mar. 15| .... |... ve | on) 295.1) | SB Washington,| April 8 : se Greenwich, April 8 | 5. | [—179.4] [—10.9] ie Sys33] || Atl & aaa noe ayes “ys —180.2 | —10.8 THE ORBIT OF URANUS. 151 CorRRECTIONS TO BE APPLIED TO THE POSITIONS OF URANUS IN THE BERLIN JAHRBUCH AND THE NavurvicaAL ALMANAC TO REDUCE THEM TO THE POSITIONS FROM THE PROVISIONAL THEORY. Date. Heliocentric. Geocentric. ba Msp 63 ol 6b n ” mr ” 1830, July 24 —18.0 +1001 + 9.0 —19.3 ak 0). Aug. 13 18.3 1007 9.0 18.8 9.5 Sept. 2 18.4 1014 9.1 17.9 9.5 Sept. 22 18.5 1020 9.1 rel 9.4 Oct. 12 18.7 1027 9.1 16.7 9.3 Nov. 1 19.0 1031 9.2 16.5 9.2 Nov. 21 —19.3 +1038 + 9.2 —6n7 aL, gil 1831, July 19 IP. +1112 SE ON5 ys +10.0 Aug. 8 22.6 1119 9.5 BT 10.0 Aug. 28 22.7 1123 9.6 2O07 10.1 Sept. 17 22.8 1128 9.6 21.7 10.0 Oct. 7 22.9 1133 9.6 20.9 9.9 Oct. 27 23.2 1138 9.7 20.6 9.8 Nov. 16 23.5 1142 Patt 20.6 9.6 Dec. 6 RG +1146 JE Oni 2k 0, 9.5 1832, Aug. 2 —26.9 +1198 a 198 —28.7 +10.3 Aug. 22 27.2 1201 9.9 27.9 10.4 Sept. 11 27.4 1202 9.9 26.9 10.3 Oct. 1 27.6 1205 9.9 26.1 10.2 Oct. 21 27.8 1208 10.0 25.4 10.2 Nov. 10 28.0 1209 10.1 25.1 10.1 Noy. 30 — 8 +1209 +10.2 9) +10.0 1833, July 28 —31.9 21933 +10.2 3455 +10.7 Aug. 17 32.2 1236 10.3 Sou 10.8 Sept. 6 32.4 1240 10.3 32.7 10.8 Sept. 26 32.6 1242 10.3 31.8 10.7 Oct. 16 32.9 1245 10.3 31.0 10.5 Nov. 5 33.2 1247 10.4 30.4 10.4 Nov. 25 — BL +1250 +10.4 — SN +10.3 1834, July 23 —38.0 +1264 +10.5 —41.2 Lila) Aug. 12 38.4 1266 10.5 40.7 mre Sept. 1 38.4 1268 10.6 39.6 Apel Sept. 21 38.7 1270 10.6 38.6 11.1 Oct. 11 39.1 1275 10.6 Sian 10.9 Oct. 3 39.4 1277 10.6 36.9 10.8 Nov. 20 39.7 1282 10.7 36.6 10.7 Dec. 10 40.1 +1282 +10.7 —36.1 +10.5 1835, July 18 —43.9 +1309 +10.8 Al 411.3 ' Aug. 7 44.5 1311 10.7 47.6 11.3 Aug. 27 44.5 1313 10.7 46.6 DAS Sept. 16 45.1 1316 10.7 45.8 TO LEy) Oct. 6 45.4 1318 NOSSO 44.8 11.2 Oct. 26 45.9 1321 10.7 44.0 10.9 Nov. 15 46.5 1324 10.8 43.6 10.8 Dee. 5 —46.6 +1327 +10.7 —42.5 +10.6 1836, July 12 —51.0 +1361 SEN — 55.3 411.4 Aug. 1 51.3 1364 11.0 55.2 | 11.5 Ang. 21 Bile 1365 uate 54.6 11.7 Sept. 10 Sil +1368 +11.0 —53.7 | -+11.6 152 THE ORBIT OF URANUS. CoRRECTIONS TO BE APPLIED TO THE POSITIONS OF Uranus— Continued. Heliocentric. Geocentric. Mdp ie 6b ce A co bo 1183 Tie 11.0 goto me HoSos ae “1387 1387 1388 1389 1391 1393 1394 1394 +1393 Hwooonwrw eooooresse +1395 396 1395 1392 1390 1388 1388 +1389 (So) as Ps PS es Pp 4 2 Dori 8.1 9) 3) 9.4 ot +1382 1381 1380 1379 1379 1379 1378 1377 +1376 ea +1376 1374 1376 1377 1376 1376 1377 1378 +1377 i — +1378 1384 1385 1388 1390 1392 1393 1395 1398 +1404 Ww mow Oe OO on o DHE ORBIT OF URANUS: 153 CoRRECTIONS TO BE APPLIED TO THE Posrrions or URANUS— Continued. Heliocentric. Mdp +1409 1408 1409 1409 1410 1412 1413 1414 1415 +1417 PANN RAS ro r 101 OU WoW arwNwawea 41422 1420 1420 1420 1419 1418 1418 1417 +1418 CH He eee LD 6 Dene PrwRar 41406 1403 1400 1397 396 1392 1389 1384 1381 +1378 +1340 13: mw mT for mor mor) nN % Oo 09 I Oe 7 Oe me I ww bo for} © 9 0 OOO hO bo bo bo 0 OO eC — i — i © how OTD OO im ee ee et et co wo co Go Co CD OO CD em oo 09 to One rwpwns nanmnmmnnn Se na +a +7 00 GO 0 o oN CO He i He CO cD Ot wma OH wb LD | eye 2} or) H es a 20 May, 1873. 154 THE ORBIT OF URANUS. CORRECTIONS TO BE APPLIED TO THE PosITIONS OF URANUS— Continued. Heliocentric. Geocentric. Mdp él ~ oa " Nov. 1 +1185 8. —145.8 Dee. 2 af 1183 : 143.7 Dec. 2. 1182 : 141.4 Jan. af +1182 8. —139.3 ~ ~ + Ge eeeenles) = bo to ox + Sept. ac +1165 : —156.0 Sept. : 1166 156.7 Oct. 8. 1164 3 156.4 Nov. 6 Be 1164 ; 155.3 Nov. 26 : 1163 6 153.2 Dee. H 2 1162 6 150.7 Jan. 5 6 1161 of 148.3 Jan. 2 31.3 +1161 zc —146.6 + —T -T ~T -T 00 © 00-7 mONoOooeomo es + +1149 ( —164.6 1150 165.1 1149 165.1 1148 164.0 1149 162.4 1148 160.0 1147 157.5 41147 5. —155.0 or Sept. Sept. Oct. Nov. Nov. Dee. Dec, Jan. > Sd OS Se a aT TT - - | NO Oran ro Sr D> WO HOOD COO Co He = co CO ge Se zie | TONG 172.6 W383 173.1 171.8 169.4 166.7 163.8 —161.8 Aug. Sept. Oct. Oct. Noy. Dee. Dee. Jan. Feb. 41135 1133 1131 1129 1127 1126 1197 1127 41124 + TWO DAT AT > Sb > Or Oe i bobo £9 MNADAARAMAAD Cd eS + Sept. Oct. Oct. Nov. Dee. Dee. Jan. Jan. +1109 5. —180.0 1105 5! 181.4 1103 if 181.9 1103 180.8 1103 : 178.7 1102 5.8 176.1 1101 x 173.4 +1098 = 1009 of OV ON OT on oc = G9 9 1 7 G0 Go =r | bo mr bo bo bo (coll ell ll el me bo oOo = He He CD 0D Do LOR TW Ot Oo ROM at at aT tT et eT -T -T + | —186.8 188.9 189.7 189.6 187.8 185.3 182.3 fr) Sept. 11 7(). Sept. 179. Oct. Nie Nov. 180. Noy. 180. Dee. 181. Jan. 181. Jan. 181. Feb. —182. 41074 1073 1071 1068 1067 1065 1064 1062 +1060 + ‘pda ata ad ella Dore OID OATAIw Ss —_ co ee bo Oo He He Or Or Or rPamnNpa+awork oOo He He eee Cl > He He He OV OLD -T — bo ao a + Sept. —— 85: Sept. 2 186. Oct. —186. +1055 1055 41055 Op @ oo go Ios oo 90 oooc a ale CorRECTIONS TO BE APPLIED TO THE Positions or URANuUs— Continued. TEE ORB EE COPE WU ReACN | UsSe cn or Date. Heliocentric. Geocentrie. on Mdp 62 él ch vr 7 Wt 77 1853, Oct. 31 —186.8 +1054 +3.6 —196.8 413.8 Nov. 20 187.1 1054 3.5 195.6 Bet Dec. 10 187.5 1053 3.5 193.5 3.6 Dec. 30 187.9 1053 35 190.7 3.6 1854, Jan. 19 188.1 1053 ot.) 187.4 3.5 Feb. 8 —188.5 +1052 +3.5 —184.5 +3.5 Sept. 16 —=1'9952 +1034 +2.9 —200.5 +3.0 Oct. 6 192.6 1032 2.8 202.9 2.9 Oct. 26 193.0 1032 25 203.7 2.8 Nov. 15 193.3 1032 2.6 203.0 2.7 Dee. 5 193.7 1029 2.6 201.3 PAH Dee. 25 194.0 1025 2.5 198.6 2.6 1855, Jan. 14 194.2 1021 9.5 195.2 25 Feb 3 — 19455 +1018 +2.4 —192.0 42.4 Oct. 1 ——98s3 +1010 +1.8 —207.9 +1.9 Oct. 21 198.7 1009 ile 209.6 1.8 Noy. 10 199.0 1008 1.6 209.7 eri Nov. 30 199.3 1007 et) 208.4 1.6 Dec. 20 199.6 1006 ES) 206.0 1.6 1856, Jan. 9 199.8 1005 11 8) 202.6 iL Jan. 29 200.1 1003 1.4 199.2 1.4 Feb. 18 —200.3 +1001 aE 3} —196.1 LIS Oct. 15 —203.3 + 962 +0.7 — 213.9 +0.7 Nov. 4 203.7 957 0.7 214.9 0.7 Nov. 24 203.9 953 0.7 214.5 0.7 Dee. 14 204.3 949 0.6 212.4 0.6 1857, Jan. 3 204.4 945 0.6 209.3 0.6 Jan. 23 204.5 941 0.5 905.6 0.5 Feb. 12 —204.7 + 934 +0.4 —202.2 +0.4 Sept. 20 —206.9 + 879 —0:2 —214.0 —0.2 Oct. 10 207.2 874 0.3 217.1 0.3 Oct. 30 207.5 868 0.3 218.8 0.3 Nov. 19 207.8 861 0.4 219.0 0.4 Dee. 9 208.0 853 0.4 217.6 0.4 Dee. 29 208.1 846 0.5 215.0 0.5 1858, Jan. 18 208.2 839 0.6 211.5 0.6 Feb. 17 208.3 830 0.6 207.7 0.6 Feb. 27 —208.4 + 823 —0.6 —204.2 —0.6 Oct 5 —210.4 + 741 —1.3 — 219.1 —I.8 Oct. 25 210.6 732 1.4 291.4 1.5 Nov. 14 210.8 723 1.4 922.9 ef) Dee. 4 211.0 713 il) 221.9 1.6 Dec. 24 211.2 704 1.5 219.7 1.6 41859, Jan. 13 211.3 695 1.5 216.6 fs Feb. 2 211.4 686 1.6 213.0 1.6 Feb. 22 —— JG + 677 ——I1,.5) —209.4 —1.5 Oct. 20 —— ONAL + 5176 —2.1 999.8 —2.2 Nov. 9 913.2 569 Del 99475 Dp) Nov. 29 —— ey! + 561 9), Il — 99478 ae, THE ORBIT OF URANUS: Gr for) CORRECTIONS TO BE APPLIED TO THE Posit10Ns or URANUS— Continued. Date. | Heliocentrie. Geocentric. ba Udp 88 al ‘t ” la 1859, Dec. 19 —= 918 5) +554 ee —293.5 —2. 1860, Jan. 8 } 913.5 546 2.2 220.9 2. Jan. 28 213.6 539 2.2 217.4 2. Feb. 17 IIB +535 —2.3 ed —2; Sept. 24 —214.7 4-456 —2.8 —219.7 —2. Oct. 14 214.8 451 2.8 223.1 2. Nov. 3 215.0 444 2.9 225.6 ah Nov. 23 215-1 437 2.9 226.8 3; Dee. 18 215.2 431 3.0 226.3 ey 1861, Jan. 2 215.3 424 3.1 223.8 3: Jan. 22 215.3 417 3.1 221.0 3. Feb. 12 | —215.3 +411 —3.1 —217.3 —3, Oct. 29 —215.6 +339 —3.7 —225.0 —3 Nov. 13 215.6 333 3.8 226.9 4, Dee. 8 215.7 329 3.8 227.4 4, Dec. 28 215.8 324 3.8 226.3 4. 1862, Jan. 17 215.8 320 3.9 223.5 4, Feb. 6 215.8 315 4.0 220.0 4, Feb. 26 —215.9 +310 —4.0 —216.0 ——4e Oct. 24 —216.0 +246 —4.4 —224.2 ——4 Noy. 13 216.0 241 4.5 226.6 4. Dec. 3 216.0 237 4.5 227.7 4. Dee. 23 215.9 232 4.6 227.2 4, 1863, Jan. 12 215.8 226 4.7 225.0 4, Feb. 1 215.8 219 4.7 221.8 4. Feb. 21 RY 213 4.7 218.1 4. Mar. 13 —215.7 +208 —4°8 —214.1 ——fe Nov. 8 —-215.3 +139 —— ee —224.6 ae Nov. 28 215.2 133 5.4 226.5 by Dec. 18 215k 126 5.4 226.8 5. 1864, Jan. 7 215.0 120 eo) 995.4 4), Jan. Q7 214.9 114 5) 229.7 5. Feb. 16 214.8 108 5.6 219.3 By, March 7 —214.3 +103 —).6 —215.3 5). Oct. 13 —213.8 + 21 —6.0 —217.8 i). Nov. 2 213.7 16 6.0 221.3 6. Nov. 22 213.7 9 6.1 224.0 6. Dee. 12 213.6 + 1 6.2 295.1 6. 1865, an. 1 Q13.4 — 6 6.2 224.7 6. dan. 27 213.2 14 6.2 222.7 6. Feb. 10 213.1 20 6.2 219.6 6. March 2 —213.0 — 27 —6.3 —216.7 —6. Oct. 8 —— IEG —103 —6.8 —213.5 —6. Oct. 28 211.4 110 6.8 217.2 TT. Nov. 17 211.2 HT 6.9 220.0 Te Dec. 7 QTL 1 124 6.9 222.0 te Dec. 27 210.9 132 6.9 229.4 Me 1866, Jan. 16 210.7 139 7.0 291.5 Ne Feb. 5 210.6 145 7.0 218.8 uf Feb. 25 210.4 151 7.0 215.4 ie Mar. 17 — 21051 DS —T.1 —211.2 —T foe) i=] = = DOOR WWI OS RR OOD OTTO DDD DHDANR SHOESSSOD HMHNWNHODS Www THE ORBIT OF URANUS. CoRRECTIONS TO BE APPLIED TO THE PosiT10Ns OF URANUs— Continued. -t or Date. Heliocentric. Geocentric. ays Mdp 63 él wt aA vr US6 Ga Oc s —208.2 — 225 en: —— AN T( 533 — Ocias23 208.1 230 7.4 211.5 Nov. 12 207.9 935 eo 914.9 Dec. 2 207.8 941 7.6 21626 Dec. 22 207.6 246 7.6 918.8 1867, Jan. 11 207.4 951 7.6 918.5 Jan. 3 207.2 256 en 916.7 Feb. 20 207.0 262 Lets 213.8 Mar. 12 —206.7 —) 263) == fatl —209.8 — Nov. 27 —203.2 —— sil So —211.4 — Dec. 17 203.0 358 Say 913.5 1868, Jan. 6 202.8 365 8.2 ONS Iat Jan. 26 202.4 371 8.3 912.7 Feb. 15 202.2 376 8.3 910.3 Mar. 6 201.8 882 8.4 207.0 Mar. 26 ——20 1.4 =O Oin =—— O-4 902.9 — Oct. 12 —198.0 nS —— eno: — 1197.2 _— Nov. 1 197.8 465 8.7 201.1 Nov. 21 197.5 73 8.8 203.8 Decs ul 197.2 481 8.8 206.2 Dec. 31 196.8 490 8.8 207.5 1869, Jan. 20 196.4 498 8.8 207.1 Feb. 9 196.0 507 8.8 205.4 Mar. 1 195.6 515 8.9 202.4 Mar. 21 ——J(9)5).3) ——n Oe. — 8.9 —'98.4 — Dee. 6 119050 evarer Sees — 1197730 Dec. 26 189.5 Les ee 198.9 SOs ean. V5 189.0 —— (Hes) —= f),¢ 199.5 — Feb. 14 188.6 676 9.4 198.7 Feb. 24 188.0 686 9.4 196.2 Mar. 16 Si) 697 9.4 193.0 April 5 —187.2 — 708 — 9.4 —189.4 — Dec. 1 IL SAG a S655) Dec. 21 LST 859 9.6 189.0 1871, Jan. 11 180.5 872 Hots 190.3 can. 30 179.8 883 9.8 190.1 Feb. 19 179.1 894 9.7 188.3 Mar. 11 178.4 906 9.7 185.6 Mar. 3 ow) —— OG — 9.8 —181.2 Dee. 16 Aral als —1083 00) Nitin! 1872, Jan. 5 171.0 1095 10.1 179.3 Jan. 25 170.5 1107 10.0 180.2 Feb. 14 169.8 1120 10.0 179.6 Mar. 5 169.3 1133 10.1 ities Mar. 25 168.7 1145 10.1 174.9 April 14 168.1 1156 10.1 ilps May 4 —167.6 —1168 —10.1 —166.3 mMmmommomm®d 7AT1HMHDMO Joi nl Sol i= ilo oli. 2) SCwnNwnwwnrwmrwos 158 THE ORBIT OF URANUS. CELA PE Re Velie FORMATION AND SOLUTION OF THE EQUATIONS OF CONDITION RESULTING FROM THE PRECEDING COMPARISONS. In the preceding chapter we have obtained from observations a series of cor- rections to the geocentric positions of Uranus resulting from the provisional theory. The further operations are as follows:— 1. To reduce all the corrections in right ascension aid declination to correc- tions in geocentric longitude and latitude. Most of the corrections are already so expressed, so that this reduction is necessary in only a few cases. To find the mean value of the correction in geocentric longitude during each opposition, and to express this mean value in terms of the correction to the helio- centric co-ordinates. 3. To express these corrections to the heliocentric co-ordinates in terms of cor- rections to the elements of Uranus and the mass of Neptune. 4. To solve the equations of condition thus formed. The first of these processes is too simple to make it necessary to present any details of it. With regard to the second I have sought, not the simple correction to the geocentric longitude, but this correction multiplied by such a factor as it was supposed would make the probable error of the correction 0’.5, The equations for expressing the error of geocentric longitude in terms of errors of heliocentric longitude and radius vector have been given on page 129. The first observation of Flamstead, p. 107, gives the equation 22” = 1.0452 + .0275p 64 being the correction to the heliocentric longitude, and dp that to the Neperian logarithm of the radius vector. From the discordance of Flamstead’s clock errors it may be estimated that the probable error of the first member of this equation 1s 10”. Therefore we divide the equation by 20, which gives siyol = 1.1 = 05252 + .001 dp. In the opposition of 1715 we have four observations. The best were those of March 4 and 10, of which we may estimate the probable error at 10”, and the worst that of March 5, of which the probable error may be estimated at 20’, while that of April 29 is intermediate in certainty. The separate observations give the equations March = él = 28” — 1.0652; Weight, 4 March 5, d1=+44 =1.0602; Weight, 1 March 10, él April 29, éJ Mean ol +36 =1.068A; Weight, 4 of : = 1.0452 + .045p; Weight, 2 + 27.6 = 1.05632 +.003%p; probable error = + 6”, ny THE ORBIT OF URANUS. 159 Applying the correction —1".1 for equinox, and dividing by 12, the equation of condition becomes 761 = + 2’.2 = 0.0889a. In this way the following equations were obtained. It is deemed unnecessary to give the details of the process, as it is one which every one can go over for himself from the data already given, and can reproduce all the results, except so far as they depend on the relative weights assigned to the different groups of observations during one and the same opposition. Number of No. Date. Equations. observations 4 in Rie AY 1 1691.0; J,2=+4+ 11 = .052),+ 0015p 1 eis ee = 2 088 4 2 PAS 2 eis = S88 ah ony 1 4 1750.8 4 =+118 = 345 4+ .010 3 Seeings) = Eile = saa 016 1 6 1756.7 4 =-+ 5.0 = .210 + .003 1 fee i690 2 ===! 4.8 = 203" O10 8 & 1782.0) ——— 9.0) 15310 21 9 1783.0 1 =+ 1.25=1.030 — .002 13 10 1784.0 1 =+ 1.92=—1.026 — _ .008 13 11 1785.0 1 =— 0.26=1.034 + .006 10 fomise.0) 2) == 1230684 = 2006 5 13ee 1189.0) 2° =" 1580504 -— 008 6 14 1790.0 4 — 0.52=0.514 — 013 4 15 1791.0 2 =— 0.66=—0.684 — .010 7 16 17920 4 =— 0,12=0.340 — .011 3 iy 1793.0 i =— 0.19 =0.512 — 015 5) 18 17940 4 =4+ 0.79=0.544 — .006 3 19 1795.0 2 — (0.66—0.683 — .019 rd 20 1196.0 5° —— 0638 — 0/514 OLS 4 21 ie Ooo Or 28 3 22 18002 = — 0.00 = 0.352 2 ao 1803.2 4 =— 0.26 =0.352 2 YM IG 1 4k ORGS 13 25 180533" 1 == ——) 0169 = 1:05 13 2% 18063 1 —4+ 0.20—0.52 5 27 1807.38 1 =4+ 2.34— 1.045 16 28 1808.3 4 =— 0.04—0.52 6 29 1809.3 2 =+ 1.80—0.70 9 30 WgSHO)33 1 ee DU He) cee 1105) 16 31 1811.38 1 =-+ 149=1.04 —Q0.01 11 32 1812.4 t — + OlSsie— loo 8 33 1813.4 t =+ 1a (Ros 9 34 1814.4 1a eee) 15 160 THE ORBIT OF URANUS. Number of No. Date. Equations. observations i in R. A. 35 1815.4 8@=-+ 21 =1.5802 20 36 L818i4* 3° == ==) 038 (= as 94 37 1si9Ay Le — 05805 i 38 18205. asa —e0S 14 39 18215 2 =-+ 0.9 =0.70 10 40) 1822.5 2 =F 0:9) 0710 7 41 1823.5 2 = 0.0 =0.70 a! 42 18245 1 = Oo 105 12 43 1825.5 2 =— 04 =0.70 al 44 1826.5 1 S03 1005 11 45 18200 25 eS 00 + 0.04 dp 37 46 18980) SY ee Olas Sean 67 47 182951 22 = S259. + 0.04 61 48 1830.7 25 =— 49 =2.56 + 0.07 73 49 ieskis 2 0.0 =2.06 + 0.05 54 50 1832-2) 220k + 0.05 65 51 1833.8 24 =— 41 =258 -+ 0.08 88 52 1884.8 24 =— 3.9 =2.57 + 0.08 oil 03 1835:8) 25 i oe 259 + 0.05 82 54 1836:8)8 0009 ——) 0 re + 0.08 157 55 183-33 — 30 — + 0.04 162 56 1838.8 . 3) == Lo (31-2006 193 57 1S39'8e oe oe + 0.06 170 58 1840:8 73) 3 —— ome LL + 0.06 124 59 8418 3 = Os — 3100 eu0ue 108 60 1s40lg 9 =e 8 10 ONG 169 61 1843.8 38 =4+ 4.7 =312 + 0.04 111 62 1844:9' 3) == oe lt + 0.03 106 63 18459 2 = 49 908) = house 55 64 1846.9 3 =+ 63 =3.10 -+ 0.04 98 65 1847-9) 32a 018) 210i + 0.03 74 66 1848.9 2 =+ 44 =2.08 -+0.02 59 67 1849.9 2) === 6.6 = 2:08 0104 33 68 1850:9 25 ==! 71.2 = 2108-0 46 69 18519 2 == 63 207-00 42 70 1852.9 2 =-—- 66 =207T = 008 54 Ci 1853:9"" 2 == 9) 2:09 + 0.01 49 72 18549" 25 = 89 = 2.09 + 0.02 49 74 18539 2 =+ 8:5 —2038 e5 0.04 48 75 1856.9 2 == 7.9) 208 + 0.04 45 76 1858.0, 25 == 10'3) = 2°61 + 0.09 66 * The results for 1816 and 1817 were omitted in this list through oversight. THE ORBIT OF URANUS. 161 Number of No, Date. Equations. observations 77 1850.0 Qb8= 410.6 = 2605, 40.03 d 58 78 1860.0 24 =+ 82 =260 + 0.05 64 79 1861.0 2 =+ 63 =2.09 +0.03 41 80 18620 21 =+ 7.0 =260 40.05 60 81 1863:0 24 =-+ 66 =259 + 0.08 88 82 1864.0 2 =+ 43 =2.09 + 0.03 35 83 1805.0 14 =+ 39 =157 40.04 37 84 1866.0 24 =4+ 11 =260 4+ 0.06 76 85 1867.0 24 =— 18 =260 40.02 83 86 1868.0 2 =— 3.8 =2.09 + 0.04 40 87 1869:0) 525) —— 91 — 261 4. 0102 68 88 1870.0 2 =— 9.6 =2.084 + 0.054 31 89 1871.0 13 =—10.6 =1.560 ++ 0.040 21 90 18i2 25 — 19s — 2600) =. .0:070 50 Total number of observations im R. A... . . . « «. . » 93163. We have next to express the values of 62 and ¢p in terms of the corrections to the elements. Differentiating the expressions A= 1+ (e — te’) sin (d — x) 4+ fe’ sin (21 — 27) 4 13e’ sine’ sin (31 — 32) 4 ete. p—v-+ qe’ + (—e + #e*) cos (J — x) — Fe’ cos (2/ — “x) — ete., with respect to ¢@, ce, and zx, and reducing the coefficients to numbers, we find a) —1 + 0.0939 cos g + 0.0055 cos 2g Co = = 1.999 sin g + 0.117 sin 2g + 0.007 sin 3g Or = < ea 2.000 cos g — 0.117 cos 2y — 0.007 cos 3g We have here put @ for the mean longitude, or l=ni+te whence on _ oA oe ol x On = =ta On ol Also, from the expression for p Ce De ad hig Ses) 7 a G 2e*) sin g + fe’sin 2g +... 5 = = be — (1 — 2°) cos g — Ze cos 2g — ete, (22 on ol 3n The values of these coefficients whieh depend only on g are shown in the following table : 21 May, 1873. 162 THE ORBIT OF URANUS. On On O& c edna 099 2.124 099 : 2.124 099 : 2), 1S} 099 5 SONI .099 158 118 099 4 114 099 23: .110 099 0. 2.105 .098 t). 2.099 098 : 2.092 098 : O85 097 : OTT 097 : 2.069 097 0). 059 096 ue 049 096 ’ 2.038 095 ‘ 027 095 .655 2.014 094 : 2.001 094 : 987 093 : 913 .092 ; 957 091 : .941 .090 OTS ~925 090 ). 907 089 0), .890 O88 0), 872 OST : -852 086 : “832 O85 : 811 O84 : .790 084 ; .769 .083 5 SANT 082 2 Tle 080 as . 700 079 : .676 078 : .651 OTT Bee .626 076 aoe 601 O74 a3 5714 .073 F 548 072 Ae 521 070 ; 494 069 P 466 .068 5 438 067 : 409 065 5 .380 064 : pop .063 ‘ .320 .061 : .290 060 : .260 .058 : 229 057 é .197 n0'55) - .165 054 Bite 2133 .052 : .100 O51 5 067 049 é 034 O4T . 002 046 ; —0.968 | | | | eoooesssssssssssssssssssssss Hee He eR OOH NOIOONOONNN E S t e OO COCO OSD AOrNW RK OATRD EWR OOD PENW RK OTD HP WOO ATU soooooo ON OV DR wWH So THe OF bo i>) -T Hm CO CO 02 CO OO OO OO OO oO ==} Je} Sosesssssssses: | OD 0D OH OO OO TT 7 = a wT Ol = 09 LO Ht SO OO ATS Ore | THE ORB eS OF WyRTAGN Ur Si. 153 .848 .862 875 588 901 .912 -922 -932 942, 950 959 967 974 980 .985 990 994 BOT 009 2.003 005 2.007 2.007 .096 2.005 2.004 2.002 .000 997 aH}8)B3 989 984 -978 912 965 958 950 5943) 933 924 914 903 .893 882 869 .85T 844 .829 815 800 .T86 .170 154 Ar(ets: 21 104 686 668 .650 164 THE ORBIT OF UR AN US? OX On OX op op Op g O¢ Oe | eda O& Oe eda 120° +0.950 +1.631 | +1.051 +0.041 +0.50 —0.86 121 0.949 1.611 1.077 0.040 0.51 0.85 122 0.948 1.592 1.104 0.040 0.53 0.84 123 0.947 1.571 1.129 0.039 0.54 0.83 124 0.945 1.551 1.154 0.039 0.56 0.82 125 +0. 944 +1.530 + 1.180 +0.088 +0.57 —0.81 126 0.943 1.509 1.205 0.038 0.59 0.80 127 0.942 1.488 1.229 0.037 0.60 0.79 128 0.941 1.466 1.253 0.037 0.62 0.78 129 0.940 1.443 NTs 0.037 0.63 0.77 130 +0.939 +1.421 +1.300 +0.036 +0.64 =O 6 131 0.937 1.397 1.322 0.036 0.66 0.75 132 0.936 WEBI! 1.344 0.035 0.67 0.74 133 0.936 1.350 1.367 0.034 0.68 0.72 134 0.935 1.327 1.388 0.034 0.70 0.71 135 +0.934 ST SiO +1.409 +0.033 +0.71 (Osun 136 0.933 1.277 1.43 0.033 0.71 0.70 137 0.932 1.253 1.451 0.032 0.72 0.68 138 0.931 1.228 1.470 0.031 0.74 0.67 139 0.93 1.202 1.489 0.031 0.75 0.66 140 +0.929 =Eenltray +1.508 +0.030 +0.76 —0.64 141 0.928 1.152 1.527 0.030 0.77 0.63 142 0.927 1.124 1.545 0.029 0.78 0.62 | 143 0.927 1.099 1.562 0.028 0.79 0.60 | 144 0.926 1.072 1.580 0.028 0.80 0.59 145 +0.925 +1.044 +1.596 +0.027 +0.81 1); 146 0.924 1.016 1.613 0.026 0.82 0.56 147 0.923 0.989 1.629 0.026 0.83 0.54 148 0.922 0.962 1.644 0.025 0.84 0.53 149 0.922 0.934 1.659 0.024 0.85 0.51 150 +0.921 +(.906 +1.674 +0.023 +0.86 —0.50 151 0.921 0.878 1.688 0.023 0.87 0.49 152 0.920 0.849 1.702 0.022 0.87 0.47 153 0.919 0.820 1.714 0.021 0.88 0.45 154 0.919 0.792 ON 0.021 0.89 0.44 155 +0.919 +0.762 1.740 +0.020 -+0.90 —0.49 156 0.918 0.733 1.751 0.019 0.91 0.41 157 0.918 0.704 1.763 0.018 0.91 0.39 158 0.917 0.674 STi} 0.018 0.92 0.37 159 0.916 0.645 1.783 0.017 0.93 0.36 160 +0.916 +0.615 +1.793 +0.016 +0.94 —0.34 161 0.915 0.585 1.803 0.015 0.95 0.33 162 0.915 0.555 1811 0.015 0.95 0.31 163 0.915 0.525 1.820 0.014 0.96 0.29 164 0.915 0.494 1.829 0.013 0.96 0.28 165 +0.914 +0.465 +1.836 +0.012 +0.97 == 0826 166 0.914 0.435 1.843 0.011 0.97 0.24 167 0.914 0.403 1.849 0.011 0.97 0.22 168 0.913 0.373 1.855 0.010 0.98 0.21 169 0.913 0.343 1.860 0.009 0.98 0.19 170 +0.913 +0.311 +1.865 -+0.008 -+0.98 —0.17 171 0.912 0.280 1.870 0.007 0.99 0.16 172 0.912 0.249 1.875 0.006 0.99 0.14 173 0.912 0.219 1.879 0.006 0.99 0.12 174 0.912 0.187 1.882 0.005 0.99 0.11 175 +0.911 -+0.156 -+1.884 +0.004 +1.00 —0.09 176 0.911 0.126 1.886 0.003 1.00 0.07 177 0.911 0.094 1.888 0.002 1.00 0.05 178 0.911 0.063 1.889 0.002 1.00 0.03 179 +0.911 +0.031 -+1.890 +0.001 +1.00 —0.02 THE ORBIT OF URANUS. 165 In the equations of condition ten years has been adopted for the unit of time, in order to make the general value of the coefficients as nearly equal as possible, and the time has been counted from the epoch 1830.0, in order to have the posi- tive and negative values of ¢in the equations more nearly balanced. ‘To distin- guish these values of de and én ney are marked with an accent. ‘This unit of time gives 0.8914 for the value of ~ in are, whence t By) a) 0 Bea ot ORO, On ol The equations of condition are now formed by putting in the preceding equa- tions for heliocentric longitude and radius vector to) an, On ayy im 2 © cr = AE OE aS a ov — =¢e oe am + Wee Oo 08 O ee bins 2s ie oy O€ =v On : O tebe OD For the coefficients oe “have been taken one-hundredth the perturbations of Ou longitude produced by Neptune, as given in the heliocentric ephemeris at the end of Chapter V. The corrected mass of Neptune will then be Ou 1 aie ty a) Finally, I remark that all the preceding comparisons are made with the helio- centric ephemeris as printed, without the correction indicated in the column adjoining it, but in the following equations this correction is for the first time introduced. Equations of condition given by the Correcticns in Longitude. 1 | 0.053 — 0.70in’ —0.108e +0.03e30 +0.123u =-+ 11 POO) = =i LOO = O19) 2099" SEE ote 2 | Wel = OR Se sey eee eae SaaS en 032) == 252) 007 | 2058 ose _ ins Pa 030) e230 EO 19) 2059 0109 == ENG CaO == 40) 0101 -4#0839 OO 5x0 T0190) = 1s) 000" 003) 02" Fa serals 2 | 1 AR = re See COG = 50le = 098 8 = 079) 1850 Sa ey soe Oiee—(h9d 2 0'86) E185) Se ReeCOS ae Seg S04 bigs 1005 m0 —=206 — hor O97 Erie) = 10 12 | OR Oy | Oi) 0 SNR eee! aio5s — 919 06s ~~ =olsl 4081 =— 0.6 ieeeeOws — 986. 082 116 2103, L2=—'0%9 fomeOsT — 139° =036 =060 =+-049 202 ieOisGe = 205 = -047 ~~ 095) E0710" == 04 fom = 133 —=026 065, F044. === 076 (ONS 59 «= 038 9 138-4082 0 166 20 46 48 49 oO Sipcalbalea ectea eae cleale cles! cleat at aclagleaiea sage ie ai=ati (aml ale siaail clam atical sulvalies(milelios[scles) "| a0 st he sales a THE 1.898’ 1.92 ORBIT —().22ke —0.13 +0.09 +0.15 -+-0.69 +1.11 +0.62 +1.38 +0.75 +1.09 1.73 +1.81 -++0.96 +1.00 +2.03 Sale 41.29 rm 09 0 Or 219 22 WD Ss) > DH WO re OF =2 W WD A 0D FW 0H 0 DW WD OW 0 t+++4++++4+4+44+ —_~ > o URANUS. —1.04eSn — jl. —=(() 7/4 —=((),'7133 alles — 191 —().89 ——1.68 —().79 (97 = Be Jl, i4. ——(())5 1! == (())4b33 = (() pl == (() iy —=((), 338} (DY 140.585 u +-0.55 +0. 24 40.21 0.50 +0.19 +0.05 0.00 —().04 Ea l() == (()) 53 —().29 mee ( lS —(()) Wil = (EUG = (Net =" 2 —(()) ND = (() Si —(()) 58) =—=()59 =*()139 (39) = (939 —=()), HY) —().38 (0) ,57/ == 1.19 == 11 de ee ST Ue SU SI TE STS ts tet +4+++4++4++44- | EI | F+++4+4+4+++ | +4+44 | | b+i+ ++ PO SE a SIO OORP EHOW PEP NHWOEUAMEERWORWHWONYNNWCOCSCSOSCOHPSCOOCONWHHOHWDWHCOCNwWCCCCCCS MNWAWASCOCHOENWNWSORH HUD HORAOMWNWODBDHNnODWoOROMHNINDWOHHUHUOAwWwoe THE ORBIT OF URANUS, 167 70 | 1.953¢ + 4.448n’ —8.043e 42.59eix —1.683u =-+ 6’.8 imebospe 472 393 roa7) 173° Sa ea ceo fos) 344 0s SG 60 Eo A Preeloue 518 = 858 119i 176 2 67 Pa O0eee= 5850 — 373° “eed 92 gg9 Bote feo 02) ) 463 is 939 = F106 fom oe st = 496 gg 69990) 10.9 Teneo 16h 507 +097 S931 29.9 fom wee 688 =i 047 =1s6 2-263 eo 825, = 520° 2021 =939 2+ 7.0 Sumeee OO e851 = 52 —0.15 999. 2 6.6 Pirie se ad 49 2051 E1g3 ek 48 Saeco 556 | —818 058 {136 “= 3.9 Seo 959 8 — 512 —13e7 099) Ea St DGS EL OGD Ae) Sc Soir Se 829 9190 7S 67 3:8 comeeyier 1061 =407 , 9.59 9 o01 9 23> 910 SUMO ESS | = 3163: 8933. 159 2S ons Seton 8 G4 958 196 1.08. = — 105 Some LlGi ~~ 404 = 361 =169 ==19.9 The following are the approximate normals to which these equations give rise. Tnaccuracies being detected in several of the equations of condition after these normals were formed, they do not accurately correspond to those equations as written. 283.645 -| 414.363n’ —151.63%e 1247.23ein —176.033u —-+193".5 414.36 +1619.44 —689.11 +-260.26 — 456.02 = + 103 .2 —151.63 — 689.11 +557.82 + 38.45 +122.88 399 .8 247.23 + 260.26 + 38.45 +618.45 —194.60 = + 267 .3 —176.05 — 436.02 +122.88 —194.60 +163.13 = — 128 .1 The values of the unknown quantities deduced from these normals were substi- tuted in the equations of condition, and a farther approximation was made by solving the equations given by the residuals. The following are the first approxi- mations given by the normals, and the finally concluded corrections Preliminary. Final. bu’, —15.00 —13.44 oor, = i O86 de, — 4,25 — 4.04 10dn = dn’,— 4.73 — 4,33 de’, — 3.718 — 3.44 Mass of Neptune ZDITO IGE SG The final values of the corrections being substituted in the equations leave the following system of residuals, or outstanding excesses of the observed longitudes over theory. Column f8/ gives the residual of the equation itself; the probable error of which has always been judged to be 0.5, while in column ¢/ this residual is divided by f to obtain the residual correction oe the longitude itself. The values of the factors f are found with the original equations on pages 159 and 160. 168 No. of Eq. 1 CNS) GS Or He CO -t 8 10 11 12 13 14 15 16 17 18 WY) ~© wo = 5S wnwww WOW w BiH Ww Year. 1691.0 ipleone 1748.8 1750.8 1753.9 17.6.7 1769.0 1782.0 1783.0 1784.0 1785.0 1788.0 1789.0 1790.0 1791.0 1792.0 1793.0 1794.0 1795.0 1796.0 Wigs) Tel 1800.2 1801.2 1802.3 1805.3 1806.3 1807.3 1808.3 1809.3 1810.5 1811.3 1812.4 1813.4 1814.4 1815.4 1816.4 THE ORBIT fol == (Fe) SEO —0.5 +0.6 = 150 aE9 sey —0.5 017 0.0 —().1 +0.5 —0.6 —0.5 (IF —1.4 —1.4 —0.6 —1.4 —1.0 (al: —1.0 ahoal = —0.9 —=()'3 +0.1 —().4 == (ks —0.8 él " 10: == iO, AL ne a aes + 6.6 1.0 1.0 0.2 0.5 0.6 1.0 | +) + 1.8 3.4 1.0 1.0 0.0 | ++ 0.2 1.5 0.9 — 1.0 Inet — 4.2 — 4,2 — 0.6 — 14 — 2.0 — 0.1 — 2.0 aE02 =O = 10:9 = 06 AE }0:2 ie =102 me) OF URANUS. No. of Eq. Year. 36 37 38 39 40 4] 42 45 44 45 46 47 48 49 55 56 57 58 59 60 61 62 63 64 66 67 68 69 70 qa 1817.4 1818.4 1819.4 1820.5 1821.5 1822.5 1823.5 1829.7 1830.7 1831.7 1832.7 1833.8 1854.8 1839.8 1836.8 1837.8 1838.8 1859.8 1840.8 1841.8 1842.8 1843.8 1844.9 1845.9 1846.9 1847.9 1848.9 1849.9 1850.9 1851.9 8 1852.9 1853.9 eee 81 —0.3 —().7 —1.4 —1.4 +0.9 mei 10.6 41.2 . Be) +2.0 41.0 42.5 +2.8 40.4 441 a8 40.5 +0.1 a —3.4 —0.6 +0.5 —0.4 —1.7 098 —0.5 +1.6 +0.7 10.6 —0.4 +0.6 = 16 08 40.3 =18 > —1.2 —0.2 él — 03 — 0.5 — 14 — 14 4+ 12 me) +. ig + i + om) => on Se a +++4++ + 0.9 + 0.2 0.0 ms ~ 08 + 08 = Of 0.1 0.2 0.5 0.2 0.3 0.1 |+++4 | 0.3 0.8 0.1 ().2 0.6 b++1+ 0.6 THE ORBIT OF URANUS. 169 No. of Eq. Year. Sl él No. of Eq. Year. fol ol me 1s549 05. + 0.2 Bi 18640 Lorn 04 73 1855.9 0.0 00 82 16650, 222 4815 18569 +05 — 02 STA SGN ek hy 8 567. 0.5 = 2 foeede5s0, 0.2 — 0.1 Suey ie Mee We 18590 109 + 04 pe Oey FimeiscnO, 11° —'0.4 Sola Von Bega Feeeisci 0G | = (08 Sore co re a gE Seu STiLO aon k= -0.8 80 18630 402 40.1 oe 1672d | ho = i09 A simple glance at the course of the residuals shows (1) that their probable value is considerably greater than the probable error attributed to the equations of condition, being more nearly 0”.7 than 0”.5, and yet larger in the later years; (2) that during certain periods they are of a systematic character. During the years 1748 to 1755 the observations show a decided positive correction to the theory of a magnitude greater than we can consider probable, amounting to about one-third of asecond of time in the mean of Bradley’s two observations of 1748 and 1753. About 1800 the correction becomes negative, and so continues for 20 years with an average value of about 1”. In 1821 it suddenly becomes positive, and so continues until 1833. From this year forward the residuals are not systematic in character. In order to show clearly the general course of the outstanding corrections, they have been divided into groups, generally including about five years each. The mean outstanding correction for each group, taken with respect to the weights indicated by the factors f, is as follows. In the column ¢ is shown what the probable error of the residual should be if the weights assigned to the several equations were strictly correct, and no systematic errors were present either ip theory or observation. Year. él € Year. od € 5.2 —10. SLO, 1824.8 + 1.50 +0.16 lifolel + 3.7 -+-0.8 1829.7 + 0.91 0.10 1769.0 — 1.0 SLL 1835.2 — 0.27 +0.09 1783.3 —— (Oils) +0,23 1839.8 — 0.17 0.07 1790.0 + 0.62 +0.40 1844.8 + 0.14 +0.03 1795.0 ——= (0). 555 -+-0.41 1849.9 —= (NII SEIU 1802.0 = 1105 +-0.45 1854.9 — 0.14 SEOUL 1806.5 —— LAND) +0.31 1860.0 aan) Biles --0.09 1810.5 —— {)),337/ +0.32 1865.0 + 0.56 SEL 1814.5 == ())337/ +0.23 1870.0 — 0:21 == (ale 1819.5 — (a7 +0.18 92 May,1873. 170 THE ORBIT OF URANUS. A simple glance at the residuals é/ shows that they are much greater than the purely accidental residuals resulting from the theory of least squares. We may divide the possible causes of these systematic errors into three classes, 1. Systematie Errors of Observation.—These may result from deviation of the line of collimation of the instrument from a true great circle, or from any pecu- liarity of the observer which leads to his registering the transit of Uranus earlier or later than that of a fixed star. If we compare the corrections derived from the work of different observatories as given in the last chapter, we shall find frequent cases not only of systematic differences between the results of different observa- tories, but between those of the same observatory in two successive years, An instance which particularly attracted my attention on first preparing the com- parisons of theory and observation is that of the Greenwich observations for 1831, which, as compared with observations at the same observatory during the years preceding and following, seem to be affected with some constant error in R, ix of about 2”. I find that this discrepancy can be attributed only to the original observations, 2. Errors in the Theory compared.—These may arise from errors in the preceding theoretical computations, from the omission of the terms of the second order pro- duced by Neptune, from the adoption of an erroneous mass of Saturn, or from the attraction of an unknown planet. With regard to the probability of these different sources of error it may be remarked that errors of computation seem possible only in the terms of the second order, that the mass of Saturn is taken from the exhaustive discussion of the Saturnian system by Bessel, in which an error sufficient to influence the theory of Uranus seems highly improbable, and that a trans- Neptunian planet large enough to produce a sensible deviation of the orbit of Uranus from an ellipse in the course of a century would be too large to have escaped detection. The choice of the elliptic elements of Uranus and Neptune is such that the terms of the second order, due to the action of Neptune, can scarcely become sensible within a century of the epoch. 3. Errors in the various Reductions by which Theory and Observation are com- pared.—In the method adopted for comparing theory and observation a number of small uncertainties incident to the imperfections of the older data of reduction necessarily creep in. In the early observations the imperfections arise principally from the uncertainty of the instrumental corrections, and the errors in the adopted positions of the fundamental stars, and indeed in nearly all the data of reduction. In the late years they arise principally from the great magnitude of the correction to Boavard’s tables, and the consequent rapid change of the corrections to the geocentric ephemerides, which make the determination of the corrections Al and cl from theory and observation somewhat uncertain. Errors from this source will necessarily be in part of a systematic character, and, in view of their possibility, I regret not having been able to completely re-reduce all the observations before 1840, and to compare all since directly with ephemerides computed from the provisional theory. In order, however, to test the question whether they are sensible, I have prepared an ephemeris from the provisional theory for the three recent oppositions of 1861-2, 1862-3, and 1872, and compared it directly with THE ORBIT OF URANUS. 171 the observations. The mean corrections in geocentric longitude for groups of observations are given in columns (2), column (1) showing the correction given by the work of the last chapter. Opposition 1861-2 1862-3 1872 1 (2) (1) (2) (1) (2) ALO WS SO SSO == RG 1n 970! DOI ee DB EO) ee 26) Bil) de () ae SO: eat (er een 2.4 2 A, 3.3 2D on —7.3 —7.1; on ee DG F MEE Sy Oh ee 186) Mean 12.79 +2.62 TQ G59) 50 Sen aie e2 A systematic difference of 0.16 would seem to be indicated, and on account of it a correction of 0’.10 was applied to the comparisons of the last few years in forming the equations of condition. In view of the possibility of systematic errors from this source it may be con- sidered that too great relative weight has been assigned to the results of the later observations. If the residuals arise from errors of comparison and of theory, their probable magnitude is nearly as great at one epoch as at another. It may there- fore be interesting to inquire what result we should get if, instead of assigning such different weights to the comparisons at different epochs, we sought only for the best general agreement with observations during the period the planet has been observed. ‘The preceding system of mean residuals will enable us to discuss this question quite easily. In the first solution we shall reject the results from Flamstead’s observations, owing to their assured uncertainty, and those from Le Monnier’s of 1769, owing to the possible maladjustment of his quadrant. The equations from the remaining residuals will be the following: 1.05% —T7.65'n' 40.88% +1.7e5%7 —0.55% =+3".7 Wt. 11 = 20. 08 | Ms) = 0g.) 2 1.1 i SS Ss iG Se ll “288)- S206 =90 ~ "4502 "2085 7a e312 LOG) 200 05) === 125), A Ile, eOGes 2E1) Ihe 0.0 =—-1.00 2 1 SO SEU SL ES a ll sei LO RY ee Ss Sa ee One edits es aL) =060 5 Obs 2 10 205 5 209 =0G asl A 1.0 OO dei ae! OR Sein 1 205 OS 4S 0B SO OD — ALOR XO SLO 08 Sai ‘LO. SR AAG = EL) ee 10 — SS Soy SSG SSG eS 10-5 2S ee “09 aa a0” ¢ 10 Ea =O) LOLs e090) 0.15 ee ee eS @ = 21 ik Ne ae wy) eects ei 18. og = 0701 12 TH EO RAB OF UR FAGN URS: Giving these nineteen equations equal weights, we have the second of the following solutions, and the*second of the series of residuals the first corres- ponding to the primitive solution. Solving them again and assigning the weights attached to the respective equations, which I judge to be tnose to which they are entitled when a liberal allowance is made for systematic errors of observa- tion and of comparison of theory with observations, adding also the equations given by the observations of Flamstead and Le Monnier, which are as follows: 1690 0.05%’ —0.75°n’—0.15%e +0.0e5?x +.0.15%u’ = —0".5; f, sp; Wt, 1 Wyle = ALT CLO OR2 +01 =—0.5 -3 il 1769 0.2 —1.2 —0.3 -+0.2 +0.2 =+2.2 4 1 we have the second solution, and the third series of residuals. Q) (2) (3) fee 0 —0".39 —().21 On, 0 —() .38 —0.19 oe 0 —=( ae —0.15 eon 0 +0 .25 -+-0.19 ou 0 —1 .02 —0().49 me 1gUtD ToSTD 19750 RESIDUALS. nee Year. Al Al Al 1691.0 ——a)i()s —14. —12. WH) 2 ——ill()s 9: — 7. ifoiee + 3.7 + 0.5 + 2.0 1769.0 — 1.0 — 2.6 — 19 1783.3 ; — 0.18 a0) — 0.17 1790.0 + 0.62 + 0.93 + 0.89 1795.0 — 0.55 — 0.09 — 0.18 1802.0 — 1.25 == (7s) — 0.87 1806.5 — 1.00 — 0.69 — 0.738 1810.5 — 0.37 = (110 — 0.05 1814.5 == a7 —— 0826 — (28 1819.5 — O37 — (),34 = ((),37/ 1824.8 + 1.50 + 1.46 + 1.41 1829.7 + 0.91 + 0.90 + 0.81 1835.2 —= OY == (43) — 0.46 1839.8 — 0.17 — 0.56 — 0.48 1844.8 + 0.14 (Pei — 0.18 1849.9 — 0.21 =n (5) — 0.52 1854.9 — 0.14 — 0.54 — 0.36 1860.0 — 0.13 — 0:23 =i (A: 1865.0 + 0.56 + 0.70 + 0.62 1870.0 — (21 + 0.80 + 0.44 IESE OR BID OF UR AN Uist is It will be seen that the effect of these changes of weights is, that the older observations are a little better, and the later a little worse represented. I conceive that our choice must lie between the first and third solutions, the first being the more probable if we conceive the outstanding residuals to be due to errors of observation only, and the third if we suppose them equally due to errors of com- putation, On the whole, I consider the mean of the two to be about the most probable, and this will give the mass of Neptune very near the round number 1 19700 which will be adopted as the definitive value. The definitive corrections to Elements III (p. 99) will then be de (1830) — 37.56 de (1850) —12 .45 10dn — 4.44 de — 4.12 eén =) 025 du == (NB 7 Corrections to the Inclination and Nide. These corrections have been derived entirely from the modern observations, the ancient ones being too uncertain to add anything to the weight of the result. The mode in which the correction to the latitude of the provisional ephemeris has been concluded from the observations has been sufficiently explained: it is only necessary to add that the immediate results from the data of the preceding chapter require two corrections, namely: (1) A correction to the theoretical latitude for the change in the adopted mass of Neptune. The value of this correction, as derived from the data of Chapter V, is with sufficient approximation 63 = 0".25T cos g. (2) A correction to the observed latitude on account of the difference between the obliquity of the ecliptic adopted in the various ephemerides compared, and that of Hansen’s Tables du Soleil, which having been adopted in the theory should be used throughout. Applying the correction (2) —(1) to all the observed latitudes, we have the following corrections to the latitude of the provisional ephemeris derived from all the observations of each opposition since 1781. The third column gives the number of observations in declination. These numbers may, however, in some cases be inaccurate. The fourth and fifth columns give the sine and cosine of the argument of latitude, to be used in forming the equations of condition. 174 THE ORBIT OF URANUS. No. of obs. oa cso) | . No. of obs. sin u cos u = > 1782. 1783. 1784 1785. 1788. 1789. 1790. 1791.0 1792: 179320 1794. 1795 1796. 1797. 1800. 1801.5 1802. 1805. 1806. 1807.: 1808.: 1809.3 1810.; Stale 1812. 1813. 1814. 1815, 1816. 1817. IS18. 1819. 1820. 1821. 1822. 1823.5 1824.5 1825.5 1826.! 1827.6 1828.6 a SSS Slee Sees SE, 1830.7 0. 1831.7 1832.7 1833.7 1834.7 1835.7 1836.7 1837. 1838.8 1839.3 1840.8 1841 1842.8 1843.8 1844.9 1845.$ 1846. 1847.§ 1848. § 1849. 1850.¢ 1851.¢ 1852. 1853. § 1854.§ 1855. 1856.6 1858. 1859. 1860. 1861. 1862. 1863.0 1864.0 1865.0 1866.0 | 0 all pil +44 4+ S92 IS wow or +P We wonae aTeT Or ® Go C2 wD — + S bo — = or fe S Etihad S DO THT MWS TITHE WOOD WHT ODH HOO Say who oo: + o > Re He bo SO > S> _ So — PO Whh WHAT WOW TRO 5 & C202 to Cece co bo KO oo oo oo 8 iS (| 0 1 bow 0 C9 SEE See ae r= + S92 2 bo Fw bo OWS awk wWlom —) =) S i=) | =) . > . OCrvOroyr Ovcr ot Hm ee Be OO S | i 1867. 1668 1869.1 | 1870.1 1871.1 1829.7 fi 1872.1 Or Oren or Cow one S It will be remembered that the observed declinations have, as far as possible, been reduced to Auwers’ standard. We have no positive proof that this standard is correct. If it be affected by a constant error, the result will be that the orbit of the planet on the celestial sphere, as deduced from observation, instead of being a great circle, as we know the real orbit to be, will be a small one, and the com- parison of a uniform series of observations extending through an entire revolution of the planet, after making the best correction to the position of the’ orbit, will leave a constant residual. Now, we can best determine this residual by including it as an unknown quantity in our equations, THE ORBIT OF URANUS. 175 Again, the error of the standard is not necessarily constant, but may contain a term proportional to the time, arising from erroneous proper standard stars. suppose it of the form a + dt. will then give the equation, sin udp — cos upd) + a+ b6T=S§p. motions of the Therefore, instead of supposing the residual constant, we shall Each observed correction to the theoretical latitude To facilitate the solution of these equations they have been divided into groups, each group usually comprehending three oppositions, and combined into a single equation multiplied by such a factor as would make its probable error half a second. The factor by which the correction of the latitude is multiplied in the equation is the same with the coefficient of a. ‘The year 1840.0 is taken as the Thus we have the following: Equations oF LATITUDE. epoch for 6. Dates of oppositions. | No. of opp. Equation. 1782.0- 1788.0- 1792.0- 1795.0- 1800.2— 1805.3— a 8é 9 9 y: 1808.3-10. HS ies— 13. 1814.4-16. 1817.4-19. 1820:5—-22. 1823.5-25 1826.5—28. 1829.6—-31 1832. 7-34. 1835. 7-37 1838.8—40. 1841.8—43 1844.8—-46. 1847.8—49. 1850.9- 2 : 1853.9-§ 1856.9-— 39 1860.0-62. 1863.0-65. 1866.0—-68 1869.1—70.0 1871.1-72.1 0.439 ).§ +1.0a 0.8 +1.0 0. 0. 0. 0. 0. 0. 0. —0. Ne oo ow i. > Lo o IR MAMOMOARWOR oO We She ROG (. ? r.¢ 2, RU aOie aie > DHHOUER ER WD H OH EH wWwWRwWwwPrDwe Feet e oe dee eae eee ed 0 Dp Ceo CUO CDH WO WW OOH bo aT OL SSSoSoOSCoSCSoOSCOoOSoSOSOMOOoOMUUMSCOCOO OM, RH RO RI DOOH PR ODOOD RWS wor OUT tH+tt+tt4444444444 bo LO 29 Go G9 Go Go Go Es Ee Se 9 Oo Co for) ’ 0 oo CO are Mo. m Wl oi aol) me bro or Treating these equations by the method of least squares, we find the normal equations " wr " " ” 63.6059 + 6.45980 — 52.102 —“ 1.086 —+ 28.33 6.4559 + 69.5993) — 52.60a — 14.25b = 4 111.02 —52.103¢ — 52.6095 + 133.50a 4+- 8.95b = — 16.55 — 1.0839 — 14.25980-++ 8954+ 4.496 = — 23.69 176 THE ORBIT OF URANUS. The solution of these equations gives op = + 07.28 + 0".75 a = + 0".54 pod = + 1.57 + 0 .686a + 0.2056 = + 1.75 a=-+ 0.39 R28 These values of a and d indicate that at the epoch 1840 Auwers’ equatorial declinations are too great, or lis north polar distances are too small by 0.35, and that this error is diminishing at the rate of 0.28 per century. If the older measures in declination had been comparable in precision with those made at the present time, and if the possible periodic error in the reduced right ascensions had been carefully eliminated, I should regard this determination as entitled to considerable weight. In view of the great uncertainty of the declinations previous to 1820, it can be regarded as little more than a rough attempt at a determination. For this reason the first two normal equations have been solved, leaving a and 6 indeter- minate, so as to show the valves of d@ and @d@ in terms of these quantities. It will be seen that had we neglected a and 6 entirely, the value of d@ would have been smaller by 0’.26, and that of oS smaller by 0”.18 than those actually con- cluded. As the observations with the Washington Transit Circle, and those with the Pulkowa Vertical Circle, both indicate an increase of Auwers’ polar distances, I shall take for the definitive corrections to the inclination and node those which follow from the above values of a and 4, or, dp = + 0".54 poo ioe The following table shows the residuals of the equations, and the mean out- standing corrections to the latitude, (1) when the concluded values of do and 950 a and b are all used, and (2) when w and d are supposed zero, and the values of dp and 30, corresponding to this supposition, are used: Year. Residuals. 63 oo} ae oy ©) 1783 —0.8 =(().53 =0:8 ==03 1789 aeale? +1.8 SE) =e 1793 +0.7 aioe +1.4 aE 1796 SHO +0.6 +0.6 eee 1801 =05 +0.3 5 +0.3 1806 == 119 == =1k2 —=(i5 1809 +0.5 JE +0.5 aie 1812 +0.9 SEINE +0.9 16 1815 4013 +0.6 =e 0.4 1818 +1.6 125 =i 41.7 1821 Oxi ==(al =A) —0.1 1824 [23 == (uv. S=1t.3) —0.7 1827 —0.6 =) 2 = 0:6 -0.3 THE ORBIT OF URANUS. 177 Year. Residuals. 63 oy oO, @) @) 1830 —().2 +0.2 —0.1 +0.1 1833 +0.5 +0.9 +0.10 +0.30 1836 0.0 +0.5 0.00 +0.10 1839 —(0.1 +0.1 —0.03 +0.03 1842 —0).5 —(.4 —0.17 —(0).13 1845 +0.1 0.0 +-0.03 0.00 1848 +0.6 +0.7 +0.20 +0.23 1851 +0.9 +0.9 +0.30 +0.30 1854 10.3 0.3 40.10 0.10 1858 +0.1 +0.5 -+-0.03 +0.10 1861 —1.4 —1.0 —0.47 —0.33 1864 —1.5 —1.0 —0.50 —0.33 1867 Bieta Sey 40.37 +0.57 1869 +0.5 41.0 SEO BIEQS3 1871 0.0 za) 0.00 1.0.23 The sum of the squares of the residuals is in the first case 17.94, and in the second 25".41, so that the introduction of a and 6 makes a decided improvement in the representation of the observations. I have not attempted a rigorous investigation of the probable error of any of these results for the reason that the values of the probable error deducible by the method of least squares would, in a case like the present, be entirely untrustworthy. It is, however, very desirable that we should be able to form some judgment of the uncertainty of the mass of Neptune. From the last system of equations of condi- tion the value of «’ comes out with the weight 3.15, or nearly that assigned to the mean result of each five years of modern observations. Regarding these results as independent, their mean error would be about 0”.5, so that the probable error of w would be 0.5, and that of « would be .005, or about 51; the entire mass of Neptune. A probable error derived from the original equations would have been much smaller, and when, in the last equations, we allow for the systematic character of the residuals, it will be larger. If we suppose the theory to be perfect, I conceive we may fairly estimate the probable error of the mass of Neptune to be zi, of its entire amount, and its possible error two or three times greater. If there is any error or imperfection in the theory, the error may be much larger, _23 May, 1873 178 THE ORBIT OF URANUS. C BEACPAT ER Va COMPLETION AND ARRANGEMENT OF THE THEORY TO FIT IT FOR PERMANENT USE. In the preceding discussions the terms of the second order due to the action of Neptune have been neglected, the elements of Uranus and Neptune being so chosen that these terms can scarcely become sensible within a century of the epoch. But this very choice will make them larger in the course of centuries than if mean elements had been chosen. ‘They will be most sensible in the case of the great inequality of 4300 years between Uranus and Neptune, an inequality which will make centuries of observation necessary to an accurate determination of the mean elements of the two orbits. The uncertainty arising from the great inequality is probably of the same order of magnitude with the omitted terms of the second order, and, such being the case, the theory would really be made but little more accurate by the addition of those terms. I conceive, however, that the theory will be made much more satisfactory by the computation of at least the largest of the terms in question, if only to arrive at a certain determination of their order of magnitude, and of their effect on the planet during the period in which it has been observed. ; The term in question, being of very long period, may be most advantageously treated by the method of variation of elements, more especially as it has im the theory been already treated as such a perturbation. The largest of the pertur- bations in question are those of the mean longitude which are multiplied by the square of the integrating factor y, which is nearly 51, but which also contain the eccentricities as factors, and those of the eccentricity and perihelion which are independent of the eccentricities, but are multiplied by only the first power of ». ‘These terms will probably comprise nearly or quite nine-tenths of those arising from the term of long period. Let us begin with the perturbations of mean longitude. These are given by the integration of the equation d?l dt? i, and k, being functions of the ratio of the mean distances, or a. If we integrate this equation, supposing all the quantities in the second member except I’ and J to be constant, and these two to be of the form nt + ¢, m and ¢ being constants, we shall reproduce the principal term of long period already found. But in the second approximation we must suppose all the elements variable. It is not, how- ever, necessary to take into account the variations of a, n, and &, because these are = — 3m'an? Sek, sin (2U—I—x) + ek, sin (27—I—z')} THE ORBIT OF URANUS. 179 of a lower order of magnitude. The perturbations to be added will be those of UU, e,é, 2, and 7. The point from which the longitudes are counted being arbitrary, we shall take the position of the perihelion of Uranus for 1850.0 as the origin, and put, as before, g for the mean longitude of Uranus counted from this point, and let U represent the mean longitude of Neptune counted from the same point. The terms of fF p within the brackets will thus become ek, sin (2U’—g—S8x) + ek, sin (2l’—g—(a’'—2)) or, if we put 2—¢g = N ésin(a—a2) =’ é cos (ax — 2) = Kt and notice, that to terms of the first order we have, sin Ja = dz, cos da = 1, we shall have ihah dt? = — 8m'an’ } (el, + Ky) sin N — (ehba + Wk,) cos N} differentiating the quantities, of which the perturbations are to be considered with respect to the sign d, we find for the terms of the second order. al 0) We = Buvan® | (Ie de + heySl! + Why5N) sin N +( (ek, + hk) 8N — ek,da — ke, Sh’) cos NY. We have now to substitute in this expression the numerical values of the quanti- ties within parentheses. Those of the perturbations of Uranus have already been given in Chapter III, but it is necessary to diminish them by the factor 0.145* for the altered mass of Neptune. Those of Neptune are taken from my investigation of the orbit of that planet (p. 38). The mass of Uranus there adopted is 551,55, while the investigation of Dr. Von Asten,t from the observations of Struve and others, shows it to be s5455. The perturbations are therefore diminished by ,1,. In accordance with the system adopted throughout both investigations, constants are added to all the perturbations to make them vanish at the epoch 1850.0. A term is also added to make also vanish at the epoch; this corresponds to the dt constant which ought to be added to éa. The numerical values thus obtained, are: * This factor was adopted hefore the mass of Neptune to be employed had been finally decided upon. Hence the difference between it and that in the preceding chapter. + Mémoires de l’Académie de St. Pétersbourg, tome xviii, vii série. 180 GEE 1B) OVI IBIEMY (iN IIR AGS G7 Se d5N= + 7260" sin N — 6658" + 4”.26¢ eon = — 414 sin V+ 380 ce = — *414 cose —- 165 che 12) 120) sine alan 6k =+ 120 coseN-+ 48 je ee ee) Aap) hi = -+ 0.00695 ki = — 0.00486 9 : a’ : : : Substituting these values in the expression for OTe and integrating twice, we find, putting 6 for the coefficient of the time in N, of which the value, taking the century as the unit, is +0.1472, and putting 7’ for the time in centuries, 3 él = = m'ayv A } 10278 5”.3 9 * 4) (411 + — = 4 2.67) sin N+ (1887" + |" 51.47) cos — 109".3 sin 2NV — 5".5 cos 2 ; — 16".5 man’? + cT + €, ce and ¢ being the arbitrary constants of integration, which are to be chosen so that both é7 and its first differential coefficient shall vanish at the epoch. Reducing to numbers, we find od = (140".70 + 0’.327) sin N + (232.60 —6 377) cos N — 18.60 sin 2N — 0.70 cos 2N — 0.087? + 34.277 — 46.76, the last two terms being arbitrary. When we carry the perturbations of the eccentricity and perihelion to quantities of the second order, we are troubled by the introduction of large terms depending on the square of the disturbing force, which disappear from the rigorous expres- sions for the co-ordinates. These may be avoided by substituting for the eccen- | tricity and perihelion the quantities A and & determined by the condition h=esnzx k —ecos If, as before, we count the longitudes from the perihelion of Uranus at the epoch 1850, we should substitute Jz for 2 in these expressions. The values of / and k will then be given by the integration of the equations dh —= | mank a vank, cos N dk , : “= T™ ank, sin NV. ~ THO MORTEM OLE SUR eACN USE 18] Differentiating with respect to ¢, we find for the terms of the second order Ui : ‘eae mank, sin NSN dt se wa ‘ant VN On TT Manhy cos NON. Substituting for éN its numerical value just given and integrating, we find th = mark, | —2895" sin N— (6658" — 4”.26f) cos N+ 1815” sin 2N} — 3630"maknt + constant ; th = mark, | —2895" cos N + (6658" — 4”.267) sin N+ 1815” cos 2N} + constant; the constants being so chosen that 64 and d% shall vanish at the epoch. Reducing the values of df and d& to numbers, they become ch = 0".82 sin N + (13.40 — 0".86 7) cos N — 3”.65 sin 2N-+ 1”.087— 2".6 dk = 5 .82 cos N — (13 .40 — 0.867) sin N — 3 .65 cos 2N + 12.1 the last two terms being arbitrary constants. Computing the values of these terms of 8/, (2, and é/, for intervals of 50 years, from 1600 to 2000, we find them to be as follows: Year él th ok 1600 — Il”. By4! +0".10 == (02 1650 —() Fl +0 .05 == (R02 1700 == (il +0 .02 —( .01 1750 (1) 1@ 0.00 —0 .01 1800 ——=() (ut 0 .00 0.00 1850 0 .00 0.00 0.00 1900 0 .00 0 .00 0.00 1950 +0 .04 0 .00 =O OM 2000 40.18 —0.01 = 0).02 We sce that although the ultimate effect of these terms is very considerable, their effect, during the period that Uranus has been observed, is insignificant. Concluded Elements and Perturbations of Uranus. The corrections found in the last chapter being applied to the final provisional elements (p. 99) give the following elements for 1890, affected by the great inequality produced by Neptune: Elements 1V of Uranus. Epoch, 1850, Jan. 0, Greenwich mean noon. 1, IGS tis’ WVU Fe D8 258 lei 05 6, Yo, 4k = fh ds 0 46 20.54 e, 0469236 182 THE ORBIT OF URANUS. e e (in sec.), 9678".69 n, 15425.752 log a, 1.2829072 log a, 1.2831044 Log a, includes, as before, the constant term in the perturbations of the logarithm of the radius vector which, with the corrected mass of Neptune, is +-.0001979, To find the corresponding elements at any other epoch, the following secular and long-period perturbations are to be applied. Those produced by Neptune are derived from the expressions in Chapter III by correcting them for the new mass of Neptune, and for the change in the value of the small divisor 2n’—n produced by the correction of the elements of Uranns. ‘The logarithms of the factors for correction are, Correction of mass of Neptune 9.93598 Correction of divisor 0.00051 Log. factor for ¢Z 9.93496 Log. factor fore, da, dx” 9.93547 Including the perturbations of the second order just found, we have, by putting N=2l'— 4, = 113° 30’ 46".0 + 8° 26’ 51”. 97, él = (—2850".41 + 0".32T) sin N + (387.67 — 6".37T) cos N 1 sin 2N — 47.28 cos 2N sin 3.N -f 4.33 cos 3N sin 4N — 0.46 cos 4N 12’18sin M+ (14.03 — 0.867’) cos N 29 .20 sin 2N — 6.09 cos 2N 3.llsn3N+ 1.19 cos3N + 0.28sn4N— 0.13 cos4N + 14.767 + 398 .33 ék = — 411".53 cos N — (13.65 — 0.867) sin N + 29 .33cos2N+ 6.17sin 2N — 3.12cos83N— 1 .21sin3N + 0.29cos4N+ 0.13sn4N — 5.4537 — 124.72 6” (in units of the 7th place of decimals). = 1963cos N+ 103 sin N — lilcos2N— 67 sin 2N + 1ldcos83N-+ 6sin3N 4 511.0. The perturbations of eSz and ée are here replaced by those of h and &, defined by the equations h =esin (1 — ™) k = e cos (1 — 7%) THE ORBIT OF URANUS. 183 mM) representing the perihelion of 1850 = 168° 15’ 6’.7.. We then have for the eccentricity and longitude of perihelion at any epoch e sin (7 — 7%) = ch € COS (%# — mM) = & + Ok. In the above terms multiplied by the time we have included the secular varia- tions produced by Jupiter and Saturn. If the perturbations of the elements due to each particular planet are required, we have Action of Jupiter, ch = “s o”.13T7; ok = — 0".608T. Action of Saturn, th = + 5.567; dk = — 4.5897. Subtracting these from the above expressions all the remaining terms will be due to the action of Neptune. ‘The values of / and dn are due entirely to the action of Neptune. For the sake of rigor, we may suppose the perturbations produced by each planet to be multiplied by a factor representing the number by which the adopted mass of the planet must be multiplied to obtain the true mass. It will add to the homogeneousness of the theory to express the perturbations of long period, which are multiplied by the product of the masses of Jupiter and Saturn, as perturbations of the elements. ‘These terms, as found on page 88, are (v.c.0) = — 07.55 sin V,— 0".03 cos N, + 40 .65 sin VN, — 10.50 cos N, (v.s.1)=+ 2.64sinN,+ 4.64c0s N, + 7.35sn N,+ 4.41 cos N, (v.c.l)=— 4.23sinN,— 3.87 cos N, + 8.06sinV,— 8 .38cos N, These terms, together with the arbitrary corrections of the elements which have been applied to make them very small at the epoch, may be replaced by the follow- ing corrections to the elements : — 0’.55 sin N, — 0".03 cos N, 40 .65 sin V, — 10 .50 cos N, 2 277.27 — 11 72T. 2.09sinV,-+ 1.94 cos N, 2.13sinN,+ 3.71 cos N, + 1”.28. 32sinN,+ 2.32 cos N, 3.68sin N, + 2.21 cos N, do= 27sin N, + 104 cos N, + 76 (in units of the 7th decimal). ~ io) = |+++ | ck | + The amount of the perturbations of the elements for every half century, from the year 1000 to 2200, is given in the following table. Column (1) gives the per- turbations by Neptune, Saturn, and Jupiter, computed from the expressions 184 THE ORBIT OF URANUS. on page 182; column (2) those just given depending on the product of the masses of Jupiter and Saturn. QI ee | | ys, | 2) Tsay eRe) aan) | @) | @ yt 1000 |+2050.31 |4160. 42.66 |-1.94 |—389.51 |4-6.08 |-1955 1050 1841.17 149.08 | 27.79 BLOOM Pomosoo | 6.68 | 1882 1100 | 1638.76 136. 14.03 | 4.99} 360.31] 6.82 1802 1150 1444.87 | 122.8 | 1.45 6.01 | 343.49 6.47 1717 1200 1260.24 108.8: 9.90] 6.60 325.27 | 5.64 | 1626 1250 |4+1085.76 |4+ 94.6 19.84 46.75 |—305.69 44.42 41529 1300 921.76 80.6 .B4 6.49 284.90 2.86 | 1426 1350 769.32 67.15 35.27 | 5.90) 252.90 |11.16] 1318 1400 629.00 54.3% 40.56 .08 239.78 |—0-57 | 1205 1450 501.39 42.63 44.11 SG || ONS GE |) aly 1086 bo bo So (eo) 1500 387.18 32. 45.83 |+3.24 |—190.66 |—3.52 + 963 SSS eS SUS ee aes 1550 286.65 1600 200.65 1650 129.43 1700 73.46 1750 33.06 1800 3.51 1850 .00 1900 .64 1950 31.47 2000 .43 | 2050 | 21.33 2100 : 2150 2200 5.63| 9.42) 164.84] 4.52) 835 3.46| 1.80] 138.31] 5.12] 704 20) A) Deon ear s0n e568 2.79 .23 83.70 .09 430 BIG) tale 2 3s b 5890) eden leases 3196 | 1.86 \— 27.93) 3.81 145 00} 1.52 0.00] 2.96} 0 .65}| 1.63|4+ 27.74] 2.10 |— 145 3.70] 1.59] 55.09] 1.34] 290 DO ee ss- sles Sls 458 SLO) WOLSS 7 tel Ofer se Gi 5 E03) 132.6h 711 15 |—0.77| 156.33} 0. 843 m5 ele: i 968 ‘19 bo > ER N OOO RP OOLN§ OT OT LO bo =T TOON Ow SOereNHRDMDWOrwWNATO OS CoOnatrr ow ea EF CH eS LO — _ o © + + esssss mae PoDwPmre i or) - _— = -T Mean Elements of Uranus. If, instead of the elements of Uranus affected by the great inequality, we wish the absolute mean elements, these are to be obtained by adding to the elements already given the constants applied to the perturbations 7, éh, 64, and §» to make the perturbations vanish at the epoch 1850.0, and also the corrections (p. 113) which we have subtracted from the elements and added to the perturbations to reduce the latter to a small quantity during the period for which the tables are likely to be used. We thus find the following mean elements: Elements V of Uranus. Epoch, 1850, Jan. 0, Greenwich mean noon, Longitude of the perihelion 170° 38’ 48”.7 + 8698". u Mean longitude at epoch, 29 «12 43.73 + 2811 -4u Longitude of the node, 73 14 37.6 + 29 .6u Inclination of the orbit, 0 46 20.92+ 0 .38u Eccentricity, .0463592 — 5236u Eccentricity in seconds, 9562.27 — 108”.0. Mean motion, 15424.797 — 0.838 Log mean distance (uncorrected), 1.2829251 + 179u The same corrected, 1,2831223 + 179% l+u True mass of Neptune, alee oat Aa 19700 THE ORBIT OF URANUS. 185 Supposing the mass of Neptune to be uncertain by one-fiftieth of its entire amount, which is quite possible, it will be seen the longitude of the mean _peri- _ helion is from this cause uncertain by more than two minutes, the mean longitude of Uranus itself by nearly a minute, and the mean motion by nearly two seconds in a century. It will be seen that the logarithm of the mean distance just given does not accurately correspond to that of elements IV plus the constant term of Sn x 0.4343, as it should. ‘This difference arises from the rejection of the terms of the second order in 6”, which can not affect the geocentric longitude of the planet by a tenth of a second for a number of centuries. It is to be remarked that these mean elements are those to be used in the general theory of the secular variation of the planetary orbits. Concluded Theory of Uranus. The elliptic longitude and radius vector of Uranus, affected by the secular and long period perturbations of the clements, will be given by the following equations. Put U, = not + &, ln + él, g=l — Toy h = 0h, k=e,+ ék, Cady (ese the zeros indicating elements LV, and ¢h, &/, and d2 being the perturbations of D 5) 2 > fo) these three elements just given. ‘Then Elliptic longitude in orbit =/ 2a Cee } | A sing —heosg | one bic, 2 oN pc eey: ‘ at =e See } | @ =F) sin 2y — 2hi: cos 2g } 183 463} a 12 64 Ps vom. | (l= Ble +) sin by — (AIP — ALI") cos Ly a 3 \ ' (&> — 32°k) sin 8g — (8h1" — h’) cos 3g I 1097 yay Neperian logarithm of =» + fe’ +. se! =[l=2A \ | boos gh sing} | (= 102? 4 Sith) sin 5g — (Sith — 10% + K eos 5g } 4 24 24 May, 1873. na (ane } | (8 =2) 008 2g + 2h sin 2g } 186 THE ORBIT OF URANUS. 17) a aa (i? — 3kh?) cos 3g + (Bi? ie) sing } a i (is! — 62h? + h') cosy + (Ale — 4i:h*) sin 4g } 96 In computing these expressions it will be sufficient for several centuries before or after 1850 to develop h, d/, and ¢/ to their first dimensions: it will, however, be more convenient to correct the mean anomaly g for the perturbation é/ before obtain- ing the equation of the centre. Developing the perturbations of and & to terms of the first order, we have for the effects of the perturbations of those elements: (v.s.1) = (2 — “) ck —2—y0) (v.8.2) = = at) dk —(o— (v.e.1)= (v.6.2) = see) s ‘he 1 (0.8.3) = qe 2k oes (aS) er) oh 0: (v.8.4) = : ee dk 103 %y (v.c.4) = — fe th 1 (p.c.0) = 60+ af dk Se Nn (p21) =—( age 2) oh 9 (p.c.1) = — (1 — ges?) oe 3 (p.8.2) = — of th 3 (p.c.2) = — of ck Tiss (0.8.3) = — go ok (0.¢.3) =— ne Sie These coefficients for p must, of course, be multiplied by the modulus 0.434294 to reduce the perturbations to those of the common logarithm of the radius vector. THE ORBIT OF URANUS. 187 Among the elliptic terms may be included the effect of the following minute constants introduced by the perturbations. (v.s.2) = — 0.144 (v.c.2) = + 0.130 0.4343 (p.c.0) = +1972 in units of the 7th place 0.4343 (p.s.1) = + 63 of decimals. 0.4343 (0.¢.1) = + 78 0.4343 (p.s.2) = +5 0.4343 (9.c.2) = + 4 This term (.c.0) is that added as a correction to the logarithm of the mean distance. To the coefficients (v.s.1), (v.c.1), ete., are still to be added the following periodic terms :— 1. The periodic terms due to the action of Jupiter, given in Chapter V, omitting the terms multiplied by 7, which are included in the perturbations of the elements. 2. The periodic terms produced by Saturn, including those terms multiplied both by 7’ and by sin A, or cos A,, but omitting those multiplied by 7’ only for the same reason as in the case of Jupiter. 3. The periodic terms produced by Neptune, multiplied by the factor 0.86294 on account of the correction to the mass of that planet, and omitting the terms multiplied by 62, de, and edg. 4. The periodic terms multiplied by the product of the masses of Jupiter and Saturn, given on page 88, omitting the terms multiplied by the sine and cosine of N, and N,, because they are replaced by the terms of 82, dh, and dk, given on page 183, and tabulated in the columns headed (2) on page 184, ‘The result will be the same whether we employ the terms of (v.¢.0), (v.s.1), ete., given at the bottom of page 88 and the top of page 89, omitting the numbers in the columns 2 on page 184 from the expressions on page 186, or whether we include the latter and omit the former. The true anomaly of Uranus will then be: Jo + dl + (equation of centre from elements LV, using for mean anomaly g, + 42) + X (v.8.7) sin ig + 3 (v.c.2) cos iy. The logarithm of the radius vector will be: log r in elliptic orbit from elements IV. + & (p.8.2) sin ig + & (p.c.2) cos ig care being taken to multiply the coefficients by the modulus where that has not already been done. All the terms in Chapter V are so multiplied. To pass from the true anomaly to the true longitude we must investigate the secular motion of the planes of the orbit and of the ecliptic. The effect of this motion on ¢?, 0, and z will be found by successive approximations from the formule 188 THE ORB Hd (OPE SUR PAGNaUE Se (34), correcting the data for the new mass of Neptune. We shall also use the same motion of the ecliptic adopted on p. 95. We have thus: dp ___ 4753 dt dp _ 45 43-4.0°387 adi = + to + ot ° CH est dt CUES AG is OF: dt As a first approximation we have 7 — eo” 14 87 S16 oT » = 0 46 20.544 2.487 Substituting these values in (34) and integrating we find o=o+ 24774 0'13T? 6 = 6, — 3168 .427+4 3 .00T? Tt =7T — 3168 .767+ 3 .00T? For tabulating we shall use, instead of § and 7, the distance of the perihelion from the ascending node, or 7—r, and the value of 6 corrected for Struve’s pre- cession. Since the mean motion has been derived without making any distinction between ¢ and 6, it will be necessary to correct the motion of mean anomaly by the difference of those quantities. We thus obtain for the values of the three principal arguments :— g = 220° 10’ 10.35 + 1542574’.867 + & o= 9 0 58.70+ 3168.767 — 3.007? 6= 73 14 8.00+ 1856 8274 4.127? If we represent all the inequalities of the true longitude by AJ, so that we shall have for the true anomaly f= 9+ Al, the argument of latitude will be u=fto. The reduction to the ecliptic will then be R = — (9.387 + 0".0167) sin 2u, the true longitude on the ecliptic referred to the mean equinox of date, A=u+0+ Rk, and the sine of the elliptic latitude, sin 3, = sin @, sin w. The perturbations of the latitude will be (4.c.0) + (6.c.1) cos g + (6.8.1) sin g + ete. j MEH OR BLE OF TU RAN US: 189 The periodic terms of (b.c.0), (6.8.1), (6.c.1), ete., are given in Chapter V, on pages 86 and 87, and are to be taken without any farther modification than the multiplication of those due to the action of Neptune by the factor 0.863. The constant, secular, and long period terms are b.c.0 =-+0"26 —0’.12T7 —0.01137 + 0.04635x (6.8.1) = —0 227 — 0 .05T? + 0.9755 + 0.2215x (el) = +2 477+ 0 127? -- 0.2218, — 0.97153x (6.8.2) = —0.06 —0.01T + 0.0463, + 0.0118x Ge2)=—0.01 0.127 40.0113, — 0.0465x The values of dy and dx to be used in these expressions are those the expressions for which are given on page 97, and which are tabulated in the last two columns of the table on page 184. The following tables are based on the elements and theory laid down in this chapter. 190 THE ORBIT OF URANUS CASPAR ER Xe. GENERAL TABLES OF URANUS. Enumeration of the Quantities contained in the several Tables. THE first six tables are designed to give the values of the three arguments of the elliptic motion, g, , and 6, and of the nine arguments of the tables of pertur- bations. The argument © is, however, diminished by 3’, the sums of the constants added to the perturbations of (v.c.0) to make these quantities positive, and @ by 10’, the constant added to the reduction to ecliptic. ‘The expressions for the argu- ments of perturbations are as follows, the mean longitude of each planet, counted from the perihelion of Uranus, being represented by the initial letter of the planet. All these arguments are expressed in units, of which 600 make an entire circum- ference, so that each unit is 36... The time ¢ is counted in Julian years from the fundamental epoch, 1850, January 0, Greenwich mean noon. Arg l= J— U. = 219.190 + 43.44028¢ 2 = Se Ue = 577.349 + 13.227178 3=—= U— N == 88.884 + 3.50055¢ 4—=— J—2S —— A Qe6 -+- 9.8445¢ 5=3S— U— J 79.8 + 3.3825¢ 6=4S—2U— J= 57.1 + 16.6108 7=2W — 3S —3U= 238.7 + 18.633¢ 8= WwW — 4S —2U= 261.3 + 5.4058¢ 9=7S —2J —3U =136.9 + 19.992¢ | Table I gives the corrections which must be applied to the values of the argu- ments at any time during the nineteenth century to reduce them to the corresponding time in any preceding or following century between the Christian era and the year 2300. Since w and @ each contains a term proportional to the square of the time, the correction for these quantities is not constant during each century, but 1s of the form otal © and @ being constant during each century, and 7’ being the fraction of the ey counted from its beginning. ‘Table II gives the value of q, o — 3’, 9 — 10", and the-above nine arguments for Greenwich mean noon of Jan. 0 of each leap year from 1752 to 1948, and for January — 1 of the years 1800 and 1900, corresponding to December 30 of the years 1799 and 1899. The corrections for the perturbations of long period are not THE ORBIT OF URANUS. 191 applied in this table. The numbers at the bottom of this table, in the line A); show the variation of the corresponding quantity in 120 days, for the epoch 1850.0. In the line “Factor 7” is given the change of this variation in a century, while Ajz) 1s the second difference for intervals of 120 days. By means of these num- bers, when the arguments are computed for any date, their values for other dates at intervals of 120 days may be found by successive addition. Table III gives the motion of the several arguments between the epochs of the preceding table and the zero day of each month in the course of a four-year cycle. The variable motions, w and @, correspond to the epoch 1850, and rigorously they each require a correction for any other four-year cycle than that between 1848 and 1852. But, owing to the small inclination of the orbit of Uranus it is not neces- sary that either @ or @ should be exact, if only their sum is exact. The column @' of this table, therefore, gives the correction which must be applied to the motion of @ at the end of a century (1950) in order that, being applied to @ alone, w+ 0 may be exact. ‘This correction is, in fact, that for the secular variation of the precession. Tables IV and V give the motion of the arguments for days and hours. The motion for hours is, however, not necessary in the case of any argument but g, as all the others can be readily enough interpolated to fractions of a day. Table VI gives the corrections to the arguments on account of the terms of long period from 1000 to 2200. The terms in question are, in the case of Jupiter, the great inequality produced by the action of Saturn, in the case of Neptune the great inequality produced by Uranus, and, in the case of Uranus, the inequalities in the mean longitude tabulated in the preceding chapter. ‘The numerical expres- sions are éJ = 0.535 sin (116° 21’ + 40° 45’ 20” 7’) Orel ON — 0a éd. The corrections to the several arguments are eg = él dary l= éJ—él _darg. 2=— cl n) arg. — cl —d)N=1.75 8 No correction to the mean longitude of Saturn is applied, all its inequalities being taken account of in the terms of the second order. The corrections, expressed in seconds, have been reduced to units of the argu- ment by dividing them by 2160”. Outside the limits of the table these corrections must be computed from their formule. Table VII gives the equation of the centre, and the elliptic part of the logarithm of the radius vector. No constant is applied to the former, but the latter is dimi- nished by .0003400, the sum of the constants added to (p.c.0) in Tables VITT, IX, X and XVII. 192 THE ORBIT OF URANUS. The formule for the Tables are Equation of centre = Elliptic log. x = + -+ + + 19352”.06 sin 67 .24 sin 23 .05 sin 3g 1.07 sin 4g 0 .05 sin 5g 1.283345 .0003400 .0203618 cos g .0007165 cos 2g .0000318 cos 3g .0000016 cos 49. Table VIII gives the coefficients (v.c.0), (v.c.1), ete., for the perturbations of the longitude and logarithm of radius vector produced by the action of Jupiter. They are computed from the periodic terms of the formule on page 83, with the addi- tion of the following constants to make all the numbers of the table positive: Constant of (v.c.0) = 55”. make it positive. not amount to a tenth of a second until after the year 2000. added to the quantities of these tables to make all the numbers positive, are: (os) Oe (@.c1) sa: 4. (v8 2)= 0.20 (vic2)\—— 40220 (p.c.0) = 1200 (p.s.1) = 150 (p.c.1) = 100 (p.8.2) = 10 (p.c.2)= 10 Table IX gives the periodic part of the coefficients due to the action of Saturn, taken without change from the expressions on page 84, together with the secular variations, the latter including only the terms of (v.s.1), (v.c.1), (v.s.2), and (v.¢.2), which are multiplied by 7’ and by sin A, or cos Ag. in the columns Sec. Var. and each number is increased by the constant 1”.50 to The term —0"’.067 sin A, in (v.¢.0) is omitted entirely, as it will The constant terms ” Constant of (v.c.0) = 30. (v.s.1) = 150. + 1.507 (v.c.1) = 150. + 1.507 (v.s.2) = 130. + 1.50T (v.c.2) = 130. + 1.507 (8:3) — "8: (@icis) — 6: (usd) = Gk @ic4) ee: The coefficients of T are given THE ORBIT OF URANUS. 193 Constant of (p.c.0) = 800 (p.8.1) = 1500 (p.c.1) = 1500 (p.s.2) = 400 (p.c.2) = 400 (p.s.3) = 100 (p.c.3) = 100 Table X gives the coefficients produced by the action of Neptune, computed from the periodic terms on pages 85 and 86 without any other change than the multiplication of all the numbers by the factor 0.863 to reduce them to the new mass of Neptune. ‘The constants added to the several quantities, are ” Constant of (v.c.0) = 92.85 @isil)\i= 20.00 (wcl)\= 31.00 @is2)— 5.00 (eie22) = 5.00 (v.s.3) = 1.00 (v.c.3) = 1.00 (v.s.4) = — 1.00 (v.c.4) — nO) Constant of (p.c.0) = 400 (p.8.1)= 200 (p.c.1) = 200 (p.8.2) — 40 (p.c.2) = 40 Tables XI to XVI give the terms of the second order and of short period which contain the products ot the masses of Jupiter and Saturn, which, with the constants added to the numbers of the several tables, are as follows: (v.c.0) = + 0.08 sin A; + 0.51 cos A,;; Table XII; const = 0".60 + 0.04 sin A, -+ 0.01 cos A,; DUDE — (2.05 — (0.01 sin A, + 0.05 cos A;; XGA EE ——a(ROo —= (j) 2) sin Al, = I! 210 Gos Ales XV; le — 0.05 sin A, + 0.03 cos A,; XaVali- = 0210 Sum of constants added to these tables 215 (v.8.1) = + 0°.26 sin A, + 0”.27 cos A,; Table XI; const = 0’.40 (R04 sin As —— OMIviicoseAbs PXGIIE ==) 20) + 0.08 sin A, + 0.03 cos A,; XIII; =—=0)a0 -—(0 .02 sin A, + 0.08 cos 4,; NGI A: =—().10 + 0 .30 sin A, — 0.58 cos A;; EXO 0.75 —|()),04 sim Aj; XGVaE 0.10 25 June, 1873. 194 AVISLID, COIR IS ICAL OPIN OPT ALIN WO IS}. (v.c.1) = + 0’.06 sin A, — 0".27 cos A,; Table XI; const = 0’.40 + 0.18 sin A; + 0.01 cos A;; ite 0 .20 — 0.03 sin A, + 0..08 cos A;; )GUUEs 0.10 — 0.02 sin A, + 0.09 cos A,; SEV; 0.10 — 0.44 sin A, — 0.61 tos A,; XV; 0.75 — 0.10 sin A, + 0.03 cos A,; NSVale 0.10 Sum of constants added to (v.s.1) and (v.c.1) in these tables 1.65 The term of (p.¢.0) 11 sin A, — 3 cos A, is omitted from the tables entirely. Tables XVIla and XVII6 give the constant, secular, and long-period terms of (v.s.1) (v.c.1), computed from the formule p. 186, with the following additions: 1. The constant terms introduced by the perturbations, given on p. 187. 2. The negatives of the constants added to the tables VII to X VI inclusive to make the numbers of those tables positive. The values of these terms are Pert. Const. Tables VIII to XVI. (1)—() (0.8.1) 0 177.65 + 1.507 — 177’.65 — 1".50T ess) 0 186 .65 + 1.507 — 186 .65 — 1 .50T (v.s.2) — 0.14 135 .20 +. 1.507 — 135 .34—1 507 (v.c.2) + 0.13 155 .20 +. 1.507 — 135 .07— 1.507 (v.8.3) 0 9 .00 — 9.00 (v.6.9) 0 7.00 — 17.00 (p.c.0) [+1972] —1000 + 1000 (p-8.1) + 63 1850 — 1787 (p.c.1) + 75 1800 — 1727 (9.8.2) + 935 450 — 445 (p.c.2) -h 4 450, — 446 The perturbation constant of (p.¢.0), being added to log @ in forming the elliptic radius vector, is not included in this table. Table XVIII gives the reduction to the ecliptic — 9".37 sin Qu. The constant 10” is added to make the numbers always positive, which constant has been already subtracted from 0. Table XIX gives the principal term of the latitude 4G’ 20.54 & sin w. Table XX gives the coefficients (6.8.1) and (b.c.1) for the perturbations of the latitude produced by Jupiter. They are given by the formule (6.8.1) = 0.65 cos (J — U + 40°) (6.c.1) = 0.65 sin (J — U + 40°) The constant 0.70 is added to make all the numbers of the table positive. TE BOOB ORR TW RAN TUES: 195 Table XXI gives the corresponding coefficients for the action of Saturn, com- > puted from the expressions on p. 87 with the addition of the following constants. Const. of (b.c.0) = 0.10 (6.8.1) = 3 .30 (b.c.1) = 3 10 (6.8.2) = 0 .20 (62) —10F-20 Table X XII gives the coefficients for the action of Neptune from the formule on p. 87, all the numbers being multiplied by the factor 0.863 to reduce them to the adopted mass of Neptune. ‘The following constants are added : Dion (O:e10) orn O06 (4.8.1) 1 .00 (6.0.1) 1 .20 (0.5.2) 0 .20 (b.c.2) 0 .20 Table XXIII gives the secular and long-period terms for various epochs com- puted from the formule of p. 189. The sums of the several constants added in the three preceding tables are here subtracted again so that these expressions become 6.2.0 —Se= 07.10 —0’.12T— 011d, 4+ .046dx (4.8.1) — Se = — 5 .00 — 0.227 — 0".05T? + .975dy + .2218% (6.c.1) —Se= — 5.004 2.4774 0.127? + .221dy — .975dx (6.8.2) —Se= — 0.46 —0.01T+ .046dy + .011dx (6.c.2) — Se = — 0.414 0.1274 0116, — .046dx Precepts for the use of the Tables. Express the date for which the position of Uranus is required in years, months, days, and hours of Greenwich mean time, according to the Julian Calendar if the date is earlier than 1500, according to the Gregorian Calendar if it is later than 1600, and according to either calendar between these epochs. Enter Table I with the beginning of the century, and take out the values of g, a, o, $, 0’, and arguments 1 to 9. Multiply w’ and ¢’ by the fraction of a century corresponding to the date, and write the products with their proper algebraic signs under @ and @. If the calendar is the Julian, the century marked J must be taken, and if the Gregorian, that marked G. Between the dates 1752 and 1951 it is not necessary to enter Table I at all. If Table I was not entered, enter Table II with the year, or the first preceding year found therein. If Table I was entered, enter Table Il between the year 1800 and 1896 as if the number of the century were changed to 18. Take out the values of g, o, 6, and the arguments, and write them under the corresponding quantities from Table I. Enter Table III with the excess of the actual year over that with which Table IT was entered, and with the month. Write the corresponding values of g, @, 0, and the arguments under the previous values. Multiply @’ by the fraction of a 196 THE ORBIT OF URANUS. cen ury after 1850, corresponding to the date with which Table II was entered, and write the product under 6, or add it to it in writing #. If Table IL was entered with a date before 1850, this product is negative. Enter Table 1V with the day of the month and write down the corresponding values of g, w, etc., under the former values. If the date does not correspond to Greenwich mean noon, the motion of g for the hours must be computed from Table V, and the other quantities must be interpolated to the fraction of a day in entering ‘Table IV. Enter ‘Table VI with the year, find by interpolation the values of g, and argu- ments 1, 2, and 3, corresponding to the date, and write them under the former values. Add up all the partial values of g, o, 0, and the arguments, attending to the algebraic signs of the products. Subtract from the arguments as many times 600 as possible, and the results will be the final values of those quantities, Enter ‘Table VII with g as the argument, the seconds being first reduced to frae- tions of a minute, and interpolate the quantities Hand log r, When g exceeds 180° the former quantity is to receive the negative sign; the latter is always positive. Enter Tables VIII to XVI inclusive with their respective arguments, and take out the values of the quantities (v.c.0), (v.s.1), (v.c.1), ete., (p.¢.0), (p.s.1), ete., so far as they are found in the tables, writing the quantities having the same desig- nation under each other. In ‘Table 1X the quantities Sec. Var. must be multiplied by the centuries and fraction of a century of the actual date after 1850, and the product must be included with the corresponding quantities, (v7 s.1), (v.c.1), ete. Before 1850 this product will always be negative; afterward always positive, All the quantities taken from these tables are positive except (v.s.4) and (v.¢.4) in Table IX, which are negative. Add up all the partial values of (v.c.0), (v.s.1), ete., thus obtained from Tables VIII to XVI, and from their sum take the corresponding quantities obtained from Table XVII by interpolating to the date. ‘The required quantities are all given in Table XVIL6; Table XVIla@ being only an expansion of a part of XVIIb for the present century. The final values of (v.s.1), (v.c.1), (w.s.2), ete., (p.s.1), (p.c.1), ete., thus obtained are to be multiplied by the sines and cosines of the correspond- ing multiples of g, im doing which four place logarithms are sufficient if the com- putation is carefully made. ‘The products are then all added together, and to g, a, Z, and (v.c.0); in the case of v, and to log. 7, (p.c.0) in the ease of p. ‘That 1s, we are to form the expressions: ; u=gto+ E+ (vc.0) + (v5.1) sin g+(vc.1) cos g + (v.s.2) sin 2y + (v.6.2) cos 2g + Cie. + etc. log r= log r (from Table VIT) ++ (p.c.0) + (p.8.1) sin g) -+(p.c.1)cos g + (p.s.2) sin 27) -+ (.¢.2) cos 2g + (p.s.3) sin3g) + (p.c.8) cos 39. THE ORBIT OF URANUS. 197 wu will then be the true argument of latitude, and log + the logarithm of the radius vector with seven places of decimals. Under wu write @; enter Table XVIII with the argument w and take out the reduction to the ecliptic. Add it to u and @, and the sum of the three quantities will be the heliocentric longitude of Uranus referred to the mean equinox and ecliptic of the date. Applying nutation the longitude will be reduced to the true equinox. Enter Table XIX with w as the argument, or, when uw exceeds 180°, with wu — 180°, and take out the principal term of the latitude, which will be positive when w is less than 180°, and negative when it is greater. Enter Tables XX, XXI, XXII, and XXIII with their respective arguments, the argument for the last being the date, and add up the various quantities having the same designation, noticing that in the first three tables all the quantities are positive, while in the last they are all negative except (b.c.0). Then form the expression, (6.c.0) + (6.8.1) sin g + (6.¢.1) cos g + (6.8.2) sin 27 + (b.c.2) cos 29, and add it to the principal term of the latitude, with regard to the algebraic signs. The sum will be the heliocentric latitude of Uranus above the ecliptic of the date. When an ephemeris of Uranus is to be computed for a series of years, some modifications may be introduced, which will save the computer labor. In the first place an equidistant series of dates being selected for computation, it will be suffi: cient to compute g, a, 6, and the arguments for every sixth, eighth, or tenth date, and to fill in the arguments for the intermediate dates by adding the nearly con- stant differences corresponding to the adopted intervals. The agreement of the numbers thus obtained for the last date with those found by the original computa- tion will prove the whole process. ‘This interval may be as great as 120 days without detracting from the accuracy with which the places for the immediate dates can be interpolated, and the differences for this interval may be deduced from the numbers at the bottom of Table II. If these numbers are used without change the values of @ and 0 for the last date may not always come out right. But these errors, if less than a second, will be of no importance if the one quan- tity comes out as much too great as the other is too small, and they may be avoided entirely by making a small change in the constant difference to be added. Tables XI to XVI, inclusive, need be entered only for every third or fourth date, and the sums of the quantities can be then interpolated to every date, and added up with the corresponding quantities from the other tables. Again, it will be found convenient to compute the sum of the small terms (v.s.3) sin 3g + (v.c.3) cos 37 + (v.s.4) sin 49 + (v.c.4) cos 4g, as well as the corre- sponding terms of the radius vector, and all the terms of the latitude, not for the dates adopted, but for every fourth entire degree of g. Having a series of values computed in this way, the sum can be interpolated to the value of g corresponding to the date. ‘To facilitate the formation of the smaller products for entire degrees of g, a table of products of numbers by the sine and cosine of every degree is appended to these tables, by which the products in question can be formed at sight 198 THE ORBIT OF URANUS. whenever the coefficient to be multiplied is less than 32’. The values of these coeflicients, (v.s.3), (v.c.8), ete., corresponding to the entire degrees of g, may be either formed by interpolation at sight from those corresponding to the dates of computation, or the values of the arguments 2 and 3 corresponding to the required degrees of g may be computed, and the values of (v.s.3), etc., correspond- ing to these values of the arguments may be taken from ‘Tables LX and X, while Table X VII must be entered with the corresponding dates. If the heliocentric ephemeris is computed for ten years at a time, the last of these modifications in the mode of computation will greatly facilitate the computa- tion of the smaller terms. We first find the date, and the values of arguments 1, 2, and 3, to one place of decimals, for some entire degree of gy preceding that which corresponds to the first date, and then find the dates and the values of the arguments corresponding to successive values of g, differing by 2° or 4°, until we pass the last date of computation. We then take out the values of (v.s.3), (v.¢.3), (v.s.4), (v.c4), (p.8.3), (p.c.3), (b.c.0), (b.8.1), (6.¢.1), (6.8.2), and (6.c.2), with these values of the dates and arguments, form their products by the sines and cosines of the corre- sponding multiples of g by means of the supplementary tables, and add the proper products together so as to form three small tables with g as the argument. These terms are then interpolated to the values of g corresponding to the original dates of computation. As a first example of the use of the Tables we will compute the heliocentric co-ordinates of Uranus for Greenwich mean noon of the date 1753, Dec. 3. In computing the arguments we shall make use of Table I, though it is not necessary to do so. The computation of the arguments is as follows: / aA Table I, 1700 5 , 30g 33 |) $8 29 11.47 456.092 Product by 0.5392 +3.¢ —4.45 Table II, 1852 28 5 2.0: 3 14 “35ei0 306.010 ER Yen Dee: é 2) 4353! 08 Q 35.59 83.252 IV, 3 days 2 ie .26 0.15 0.357 VI, 1753.92 ‘ see 0.507 1753, Dec. 3 .48 8 -O¢ 2 44 17.87 246.218 Table I, 1700 *31¢ 249.975 II, 1852 8.7.85 95.880 10M, Ws ie IDE, .o46 6.709 IV, 3 days BIO -029 VI, 1753.9 A +.024 1753, Dec. 3 506.548 } { (v.c.0) (v.s.1) THE Table VIII 83.58 9.97 Ix 21.32 22.24 22 xX 71.08 19.02 1 ORBIT OF URANUS. 199 (u:c.1) (v8.2) (v.c.2) (v.s%8) (v.c.8) (u:s.4) (v.c.4) 6.61 0.37 0.31 3.49 60.92 238.89 1.42 11.07 0.32 1.64 8.30 6.64 9.38 0.70 1.19 —0.95 —1.04 XI a AT XIE 13 36 10 XII 06 1 Ol XIV 01 04 03 MNS (40 12 XVI 12 09 Bi > 178.68 53.36 250.24 67.93 244.58 2.12 12.26 Table XVII —292.55 —140.91 —140.73 —131.03 —9.41 —6.85 — 239.19 4109.35 — 72.80 4113.55 —7.29 5.41 —0.63 £0.60 . (Gc 0) (nea) (p.¢.1) @s2)) (pre2)) Gre3) Gp-€23) Table VIII 251 188 32 DX: 1086 741 360 621 597 154 111 x 326 259 222 21 42 1663 1188 614 642 639 154 Tull XVII 1104 — 1321 —500 —412 —360 —98 —94 2767 — 133 +114 +230 +279 +56 il, log (v.s.1) — 2.3787 log (v.¢.1) log sin + 9.2994 log cosg log (p.s.1) — 2.124 log (p.e.1) ° / tad g 168 30 380.48 ray Ste E59 E en Ome Geil (v.¢.0) 2 58.68 (v.s.1) sin g —47.66 (v.e.1) cos g ls (ED (v.s.2) sin 2g 28.43 (v.¢.2) cos 29 1 44.55 (v.s.3) sin 3g aS (v.€.3) cos 3g — 4.46 (v.s.4) sin 4g 0.47 (v.0.4) cos 4g 0.41 u 263 40 57.42 6 72 44 17.87 R 7.95 Long. mean Eq. 336 25 23.24 Nutation Porn Long. true Eq. 336 25 29.75 + 2.0388 log (v.s.2) — 1.8621 log (v.e.2 + 2.0552 — 9.9912 log sin 2g — 9.5916 log cos 2g + 9.9641 + 2.057 log (p.s.2) + 2.362 log (p.c.2) + 2.446 (b.s.1) (b.c.1) (6.8.2) (b.€.2) Table XX " 06 0.62 xXXxI 3 .91 5 .88 0.41 0.17 XSXeT IT .84 I 8X5 0.18 0.17 4.81 7 .86 0.59 0:34 XT $428 —6 .88 —0.43 (I) fl J ee esa ON log r Table VII 1.3023222 oa Table PXoXe —— OF 4 Geese (p.¢.0) 2767 (b.c.0) Table XXT 0.16 (p-S-1) sing. — 26 XXII 0.09 (p.c.l) cosg —I112 XXIII 0.20 (p.s.2) sin 2g — 90 (6.s.1) sin g +0) .11 (p.c.2) cos 2g +257 (b.c.1) cos g —0 .96 (p-8.3) sin 8g + &l (5.s.2) sin 2g —0 .06 (p.c.3) cos 83g — 14 (b.c.2) cos 2g —0 .16 log r 1.326035 Latitude —0 46 4.35 As a second example we will take the computation of an ephemeris for the years 1876 and 1877. We take as the extreme dates 1875, December 15, and 1878, April 3, between which are seven intervals of 120 days each, which we adopt as those We first torm the arguments for the extreme dates as follows: of computation. 200 THE ORBIT OF URANUS. I Bor sid. Deck ld) —isia.0G6: Arg. 1 5 Table II, 1872 : t De 3 574.815 HS Yer Dec: ) ’ 06 2. 170.072 IV, 15 days 3 1.784 VI, 1875.96 + 0.426 For 1875, Dee. 15 3¢ : é t wl 147.097 Table II, 1872 68.32! 165.888 100G Bx, Wee, Tht 13.705 IV, 15 days 4: 0.144 WAI 1875.96 — For 1875, Dee. 15 320.65! Wise Il. For 1878, April 3 = 1878.26. g @ ) Arg. 1 ° / ” / ” ° / W Table II, 1876 331 34 18:70 95 11 42.33 To) 222) DROS 148.576 Tit, 2 ¥. April 9 37 53.62 OF al ak23 O WO chins 97.643 Vv, 3d 2 6.70 0.26 0.15 .B5T VI, 1878.26 DIG Se dactewaeey | alll Me ee aes 421 For 1878, April 3 341 14 21.64 95 12 53.82 73 22 42.93 246.997 | Arg. 2 3 4 5 6 7 8 9 Table II, 1876 321.237 179.889 153 168 489 123 401.9 57 III, 2 Y. April 29.732 7.868 22 7 37 42 12.1 45 IV, 3d 109 -029 0 0 0 0 0 0 VI, 1878.26 == 00 Hpst= 002 For 1878, April 3 351.077 187.788 175 175 526 165 414.0 | 102 We now fill in the values of g, the arguments 1—9, and the times with which Table XVII is to be entered, for the intermediate dates, by adding the nearly constant differences deduced from the numbers at the bottom of Table II, The seconds of g are first reduced to fractions of a minute, with which to enter Table VII. In making the subsequent computation we have used none of the devices previously described except in the case of the small longitude terms, as follows: THE ORBIT OF URANUS. 201 ° ° ° ° ° ° ° g 330 332 33 336 838 340 342 3g 270 276 282 288 294 300 806 4g 240 248 256 264 272 280 288 ” " 7 fr ” wr r (v.s.3) —1.14 —1.12 —1.13 —1.16 — 1.23 — 1.33 — 1.48 (v.c.3) — 2.46 — 2.56 — 2.66 — 2.76 — 2.88 — 3.01 — 3.16 (v.s.4) 0:59 40.60 +4059 4055 +4050 +043 +0.36 (v.¢.4) — 0.10 — 0.20 — 0.30 — 0.40 — 0.47 — 0.57 — 0.64 Graspsamieg “U4 2119 LI FLV fig 2115 +£1.90 (v.c.3) cos 3g 00 — 0.26 — 0.55 — 0.85 —1.17 — 1.50 — 1.86 (v.s.4) sin 4g — .51 — - .56 — dT — .55 — .50 — .42 — .34 meecos gam 05) tecO% ate 0% | to04) == 08) 22 510 =, 119 Sum + 0.68 + 0.37 + 0.06 — 0.25 — 0.57 — 0.87 — 1,19 It will be seen that we have here computed twice as many numbers as are necessary to interpolate with all attainable accuracy. The rest of the computation is fully given on the four following pages. First we have the values of g and the nine arguments for the intermediate dates, filled in by successive addition of the nearly constant difference. ‘The arguments thus obtained for the last date may be compared with those just computed on the pre- ceding page. The numerals in the first columns of the sections of computation following indicate the arguments with which tables are entered to obtain the separate values of the quantities (v.c.0), (v.s.1), (v.c.1), ete. The negative terms in Table XVII being taken from the sum of all the periodic terms from Tables VIII to X VI with argument 1 to 9, we have the final values of (v.c.0), (v.s.1), ete. The final computation of the products (v.s.7) sin ig, etc., and the addition of the separate terms which make up the three co-ordinates, are shown on page 205. ‘The expressions c.0, s.1, etc., are employed for brevity, instead of (v.c.0), (v.s.1) sin g, ete. The longitude finally given by the tables is referred to the mean equinox, and must therefore be corrected for nutation before being used to compute the geocentric place. 26 June, 1873 902 THE ORBIT OF URANUS. Dat 11875, Dee. 15 1878, Apr. 13) Aug. 11 Dec. 9 1877, Apr. 8 Aug. 6 Dee. 4 1878, Apr. 3 ares { 1875.955. | 1876.28. 1876.612. 1876.940. 1877.269. 1877.597. 1877.925. | 1878.254. = / ° / ° / ° / ° / ° / ° [/ ° , 331 23.086'332 47.554/334 12.021/335 36.489/337 0.957/338 25.425/339 49.893/341 14.36] Arg. 1 147.097 161.368 175.639 189.910] 204.182} 218.453} 232.794] 946.995 2 320.655 325.001 329.346 333.692 338.038 342.383 346.729 351.074 | 3 179.739 180.889 182.038 183.188 | 184.338 185.488 | 186.638] 187.788 4 153 156. 159. 163. 166. 169. 2, 175, 5 UGies J Bee eases? lh eee i el eee ag | 175. 6 489. 494, 500. 505. 511. 516. 521. 597. 7 123. 129. 135. 141. 147. 153. 160. 166. 8 401.6 403.4 405.2 407.0 408.7 410.5 412.3 414.1 Oe 56. 33. 69. 15. 82 S85 95. 101. I (v.c.0) 1 108.01 107.74 106.29 103.68 99.98 95.25 89.62 83.21 2 1E.80 14.71 13.66 12.65 11.68 10.76 9.90 9.09 3 72.43 73.60 TAT 15.94 77.10 78.26 79.42 80.56 5 59 58 58 57 56 56 55 54 6 02 02 .02 .03 .03 .03 .03 03 7 07 07 .06 .06 .06 .06 .05 0 8 1.68 1.65 1.63 1.60 1.58 1.55 1.53 1.50 9 09 Ooee 09 09 08 08 08 07 (v.¢.0) 198.69 198.46 197.10 194.62 191.07 186.55 181.18 175.05 I (v.s.1) 1 6.84 7.68 8.40 8.99 9.44 9.74 9.91 9.98 2 132.20 125.48 118.82 1D. 105.67 99.24 92.89 86.69 sec. 2 18 .20 99 23 25 Oh .29 31 3 1.09 1.07 1.08 1.12 MI) 1.29 1.41 1.56 4 13 AG lil 10 10 09 09 .08 5 19 .19 19 19 19 .20 .20 20 6 04 04 04 .05 .05 .06 .06 07 7 14 14 13 3 12 19 sill 10 8 1.28 1.28 Oy 1.26 1.26 1.25 1.24 1.23 9 08 -08 .08 .08 07 07 07 07 > 142.17 136.28 130.34 124.36 118.34 112.33 106.27 100.29 Tab. XVII] —154.21 ]—153.84 | —153.47 |—153.10 | —152.73_ | —152.36 |—151.99 | —151.62 (v.s.1) — 12:04 |= 17.56 |= 93:13) | —= 28274 ||| —= 34°39 3|/== 40.03 || 45s 72a eee (v.c.1) 1 5.64 6.03 6.38 6.67 6.85 6.91 6.82 6.59 2 5.01 6 00 1-3 8.90 10.80 13.00 15.49 18.25 sec. 2 69 iil 12 14 o15 sitit af) .80 3 23.20 24.09 24.95 25.82 26.68 27.538 28.37 29.19 4 AT 48 49 50 .50 51 52 5g 5 esi 3 3 a3 iT 37 BT 53 81 6 16 sty ally sie 18 18 sil 18 1 15 14 .13 13 1) 19 ala 10 8 0.66 0.64 0.63 0.61 0.60 0.59 0.57 0.56 9 07 07 .06 06 05 04 04 03 > 36.42 38.70 41.21 43.98 46.90 50.02 53.26 56.59 Tab. XVII} —205.86 | —206.08 | —206.29 | —206.51 | —206.73 | —206.94 | —207.15 | —207.37 | (v.c.1) —169.44 | —167.38 | —165.08 | —162.53 | —159.83 | —156.92 | —153.89 | —150.78 H(v.8.2) 1 06 10 15 21 26 ell 34 37 2 54.19 49.37 44.78 40.37 36.15 32.11 28.30 24.70 see. 2 .B9 42 44 AT 49 51 53 56 3 2.05 2.19 2.31 9.45 2.60 2.76 2.91 3.07 fits 56.62 52.08 47.68 43.50 39.50 35.69 32.08 28.70 | Tab. XVII} —134.34 | —134.32 | —134.31 | —134.29 | —134.27 | —134.26 | —134.24 134,22 | (v.8.2) — 77.72 | —= 82:94 | — 86.63° | —— 90/79 |= 94 98 br Soo NG ee alome THE ORBIT OF URANUS 2038 1875, Dec. 15/1876, Apr. 13) Aug. 11 Dec. 9 1877, Apr. 8 Aug. 6 Dee. 4 1878, Apr. Date, { 1875.955. | 1876.283. | 1876.612. | 1867940. | 1877.269. | 1877-597. | 1877.995. 1878-64. ° / ° / fe} y ° / fo} / ° Lf ° / : fe} / g 331 23.086)332 47.554/334 12.021335 36.489/337 0.057/338 25.425.339 49.893/341 14.361 (v.€.2) 1 Zoi .30 .29 ~29 29 .B0 .30 On . 2 25.34 28.93 32.71 36.73 40.93 45.35 49.93 54.69 . sec. 2 -16 -16 att -18 Silt) Site) .80 .80 | 3 6.23 6.40 6.56 6.70 6.85 6.99 7.10 7.20 | lie ZOOS aes TS lg ile sal | > 32.64 36.39 40.33 44.50 48.86 53.43 58.13 63.00 Pe NOVeN === S Oro m= 136209) —=—'36560) \-—SG.62) | =136564 || —=196)65 | 136.67 136.69 @e.2) | —103.93 | —100.20 |— 96.27 |— 92.12 |— 87.78 |— 93.99 |— 79.54 |— 73.69 92 (v.s.3) 2 6 6.89 6.84 6.78 6.70 6.61 6.51 6.3 3 0.87 0.91 0.94 0.98 1.01 1.04 1.08 1.11 1.79 7.80 7.78 7.76 iva 7.65. | 7.59 7.48 Tab. XVII| —8.92 =) — 3} Oi Ol Oh 3.0011 == Sail {3-9 (v.8.3) BE || Ta =I |Swearoon ln maieacn |) wEakSon eens (v.¢.3) 2 3.19 3.12 3.05 2.98 2.90 2.82 915 2.67 3 1.34 1.3 1.84 1.34 _ 1.33 1.33 1.3 1.29 4.538 4.46 4.39 4.32 4.93 4.15) | 4.06 3.96 PAS eVell| ==7:06 —7.06 —= 06 == 0.0 —— 06 ir OR == 7e06 106 (v.c.3) ES | La a aa eS a aa (v.s.4) 2 1.48 1.48 1.47 1.45 1.43 1.40 3.36 1.33 3 N33 88 — (E83 —(.89 —0.89 (al == 0598 = on (v.s.4) +£0.60 £0.60 +0.59 10.56 40.54 40.49 +0.43 40.39 (v.c.4) 2 0.83 0.77 0.72 0.66 0.61 0.57 0.51 0.47 3 il 0) 0) — 1202 —— 05 —— eA: == 05 10) G —1.07 — 108 ee eo ||) 20185) ors | 20138) | |e” | Soe | Sone | re (p.c.0) 1 1230 1063 899 741 594 460 344 246 2 98 99 101 104 108 113 118 126 3 ll 10 9 9 8 8 8 9 Tab. XVII 968 968 968 - 967 967 967 966 966 (p.c.0) 2307 2140 1977 1821 | 1677 1548 1436 1347 (p.8.1) 1 248 240 230 221 O11 203 194 187 2 2835 2822 2807 2789. 2769 2748 2723 2696 3 173 168 165 161 158 155 151 148 S 3256 3230 3202 3171 3138 3106 3068 3031 Tab. XVII] —1984 —1987 —1989 —1991 —19938 —1996 —1998 2000 (p.8.1) 41272 1243 1213 1180 |. 1145 1110 1070 1031 (p.¢.1) 1 123 112 97 83 69 55 41 31 2 1266 1202 1138 1074 1009 947 886 827 3 66 66 67 67 68 70 72 "4 z 1455 | 1380 1302 | 1224 1146 1072 999 932 Mab exavell| e197 7 ——198)] —1986 —1990. —1994 —1998 —2002 2006 (p-c.1) — 592 | — 601 | — 684 | — 766 | — 848 | — 926 | —1003 | —1074 204 THE ORBIT | lists, Dec. 15,1876, Apr. 13 OF URANUS. Dat Aug. 11 Dec. 9 4877, Apr. 8 Aug. 6 Dec. 4 1878, Apr. 3 Be) { 1875.955. | 1876.283. 1876.612. 1867.940. 1877.269. 1877.597. 1877.925. 1878.254, ° / ° 7 F am Ss ; ae = / a Oo ’ = / ° , °o / g 331 23.086/332 47.554|334 12.021/335 36.489|337 0.957/338 25.425/339 49.895/341 14.361 (p.s.2) 2 174 181 189 196 204 213 293 232 3 26 | 25 25 24 24 24 23 23 = 200 206 214 220 228 233 246 255 Tab.XVII| —459 —459 —459 —459 —460 —460 —460 —460 (p.s.2) —259 —253 —245 —239 | —939 —293 —214 —205 (p.c.2) 2 559 568 516 584 592 60 607 3 33 33 34 35 36 37 39 > 592 601 610 619 628 637 646 Tab.XVII| —464 —464 —464 —A465 —465 —465 ‘—465 (.c.2) 4128 137 146 154 163 172 181 (p.8.3) 2 28 31 34 39 43 47 52 Tab Xcvilill: 100 ==101 ==100n iil 101 —101 —101 (p.s-3) BATS == 70 eT Go es ayy aay (p.c.3) 2 155 159 162 166 169 172 175 Tab. XVII) —102 —102 —102 —102 —102 —102 —102 (p.e.3) + 53 57 60 64 61 70 73 (b.c.0) 2 0.12 0.12 0.13 0.13 0.14 0.14 0.15 3 0.11 0.11 0.11 0.11 0.11 0.11 0.11 Tab.X XIII 0.07 0.07 0.07 0.07 0.07 0.07 0.07 (b.c.0) 0.30 0.30 0.31 0.31 0.32 0.32 0.33 (b.s.1) 1 0.30 0.22 0.16 0.12 0.08 0.06 0.05 2 0.12 0.09 0.06 0.05 0.04 0.04 0.05 20 eine Deli eee 1.12 Hats Tals 1.14 z 1.53 1.42 1.34 1.29 1.25 1.23 1.24 Pab/X XM 5.93 | 524 e584 oa ob epropes lume aie Mme (b.s.1) ET (nes | En ee ere ets | aamn (b.c.1) 1 1.21 1.14 1.06 0.98 0.90 0.80 0.70 2 0.96 1.06 1.17 1.28 1.39 1.51 1.63 3 | 1.01 | 1.00 1.00 0.99 0.99 0.99 0.98 ie 3.18 3.20 3.23 3.25 3.28 3.30 3.31 Tab.XXUIN] —4.45 | =—444 | —443 | =443 | —a4a | —4a at (b.c.1) | —1.27 | —1.24 —1.20 S18 Si Shi Tei) (b.s.2) 2 0.02 0.02 0.03 0.04 0:04 0.05 0.06 3 0.16 0.16 0.16 0.15 0.15 0.15 0.15 Tab X XU) 0149" || 0148 | 014s | = 94g 04a) ne as (b.s.2) 0.30 | —0.30 | —0.29 | —0.29 | —0.29 | —0.98 | —0.27 (b.c.2) 2 0.31 0.32 0.33 0.34 0.34 0.35 0.36 22 0.10 0.10 0.10 0.10 0.11 0.11 0.11 Tab. XXIII} —0.38° | =0.38) ==0.38 | 0:98 ||) =-088 | Storeen |) =o1as (b.c.2) +0.03 +0.04 | $0.05 | +£0.06 40.07 40.08 40.09 | 1875, Dec. 15 1875.955. 1876, April 13 | 1876-283. THOR ORB ee Ok UEC ACN UiS® Aug. 11 1876.612. Dec. 9 1876.940. 1877, April 8 187.269. Aug. 6 1877.597. Dec. 4 1877.925. 1878, April 3 1878.254., g log (v.s.1) sin g log (p.s.1) log (v.e.1) cos g log (p.c. 1) log (v.s.2) sin 2g log (p.s.2) log (v.c.2) cos 2g log (p.c.2) b ) DD RHHoeS Cc s. Cc. s. c. ww +. a ~— u 6 R ° , 331 23.086 —1.0806 —9.6803 +3.1045 —2.2290 9434 2.718 .8905 9247 413 —s-. ( +-9.7335 29), , 331 23 5.16 95 11 41.00 —2 42 49.15 3 18.69 5.17 24 28.065) 1 ° } 832 47.554 | —1.2445 —9. 6601 3.0944 .0009 . 1649 SBI fo} 25.066 | ° ! 334 12.021 —1.3642 —1.6386 3.0838 °o 1 33.225 334 40 | Suse] 8.46 3.03 | 8.86 3.88 8.3 20 4 95 2 ¢ 36.489 —1.4585 —9.6159 3.0719 —2.2109 9.9594 —2.884 -9580 | .8763 .318 - 9644 .8189 -LST " 8.067 301 21 -16 -62 -8T 8.00 3.28 yl 19 bo RSS no bo 6) LOWS He b+ bo me ho 3 / 0.957 .9365 —9.5916 3.0588 ° 337 —2.2036 9.9641 —2.928 —1.9766 -8566 —2.365 —9 —1.9434 9.8421 2.212 ° Uy ” 1 24 28.068 337 0 57.424 95 12 22.61 116; G28)33 | 3 11.07 13.48 27.13 8.12 1.03 0.41 =9 n = ° , 338 25.425 6024 .9655 0453 1957 -9685 967 -9937 9.8350 .348 .9202 .8630 .236 ” -068 no ~~ bo ive) ee bo ~ Co Or bo or bo GC bo noe SOMMRAB SS ATR OATS Woownrmnwrsd 7) 6353 2.28 73 22 0.15 2.58 89 3.26 259i 66 56 73 22 59.1% 12.37 0.20 68 8.48 3.61 GO) Te Wr galb} 73 22 24.60 3.98 TL 33 33.45 73 22 ¢ ° ! 339 49.893 —1.6601 —9.5375 3.0294 —2.1872 9.9725 —3.0013 —2.0093 —9.8111 —2.330 24 28.069 49 53.560 12 43.42 57 40.55 Seles 15.76 —2 24.48 I G18} — 59.87 — 0.83 73 5 54.37 73 22 36.82 4.79 Longitude | 37 15 5.01 | 847 7.06 |1 40 19 14. ol 27.77 1143 93 45.13 144 56 8.56 146 28 35.98 log r, c.0 Rall all 8.2 1.2647392 2307 —609 —458 +218 -- 69 + 73 sea .2644735 2140 —568 —)35 206 80 69 8 1.2642196 1977 —528 —615 192 91 65 13 .2639778 | 1821 —457 —b9T 179 101 60 19 1.2637486 1677 —447, —180 167 113 54 2 23 1.2635319 1548 —408 —861 152 126 49 29 1.2633280 1436 —369 —941 138 138 43 36 1.2648996 .2646135 1.2643391 2640774 1.2638293 1.26385954 ° 7 | 341 14.361 = 7104 —9.5073 3.0132 —2.1784 9.9763 —8.0310 — 2.0233 —9.7846 —2.312 —1.8674 9.8994 2.274 341 14 21.63 95 12 53.82 —1 49 49.66 2 55.05 16.51 —2 22.80 | 1 4.26 | — 58.45 | 1.00 74 38 19.36 | 73 22 42.93 5.22 148 1 7.51 -2631370 1347 99 — 332 —1017 nom Canin aaa (oe tl a} {? ” +41 36.64 +0.30 Sei Sant +40.25 40.02 on oasTOU MN wWwHwnndc= >» | Latitude | +41 37.87 -T 18 .63 07 22 04 ! UA 43 57.62 0.32 1.48 03 0.19 0.06 bo —) ae TAI WW OD oro WwW bh 44 41.19 0.33 1.29 —1.00 0.16 0.07 ia) 43 58.64 | 44 21.15 44 42.04 206 THE ORBIT OF URANUS. TABLE I.—Correcrions or ARGUMENTS FOR PAST AND FUTU TABLE II. Century. g @ ° Ud ” ro) ' ” 0J | 207 15 59.32 | 848 52 17.36 100 | 275 45 34.18 | 344 46 54.12 200 | 344 15 9.04 | 345 41 24.88 300 52 44 43.90 | 346 35 49.64 400 | 121 14 18.76 | 347 30 98.40 560 | 189 43 53.62 | 348 24 21.16 600 | 258 13 28.48 | 349 18 27.92 700 | 326 43 3.34 | 350 12 28.68 800 35 12 38.20 | 351 6 23.44 900 | 103 42 13.06 | 352 0 19.90 1000 | 172 11 47.92 | 352 58 54.96 1100 | 240 41 22.78 | 353 47 31.72 1200 | 309 10 57.64 | 354 41 9.48 1300 17 40 32.50 | 355 34 97.94 1400 86 10 17.36 | 356 27 46.00 1500J | 154 39 42.29 | 357 20 58.76 1500G) 154 32 39.89 | 357 20 57.89 1600 | 223 92 14.75 | 358 14 4.65 1700 | 291 31 7.87 | 359 4 5.33 1800 0 0 0.00 0 0 0.00 1900 68 28 52.63 0 52 48.67 2000 | 136 58 27.49 1 45 31.43 2100 | 205 97 20.11 238 8.11 2200 | 273 56 12.74 3 30 38.78 RE CENTURIES, | a ; 6 are | = i ” ° , ” ” +108.00 351 6 26.89 | —148.32 408.924 102.00 351 34 55.39 | —140.08 552.952 96.00 aa 3) SPL iR —131.84 96.980 90.00 352 32 17.11 | —123.60 241.008 84.00 BOOM OsS Sie alonS.o 385.036 + 78.00 353 30 11.79 | —107.12 529.064 72.00 353 59 21.49 | — 98.88 73.092 66.00 354 28 39.43 | — 90.64 217.120 60.00 354 58 5.61 | —- 82.40 361.148 54.00 3855 27 40.03 | — 74.16 505.176 + 48.00 355 57 22°69) | — 65.92 49.204 42.00 356 27 13.59 | — 57.68 193.232 36.00 SOOM O enlace ead: 337.260 30.00 Sir 2 SOT |) as ZN) 481.288 24.00 357 57 35.73 | — 32.96 25.316 + 18.00 358 27) 59.59) || — 24.72 169.344 18.00 398 21 59.08 | — 2472 168.155 12.00 Sioie) hes) BIS) |) a 1 Es} 312.183 6.00 359) 29 1147 | — 8224 456.092 0.00 0 0 0.00 0.00 0.000 — 6.00 0 30 56.77 | + 8.24 143.908 — 12.00 Wo ila? 16.48 287.936 — 18.00 1633) L5s08 24.72 431.845 — 24.00 2 4 36.57 | + 32.96 575.755 —ARGUMENTS FOR THE BEGINNING OF EACH FourtH YEAR 1752—1948. Year. gy 0 Arg. 1 1752 1756 1760 1764 1768 1772 1776 1780 1784 1788 1792 1796 1800 160 177 194 211 228 245 263 280 297 314 331 348 0 15 23 8.10 10 54.09 17.09 40.08 26.07 49.06 12.06 05 58.05 21.04 1.80 or oo. 94 94 94 94 94 94 94 94 94 94 ' 6 10.48 8 46 26.74 335351533 bo bo bk bw bv Lo Lo Cap ce Capea CR cee ey ee Lo — bo ' 43 44 46 bo rss bp bw pow Ww Ss ma So oo —) Co — Lo ee w ow bo to to oO Bm at —_ aS bo Oe ~ 59 162.101 335.862 509.623 83.384 257.145 430.907 4.668 178.429 352.190 525.951 99.712 273.473 447.115 THE ORBIT OF URANUS. 207 TABLE I.—Continued. Century. 2 3 4 5 6 a 8 g OS 191.528 299.485 280 512 102 61 470.0 5 100 314.245 49.520 64 250 563 124 410.6 | 214 200 436.962 | 399.555 449 589 424 188 351.2 | 418 300 559.679 149.590 233 32T 285 251 291.7 3 400 82.396 | 499.625 18 65 146 314 | 239.3 | 9219 500 205.113 | 249.660 402 403 ‘( 373 | 172.9 | 411 600 | 327.9g@ | 599.695 187 142 468 441 | 113.5 10 700 450.547 | 349.730 571 480 329 504 54.1 209 800 573.264 99.765 356 218 190 567 | 594.6 | 409 900 95.981 | 449.800 110 556 51 31 535.2 8 1000 218.698 | 199.835 524 295 512 94 475.8 | 207 1100 341.415 | 549.870 309 33 373 157 | 416.4 | 406 1200 464.132 | 299.905 93 371 234 291 | 357.0 5 1300 586.849 49.940 478 109 95 984 | 297.5 | 905 1400 109.566 | 399.975 262 448 556 347 238.1 404 1500J | 232.283 | 150.010 47 186 417 411 178.6 3 1500G| 231.921 149.914 47 186 417 410 178.3 2 1600 354.638 | 499.949 431 524 278 473 118.8 | 201 1700 477.319 249.975 216 262 39 537 | 59.4 401 1800 0.000 0.000 0 0 0 0 0 0 1900 122.681 | 350.025 385 338 461 63 | 540.6 | 199 2000 245.398 | 100.060 169 16 329 127 | 481.2 | 398 2100 368.079 | 450.086 554 415 183 190 | AQ TES ee 59 2200 490.760 | 200.111 338 153 44 253 | 362.3 | 196 TABLE IIl.—Continued. Year. 29 8} 4 5 6 7 8 9 1752 | 481.104 | 345.855 133 348 929 213 331.6 578 1756 | 534.013 | 359.856 172 362 296 287 353.2 58 1760 | 586.921 | 373.858 212 375 362 362 374.8 138 1764 39.830 | 387.859 251 389 429 436 | 396.4 218 1768 92.739 | 401.860 290 402 495 511 418.1 298 1772 | 145.647 | 415.862 330 416 562 585 439.7 31T 1776 | 198.556 | 429.863 369 429 28 60 461.3 47 1780 | 251.465 | 443.865 408 443 95 134 482.9 537 1784 | 304.373 | 457.866 448 45% 161 209 504.6 17 1788 | 357.282 | 471.868 487 470 227 283 526.2 97 1792 | 410.191 | 485.869 527 484 294 358 547.8 177 1796 | 463.100 | 499.870 566 497 360 433 1800 | 515.972 | 513.862 5 511 427 507 208 PH E OR BILTOR SU RAN US: TABLE II.—Continued. Arg. il. 1800 5 55 ~=1-80 94 31 33.53 1804 23 3 24.80 94 33 40.39 1808 40 11 47.79 94 35 47.25 1812 ai PAN Tics) 94 37 54.09 1816 T4 28 33.78 94 40 0.93 1820 91 36 56.77 94 42 7.16 1824 108 45 19.77 94 44 14.58 1828 25 53 42.76 94 46 21.38 1832 143 2 5.76 94 48 28.18 1836 160 10 28.75 94 50 34.97 1840 Wt 18 51.75 94 52 41.75 1844 194 27 14.74 94 54 48.52 1848 211 35 37.74 94 56 55.28 1852 228 44 0.73 94 59 2.03 1856 245 52 23.15 DOr Seri 186 263 0 46.72 95 3 15.50 1864 AY) Batfl 95 5 22.22 1868 297 17 32.71 95 T 28.94 1872 314 25 55.70 95 9 35.64 1876 331 34 18.70 95 Il 42.33 1880 348 42 41.69 95 13 49.01 1884 oy Ol 2h) 95 15 55.69 1888 22 59 27.68 95 18 2.35 1892 40 7 50.68 95 20 9.01 1896 of 16 13.6% 95 22 15.65 1900 T4 23 54.48 95 24 22.20 1904 91 32 17.43 95 26 28.83 1908 108 40 40.42 95 28 35.44 1912 125 49 3.41 95 30 42.05 1916 142 57 26.41 95 32 48.64 1920 160 5 49.40 95 34 55.23 1924 — 177 14 12.40 9> 3) LS 1928 194 22 35.39 95 39 8.38 1932 >» Al 1936 228 39 21.38 95 43 21.48 1940 245 47 44.38 95 45 28.02 1944 262 56 1.37 95 47 34.55 1948 280 4 30.36 95 49 41.07 Avian 1 24 28.007 10.411 Factor T a2 + .0007 —.0001 72 72 58 |=) or oo CO TO FP WH PD SC — ee a ea ee oO To em O19 20 22 30.59 44.41 58.84 12.98 27.13 41.30 55.48% 9.67 23.88 38.10 52.33 6.58 20.84 35.11 49.40 3.70 18.01 32.34 em bo _ bo © Ww co mm oO _ o Jt ~ 25.22 39.72 54.23 8.75 37.84 52.40 6.97 21.56 6.100 +.0001 447.115 20.876 194.638 368.399 542.160 115.921 289.683 463.444 37.205 210.966 384.727 558.488 132.249 306.010 479.771 53.532 227.293 401.054 574.815 148.576 322.337 496.098 69.860 243.621 417.382 591.024 164.785 338.546 512.307 86.068 259.830 433.591 7.352 181.113 354.874 528.635 102.396 276.158 14.2715 —.0012 0 THE ORBIT OF URANUS. 209 TABLE II.— Continued. 2 3 4 5 6 7 8 9 Ren re | (i ata : | 1800 515.972 | 513.862 5 511 427 507 591.1 33 1804 568.881 | 527.863 45 524 493 582 12.7 417 1808 21.790 | 541.865 84 538 560 56 34.3 497 1812 74.698 | 555.866 124 551 26 131 55.9 517 1816 127.607 | 569.868 | 163 565 93 205 ites 57 1820 180.516 | 583.869 202 578 159 280 99.2 137 1824 | 933.424 | 597.870 242 592 225 354 120.8 217 1828 | 286.333 11.872 281 6 992 429 149.4 | 297 1832 339.249 25.874 320 19 358 503 164.0 317 1836 | 392.150 39.875 360 32 5 578 S540 457 1840 445.059 53.876 399 46 491 52 207.3 537 1844 497.968 67.878 439 59 557 127 228.9 17 | 1848 550.876 81.879 478 13 24 201 250.5 97 1852 3.785 95.880 Sali 87 90 276 NPI 177 1856 56.694 | 109.882 5aT 100 157 350 293.8 257 | 1860 109.602 | 123.883 596 114 223 425 315.4 337 1864 162.511 | 137.885 35 127 290 499 337.0 417 | 1868 215.420 | 151.886 75 141 356 574 358.6 497 | 1872 268.328 | 165.888 114 154 493 49 380.3 517 1876 321.237 179.889 153 168 489 123 401.9 57 1880 374.146 193.890 193 181 555 IG} |) ABS 137 1884+] 427.054 | 207.892 232 195 22 272 445.1 Q17 1888 479.963 | 221.893 271 208 88 347 466.7 297 1892 532.872 | 235.895 311 229 155 421 488.4 376 1896 585.780 | 249.896 350 235 221 496 510.0 456 1900 38.653 | 263.889 390 249 287 BO |) BBA 536 1904 91.562 | 277.890 429 962 354 45 553.2 16 1908 144.470 | 291.891 468 276 420 119 574.9 96 1912 197.879 | 305.892 508 290 487 194 596.5 176 1916 250.288 319.894 54T 303 553 268 18.1 256 1920 303.197 | 333.895 587 317 20 343 39.7 336 1924 356.105 | 347.897 26 33 86 417 61.4 416 1928 409.014 | 361.898 6 344 152 499 83.0 496 1932 461.923 | 375.900 105 357 219 566 104.6 576 1936 514.831 | 389.901 144 iil 285 4 126.2 56 1940 567.740 403.902 184 384 352 116 | 1417.8 136 1944 20.649 | 417.904 293 398 418 190 | 169.5 216 1948 73.557 | 481.905 262 411 485 265 | 191.1 296 4.3457 1.1500 302 eal 5.4 = -0001! | == 0002 0 0 0 0 0 0 0 0 27 June. 1873. 210 THE ORBIT OF URANUS. TABLE III.—ReEpvucrion of THE Epocus AND ARGUMENTS TO THE BEGINNING OF EACH MonTH IN A CYCLE OF FOUR YEARS. g @ 0 6 Arg. 1 Year O oan Hf OF " oi ” " January 0 OR Oe 0200 0 0 0.00 0 0 0.00 0.00 0.000 February 0 0 21 49:94 OOS 2269 ORO Rios 0.00 3.687 March 0 0 42 14.00 OQ @ Bl OO S205 0.00 7.136 April 0 We gh Be i) © got) OD O 4h633 0.01 10.823 May 0 1 25 ~Lor24 OO ORS OO TERI 0.01 14.391 June 0 1 46 59.48 OOM Salo OOM eaente 0.01 18.078 July 0 BY 3 acs} YO Tey © @ 25 0.01 21.647 August 0 2) 29) oon h 0 0 18.48 0 0 10.83 0.01 25.333 September 0 2 5k 44595 (loi 0 0 12.40 0.0L 29.019 October 0 SL 95 (i) PRT Om Omerse93 0.02 32.587 November 0 ay Bx Ol is) 0 0 26.46 Oe Seal 0.02 36.274 December 0 SE DONN AS aLO 0 0 29.06 ORO Me R03 0.02 39.842 Yearl January 0 4 1% 37.42 ) @) Siloyta OOM ESR 0.02 43.529 February 0 4 39 26.66 0 0 34.44 0 0 20.18 0.02 47.216 March 0 4 59 9.19 0 O 36.87 OPOPe21eG} 0.03 50.546 April 0 5 -20) 58:43 0 0 39.56 0 0 23.18 0.03 54.233 May 0 5 42 5.43 0 0 42.16 O10) 245i: 0.03 57.801 June 0 6 3 BLOT 0 0 44.85 0 0 26.28 0.038 61.488 July 0 6 25 1.67 0 O 47.46 0 0 27.81 0.03 65.056 August 0 6 46 50.90 0 0 50.14 0 0 29.38 0.04 68.742 September 0 T 8 40.14 0 0 52.83 0 0 30.96 0.04 72.429 October 0 if PAS) ebay ale! ORO mabe 0 O 32.48 0.04 75.997 November 0 ft Sil BG583 OO MED Sells} 0 0 34.06 0.04 79.684 December 0 8 12 43°38 ORION OOS a259) 0.04 83.252 Year 2 January 0 8 34 32.61 0 1 3.43 OPO RR SelG 0.04 86.939 February 0 8 56 21.85 OQ i Geil 0 0 38.74 0.05 90.626 March 0 9 116 4.38 Ot 8.54 0 0 40.16 0.05 93.956 April 0 ) St BR Be Osos 0 0 41.74 0.05 97.643 May 0 9 59 0.62 ) i eRe ) © eon 0.05 101.211 June 0 10 20 49.86 OPI liGro 0 0 44.84 0.05 104.898 July 0 10 41 56.86 OO eS 0 0 46.36 0.06 108.466 August 0 ll 3 46.09 Oe TES i 0 0 47.94 0.06 112.153 September 0 i 2ommoosoo: 0 1 24.50 O10) 49552 0.06 115.840 October 0 Il 46 42.33 OO) 0 0 51.04 0.06 119.407 November 0 12 Soleo is ih Wey) 00) 52262 0.06 123.094 December 0 12 29 38.57 0 1 32.40 0 0 54.14 0.07 126.662 Year 3 January 0] 12 51 27.80 0 1. 35.08 0 0 5ds72 0.07 130.349 February 0 | 13 13 17.04 Mi Syst 0 0 57.29 0.07 134.03 March 0 le} Be ).sy7( OPIS yA0220 O M0) o8si2 0.07 137.366 April 0 138 54 48.81 0 1 42.89 0 1 0.29 0.07 141.053 May 0 14° 15 =55.81 OR lee 250 Oa Seelego 0.08 144.621 June 0 14 37 45.05 0 1 48.18 On 3.40 0.08 148.308 July 0 14 58 52.05 On ORT @ I 492 0.08 151.876 Aucust ON lee Ones @) ak 8h 4k} Oa Geet 0.08 155.563 September 0 15 42 30.52 kya 0 AG 8207 0.08 159.249 October 0] 16 3 87.52 | 0 1 58.77 | 0 1 9.60 0.08 162.817 November 0] 16 25 26.76 | 0 2 1.46 | 0 1 11.17 0.09 166.504 December 0 16 46 33.76 0 2 4.06 OL 2270 0.09 170.072 THE ORBIT OF URANUS. PANY TABLE III.—Continued. Year 0 January February March April May June July August Dpwro cows owo a15 5 wone ao o Tee oOonmo -I (Je) Roa oe SiOisr) Hw Orr bo OreT OO pO Ort © eT on Or 09 Aomo wnwore& DOE NOT ll el lll oe) Ww trom Koos oO mM -11o co cS CO Oo re let mR Ob % (Sy) o S02 eco © September October November December Year 1 January February March April May June July August September October November December Year 2 January February March April May June July August September October November December Year 3 January February March April May June July August «da —“ = S rss for aororw wPonast UWNo oqooococ ooocooco eoocooe ) (oy) Oo bo bo bo He te He OO ee eo Go Oo bo 132 3.254 SBSH .391 3.514 .600 2123 9.809 932 054 140 .263 25.349 T0960 MOTT OA wBomT awwTe co i=) Wor Oo OR ob -s DAP OOP Ee BB CO CO eet co con bh NN bb APAADSD OOO Hm Hm HH OD wT ove eoooc cooce eoooco 26.472 27.595 -609 132 30.818 31.941 33.027 34.150 35.272 Ht =7 09 00 ~) oS bo bo BDHAD ANH Bb He C2 09 9 CO OD COW -1 St Co CoM MO -TeT-3-7 — coves Cw cb b Die OOO oS ~ = ) 36.3538 387.481 38.567 ~ Qn pprn whbwr st vo oS Ss) Soe S SO SO = Spee OOOM -TeT-T-T ci eo ee OOo a (one 1 39.690 40.813 41.827 42.950 44.036 45.159 46.245 47.368 September 48.490 October 49.576 November 0 50.699 December 0 51.785 oO 09 9 =o Ro OHH eo oooco coco eT wD ws 212 THE ORBIT OF URANUS. TABLE IV.—Morion or ARGUMENTS FoR Days. (or) co 2 Or ees eo ~ So — 0.036 | 0.010 0.072 | 0.019 0.109 | 0.029 0.145 | 0.039 0.181 | 0.048 0.217 | 0.058 0.253 | 0.067 0.290 | 0.077 0.326 | 0.086 0.362 | 0.096 0.398 | 0.105 0.434 | 0.115 0.471 | 0.125 0.507 | 0.1384 0.543 | 0.144 0.579 | 0.153 0.616 | 0.163 0.652 | 0.173 0.688 | 0.182 0.724 | 0.192 0.760 | 0.201 0.797 | 0.211 0.833 | 0.220 0.869 | 0.230 0.905 | 0.240 0.941 | 0.249 0.978 | 0.259 1.014 | 0.268 1.050 | 0.278 1.086 | 0.288 1S 297 oo oS OTN ROS S O-1OO aT Or 02 G2 eco Co me OF MTT DOF KS CD Lob t BG le ee ee ee ee ee ee eo essss sesss sssss cesses sesssesse Om; PRR RR eR COO tOtON Ww PD WNW NNR PR eee eR Ce ee ee ee ee ee) Sno bo bo bo bo BO BO ND ORR See ee Oe ee ee om mM OoOOoCOomlCUCCOCCCOCcCcCOOCOULUCOCcCOCcCOC OO te OR OR RR Re RR RK OOO OO OOO COO TABLE V.—Motion or g ror Hours. Hours. J Hours. g llours. g Hours. yr Wt y} DOm WLS The period of arguments 1 to 9 is 600. In January and February of those years which, though divisible by 4, are not leap years, namely, 1700, 1800, 1900, 2100, etc., Table TV must be entered with a number 1 greater than the real day of the month. THE ORBIT OF URANUS. 213 TABLE VI.—Correcrions or ARGUMENTS FoR TERMS oF LONG PERIOD. Year. g 1 Dy Year g it 6) 3 / dd ; / Wt 1000 |+36 51.00 |—0.614 |—1.024 | +1.792] 1550 | +5 9.73 |—0.200 |—0.144 | 40.952 1010 | 36 6.49 0.621} 1.003} 1. 1560; 4 49.70 | 0.153] 0.134] 0.236 1020 | 35 29.14 0.629 | 0.983 ue 1570 4 30.32 0.106} 0.125 0.219 j 1030 34 37.97 0.638 0.962 12 f 1580 4 11.57 0.060 0.117 0.205 1040 33 54.00 0.648 0.942] 1.6 1590 3 53.48 |—0.013 0.108 0.189 H 1050 33 10.25 |—0.659 |—0.922 | +1.614} 1600 | +3 36.03 |+0.034 |—0.100 | +0.175 1060 32) 26.71 0.672 05902) 1.5 1610 3 119596 0.078 0.092 0.161 O70} 381 43.41 0.686 | 0.882) 1.5447 1620 3 Bid! 0.120! 0.085 0.148 1080 3 0.37 0.702 0.862 1 f 1630 2 47.69 0.162 0.078 0.1385 1090 30 17.61 0.718 0.842) 1. 1640 2 32.82 0.203 0.071 0.123 1100 |+29 35.15 |—0.735 |—0.822 | +1. 1650 | +2 18.77 |-+-0.242 |—0.064 | +0.112 1110 | 28 53.01 0.752 | 0.803 ile 1660 Jeo? 0.278 | 0.058 0.101 1120 28 11.20 0.770 OxiSSa eee 1670 LD 2255 0.312 0.052 0.091 #1130 | 97 29.71 ONT90N|) OlTC4 |) Ae | 1680 1 40.46 0.345 | 0.046 0.081 11140 | 96 48.56 | 0.809] 0.745] 1.304] 1690 1 29.05 | 0.375} 0.041] 0.072 1150 |426 7.74 |—0.829 |—0.726 | +1. 1700 | +1 18.32 |+0.403 |—0.036 | +0.063 } 1160 | 95 97.97 0.848 | 0.707 1, f 1710 18398 0.429! 0.031 0.054 1170 | 94 47.16 0.866 | 0.689 1.206 # 1720 0 58.92 0.452 | 0.027 0.046 1180 | 94 1.49 0.883 | 0.670 il 1730 0 50.24 0.472} 0.023 0.089 1190 3 28.06 0.899 0.652 il, f 1740 0 42.26 0.439 0.019 0.082 1200 |+92 49.09 |—0.914 |—0.634 | +1. f1750 | +0 34.96 |+0.503 |—0.015 | +0.026 1210 22 10.51 927 | 0.616] 1.0731 1760 0 28.36 0.514} 0.019 0.021 p 1220) 91 32.3 .939 | 0.598] 1.046} 1770 0 22.44 0.522 | . 0.009 0.017 1230 20 54.57 .950 | 0.581 le 1780 OW LE22 0.527 | 0.007 0.013 1240 20 17.23 -960 0.564 0. 1790 | +0 12.68 0.529 0.005 0.010 dd 0 0 0 0 1250 |4+19 40.31 |—0.967 |—0.547 0 0 0 0 +0. | 1800 |+8.84 —— |+0.528 |—0.004 | +-0.007 1260 | 19 3.83 972 | 0.530) 0.928] 1801 | 8.497~39 |] 0.528] 0.004] 0.007 1270 | 18 27.80 | 0.975] 0.513; 0.898] 1802 | 8.15 -34| 0.528] 0.004) {0.007 1280 | 17 52.21 977| 0.497] 0.870] 1803 | 7.82 “33| 0.527) 0.004) 0.007 1290 | 17 17.09 | 0.977] 0.481| 0.842] 1804 | 17.50 = 0.527 | 0.004} 0.007 1300 |4+16 49.43 |—0.974 —0.465 | +0.814] 1805 |+7.18 | |-+0.527 | 0.003 | +0.006 1310 | 16 8.25 | 0.967] 0.449] 0.786] 1806 | 6.873" | 0.526] 0.003} 0.006 1320 | 15 34.55 | 0.959| 0.438] 0. 1807 | 6.57 *3°) 0.526} 0.003] 0.006 1330 | 15 1.3: 0.948 | 0.417] 0.730 1808 | 6.27 “3°| 0.525] 0.003) 0.006 1340 | 14 28.63 | 0.936 | 0.402) 0.704] 1809 | 5.98 a 0.525 | 0.003} 0.006 1350 |413 56.44 |—0.921 |—0.387 | +0.677] 1810 |45.69 _, |+.0.524 | 0.003 | 40.005 1360 | 13 24.76 | 0.903] 0.372] 0.651] 1811 | 5.41~~*°°| 0.523} 0.003} 0.005 1370 | 12 53.61 | 0.883] 0.358] 0.6 USUI) ao gl 0.523 | 0.003 | 0.005 1380 | 12 22.99 | 0.861| 0.344] 0.602] 1813] 4.88 “7° | 0.522| 0.003} 0.005 1390 | 11 52.90 | 0.837 | 0.33 0.5 1814 | 4.62 °72| 0.522) 0.002| 0.005 .25 1400 |411 23.35 |—0.810 —0.317 | +0.555 | 1815 |+4.37_ ,, |40 521 |-0.002 | +-0.004 #1410 | 10 54.35 | 0.780] 0.303} 0 11816 | 4.13724] 0.520] 0.002] 0.004 1420 | 10 25.92 | 0.748| 0.290] 0.508] 1817] 3.89 “°4| 0.520] 0.002] 0.004 1430 9 58.04 | 0.714] 0.277] 0.485] 1818 | 3.66 “| 0.519] 0.002] 0.004 1440 9 30.74 | 0.678) 0.264) 0. 1819 | 3.44 “77 | 0.518) 0.002] 0.004 1450 |+ 9 4.01 |—0.641 |\—0.252 | +0 1820 |4+3.23 || |+0.517 | 0.002 | +0.003 1460 8 37.88 0.601 | 0.240 ; 1821 | 3.02") ) | 0.516] 0.002 0.003 1470 8 12.34 | 0.560] 0.228] 0.399] 1822 | 2.82 “"— | 0.515] 0.002/ 0.003 1480 A389 Oss) 01216) 0:8 1823 | 2.63 19) 0.514] 0.002} 0.003 1490 7 23.04 | 0.475 | 0.205 | 0.359] 1824 | 2.44 oa 0.513 | 0.002] 0.003 1500 |+ 6 59.29 |—0.431 0. 194 | 40.340] 1825 |+2.26_ ,, |+0.512 |—0.001 +0.002 1510 6 36.14 0.386 ).183 0.322 § 1826 2.09 17 0.511 0.001 pone 1520 6 13.61 0.340 | 0.172 0.303 § 1827 1.92 “76 | 0.510 0.001 | 0.002 1530 5 51.69 | 0.293] 0.162] 0.284] 1828] 1.76 “7. | 0.509) 0.001 | 0.003 1540 |+ 5 30.40 |—0.247 |—0.153 | +0.268 | 1829 +1.61_ "72 —0.508 |—0.001 | +0.002 a 214 THE ORBIT OF URANUS. TABLE VI.—Continued. 3 1830 ; +0.507 ee. +0.002 | 1885 : 0.404 |—0.002 183 32 0.506 0.001 | 0.002 7 1886 0.402) 0.002 1832 ; 0.505 0.001 0.002 } 1887 0.399 | 0.002 1833 ; 0.503 0.001 0.002 | 1888 0.397 | 0.002 f 1834 Bu : 0.502) 0.001 0.002 { 1889 0.394 | 0.002 1835 if +0.501 +0.001 } 1890 391 |—0.003 1 1836 ; ; 0.500 0.001 § 1891 388] 0.003 | 1837 : ; 0.498 | 0.001 § 1892 386 | 0.003 1838 22 0.497 1893 333] 0.003 i 1839 0.495 1894 380 | 0.003 7 1840 +0.494 1895 —0.003 H 1841 0.493 1896 0.003 1842 491 | 1897 0.004 | 1843 490 | 1898 0.004 | 1844 488 1899 0.004 487 1900 —0.004 485 1901 0.004 483 | 1902 0.004 482 | 1903 0.005 481 1904 0.005 479 1905 = 07005 H 1851 ATT 1906 0.005 B 1852 ( ANG 1907 0.005 | 1853 |4-0.01 7° ATA 1908 0.096 Y 1854 OB SS Ose | 1909 0.006 f 1855 (0 s at 1910 —0.006 H 1856 ‘ : 469 | 1920 0.008 | 1857 ; : 0.467 1930 0.010 } 1858 BO eae mOs4G om 1940 0.012 F , | 1859 : ; 463 1950 0.015 1 1860 - 461 1960 79 |+0.15! 018 1 1861 HoT aa 0.459 | 1970 Bi53) 1a: | 1862 Deka ). 457 1980 56.93 0.074 1863 Lise 0.455 1990 5.98 | 0.032 1864 02 .453 2000 68 |+0.010 0 H 1865 : 451 .OO1 f 2010 —0.053 0 1866 e , 0.449 0 -OOL F 2020 0.096 0.048 1867 se = 0.446 0 .OOL # 2030 0.140) 0.0% 0 0 0 NNN NN So OWO Com SF QUIS (=) cS oS = i) = ® MwNN N -T =e OW ke -T + wow 710 oosss ssosss secs or} ce Uo Co OS US hm bo bo OD OO aoc O) C2 Go Go Wo NON WO NH A + F 1845 a S or aI Co Ol ee ee Re OF OI in G 1 1846 H 1S47 } 1848 f 1849 fH 1850 PeSeee SSLSe w W Os Go UD 2 Wo WW as + oo 1 Srey CWT —) —) coo Gow wou' | = S22 = + oNpre ee 1868 04 0.444 —0.001 001 f 2040 0.183 001} 2050 0.226 1869 ; : 0.442 0.001 Secoes 1770 |+1.29 , | |-+0.440 |—0.001 | +0.001 | 2060 | +2 27.38 |—0.268 |—0.068 | 40.119 is71 | 1.437724] "0.438| 0.001] 0.001) 2070 2 41.53 | 0.309] 0.075} 0.130 1872 | 1.57 “74] 0.436] 0.001/ 0.001} 2080 2 56.29 | 0.349] 0.082] 0.142 Fes 22 0.433) 0.001] 0.001} 2090 311.65 | 0.388] 0.089] 0.155 11874 | 1.88 fe 0.431} 0.001} 0.0024 2100 3 27.61 | 0.427] 6.096] 0.168 [1875 |4+2.04 +0.429 |—0.001 | +0.002f 2110 | +3 44.17 |—0.464 |—0.104 | + 0.182 | 1876 Batata! 0.426| 0.001} 0.002] 2120] 4 1.32 | 0.499] 0.112] 0.196 11877 | 2.39 = 0.424} 0.001} 0.002} 2130 4 19.07 | 0.532] 0.120] 0.210 1878 | 2.57 “T°) 0.422| 0.001} 0.0024 2140 4 37.40 | 0.563! 0.128] 0.995 11879 |12.76 Pes '+0.419 |—0.001 | +0.002 | 2150 4 56.31 | 0.593] 0.137] 0.240 | 1880 |+2.96 , ,, |+0.417 |0.002] +0.003} 2160 | +5 15.74 |—0.620 |—0.146 | +-0.256 1 18g1 | 3.16%°2°) 0.414] 0.002] 0.003] 2170 5 35.72 | 0.645/ 0.156] 0.273 1882 | 3.37 °2"| 0.412] 0.002] 0.003] 2180 | 5 56.24 | 0.668] 0.165] 0.289 1883 | 3.59 °27| 0,409] 0.002] 0.003] 2190 6 17.29 | 0.689} 0.175] 0.306 1884 |+13. Os +0.407 |—0.002 | +0.003] 2200 | +6 38.88 |—0.709 |—0.184 | +0.322 SS A SS A o~ THE ORBIT OF URANUS. Dili TABLE VII.—EQuarion oF CENTRE AND PRINCIPAL TERM OF Loa. 7. Log. r j EL Log. 1.26 18915 18916 18919 18924 18931 3 : 22977 18940 19.8% : 23104 18951 By als}.8 23234 18964 50 18979 2 20 15 18996 3 Smlte 19015 19036 19059 19084 H L911 | t : 4 svi 24339 | 19140 8 A 30 3.15 : 24485 19171 20 § f RB LS 24633 19204 5) Oke : 24784 19238 113° 547.32 24937 epi Ce i & | 55.28 ; 25091 19314 i 8 53.20 : 25247 19355 30 |: 9 51. i 25405 | 19397 { Sas 25566 19442 bE 6: 3 25728 19489 | 95892 19538 26057 | 19589 26225 | 19642 26395 19697 26566 | 19754 26740 19812 26916 | 19872 27093 19934 2727 “isi 1 19999 27451 20065 27684 20134 27819 20205 28007 = Sanlar 20278 eae | 20352 25386 20428 Zoe 20507 2B 12 20588 28968 20670 29167 20755 29367 inte 20841 29568 | 20929 | 29771 21020 29976 21112 30183 21206 & ~ ~ a 1.26 22488 25. e.. | 22608 22729 22.71 25° 9 | 22852 Rmowbpreowo > _ como MmeTOON FPWNH oS TRCOTLCHCHUe + bo bo bo bo bo _ 54 Or or or Or 01 Or SOR HR Rw www He He OF OF OV On bo bo bo bo bs to Ss OV He WD bo Or Or or Or Or Or - Or or or n Go G9 Go OS bo LO wWNweooa HN NG CHW OUdnw > Oe 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1! 0 0 0 0 0 0 0 0 0 0 0 0 3 OR 0 0: 0: OF: 0 Go Go G2 OO O92 YO ae a I eo po) 10) ° fo) b NN DYN ° “TOG am ee He ©. wWrores Cournsi (oemre) 2, hPNDD aaa cs Dw 80392 921302 ; | so) = : 5 30603 21400 E | : 2G, 30816 | 21499 3 tl 99 26 elie? 21601 | F 918: = t ee 21705 2 F hat 31465 | “| 31686 31908 > | 21811 num mnn41 ono U1 U1 U1 U1 RNN HN N ® WN NHN me Om A+ N oococoooo coococeco oeoeococe 216 THE ORBIT OF URANUS. TABLE VII.— Continued. g Li Log. r g Log. r c U tHe /y 1.26 j / aA en 2 1.26 H 20° 56 44.47 33045 840° 130° | 2 51.9 HOLS f 10’| 1 5% 39.96 49 33278 233 Set le 2 50 tune 5°-35 zolee 336 ae | 20 | 1 58 35.39 43 | 33513 735 | 40 [90 12 51 31.87 5°24 50795 338 a | 80/1 59 30.74 55°35 | 3749 735] 39 [30 [9 59 92:01 5°!4 | biiga 3391 ae F 40/2 0 26.02 55-2 | s39g7 2381 99 [40 |9 53 19.06 5°25 Baye 34? oF F 50/2 1 21.93 . 34227 lo 750 |2 2.00 49°94 | 5181s 343] 49 { 9.02 d }21°/2 2 16.36 34468 age° | ar] 54.51.83 > | saig2 2" | gage 10'}2 3 11.493 55-99 | 34777 50’ F107 | 2 41.56 49°73 | 59507 345 ! 20/2 4 6.41 2 34956 40 120 |2 56 31.13 49:02 Boned 347 i 30/2 5 1.33 54-97 | 35004 30 130 |2 57 20.70 49°52 | 53903 349 40/9 5 56.17 54-84 | Seana 30 % 2: jae Bae oa 350 | $9 50/2 6 50.93 54-70 | 35705, 255 | 10 [50 |9 88 59.43 49:3! | saggs oz Oe 5g PEGE sre 0 750 | 2 58 59. Ko conto eee 10 22°/2 7 45.61 35957 338° | 32°| 2 59 48 | 54958 eee 10'|2 8 40.22 546! | 3607; 254] se Tigris ¢ 37.73 491° | 54613 355 a 20/2 9 34.76 5454 | seag; 25°! 4o [50 13 1 dere 42°° | 5agnn SST 30 2 10 29.99 48 | 36795 7591 30 130 13 2 15 69 49:89 55399 359 ae 40 | 2 11 23.60 38 | 36986 76°] 99 |40 | 3 n 25719 55689 38e a 50 | 2 12 17.90 543° | sro4g 10 150 |3 353.09 4°98 | 56050 3° | a9 Poll ain sos an 48.57 363 32/12 13 1212 37511 837° 1332/3 4 41. 56413 327° 10’'|2 14 6.96 14 | 37446 50’ {107 | 3 30 48.46 | 56x 364 G 20/215 0.39 54°5 | se943 40 J20 13 6 1847 49:35 pag SOO 30 | 2 15 54.30 53:98 | 39374 30 130 13 oa Ze -25 oie 367 eae 40 | 2 16 48.20 53°99 | ge5gi 20 [40 [3 754.85 4833 | 5ig79 369 20 50 | 217 42.02 53:8? | 3ge53 10 750 |3 8 42.87 4°? | 58949 37° | 49 94°) 9 aL a “73 ; ap H 5 41-94 37? E°| 218 35.75 6 39128 38e6° 384°/3 9 30.75 58621 326° 10'| 2 19 29.40 53:65 | so404 50’ [10 |3 10 18.58 47-89 | 5g994 373 ; 20 |2 20 92.97 53°57 | 30¢8] 40 $20 13 6a¢ 47-58 | 5939 375 | ap 30 | 2 21 16.46 53°49 | sq9¢e9 30 130 13 11 53.83 47°57 59746 377 au 40 |2 22 9.87 53°42 | 4o941 20 [40 13 19 41.99 47-40 60124 378 a 1 50/223 3.19 53-3? | yo504 10 450 |3 13 98.64 47°35 | 60503 329 | 49 }25°/2 93 56.42 ~~ | aogos 325° 135° 15)gtal. 1eeORes ae a se 4 Our — H ede 3 5 08 2 | 10'' 9 24 49.57 53-75 | 41094 50’ Vio’ | 3 agutit2 nee 362 | 825 20 |2 25 42.63 53:6 | 41989 40 $20 13 15 50.00 47-22 Bie 384 | 3) 30 | 2 26 35.61 52-98 | ajery 30 30 13 16 3689 40°89 eagce 385, |e 40 | 2 27 98.50 52-89 | 41969 20°-[40 13.17 3867 40-78 | 6oaon Somme 50: oa hon sone Coula sce aoe | SET 28:60 * 6 66.| 02822 aaa 2 28 21.3 gx | £2255 10 750 /3 18 10.33 46°20 | 62810 3 ale $26") 229 14.01 1) | 49550 334° 13613 18 56 | gso00 2 324° 1 10'12 30 6.63 52:52 | sogue 50’ £10713 19 43.31 49-43 | 63592 392 | “poy 2012 30 59.16 52°53 | 43ya4 40120 |3 90 99.63 40-32 |esgns | oon 30 | 2 31 51.60 52-44 | g3444 30 130 13 91 15.83 4°20 | easy 395 | aq 40 | 2 82 43.95 57°35 | 43745 20 P40 1392 1.91 46°8 | 647z¢ 39° 1, 5 9 92 90 +2 a : ae J 50/2 33 36.21 92°70 | atoss 10 |50 /3 22 47.83 29°37 | 65173 397 | 10 J 27°) 9 34 28.38 g | 44353 333° 137°] 3 93 33 ot Tesora ° oo igogem 10’| 2 35 20.46 Oo | AdE59 50’ #10 13 19.44 45-72 | 65y7Q 49° ! 20) 2 36 12.44 51-98 | groggy 40 }20 |3 5.04 45-60 aie 13 ie 30 2 on 3 -9O =o E H ee eo [at ara 2 euie H on oe an a jB0 3 25 50.: - 46 pote ie 30 2 37 56.14 = 558i 20 | 3 Bat 2 7180 20 50 | 2 38 47.85 STZ | a5oo1 10 50 |3 97 21 ee 67585 4°95] 10 28°/ 2 39 39.46 , | 46216 832° | 387| 3 Sonera 67992! | g990 10'} 2 40 30.98 315% | 46532 31° | 507 fio’ | 3 98 51.95 45-2 | 68400 4°8 | 597 20 | 2 41 99.40 57-42 | gegsg 3781 4g foo | 3-09 gei3 44-88 BEaI0 Are ie 30 | 2 42 13.72 5°37 | ay179 370] 39 3 30 20.90 4477 | 69991 411 39 40 |2 43 4.95 27°73 | 44491 372! 99 331 5.54 4464 | 69633 472 | 99 50 | 2 43 56.08 5I-13 Birds 8 23h ea 3 31 50 52 : 56. 47814 10 13 315 44.52 | 4 ALA ei So 150 |3 31 50. oo47 474) 10 29°| 9 4. 9 eae | Ba aa ;| 2 44 47.12 48138 46 | 881° 739°) 3 32 34. 70463 321° 10’/ 2 45 38.06 5°94 | 4gag4 32 50’ | 10’| 3 33 18 44:27 | 7oggo 42 4)\ aba | 20 | 2 46 98.90 5°-84°| aergo .328 | 49 [a9 13 34 966 4424 | q1o93 428 1 ag | 30/2 47 19.64 oe 49122 33° | 30° f30 |3 34 46.88 44°? | qiqiy 479 [ap 5012 ae oes gog4 | 22458 29. | 20° [403 85 0.18 943-2° | Toes econ ee ere a Ol ie (Sg ie 72560 737 | 10 > C e a 30°) 2 49 51.98 50121 830° [40° 3 36 58.19 72984 320° 126 Chi vg 1.26 g THE ORBIT TABLE VII.—Continued. OF URANUS. June, 1873. #E Log. r g B Log. r ° ” 1.26 yr 1.27 3 36 58.19 72984 | 820° 150°! 4 16 34.97 00687 3 87 41.71 73409 473) 50’ |10’| 4 17 9.62 35-35 | o1ig3 496 3 88 25.10 73835) 4? 40 ]20 | 417 44.82 35-79 | g1gg1 49° 3.39 8.37 74262 477 | 39 [30 |4 is 19.88 35°°° | oa1go 499 3 39 51.51 74691 4°9 | 90 [40 | 4 18 54.80 34-9? | 02680 5°° 3 40 34.52 Tolan 432°) or 150) | 4 1929.57 34°27) ogig 5° 432 34.62 502 3 17.40 75553 319° 751°) 4 20 4.19 03683 342 0.16 75936 433 | 50’ [10’| 4 20 38.67 3445 | o41g6 53 3 42 42.79 76420 43%) 40 720 | 4 21 13.00 3433 | 4690 504 3 43 25.29 76856 43°-| 30 130 | 4 21 47.18 34-25 | 95195 595 3 7.66 77293 as 20 | 40 | 4 22 21.99 Soe 05701 ae 7 5 99 BF ds on 3 44 49.91 T1181 49 | 10 [50 | 4 22 55.11 330% | 06207 52. 3 45 32.02 78171 318° [52?| 4 93 98.85 | 06714 3 46 14.01 T3611 50’ J 10’| 4 94 9.44 33°59 | g7999 528 3 46 55.86 79053 40 [20 | 4 24 35.88 33°44 | ox739 91° 3 47 37.58 79496 30 730 | 425 9.18 33°3° | ogg4g 357? 3 48 19.17 T9941 20 [40 | 4 25 49.39 ae 08754 ore © ag 2Q ix oF A ae J DFR = 3 0.63 80387 19 [50 | 4 26 15.32 39°37 | 09266 3F) 3 49 41.96 80835 817° |53°| 4 26 48.16 09780 3 50 23.16 81284 50’ J 107) 4 27 20.85 32-69 | 19994 514 3 4.99 81733 40 }20 | 4 27 53.39 37°54 | 10809 5175 3 51 45.15 82183 30 [30 | 4 98 95.79 324° | 11395 51° 3 52 25.94 82635 20 $40 | 4 98 58:03 37°74 | 11849 977 3 6.60 83089 10 [50 | 4 29 30,12 37-°9 | 19361 979 3 31-94 520 3 53 47.13 83544 B16° 154°|4 30 2:06 | | 1egsi > 3 54 27.52 4°39 | s4svo0 50’ | 10’| 4 30 33.85 3-79 | 13401 57° 355 1.78 42-29 | 94457 40 720 |4 31 5.49 31:94 | 13999 5?! 3.55 47.90 4°? | g4a914 30 130 | 4 31 36.98 37°49 | 14444 577 3 56 27.89 39°09 | 85373 20 $40 |4 32 8.31 3733 | 14966 977 3 7.75 39°°> | 85834 10 150 | 4 32 39.50 = 15490 te 72 2 3 57 47.47 -g | 86297 315° | 55°) 4 33 10.53 ee) |) LegLon oe 3 58 27.05 39°5° | 86760 50’ | 107| 4 33 41.41 3°°° | 16540 579 359 6.50 39°45 | g7994 AO 204 states 2509 i a06n, af 3 59 45.81 39-3" | 87690 XN (EOE LTO eo ais ee 4 0 24.93 39°17 | 98156 20 140 |4 35 13.13 3°47 | igi93 5? 4 4.01 39°93 | gs6o4 10 150 | 435 43.40 3°77 | 18659 279 89 12 53° 4 1 42.90 _ | 89093 14° [156° 4 36 13.52 |, | 10182, 4 221.65 32°75 | 89564 BOW 0g| 2360459485 729 agp ae 282 4 8 0.27 g | 90036 QO) | 4°81) 189998 s 70s | 202tun ee £3 38:75 32°" | 90507 30 }380 |4 37 42.94 79°°5 | 90776 33 4 417.09 32°34 | 90980 20 140 | 4 38 12:43 72°49 | 91309 233 4 4 55.3 91455 10 [50 |4 38 41.77 29°34 | o1g4g 933 , 19 534 4 5 33.36 91931 318° [57°| 4 39 10.96 |, | 22876 _. 4 6 11.29 92408 50’ | 10’ | 4 39 39.99 22°83 | ao911 535 ae 649,07 92886 409 1120) 440) 8387 =. | 03448) 287 4 7 26.73 93365 30 |30 |4 40 87.59 22°72 | 3986 53. 4 8 4.99 93845 20 [40 |4 41 6.16 79°! | 24504 23° 4 8 41.58 94327 10 [50 | 4 41 34.57 25°55 | 25063 330 4 9 18.80 94810 312° [58° 4 42 2.83 25603 4 9 55.88 37- 95294 50’ |10’| 4 42 30.93 2°-T° | 96143 54° 4 10 32.81 3° 95779 40 [20 | 4 42 58.87 77-94 | o66s4 54° 1 9.60 3° 96265 30 [30 | 4 43 26.66 77-79 | o7296 37° 4 11 46.95 3° 96752 20 | 40 | 4 43 54.29 27-3 | 97768 54? 4 12 29.76 3° 97240 10 [50 | 4 44 21.76 27-47 | 98311 by bre) 4 59.12 97730 811° §59°) 4 44 49.07 6 | 28855 4 13 35.34 98220 50’ |10’ | 4 45 16.23 19 | 99399 544 4 14 11,41 98711 40 | 20 | 4 45 43.22 77°09 | 20944 545 4 14 47.34 99203 30 30 | 4 46 10.06 22—* | 30491 547 415 23.13 99696 20 440 | 4 46 36.75 °7°29 | 31038 ae 0 9 ¢ Oa) 215 Q7 J 4 15 58.77 *00191 10 50 [4 47 3.27 Fe5" | 81587 222 416 34.27 *00687 310° [60°| 4 47 29.64 32137 *1,27 iL O27 15 THE ORBIT OF URANUS. TABLE VII.—Continued. E Ibe, P| g EB Log. r Oo FF it ” 1.27 | ousrw#w ” 1.27 4 47 29.64 32137 800° | 70°|5 8 57.01 66156 447 55.85 22% | 3agg7 55° | "507 | 10" |5 9 13.48 7647 | corsa 532 4 48 21.90 2°25 | 33937 55°] 40 |20 |5 9 99.79 1°9) | 67320 36 4 48 47.79 25°89 | 33787 55°] 30 [30 |5 9 45.93 7274 | 6r903 353 4 49 13.53 75°74 | 34338 55% | 20 | 40 |5 10 1.91 122° | 68487 ae 4 49 39.10 oe 34890 oe 10 |50 |5 10 17.72 6 69071 387 450 4.52 . 9, | 35443 __, | 299°]71°|5 10 33.36 |. 45 | 69655 <3, 4.50 29.78 75°7° | 35996 593 | 50/ | 10’|5 10 48.84 (23, | 70289 Zee 450 84.88 752° | 36550 554 | 40 [20/511 4.15 15°35" | 70824 Be 451 19.82 2494 | 37108 555] 30 |30 |5 11 19.30 75°73 | T1409 25 451 44.60 7 he 37660 225 | 20 | 40 |5 11 34.28 ae 71994 oe 452 9.92 ¢ 7G 38216 aoe 10 ]50 |5 11 49.09 7°6. | 12579 246 4 52 33.68 38773 o98°| 72/5 12 3.74 1, 4. | 13165 296 4.52 51.98 243° | 39339 557 | 50’ | 10'|5 19 18.23 14°49 | 13751 386 453 22.11 2413 | 39388 558) 40 | 20 |5 19 82.55 173" | 74387 23, |4 53 46.09 23-98 | 40446 55° | 30 |30|5 19 46.70 [70> | 74924 2a) 454 9.90 238 | 41005 559 | 20 [40 [5 13 0.69 133? | 75511 2, 4 54.8356 230° | dinep 5° | 10 450 | 5 13 1451 Crees meee 23.49 561 13.66 B27) 4 54 57.05 42126 297° | '73°| 5 13 28.17 76685 ge 455 2038 73°33 | 49687 55% | 507 | 10’|5 13 41.66 73-49 | T7973 2Oe 455 43.56 23-18 | 43949 52 | 49 [20 |5 13 54.98 3-37 | 77861 a 456 657 232° | dsg11 202} 3p 80/5 14 Bliss Sai istpo 456 29.49 22-85 | 44374 53 | 20 | 40 | 5 14 21.12 eee 79039 eA 456 52.11 a 44938 oe 10 [50 |5 14 3.94 1262 | 19628 390 4.57 14.64 -. | 45508 5. | 296° | 74°) 5 14 46.60 |, 80217 60, 4 57 81,01 27:37 | 26068 5°5 | 50’ 10’ | 5 14 59109 <= | 80807 2 Ze 4 54 59.91 222° | 46633 505 || 40 | 20/5 15 tat aoe 81397 ae 4.58 21.95 27-04 | azyog 56° | 39 130 |5 15 938.5% (77° | 81987 222 453 43.13 27-88 | arzgg 567 | 20 | 40 |5 15 85.56 11°29 | s2577 Oo , PA 7/4 a6 i 9 93 wo) 459 4.84 aie 48333 268 10 a 5 15 47.39 11768 | 83168 ZO, 4 59 26.40 43901 69 | 295° 75°) 5 15 59.05 |, 83759 205 459 47.79 27-39 | 49469 9 50’ | 10’|5 16 10.54 11°49 | 84350 27° 5 0 9.02 27-23! 50038 5°9 | 40 | 20 | 5 16 21.87 77°33 | s4g41 27! 8 0 30.09 27°? | 50607 5°9 | 30 130 |5 16 93.03 ** 7° | 85532 22° 5 0 50.99 229° | 51177 57° | 90 | 40 |5 16 44.03 (OQ | 86124 one Sv itis: 22 0 | bliay 27) to 500m Te otee > cog ee ae 5 1 32.31 52318 394°)|"767)5 17 552 1) 2 | Sioa 5 159.72 20:47 | soggg 277 | “50% 10" | 5 17 16.02 ae 87899 20. 5 219.97 7°25 | 53461 372 | 4o | 20 |5 17 96.85 °° 33 | 88491 253 5 2 33006 = 2 || 54033 247 | 30 4) 30))| 5 iy 36751 =e cau) S004 ace 5 252.98 179% | 54606 573 | 90 | 40 ]5 17 46.51 “5/3, | 89677 2a; 5 8 1a.74 19-7) | 55179 373} 10 [50/5 iy 56.84 O65 | 90270 2a, 5. 3 32.3 55153 293° | 77°|5 18 6.00 > | 90863 co, 5 8 bltr 1243 \4peso7 224 Vigo” 10" lib ig 15250 | ee 5 411.04 7927 | 56901 274 | 40 $20 |5 18 24.82 23" | 92049 207 /5 430.15 727" | 51476 575 | 30 130 |5 18 33.99 307 | 92643 20) |5 449.09 1894 | 53051 373 | 20 [40 |5 18 42.98 99? | 93237 Zo. }5 5 7.87 isk) | 5scar 57° | 10 |50 |5 18 51.81 5.65 | 93830 50) [5 5 26.49 59204 992° |'78°|5 19 0.47 g.. | 94424 0, 5 5 44.94 1845 | 597g1 577 || 50" 1102/5 19 8.9%) > oD Onemme 5 6 3.93 1829 | go35g 577 | 40 [20/5 1917.30 52 | 95612 297 5 6 21.35 7822 | 60936 578 | 30 130 |5 19 25.46 50 | 96206 277 15 6 39.30 27:95 | 61514 578] 90 140 |5 19 33.46 7.83 | 26800 ce |5 6 57.09 eee 62093 os 10 |50 /5 19 41.29 7°65 | 97395 202 7 14.72 62673 __ | 291° |'79°) 5 19 48.95 97990 7 32.18 17-46 | 63952 579 | 50’ | 10’|5 19 56.45 75° | 98585 929 7 49.48 173° | g3gg2 58°] 49 J20 |5 20 3.18 7°38 | soit9 29° 8 6.61 273 | gaara 582 | 39. 30) 5 20 10sade eae ogi iease se g 23.57 789° | eag93 S57 | 20 | 40 |5 20 17.94 Fe" j*00369 a 2 se 5 c 5 Fane OA NY Oe 35 840.37 Te, | 65574 55) | 10 [50 [5 90 24.77 Gre> j*00965 CO, 8 57.01 66156 290° | 80°, 5 20 31.44 pues 1.27 g : THE ORBIT OF URANUS. 219 TABLE VII.—Continued. g EH Log. Ue g E Log. r ° , wt ust 1.28 |o t wt uw 1.23 80° | 5 20 31.44 01560 280°] 90° |5 22 9.05 _ | | 37187 270° 10’ |5 20 37.94 aoe 02155 595 | 50’ | 10’ |5 99 5.68 337 | 37776 59 50 20 |5 20 44.28 ae 02750 393) 40 20 |5 92 215 353 | 38365 599 | 40 30 | 5 20 50.45 ae 03346 99 30 30 |5 21 58.45 SH 38954 wee 30 40 | 5 20 56.46 2a 03941 ae 20 | 40 (5 21 5459 3 o0 | 39543 288 20 2.30 > 5 25058 sn) 4013 10 50 |5 21 2.30 2°6f | 04536 399 | 10 | 50 |5 21 50.58 495 31 es gle |5 21 7.96 05131 279°] 91° |5 21 46.40 |, | 40719 14, | 269° Mi moteisthe > || 05726) 22>) 50’ Ff 10 5 21 49:06 94 | 1307 3, | 80 20 |5 21 18.80 234 | 06322 59°} 40 | 20 |5 21 37.57 L6G 41894 597 | 40 30 |5 21 23.97 27 | 06917 59? | 30 | 30 /5 21 32.91 4g, | 42482 32° | 30 40 |5 21 28.97 577 | 07512 395 | 290 | 40 |5 21 28.09 703 | 48069 537 | 20 50 |5 21 33.81 ee 08107 bee 1) || 0 pb Oil PEau ae 43655 287 10 : 5 82° | 5 21 38.48 08702 278? | 92° |5 21 17.97 44949 | 268° To’ |5 21 42.99 457 | 09297 595 | 50’ | 10’ |5 21 12.67 93° | 44898 Soe | 50 20 |5 21 47.33 +34 | 09893 59°] 49 | 20 |5 21 17.20 See ces eo 30 |5 21 51.50 4 °7 | 10483 595) 30 | 30 [5 21 1.58 3° ° | 46000 509 | 30 40 | 5 21 55.51 *°% | 11083 3595 | 20 | 40 |5 20 55.79 ~2°79 | 46585 ee eee 50 |5 21 59.36 oe 11679 59°] 10 | 50 |5 20 49.85 Se || Sali 8. 10 eso |522 3.04 > | 19974 > | 977° | 98° |5 20 43.74 be | 28> 25) Doe 10’ |5 22 6.56 357 | 12869 595 | 50° | 10’ |5 20 37.48 27) | 48339 534) 50 20 |5 22 9.91 335 | 13465 59°] 40 | 20 | 5 20 31.05 Pars RPE el 20 30 |5 2 13.09 37° | 14060 595] 30 | 30 |5 20 24.47 65° | 49507 oe ee 40 |5 22 16.11 30° | 14655 595) 20 | 40 |5 20 17.72 (75 | 50090 395 | 20 50 | 5 22 18.97 72° | 15250 595 | 10 | 50 |5 20 10.82 2-96 | 50c73 553] 10 : 2-09 59 2 : ane 3 84° | 5 22 21.66 15346 |) | 276° | 94° |5 20 3.76 | ,, | 51256 (9 | 266° Jo’ |5 22 94.19 753 | 16441 595) 50’ | 10’ [5 19 56.54 77) | 51838 52° | 50 20 |5 29 26.55 73° | 17036 595] 40 | 20 |5 19 49.16 73° | 52490 289 40 30 |5 22 98.75 772 | 17631 595| 30 | 30 |5 19 41.62 7-34 | 53002 epee 40 |5 22 30.78 7°23 | 18226 595 | 90 | 40 |5 19 33.93 136 53583 355 | 20 me tion) so 64 ae 18820 ace 10 | 50 |5 19 26.07 go, | 54163 25, | 10 ‘ R ° 85° |5 22 34.34 19415 _|, | 275°] 95° |5 19 18.06 .,, | 54743 4. | 265 Moey)5) 2235387 © 23 | 20010' 275 | 50’ | 10’ |5 19 9.89 2.) | 90828 2° | 80 30 | 5 22 a7.24 © 37 | 20604 277 | 40 | 20 |519 156 ges | 55902 2° | 40 BO) /5.22/38.44 7° | a1198 594) 30 | 30 |5 18 53.06 52) | 56481 277 | -80 #0) 5 22.39.48 © 5* | 21792 577 | 20 | 40 |5 18 4442 ot | 5r060 273 | 20 50 |5 22 40.35 °°7 | 99386 oe 10 | 50 |5 18 35.61 5°05 | 57638 27, | 10 E 0.71 9 fi tee Sein : £4 R 86° | 5 22 41.06 22980 974°} 96° |5 18 26.65 9 |, | 58216. 264 | Hogeowe2 4i,60 9" || o3574 274 || 50’ | Voy 5 18 Taos 5). | 58194 27 | 50 20 |5 22 41.99 °39 | 24167 593 | 40 | 20 |5 18 8.26 Ae p97 277 | 40 Eom 22 42.01 2-2 | 24761 397 | 3 30) |i 1h 58.83) >| 59048 27 80 49 | 5 29 49.98 °° | 25354 593 | 90 | 40 [5 17 49.94 077 | 60525 277 2 ‘ont ES ¢ Gy i : 7 50 |5 22 42.15 ©). | 25948 zee 10 | 50 |5 17 39.50 Q'5 | 61101 276 B7° | 5 22 41.88 26541 278” | 97° |5 17 29.60 5.96 | 616tT 22 Mo” |/5 92 41.45 643 | avis4 593) 50’ | 10’ |5 17 19.54 10°,, | 62262 375 | 8 Sos 22 408; 0 °° 97796 592) 40 | 20 |5 17 9:33 7° | 628a7 375;| 40 3 92 40.09 27° 319 593) 3 3 5 16 58.96 1-3! | 63401 574] 30 30 | 5 22 40.09 28319 593 | 3 SO Pe NERO EO rotean Meee arr pills Bigs 22 30.16 22% | 2s9n1 59° | 90 | 40 | 5 16 4844 ogg | B801o 2 | 2 a aieteke 1.0 zs 3 — 1p 2h He : 54S 50 |5 22 38.07 ae 29504 Be 10 | 50 |5 16 37.76 56g, | pa ieee 88° | 5 22 36.82 30096 DBs OBrh| oe, 26.92) Foe 6120 ee 71) 26 M522 35:40 = 4 930688 592 | 50! 10’ 5:16 15.98 7 | 60602 2 99 29 Rg 1.58 31280 592 40) 90 5 16 4.78 ee) 66264 9 Z 40 20 |5 33.8 3128 : : 2 Sean 592 | 94 ‘ 5 15 53.48 12:32 | 66886 272 | 30 30 |5 22 32.08 174 | 31872 297 | 30 30 |5 15 53.48 7732 | 66836 27 30 40 |5 22 30.17 1-9 | 32464 592] 20 | 40 [5 15 42.02 17-40 | Se | ae 50 | 5 22 28.10 ae 33055 ae | 10 | 50 [5 15 30.41 1F ont cE ae 89° | 5 22 25.86 33647 Vrmeyy BOee (515) Wey | Ober zor gga ion 2293.47 7-32 | 34238 59° | 50! fF JO’ 515 6.12 73 og | CONT ceo | OF 20 |5 22 20.91 75° | 34898 59° | 40 | 20 |5 14 54.64 (33 | 69686 OE BS 30 |5 92 18.19 772 | 35418 59° | 30 BO) 5 14 4941 73 | 1022) 68 ao Mg 5 2315.31 28° | se00s 22° | 20 | 40 | 5 14 30.03 7c) | 70828 cag | 20 50 15 92 12.96 3°5 | 36597 ae 10 50 [5.14 17.49 172" | 11391 (26, = 7 BN2i6 ¢ : rae, ~10RQ EQ? 90° 5 22 9.05 31187 270°} 100° |5 14 4.80 es ae g ee 123 2) THE ORBIT OF URANUS. M TABLE VII.— Continued. g E | Lorr (| @ || i | Log. r ° / aA Mt 1.93 | ° ' ” " 1.29 100° 5 14 4.80 71958 260° 110° | 4 56 49.88 04888 10’ |5 13 51.96 784 | 79525 507)” 50’ | 10" | 4 56 28.90 27-88 | o54ig 528 20 |5 13 38.96 73° | 73091 52° | 40 | 20 | 456 6.39 2h | o5944 9° 30 |5 13 25.81 1375 | 73656 52> | 30 | 30: |4 55 44.44 77-99 | 064i1 oa! 40: | 5-13 1pibt "3/30 7) doo S25)\) BOF Ie 40m) 4 by 221k ea 0Goon 5? 50 | 5 12 59.06 Pe T4785 oe 10 | 50 |455 0.10 ae 07522 ee 101? |5 12 45.45 75348 | 259° Filye | 4 54 37.72 08047 10° |5 12 31.69 %3°7© | y591n 53 | bor | to: | 4 54 15.91 225" | ossya a8 90 |5 12 17.78 73°97 | y6474 53 | 40 | g0- | 4 53 52.55 226° | ogo9a 923 30 (5.12 3.78 4° | 7036 322 | 3 30 | 453 99.76 77°79 | og616 30° 40 |5 11 49.50 eee 77598 26, | 20 | 40 | 4 53 6.83 ee 10137 ree py ON) He 8 p RC ¢ > 0 2 FC 60 |5 11 35.14 143) | 78159 26, | 10 | 50 )4 52 43.76 337 | 10658 Fo, 102° | 5 11 20.62 78720 | 258° $112° | 4 52 20.55 11178 10’ 15 11 5.95 14:67 | 79989 56° | 507 | 10’ | 4 51 57.20 23°35 | 11697 579 90 |5 10 51.13 74-82 | 79839 559} 40 20 | 4 51 33.79 23-48 | 19915 578 30 15 10 36.16 14-97 | go398 559 | 30 | 30 |4 51 10.10 23-62 | 19739 S57 40 |5 10 21.04 75-22 | go95¢ 558] 90 | 40 | 4 50 46.34 23-76 | 13948 57° 50 1510 5.77 15:27 | gi514 55° | 10 50 |4 50 92.44 23-99 | 13764 526 15.42 557 24.03 515 103° |5 9 50.3 82071 —_ | 257° 118° | 4 49 58.41 14279 10° |5 9 84.78 15°57 | gog98 957 | 50 10° | 4 49 34.94 24-77 | 14793 324 20 15 919.06 15-7? | gsig4 55°} 40 20 |4 49 9.94 24-32 | 15306 973 30 15 9 3.19 15°87 | g3z39 555 | 3 30 |4 48 45.50 24-44 | 15818 97? 40 |5 8 47.17 6-2 | gao94 555 | 90 | 40 | 4 48 20.92 245% | 16399 577 50 |5 8 31.01 7°%® | gagag 555 | 10 | 50 |4 47 56.21 74-77 | 16840 272 16.32 554 24.85 510 104°|5 8 14.69 85403 | 256° } 114° | 4 47 31.36 17350 ria 5 758.92 76-47 | 95956 553 | 50’ | 10’ |4 47 6.38 249° | 17859 202 30 |5 7 41.61 120% | s6508 257 |. 40 | 20 | 4 46 41.26 75°72 | isa6r 598 30 15 724.85 10-76 | sros9 557] 30 | 30 | 4 46 16.00 25-26 | issta 927 40 15 7 7.94 159% | gze19 55% | 20 | 40 | 4 45 50.61 75°39 | 19380 Oe 50 |5 6 50.88 17:0 | gsigo9 55°] 10 | 50 |4 45 25.09 75-52 | 19886 5° 05°15 633.67... | ssz09 ote 255° | 115° | 4 44 59.44 ee 20301 105 » 6 33.6 tor : og. 2038 10’ |5 6 16.32 27°35 | g9a5a 549 | 50’ | 10° | 4 44 33.66 75°78 | go895 304 20 |5 5 58.82 175° | g9s0g 548] 40 | 90 14 44 17.74 75:9? | 1398 203 30 |5 5 41.17 7765 | yo354 548} 3 30 | 4 43 41.69 7°°°5 | 91900 308 40 |5 5 28.37 178° | go902 548] 90 | 40 | 4 43 15.50 27°79 | o9401 355 50 5 5.49 i085 | oi4gg S47 | 10 | 50 [4 42 49.19 26g | 2200 oa 106° |5 4 47.838 ,., | 91995 _ | | 254°] 116° | 4 42 22.74 1, | 23801 Jog 10? 5 4 29°10 8.28 92540 545 | 50! 10’ | 4 41 56.16 are 23899 8 20° |5 P10 “33. 93005 34) 40 80 |4 41 29.45 oo 7) | 94307 Be 30 |5 359.19 75°93 | 93699 544] 30 | 30 [441 9.60 52°°3 | 24894 ics A Wo: 8 ses. oelira Be 20 | 40 | 4 40 35.62 2°°9° | 25390 Jo. 60 [5 81470 te°06 | 94715 34S) 10 | 50 [4 40 8.52 37°24 | 25885 Jos 107° |5 2 55.74 95257 258° [117° | 4 39 41.28 26380 10’ [5 2 36.63 792% | 95798 547 | 50” | 10’ |4 39 13.91 27-37 | oG87a 28 20 |5 217.38 57°73 | 96338 24° | 40 | 20 | 4 38 46.41 77-3° | 97366 ae 30° |5 1 57.99 19°39 | o6s7g: 54°] 30: |. 20: | 4 38 ig.z8 27-03 | aye5rem 40 |5 1 38.45 mee 97417 ae 20 | 40 | 4 37 51.02 sie 28348 489 | = | ad 2 M4 OQO7VGAAR Je 2 ' S vy 2 ~ 4S 50 |5 118.77 76's. | 97955 23. | 10 fF 50 | 4 37 2313 27-09 | 28837 Jeg 108° |5 0 58.94 98493 252° | 118° | 4 36 55.12 293825 9 10’ |5 0 38.97 19-97 | 99030 537 | 50’ | 10° | 4 36 26.98 2874 | gosig 437 20 |5 0 18.86 2°! | 99566 53°] 40 | 20 |4 35 58.71 28:27 | 30299 7e¢ 30 | 459 48.60 2°26 lxoo101 535 | 30 f 30° |4 35 30.31 72°4° | 30785 Je 40 | 459 38.91 20-39 |xoos3e 535| 20 | 40 |4 35 1.78 78:53 | 31970. 76> 50 | 459 17.67 2°54 j*o11z0 534] 10 | 50 |4 34 33.13 28-65 | git54. Jo) 109° | 4 58 56.98 ose 01703 >> | 951° Jag? | 4 34 4.35 gore 32937 ee 10’ | 458 36.15 20°83 /*og936 533) 507 | 10’ [4 33 a5.44 2892 | gor1g 462 20 | 4 58 15.18 20-97 |*oa7gs 532 | 40 | 90 | 433 6.41 79°°3 | 33900 79: 30 |4 57°54.07 27-12 |*o3999 537) 30 | 30 | 4 39 97.95 292° | 33681 Jaq 40 | 4 57 32.81 2726 |xosgo9 53°] 90 | 40 |4 39 7.96 29°29 | 34161 fo0 50 | 4°57 °11.42 ona *04359 28 | 10 | 50 | 4 81 38.55 ck 34640 a5 110” | 4 46 49.88 #04888 250° | 120° | 4 31 9.01 35117 | | #129 g 1.29 ‘i | A A ES A a THE ORBIT OF URANUS. 291 TABLE VII.—Continued. B Log. r g ii Log. r 1.29 1.29 | 431 9.01 35117 |. | 240°9 180° | 3 57 57.75 61903 230° 4 30 39.35 79-6 | 35593 47° | 50’ | 10’ | 3 57 21-10 36-65 | gasiz 44 | “50° 430 9.56 pee 36069 47 40 20 |3 56 44.3 oan 62729 41? | 49 4 29 39.65 °2°9) | 36544 475 | 30 30 356 7.48 3e°7 | 63140 477 | 30 429 9.61 paee 37018 oe 20 40 |3 55 30.51 ae 63550 ee | 20 qc QC Ee : 2+ 4C RS F 2 Ae € Oe P2 OF G | 428 39.45 3055 | S491 772 | 10 | 50 [3 54 53.43 379° | 63959 4320 428 9.16 ,,,, | 37963 |, | 239° | 181° | 8 5416.95 | | 64366 229° £97 3875 3" | 38434 477 | 50’ | 10 |3 53 38.97 3725 | Gar73 497 | 50! 427 8.22 3°33 | 38004 47° | 40 | 20 [353 1.58 37°39 | G5179 406] 40 4 26 37.56 Bras 39372 168 3 30 |3 52 24.08 ote 65583 494] 3 426 6.78 oes 39840 765 | 20 | 40 |3 51 46.48 as 65987 se 26 95 35 8 > V206 5 9 5 2 : APOE 2 425 35.88 375, | 40306 J6.| 10 | 50 |3 51 8.78 372° | cossg fees | 10 495 4.85 40771, | 238° 11382" | 3 50 39.97 66790 | 228° 4 24 33.71 31-14 | 41936 495 | 50’ 10’ | 3 49 53.06 37-91 | 67190 49° | 507 Aroae toa 32-27 | 41700, 494 | 40 20 |3 49 15.04 38-02 | 67588 395 | 40 4 23 31.05 3%-39 | 49163 493} 30 | 30 |3 48 36.92 3°22 | 67935 397] 30 4 99 59.54 37-51 | 49695 492 | 99 40 |3 47 58.70 38-22 | gg3gg 397 | 90 492 27.91 31-3 | 43086 492 | 10 | 50 {3 47 20.38 38:32 | es777 395 | Jo 31-75 460 38.43 394 4 21 56.16 43546 = | 937° 1138° |3 46 41.95 —, | 69171 ~~ | 297° 421 24.99 31-87 | 44005 459] 50° | 10’ |3 46 3.42 38:53 | 69564 393 | 50’ 420) 52.99 32°22 | 414463 oe 40 20 )3 45 24.79 38-63 | 69955 39% | 40 4°2020.18 37-1t | 44919 45°) 3 30 | 3 44 46.06 3°°73 | 70345 39° | 30 419 47.94 32:24 | 45375 45 20 40 13 44 7.93 3°83 | 4073 oe 20 419 15.59 32°35 | 45899 4541] 10 50 |3 43 28.30 3°93 | q1199 38 10 32-47 453 39-03 387 418 43.12 ~ | 46982 | 9367 [134° | 3 42 49.07 _ 71509 226° 418 10.53 3259 | 46735 453] 50° | 10’ |3 42 10.14 39-13 | 71895 386 | “ 50" 417 37.83 377° | 47186 45° | 40 | 20 | 3 41 30.91 3973 | 72079 384 | 40 417 5.01 32-5? | 47636 4°] 30 | 30 |3 40 51.58 39°33 | 72669 383 | 30 4 16 32.07 37:94 | 48086 ee 20 40 | 3 40 12.15 39°43 | 73044 352 | 90 415 59.01 33:0 | 4534 44° | 10 | 50 |3 39 39.62 39°53 | 73495 38% | 10 33-17 447 39.02 380 4 15 95.84 48981 235° 1185° | 3 88 53.00 , | 73805 295° 4 14 59.55 33-79 | 49408 447 | 50° | 10’ |3 38 13.98 39-7? | 74184 379 | 50’ | 414 19.14 33-8" | 49873 445 | 40 | 20 | 3 37 33.46 39°? | 74561 377) 40 pepieedeca) 99/971) 5031 tt |) 80 80) 836 53154. 97 7° | m493h 372) 30 4 13 11.98 ae 4 | 50761 7 20 | 40 | 8 36 13.59 bone 75312 oe 20 ¢ s OF 3-7 ATS < € 9R 99 . rFEe oR ii 412 38.22 337" | 51204 443 | 10 | 50 | 3 35 33.41 1025 -| 7686 37%) 10 412 4.3 51645 234°1186° |3 34 53.21 ,, ., | 76059 |, | 224° 411 30.37 33:98 | 59085 44° | 50’ | 10” | 3 34 12.90 4°37 | 76430 372 | 50’ 410 56.27 347° | 59594 432 40 | 20 |3 33 32.50 *°°4° | re6800 SE 2) 410 29.06 342% | 52962 43°) 30 7 30 |3 32 52.01 40°49 | rr169 373 | 30 4 9 47.73 3433 | 53309 437 | 20 | 40 | 3 32 11.48 neice T7537 See | 20 © 9 3 . 5385 2) f °° » s a(S - & ( « 10 ee Ga | es Gk eee iS} Byles 04268 2 Sie 9.9% S26? 4 8 4.07 3497 | 54702 434] 50’ | 10° |3 30 9.08 ei 78632 a 50! 4 729.99 3478 | 55134 437 | 40 | 90 |3 29 98.11 4°97 | 78995 3°3 |) 40 Bysshe) || Seen cist 5 a BD GAM An Gpoel amen go 4 6 54.40 55565 45, | 30 | 30 | 3 9847.05 4,7. | 19356 36°] 30 4 6 19.40 ee 55995 ee 20 | 40 (398 5.90 {753 | T9716 39° | 20 ¢ oo° RAC F is Q ¢ 2 ie 3 () 5 J 4 5 44.29 32"7, | 56424 J5g| 10 | 50 | 3 27 24.66 77°27 | 80075 $23 | 10 4 5 9.06 56852 232° 138° | 3 26 43.32 5 | CUS) Seabee 4 4 33.72 35°34 | 57279 427 | 50’ | 10’ |3 96 1.89 41-43 | gors9 35° | 50 4 3 58.97 35:45 | 57705 479 | 40 | 20 |3 25 20.37 47-5? | si144 Set 4 3 29.72 35:55 | 58130 425) 30 | 30 | 8 24 38.76 47% | si4gg 35% | 30 4 2 47.05 35-97 | 58554 4241 90 40 13 93 57.05 47-7? | 91861 Bee if 20 4 9 11.97 35-78 | 5g977 423] 10 50 |3 23 15.96 41°79 | g9002 Y» 10 35-89 422 41.88 i 35° E Bees538 59399 | 981° [139° | 3 22 33.38 82552 221 40) 9.38, 37-29 | 59820 477 | 50 | 10’ |3 91 51.41 47°97 | gag901 349 | (50 4 0 93.97 3°71 | go939 479 | 40 | 290 |3 21 9.35 42:09 | ggo49 34° | 40 859 47.05 3°22 | 6oes7 478) 30 | 30 | 3 20 27.20 47-25 | gs505 346] 30 3 59 10.73 3°37 | gioza 477 | 90 | 40 | 319 44.96 4274 | ga940 345 | 90 3 58 34.29 Sp: 44 61489 ae 10 | 50 |3 19 263 42°33 | s4osa se 10 ; ae 3 3 HD STD ae 61903 230° | 140° | 8 18 20.22 84627 220° 1.29 g 1.29 q 999 THE ORBIT OF URANUS. TABLE VII.—Continued. Log. r g Log. 1.29 1.380 20.22 84627 150° : 02798 37.72 ; 84968 10’ 03060 55.13 "22 | 85308 20 03320 12.45 9 85647 30 03579 29.68 : 85985 40 03837 46.83 5 86321 50 04093 3.89 86656 151° 04348 ZOE Sind 86990 10’ 04602 37.76 ; 87322 20 04854 54.56 ; 87653 30 05105 WL Pes eS 87983 40 05355 Paes 88312 50 05604 05851 06096 06340 . 06583 ho @ 06824 ia OT064 . 07302 f 07539 4 OTTT5 08009 08249 08473 08703 08951 09158 09384 09608 09831 10053 10273 10492 10709 10925 11139 11352 11564 11774 11983 12191 12397 12602 12805 13007 13207 13405 13602 13798 13992 14185 14377 14567 14756 14944 15130 15315 15498 15680 ae OH bo bo £9 bo bo bo G2 Gs Go Sup Or DOO . h t } (oe) WWwWWWWW WW WW WwW bo bo bo bo bg bo NN WWW ae) 44.4% 88639 152° 0.94 43: 88965 f 17.32 43: 89290 33.62 43° 89614 § 49.83 43: 89936 5.97 : 90257 29.02 90577 oO 10 20 30 40 bh bw NNN N ae oy oO bo bo bo bo bo bo O° 37.99 : 90895 10/ 53.88 F 91212 9.69 : 91528 5.42 ; 91842 .08 i 92155 .65 92466 .14 ; 92776 55 19. 93085 2.89 ; 93393 3 3 Se4 ; 93700 ( 3.32 ¥ 94005 50 8.42 94309 44 ; 94612 10 Spa) Te 94913 3.26 % | 95213 05 ; 95511 IT : 95808 .41 96104 Oh ; 96398 46 : 96691 87 “2 96983 5.21 ; 97273 -48 ' 97562 ‘ 97850 37.19 : 98136 ; 98421 98705 98988 99269 99549 99827 *OOLO4 \*00380 KOOGH4A *()0926 *O1197 *O1467 \*01736 *OQ2004 *(02270 * (2535 15860 *()2798 16039 *1.80 ‘ 12¢0 40 50 WwWwwwww wwwwww wwwwiodcd ot ts t tot) bo bo bo bo bo bo WwwWWWW WHWW WwW Ww oom Hwee Ans CO io) H [e) 10! bo bo bo Wo bo bo bo bo bo bo WY OO bo bo bo bo bo bo bo bo bo bo bor bo bo bo bo bo bo bo bo bo bo bb bo bo bo bo LO bo Lo bo bo bo bo bo bo NNN NNN bo bo bo to b bo NN NNN N ee ee ee en ee ee bo HOR OURS Ml TOUR UeAU IN Urls: 223 TABLE VIL.— Continued. E Toe vy} Log. r meet co 1.30 heer i 1.30 1 44 33.14 16039 52 57.82 24087 190° 1 43 49.65 5°°49 | 16916 52 4.81 575% | oaryg 89 | 507 1 42 52.11 ae 16392 51 12.98 57:53 | o49¢4 88 40 142 1.53 5°°5° | 16566 50 19.72 525° | 94359 8° | 3 1 41 10.91 eee 1673 49 27.14 aes 24434 = 20 1 40 20.24 2°°7 | 1691 Sy, BELO ovis Se aes 50.71 e : oS Gite 52.62 aout 82 ne 29.5: - 080 AT 41.99) 24599 189° 1 38 38.78 20°75 | 17248 46 49.98 52°94 | o4ezg 8° | 50 tay 4798 2-5) | 17416 iy Bee) eye) Orit: | 70) apoio: 3 | 11581 45 3.93 522°) | o4sa5 77 | 3 1 36 6.25 30°) | 17745 44 11,93 23°72 | 2410 13 | 20 iesote.s3) 2 27 |, 17908 ASS. sees | 2t9se. > 50.96 : oO 52.74 4984 73 10 1 34 24.37 18070 0 42 25.7 5 | 25057 188° ga) ase3% 222° | 18230 ATS S00 227 opnae: ae 50! fe3220933° 22° | 18388 LOMAOR SE Sagi soar eh eA0 P31 51.24 27-99 | 18545 3947-45) 2719" | 25267 6? |. 380 1 31 0.12 517? | 18700 38 54.64 ae 25334 2!) 920 c a Gyto 885 82 2.02 253° 5 130 8.96 3075 | 18854 0.38 1.82 375, | 25399 G2 | 10 2917-76 | |, | 19007 | 197° J1738° | 0 37 8.98 _ 4, | 25463 ¢, | 187° 198) 26152° 2" 7* | toda "2" | 50’ | 10" |0 86 16:12 2")? | 25525 CF || 50! 1 27 35.25 2°77 | 19308 152 | 40 | 20 | 0 35 23.25 5° )7 | 25586 40 1 26 43.93 51-32 | 19456 74° | 3 30 | 0 34 30.36 2°°°9 | 25645 22 | 30 1 25 52.58 5135 | 19603 147 | 20 | 40 | 0 33 37.45 329% | a5708 26 | 20 1 95 -1.19 5739 | 19749 10 | 50 |0 32 44.53 279? | 9575 10 1 tale ne ale 0 aa ee Bs - 24 9.77 9893 _, | 196° | 174° | 0 31 51.60 _. 25814 ., | 186° teosrisisi) 5" > | 90035, 2478) 50! |) 10° |0'30 58.66 522%" o5sg7 23 | 50/ moose 2 20176 21) 40 | 20) || 0930 15.70) 2272 | gho1g 2e | 40 Miss 23) 23) 2036, <4) 930)-|) 3p || 0/29 12.73. 2727 | ebae9) 92 | 30 1 20 43.71 eee 20454 2. | 20 40 | 0 28 19.75 oes 26017 44 20 ¢ 51. 905 3} F Om (OPRurian Js 9000 119 52.11 306 ata reg | 10 | 50 |0 27 26.76 27°22 an MG ty tomo 2) 7 | 20795 195° | 175° | 0 26 33.76 _. 26110 185 118 8.80 51°97 | 90358 133 50’ | 10’ | 0 25 40.75 UES 26154 be 50! it elelON 221201) 90990) 232.12 40) fF) 900 | O24 .4Te72 25 28.| Sorae OS | 40 fiestas 2) 7* | onnon 23) 3 80) hea Sae68 224) | 96080) 7 FIs 1 15 33.59 ee 21250 7 20 40 |0 23 1.64 Bee 26279 36 20 een sse ||) 203 5 5 22 8.58 «22% 26318 114 41.79 305) | ustt Tog | lo | 50 | 8.58 23 57 | 26818 3, | 10 1 13 49.96 21503 *| 194° J176° | 0 21 15.51 g | 26355 184° 1 12 58.10 575° | o1¢93 725 | 50’ | 10’ | 0 20 22.43 Sees 26390 35 | 50° 112 6.20 21% | 21751 773) 40 | 20 |0 19 29.35 22°75 | 26494 32 | 40 ipl taeay 2778 | 21873, 277 | 30 || 30) | 018 36:95 2-174 | 26406: 2.“ 30 110 22.31 203 | 21993 Tro] 20 | 40 [0 17 43.15 35°,, | 26486 eo le 20 96° “Qs 96 ¢ D 0 ORATA 1 9 30.33 372° | 22112 1121 10 | 50 10 16 50.03 2°) | 26515 23 | 10 1 8 38.31 22930 193° }177° | 0 15 56.91 ., 26543 123° 1 7 46.27 52-04 | 9346 17°) 5 | 10" [015 378 S313 | a65c9 22 | 507 1 6 5419 2° °° | 99460 774] 40-] 20 [0 1410.65 2253 | 26504 5 | 40 ih 1G PLUS eras 9a Cy ead ae 30 |0 13 17.51 277. | 26617 35 | 30 1 5 9:94 2° (2 | 29688 roy | 20 | 40 | 0 12 24.86 53°,5 | 26639 4, | 20 y S 997 9 i 966 = 1 417.78 377, | 22798 15g | 10 | 50 |0 11 81.21 52°72 | 26659 1, | 10 1 8 25.59 22901 192° |178° | 0 10 38.06 |, | 26678 ,, | 182° Y 2 33.37 5222 | 93008 707 | 50’ | lo’ |0 9 44.90 237" | 26605 16 | 50° We (Se De eee lon iis) =e) 40 Mhy20) Oe Svolyide 2857s | 6 ug ve 720 Hen OrasigG, 22-270) 930Ti tO 3 30 |0 758.58 25°? | 26726 72 | 30 059 56.57 57°79 | 23390 193) 90 | 40 |9 % 5:41 53°77 | 96739 =2 | 20 059 4.25 3°-3% | a34o1 (°* | 10 | 50 |0 6 19.24 ae 26750 5, | 10 52°39 10 53- 058 11.90 . ~. | 23591 |u9ae J179° |0 519.07 - , | 26760 , | 181° 0 57 19.53 52°37 | 23619 98} 50’ | lo’ jo 4 25.89 53: 26768 ° 50! 0 56 27.13 574° | 3716 97 | 40 | 20 [0 3 32.72 2317 | o6775 7 | 40 0.55 34.71 52-4? | o3g11 95) 30 30. |0 2 39.54 53:1. | 26780 5 30 0 54 49.97 57-44 | 93904 93 | 90 | 40 |0 1 46.36 531) | o67s4 # | 20 0 53 49.81 57:46 | 23996 92] 10 | 50 |0 053.18 53°'° | 26786 7 | 10 52.49 gt ; 53-15 I 0 52 57.32 24087 150° }180° |0 0 0.00 26787 120° 1.30 g 1.30 g 224 THE ORBIT OF URANUS. TABLE VIII, Ara. 1.—Acrion or JUPITER. Arg. | (v.¢.0) Diff. (use) | Cosel) |) \(urss2) ie) @oses2)i | (eses0))i i) (pase) aie (Gace) ” A ” ” Ww ” 0 55.03 ; 0.14 4.42 0.10 0.11 2331 164 134 1 55.53 + 2°59 0.13 4.42 0.10 0.12 9331 166 134 2 56.13 eee 0.12 4.41 0.10 0.12 2330 167 134 3 56.68 eee 0.11 4.41 0.10 0.12 2330 169 134 4 57.23 See 0.11 4.41 0.10 0.12 2330 171 135 5 DiOUCN a) 0.10 4.40 0.10 0.13 2329 172 135 6 58.32 es 0.09 4.40 0.11 0.13 2328 174 135 1 pie | See 0.09 4.39 0.11 0.13 9398 176 135 8 59.42 a 0.08 4.39 0.11 0.13 2327 177 135 9 59.97 Se 0.08 4.38 0.11 0.14 2326, 179 136 10 60.51 4 5. 0.08 4.38 0.11 0.14 2394 180 136 11 61.06 oe 0.07 4.3 0.11 0.14 2323 182 136 12 61.60 0.55 0.07 4.37 0.11 0.15 9391 184 137 13 6215 Sey 0.07 4,36 0.11 0.15 2390 185 137 14 62.69 24 0.08 4.36 0.11 0.15 2318 187 137 15 63.23 0.08 4,35 0.11 0.15 2316 188 137 16 63.78 2 705 0.08 4.35 0.11 0.16 2314 190 137 17 64.82 0:54 0.09 4.34 0.11 0.16 2312 191 138 18 GL eon 0.09 4.34 0.11 0.16 2310 193 138 19 65.40 ae 0.10 4.33 0.11 0.17 2308 194 138 20 65.93 : 0.11 4.33 0.11 0.17 2305 196 138 21 66.47 + 0:54 0.12 4.32 0.11 0.17 2303 198 139 22 67.00 253 0.13 4,32 0.11 0.17 2300 199 139 23 67.54 O94 0.14 4.31 0.11 0.18 9297 200 135 24 68.07 a 0.15 4.30 0.11 0.18 2294 202 140 25 68.60 0.16 an3 0.10 0.18 2291 203 140 26 Golde 0.18 4.29 0.10 0.19 2288 205 140 27 GO8G ae 0.20 4.29 0.10 0.19 2984 206 140 28 70.19 2:93 0.21 4.28 0.10 0.19 2281 208 140 29 70.72 see 0.23 4.28 0.10 0.20 | 2277 209 141 30 71.24 0.25 4.9 0.10 0.20 29074 210 141 31 41.76 + 0-52 0.27 4.97 0.10 0.20 2470 212 141 32 72.98 09? 0.29 4.26 0.10 0 21 2266 213 141 33 280 pale 0.31 4.26 0.10 0.21 | 2262 214 142 34 13.32 ee 0.33 4.95 0.10 0.21 2258 216 142 35 73.83 0.35 4.25 0.10 0.22 2254 217 142 36 14.34 ase 0.38 4.94 0.10 0.22 2949 218 142 37 14.85 a3 0.40 4.24 0.10 0.22 2945 219 143 38 15.36 ae 0.43 4,23 0.10 0.23 2240) 221 143 39 75.817 neo 0.46 4.23 0.09 0.23 2236 222 143 40 16.3 a 0.49 4.22 0.09 0.23 2931 223 143 41 16.87 + 25° 0.52 4.29 0.09 0.28 2996 2924 144 42 ny.37 | Owe 0.55 4.99 0.09 0.24 2221 925 144 43 rT Sit Pe 0.58 4.21 0.09 0.24 2216 927 144 44 18.3 oe 0.61 4.21 0.09 0.24 2210 298 144 45 78.86 0.64 4.21 0.09 0.25 “| 2205 229 144 46 79.35 + 2-49 0.68 4.20 0.09 0.25 2200 230 145 47 Ones 9» ka 0.71 4.20 0.09 0.25 2194 231 145 48 80.33 048 0.75 4.20 0.09 0.26 2188 232 145 49 80.80 eae 0.79 4.20 6.09 0.26 | 2182 233 145 50 81.28 0.82 4.20 0.08 0.26 2176 234 146 51 81.76 + °48 0.86 4.19 0.08 0.26 | 2170 935 146 52 g9i9g O47 0.90 4.19 0.08 0.26 2164 236 146 53 S70) postd 0.94 4.19 0.08 0.27 2158 237 146 54 ay au 0.98 4.19 0.08 0.27 2151 238 146 55 83.63 1.02 4.19 0.07 0.27 2145 239 146 56 84.10 +°-47 1.07 4.19 0.07 0.28 2138 240 146 57 84.55 045 eal 7)! aks 0.07 0.28 2132 241 146 58 Prirl | She 1.16 | 4.19 | 0.07 | 0.98 | 2195 242 147 59 85.46 ee 1.20 4.19 0.07 0.28 2118 243 147 60 85.91 1.25 4.19 0.07 0.29 2111 244 147 THE ORBIT OF URANUS. 229 nnn TABLE VIII, Arc. 1.—Continued. (v.c.1) | (v.s.2) Arg. (v.¢.0) Diff. (v.s.1) ” wr w" ” 60 85.91 1.25 4.19 61 86.36 10°45 1.30 4.19 62 86.80 2-44 1.34 4.20 63 87.24 ee 1.39 4.20 ae) (0. 3 | eae 8. rt: 4.20 66 88.53 +042 Wet 421 67 88.96 °°43 1.60 4.21 68 goias | 2 * 1.65 | 4,91 69 89.30 ue 1.70 4.22 70 90.21 1.75 4.22 TL 90.62 12-44 1.81 4.23 72 il g3 Char 1.86 4.23 13 91.43 ©-4° 1.92 4.24 Tt 91.83 ae 1.98 4.24 Th 92.99 > 2.03 4.95 16 92.42 +24? 2.09 | 4.26 TT 93.00 2:32 2.15 4.26 3 piss, 0-38 2.21 4.27 9 93.76 038 2.27 4.28 0. 38 80 94.14 2.33 4.29 81 94.51 19°37 2.39 4.30 82 Mi 82 2.45 4.31 33 95.93 2-36 951 | £39 84 559) 230 2.57 4.33 0. 36 85 95.95 ; 2.64 4.34 86 93.30 1°35 2.70 4.35 87 NRRL sky 2.76 4.36 838 96.93 °-34 2.88 | 4.37 89 97.32 ae 2.89 4.38 90 97.65 2.96 4.3¢ 91 OTEGT eos 3.02 4.41 92 ggia9) 2°38? 3.09 4,49, 93 gavel, 2:92 3.15 4.43 94 98.92 03" 3.22 4.45 0.30 95 99.23 3.29 4.46 96 99.53. °0°3° 3.35 4.48 97 99.83 3° 3.42 | 4.49 98 GON, be eee 3.49 4.51 99 190.41 ae 3.56 4,52 100 100.70 3.63 4.54 lol ROgigT 2h.) 18169, || | 4:56 102 NOUS25, 6 = 3.76 4.57 103 IOUS 2 tsa 3.83 | 4.59 104 101.78 3 56 3.90 Ase 105 102.04 : 3.97 4.63 106 102.29 F275 | 4.04 | 4.65 107 102.54 ey 4.11 4.67 108 LOD TBs, 4.18 4.69 109 103.02 6 53 4,25 4.71 110 103.25 4.32 4.73 111 IGE ae 4.39 4.15 112 103870) sense 4.46 417 113 NORD SREY 4.53 4.79 114 104.13 ee 4.60 4.81 115 104.34 4.67 4.83 116 10M 4 toe 4,44 4.85 117 Voaua 222° 4.81 4.87 118 1104493) so722 4.88 4.90 119 MO5Msle Cau 4.95 4.92 0.18 120 105.29 5.01 4.94 ” 0.07 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.05 0.05 0.05 0.05 0.05 0.05 0.04 0.04 0.04 0.04 0.04 0.04 04 03 .03 .03 .03 03 03 .03 .03 02 02 .02 02 .02 02 0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.02 SeSeS SS) ooo oo (v.¢.2) | (p.c.0) (p-s.1) | (p:e:1) u" 0.29 0.29 0.29 0.29 0.29 0.30 0.50 0.30 0.50 0.31 S Us oS bo bo bo to bo Guwwuwse S 20S UD US Ww w 02 0 69 LO So (ey) oo ww Ww vw oo © Oo OS Go oo ww cy ww Ww oD ts He He He He CO www & ww ww Gs G2 Us bl en GS to OO OS tO eo 9 to OS UO - pears Se) oo Ww oO Oo Oo LS = wo oo 2111 2104 2097 2090 2082 2075 2067 2059 2052 2044 2036 2028 2020 2012 2003 1995 1986 1978 1969 1960 1952 1943 19384 1925 1916 1906 1897 1888 1878 1869 1859 1850 1840 1830 1820 1810 1800 1790 1780 1770 1760 1750 1739 1729 1718 1708 1697 1686 1676 1665 1654 1643 1632 1622 1611 1600 244 244 245 246 247 248 248 249 250 250 251 252 252 253 253 254 954 255 955 256 256 257 257 257 258 147 147 148 148 148 148 148 148 148 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 148 141 140 29 July, 1873. 226 THE ORBIT OF URANUS. TABLE VIII, Ara. 1.—Continued. (oo) Diff. (uv.s.1) | (v.e.1) | (rs.2) | (.e.2) | (p-¢-0) | (p-s-1) (p.e.1) OL 4.94 0.02 0. 08 4.97 0.02 0.3: 15 4.99 0.02 0. 22 5.01 0.02 Ok: 29 5.04 0.02 0. 0.02 0.02 0.02 0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.03 0.04 0.04 0.04 0.04 N w” 105.29 a 105.46 FO 07 1d, 6 Ose, ee 105295) 7 ee 0.15 106.10 106.25 +2 *5 106.39 ae N0c'5 3) ane 106.6 oie aig} 106.78 OGRS Ommoues 107. se 107. “EO NOf22) ee -09 107.3 107. 107.49 107. 107.6 107. 107. 107.82 107.8 ORES TOME NOAk 108. 108.0¢ 108.0! 108.0 108. 108.0% 108.0 108.0: 108. 107.98 LOM. 9% 107.§ 107.8 107.82 107. 107. 107. 107. 107.4 107. 107.: 107.2 107. 107. 106.85 106. 106.65 106.£ 106.35 106.2 106. LODE 105.1 105.6 wy) 1544 1533 1521 1510 1499 1487 1476 1464 1453 1441 1430 1418 1407 1395 1384 1372 1560 1349 33 1325 1314 1302 1290 1278 1267 1255 1243 1231 1220 1208 1196 1184 1172 1161 1149 1137 1126 1114 f ee 1102 1090 1079 1067 1055 1044 1032 1020 1009 997 986 974 963 951 940 928 917 | | | 906 ) 4 . 140 140 139 139 138 138 137 137 136 135 135 134 13 133 -) Or on —_ G2 G2 Co bo bo bo bo bo on on on DAamamws bo bo LO Lo bo Nt Oo LO ye ra) on S 36 43 50 OT 64 S(t 18 85 soil 98 05 eelalt 18 25 31 38 44 oil 5T Sau Sareea es co cs oo bo bo to tO Sy war a Wwwww Www oo NNN SS DD Tat ee C bo oOo boro p rr ww bs bt to Ww G2 WwW YD Hm CLR Oo ew oo G2 Co OD bo bo bo bo bo 1 U2 GO OO OO ra) Or Or St Or Ov Or Or Or EN Or bp Wt Ee Fe OLS bo bo bo bo bo PS peeyayeyay ww (Toile ot § bo bo bo bo bo ee i Lk) _ ee He eo bo wWweowt cw th % Ww vs DPF ADAPRARDD. AAMBAMD oOomoenc Or or Or or or Ov Or Or Or or qo co 02 0) Go Oo to GD 09 G2 G9 Go OG OD S i. oa) 0 CO > SS ene > soo 8 b Ne >> Ol H Co 02 poate OOo FU wWwat Re Os 02 Ssooocoo ocooocooo COP KR HB BR Ree eee — wmT-T bo oo 05 bo bo bo bo tO WHrNRrH oop ow bo bo bo bo LO (oy) wwmo wewwuwevnvno wewos 894 883 872 861 850 — <2) sooso Sees SSse5 SSseS9 Sees Sess SSsses Ssess SSSSS SSeSsSs SSsss bo bo bo ©0)160)100100 1008 1001160110000 100) (GON C0 Ti TT) TT TTT eet Tite ToT oo or) THE ORBIT OF URANUS. TABLE VIII, Ara. 1.— Continued. | \ Arg. (v.c.0) Diff (v.s.1) | (v.e.1) | (v.s.2) | (v.c.2) | (p-c.0) i" " u 7m "” 7 Hin : ” : 180 105.62 8.60 6.47 0.17 0.29 850 181 105-4528 8.64 6.50 0.17 0.29 839 182 105.27 nee 8.68 6.52 0.18 0.29 82) 183 105.09 ae 8.72 6.54 0.18 0.29 816 184 104.90 ous 8.76 6.56 0.19 0.29 805 185 MO Lely 3s = 8.80 6.57 0.19 0.29 794 186 NOL 51 8.84 6.59 0.19 0.29 184 187 OST Ee 8.88 6.61 0.20 0.29 113 188 | 104.10 Or. 8.92 | 6.63 | 0.20 | 0.29 762 189 10389) 8.96 6.65 0.21 0.29 15] 190 M0367) 8.99 6.67 0.21 0.29 740 191 103.440" 9.03 6.68 0.21 0.29 730 192 103.21 Bee 9.07 6.70 0.22 0.29 719 193 102.98 ae 9.10 6.71 0.22 0.29 708 194 102.74 ae 9.13 6.73 0.22 0.29 698 195 102.49 oe 9.17 6.74 0.23 0.29 688 196 102.24 aor 9.20 6.76 0.23 0.29 617 197 ROIGG8) oe 9.23 67% 0.24 0.29 667 198 LOE) 9.26 6.78 0.24 0.29 657 199 TORS. 9.29 6.79 0.24 0.29 646 200 101.18 9.32 6.81 0.25 0.29 636 201 100.90 —°- 28 9.35 6.82 0.25 0.29 626 202 100.62 228 9.38 6.83 0.25 0.29 616 203 NONS3y tapas 9.41 6.84 | 0.26 | 0.29 606 204 100.04 77? 9.43 6.85 0.26 0.29 596 205 Cou ae & 9.46 | 6.86 | 0.26 | 0.99 586 206 99.447 - 3° 9.48 6.86 0.2 0.29 517 207 Goud aers 9.51 6.87 0.27 0.29 567 208 98.82 °-3? 9.53 6.88 0.27 0.29 B57 209 98.50 Be 9.55 6.88 0.28 0.29 548 210 98.13. 9.58 6.89 0.28 0.29 538 211 97.85 2°33 9.60 6.89 0.28 0.29 529 212 BIO) Sees 9.62 6.90 0.29 0.29 520 213 97.18 °34 9.64 6.90 0.29 0.29 510 214 96.84 ae 9.66 6.90 0.29 0.29 501 215 96.49 j 9.683 6.91 0.3 0.29 492 216 96.14 °° 39 9.70 6.91 0.30 0.29 483 217 goon a> 9.71 6.91 0.30 0.29 474 218 Ci) oe 9.73 | 6.91 0.31 0.29 464 219 95.06 a, 9.75 6.91 0.31 0.3 456 220 94.69 9.76 6.91 0.31 0.30 448 921 94.327 — 37 9.78 6.90 0.31 0.3 439 223 93.94 oe 9.79 6.20 0.32 0.30 430 223 93.56 ae 9.81 6.90 0.32 0.30 499 924 93.17 oe 9.82 6.90 0.32 0.30 413 225 92.78. 9.83 6.89 0.33 0.3 405 226 92.39 ae 9.85 6.88 0.33 0.30 398 227 gie99) Fae 9.86 6.88 0.33 0.30 390 228 O1-DON ier 9.87 6.87 0.33 0.30 382 229 118 Sia 9.88 6.86 0.34 0.30 374 230 90.77 9.89 6.85 0.34 0.30 365 231 90°36im sate 9.90 6.84 0.34 0.30 357 232 Rx) yy) ease) 9.91 6.83 0.34 0.30 349 933 89.51 243 9.91 6.82 0.34 0.30 849 23 89.09 See 9.92 6.81 0.35 0.30 334 235 88.65 9.93 6.80 0.35 0.30 327 236 88.29 —°°43 9.94 6.79 0.35 0.30 320 237 ha) aa 9.94 6.78 0.35 0.30 312 238 Cie aeons 9.95 6.76 0.35 0.30 305 239 86.89 ee 9.95 6.74 0.36 0.30 298 240 86.44 = 9.96 6.73 0.36 0.31 291 (p.s. 1 ) | (ea 1) 227 226, 226 bo bo bo bo Ht OF bo bo Ww bo bP bo bo bo bo me bo bo bo 200 199 198 198 197 197 196 196 195 194 194 194 193 192 192 191 191 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 58 57 56 54 53 52 51 49 48 47 46 46 44 43 42 41 41 40 39 eo oc 38 Qy ol 36 228 THE ORBIT OF URANUS. e TABLE VIII, Ara. 1.—Continued. ee Arg. (@ie50)) itt (ws.1) | (v.e.1) | (vis.2) | (v.e.2) | (p.c.0) | (p-s-1) || (pse2t) wr dd A ” wr ” 240 86.44) 9.96 6.73 0.36 0.31 291 191 36 241 85.99 °° : 9.96 6.71 0.36 0.31 284 190 35 242 |. 85.53 oe 9.96 6.69 0.36 0 31 278 190 35 243 85.08 ok 9.97 6.67 0.36 0.31 271 189 34 244 84.61 ae 9.97 6.66 0.37 0.31 265 189 33 245 S415) 9.97 6.64 0.37 0.31 258 188 33 246 SELGoins aL Ona || NG .Gian| Orie Or ad 252 188 32 247 83.21 eon 9.98 6.59 0.37 0.31 246 187 3 248 82.73 SS 9.98 6.57 0.37 0.31 240 187 81 249 82.25 Boe 9.98 6.55 0.37 0.31 234 186 30 250 Sey 9.98 6.53 0.37 0.31 298 186 29 251 SPOS ieee 9.98 6.51 0.38 0.31 922 185 29 | 252 80.80 248 9.98 6.48 0.38 0.31 216 185 28 i 9 253 80.30 25° 9.97 6.46 0.38 0.31 211 184 28 254 79.81 Bhs 9.97 6.43 0.38 0.31 205 184 27 25 79.31 9.97 6.40 0.38 0.31 200 183 QT 256 SO Se 9.97 6.3 0.38 0.30 195 183 26 257 Tao So! 9.96 6.34 0.38 0.30 190 182 26 258 Hien “oe 9.96 6.31 0.38 0.30 184 182 25 259 77.30 Bee 9.96 6.28 0.38 0.30 179 181 25 260 16.19 9.95 3.25 0.38 0.30 175 181 24 261 O29 20: 9.95 6.22 0.38 0.30 170 180 24 262 osu ses 9.95 6.19 0.38 0.30 165 180 24 263 am De 9.94 6.16 0.38 0.30 161 179 23 264 74.73 ee 9.94 6.13 0.38 0.30 156 179 23 265 74.21 9.93 3.10 0.38 0.30 152 178 22 266 13.69 ee 9193 | 6.06 | 0:38 | 0130 148 178 22 267 13.16 2-93 9.92 6.03 | 0.38 | 0.30 144 177 22 268 79.64 0-52 9.92 5.99 0.38 0.30 140 177 21 269 72.11 oy 9:91” | 5296" | 0.88 |” 0130 137 176 21 270 71.54 9.90 5.92 0.39 0.30 133 176 21 271 71.047 0-93 9.90 5.88 0.39 0.30 129 175 20 272 70.51 253 9.89 5.85 0.39 0.30 126 rt 20 273 BOLO oe 9.88 | 5.81 | 0.39 | 0.30 122 174 20 274 69.43 ee Fe" | Sir esse Ose 119 174 20 275 68.8! 9.87 5.73 0.39 0.30 116 174 20 276 68.35 0°94 9.86 5.69 0.39 0.29 113 173 19 277 67.8) ot 9.8 5.65 0.39 0.29 110 172 19 278 Gie26) 25) ) |e Os84" (i156 1 | 0s sOnnl| mioeae 107 172 19 279 66.72 ae 9.84 5.54 0.39 0.29 105 171 19 280 66.17 9.83 5.52 0.39 0.29 102 171 19 281 65.62 —°°55 9.82 5.48 0.39 0.29 100 170 19 282 65.07 255 9.81 5.44 0.39 0.29 97 170 18 283 64.52 2-55 9.80 5.40 0.39 0.29 95 169 18 284 63.97 2ae 9.79 5.35 0.39 0.29 93 169 }| 18 285 Est) ae 9.78 5.31 0.39 0.28 91 168 18 286 62.86 —2-55 9.77 5.26 0.39 0.28 90 168 18 287 GF C55 9.7 5.22 0.39 0.28 88 167 18 288 Gig SSS eG WN ate I Oe) I) O88 86 167 18 289 61.19 oe 9.75 5.13 0.38 0.28 85 166 18 290 60.64 9.74 5.08 0.38 0.28 84 166 18 291 60.08 —°: 56 9.73 5.03 0.38 0.27 82 165 19 299 59.59 2-56 9.72 4.99 0.38 0.27 81 164 19 293 58.96 05° 9.71 4.94 0.38 0.27 80 164 19 294 58.40 cer 9.70 4.89 0.38 0.27 80 163 19 295 51.84 9.69 4.84 0.38 0.27 19 163 19 296 57.28 —0-5® 9.68 | 4.80 | 0.38 | 0.97 48 162 19 297 56.79 2-56 9.67 4.75 0.38 0.26 78 162 19 298 Seg O58 9.66 4.70 0.38 0.26 7 161 20 299 55.60 a 9.65 | 4.65 | 0.33 | 0.96 “7 | Y6l/eias 300 | 55.04 9.64 4.60 0.38 0.26 | 7 160 20 PR OR BILL OH URANUS 229 TABLE VIII, Ara. 1.—Continued. Arg. (v.¢.0) Diff. (v.s.1) (v.¢.1) (w.s.2) (w.0.2) (p.c.0) | (p.s.1) (p.c.1) " " ” ” ” ” = 300 55.04 6 9.64 4.60 0.38 0.26 ra 160 20 301 54.48 9.63 4.55 0.38 0.26 fy 159 20 302 53.92 a6 9.62 4.50 0.38 0.25 ra 159 20 303 53.36 aes 9.61 4.45 0.38 0.25 11 158 21 304 52.80 Be 9.60 4,40 0.38 0.25 13 158 21 305 52.24 6 9.59 4.35 0.38 0.25 78 157 21 306 51.68 are 9.58 4.30 0.38 0.25 79 156 22 307 51.12 ace 9.57 4.25 0.38 0.24 80 156 22 308 50.56 2. 9.56 4.20 0.3 0.24 81 155 22 309 50.00 Hee 9.55 4.15 0.38 0.24 82 154 23 310 4944 9.54 4.10 0.38 0.24 83 154 23 311 48.88 05° 9.53 4.05 0.38 0.23 84 153 24 312 48.33 oes 9.52 4.01 0.38 0.23 85 152 24 313 Ante 2s 9.51 3.94 0.38 0.23 87 152 24 314 47.29 Be 9.50 3.89 0.38 0.23 88 151 25 315 46.66 9.49 3.84 0.38 0.23 90 150 25 316 46.11—°°55 9.48 3.79 0.38 0.22 92 150 26 317 45.56 255 9.47 3.74 0.38 0.22 94 149 26 318 45.01 955 9.46 3.69 0.38 0.22 96 148 27 319 LAG) >? 9.45 3.63 0.38 0.22 98 148 27 320 eecnte 2 9.44 | 3.58 | 0.388 | 0.91 | 100 147 28 321 43.36 0°55 9.43 3953 0.38 0.21 103 146 28 322 A i 25S 9.42 3.48 0.38 0.21 105 146 29 323 49.97 9-54 9.41 3.43 0.38 0.21 108 145 29 324 41.72 ae 9.40 3.38 0.38 0.20 111 144 30 325 cies 9.40 | 3.33 | 0.38 | 0.20 | 114 144 30 326 40.64 2°54 9.39 3.28 0.38 0.20 117 143 3 327 40.10 °54 9.38 3.23 0.38 0.20 120 142 32 328 39.56 2:04 9.37 3.18 0.38 0.19 123 141 32 329 39.03 ee 9.36 3,14 0.37 | 0.19 127 140 33 330 38.49 9.35 3.07 0.37 0.19 130 140 34 331 37,9623 9.34 3.02 0.37 0.19 134 13 34 332 Si isy SPOS 9.33 2.98 0.37 | 0.18 138 138 35 333 36.90 293 9.32 2.93 0.37 0.18 141 137 36 334 36.37 ae 9.31 2.88 0.37 0.18 145 136 36 335 35.85 | 9.30 2.83 0.37 0.18 150 136 37 336 S5583ha oe 9.29 2.78 0.3 0.17 | 154 135 38 337 34.81 25? 9.29 2.73 O37 | Oly 158 134 38 338 34.99 295? 9.28 2.69 0.37 0.17 162 18% 39 339 133.78 aes 9.27 2.64 0.37 0.17 167 132 40 340 $396 9.26 2.59 0.37 0.17 172 132 41 341 3975 a. 3t 9.25 2.54 0.37 0.16 176 131 42 342 32.94 °-5" 9.24 2.50 0.37 O16) | Ist 130 42 343 saya 5 9.23 2.45 0.37 0.16 186 129 43 344 31.23 cee 9,22 2.41 0.37 0.16 191 128 44 345 NOG 9.21 2.36 0.37 0.15 197 127 45 346 Dey ee 9.20 2.31 0.37 0.15 202 126 46 347 2ONT4) ve 9.20 2.27 0.37 0.15 207 126 46 348 29.25 bee 9.19 2.93 0.37 0.15 213 125 47 349 28.76 ane 9.18 2.18 0.37 0.15 218 124 48 350 28.27 9.17 2.14 0.37 0.14 224 123 49 351 27.79 —2: 48 9.16 2.09 0.37 0.14 230 122 50 352 97.31 48 9.15 2.05 0.37 0.14 | 236 121 51 353 26.93 °48 9.14 2.01 0.37 0.14 | 249 120 51 354 26.36 ae 9.13 1.97 0.37 0.14 248 119 52 355 25.89 9.12 1.93 0.37 0.14 255 118 53 356 95.49 0-47 9.12 1.89 0.37 0.13 261 117 54 3517- 94.95 0-46 9.11 1.85 0.37 0.13 | 268 116 55 358 24.50 a) 9.10 1.81 0.37 0.13 274 115 56 359 24.04 ae 9.09 ent 0.37 0.13 281 114 | 57 360 23.58 9.08 1.78 0.38 0.13 288 1 | BS I PE SE — EE ES SS SL 230 THE ORBIT OF URANUS. TABLE VIII, Ara. 1.—Continued. Arg. (v.c.0) Diff. (urs) | (se) | (es: 2)) |) (@rxe52)) (p2e20)) | Ges: 1) | (pean) ” wt ” a” Ww ” 860 93.58 x 9.08 1.73 0.38 0.13 288 113 58 361 93.137 0°45 9.07 1.70 0.38 0.13 294 112 58 362 99.69) Oatt 9.06 1.66 0.38 0.13 301 111 59 363 99.94 245 9.05 1.62 0.38 0.13 308 110 60 364 21.80 a 9.04 1.59 0.38 0.12 316 109 61 365 21.3 9.03 1.55 0.37 0.12 323 108 62 366 20.9443 9.02 | 152 | 0.8% | ‘oe 330 107 63 367 Nh es 9.01 148. | 08% |) One 338 106 64 368 90.08 — #5 SON ep Oe Ie Oe 345 105 65 369 ORCS paige 8.99 1.42 0.37 0.11 353 104 66 370 19.24 8.98 1.39 0.37 0.11 361 103 66 371 GNSS moat 8.97 1.36 0.37 0.11 369 102 67 372 Ip iey eae 8.96 1.32 0.37 0.11 377 101 68 373 IG.Gg;, Cha 8.95 1.29 0.3 0.11 385 100 69 374 17.62 ae 8.94 1.27 0.3 0.11 393 99 70 375 17.22 8.93 1.24 0.37 0.11 401 98 1 376 GES meee 8.92 1.21 0.37 0.11 410 97 72 317 16.44 °39 8.91 1.18 0.37 0.11 418 96 13 378 16.06 23° 8.89 1.16 0.37 0.11 496 95 14 379 15.68 O38 | 88g | 113 | 0.87 | 0.11 435 o4 5 Be || ee 8.87 | Walt |! 005m | Ol0N |) eae 93 | 16 381 1403 pee St 8.86 1.08 0.37 0.10 452 92 77 382 Ue Se Ee ie OS Ose || Old 461 91 78 ges | adn, 230) gies oes | oan) oul een 90 | 178 384 3.85 cae 8.82 1.02-| 0.37 | 0.10 479 89 79 385 3.50 a 8.81 1.00 0.37 0.10 488 88 80 886 13.15.39 8.79 0.98 0.37 0.10 497 87 81 387 12.80 °35 8.78 0.96 0.37 0.10 506 86 82 388 Te eS a7 | 0.94 1, 0187 | ono 514 85 83 389 123 oe 75 1] 0b92) | s0:si | SOG 525 84 84 390 11.81 ; 8.74 0.90 0.3 0.10 534 82 85 391 Tsim 89 8.72 0.88 0.3 0.10 544 81 86 392 Gan oe 8.71 0.87 0.3 0.10 554 80 86 393 Sa 2 8.70 0.85 0.36 0.10 564 79 87 394 LO ee ee 8.68 | 0.84 | 0:36 | 0.10 513 78 88 or) 395 10.23 | 8.67, 0.83 0.36 0.10 583 TT 89 396 Oho 3 mee 8.65 0.82 0.36 | 010 593 76 90 397 Ce ee 8.63 | 0.80 | 036 | 0.10 603 15 91 398 9.35 Ae 8.62 0.79 0.36 0.10 613 14 92 399 9.07 nae 8.60 0.78 0.36 0.10 623 13 93 400 sig. | 8.58 0.77 0.36 0.10 633 72 93 401 Cp) eed $5 |, OlTG™ | w0s86 | s0ul0 643 v1 o4 402 S205 pas 8.55 0.76 0.36 | 0.10 653 70 95 403 ieee oe 8.53 0.75 0.36 0.10 664 69 96 404 Was wens 8.51 0.74 0.35 0.10 674 68 97 405 147 8.49 0.74 0.3: 0.10 684 67 98 406 opr aoe) 8.47 0.74 0.35 0.10 695 66 98 407 6598) eoo44: 8.46 0.73 0.35 0.10 706 65 99 408 Baits PES) 8.44 0.73 0.3: 0.10 716 64 100 409 Gol 2 es 8.42 | 0:73 | 0.84 | 0.10 727 63 | 101 410 6.29 8.35 0.73 0.34 0.10 137 62 101 411 GiOTancsee 8.37 0.73 0.34 0.10 748 61 102 412 Bagh.) O22 8.35 0.73 0.34 0.10 159 60 103 413 5iG4 o3t 8.33 0.73 0.3 0.10 770 59 104 414 6.44 8.31 Oy i CS) O30 781 58 | 104 415 5.4 8.28 0.73 0.34 0.10 792 5Y 105 416 50d case 8.26 0.74 0:33 0.10 803 56 106 417 £36 228 8.24 0.74 0.33 0.10 814 55 107 418 Ayre SPLS) 8.21 0.74 0.33 0.10 825 54 107 419 4.50 re 8.19 0.75 0.33 0.10 836 53 108 420 4.33 8.17 0.76 0.33 0.10 847 52 109 eR A AR SA I SS STL THE ORBIT OF URANUS. 231 TABLE VIII, Ara. 1.— Continued. Dif. | (v.s.1) | (v.c.1) | (v.s.2) | (v.c.2) | (9.0.0) | (8.1) 8.17 0.76 14 0.76 ell 0.77 09 0.78 06 0.79 04 0.80 OL 0.81 -98 .82 95 .83 -92 85 89 86 .83 80 oUt 74 offll 67 64 60 25it 54 .50 AT 43 = = ~ ” 10 847 52 10 858 51 lel 870 50 11 881 50 892 49 904 48 916 47 927 46 938 45 950 44 961 44 973 43 984 42 996 41 1008 40 1019 40 1031 39 1043 38 1054 38 1066 3T 1078 1089 1100 1113 1124 1137 1149 1160 1172 1184 1196 1208 1220 1232 1243 1255 1267 1279 1291 1302 1314 1326 1338 1350 1361 1373 1385 1396 1408 1420 1451 1443 1455 1466 1478 1489 | 9 0° ©) G2 Go G2 GO OO Lo bo bo bo Oo ) | 9 99900 ee oo) 99000 We ct wo vd cor 87 89 90 2 cs o TTT ATT OTA HATO WOOHOO 94 | 2) eos sesss sesses sssse bo po bo bo bo bo bo bo bo bo wmocqw$we ww rk» FDNONWFE ODAADDS Crew RO woo H oo Pwoarn WWWNWNwW RRO ND COMORW 9900 — —_ iS for) ° i=) boc 909999 9900 He Oro eo G2 03 SO 0D (ee) 9 0 36 32 28 D4 20 16 .12 08 04 00 Oc cc conmp oo m= bo bo Go oo 0 09 OD 09 bo ke Or f=) aT a aaa aa a a a On a TT -T bo bo bo bo bo On ae Oo 2. 2. 2 11st ile ek 1 ik, 1’; ile 1.8 1.8 Ie iak 1. He bo bo bo Ww We CD eososos sesss ssses sesso ss bo Six rE toto tok bo bo bo b = S bo bo bo bo LO ooo bo So Oo er bo b essos ssese sssss Ssees5 Ssess5 SS2995 SS955 SSSe5 Soses Ssose Hm LObo rR bo CO ar fo} CROBORORS + O° ©} © f) 2) NPD CO MOATS bo ww bo bo b> bo bo bet Fb DO boa bo Tb bo bo a + ©) —) to BO BO BO DO RO i et OOOO OOO COCO COCO SLO BD RRS Tey ie oy - x . Whole eoatre ioe a 5 Syauenitenee s 5 bo bo LO 19 LO Depmw oe won S = more o wo 99090 Lo bo bo LO LO bo w PR RR 9222 g2 9 99 99 92 99 tO LO tO tO SS N29292 2eloPe aS on bo 232 THE ORBIT OF URANUS. TABLE VIII, Ara. 1.—Continued. Arg. | (v.c.0) Diff. (v.s.1) | (v.c.1) | (v.s.2) | (v.c.2) | (p-¢.0) | (p.8.1) | (p.¢.1) ” Wt A ” | wu ” 480 £65 10a 5.81 2.18 0.15 0.12 1546 23 132 481 Tee: 5.15 2.21 0.15 0.12 1558 23 133 482 5.02 ate 5.70 2.24 0.15 0.12 1569 23 132 483 5.21 ee 5.65 2.28 0.14 0.12 1580 23 132 484 5.40 ee 5.59 2.31 0.14 0.12 1591 23 133 485 Slee 5.54 2.35 0.14 0.12 1602 24 133 486 Beg Wee 5.48 2.38 0.13 0.12 1613 24 133 487 G03 aoe 5.43 2.42 0.13 0.12 1624 24 133 488 (24 pe 5.37 9.45 0.13 0.11 1635 24 134 489 6.47 ae 5.32 2.48 0.12 0.11 1646 25 134 490 6.70 5.26 2.52 0.12 0.11 1657 25 133 491 6.93 +923 5.20 2.55 0.12 0.11 1668 25 133 492 elle) See 5.15 2.59 0.12 0.11 1679 25 133 493 lee see 5.09 2.62 0.11 ONT | 690 25 133 494 T66 5.03 | 2.66 | -0.11 0.11 | 1700 26 133 495 (CO 4.98 2.69 0.11 0.11 1711 27 133 496 8.18 eae 4.92 2.72 0.11 0.11 1721 27 133 497 Le Se 4.86 2.76 0.10 0.11 1732 28 133 498 SoTL Bee 4.80 | 2.79 0.10 0.11 | 1742 28 133 499 8:99) | 4.75 2.83 0.10 Ota) alz53 28 133 500 912600 8 4.69 2.86 0.10 0.11 1763 29 133 501 9155 F°:29) 9) 4463") areQ ee O-lO |) Ose ra 30 133 502 O84 222 4.57 2.93 | 0.09 0.10 | 1784 30 133 503 HONIS) Beas 4.51 2.96 0.09 0.10 | 1794 31 133 504 10.43 03? 4.44 2.99 0.09 0.10 | 1804 31 133 re) 505 10.74 4.39 3.02 0.09 0.10 1814 32 133 506 11.05 7 °3! 4.33 3.06 0.09 0.10 1824 32 133 507 WeSGue oe 4.28 3.09 0.08 0.10 1834 33 133 508 TAG Seer 4 22 Sale 0.08 0.10 | 1844 34 133 509 12.00 = 4.16 3.15 0.08 0.10 | 1853 35 133 ne) 510 12.33 | | 4.10 3.18 0.08 0.09 1863 35 133 511 12.66 + 0°33 4.04 3.21 0.08 0.09 | 1873 36 133 512 13.00 oe 3.98 8.25 0.07 0.09 1882 37 133 513 13.34 ner 3.92 3.28 0.07 0.09 1892 38 133 514 13.68 ae 3.86 3.31 0.07 0.09 | 1901 39 133 are) 515 14.03 | 26 3.80 3.34 0.07 0.09 1910 40 133 516 14-39 Pe 3.74 88 0.07 0.09 | 1920 40 133 517 MS 22 3.68 3.40 0.07 0.09 | 1929 41 133 518 Ida eS 3.62 3.43 0.07 0.09 | 1938 42 133 519 15.48 Bee 3.56 as 0.07 0.08 | 1947 43 133 520 15.85 4. 4g 3.50 3.49 0.06 0.08 1956 44 132 521 1G 230 ce 3.44 3.51 0.06 0.08 1964 45 132 529 16.61. 5 3.38 3.54 0.06 0.08 1973 46 132 523 G99 2 3.32 3.57 0.06 0.08 1982 47 132 524 17.38 55) | 3.27 | 3.60 | 0.06 | 0.08 | 1990 48 132 525 ena 3.21 3.62 0.06 0.08 1999 49 132 526 al are 3.15 3.65 0.06 0.08 2007 50 132 527 IG ae 3.09 3.67 0.06 0.08 2016 51 132 528 1G See 3.03 3.70 0.06 0.07 2024 52 132 529 19.38 ue 2.97 Sige |) e805 0.07 | 2032 53 132 530 IE) 2.91 3.75 | 0.05 0.07 | 2040 55 132 531 90.91 +°-42 2.86 3.1 0.05 0.07 | 2048 56 132 532 90.69. te 2 80 3.80 0.05 0.07 2056 5T 132 533 21.05 °43 2.75 3.82 0.05 0.07 2063 58 132 534 21.47 ae 2.69 3.84 0.05 0.07 2071 59 131 535 21.90 | | 2.63 3.87 0.05 0.07 2079 61 131 536 a2.34 0 38 2.57 3.89 0.05 0.07 2086 62 131 537 22.77 ae 2.52 3.91 0.05 0.07 2094 63 131 538 23.21 eee 2.46 3.93 0.05 0.07 2101 64 131 539 23.66 ae 2.40 3.96 0.05 0.06 2108 66 131 540 24.10 2.35 3.98 0.05 0.06 2115 67 131 THE ORBIT OF URANUS. 233 TABLE VIII, Ara. 1.— Concluded. (v.c.0) Diff. (CESS Ges DY (Cae) (v.c.2) (p-c.0) | (p.s.1) | (p.e.1) 7 ”" = > “ur ur “ 3E 3.98 29 4.00 24 4.02 18 13 08 .02 97 92 87 24.10 24.56 10-48 Doty 24> 25.47 oe 25.93 3.46 fen to 46 97.32 0-47 HC, ee 98.97 0-4 0.48 28.75 99.93 +0-48 29.71 ° 8 30.20 30. 31.1: 31.6 32. 32.6 33. 0.06 2115 6T 131 0.06 2122 68 131 0.06 2129 70 131 0.06 2135 71 130 0.06 2142 73 130 0.06 2149 74 130 0.06 2155 75 130 0.06 2162 TT 130 0.06 2168 78 130 0.06 2174 80 130 0.06 2180 130 0.06 2186 5 130 0.06 2191 13 0.06 2197 3 0.06 2203 0.06 2208 0.06 2214 0.06 2219 0.06 2224 0.06 2229 0.06 2234 0.06 2238 0.06 2243 0.06 2248 0.06 2252 0.06 2256 0.06 2261 0.06 2265 0.06 2269 0.06 2272 0.06 2276 0.06 2280 0.07 2283 0.07 2287 0.07 2290 0.07 2293 0.07 2296 0.07 2299 0.07 2302 0.07 2304 0.07 2307 0.08 2309 0.08 2312 0.08 2314 0.08 2316 0.08 2318 0.08 2319 0.09 2321 0.09 2322 0.09 2324 0.09 2325 0.09 2326 0.09 0.10 0.10 0.10 0.1u 0.11 0.11 0.11 0.11 Se) SSS SSS Se Sar) Cror Grower or Or St Or Or Or Or 17 "9 OPS SSeSee SSLSee 2 bo wo Go 0S 9) Oo bo uw 9 1 2 2 3 4 5 6 1 8 G5 Gs Go te (al aaa al al etal allt al-alal-al-alllalebal-nl gilt al alata alll alata al all ot ol pl pl ofl ol of al ol oll al otal ate Seas WNwWMwWNww BWR Re WoOIToe WrADWwWow <3) o oeoeoo esscocoo eceocesc ceoseoscoc ecseosoeoo eseososo COOH HH Tess a Ftd ah ee eet Pe Pl eg Peed re oe Pe Pek Fee Feet Peed ONO OU RON EROTRO) crore > mom G wo. wo uo o Go Go tO ww vo ww OD bo bo bo bo LO eS) a (er) bo (oe) Oo = le cs H bo 30. July, 19873 234 THE ORBIT OF URANUS. TABLE IX, Area. 2.—AcTION oF SATURN. im re. (v.c.0) Diff. |(v.s.1) Diff. See.var.|(v.e.1) Diff. Sec.var. (v.s.2) Diff. See. var.|(v.c.2) Diff. See.var, me ” ” ” “ " ” 182.62 UT 1244.67 0.12 1g2.80°t ace 16 (244.10 297 oi 184.96 tte 1.74 [243.52 a 0.12 186.12 1° 73 [242.93 ae 0.11 187.27 775 1.71 | 249.32 22 game T.15 0.6% 188.42 241.71 - 0.1m 189.56 +114 241.09 02. 0.1mm 190.69 113 240.46 723 O0mm HEN 239.81 ° 22 O10 19gige ace 239.15 ©. Om 194.05 238. 48 0.10 195.16 +1 TE 237.80 —° 08 196.26 3-10 937.11 269 NOKS5I Ce 936.41, aie 198.43 ae 935169) Cue -08 0.73 199.51 96 200.58 11-07 99 C14 | e064 1-00 43, 0:72am ) 202.69 1:05 13 ee : | 203.74 %:Cd 96. O77 1.04 0.78 204.78 205.81 1-93 BOGKSS) oe D084 sect B0ee85) eae 1.00 209.85 910,84 +299 Osa ees 212.79 °-97 F176) ee ie 215.66 +2°95 216.60 094 217.53 0-93 O1Grda) ae 0.91 219.35 220.25 221.14 222.03 D299 “as. ” " ” ” ” 38.54 139.94 2.53 38.63 +009] 141.38 +744 9.59 38079 1 0-CO Tas soe = B8lsi 2°29) 14496 38.90 °°2)145.70 .08 38.98 147.14 39,07 Ne oo) 148.58 os 39.16 ee 150.01 39.24 a 151.45 | 39.33 ve 152.89 =i 39.41 32 50 + 0°09 1155.76 +2 Bee yloteloye -67 nell 58.62 ; 5 fi) 09 .05 =} 84 48 93 +998) 169.99 + Opes calli ey) bs -09 2 te 08 ay $0.08 mg 28 Ai 62 Ov or on on Or wRewran OV Or — So On RwWhHS ces esess bo bo bo bo Hm ¢ nS oo He He a0 SS) He or} pow ro bo ws) reg Go co C2 bo bo bo bo bo bo bo bo bo bo aAamoor mm bo Ol > A be wwwww Wt mmm WWERAD WOOD & 02 0 He He me He hon poe (Sy) 18 a0 .09 Broo SSO ee he Rin ee a -09 “bn $0-09 35 -10 om -09 “Bd LO 2 bo po wo bo bo bo bo bo G2 CO alto Oo = Co Co Oo Oo 09 WWEUD AADDMMD CODORrNwW FANGS T ODUOorehby bo bo bo bo bo bo bo bo & OO af CS) Oo He OR bo b bo bo bo bo 19 C} Ww bo bo 69 bo bo 19 bo bo WO 1 me we ann oS yf = On bo bo bo bo bo a bOLO POW Ll 3 Co Co W& OD 02 bo bo bo bo bo a re 0 0.90 bo bo to bo bo bo bo bo Ge G2 G2 NHOWAR An QD bo bo bo bo LO oo OFF WLS bo bo bo bo bo bo bo bo bo bo bo TT AIDS _ w 206.64 205.61 204.57 Oooo eee OOOO WO bob PO WO = = N ~ we a) o_— m< ng D Es 903.52 902.47 7° 201.41 7 200.34 199.26 a 198.18 197.09 = Moen = 194.90 193.79 | 192.67 N Oa GW 02 G2 Gn G2 Go OG Go Oo Ko) _ oS bo bk bo bo bo eo Oo bo bo bo nN Ne) nN ee OR Oe OH OR ee : © « eprIcnto aigin (G15 > A - ; sie O20 Ny oo CO N N Nn fon BO BO DO DRO eR Ot T. I. hon NNN Oo POU 2 oSs998 SSeS5 SSS99 SSoSsSeS SSSS5 SSSsSes SSSSsS SSsssSs sssss Soe: co coesss Ssssss ssess Ssesss sssss SSsse: ja ee ree ee ee rs ; Dm wD: Db BPH SSS WMO1-1D bo ESE ORB SOE UEReAGN Wise TABLE IX, Are. 2.—Continued. Arg. | (v.s.3) | (v.c.8) | (v.8.4) | (v.c.4)| (p.c.0)] (p.8.1)] (.c.1)] (p.8.2)| (p-c.2)| (9.8.3)| (p.¢.3) ” ” ” | ” | 0 11.40 14.53 1.58 | NS | TGS 4) ER |) ales |) Yala |) SYR | AY 1 11.52 14.48 159) 9 | E66) G9) 182" 1526) | ve S0on | Lai 2 11.64 14.43 1.60 | 1.65 | 1680 | 183 | 1539 | 710 | 998] 137 3 11.75 14.38 1.61 | 1.64] 1680 | 184] 1551] 709 | 9951 137 4 11.87 14.33 162) 163) West |) 185) | W564 |) 708) |) 291 | 137 5 11.98 11.28 1.63 | 1.62 | 1682 | 187 | 1577 | 707 | 988} 187 6 12.10 14.22 164 | Wel |) 1682 | iss | 1590) 706) || 984 |) 137 q 12.21 14.16 1.64 | 1.60 | 1683} 190] 1603 | 704] 981 | 1388 8 12:32 14.10 1.65 | 1.59 | 1683] Lor | 1615 | 03 | 978 | 138 9 12.43 14.04 1.66 1.58 | 1682 | 193 | 1698 | OL | 9%4 1 138 10 12.54 13.97 1.67 1.56 | 1682 | 195 | 1641 TOON eeoilullalss 11 12.65 13.91 1.68 1.55 | 1681 | 197 | 1654 |} 699 | 267) 138 12 12.76 13.84 1.69 1.54 | 1680; 199] 1666 | 698] 264] 138 13 12.87 13.77 1.69 1.53 | 1679 | 201 | 1679 | 696 | 960] 188 14 12.97 13.70 1.70 1.52 | 1678 | 203 | 1691 | 695 | 257] 137 15 13.07 13.63 Ten 1.50 | 1677 | 906 | 1704 | 694 | 953 | 137 16 13.18 13.56 neil 1.49 | 1675 | 208 | 1716 | 692 | 9250; 137 17 13.28 3.49 1.73 1.48 | 1674 | 211 | 1799 | 691 | 246] 13 18 13.35 13.41 1.73 1.47 | 1672 | 214 | 1741] 689] 243] 13 19 13.48 13.3: 1.74 1.45 | 1670 | O17 | 1754) 687 | 239) 137 20 13.58 13.25 1.74 1.44 | 1668 | 220] 1766] 685 | 236 137 21 13.68 13.17 1.75 1.43 | 1666 | 293] 1778 | 683] 233] 13% 22 13.78 3.09 1.75 1.42 | 1664 | 227 ]1790| 681 | 230] 1387 93 3.87 13.01 1.76 1.40 | 1682 | 230] 1802 | 679 | 296] 1837 24 13.97 12.92 1.76 1.39 | 1660 | 234 | 1814] 677] 223) 137 25 14.06 | . 12.83 NET 1.37 | 1658 | 237 | 1896 | 675 | 220] 137 26 14.15 12.74 1.78 1.36 | 1656 241 | 1838 673 Q17 13 27 14.24 12.65 1.78 1.35 | 1653 | 245 | 1850 | 671 | 214 | 137 28 14.3: 12.56 1.79 1.34 | 1651 | 949 | 1862) 66s | O11} 18 29 14.41 12.47 1.79 1.32 | 1648 | 253 | 1874 | 666 | 208 | 137 30 14.50 12.3 1.79 1.31 | 1645 | 9258 | 1886 641 905 | 137 31 14.58 12.28 1.80 1.30 | 1642 | 9262] 1898] 662] 202} 137 32 14.66 12.19 1.80 1.28 | 1638 267 | 1910 659 199 137 33 14.7 12.09 1.80 1.27.| 1635 | 272 | 1992 | 657 | 196 37 34 14.82 11.99 1.81 1.25 | 1631 | 977 | 1984 | 654] 193] 137 35 14.90 11.89 1.81 1.24 | 1627 | 289 | 1946 | 652) 190] 137 36 14.97 11.79 1.81 1.22 | 1623 | 287 | 1958 | 650] 187 | 187 37 15.05 11.69 1.81 1.21 | 1619 | g92 | 1970) 647 | 184] 137 3) 15.12 11.59 1.82 1.20 | 1615 | 297 | 1982] 645] 181] 18 39 15.19 11.49 1.82 1.18 | 1611 | 302 | 1993] 642] 179] 137 40 15.26 Tel ey 1.82 1.17 | 1608 | 308 | 2005 | 640] 176] 1837 41 15.3: 11.28 1.82 1.15 | 1602} 313 | 2016 | 6387 | 173 | 187 42 15.3 Heaer 1.82 14 |) 59% |) S19) 20285) 635) 15 Wid 13 43 15.46 11.06 1.82 1.13 | 1592 | 325 | 2039 | 632] 168] 137 44 15.52 10.95 eo 1.11 | 1587 | 331 | 2051} 630] 166) 137 45 15.58 10.84 1.82 1.10 | 1582 | 3387 | 2062} 627} 163 | 137 46 15.64 10.73 1.82 1.08 | 1577 | 343 | 2073 | 624 | 160] 137 47 15.70 10.62 1.82 1.07 | 1572 | 349 | 2084] 621 158 | 137 48 15.76 10.51 1.82 1.06 | 1566 | 355 | 20951 619] 155] 13 49 15.81 10.40 1.82 1.04 | 1561 | 362 | 2106] 616] 153) 137 50 15.86 10.28 1.82 MOS ISSE I eSESu ella Gls: |. 150N | 137 51 15.91 10.17 1.82 1.01 | 1550 | 375 | 2128] 610] 148 | 137 52 15.96 10.05 1.82 1.00 | 1545 | 381 | 21881 607 | 145) 137 53 16.01 9.94 1.81 0.98 | 15389 | 388 | 2149] 604] 143] 137 54 16.05 9.82 1.81 0.97 | 1583 | 394 | 2159 | 601 | 140] 137 55 16.09 9.70 1.81 0.96 | 1597 | 401 | 2170 | 598} 138] 137 56 16.13 9.59 12a 0.94 | 1521 | 408] 2180] 595 | 135 | 187 57 16.17 9.47 1.80 0.93 | 1515) 415] 2191 | 592] 133] 137 58 16.20 9.35 1.80 0.92 | 1509 | 422 | 2201] 589] 138 137 59 16.23 9.23 1.80 0.90 | 1502 | 430] 2212] 586} 129] 137 60 16.26 9.11 1.80 0.89 | 1496 | 437 | 2292 | 583] 127 | 137 236 THE ORBIT OF URANUS. TABLE IX, Ara. 2.— Continued. Arg. (ey) De (v.s.1) Diff. See.var. (v.c.1) Diff. See.var./(v.s.2) Diff. See.var.|(v.c.2) Diff. Sec.var. hd Ae: ” pea ” : ” ” ” ” ” ” N ” " 30 | 44.35 _ (299. 78 |270.69 0.14 | 238.85 0.91 |192 sony |222011 eee oi 238. 92.67 0.22 61 a4.a7 eee 223.30 733 1.76 |269.89 eee 0.13 939.49 +°-64 0.90 |191.55—"°!? 0.93 62 pees 012 (224-52 715, 1-75 |269.09 Co 0.13 [240.12 we 0-98 [190.42 toeiaae oi ee 225,73 13, 173 [268.28 fig, 0.13 [240.74 Cig 0.87 |189.29 1-13" \0iom ; 4.83 27 |226.94 "7. 1-72) |2e%i45 Big 0.13 ]241.35 ° 7° 0.86 |188.14 TTS 9.95 65 | 44.95 298.14 1.70 |266.61 .. 0.12 .\241.95 ie Baar Ee, oa, 241.95 0.84 | 186.99 0.25 & pa ora [229-33 77g 1.69 | 265.16 ee 0.12 249.53 12:58 0.83 |185.83—2°2° 9.96 a oak o.12| 200-51 yg 1.67 1264.89 f'g7 0.12 | 243.10 ood 0.82 |184.67 110 0H G8 | 45.51 375/281-6977) 1-66 /264.02 O57 0.12 |243.67 °-57 9.81 [183.51 <7? OE 5.44 ~ $753 /282.86 112 164 [263.13 O'? 0.11 |244.09 ey 0.79 [182.34 7 22 900e 10 | 45.56 234.02 1.63 }262.23 0.11 {2 , Ne 5.56 4, 1, {234.02 , ,. 1.63 |262.93 11 |244.76 , 0.78 |181.16 0.29 ie 45.68 O12 [280-17 ae 1.62 |261.32—°97 0.11 245.987 °°? 0.77 1179.97. 2 0.0 72 Maal o12 [286-31 7/74 1.60 1260.39 993 0.11 245.79 c 35, 0.75 [118.78 — a2 | 237.45 73 1.59 1259.46 ‘93 0.11 |246.29 °° 0.74 [177.59 =| 22am 604 iy2/288-58 112 LST [258.52 5-94 0.11 (246.78 o 4g O73 [176.40 3°22 0.88 "5 | 46.16 239.70 1.56 [957.56 0.11 |947.96 e 75 | 46 9. 56 | 257. 11 | 247. 0.71 }175.2 16 | 46.29 + 5 13/940 8175 5 1.55 |256.59—°-97 0.11 Joan ia toa? 070 [173.9922 0.34 i ee ae 241.91 y/o 1.53 |255.61 FP" 0:11 [248.18 8450.69 |172.78 =yaaumine ie Oe Bee 243.00 tog 1-52 | 254.62 ae 0.11 |948.62 9:44 0.68 1171.56 7°22 0.36 ( 6.65 ¢1>/244-09 5/49 1.50 | 253.62 Ase 0.11 |249.05 °43 0.66 1170.34 7:2? 0.37 80. | 46.77 245.17 1.49 [05961 0.1 Io ee ae BIT ope] 245.17 4 yo, 1.49 ]252.61_ 11 }249.47 0.65 |169.11 0.38 a Hi o12| 246-24 "y 156 1.48 /251.59 see eau 249,87 +°°4° 0.64 |167.88— "23 0.39 BoM an 247.30 y'og 1.46 |250.56 1°03 0.11 1250.26 33g 0.63 ]166.65 1°73 0.40 Bh) a 248.35 593 1.45 /249.51 1°03 0.11 |250.64 °°3° 0.62 |165.41 7°24 0.41 1.24 6 12 (249.38 5, 1.43 |248.45 ee 0.11 |251.01 ae 0.61 |164.17 *°24 0.49 85°| 47.36 250.41 1.42 |247.88 | 0.11 |951.37. 0 ‘3 5 Bo r2 (200-41 ay og 142 [247.88 ay Wom 60 |162.92 0.43 86 | 414s t-'? |a51.43 00% 1.40 |246.20- 10° 0.12 951.71 16-34 0.58 |161.67 129 0.48 87 | 47.59 OTT )252.44 o°* 1.89 |245.21 1-99 0.12 1959.04 °-33 9.57 1160.42 7°72 0.44 : ae é = : 1.0 : . a . DU, 2a - as oe err feet Sigg 138 [24412 | 29 0.12 | 252.36 °-3? 9.56 |159.16 1-2000gma 8° 1.82 31, [254-48 63g 1-86 [243.01 1e 0.12 1959.67 °3% 0.55 |157.90 22° 0.46 52 5 : 29 ; 1.2 ; 90 | 47.94 255.41 1.35 |241.89 0.12 |259 - 5 ae ei, BO) |, .12 |952.96 - . 0.54 [156.63 0.47 1 ss or [206-38 og 188 [240.7613 0.12 953.24 +228 0.53 |155.836- "> 0.48 ae 257.34 Si), 1.32 [230.62 TTF 0.12 [253.50 fo. 0.52 |154.09 1-275 04g 48 | 48.27 crt [22829 Gogg 1-30 [238.48 714 0.13 253.75 "25 09.51 [159.82 7°77 0.50 H4 | 48.38 $17 /259.23 C9) 1.29 [287.83 113 0.13 /253.98 $33 0.50 1151.55 en 95 | 48.49 260.16, .. 1.98 1286.16 __, 0.18 954.91, 0.49 {15 a 96 | 48.59 +81 [e618 Toor 1.26 934.982 23 0.13 leeds $222 ne has 99 28 oe 9 Ear ) 9) ¢ 99 1.1 qe ae ae +2 ta ea +2 ; or | 48.10 31, |261.99 Og) 1.25 [293.79 119 0.18 |254.63 O70 0.4% Lat. T1 3°20 0.53 BO i Bw 1.23 |232.60 1°)? 0.13 |254.82 279 0.46 [146.42 172 0.56 f 99 | 48.91 S72) 263.76 Sig, 1.22 ]281.89 727 0.14 | 254.99 ry 0.45 145.13 Fee 1100 | 49.01 264.63 1.20 |230.18 0.14 ; He : 40.10 | 264-63 4.0.36 230.18 0.14 |255.15 0.44 |143.84 0.58 101 oi oro | 205-49 "os 1-19 ]228.96—%2? 0.14 256.30 °°75 0.43 [142.5579 0.59 [02 P21 266.34 oig4 1.17 /227.73 1-23 0.15 1955.43 5113 0.42 |141.26 von ned Be as 267.18 o1g2 1.16 [226.50 1-93 0.15 |255.55 ,, 0.41 |189.97 4/45 0.61 40 3772 |268.00 Gigy 1.14 ]225.25 %73 0.16 |255.66 55 0.40 |138.67 pe 0.62 J 105 | 49.50 268.81 1.13 }293.99 75 te nad. 106 | 49.59 T°°9|o69.61 +282 111 |o09 73-126 O16 [oerres 7°32 oa 1180 0E eect ee ooo es 222.731 0.16 255.83 9-0" 0.39 |136.07 3° 0.65 ee nee se 270.41 gpg 1.10 [221.46 12% 0.17 |255.90 ‘O7 0.38, | 13430 amu | tee i eecs 271.19 gi77 1.09 [220.18 17° 0.17 ]255.95 > 0.37 |13347 isda | 109 | 49.85 6°55 |271.96 G7, 1.07 |218.88 1°59 0.18 | 255.99 c-04 0.86. |182:0 ep ammiaee “a ae .03 1.30 re oes noes OE ach 1.06 /217.58 | 0.18 | 256.02 , 0.35 |130.87 0.69 ae 09 Gog [273.45 TS 7% 1.05 [216.2839 0.18 256.04 +00? 0.34 {199.57 773° 0.70 ie ee ee 1.03 {914.97 1-3! 0.19 | 256.04 °°°° 0.83 |198.26 1°32 0.71 J 113 | 50.19 515, /274.90 5-72 1.02 [218.65 13° 0.19 256.03—0°°? 0.38 |196.96 “73: Og 114 | 50.26 5759]275.60 tee 1.00 | 212.33 eon 0.20 }256.60 °°3 0.32 |125.66 773° 0.74 : 5 0.04 1.30 115 | 50.3 276.29 0.99 |210.99 0.20 |255.96 0.31 |124.36 0.95 j its 50.41 Fo oF 276.91 Fo. 0.98 |209.65—"'34 0.91 |255.91—2°°5 0.80 |123.05— 73> Oemb eae 48 O01 /277.63 S og. 0.97 |208.31 -34 0.91 | 255.84 °°2 0.29 |121.75 * Soa J 1s | 50.55 Foy [278-28 S15, 0.96 | 206.96 a8 0.22 1255.76 °° 0.98 |120.45 73° 0.78 J 119 | 50.62 S07 /278.92 7g, 0.94 [205.60 13, 0.22 (255.67 2-09 0.28 |119.15 7 /Saeesey | 120 | 50.69 279.55 0.93 | 204.23 0.93 1255.56 0.27 | 117.85 08 76 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 THE ORBIT OF URANUS. OR SS ACS TABLE IX, Ara. 2.— Continued. (v.€.3) | (v.s.4) |(v.¢.4)} (p.c.0)] (p.8.1)} (p.c.1)] (p.s.2)| (p.c.2) (p.8.3)) (p.c.3) ” ” ” Peal 1.80 0.89 | 1496 437 | 9999 583 | 127 137 71 9.00 Wey) 0.88 | 1489 445 | 2233 580 | 125 137 70 8.88 aes) 0.87 |} 1483 453 | 2943 ot |) 123 37 70 8.76 1.78 0.85 | 1477 460 | 2254 574 | 121 137 70 8.64 1.78 0.84 | 1470 | 468 | 2264 570 | 119 137 69 8.52 1.78 0.83 | 1464 476 | 2275 567 | 117 136 69 8.40 We tH 0.81 | 1457 484 | 2285 564 | 115 136 68 8.28 at 0.80 | 1451 492 | 2295 560 | 113 136 68 8.16 1.76 0.79 | 1444 501 | 2306 557 | 112 136 67 8.04 We) 0.78 | 1488 509 | 2316 553 | 110 136 67 7.91 1.75 0.76 | 1431 518 | 2326 550 |} 108 136 66 ot) 1.74 0.75 | 1424 527 | 2336 546 | 107 136 66 Reo 1.74 0.74 | 1417 536 | 2346 543 | 105 136 66 7.55 1.73 0.73 | 1410 545 | 2356 540 | 103 135 65 7.45 1.72 0.71 | 1403 554 | 2365 536 | 102 135 65 7.3 1.72 0.70 | 1396 563 | 2375 33 | 100 135 | *64 eg) Mel 0.69 | 1889 572 | 2384 530 99 135 64 7.07 1.70 0.68 | 1382 581 | 2394 526 98 135 64 6.95 1.70 0.67 | 1374 590 | 2403 523 96 135 63 6.83 1.69 0.65 | 13867 599 | 2413 519 95 135 63 6.71 1.68 0.64 | 1360 608 | 2429 516 94 135 62 6.60 1.67 0.63 | 1353 618 | 2431 512 93 135 62 6.48 1.66 0.62 | 1346 627 | 2440 508 92 134 61 6.36 1.65 0.61 | 1339 637 | 2449 505 91 134 61 6.24 1.64 0.60 | 1332 646 | 2457 501 89 134 | 60 6.13 1.63 0.59 | 1325 656 | 2466 498 88 134 60 6.01 1.62 0.58 | 1818 664 | 2475 494 87 134 59 5.89 1.61 0.57 | 1311 676 | 2483 490, 86 13 59 5.78 1.60 0.56 | 1304 684 | 2492 487 85 13 58 5.66 1.59 0.55 | 1296 696 | 2500 483 84 33 58 5.90 1.58 0.54 | 1289 706 | 2509 480 84 133 57 5.44 1.57 0.53 | 1281 TL | 257 476 83 133 57 5.32 1.56 0.52 | 1273 727 | 2595 | 472] 82 133 | 56 5.21 1.55 0.51 | 1266 738 | 2534 469 82 132 56 5.09 1.54 0.50 | 1258 749 | 2542 465 81 132 55 4.98 LOS, 0.50 | 1250 760 | 2550 461 81 132 55 4.87 1.52 0.49 | 12492 (00 | Dyers} 457 80 131 54 4.76 15 0.48 | 1234 782 | 2566 454 80 131 54 4.64 1.50 0.48 | 1226 793 | 9574 450 80 131 | 53 4,54 1.49 0.47 | 1218 804 | 2582 446 79 130 53 4.44 1.48 0.46 | 1210 815 | 2590 442 79 130 | 52 4.33 1.46 0.46 | 1202 825 | 2598 438 79 130 | 51 4.93 1.45 0.45 | 1194 838 | 2605 455 79 129 51 4.13 1.44 0.44 | 1186 849 | 2613 43] 78 129 50 4.02 1.43 0.44 | 1178 | 861 | 2620 428 78 129 50 3.92 1.42 0.43 | 1170 | 872 | 2627 424 18 128 | 49 3.82 1.41 0.43 | 1162 883 | 2634 420) 78 128 49 3.72 1S 0.42 | 1154 895 | 2641 416 78 128 48 3.62 1.38 0.42 | 1146 907 | 2647 413 78 127 48 3.52 1583 0.41 | 1138 919 | 2654 409 78 127 AT 3.43 1.36 0.41 | 1130 931 | 2661 405 78 126 46 3.8 1.34 0.40 | 1129 943 | 2668 402 78 126 46 3.24 1833 0.40 | 1114 | 955 | 9674 | 398) 79 125 | 45 8.15 1832) 0.40 | 1106 967 | 2681 394 79 125 45 3.06 13 0.39 | 1098 | 979 | 2687 | 391 | .79 124) 44 2.97 1.29 0.39 | 1090 | 991 | 2693 | 387] 80 124 | 44 2.88 1.28 0.38 | 1082 | 1003 | 2699 383 80 123 44 2.80 1.26 0.38 | 1074.| 1016 | 2705 880 80 122 | 43 271 125) 0.38 | 1066 | 1028 | 2711 376 81 122 43 2.62 1.24 0.38 | 1059 | 1041 | 2716 373 81 121 42 2.54 1.23 0.37 | 1051 | 1053 | 2722 370 82 120 | 42 Ww 16. 16. 16. 16.: 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16.! 16. 16. 15. 15. 15. 15. 14. (v.8.3) 26 29 32 35 38 41 43 45 47 49 50 51 52 53 54 54 54 54 54 53 53 52 24 20 ollity 13 09 5.05 01 -96 .92 87 .83 78 -13 68 62 -56 ol 45 5.39 .33 27 21 14 08 01 94 238 THE ORBIT OF URANUS: TABLE IX, Ara. 2 .— Continued. Arg. (v.c.0) Diff.|(v.s.1) Diff. Sec.var.| 2 | (v.c.1) Diff. Sec.var.|(v.s.2) Diff. Sec.var. (v.c.2) Diff. See.var, 120 121 122 123 124 125 126 127 128 129 a ee co 0) Co Co CO mW RO G2 G2 G9 CO co oro! 139 140 141 142 143 144 145 146 ad I -1 -T-1 H GO bo FS oom Morn) | (to) ee oo) CSpCap peep cer 92) a) ” ” 50.69 : BO Te moo! 50.82 0:06 50.88 2:08 50.98 ee =) 50.98 RlaQe ares ENS HPS MLN) ES) Slit r maces 0.05 51.22 51 51.3 51 51 on e o on fo} ou = on oO 00 Gror Bee oo Srp w [oume) _~ co ee RemeTOowW oer or me ow a © 2 22 [o} oO - — ee boat OSS Oe a 50. 50. 50. 50. 50.69 50 50. 50 50 50.28 50 50. 50. 49 49.81 iva ito) l o 990 0900 | CVONNONONONO lop ol or Ol .0O0 -0O .00 -OI -O1 -OL 08 09 09 os) 9) Oy) ie) Io —— " ” 279.55 280.16 +? 280.76 — OS1eS5e ee 981.92 ° 289.48 283.02 1°: 28355 © 93 284.07 2 284.57 =, 285.06 285.53 +° Oma 44 87 29 Dias 286 286 287 287.69 +°- 288.08 ° MQ. SS BIE © 289 289 289 290 290 290 290 291. 291 291. 291. 291 292 292 292.42 292. 292 292. 292 299. 292 292 292 292 292.92 =2 292 292. 292 292. 292 fe} fo} fe} 16 49 +° 80 10 39 292 299.34 =? 299 292 291.86 291 29) 291 291 990 290 08 10 Il 4 Tw 4 Oo HoH we onan NNN NN INQ NH 0. 0. 0. .89 .88 87 SESESUS (=) _ be " 93 92 90 85 S4 83 81 Ww oos ADO = bo or oon bo Oo rH OID (Sea on — eee on aos tm ee he Hm Ol SD co re) _ — ae oo 38 ” 204.23 202.86.” PX) ae |: a 200.10 7 Rsv 1 197.31 195090 meee 194.50 193.09 191.67 190. 188. 187. 185. 184 183 181. 180 178. Tat 175 174 172 Ib7At 169 168 166 165. 164. 162. 161. 159. 158. 156 155 153 152 150 149 147 146 144 143 141 140 bom OlaT oO G2 Go CO Co Ww co Low Wl wb to to wb CONnNW OD ee ee _ = Ws} 118 116. I 20s ¥ " iy es a = Ge I 80 33—! 86 3902 2a 44 96—1 Ag 00 ils 03 RAL" 05 1 56 58 Oo ame 60 ble 62 7 12 Sy 13) 2 > oO ip 45 I 52 I. I ip ” 0.23 |255.56 37 0.94 |255 38 9.94 | 955.3 38 0.95 | 255.16 39 0.25 |254.00 -40 0.26 |254.83 4° 9.97 |954 4l 9 9% |954.44 4T 9.98 | 954.93 2 0.28 | 254.00 42 0.29 |253.76 430.30 |95 43 0.31 1953.94 44 9.32 1959.96 44 09.33 | 959.67 33 | 2 3b4 |2 35 36 37 38 39 40 41 42 42 43 44 45 46 AT 48 49 50 sll 5 54 D5 | 249.33 By eile Pole paae 58 |240.49 Ms) | BY). 60 | 239. 61 | 238.54 62 | 237 64 65 66 BOM .68 250 250 249 249. 249 248 248 247. 247 246. 246. coe hm Co 57 66 ao oo ooo 245 244 244.06 243.50 I> oo Oho bo bo bo bo LO Co Go Go to US SRewmwWwwW Bm OOD -T or wm bo we Os Go GO 05 tS -I rey bo 229 228 228 2277 226.3 225.51 So oo cscso Soo coo *ooosas cqoooce cqococ kono 44 coo — SJ) 945.68 249°99'—0 7 oO. 9990 90900 Www &WNNNN BRO COnmU Ww Dnw nv w nx 39 00000 EES 0 0000 co J bo bo bo bo 9 lor Dow wWwre OOS bo bo bo bo 0 ” ” 117.85 116)55 eee 115.25 2732 113.95 2232 112.65 ease 1.29 111.36 110/07 mie 108.77 ae 107.48 oe 106:19 "aes 104. 103 102 lO, = 9987 9m I 98.52 97.251? 95.99 1 94.73 1 93.47 7 I 92.21 90.96 —1 89.71 88.46 7 87.22 1 85.98 84.752 83.52 7 82.99 1 81.07 : (i) * ” 0.81 0.82 0.84 0.85 0.87 wpeownram Monwcs Ww wWowew pwporwmnpwmnwrn wPmorReH BREE HES EE eet et te ee ee et pp tk et tp ek fe pe ee ep pet RWS OTD RO HOOT wre os a OrW RO OATS OO SAPASADPD AMO KOO Re — Se PS OO THE ORBIT OF URANUS. 239 TABLE IX, Ara. 2.— Continued. cS 3 ww w (v.s.4) (v.c.4) (p.c.0) (p.8-1) (p.c-1)| (p.8.2) (p.¢.2) (°.8.3)] (p.0.3) ” ~ 1051 |.1053 | 2722 370 82 120 1043 | 1065 | 2727 366 82 120 1035 | 1078 | 2732 362 Be 120 1028 | 1090 | 2737 359 3. 119 1020 | 1103 | 2742 355 3 119 1012 | 1116 | 2747 | 359 5 119 1004 | 1129 | 9751 3: 118 996 | 1141 | 2756 | 3 36 «| 118 989 | 1154 | 276 : 8 117 951 | 1167 | 2765 | 938 38 | «117 Oise lS OM OMG 3 116 965 | 1193 | 2773 : g 115 958 | 1206 | 2777 2 115 950 | 1219 | 2781 p 114 942 | 1282 | 2785 3% ‘ 113 935 | 1245 | 2789 Blt 9 112 927 | 1258 | 2793 9 112 920 | 1271 | 2796 111 912 | 1284 | 2800 0 9 110 905 | 1298 | 2803 ‘ 109 897 | 1311 | 2806 108 889 2809 298 108 882 338 | 2811 29: : 107 875 352 | 2814 292 106 867 366 | 2816 8 105 859 379 | 2819 852 393 | 2821 844 2823 837 : 2825 829 33 | 2826 822 2828 815 ) 2829 807 2831 800 88 | 2832 793 2 | 2834 786 515 | 2835 178 529 | 2836 771 943 | 2837 764 556 | 2837 756 56 2858 749 58: 2838 742 5¢ 2838 135 j 2838 728 52 2837 721 538 | 2837 714 55 2836 107 16 2835 700 57 2834 229 693 3 | 2833 219 686 706 33 216 679 2 83 214 672 i 29 211 665 82% 208 659 76 82 206 652 oD | 282 204 645 88 | 902 638 | 1802 199 332 815 | 2817 197 625 | 1829 | 2815 | 195 618 S4é 9812 ) 192 611 $55 | 2810 | 190 bo bo bo bo bo nw we ON wot sd > ay a 1 3-7 Os 08 G2 2 Oo cor aro eo vs 09 tO arts oo Om a Go Go Co te CO Dw co wo 09 U2 OO OOH oo ee ee oe ole) — e coos He cee iS ai ~ WwW bok Hee Hm He OO = 0 0. 0. 0. 0 0 0. 0. 0. OF 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.42 0. 0. 0. 0. 0 0. 0. 0. 0.43 0.46 0.45 0.5 0.6 0. 0. 0.55 0.! 0.55 0. 0. O25: ORS 0. 0. 0. Go G CO OD CO [I CO el Ol Co) ese cesses cesses sesss Seees Seses SSSSS SSSSS SHEE He EEE Bee Ee pee Sa > CO CO 000 MH O;°o 0 CO ¢ S S S >> eo ossss Sese5 Ssess Ssssoss Sssses sssss os oo 5 09 OD CO CO h CO 240 THE ORBIT OF URANUS. TABLE IX, Ara. 2.— Continued. Arg. | (ey) Diff. (v.s.1) IDES SORTER Oke!) Diff. See.var.|(v.s.2) Diff. Sec.var.|(v.c.2) Diff. See.var, ; : ” wt ” ” ” ” "” ” ” "” ” ” ” a” 80 | 49.81 290.49 0.29 | 116.45 0.81 |225.51 0.11 | 46.95 Ls] ae 290.200 28 0.28 114,99 ae 0.89 |924.64—9°87 9.11 45.9489} ee = 9.60 14|289-90 0.3 0.28 |113.53 [4° 0.84 |223.77 gy Ol | 44.94 (2s aie 49.49 1, [289.59 53, 0.27 [112.08 0.85 |222.89 °°° 0.12 | 48.95 299 147 184 | 49.37 012 )289.26 C33 0.26 | 110.63 Ve 0.86 [221.99 °° 0.12 | 42.98 097 1.73 a site : 5 : 9 4 O\07—am 9 C 9r VQ ee ee 288.91 _ 46 0.26 |109.18 | ,, 0.88 221.08 |, 0.12 | 41.31 1.74 5 | 49.13 717] 288.55 S33 0.25 [107.74 7 0.89 990.16 97 0.12 | 40.3510 2 mie ee oe on 238.17 oa 0-24 |106.80 77 0.90 | 219.28 ne 0112'| $9.40 25 eaiae ae 8.87 S75/28T-78 637 0.24 [104.87 [2 0.91 | 218.29 coe 0.13. |°38:46 |e : 48.74 ee 987.37 ace 0.23 | 103.44 hie 0.93 | 217.34 ae 0.13 | 37.53 293 1.80 ni | sc: 5 bene : ei : 0.92 i 48.61 oy 286.95 _. 44 0.22 102.02_, ,, 0.94 |216.38_, 9, 0.18 | 36.61 1.81 a 48.41 o.14| 286-51 6.45 0 21 100.60 “14% 0.95 | 215.41 727 0.13 | 35.70 ooaamiaaa 192 noe a 280 oy) Oak | 98O a: 0.97 |214.44 ae, 0.14 | 34.80 0:98 1.84 oe 19 §774]285.58 3.43 0.20 | 97.78 7745 0.98 [213.46 FO, O14 | 33.91 Paes: 43.05 9.12/285.10 O75, 0.20 | 96.38 y745 1.00 |212.46 Jo, 0.14 | 83.02 ose lee 195 | 47.90 284.60 0.19 | 94.98 1.01 |911.45 _ Sid 0 234.60 19 | 94.98 01 | 211.4! 0.15 | 32.15 1.88 9( r o.T O.o1 5 —TI. ¢ Se ise Ae aan ue) ees 0.19 93.59 ae 1.02 |210.43 rte 0.15 31.29 2-80 1.90 1 AO eae eer ene 92.20 1°39 1.04 [209.40 103 0.15 | 30.44 Csg> 1.91 195 fs 316 (28e0L O38 0.18 90.82 13) 1.05 | 208.37 T-03 9.15 | 9960 °2* dige 9 1.28 o16| 282-45 Ge, 0-17 | 89.44 oe 1.07 | 207.33 ee 0.16 | 28.77 oes 1.94 200 | 47.12 281.88 0.17 | 88.07 —-, 1.08 |206.28 i “a 1.12 | (281.88 _ 0. 8. 08 | 206.21 0.16 | 27.9: 1.9 201 | 46.9517 |981.99-259 O17 | 86.7173 1.09 |205.92—F°6 0.17 oid 8 19% 202 | 46.78 °47/930.68 2-88 5.36 1°35 Bel eS) Be eae ee 0.16 | 85.36 1°33 1.11 [204.15 1% 0.17 | 26.34 "OO. Waeg ate }.61 517 /280.06 C5, 0.16 | 84.01 #35 1.19 |203.07 "°° 0.18 | 95.55 2 77a aniM 204 | 46.44 S17 /979.43 O63 0.16 | 82.66 139 1.14 1201.98 1-09 9.18 | 94.73 2-10 ot a: F io -65 3 E 1.09 : 0.77 one fe aes r8 BUBB 5.96 0.15 eae 1.15 |200.89 , 0.19 | 24.01 2.03 208) 16.08/20 18 See Cee ee ee 70° 0.20 | 93.96 1/2 2m 207 a erg ett dt Sigg 9-15 mse +3? 1.18] 198.68 =)** 0120 || 9n.59 °-74 9105 a 5.11 319 (216.15 So? 0.14 Wra6 | 6:37 Tilo) 1956) eyo orem oieas °-73 906 BOG 45.02) ee | 2UCLOk 2 gO tee 0 eto te eae ™-*3\ 0.91 | 21.07 ema 210 | 45.33 275.32 Los ey a 211 45 14—-0:19 Seg eure pas Cae ee ee Oe ah 0.22 20.36 __, 70 2.08 oe meee re 13 | 13.4675 1.23 194.16 T1% 0.23 | 19.66 6.6 2.10 913 | 4474 -22°lo73 08 O77 ea | a ae eh LEN ee ay cep ilebeld oe, zie B18) 44.74 og [2t8-08 org 0-18 70.91 157 1.26 |191.85 105 0.24] 18.30 pee 2 44.54 5-25 /272.28 $7) 0.18 | 69.64 | >, 1.28 |190.69 ee 0/250 ae6e oe 2.14 ae ie 2 ; ; 5164/ z 0.0) a 5 | 44.34 (27149 4g, 0.12 | 68.87. 1.29 |189.52 __ 0.26 | 17.00 2.16 216 | 44.13 270.68 Ono" Gvele menaen alk eine race 917 | 43.93 o-2% Braise 0:82) Waa es. a oad “at soe 1.19 0.27 | 16.37 96, 21 mir] aoa) oy] se Si, one] teas 2g Lan ere oop a8 ce &:27 1969.03 O33 0.19 | 64.65 7°23 1.34 185.96 709 0.98 | 15.18 G65 2a 219 | 43.50 °° 1268.18 ° 2 0.19 | 63.42 tgp 1:85 1184.76 10° 0.29 | 14.58 eee 2.21 920 | 43.28 267.32 0.11 | 62.90. 1.87 |188.56 -e a ee AN oe xy ZEA | Loe 83.56 f 03295 iseo>aas 2.29 22 oe alee i! 60.99 175 1.38 /182.35 Ze 0730 Nn IIee °-57 9.98 BO nee 0.11 | 59.80 {712 1.40 [181.14 13) 0.81 | 12.82 oS ae 293 | 49.62 °2?) 964.65 0.11 | 58.61 1.41 |179.92 72? 0.89 | 19.97 ~ 2) @e 224 | 42.40 265 0.92 1.18 1.2 Beil : 224 | 42.40 $23/263.73 297 ol | 51.43 Yi, 1.48 1178.69 123 0.83 | 11.73 og 2a 25 5 : ao | c 252 ate Po BOF ES os 0.11 56.26 | 36 1.44 |177.46 reg Oot | 11.21 2.29 ae eee ec: Os foo tO ee |176.22—-*-24 0.34 | 10.7023" ae Pe oll AO cs 0.11 | 58.96 O07 1.47 |174.97 | 7250.35 | 10:20 23 alae ga3 | 4148 9:23)250.92 99) O11 | 5282 773 1.88 |113.78 125/036 | 9.72 ° 8 aee 229 | 41.24 °-?4/958.94 9-99 0.11 | 51.69 1-13 1.50 |172.46 126% .37 | °9.36) naan 230 25 | ei ee 2 ee BN 0.11] 50.57), 151 171.19 |) 0.38] 8.79 2.35 232 | 40.59 0.24 355.99 1.01 Mee ee 1.09 hee pee ee Wee 8.35 ae Se es )255.92 TO, O11 | 48.37 70 1.54 168.65 ae 0.40 | 7.92 °43 oF3m Be Bele 8) pon Olt ets Tog be eis 7, O41 | 1.50.) 2 eae 234 | 40.03 5°72 /258.85 og 0-11 46.20 10. 1.57 |166:09 72° “0s42 | 7.09) eee 5 235 | 39.78 laser) 0 vokatll) data | eS Sh newken name ae 235 | 39.78. | 252.79 5.13 1.58 |164.80 0.42 | 6.70 2.41 a 39.53 ae 251.72 i 0.11 | 44.08 ne 1.60 |163.51— °°? 0.48 6.39 2°38 9.49 23 pe 6. 22| 200-64 yc, 0-12 43.05 nee 1.61 |169.23 1:29 0.44| 5.96 °3° 1a 2 sue 326 249-55 772 0.12 | 42.02 1-03 1.63 1160.99 13° 0.45 | 5.61 5 aaeee BH | 0.26 248-45 fy, O12] 41.00 To 1.64 /159.62 13) 0.46 | 5.27 see 2.45 240 | 38.51 | 247.38 0.12 | 39.98 1.66 158.31 0.47 | 4.95 ai THE ORBIT OF URANUS. TABLE IX, Ara. 2.— Continued. Arg. | (v.s.3) | (v.c.3) | (v.s.4) |(v.c.4)| (p.¢.0)] (p.s.1)] (p.¢.1)] (p.s.2)] (p.c.2) (».8-8) (p.c.3) ” " i ” 180 | 10.01 0.33 0.62 0.71 611 | 1855 | 2810 | 190] 178) 172 28 181 9.93 0.34 0.62 0.72 604 | 1869 | 2807 188 180 Tl 28 182 9.85 0.35 0.61 0.73 598 | 1882 | 2804 186 183 70 28 183 9.78 0.36 0.61 0.75 | 591 | 1895 | 2801 184] 185] 69 28 184 9.70 0.37 0.61 0.76 | 584 | 1908 | 2798} 182] 188] 68 28 185 9.63 0.38 0.60 OTT One eLOATes e219 45 = LS One ION GT 29 186 9.55 0.39 0.60 Ost] Sal |) Tesi") BBL WSs], ISR BG 29 187 9.47 0.41 0.60 Onno 564 | 1948 | 2787 176 195 65 29 188 9.40 0.42 0.60 0.80'| 557 | 1961 | 2783 | 114) 197 64 29 189 9.33 0.44 0.59 0.81 651 | 1974 | 2779 172 200 63 30 190 9.26 0.45 0.59 0.83 | 544 | 1987 | 2775 171 202 | 62 30 191 9519 0.47 0.5 0.84 | 538 | 2000 | 2771 LEO iee205) |e 16 30 192 9.12 0.50 0.59 0.85 531 | 2013 | 2766 167 207 60 3 193 9.05 0.52 0.59 0.86 525 | 2025 | 2762 165 210 59 31 194 8.98 0.54 0.59 0.87 519 | 2038 | 2757 163 | 212] 58 3 195 8.91 0.56 0.59 0:88 | 512 |) 2051 | 2759) 161) 215.) 5% 32 196 8.84 0.59 0.5§ 0.90 506 | 2064 | 2747 159 218 56 32 197 8.78 0.61 0.59 0.91 | 500 | 2076 | 2742 | 157 | 220] 55 32 198 8.71 0.64 0.59 0.92 | 494 | 2089 | 2737 156 | 223] 54 3: 199 8.65 0.66 0.59 0.93 488 | 2101 | 2731 154 225 3 33 200 8.59 0.69 0.59 0.94 | 482 |} 2113 | 2726 152 | 228) 52 3 201 8.53 0.72 0.59 0.95 | 476 | 2126 | 2720 151 | 231 51 34 202 8.47 0.75 0.59 0.97 470 | 2188 | 2714 150 233 50 39 203 | 8.41 | 0.73 | 0.59 | 0.98| 464 | 2150 | 2708 | 148] 236] 49 | 35 204 8.36 0.81 0.60 0.99 459 | 2162 | 2709 147 239 48 36 205 8.30 0.84 0.60 1.00 453 | 2174 | 2695 146 242 47 36 206 8.25 0.87 0.60 1.01 447 | 2186 | 2689 144 244 47 3 207 8.19 0.90 0.61 1.02 441 | 2198 | 2689 142 247 46 37 208 8.14 0.93 0.61 1.03 436 | 2210 | 2675 141 250 45 38 209 8.08 0.96 0.61 1.04 430 | 2222 | 2668 140 253 44 39 210 8.03 1.00 0.62 1.05 425 | 2934 | 2661 138 256 3 39 211 7.98 1.03 0.62 1.06 A419 | 2245 | 2654 13 259 3 40 212 1.93 1.06 0.62 1.07 413 | 2257 | 2646 135 262 42 41 213 7.88 1.10 0.63 1.08 408 | 2268 | 2639 134 265 41 42 214 7.84 1.18 0.63 1.09 402 | 2280 | 2631 133 268 40 43 215 UctY) Waly 0.64 I || Oye |) Cagik | SAA) sisy} || Brak || sk 43 216 7.74 1.21 0.64 Wow 392 | 2302 | 2616 13 274 39 44 217 7.70 1.24 0.65 1.12 | 386 | 2318 | 2608 | 129) 277 38 44 218 7.65 1.28 0.66 1.13 380 | 2324 | 2600 128 280 37 45 219 7.61 1.32 0.66 114 || 875) 2335) |) 2592) 127 283 | 36 46 220 7.57 1535 0.67 1.15 | 870 | 2346 | 2584] 126 | 286 35 46 221 7.53 1.3 0.68 1.16 | 364 | 2357 | 2575 | 125] 289) 35 47 222 7.49 1.43 0.69 elt 859 | 2367 | 2567 125 292 34 47 223 7.45 1.46 0.69 1.18 854 | 2378 | 2558 24 295 35 48 224 7.41 1.50 0.70 1.19 | 349 | 2389] 2550) 123] 298} 32 49 225 7.38 1.54 0.71 1.20 844 | 2399 | 2541 122 301 32 50 226 7.34 1.58 0.71 1.20 | 389 | 2409 | 2532 LOT 8045) 31 50 227 7.31 1.62 0.72 1.21 334 | 2420 | 2523 121 307 30 51 228 7.28 1.66 0.73 1,22 329 | 2430 | 2513 120 310 30 52 229 7.25 1.70 0.74 1.23 | 324 | 2440 | 2504 | 119) 313! 29 53 230 7.22 1.74 0.75 1.94} 319 | 2450 | 2494] 118] 316) 28 54 231 fall’) 1.78 0.75 1.24 | 3814 | 2460 | 2484) 118] 318 28 54 32 Tally 1.81 0.76 1.25 310 | 2470 | 2474 117 321 27 55 233 7.14 1.85 0.77 1.25 | 305 | 2480 | 2464 | 116] 324) 27 56 234 7.11 1.89 0.78 1.26 | 300 | 2489 | 2454 | 116 | 327 26 57 235 7.09 1.93 0.79 1.27 296 | 2499 | 2444 115 | 330) 25 58 236 7.07 Teh 0.80 1.27 291 | 2508 | 2433 114 333 25 59 237 7.05 2.00 0.81 1.28 | 287 | 2518 | 2423) 114] 336 24 60 238 7.03 2.04 0.82 1.98 | 288 | 2527 | 2412 | 113 39 | 23 61 239 7.01 2.08 0.83 1.29 278 | 2536 | 2401 113 | 342 22 62 | 240 6.99 2.12 0.84 1.30 274 | 2545 | 2390 113 345 21 63 31 July, 1873. 949 THE ORBIT OF URANUS. TABLE IX, Area. 2.— Continued. Arg. | (v.c.0) Diff. |(v.s.1) Diff. See.var.|(v.e.1) Diff. Sec.var.|(v.s.2) Diff. Sec.var./(v.c.2) Diff. Sec.var. ” ” ” ” ” ” ” ” ” " ” nw “” wr 240 | 38.51 247.33 0.12 | 39.98 1.66 |158.31 0.47 | 4.95 2.46 |. 241 38.25 0°20 | 946.90" 33 0.12 | 38.98 5°°3 1.67 |157.00- 7°31 0.48 | 4.6403! 9.4y 942| 37.99 © 72 \945.06 Fat 0.19 | 38.00 322 1.69))|155.69) =79) (049) ana 3 35 aoe 243 | 37.73 ee 243.91 773. 0.18 | 37.08 oe 1.70 |154.37 Ae 0.50 | 4.06 Oo, 2.49 244 | 37.46 056 /242.75 Tig 0.13 | 36.07 SO. 1.71 /153.05 [35 0.51) 3.79 O77 2.50 245 | 37.20 241.57 , 0.13 | 35.12 1.73 |151.73 0.53 | 3.54 2.51 246 | 36.93 —°77|340.39- 777° 9.13 | 34.19 2°93 1.74 1150.40.23 0.54 | 3.300 072 o aaiae 247 | 36.66 ©! |989.20 © 59 0.18) 33.0% 297 1.751/149.07, 7°29 0.55) «3.07 ogame 248 | 36.39 5137/237.99 722 0.14 | 32.36 09% 1.76 [147.74 139 0.56 | 2.86 5.05 2.54 249 | 36.12 5°37 /236.77 155 0.14 | 31.46 O30 1.78 /146.41 54 0.57 | 2.66 C1 2.55 250 | 35.85 _ ,|935.54 0.14 | 30.57 1.79 |145.07__ 0.58 | 2.48 2.56 251 | 35.58 0-27 | 93430724 0.14 | 29.70~ 2°87 1:80 [143.731 34 0.59 | 231-277 2.54 252 | 85.31 $754 /283:05 73 0.15 | 28.84 ¢'3) 1.82 |149.39 137 0.60 | 216 075 Bibs 253 | 35.03 ¢75g/231.80 {722 0.15 | 27.99 35 1.83 |141.05 773) 0.61 | 2.02. 01) 2.59 254 | 34.75 55g [230.54 13g 0.16 | 27.15 Og) 1.85 /139.71 {5° 0.62] 1.89 677 2.60 255 | 34.47 | 2 |229.26 0.16 | 26.33, 1.86 |138.36 0.64 | 1.78 2.61 256 | 34.1902. 227.98—7-28 0.16 | 25.5208! j.88 137.01 1°35 0.65 1.68 0-og 2.61 257 | 33.91 0°75 1226.68 73° 0.17 | 24.73 © & 1.89 | 185.66 7732 0:66 | 1.60 con me 258 | 33.63 nee 225.37 me 0.17 | 23.95 sa 1.91 | 134.31 ee 0.67 | 1.53 © 9% 9.63 5 G : 6 9 ¢ © C 259 | 33.34 0 .23/224.06 13) 0.18.) 23.18 S/7 1.92 1182.96 133 0.68 | Lat Co) 2.64 260 | 33.06 | ..|222.74 0.18 | 29.43 1.94 |181.61_ |, 0.69 | 1.43 0am 261 | 32.787" |921.41—"-33 0.18 | 21.697 74 1.95.1130.96 1°33 0.70 | 1.40: omen 262 | 32.49 °791990.07. 7-34 .0.19 | 90.97 °7? 1.97 198.90 *3° o.71 | 1.3950 oeumonam 263 | 32.21 ae 218.72 ree 0.19 | 20.96 ae 1.98 | 127.55 ne 0.73 140ne 2.67 REO. olllc ; 5 aes 264 | 31.92 $35 /217-36 5) 0.20 | 19.56 Ogg 1.99 |126.20 732 0.74] 1.42 Oo. 2.68 265 | 31.63 215.99 0.20 | 18.88 2.01 |124.85 0.75 | 1.45 2.69 266 | 31.34 229] 214.62—137 0.21 18.21—°°67 9:02 |128.50—7°35 0.76 | 1.50+2°S 9x0 267 | 31.05 °29}213.94 1-38 0.91 | 17.56 TG, 208 [122.14 230 O77 | 156 Seam 268 | 30.76 0°29} 911.85 132 0.22 | 16.92 eee 2.04 |120.79 ne 0.79 | 1.63 oe 2.71 oe 6 2 99 ‘A 2 s 5 - 9 269 | 30.47 9757 /210.45 0° 0.22 | 16.29 ogy 2.06 [119.44 732 0.80) 1.72 579 2.78 270 | 3018 209.05 0.23 | 15.68 2.07 |118.09 0.81} 1.82 2.73 ami | 29.88 —°-3°/ 207.6447 0.24 | 15.08 2°00 9.08 |116.74-7°35 0.82 | 1.947272 oe a72 | 29.59 © °2/206.92 4? 0.24 ):14.50 2° 2.10 |115.39) 73) 0.84 | 9107 9 ee 273 | 29.29 °3°)904.79 oe 0.25 | 13.94 ne 2.11 |114.04 oe 0.85 | 2.21 2 eae 274 | 29.00 51,5 /203.36 143 0.25 | 13.39 523 2.13 112.70 79" 0.87 |* 2.31 og 215 275 | 28.71 201.92 0.26 | 12.85 9.14 |111.35 0.88 | 9.55 2.76 976 | 28.41 —°3°]200.48—1-44 0.27 12.38 2°52 9.15 |110.01-.34 0.89 | 9.74% > gig 277 | 28.12 3759 |199.03 145 0.27 | 11.83° 22° 2.17 108.67 137 O.o1 | 9.95 3 coma 278 | 27.88 5120 /197.57 346 0.28 | 11.84 $73 2.18 |107.88 132 0.99 | 3.17 65, 218 279 | 27.54 6 .25/196.11 7) 0.28 | 10.86 946 220 1105.99 755 0.94 | 3.40 oo: 2.78 280 | 27.25 194.64 0.29 | 10.40 2.21 |104.66 0.95 | 3.65 2.79 981 | 26.95 219d ead s0lS0N| sob ae 99.05) loess anos 3.91702) ong ga2 | 26.66 © 5/191.69 “4° 0.31] 9,52 73 9.93 |102.00 — 99 0.98 | 4.19) oem 283 | 26.86 ¢/25/190.20 1:79 0.32 | 9.10 ¢.75 2.24 (100.67 733 0.99| 4.48 ¢ 149 2.80 284 | 26.07 6 .55/188.71 77) 0.88 | 8.70 5.44 2.25 | 99.34 [52 1.01] 4.78 535 26 285 | 25.78 187.22 0.34 | 8.31 2.27 | 98.02 1.02 | 5.10 2.81 286 | 25.48 0°3°/185.72—1-5° 0.34 | 7.94037 9.98] 96.70—1-3? 1.03 | 5.43 +°33 9.89 287 | 25.19 609 |184.21 15% 0.35 | 7.59' 295 2.293] 95.39) 2°35 1:05) baie eiae me 288 | 24.89 °°3°/182.70 757 0.36 | 7.25 O34 9.30 | 94.08 73% 106] 6.14 S800 gias 989 | 24.60 °°°9/181.19 7:5t 0.387] 6.93 2-3? 9.31,| 99.77 7:3% 3108] G:5a Sueno 0.2 1.52 0.30 1.30 0.39 290 | 24.81 | |179.67 0.38 | 6.63 2.32 | 91.47 1.09 | 6.90) jonas 291 |° 24.02 291781532 0.89)|) 168d pene 122851) OOM Tan >>) Malo anon ee 9.84 992 | 23.78 °°°7|176.68 15 0.40) 6.0% © 77 9.34:// 88.87. 23°) Lio aere eg eee 293 | 23.44 ©°91175.10 793 0.41.) 5.8) 22? 9.36 | 87.58 ° 79 113 |) 8b ee 994 | 98.15 °?9/173.57 753 0.49] 5.57 224 (9.37 | (96.995 2:79 195 | 8169 amen 0.2 nes 0.23 1.28 0.46 995 | 22.86... |172.04 0.43 | 5.34 2.38 | 85.01. ,, 1.16] 9.05, , 4, 286 296 | 22.57 09 |179.50 8 0:43)|1 9 5.18e > + 12:39) 88iidaee etal | Oise oases 297 | 99.99 >" 1168.96 754 0.44] 4.93 °?°2 9.40] g9.47.%27 119] 10.01 -s2oiam 298 | 22.00 5755) 161.42 reg O45] 4.75 O18 9:43} (81:00 ray 1.20 | 10.51 [37 a8 299 | 21.71 5125/165.88 13% 0.46] 4.59 Ci? 2.43 | 79.93 7 1.92] 11.08 $3) 2.88 300 | 21.42 164.33 0.47 | 4.44 9.44 | 78.67 1.23 | 11.55 9.88 THE ORBIT OF URANUS. 243 TABLE IX, Ara. 2.— Continued. Arg. | (v.s.3) | (v.c.3) | (v.s.4) | (v.0.4) (p.c.0) (p.s.1)| (p.c.1)| (p.s.2)| (p.c.2)} (p.s.3)} (p.¢.3) ” ” ” ” 240 6.99 2.12 0.84 1.30] 274 | 2545 | 2390 | 113 | 345] 21 63 241 6.97 2.16 0.85 1.31 | 270 | 2554 | 9379 | 113] 348] 21 64 242 6.96 2.20 0.86 1.31 | 265 | 2562 | 2368 | 113} 350] 20 65 243 6.94 2.24 0.87 3 V6 | ose | 235% |) Wen S53 19 66 244 6.92 2.28 0.88 1.31 | 257 | 2579 | 2346 | 112] 356] 19 67 245 6.90 2.31 0.90 1.32 | 253 | 2587 | 2334] 112] 3591] 18 68 246 6.88 2.38 0.91 1.32 | 249 | 2595 | 29322 | 112) 362) 18 69 247 6.87 2.38 0.92 1.32 | 245 | 2603 | 2311 | 112] 365] 17 70 248 6.85 2.42 0.93 1.32 | 242 |"2611 | 2299] 112] 3868] 16 12 249 6.84 2.45 0.94 1.3 938) |) 2618) || 2287 | | si) 16 73 250 6.83 2.48 0.95 1.3 234 | 2626 | 2275 | 111] 874] 15 14 251 6.82 2.52 0.96 1.3 230 | 2683 | 2263 | 111! 3877} 15 75 252 6.82 2.55 0.97 1.33 | 227 | 2641 | 2251 | 111] 380] 14 16 253 6.81 2.58 0.98 1.3: 993 | 2648 | 9239 | 111 | 383] 14 TT 254 6.80 2.61 0.99 1.34 | 219 | 2656 | 2226 | 111] 3886] 13 78 255 6.79 . 2.64 1.00 1.3 215 | 2663 | 9214 | 111] 3889] 18 19 256 6.79 2.67 1.02 1.3 212 | 2670 | 2201 | 111 | 392] 13 80 257 6.78 2.70 1.03 13 208 | 2677 | 2189 | 112) 3895] 12 82 258 6.78 2.73 1.04 1.3 204 | 29684 | 2176 | 112] 398] 12 83 259 6.77 2.76 1.05 1.3 201 | 2691 | 2164 | 112] 401] 12 84 260 6.77 2.79 1.06 1.3 198 | 9698 | 2151 | 112) 404) 11 85 ) 2.82 i. TOA 195 OS TO4 POTS S | oe 40a 86 2.85 it 1.3: 192 | 2711 | 2125 | 113] 410) 11 87 2.88 i. 1.3: 189 | 2717 | 2119 | 113} 413) I1 89 2.90 il. 1-3: 186 | 2723 | 2098 | 113 |. 416] 11 90 2.93 ike 1-33) 183) 9729 | 2085 | 14 | 419) I 91 2.95 1. 1.82 | 180 | 2735 | 2072 | 114 | 422) I1 92 2.98 Te 1.32 | 177 | 2740 | 2058 | 115 | 425] IL 93 3.00 1.1: 1.32 | 174 | 2745 | 2045 | 115) 427] 11 94 3.03 il. 1.32 | 172 | 2751 | 2031 | 116) 430} I1 96 3.05 ile 1.31 | 169 | 2756 | 2018 | 116] 433] 11 97 3.07 atk 1.31 | 166 | 2761 | 2004] 117 | 436] I1 98 3.09 i. 1.31 | 164 | 2766 | 1990 | 117} 439) 10 99 3.11 10 1,83 161 | 2771 | 1977 | 118 | 442] 10 | 100 3.13 il 1.30 | 159 | 2776 | 1963 | 119] 444] 10 | 102 3.15 1.2: 1.99 | 156 | 2781 | 1949 | 119] 447 | 10 | 103 3.17 1. 1.98 | 154 | 2785 | 1985 | 120) 450] 10 | 104 3.19 ile 1.28 | 151 | 2790] 1921 | 121 | 452] 10 | 105 3.20 1. 1.27 | 149 | 2794 | 1907 | 121) 455) 10 | 106 3.22 Ie 1.27 | 146 | 2798 | 1892 | 122| 457] 10 | 107 3.23 Th. 1.26 | 144 | 9802 | 1878 | 123 | 460] 10 | 108 3.25 1. 1.95 | 142 | 2806 | 1864 |} 1294) 463] 10 | 110 3.26 1.3 1.25 | 139 | 2809 | 1849 | 124 | 465} 10 | 111 3.27 1. 1.24 137 | 2813 | 1885 | 125 468 10 112 3.28 te 1.23 | 135 | 2816 | 1820 | 126] 471] 11 | 11s 3.29 1. 1.22 | 133 | 2819 | 1805 | 127 | 474] IL | 115 3.30 1.3% 1.22 | 131 | 9829 | 1791 | 127] 476 | 11 | 116 3.31 Le 1.21 | 199 | 9895 | 1776 | 128] 479 | 11 | 117 3.32 il. 1.20 | 127 | 2828 | 1761 129 | 482] 11 | 118 3.32 il. 1.19 | 126 | 2830 | 1746 130 | 484 | 12 | 120 3.33 il 1.18 | 124 | 2833 | 1731 131 | 487 | 12 | 121 3.33 1: 1.17 | 193 | 2835 | 1716 | 132] 490 | 12 | 122 3.34 1 1.16 | 121 | 2837 | 1702 | 133} 492] 12 | 123 3.34 il. 1.15 | 119 | 2838] 1686 | 134| 495 | 13 | 125 3.35 1. 1.14 118 | 2840 | 1671 | 136) 497 13 | 126 3.35 le 1.13 | 116 | 2841 | 1656 | 137 |) 500 13 127 3.35 il, 1.19 | 115 | 2843 | 1641 | 138 | 503 | 13 | 128 3.35 1 111 | 114 | 9844 | 16296] 139] 505] 14 | 130 3.36 1. 1.10 | 112 | 2845 | 1611] 141 | 508 | 14 131 3.35 il 1.09 | 111 | 9847 | 1596 | 142| 510) 14 | 182 3535 ie 1 9848 | 1581 145 512 ‘ 944 THE ORBIT OF URANUS. TABLE IX, Ara. 2.— Continued. re. |(v.c.0) Diff. |(v.s.1) Diff. See.var.'(v.c.1) Diff. See.var.|(v.s.2) Diff. Sec.var.|(v.c.2) Diff. Sec.var. " pia ay Tgp ” ” " ” ” " ” ” ” ” 300 | 21.42 164.33 Oat |) aia 9.44 | 78.67 1.23 | 11.55 2.88 301 | 91.13 —229/162.78 2°55 0.48 | 4.317073 9.45 | 17.427125 1.94 | 12.09+°-54 9g 302 | 20.85 °:22|161.93 '5>5 0.49 | 4.19 2%? 9.461 46.17 775 1.96 | 12.64 S5vaiae 303 | 20.57 tes 159.68 Pee 0.50] 4.09 °°) 2.47 | 74.93 apeeeieeya| ie. ees 2.89 304 | 20.29 325/158.12 72. 0.51 | 4.01 00) 2.48 | 73.70 173 1.29 13.79 oeg 288 305 | 20.01 (156.57 0.53 | 3.94 9.49 | 19.47 - 1.30 | 14.38 2.89 306 | 19.73 0:28 1155.01 275° +0.54 | 3.897025 2.50 | 71.25- 72? 1.31 4.99 9-61 2.89 307 | 19.45 °°291153.46 222, 0:55) |" 18.86 eo 2.5L | 70.03 1". 1.33) 15.61 - yeaa 308 | 19.17 cee 151.90 "6 0.56 | 3.88 °°. 2.52 | 68.82 ape 1.34 | 16.9% ate 2.90 309 | 18.90 5737 )150.34 a 0.57 | 3.8400, 2.53 | 67.62 13> 1.86 | 16.88 Oe 2.90 310 | 18.63 148.79 0.58 | 3.85 9.54 | 66.42 1.87 |) Wb 2.90 Sie St 36 aes ae eo ae D 0.59 Ege 9.55 65.03 me 1.39 i891 +o oT 2.90 312 | 18.09 ©27|145.67 15° 0.60] 3.92 oa 2.56 | 64.05 (5.1.40 | 18.90) “caimaame 313 | 17.82 °27|14419 7°93 0.61 | 3.98 Gog 2.57 | 62.87 7). 1.42 | 19.60) 2g oem 314 | 17.55 ee 142.56 AG 0.62] 4.06 C7 2.58 | 61.70 el 1.43 | 20.31 oa 2.90 315 | 17.28 141.00 O64) vaste 2.59 | 60.54 1.45 | 21.03 2.91 316 | 17.02—02|139.44—1:5© 0.65 | 4.97 +O22 9.59 | 59.39—1-15 1.47 | 21.76+°-73 9.91 317 | 16.75 °271137.88 1:5° 0.66 4.40 °'3 9.60 | 58.24. 2-15. 1.48 | 99.51 9-75) eoron 318 | 16.49 °261136.33 3:55 0.67] 4.55 © 15 2.61] 57.10. 1-14 1.50] 93.97 0-76 9 94 319 | 16.23 °25l134.77 31:55 0.68| 4.71 22 2.69] 55.97 1-13 1.51 | 24.04 77 9.91 0.26 1.55 0.18 Tigh 0.78 320 | 15.97 133.22 0.69 | 4.89 2.63 | 54.85 1.53 | 24.82 2.91 391 | 15.71 —°-261131.67 1:55 0.70 | 5.08+° 19 2.64 | 53.74—2-12 1.54 25.69 +0-80 9 94 399 | 15.46 °251130.12 1:55 0.71] 5.99 22" 2.65 | 59.64 1-19 1.56 | 26.42 °8° 9 91 393 | 15.91 °751198.5% 1:25 0.73 5.51 22 9.65 | 51.54 3-19 1.54 | 97.94 2-82 9.99 394 | 14.96 25\197.02 155 0.74] 5.75 O24 2:66 | 50.45 1:99 1.59 | 28.07 283 2:99 0.25 1.54 0.25 1.08 0.85 325 | 14.71 125.48 0.75 | 6.00 2.67 | 49.37 1.60 | 28.92 2.90 396 | 14.47 —°-241193.94 854 0.76 | 6.97+°27 2.68 | 48.30—-1-°7 1.61 | 29.77+9-85 9.99 Bom 1499 os oo de OS ROnTER 6.56 °29 9.69 | 47.93 1-97 1.63 | 30.63 °86 9.90 328 | 13.99 °%°24\190.88 1:53 0.79] 6.87 23% 9.69) 46.17 1-09 1.64 | 31.51 9°68 —guRe p09 | 13.75 °24\119.35 153 0.801 7.19 °3? 9.70 | 45.13 1-94 1.66 | 32.40 289 gag 0.24 1.53 0.33 1.03 2.90 330 | 13.51 117.82 (ese, Say 20 1.67 | 33.30 9.89 331 | 13.98—°23(116.30-1°22 0.82 | 7.87 %2 3) 9.72 | 43.07©-°3 1.68 | 34.917-0-Oxuaamn 332 | 13.04 °?4)114.78 75? 0.84 | 8.94 °32 2.79 | 42.06 7-°F 1.70 | 35.13 29 eiaaim 333 | 19.81 °23/113.96 1°? 0.85 | 8.62 °3° 92.73 | 41.06 1:-°° 1.71 | 36.06 °-Smuamem 334 | 19.58 °23l11i.74 '5? 0.86] 9.02’ S42 274) 40.06 "°° 1.73) 30> oOo 0.22 Tessie 0.42 0.99 0.96 335 | 12.35 110.23 0.88 | 9.44 2.75 | 39.07 Leva | Seo 2.88 336 | 12.137°-221108.72—2°5! 0.89 | 9.87 1°43 9.75 | 38.10—-2-97 1.75 | 38.94+°-97 B16 337 | 11.91 °22{|107.22 %2° 0.90 | 10.32 nee 2.76 | 37.14 99° 1.77 | 39.91 °97 9.88 g93 1 11.69 C22 l0575) a 22 0.90 | losTs ae 2.77 | 36.19 995 1.78 | 40.89 2-98 9.87 339 | 1i.47 °° 27\104.94 7:49 0.93 | 11.26 24° 9.7% || 35.94 °-95 1.80] 41.89 =-CQuemen 0.21 1.49 0.49 0.94 1.01 340 | 11.26. | 102.75 g 0-94 | 11.75, ., 2.78 | 34.20 1.81 | 42.90 2.87 341 | 11.0522" |101.27 4° 0.95 | 19.9679) 9:73 | 33.38=°°92- 1.82 | 43/GRmtC ame 349 | 1o.s4 °2!| 99.80 47 0.97 | 12.79 222 2.79) 32.47 092 1184 | 44 gps Cee 343 | 10.64 °° | 98.33 oe 0.98 | 13.33 ae 2.79 | 3157 O-9° 1.85 | 45.99 roe 2.86 “se oR = 9 © 2 G a «OC © . 344 | 10.44 $20) 96.86 17% 1.00] 13.89 52, 2.80 | 30.68 S99 187 | 47.03 bos 2.86 345 | 10.24 95.40 1.01 | 14.4 2.80 | 29.80 1.88 | 48.09 2.85 346 | 10.05 0-29 |~ 93.95 1:45 1.02 ee. 59 9.81 | 28.93—°-87 1.90 | 49.15 +!-°% 9.85 347 | 9.85 2°] 92.50 1:45 1.041 15.65 °° 9.81 | 98.08 85 1.91 | 50.92 1:97 9.85 318 | 9.66 29! 91.06 1:44 1.05 | 16.96 8! 9.99 | 97.93 85 1.93] 51.30 1:28 9.85 349 | 9.47 29! 89.63 1:43 1.07] 16.89 °°3 9.82 | 26.39 °84 194] 59.40 T° 9.84 0.19 1.42 0.65 0.82 1.10 350 | 9.28 88.21 1.08 | 17.54 9.83 | 25.57 1.96 | 53.50 2.84 351 | 9.107-°-28| 86.79-2°42 1.09 | 18.207°O° 9.83 | 2476-082 1.97 || 546112 soe 352 | 8.92 .°°18| 95.38 2°42 yur] isga 29 o4 | 93.97 °79 799 | 155.73) Soon 353 | 8.74 28] 83.98 14° 1.19] 19.57 299 9.84 | 93.18 °79 9.00! 56.86 1:33 9.99 354 | 8.57 O17] 89.58 14° 1.14 | 90.98 272 9.84 | 99.40 278 9:01 |-57.99 1°13 2189 0.17 1.39 5 9/e3 0.76 1.15 855 | 8.40 81.19 1.15 | 21.00 2.85 | 21.64 2.03 | 59.14 9.81 356 | 8.937017] 79.8171°38 1.16 | 91.74 7°74 9.85 | 90.897°75 9.04] 60.2977 15 2.80 35% | 8.0% 2°] ysi44 %:3% Tis | 9949 [95 9185 | 20 ee tts 05 | Gieds 1.16 9 g9 S59 Weole co olmoy a 1.19 | 93.95 °7° 995 | 1949 2723 20) 62.62 12 amie 959 | 7.75 2] 4571 1:3 3191 | 94.03 °78 9/86 | 18:71 °7? 9.08 | €3.80 722 9.79 0.15 1.35 0.79 0.70 1.18 360 | 7.60 74,36 1.22 | 24.82 9.86 | 18.01 2.09 | 64.98 2.78 OE EE ES THE ORBIT OF URANUS. TABLE IX, Area. 2. — Continued. (v.c.4) (p.e.0) (p.s.1)} (p.€.1) 300 301 302 303 304 305 306 307 308 309 310 311 312 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 6.92 6.92 6.93 6.93 6 6.94 6.94 6.94 6.94 .95 6 6.95 6.95 6.95 6.95 6.94 6.94 6.94 atiles 93 92 92 ror =r or ior) Ye) = DAA S fo’) -r o2 09 9 9 o9 co Go G9 U9 CO r Rept Coe He OVO or go 29 99 2 98 Go 03 te OO ©9 oo G2 2 09 bo To) co 9 oo go oo 99 Ge O9 O9 O Lo So 12 10 08 .06 04 03 OL 99 97 96 94 92 90 88 87 85 83 81 19 78 76 74 bo (a 70 69 67 brwmeweow wt cw co roprrwrn wnwMwlht bo bo bo bo bo 64 62 60 59 bo bo bo bo bo =r) oO bo bo bo 56 ae om on 0 -46 46 32 .28 .26 +20 24 .23 .08 07 06 05 04 COM eS Heep —) ow Oooo S Socoos Socec]o 49 42 eo oscss ssses ssess sSsse ? aie > - - en = wa 110 109 108 107 106 105 104 103 103 102 102 101 101 100 100 99 99 99 98 98 98 98 98 98 99 99 99 100 100 101 101 102 103 103 104 105 106 107 108 109 110 111 112 114 115 116 117 119 121 122 124 126 128 13 132 134 136 39 141 144 146 2848 2849 2849 2850 2850 2850 2850 2850 2850 2849 2849 2848 Q847 I847 2846 2845 2843 2849 2840 2838 2836 2834 2831 2828 2825 2829 2819 2815 2812 2808 2804 2800 2796 2792 2788 2784 2719 2779 2770 2765 2760 2755 2750 2744 2739 2733 2727 2721 2715 2709 2703 2696 2689 2682 2675 2668 2660 2653 2645 2637 | 2629 1581 1566 1551 Lins 45) 1520 1505 1490 | 1474 1459 1444 1429 1413 1398 13838 1368 1352 1337 1322 | 1307 1291 1276 1261 | 1246 1231 1216 | 1202 1187 1172 ia Gate 11438 1128 1113 1098 1084 1069 1054 1039 | 1025 1010 996 981 967 953 938 924 910 896 882 868 855 841 828 814 801 788 TT5 762 749 736 723 710 143 512 515 517 519 522 524 526 528 531 533 535 538 540 542 545 547 549 Obi 554 556 558 560 562 564 566 568 570 572 573 575 5TT 579 581 583 585 587 589 59 592 594 596 598 599 601 603 605 606 608 610 611 613 614 616 617 619 620 621 623 624 626 | 627 65 67 | 246 THE ORBIT OF URANUS. TABLE IX, Ara. 2.— Continued. (v.c.0) Diff. |(v.s.1) Diff. See.var. (v.c.1) Diff. See.var.|(v.s.2) Diff. See. var. |(v.c.2) Diff. See.var, ” “wr ” ” ” ” ”" ” ” ” ” wu” ” ” 7.60 74.36 1.22 | 24.89 8 2 1452-15] 43.09—%34 1193 | 95.63 +081 ane ae ati cee ae ts o.15 oe 1.33 = 20: Aon ee 1.33 oGF : 56.17 ae QT ae ee 69 733 1.95 | 26.45 Cig) 2.87 | 16.66 Sef 2.12 | 67.37 TOP 2.TT 7.15 O54| 10.37 732 1.26 | 27.29 o'ge 2.87 | 16.00 ee 218 | 68.58 “2. ae 7.0L Fy] 69-05 35 1.28 28.14 © 92 2.87 | 15.35 ae. 2.15 | 69.79 ae 2.75 6:87 0 BTS 1.29 | 29.00 2.88 | 14.71 26 |) TsOlnae 6k os 66.46—7°29 1.31 29,88 +0-88 2.88 100 ere 217 | 79.93 +322 A 6.61 513] 65.18 eg please |ns0ni8 aoe 2.88 | 13.48 es 9.18 | 73.46 7°79 ol%e 6.48 $13] 63.90 [57 1.34 | 31.69 Aa 2.88 | 12.88 zi ° 9.19 | 74.70 124 998 6.36 O13] 62.63 73, 1.85 | 32.61 99° 2.89 | 12.29 ous 2.21 | 75.95 aoe 2.71 6.24 | 61.38 1.37 | 33.54 2.89 | 11.72 2.99 | 17.91, . 3 eae a 604 Ties Tees | By Wag ee hei Unig ae ees 78.47 11-20 210 6.00 °77) 58.91 773 1.40] 35.45 29° 2.89) 10.63 224 2.85 | 18.74 Fete 5.89 57.68 li | 36.42 2:27 9%39 1 10.10 6235 9.96 | Stlon mame 573 CI! 646 1:22 1.43 | 37.40 0.98 : Og Sse : 1.27 2.68 S18) | Sel DO Cera gle 40 Too 2-89 | 9:59 25 227 | 82.28 78 2.67 5.68 55.26 1.44 | 38.40 2.89 | 9.09 2.29 | 83.56 2 ORO Tea BAAOiimeeine 1.45 | 39.41+!-°% 9891] 8 60-49 9.30 | 84.95 +2-29 ae BAT eee SO oe tot (0283 eg 28] 8.12 ae 2.31 | 86.14 7-230 2ag 5.38 ee 51.72 ire 1.48 | 41.47 zee 2.89 | 17.66 ae 9.32 | 87.438 1°79 9.64 Bed ae | OOO ine Os (aaea2 Se OE) | eel ae 2.34 | 88.73 73° 2.68 5.22 49.49 1.51 | 43.58 2.89| 6.78 2.351! 9 4 p.14—208) 4g.o9—T13 1.59 44.65 41-°7 989 | 636-042 aes ass as ae 5.07 oes ANT inn leok a ood 1-09 9.89 | 5.96 °°4° 9.37 | 99767 <-S2aRoumm 4.99 ©:°8| 46.06 27! 1:55 | 46584 7° 9.89] 5.57 © 2°32 “0°38 | 93199) t7Semmeeee 4.91 0.08 44.96 1.10 1.57 47.95 Tssst—ore 89 5.19 0.38 : ES Hee i ee He a ee see 8 19 535 289 | 95.31 73° 2.58 ‘ 35 LS 9. 2.89 | 4.83 2.40 | 96.64 2.5 L727) Adigg mec! 16601) 5020 WLS 9189) 14 4emc 35> 949 91 97 11-33 or 4 11 0.06 4] 14 1.06 1-61 51.34 WoL Se 0.33 2.44 ‘ I 3 4. TL eg Mid Zige WSU) olrse ee 2 Be) V4 To eis oa 335 o"ae 4.65 ae 40.69 192 1.63 52.50 11? 9.88 | 3.83 °-32 9.44 1100.64 32 iDieS 4.60 °% 39.65 ~°4 1.64 |.53.67 ':I7 9.88 | 3.59 23! 9.45 |1ol.93 * 3tepie ieee 0.04 1.02 1.18 -29 1.35 BG 38.63 1.66 | 54.85 2.88 | 3.23 2.46 |103.33 9.53 1597 O04) Brig, NG SOLOL 82 9.86.) to 0G mci! 24 104.68 72°35 9.59 4.43 004 6 0.99 fc Seiya | sei : ete 0.26 Fs 2 Te Soa 48 G04] 36.68 Gog 1.69 | 51.25 T2r 2.88] 2.70 2.48 |106.03 7°25 9.51 4.45 9°03) 35.65 Og 170 | 58.46 12) 2.81 | 2.45 °-25 9.49 |107.38 = 5) amy 4.41 ae 84.67 6.96 1.71 | 59.68 ee 2.87 | 2.22 ae 2.50 1108.74 ie 2.49 4.38 33.71 1.73 | 60.91 2.8 ; eam SEC heme eT) ge g87 | 1817222 959 (111g tt36 046 4.33 207 | 31.85 Lee 1.75 | 63.40 125 987 | 1.62 °29 9.53 |119.89 13° 9.45 431 97] 30.94 29% 1.76] 64.67 127 986] 1.45 O22 954 |114.18 3° 9.44 43 0.01] 2 03 0.91 1.78 65.95 2 28 aya . 16 1B 4/ rs0) eae CE ae 5.95 3g 286] 1.29 Sy, 2.55 (115.55 [57 2.48 4.29 29.14 1.79 | 67.23 OMe iy aie 5 6.92. _ 9 4.98—2°°!| 98.97—°:87 1.80 | 68.52+%-29 9.86] 1.01~°°73 ane lig eg hea aa 4.98 22°) orgy cae | eees|aeotea pace 2.85 | 0.90 22 958 |119.66 137 9.40 49g 22° \96iT) ie) | Tes) | lesa aoe 285A LOgD a ae 9.59 |191.03 132 9.39 5 j : : 3 : BE as : Te 4.28 Or | 25-73 ogg 1:85 | 1245 (3) 284) 0.72 5.5, 2.60 |122.41 o 2.38 4.29 24.91 1.86 | 73.78 2.84] 0.65 2.60 |123.78 36 £530) See Soe Uigs! | tople Ss) 9 ei 0 etme 261 ie ete ae 4.39 0 0,| 23:83 Gy, 1.89 | 76.47 35 2.83 | 0.56 C04 2.68'|126.58 [ama aad 2°?| o056 277 1on| vrs 138 283 0.53 203 a.6s lnotor seems 4.36 Oo5| 21-80 oq 192 | 1919 73, 282) 0.52 oo, 2.64 |129.28 13109538 4.39 a) petie8 1.94 | 80.56 2.821 0.52. 9.65 |130.65 a 9.31 4.490 20.33 -0-73 1.95 | Sirgdts 23° 9.69.) Vomacc 22 woree 132002 ae 2.30 4.46 ONG OPece 1G) BERS) Hes) orien Nar SoS Gay eeEAD Les! 9.99 4.50 a 18.93 mee 1.98 | 84.73 Ba 2.81 0.62 a 9.67 (134.77 137) 797 4.54 mee 18.25 Ge 1.99 | 86.14 ee 2.80 | 0.68 nee 9.68 |136.14 a: 2.26 4.59 ~ | 17.58 2.01 | 87.55 90.30 | 0:76... 269 |137.51. | samme 4.64 10°05 16:93 2.02 88.97 7%: 2 979 | o.33t%°9 9.70 |138.88+%-37 9.94 eat hao) ee ee 14? 9.79] 0.96) £22 72.11 |140,95 Saeaee 16 O22 | WereT: ize st04" Bones lee eared alec 2 941 1141.62 '37 9.91 4.82 O02 | 15.06 aah 2.06 | 93.26 a 2.78 | 1.22 are 2.72 | 142.99 poe 2.20 4.88 14.47 2.07 | 94.71 277 | 1.37 9.73 |144.35 9.19 THE ORBIT OF URANUS. 247 TABLE IX, Ara. 2.— Continued. Arg. | (v.s.3) | (v.c.3) | (v.s.4) | (v.¢.4) (p.c.0), (p.8.1) (p.c.1)) (p.s.2) (p.c.2)| (p.8.3)} (p-¢.3) ” ” ” " 7 gre | 360 6.07 2.53 1.23 0.38 146 | 2629 710 252 627 67 181 361 6.02 2.51 1.22 0.38 148 | 2621 697 254 628 68 181 362 5.98 2.50 1.21 0.37 151 | 2613 685 256 629 69 182 363 5.94 2.49 1.20 0.36 153 | 2605 672 259 631 70 182 364 5.90 2.48 Weil) 0.35 156 | 2596 660 261 632 71 182 365 5.85 2.46 1.17 0.35 159 | 2588 647 263 633 72 183 366 5.81 2.45 1.16 0.34 162 | 2579 635 265 634 74 183 367 5.76 2.44 1.15 0.34 164 | 2570 623 267 635 75 183 368 5.72 2.44 1.14 0.33 167 | 2562 610 270 636 76 183 369 5.67 2.43 1.12 0.32 170 | 2553 598 272 638 17 184 370 5.63 2.42 tel 0.32 173 | 2544 | 586 274 639 78 184 371 5.58 2.41 1.10 0.31 176 | 2535 574 276 | 640 79 184 Sie 5.53 2.41 1.08 0.31 180 | 2526 | 562) 279 641 80 184 373 5.48 2.40 1.07 0.30 183 | 2516 550} 281 642 | 81 184 374 5.43 2.39 1.05 0.3 186 | 2507 539 | 283 643 | 82 184 375 5.38 2.39 1.04 0.29 190 | 2498 | 528 285 | 645 83 184 376 5.33 2.39 1.03 0.29 194 | 2488 oat 288 646 84 184 377 5.28 2.38 1.01 0.29 197 | 2479 506 290 647 85 184 378 5 22 2.38 1.00 0.28 201 | 2469 495 | 292] 648 86 184 379 5.17 2.38 0.98 0.28 205 | 2460 485 294 649 87 184 380 5.12 2.38 0.97 0.28 208 | 2450 474 297 650 | 88 184 381 5.07 2.3 0.96 0.28 212 | 2440 463 299 651 89 184 382 5.01 2.38 0.94 0.27 216 | 2429 453 302 652 90 184 383 4.96 2.39 0.93 0.27 220 | 2419 443 304 653 91 184 384 4.90 2.39 0.91 0.27 224 | 2408 433 | 306 654 92 184 385 4.84 2.39 0.90 0.27 228 | 2397 423 309 655 94 184 386 4.79 2.40 0.88 0.27 232 | 2386 413 312 656 95 184 387 4.73 2.40 0.86 0.27 236 | 2374 404 314 657 96 184 388 4.67 2.41 0.85 0.27 240 | 2363 394 317 657 97 183 389 4.61 2.42 0.84 0.27 245 | 2351 384 | 319] 658 98 183 390 4.55 2.43 0.82 0.27 249 | 9340 | 375 3822 | 659 99 183 391 449 2.44 0.81 0.27 253 | 2328 366 324 660 | 100 183 392 4.43 2.46 0.79 0.27 258 | 2316 35il 327 661 | 102 183 393 4.37 2.47 0.78 0.27 262 | 2305 348 329 662 | 103 18% 394 4.31 2.48 0.76 0.27 266 | 2293 339 332 662 | 104 182 395 4.24 2.49 0.75 0.27 271 | 2281 330 334 | 663 | 105 182 396, 4.18 2.51 0.73 0.28 276 | 2269 322 336 664 | 106 182 39T 4.12 2.53 0.72 0.28 | 280 | 2257 313 339 665 | 107 182 398 4.06 2.55 0.70 0.28 285 | 2245 304 341 666 | 109 181 399 4.00 2.57 0.69 0.28 290 | 2232 296 344 667 | 110 181 400 3.93 2.59 0.67 0.29 | 294 | 2220 288 | 346 667 | 111 181 401 3.87 2.62 0.66 0.29 | 299 | 2207 280 | 348 668 | 112 181 402 3.80 2.64 0.65 0.30 304 | 2195 273 |} 351 668 | 113 180 403 3.74 2.66 0.63 0.30 | 309 | 2182 265 353 669 | 114 180 404 3.68 2.69 0.62 0.31 314 | 2170 258 | 355 669 | 115 180 405 3.62 2.72 0.60 0.31 320 | 2157 251 358 669 | 116 179 | 406 3.55 2.75 0.59 0.31 325 | 2144 244 361 669 | 116 179 407 3.49 2.78 0.57 0.32 | 331 | 2131 237 363 | 670 | 117 179 408 3.43 2.81 0.56 (883 || Bay |) Cable) 230 365 670 | 118 178 409 3.36 2.84 0.55 0.38 | 342 | 2106 224 368 | 670 | 119 178 410 3.30 2.88 0.53 0.34 348 | 2093 217 371 670 | 120 178 411 3.24 2.92 0.52 0.35 | 354 | 2080 211 373 670 | 121 1i7 412 3.17 2.95 0.50 0.36 | 359 | 2066 205 | 376 670 | 122 177 413 3.11 2.99 0.49 0.36 365 | 2053 199 378 671 | 122 176 414 3.04 3.03 0.48 0.37 | 3871 | 2039 193 | 381 671 | 123 176 415 2.98 3.07 0.46 0.38 | 377 | 2026 187 882 |. 671 | 124 175 416 2.92 3.11 0.45 0.38 383 | 2012 181 386 671 | 125 174 417 2.85 3.15 0.44 0.39 389 | 1998 176 388 671 | 126 174 418 2.79 3.20 0.42 0.40 | 3895 | 1985 170} 391 671 | 126 173 419 2.73 3.24 0.41 0.41 401 | 1971 165 393 | 672 | 127 173 420 2.67 3.29 0.40 0.42 | 407 | 1957 160 | 396 672 | 128 172 248 THE ORBIT OF URANUS. TABLE IX, Ara. 2.—Continued. rg. |(v.c.0) Diff. |(v.s.1) Diff. See. var.|(v.c.1) Diff. Sec. var. |(v.s.2) Diff. See. var. (v.c.2) Diff. See var, uv wu ” ” uw uw " " 4.88 AY 137; 144.35 2.19 HO to TSG 0m pled 145.72+1-37 918 02 aA 3.35 ake 147.09 ap 2.16 S10) Ha aes “4 CP ae 148.45 73° 915 ng cola 3 j1o0.56 142 iQ Oe 149.80 35 94g 151.15 0.2 26 35 +°-09 1.48 152.50 +2: Ae tee? Bee) 153.84 7:3 53 hes ee 155.19 6504 ee 156.54 : 157.88 ag te: oF 159.99 +7 94 zoe 160.55 © 5 oe 161.88 e 163.20 Leas 53 165.8 Toby 167. “54 168. oe 169. 171. ZO) UO. Heb) 173. 174.6 176.2 177.48 178. ~45 1.46 Dee ee bo bo bo bo bo bo bo bo bo bo 35 59 +o bo bo bo bo bo bo bo bo bo bo fesse Gere earesy bobo tO CO Re OL OU S aT bo bo bo bo bo bo bo bo bo bo I. 34 -~T —_ bo bo bo bo bo bo bo bo 9 bo DPDAMQDD DAO COIN omn Nog bor po bo bo bo bo bo to bo bo bo bo po bo woN & GH Ww CO ie) ot) > He OY LO [ooo oMo*o) ON © CO CO -T -T TID OVO Ol me OD CO CD bo bo bO ou t2 nnn bo be bo bo bo bo po bo bo Po os 0 oD CD oer co Oo ao -! oo GO Wie S oO > SD COTS Om CLO WNmowwW WwWwWwwW Wwe BP O° ho bo bo bo bo Or Or Or Or or mh yy HN oO I I. i I I. Lawl = Loan! Cn UUW or on r=) bo bo bo bo bo oro ro 9 bo it 15 206.00 207.09 208.17 +7: 209.24 7: 910580) 23 211.36 212.40 913.43 bo: 914.45 ™ 215.47 216.48 217.47 RRR NN NAH ee NNN OW N ty AN ATT FREE PRR RR PRR PRR Ro z ‘ ORS ; N moO CONN bo to Aa TSCor NOWF OD WO bo bo bo bo bo bo 19 19 bo to I. 18 NN KN N Oo NNN ® NN Lo OV I. I. 0.99 Te Leet pele ced ce ce cl ce ce ce ce cl oe ee el oe ee le oe Ol Ol Ol CM OM OM Ol Ol U) 02 CORR RON Om Aaaci Liisi Como Oooo CORES BKK HH COCO S COCO SS OOOO WO WHT ato OD KWH DAT COCONWWN ADO HW BROATOSO Kw RS bo bo bo bo bo bo 19 £0 10 bo NN HNN p> G2 WW RW WH GH Go ONT ANN Atle RD POLLO LO WP PLOION POW PION POTN POLS PTI POIO ppt wpe to pw toto MOM mms ess Oooh OOo OHO Occ bo = _ ite) od eo oo Oo Arg. 420 421 422 423 424 425 426 427 428 429 430 431 432 434 435 35 437 438 439 440 441 442 443 444 445 446, 479 480 THE ORBIT OF URANUS: 249 (v.s.3) | (v.e.3) ” _ mr 2.67 3.29 2.61 3:3 D55 3.3 2.49 3.44 2.43 3.50 2.37 3.55 273 3.61 2.25 3.66 2.20 3.72 2.14 3.78 2.08 8.84 2.03 3.90 Tey 3.96 1.92 4.03 1.86 4.10 1.81 4.16 1.76 4.23 1.70 4.30 1.65 4.37 1.60 4.45 1.55 4.52 1.50 4.60 1.46 4.67 1.41 4.75 IL Siri 4,83 1.32 4.91 1.28 4.99 1.24 5.07 1.20 bald 1.16 5.94 1.12 5e3o 1.08 5.40 1.04 5.49 1.01 Hah 0.97 5.66 0.94 (5) 0.91 5.84 0.88 5.94 0.85 6.03 0.82 6.13 0.80 6.23 0.77 6.32 0.75 6.42 0.73 6.51 0.71 6.61 0.69 6.71 0.68 6.81 0.66 6.91 0.64 7.01 0.63 alt 0.62 (Gani 0.62 1183 0.61 7.42 0.60 (ode 0.60 62 0.60 7.72 0.60 7.83 0.60 7.94 0.60 8.04 0.61 8.15 0.61 8.26 32 July, 1873. TABLE IX, Are. 2.— Continued. 2 Wiansies | “ 7 | (v.s.4) | Shee) a) (p.8.1), (p.e.1) (p.5.2) (p.¢.2) (p.s.3)} (p.€.3) id ” 0.40 | 0.42 | 407 | 1957 | 160] 396] 672] 198 0.39 0.43 | 413] 1943] 155 | 399 72) 1929 0.38 | 0.44) 419 | 1980) 150) 402] 6731 199 0.36 0.45 | 425) 1916 | 146 | 405) 6731 130 0.35 0.46 | 431 | 1903} 141 | 4071] 673! i131 0.3 0.47 437 | 1889 136 410 673 132 0.3: 0.49 | 443 | 1875 | 132] 413! 673 | 133 0.32 0.50 450 | 1862 128 416 673 133 0.31 | 0.51} 456 | 1848] 194] 418] 673 | 134 O30) 0552) 463) 11834 190) 491 |) 673} 135 0.29 0.53 | 469 | 1820 | 117 | 494] 673] 136 0.28 0.54 476 | 1806 114 496 | 673 136 0.27 0.55 | 482) 1791 | 111 | 499] 673) 187 0.26 0.57 489 | 1777 108 432 | 673 138 0.25 | 0.58 | 496 | 1762 | 105 | 434] 673] 189 0.24 O15 9) | 503s | Ss 09) asi lero 136 0.23 0.60 | 510 | 17383 | 100! 440] 672! 140 0.22 | 0.62) 517] 1719} 98] 442] 672 { 141 0.29 0.63 | 524 | 1704 96 | 445 | 671] 149 0.21 0.64 | 531 | 1690 94 | 447 | 6Y1 | 149 0.20 0.66 | 539 | 1675 99 | 450) 671] 148 0.19 0.67 546 1660 9] 453 670 144 0.18 0.68 | 553 | 1645 90 | 455 | 670] 144 0.18 | 0.70 ; 560 | 1631 89 | 458 | 669] 145 0.17 0.71 | 567 | 1616 88 | 460 | 669] 145 0.17 0.73 | 574 | 1601 87 | 463} 669] 146 0.16 0.74 | 582 | 1586 87 | 466 | 668} 146 0.16 0.75 | 589 | 1571 86 | 468 | 668] 147 0.15 | 0.77 | 596 | 1557] 86) 471 | 667] 147 14 0.79 | 603 | 1549 86 | 473 | 667] 147 14 0.80 | 610 | 1527 86 | 476 | 667] 148 0.13 0.82 | 618 | 1512 87 | 479] 666} 148 0.13 0.83 | 625 | 1497 87 | 481 | 666] 149 .12 | 0.85 | 633 | 1483 88 | 484] 665, 149 0.12 0.86 | 640 | 1468 88 | 486] 665] 149 0.12 | 0.88 | 648 | 1453 89 | 489] 664] 150 0.11 0.89 | 656 | 1438 90 | 492] 664} 150 0.11 0.90 | 664 | 1423 92 | 494] 663] 150 0.11 0.92 | 671 | 1409 93 | 497 | 662] 151 0.10 | 0.94 | 679 | 1394 95 | 499] 661 | 151 0.10 | 0.95 | 687 | 1379 96 | 502] 661} 151 0.10' | 0.97 | 695 | 1364 93 | 505] 660] 151 0.10 | 0.99 | 703 | 1349] 100] 507] 660] 152 0.10 | 1.00 | 711 | 1335} 102! 510/] 659] 152 Osi) |) TO) | als) |] ts) || as} |) oul) || Gash || le 0.09 1.03 | 727 | 1805] 108] 515] 657] 159 0.09 Oa) 1 |) HO | a tS | Gas) 10583 0.09 1.06 | 743 | 1275 113 520 656 153 0.09 1.08 | 75 | 1261 | 116 |) 523) 655 | 153 0.09 1.10 | 759 | 1246 | 119 | 525] 654 | 154 0.09 oi |) Gy |) 1B |) eB) PY) G8} |) By! O10 I) WN eg I Mee || ee |) BBL || (REF |) ee O00 |) ahatis |} geet || ets |) 8 583. | 651 | 155 0.10 | 1.16] 792 | 1189 Bar| 556) e649 tba 0.10 1.18 | 800 | 1175] 1389] 539| 648 | 155 0.10 1.19 | 809] 1161 | 143 | 542) 647 | 156 0.11 1.91] 817] 1147} 148] 545) 646) 156 0.11 1.23 | 826 | 1133 | 153) 548 | 644 | 156 0.11 1.94 | 834] 1119} 158} 551 | 643-) 156 0.12 | 1.25 | 843|1105| 163] 554] 642] 156 0.12 1.27 | 851] 1091] 168] 556] 640 | 156 142 142 141 140 139 138 137 136 136 135 134 133 133 132 131 130 130 230 THE ORBIT OF URANUS. — Continued. ce a | TABLE IX, Ara. 2. Arg. | (v.c.0) Dif! v.s.1) Diff. See.var. v.c.1) Diff. Sec.var. (w.s.2) Diff. See.var. (v.c.2) Diff. Sec.var. ” ’ 480 | 14.65 ASi |) tasdeacies Wepaleliies Soo AS3) Selo on eae FeV) GG See 0.24 485 | 15.85 486 | 16.09 °° 24 Ast | leisaroes 488 | 16.58 °° AS9)| 16088) cece 0.25 490 | 17.08 A |) Wi BBar 225 493 | 17.58 25 AGH || iisey e285 494 | 18.09 92 0.2 495 | 18.34 496 | 18.60+0-26 497 | 18.85 °25 Teo || Way S20 499 | 19.36 ° 75 0.26 500 | 19.62 501 | 19.88 +2: 26 502 | 20.14 2-26 503 | 20.40 0.26 504 | 20.66 22° 0.26 505 | 20.92 506 | -21.18 +2: 26 507 | 21.44 226 508 | 21.69 °25 509 | 21.95 2:26 0.26 510 | 22.21 Bib) |) Pegs ale) || PPR eee 513 | 22.98 26 514 | 23.24 0:28 0.25 515 | 23.49 Bile | Qeuip acess 517 | 24.00 ° a SiS || Daas 28 519 | 24.51 279 Oo. 25 520 | 24.76 591 | 25.02 1°20 592 | 25.27 2-79 593 | 25.52 079 594 | 25.77 2:79 (Mo 25 525 | 26.02 1596 | 26.27 10:25 1 597 | 96.52 0-25 RO) || DE Coes 59) 5 2TH00) sous 0.24 A530 OT 94 153 on 4e oct 539) Oe cee 1533) [OT Ome cee 1534 | 28.19 973 i 0.23 535 | 98.42 136 | 28.65 2°23 1 537 | 28188. =-3 aS || Gj Ces BEG) || D,35 See 0.22 540 | 29.55 A dA dd 9.02 2.71 Q.aSmc nt eo tue Ben ee 2 10.29 ave 2.73 : 9 ee 11.21 2.74 11.697 *® 9.75 1919) 22 Oh 12.71 aoe 2.47 Tenor PI Ty eo alas 13.79 2.75 14.35 7°59 9.79 14.93 25% 9.79 15.52 °29 2.80 16.13 °° 2.80 0.62 16.75 2.81 Ty Sdaee enece Wee: Pe 9.83 1.71 °97 9.89 19.39 ae 2.83 20.09 9.83 90.80 To 7! 2.83 91.52 27? 9.84 99.96 274 9.84 92 0.75 9 23.01 0.76 2.84 93.71 9.85 455 TO 78 9.85 Sromeey Se) 5) is 26.15 one 2.85 ae 82 6 26.97 Gig, 2-86 Hil oe 2s 28.667 "08 2.86 29.52 ee 2.87 30.40 ae 2.87 31.28 2 og 281 32.18 9.88 soe 2. 2.88 34.01 ae 2.88 34.95 ae 9.88 35.90 619g 2.89 36.86 9.89 Set oe 2.89 B8.8O) oe 2ne9 9.82 2 2189 40.83 7, 2-89 41,85 - 2.89 49.88 11:93 9.89 43.92 1:04 9.89 Aan Bsc 5189 AGLOL see 1.08 47.19 2.89 48.91 +1°°9 9.89 AG Bt Poe OReo 50.42 %°'t 9.89 ese weet EC) 1.13 52.66 2.88 53.80 Te 288 54.95 ep 2.88 56.11 nee 2.88 RY OF 2 98 o1.21 aoa 2.88 58 Te on J to f co co 263.85 ” wt wv 187.06 2.19 (Ss 570 2s 190.08 795" 2.16 191.59 '9 9.15 198209) so 2 Ohi4 1.50 194.59 9.12 196.08 +249 9.11 Lge sia G20 199.05 ae 2.09 900.5 Pst) 200.53 Yt, 2.07 202.00 2.06 203.46 +146 9.05 204.92 14° 9.03 Ay wo Dy: Ail VA DV 1.43 209.24 1.99 06 tests 9s FLONOIe 222 kG TR) Loa 715) HVA AS TOs 1.40 916.30 1.92 Dil eou 32 Ok 219.07 138 1.89 990.44 1°37 1.88 991.81 - »32- 1286 1.36 223.17 1.85 994.59 1%7°35 1.84 925.85 1:33 1.89 Die) 9 IGT 928.50 3° 1.79 21) 929.81 1.78 Ot ime se deny pease) ee Tae Pea ae i 934.94 277 1.78 236.20 i 937.45 + 1°29 Pi O58.60. sean 168 239:92)) wine 166 OUST aiaenlecs 249.33 1.6: 243.59 1118 ee 244.70 ee 1.60 OY Cee ale) 247.03 Tyg 1.57 248.18 1.56 SAOUS1 tee ss 15> O50M3 Bees 53 Nite, See i ee 252.63 eae 1.50 253.73 149 954.49+1-°7 1.48 955.85 1° 1.46 956.90 1:99 1.45 DOR? LSS TAs} 1.02 258.95 1.42 959.96 bo. 1.40 960.95 2°22 1.39 Hace, ES 1S 962.90 297 1.36 0.95 1.34 105.56 ld Mr ” 35.52 2.89 36.46 7°94 9.89 37.41 2.89 38.37 a 2.88 39.35 Cg 2-88 40.33 9.88 41.39 7°99 9.88 ATSO 21 eaeoNsS 4133) 21 ONE 44.35 15, 2.87 45.38 9.87 Ne Thame Gy ele ATLG TiO ONG 48.51 1°05 9.86 49.58 ee 2.86 50.65 2.85 Siar OSH Bee) 2 OE RSCQO) a 8 a ONSD 55.03 ee 2.84 56.14 2.84 Megas OLR BSL3S) . os mOKeS 59.52 14 9.89 60.67 me 2.82 61.82 9.81 ga.ggt tte 2.80 Tote SES Oe 65.3 rie 2.79 66.49 | 5 2.79 67.68 2.78 egy ee aii HOMOTe cecee 2807 128 2. 276 DAG gee 73.71 9.74 Waco wie ieeen oa T6c1G) ais 2ni3 17.39 2°73 9.59 aig) : 78.63 \ 5, 2-72 19.87 2.71 SLD ait Se, Geese (oN 83.62 1°02 9.68" 84.88 © 6 2.67 86.14, ,, 2.66 Silas ak 2.66 88.69 aes 2.65 89.97 ne 2.64 91.25 ee 2.63 92.54 2.62 93.83 +129 9.61 95.12 179 2.60 96.42 13° 2.59 97.72 3° 2.58 I.%0 99.02... 9.58 100.32 713° 9.57 eGo eee SLAG 102 94 23% 9.55 104.95, ~2' 2.54 Tar bo on h cS 7 ” ” 217.41 1.33 O18 46 ee 1.32 219.44 22 ieen 220.40 oe 1.29 291.35 °95 19 2929.99 ee 222.29 1.26 293.99 +°-93 4.95 994.13 © 2) ieee 925.04 oes 1.22 925.94 073, 1.20 296.83 1.19 997.71 +988 1.18 998.58 087 1.16 999.43 085 1.15 230.27 © St) siaie 0.83 231.10 1.12 931,99 7° 2a gal 939.73 °° Tagg 933.52 oe 1.07 234.30 27° 1.06 025.07... ane 935.83 +278 1.03 936.58 275 1.01 937.31 i Sanaa 238.03 27? 0.99 0.71 238.74 0.97 939.44 TOT" 0.96 240.12 a 0.95 940.79 cis 0.94 a 241.45 fg. 0.92 242.10 0.91 242 73 +o-03 0.90 943.36 © 2 0.88 243.98 O° 0.87 244.58 07.3 0.86 245.16 0.84 245.71 1022 0.88 246.26 se 0.82 946.80 oe 0.81 247.83 Gay Od 247.84 0.78 948 34 Toe 0.77 248.83 Be 0.75 249.30 646 0-1 z j 249.76 G4 0.18 250.21 0.71 950.64 +043 0.70 251.06 C42 (0:69 951.47 °°4) 0168 251.86 ° 32 0.66 0.38 959.94 0.65 952. ¢0+°3° 0.64 959.95 235 0.63 953.99 234 0.62 953.61 °'37 0.61 “31 953.92 0.60 254,99 F0°3e 0.58 954.50) as auOnait 954.77 277 0.56 955.02 1 Geog 0.24 955.26 0.54 THE ORBIT OF URANUS. 251 TABLE IX, Ara. 2.— Continued. Arg. | (v.s.3) | (v.c.3) | (v.s.4) (v.c.4)| (p.c.0) (p.s.1)| (p.c.1)} (p-s.2)| (p.e.2)) (p.8.3), (p.e.3) wu “"” ” " 480 0.61 8.26 0.11 1.27 | 851 | 1091 | 168! 556] 640] 156] 130 481 0.62 8.36 0.12 1.28 | 860] 1076 | 173] 559 | 639] 156] 1929 482 0.64 8.47 0.13 1.30 | 868 | 1068 | 179] 561 | 637 | 156] 198 483 0.65 8.58 0.13 1.31 | 877 | 1049 | 184} 564] 636] 156| 198 484 0.66 8.69 0.14 1.83 | 885 | 1035 | 190] 566] 634] 156) 194 485 0.67 8.79 0.14 1.35 | 894 | 1021 | 196] 569] 633) 156) 196 486 0.69 8.90 0.15 1.36 | 902 | 1007 | 202) 571} 682) 156] 195 487 0.71 9.01 0.15 1.37 | 911} 994] 208] 574| 630) 156) 194 488 0.73 9.11 0.16 1.39 | 919 | 980] 214] 576} 629] 156] 194 489 0.75 9.22 0.17 1.40 | 927 | 966] 220] 579) 627] 156, 123 490 0.77 9.33 0.17 1.42 | 936 | 953 227 | 581) 626] 156] 199 491 0.80 9.43 0.18 1.43 | 945 | 940] 234] 584] 624] 156] 191 492] 0.82 9.54 0.19 1.44 | 954 | 926] 241] 586] 623] 156] 120 493 0.85 9.65 0.19 1.46 | 963] 913] 249] 589] 621 | 156} 120 494 0 88 9.75 0.20 1.47 | 972 | 900] 256] 591] 619] 156] 119 495 0.91 9.86 0.21 IPAQ OSM SSil 264ul SOs GIlst | el5 en litle 496 0.94 9.97 0.22 1.50] 990] 874] 272] 596] 616] 156] 117 497 0.98 10.07 0.23 ISL) 9990) Séit ) 280))) 599)" 615) |) W561 al; 498 1.02 10.18 0.23 1.53 | 1008 | 848] 288] 601 | 613] 155] 116 499 1.06 10.29 0.24 1.54 | 1017 | 835 | 296) 604] 611) 155] 115 500 1.10 10.39 0.25 1.55 | 1026 | 822] 304] 606] 610] 155] 115 501 1.14 10.50 0.26 1.57 | 1035 | 810 | 312) 608] 608 | 155] Il4 502 1.19 10.60 0.27 1K |] TOA |) OCH || SNL aL | Gaye |) a |] 503} 1.24 | 10.70 0.28 | 1.59 | 1053’) 785 | 329 | 613] 604 | 154] 113 504 1.29 10.81 0.29 1.60 | 1063 772 | 338) 615 | 602] 54) 113 505 1.34 10.91 0.30 1.62 | 1072 | 760) 346) 617] 600 | 154) 1192 506 1.39 11.01 0.31 1.63 | 1081 | 748) 355) 620] 598) 1541] Ile 507 1.44 Wile 0.32 1.64 | 1090 | 736] 3864] 622] 59¢ | 154] 111 508 1.50 Ine, 6.33 1.65 | 1099 | 24) 873 | 624 | bos | 154] 110 509 1.56 11.32 0.34 1.66 | 1108 | 712] 382] 626} 599} 153) 110 510 1.62 11.42 0.35 1.67 | 1117 | 700] 391] 628] 590 | 153) 109 511 1.68 11.52 0.37 1.68 | 1126 | 688 | 401] 631} 588] 153] 109 512 1.74 11.62 0.38 1.69 | 1135 | 677% | 410] 633] 585 | 153] 108 513 1.80 tial 0.39 LTO Lela G6 GGul) 420) 6S5N5Sanie wos els 514] 1.87 | 11.81 0.40 | 1.71 | 1153] 654] 430| 637 | 581] 153) 107 515 1.94 11 91 0.41 1.72 | 1162 | 643] 440] 639] 578 | 153] 106 516 2.01 12.00 0.42 1.73 | 1171 | 682] 450] 642 | 576) 153) 106 517 2.08 12.09 0.43 1.74 | 1180 | 621 | 460] 644] 573 | 152] 106 518 Dal 12.19 0.45 1.75 | 1189 | 610] 471) 646] 4571) 152) Os 519 2.23 12.28 0.46 1.76 | 1197 | 599 | 481] 648] 569] 152 | 105 520 2.31 12.37 0.47 1.77 | 1206 | 588] 492] 650] 566) 152} 104 521 2.39 12.46 0.49 1.78 | 1215 | 577 | 503] 652 | 564 | 152) 104 529 2.447 12.55 0.50 1.79 | 1224 | 567 | 514] 654] 561 | 151) 103 523 2.55 12.63 0.51 1.79 | 1283 | 557 | 525) 656] 558) 51} 103 524 2.63 12.72 0.52 1.80 | 1242 | 546] 5386} 658} 556] Id51 103 525 2.71 12.81 0.54 1.81 | 1250 | 536] 5481] 660] 553 151 102 526 2.80 12.89 0.55 1.82 | 1259 | 526] 559] 662] 550] 151] 102 527 | 2.88 | 12.98 0.57 | 1.88 | 1267 | 517] 571 | 664) 548) 150) 101 528 2.97 3.06 0.58 1.83 | 1276 | 507 | 583 | 666] 545} 150] 101 529 | 3.06 13.14 0.60 1.84 | 1284 | 498 | 595 | 668] 542 | 150] 100 530 3.15 13.22 0.61 1.85 | 1293 | 488] 607] 670] 540 | 150] 100 531 3.24 13.3 0.62 1.85 | 1301 | 479] 618] 672] 537] 150] 100 532 3.34 13.37 0.64 1.86 | 1310 470 630 | 673 535 149 99 533 3.43 13.45 0.65 1.86 | 1318 | 461 | 642) 675 | 532] 149 99 534 3.53 13.52 0.67 1.87 | 1327 | 452 | 654] 677 | 529 | 149 98 535 3.63 13.59 0.68 1.87 | 1335 | 448 | 666 | 678 | 527] 149 98 536 Shia 13.66 0.69 1.88 | 1343 | 43 678 | 680 | 524) 148 98 537 3.83 13.73 0.71 1.88 | 1352 426 690 | 682 521 | 148 98 538 3.93 13.80 0.72 1.89 | 1360 | 417 | 1702] 683 | 519] 148 97 BG) || ALaBy 1) ey 0.74 | 1.89] 1368] 409 | 714] 685] 516 | 148 97 540 4.13 3.93 0.75 | 1.90| 1376 | 401 | 726 | 686} 513] 147 252 THE ORBIT OF URANUS TABLE IX E IX, Are. 2.— Concluded. Arg. | (v.e.0) Di eee oe pans ) iff. |(v.s.1) Diff. See.var.|(v.e ji SSS = - - -c.1) Diff. Sec.var.|(v.s.2) Diff. See.var./(v.c.2) Di 540 | 99.55 Be " r i 7 7 evar./(v.c.2) Diff. Sec.var, 541 | 99.77 +°-221 59,6342 2.88 | 263.85 ‘ y ” fi 542 | 29.99 07? So Gsveaie 2.88 |964.79 +094 1.34 | 105.56 2.53 | 255 ee. 60.82 1-19 79 1.3: 41.32 255.26 543 20.21 22 a a Te 2.88 265.71 0.92 ee 3 106.88 +3 9.52 255 +o 2 0.54 54 0.22| 02-02 9° 1.32 {108.19 *°3% -49 0 544d 1 80143) aaa A63tOe aaaes 2.87 |266.62 °:9? an ae 9 12, 2.51 | 255.70 0.21 0:98 alas iene dds See || 208i 6c 52) eee ie 109.51 573) 2.50 [255.90 P70 0.52 5 5 3) O26 x Gande 5 0.88 .29 |110.83 -32, 9 49 | 95 5 0.18 0.51 516) 230 Sowicac alle skG mesic 2.87 | 268.40 12 132 eel eOb-RE 5 0.50 sat | 308 224] gags ©2484 BT OED. MOL ipe et mee lee wal e 548 | 31.97 22] 631g 1:25 2.87 /970.19 °85 y/95 US.47/™ See 246 256.41 +o: 16 0.49 No oleh ee 18 732 2.86 |270.96 Bea Voy 114.80 2°33 9.45 |2 0.14 0-48 550 Seo) 604s ee paeiGraina 273 1993) | Mileatia eae oeeen SS aeae 50 | 31.67 b 220) fet 7 ie oe (oa iy 45) 1 oe oe ee 0.4 eI Bley ae CO iearag Ae | att Sele Ta) bates [ene eae 0°45 552 | 32.06 229! 73.93 2 2.86 |273.49 + 2-59 1.91 |118.77,, ,, 2-42 {256.8 oe BBS 0.1 13.23 ORS 1.20 |12 1.32 3.89 0 Ree) aes Sieve 9.85 |o74.ig 278 47 UI epee a 256,98 +°-°9 gag 554 | 39.44 279! 7599 1°29 2.85 [274.95 277 He Hae nee 2.40 |257.05 °° 24 oe Pali. ¢ 280 Gee, | 2060) 2Tos70i eae eA sae ae 32 9.30 |25701d «waa nee ae od. bo 77.10 é 0.74 a5 124.07 Ooo) 9 38 Or. 0.0 0.41 556 | 39.81 +o-18 48-49 123° 2.84 1976.44 2 7.15 2 0.40 Do Ts || Bas09 meee Da SPS ir iG tie U4) 125-39 9.36 (257.18 Bay eas CaCl Gate 31 9°83 |977.3g 7° 1.19 1196.71 +2°32 9.35 | 95 oe 0.40 55 Bie S103 i 228 oy Po seala | aes} 1.22 57.20 0.3 59 | 33.35 218] go Lap 2783 | 278.55 0.69 7 98080 73? (2134 | 25e 200 eee eg 0.1 82.35 9* 9.89 |9% : 68 Oe) OG. Se Oe é -29 0.38 560 | 33.52 i, 133g) oa Die). Selo 7) See, Uae 2735) ||: 25 ial Ome case : 56 Saeed 83.68 9 G6, 08 | LaOrGl aie 2.32 [25 0.03) eee 561 | 33.69 +°:17| 95 91 +!-33 2.82 |279.89 Ree sa 257.16 © °3 0.36 5620 335860 ned Boe 1.34 2°82 |280 54 +0-65 1.06 |132.01 o.31 25719 Peat Si Gel eine canoe le eee Gg Ue Pes 3 +1-32 9.39 [257.07 2-0 0:35 56 ° 6 87.69 -34 9 @ 5 1 03 134.65 T3212 201.07 5 0 34 540 SSauTON ages 89.04 1:35 2.81 281.78 -O1 1.09 g -65 I oa 229 | 257.00 Chey 0. eines Sri ees Wing osu Peete) 0.60 49] rae sion 22H 256.99) (oe a 566 | 34.51 1° 16 sees 9.80 |989.97 On50/an 37.28 1.32 9.96 |256.89 22e 0.39 p67 | 84.67 275] 9313 137 ona [anaieutorsy Peel io ete wn gulga oeNae eG on RES)! gdngo) On. MLO 1.37 2.79 |284.09 2955 pe 139.91 =o Sian \aseisd ane 0.31 Beall aor 251 ycteel me Pig lesa e540 eed aaoiee 31 -9.93 |56\45 som On He pete ea a 26: nine (geste 8830 cee Ieee Cen erate ee O55) ge 57 les) 97.26 5) 0.51 .94 |143.85 “31 9 99 | 25 0.16 0.29 Pesaran Separate: aul eee 1 jae Teale 0.28 pr Tetouan cory ON 40.49 0:98 {145.16 oes 12 | 35.41 Thi ee. ae 2986.16 + °° 49 0.99 8 Ts) Lays) 9.19 | 255.96 0.2 573 | 35.55 O14! t 97.49 1:39 2.76 |986.64 248 9 99 146.46 "9° 2.18 | 255 Hy eae a 574 35.69 0.14 ose 1.40 2.75 |987.10 0.46 .90 |147.76 +39 2-16 ae 0.20 0.26 aye lokaae 102.82 7745 2.75 [287.55 Ot 0.89 [149.05 129 2.15 955.36 0-21 ge 575 | 35.82 ate > 0.87 |150.35 ~~ 3° pera neces 0.25 576 | 35.95 7o13 104.22 3.40 2-14 | 287.98 is 1.29 9.13 |255.13 279 O:as BT 2 0.12 | 109-62 49 948 : 0.86 |151 0.24 77 | 36.08 °73l107.03 142 9:73 988.40 t°42 9. Gere 9.12 | 254.89 B78 | 36.91 23 07.03 Ba lbeacy ee 0.95 1159.9372°29 9.11 |954.63 0 2° 0.24 Bro | 30083 272 |109.85 ener sea Poseuree Ieee 1-29 5°99 lo5.36 227 0.98 2 Gussie Scaler emONS aur Tae pes alee gare Bas, 02 0.12 9.79 1989.55 0-37 2 |155.50 F 0.23 0:28 580 | 36.45 1.41 2 | 289.55 37 9 89 115 Loy 2:08 | 254.08 3 VAL 2 .f 6.7 27 Ong Boe | Cader oie ea? ont loso.o1 TT 5.23 2-06 | 258.79 0-29 Onn 582 | 36 0.12 | 112.68 9.70 1990 on +23 0.79 | 158.05 Ob SH 5.69 114 1.42 2:10 1290.28 SA ete 9.05 | 253.48 58311 86.8 20. ae ie 602908 Vaeee Onis 159.397 1°27 9.04 |953.16 0°32 Mee 584 | 36.93 227 1116.95 1°43 2.68 |290.88 °3! pine 160.59 ah 2.02 252.83 0°33 Ae Be Sea ee ig (2280 | 20iln8y) aes 0.75 eae 17 2.01 [962148 mee 0.20 587 | 87.26 oT ae Se ds POs 0.73 |164.37 1.9 {251.75 ey || Syicky ee 199. 67 1.43 9.65 1291.96 224 see 165.62ih 22 1.97 951.34 0°39 me: Eat remodels Maule Cain aed 1-25 1.95 |250.96 282 ei: ‘ Onno! bean 2.63 | 299 Oueos OE i LOR TL @yk |S ope (0: ; 590 | 87.58 95 Tats ee 0.69 1169.35 1:24 94 | 250.55 4 0.1% polll area tooo liseloat 4s (aie legates ie DEO og alee oa Oo Boag iseivs mel vacua wean 990. 79+ 2-18 O67 USS Ares) 25k ie 0 593 | 37.88 ©%°|j90,95 1-44 2.60 |292.96 O77 hee Liles a 53 S080 249,94—0-45 me 594 | 97.98 1213199 1:44 2.59 | 293.12 0.16 6 64 ceeeUy ae 1.88 |248.78 24° 0.15 son || Sage 29 44 2-58 1293.26 T4 64 Lit. 25 sig eset 0.47 9, 95 | 38.08 132.73 a or, 0:63 | 175.46 tT 2L esp 2 0.48 fm 59G'| 38.18 F°"° | 194 nytt 44 2.58 | 293.88 ara See ba 0.14 597 | 98.97 2°9|13 AG 2.57 |29: ocrz 0°62 [176-67 0.50 38.27, 911395 61 1:44 “ 293.49 LE ove 441,20 1.84 | 947.33 0.14 598 | 38.36 °°9|137.05 1-44 2.56 |293.58 °° 0.6 177s7t%22 1.93 |946.82—°5! 0. 599 | 98.45 9:°9 Teese ak 2.55 |993.eq 22° se TOO tase IASI BIEN) 0.52 ee 5 |138.5 “45 2d9.0 58 THE . v4 ate a 600: | 9854 | || rag Pd4 [293.73 8°70 38 1180.96 712 1.80 245.77 0.53 ime 54 139.94 6 0.07 5% | VSils44 Ic 1.48 |2 ae 0.54 0.13 0.56 | 182.69 t 0.56 TT | 244.67 0.12 THE ORBIT OF URANUS. TABLE IX, Ara. 2.—Concluded. Arg. | (v.s.3) | (v.¢.3) | (v.s.4) | (v.e.4)] (p.¢.0)] (p.s.1)| (p.€.1)] (p.s.2)] (p..2> (0-8.3)] (p.0.8) ” ” ” ” 540 4.13 13.93 0.75 1.90 | 1376 401 726 686 513 147 97 541 4.23 13299 0.76 1.90 | 1384 393 738 688 510 147 97 549, 4.34 14.05 0.78 1.90 |} 1392 386 751 689 5OT 147 96 543 4.45 14.11 0.79 1.90 | 1400 278 763 691 503 146 96 544 4.56 14.17 0.81 1.91 | 1408 371 176 692 500 146 95 545 4.67 14.23 0.83 1.91 | 1416 363 788 693 497 146 95 546 4.79 14.28 0.84 1.91 | 1424 356 801 695 494 146 95 5AT 4.90 14.34 0.85 1.91 | 1431 349 813 696 491 146 94 548 200 14.39 0.87 1.92 | 1439 343 826 698 487 145 94 549 5.12 14.44 0.89 1.92 | 1446 336 838 699 484 145 94 550 5.24 14.49 0.90 1.92 } 1454 329 851 700 481 145 93 551 Deoo 14.54 0.92 1.92 | 1461 323 864 702 478 145 93 552 5.47 14.58 0.93 1.92 | 1468 316 87 703 475 145 93 553 5.58 14.63 0.95 1.92 | 1475 310 890 704 471 144 93 554 5.70 14.67 0.96 1.92 | 1482 304 903 705 468 144 92 555 5.82 14071 0.98 1.92 | 1489 298 916 706 465 144 92 556 5.94 4075 0.99 1.92 | 1496 292 929 708 462 144 92 557 6.06 14.79 1.01 1.92 | 1502 287 943 709 458 144 92 558 6.18 14.82 1.02 1.92 } 1509 281 956 710 455 143 92 559 6.30 14.85 1.04 1.92 | 1515 276 970 711 452 143 91 560 6.42 14.88 1.05 1.92 | 1522 270 983 712 449 143 91 561 6.54 14.91 OM 1.92 | 1528 265 996 713 445 143 91 562 6.67 14.93 1.08 1.92 | 1534 260 | 1010 714 4492 143 91 563 6.79 14.96 1.10 1.91 | 1541 955 | 1023 715 438 142 91 564 6.91 14.99 etal 1.91 | 1547 250 | 1036 716 435 142 90 565 7.04 15.01 alts 1.91 | 1553 246 | 1049 716 431 142 90 566 7.16 15.03 1.14 1.91 | 1559 242 | 1063 TT 428 142 90 567 7.29 15.05 1.16 1.90 | 1564 238 | 1076 T1T 424 142 90 568 7.41 15.06 eal 1.90 | 1570 234 | 1089 T17 421 141 90 569 7.54 15.08 1.18 1.90 | 1575 230 | 1102 718 417 141 89 570 7.66 15.09 1.20 1.89 | 1581 226 | 1116 718 413 141 89 571 779 15.09 1.21 1.89 | 1586 223 | 1129 718 410 141 89 572 7.91 15.10 1.23 1.89 | 1591 219 | 1142 718 406 141 89 573 | 8.04 15.10 1,24 1.88 | 1596 216 | 1155 719 403 140 89 574 8.16 ils itil 1,26 1.88 | 1601 213 | 1169 719 399 140 89 575 8.29 U5) Til 1,27 1.87 | 1605 210 | 1182 719 395 140 88 576 8.42 ayy 1.29 1.86 | 1610 207 | 1195 719 392 140 88 517 8.54 15.12 13 1.86 | 1614 205 1208 719 3888 140 88 578 8.67 baal 1.32 1.85 | 1618 202 | 1221 719 885 140 88 579 8.80 15.10 1.33 1.84 | 1622 200 | 1234 719 381 139 88 580 8.93 15.09 1.34 1.84 | 1626 198 | 1248 719 378 139 88 581 9.05 15.08 1.35 1.83 | 1630 196 | 1261 719 374 igs 87 582 9.18 15.07 1.36 1.83 | 1634 194 | 1274 719 371 139 87 583 9.31 15.05 1.38 1.82 | 1637 192 | 1288 719 367 139 87 584 9.43 15.04 1.39 1.81 | 1641 191 | 1301 719 364 139 87 585 9.56 15.02 1.40 1.81 | 1645 189 | 1815 718 360 139 8T 586 9.68 15.00 1.42 1.80 | 1648 188 | 1328 718 357 139 87 587 9.81 14.98 1.43 1.79 | 1651 186 | 13842 418 853 139 87 588 9.93 14.96 1.44 1.79 | 1654 185 | 1356 elsi 350 138 86 589 | 10.06 14.93 1.45 1.78 | 1657 184 | 1369 T1T 346 138 86 590 | 10.18 14,91 1.46 1.77 | 1660 183 | 1382 TL 343 138 86 591 | 10.31 14.88 1.48 1.76 | 1662 182 | 1396 716 | 339 138 86 592 10.43 14.85 1.49 1.%5 | 1665 182 | 1409 716 335 138 86 593 10.55 14.81 1.50 1.74 | 1667 181} 1499 716 332 38 86 594 | 10.68 14.78 oil 1.73 | 1669 181 | 1436 715 | 328 138 85 595 | 10.80 14.74 1.52 1.42 | 1671 181 | 1448 715 324 138 85 596 | 10.92 14.71 1.53 | 1.72 | 1673 | 181 | 1461 | 714] 3821 | 138) 85 597 | 11.04 14.67 11,65) 1.71 | 1674 181 | 1474 714 317 137 85 598 11.16 14.63 1.56 1.70 | 1676 181 | 1487 HBX |) BES} EH 85 599 11.28 14.58 ILENE 1.69 | 1677 182 | 1500 712 310 137 85 | We78 | 182) 1518 | T11 |} 306 954 THH ORBIT OF URAN US TABLE X, Ara. 3.—Acrion or NEPTUNE. re. |(v.c.0) Diff. |(v.s.1) Diff. |(v.c.1) Diff. |(v.s.2) Diff.'(v.c.2) Diff) (v.s.8) (v.c.3) (v.s.4) 4 | : | =| ” ” ja ” id ” ” ” " " 0.80 0.81 0.83 0.84 0.86 0.88 0.90 0.91 0.93 0.95 0.97 1.00 1.02 1.05 1.07 1.09 1.12 —0.85 0.86 0.87 0.89 0.91 —0.93 0.95 0.97 92.96 0.87 84. 92) Ciun.. 0.93 +o ra) 84 > eon aa co 3 92.38 pe 25 § : "39 O° 92.07 -13] 95, ¢ Shir ie 36.5 541716 o1| 91.46 Be ea 91.15 3 el 90.84 ; ey) .20 .10 5D 90.52 = ae ~22 LE 2) 9 0 Ln a | IO _— _— > ON © oe 09 OD co OO mat ws aow on bh [e) fo) 4 mh _~ a We Ge Os ty Nv 4 - wy) for) fe) — 4 bo 7 io} 4 2 0 Se lore) ON Sl ll od oie Ie ae a S o bo 02 CS Oo tO _ 1 (te) oO Wm ur on = NN NN WwW New -24 .10 -24 Ai) .28 -29 Ou ow WR WwW Go Ge Oy Go Go D1 1 Gs bo bo bo bo bo a a wo ) L Oo Ge Go Oo bo bo bo ro oo 09 rad = ~~ <} ds Go Oo moO tN ARR HW WD WQ HHO C&O © TIP DO Om RR LD G8 Go BO TO tO bo bo bo bo pO He DO NOWNwNWwnhPpb ee = + -~I-I 83.92 82.7 82.17 81.62 81. 80.48 79.8 79.2 78. =I @. 5 spo’ 4 bo bo bo bd bo SCNHENT WDHSSCS SSHHH AAMWH OTMNWH ot Wwwwe WwokhRhr RPRwWwWe wwwmc bo bo bo bo tO bo os co esses cesses sso: em Ree POO HONDAS DDT bo bo bo bo bo Ce ee ee ee el el el le ee ee THM SeE DO FPAwWoONW NATO WH To he oa ee = eae mp CWwaTor TOWN O Pe (ae Se Sak sve 0 00 6 mmono CoCr FW bh Ww oo me Om n-r-t aU nN Cn! — a bo < 0000¢ Nw WwW N Wwwn nd aS oS oo re OO So Ts THE O RBI OF URANUS, TABLE iz 255 Arg. |(v.c.0) Diff. | ( : SLE X, And. 3.—Conti i v.s.1) Diff. |(v.c.1) inued, " F ? Gs Diff. | aS el Bel: ' 7 ; \(v.s.2) Diff.|(v.c.2) Diff | 61 9.30 ain 30.17 uf alt ei 2) Diff. (v.s.3) (v c.3 | > ee eeules0.6e 48.07 I ee Peele EOS eG 63 5 67 eu 81.59 0.71 LRGs List? 06 6.36 Ui " Reg QU seo ° Ain Gy adits 1g 10:09 | @ 54 +0-18 Ont 110 i Bee 56:05 Oot 32.29 “72! 46 Bor illest eek eee “1 0.73 0.49 |--0.83/—1.0 BB aan 39.97 2 8 SG) SPSS PSY Gin Ceid 016s 0.50 0.84 | 8 ie 55. O4 ee 0.67 46.31 ee aie ams 6.89 0.18 ies 0.52 0:85 my = 3 i a“. es ro le 36 i= whee ° or | 53.04 29) aiuaes Bel dle Ee, 063 | 0.36 | 0.88 113 68 52. 8: 0.80] 24:99 0.65 45.98). 23 39g 0-12 7.23 5 sory eOL88) aad Te 69 59, 4 iS $5.58 0.63 Adie} Ces 04 ae 7.39 +o 16 0.59 | 0.59 I —0.8¢§ ze aus 36.91 °° °3 UG ee Pel eel laee Gina wae | ees el ae a | ae Se Vee Bons eee ean) Seeea ee oh) a8 71 = 36.82 0.62/29 5 5 0.50 0.93} 1.16 ra 50.47 0.79 3" Bs = LG) 42.93 S| O.1c 7.83 eu 0.4 : 0.68 0.95 6 79, 5) ani sd Pian 09 Zoom (Oe 5) 0.12 ty || eel 95) 1.17 3 37.93 5 49.98 © 293.05 7.96 : 0.97| 1 a as93) 7? amdse ao! oe B00 reed aK 3 0.45 ue falas, °7° 38.53 095 ten ee Ole 19 O17 suogiis ee ee 0.75 |—0.99|—1.19 : 290 0 = 3 0:9 O31 Braye 8.5 Srila 7 SiMe amet cea 6s | 4T43 39.06 625 Tose cules ae 331 oar| oa Peete m6 | 46.697 39.56 aoa lie SO meee atoll ken aval aie a. Areca] ce ee ete ole mses AEN eae leas Mia 78 fee ©-73) 40.52 2-47 Sees Bigg ot 8.49 0.3 usu) calle ig | 44.53 Bie t0 8 a 38.01 279 416 22° Bee oes ca oes neal eo eae 52. O72] 41.38 242/53 1993 278/437 °°? 8.64 O-C7 36) 1.00) 11 rt go | 43.92 | Sal oy See ie ou lekeg 25 0.36 | 1.05} 1 Hee 81 43 AGS 41.78 0.81 .o9 2/373 Cue) 0.36 1.10 : 1.16 82 oe 0.68 Aone dc 3° Bee 3, |4:80 DBA ates 0.36 | 1.14 cia ple 8 a esos 2:35 81 —2-82 b5,03 a2 |e 6 13] 1.13 Beaeiiesn, 2O°| 40 BL 833 | 33-98 o34 (00) 6 22) "77 $0.01 O83 ile a 84 || 41.15 °-95 49.83 °3?) 33.13 moored fe 2 8 Teer] 0:38 | 1.2: 1.14|—1.12 Ben) 39.90 13.38 26) NBO" ae eel EA ea) UR at) Pols) | Bel) 7 9.90 0-02] 43 Bruioizy'| oye ie etre a te BDy ce | Oe peel aie 1.06 ee eee me) ot ONE = OrBr leg een22 ee) me hare .36 | 1.17] 1.04 : 38.71 °59| 44.00 °-78 aan Oeila as raed NCR cee 46 | 1.41 |=1.18|— 9 | 3814 %5 00 6.35 2 0.49 .18 |—1.02 i OS ee 93.89 088) @57 °22 8.63 2:99) 9.58 1.45) 418|) Lot 90 | 37.58 ee 6 or2| 27-91 |e oan Ser ee | ek ae Roe BM es2| aa ae zie Seco eae oat Sil(055:) 5) eae Slee Pe ctics 0-521 a4 8 owned hh Hi od Bide eo eee Te ee 93 Boies =e 44.44 0.06 26.192 89 | il eee 8.3% 0.62 ; pauses o | 353 49) 44 ig +204] 94°33 Oe lire eee 8.27271] 0.66 feito aes =: ane 44.48 °°] 93.43 Se lteaOn eae Se eon ie ae ? 35.07 A Spa DT ee tate aia ShO2 Tae | OUIe 64 1.14) 0.91 ees Be ae. 28| or ec a) SD eae Pe li eeon ante 0.90 ; : r -t b oT -O OY 98 33.79 o.41 Tee) 21.656. 58 fe eo WE a 0.85 ae 1.12] 0.89 99 See 44.99 ° 22 20.77 ote 83 0.16 7.58 -10 aiek 10 Santo fe 33.41 ee 44.07 213 19.90 0-87) 9 45 0.15 7.41 eon eae 1.71 1.10 xe 101 33.04 43.90 0.17 19.03 5 3. | 8-60 0.15 7.24 se 1.00 ee 1.08 0.86 103 | 32.38 2°34 43:10 22° ee ee eau is eae 13 HN Ge eee 0-30] a5. Bee |S TS ge BO rota eee 1 El ies 104 Pm 5 eal eee ©.26| 15 65 a sisi coe ad oat 115 ree aa eUS 0-88 : aong9) 229 5.62 284/906 29/6. SEA oy 72 | 1.02] 0.8 105 | 31 e228 an 14.80 °°? 9.06 6.93 21 1.21 | 1.70 83 54 0.31 .80 9 0.09 9.23 1.26 oil 1.00 0.8: eee 2-231 ao oy esuleeae Boe ie 28a ace le yice oo ie 23| 49.97 3.9 0.08| 9-01 3 69 | 0.98} 0.8 107 eo boy oe) 9.2 0.22| 1-31 | 1.6 3) 0.83 is | eace eet 2 3 3.38| 13-207 6.78 pe8-+0.07 | ofa 1.35 bela: Nees 109 a 2| 41 Gixol| 12-42 Tales melon ae EN eee Te aC 30.7 0.16 49 0.76 | 9.39 Rs) 0.2 1.40 pe 0.83 Re 22° || 41.08 S|) EGS -76| 9°39 0-04 aay SIM FL 1.63 | 0.94| 0.84 ne aE. 4 a 0.44| 10-92 2.74| 9 495 923 Ben ee ae | 1.60 | 0.92 nee | | eres NOTE I eee 8] 1.56) 0.90] 0.86 112 ; 40.1704 10.19 9 on o.22| 1-33 | 1.5: a 0.86 113 | 30.35 Sone 03 pe tO ae oo ore aes Aeisen lel 2 Cees ee Boe. o10s| oe. 52a EEO ae ee, Fulaeas —0.88|—0.88 Be 0.30 03 38.61 055 Beale led ese Teall tee 0.87| 0.89 30.28 0.56 fa) : 4/9 ayy -Cxlevn 3.94 -23 1.65 41 0.86 0.91 116 | 30.99 +°-° 38.05 pa re162 08 Aer 223) 65 | 1.37 | 0.86 4 117 ae or O88, eae es Sola tuet) | tes2 86) 0.93 ae 30132) 2:3 ee Sal Goliae 60 er prone oe Solan 382 | 0.85} 0.95 2 0.0¢ 00.0 = Oy Oy SY aia ae || .10 ; Me [sor oss) gaat oe] fu San oe Shon ES) 1S Paella 120 6.10] 39°80 0.64) 4 u 3/34 | 9.08 CHOY) ac. Sips aeaks 111 0.84) 0.97 30.56 0.65 .65 2-/8,.99 0.09 2.82 ox 1.7 : 0.84) 0.96 34.95 eee SO eet ble ea 1.12 | 0.84 a .16 8.88 Av base ‘44 | 1.07 | 0.84 ea 1.74 | 1.02 |0.84 ae TE ORR: BORE Us PAGN Use TABLE X, Ara. 3.—Continued. b> 3 g pet pee bo bo bo bo bo m Oh eS bo bo bo bo bo co wo 09 09 co CO OT RWND ES SWHITAM > aor) ee co ©) G2 2 Gd vO elie! _ [—i=) WS) = rc So) b ton cs ww (v.c.0) Diff. | (v.s.1) Diff. |(v.c.1) Diff. |(v.s.2) Diff.|(v.c.2) Diff. | (v.s.3)) (v.c.3) (v.s.4)| (v.e.d) ” ” ” ” ” dd ” ” ” ” u u = 30.56 34.95 - 4.16 8.88 2.41 — 1.74 | 1.02 0.84/14, 30.70 to: 14 eae 3569 SU BG e201 yo eo 74 Oo ce epee ng 30.85 O73 | 33.58 22] 8.26 ~ [3/863 - 7312.02 © 22] 1.73 | 0:91 | Oso mee 31.03 een 32.88 ae 2.85 075 |8.49 ae 1.84 ae 1.72 | 0.86 | 0.86] 1.09 DIODE J © A ae) ¢ Oro 9 v a a) Ld 81.23 $39) 82.15 STS] Bar SS ]8.84 Crs [to S74] 171 | O81 | 087] 1.20 31.46 B42) | meas 8.18 1.51 1.70 | 0.77 |0.88|—im 31.71 +°75| 30.67 -075| 1.81 2°3218.03-°°7°|1.367 0° 2°) 1.67 | 0.73 |) 0lgo) aaa g1.98 °77) 29.99 573] 1.52 077 l7.85 721.21 2 >| 1.65.) 0.68 | 10a7 aa 32.98 © 3°) 29.16 ae TS as ae Gi Bae 1.08 © 13) 1.62 | 0.63 | 0.93) sige 5 *33| 9292 avi ; om = 5 5 82.61 533) 28.39 O78] 108 5123/7483 G75] 0-96 or2| 159 | 0.59 | 0.04) 1.14 32.96 27.60 0.83 7.29 0.84 1.56 | 0.55 |—0.95|_1.15 33.3471 °°38| 96.327 279] g BT tot 092° | 0:747 22°]! 1.53 | 0.51 | 10/96) an g3.74 4°! 96.03 3/71 0.54 215/688 901/065 2°02! 1.49 | 0.48 | 9 0.08) aaiem 84.16 347 | 25.24 rea nOed 007 [8:87 oor |0-57 6.06) 1-45 | 0-45 | 0.99) dis 34.60 9.46, 24-44 Cae| 0:37 oon (648 Gx |0-5l Gog| L4U | 0.42 | 1.00) eam 35.06 |. 3 | 28-65 0.33. 6.25 0.46 1.37 | 0.40 |—1.03]—1.16 35.54 1°49! 99. 86—©8-79] 9.39998 16,9322 10.41 —©-°5 || 1.32 | 0.38 | 05) nie 36.05 °'5'! 92.06 °8°! 0.35 +903 15.81 °22/0.38 °°3! 1.98 | 0.37 | 1.06) aiame 86.58 °53! 91.97 79! 0.49 °°515.59 2210.36 0-07 1.93.1 0.35 | T07/mimiee 37.13 235]. 90.48 279) 0.49 °°9'15.38 2-22 )0.36 98°) 1.18 10:34 | INOS ima 0.57 0.79 0.12 0.22 0.00 37.70 19.69 0.61 5.16 0.36 1.13 | 0.34 |—1.09|—1.10 38.30 +2-60] 13,.99—©-79| 0.76 -+2-25|4.95—°-22 | 0.38 +022] 1.09 | 0.33 | 1.09) Menem 38.92 262! 1813 77) 0.94 %78\443 9221041 °°3) 1.04 | 0.34 | Tol0l mein 39.55 993! a7.36 O77) 1a5 281d 5a S82" 10.45 9-04) q'00) | 0.84 | eee 40.21 269)" 7¢.69 7) a.g9: %24)4-37--°:2110.59\ | 05 0.95 | 0/35 | anieie e 0.67 ae 0.75 0.27 0.21 0.07 rel ers oe on 40.88 15.85 1.66 4.10 0.57 0. 36 |—1.13) 41.56 +°-68| 15,19—0-73| 1.95 +°29)|3.90=°-2°0.65 +2-°*| (0.86 | 0.88 | Nile rn 42.97 °7T| 14.39 9-73] 9.99 °%33)3 41 27910.73 2°3| 0.82 | 0140 | ial nn 3.00 °73) 13.67 °-77! 9.63 2333.59 2790.89 © 9°99) 0.78 | 0141 | dale 43.75 O75! 19.97 27°) 3.917 ©3%l3.33 22919.93 2:27) ol74 | (0:44 |) Sete 0.7 0.69 0.41 0.18 Gy Osun toe vile 44.51 12.28 3.42 3.15 1.04 0. 46, |—114| ean 45,30 +°-79| 1160-0708] 3.96 +944 1993—°'271.16+° 27) 0.67 || 0.50|) aa) 46.09 2-79| 10.94 2-01 4.39 ©4898 2791199 °73) 0.64 | 0.53 | Tos) eae 46.90 31) 10.30. O74) 4.80 aa 2.66 O12 [1.43 ae 0.62 | 0.56 | 1.13} 0.94 ho re oP -O2 . ¢ . Q l 47.73 S53) 9.68 Sooo] 6.29 Ceo /2b1 S15 /1.58 C7] 0-59 | 0.59 | 1.12) 0.93 48 57 9.08 5.82 2.37 1.74 0.57 | 0.63 |—1.11|—0.99 49,49F°-85| 3.49059] 61367 9°5419.95—°'82] 1.99 +226! 0155 | 0166 /mIntO amen 50.29, 087) 7.92 °57! 6.92 252la13 2221907 227) 053 |) 0170 |G 51.17 are 7.3 ee TB 0°29 |2.01 1° |2.e4 © 92) 0.52 | 0.74 || 1 08am 5S 2 I3)3) 5 . ¢ 5 52.07 Spe] 6.84 Ges] 8.12 Sigaf ll Sioo]242 org] 0-51 | 0.78 | 1-06) 0.88 52.97 6.3 8.74 1.82 2.60 0.50 | 0.82 |—1.05|—0.87 53.89 O92 5.857249] 9.39 + °-85 11 74-08 19.48 T2279) 0.50 | 0.86 | 04 ion 54.82 293) 5.39 249) 10.05 20°11 66 o-8 12.97 279) 0.50 | 0,90 | 1-03] iudeam 65.75, 2.93) 4.95) 44) lod 2 2el160" 22 /3.06 | 2-7) 10.50) | nOLe4 | ale 56.10 O08] 4.54 Soyo] 1E40 Giro [1-5 '3/8-38 Gizo| 0-51 | 0.97 | 1.00) 0.86 57.66 4 414 | .12.10 1.51 _|8.55 0.52 | 1.01 |—0.99|—0.86 58.63 9/| 3.77237) 12.30 F270 |] 48—993 13.75 1°29 0.53 | 105") Oo meee 59.60, 2°92) 3.42 ©35| 13.52 277/146 °-92/3.94 2-79! 0.54 | 1.08 || 0.96 /aaiae 60.58 67) | B10 237! 14.95 O73) 1457 oOl la 14 220) 0:56 || Uae Oso ee 61.57 —°99) 9.80 93°! 15.00 2751.45 20°]4.34 9:29) (0.5% | Tbe Oso2) ae 0.99 0.28 0.74 +0.01 0.19 62.56 2.52 15.74 1.46 4.53 0.59 | 1.18 |—0.93|—0.89 63.56 11-00] 9.94 —2-25|| 46.49 F075 | 1.48 £202) 4 yo Fo 291) Olga) | aoa a) nse) ene 64.56 12°] 9.05 22) 17.95 O78)71.51 2934.91 2291 (0:65 | 1930) 0a 65.57 TF] 1.g4 27) 18.99, 27711 55 2415.09 =22°! oey | ieo5) mOsm 66.58 POT) a.67, 22) te.7g 2791.60. 9 2515.98., 220 glo IF 1:27 | 0seg) eee 1.02 0.16 0.77 0.06 0.18 67.60 1.51 19.55 1.66 5.46 0.72 | 1.29 |—0.89|—0.95 68.61 F2-9F] 1.38=0:23] 90.33+°-78 11 7340-0715. g4 19-25] 0.75 | 1.301] O.Sa/mee 69.63 1°?| 1.97 S72! on99 O70ly gi «O38 l5igp | OT sons | ust) etc n065 1°?) 1.19 98) of.3¢ 771.89 2°8l5.97 $29) ogi | 1.92))esOsec meee "1.67 1:°2|' 1.13 0-9°| 99:63 77 |1.93° 29149 °%5)| olga || 1.33 | a0es) ee 1.02 0.05 | Tel 0.10 0.15 72.69 1.08 23.40 2.08 6.27 0.88 | 1.34 |—0.88|—1.00 THE ORBIT OF URANUS. Q57 oO TABLE X, ARG. 3. — Continued. (v.s.1) Diff.|(v.c.1) Diff a 2 -| (v.¢. . |(v.s.2) Diff. 9) Diff | (v.s.3)| a - 5 - - ) Ce) Diff.| (v.s.3) (v.c.3) (v.8-4) Gey 180 72.69 Us Me " ” a —| —— ae 1.08 q " " ni Wet | ys7l eecose _ | 2-08 6.2 : : means 62 1.07 6 or 94.17 to 77| 9.90 +O 2? B21 40.15 | Oe ee —1.00 183 | 75. Bal po?" 6.03 Sch) US acl se ee gOnU3)l Ont ee 0.88| 1.02 ; GE ae 1.11 203) 25.68 AE Ollaren, Cole 6.99 3| 0.94 | 1.34 | 0.88) 1.03 se) ro.76 Fert 116 O08 96.43 075 Fac 0-13| 9.97 | 1.34 | 0.89 1.04 git ees Be 2.56 -*3/1e.8 0.13 ris se -On : 185 | 77.77 Re earl CTE ars Grate Sh COs eal 237 ee a Po atete | 1s4at ol UP ssi 2.69 1 o.44 1.93 1.03 | 1.33 |—0.90! 187 | 79.78 .00 145 eonnll eles oe 9:83 aes Odo | 106 ee eae Neaee ee Pee e29| i552 74| 39 680s a lane malls. Seley Naan sora £06 elecive 099\ ra oS 29.34 gai (o 0 14\y 99 299] vat | 1.3: ee aa 0.99 ee 30.05 4 (3.25 eis Fens RCS SIS 29 | 0.94) 1.08 190 | 82.75 | 1.91 30.75 Sie Wei t cte ESO I ett 0.95] 1.09 lest ae | e108 8 30.75 4 4.68 |5 -|7.37 : 192 | 84.71 2292 a 0.22 3143" 65 Bo ae ie sane ce ee ee ene oe 193 | 85.05.95 2.89 C55] S210 0.6615 55 O85 | tis ©:95| 1.90 | 1.22 neal ay 86. =95 a uO! tae =| 2-99 SW foie w3) “79 a : 361g 98 2.79 ae 33.41 oe FeGeie Coe Oe onllene es 1.19 | 0.99] 1.10 195 | 87.56 a5 aie een Ohrsi| ue wore 1.23 | 1.17 | 1.00) 1.10 196 | 88.507 °-94 3.337028 34.6710 O? eee 764 00 1.94] 1.15 |—1.01|—1.11 een 0.92| 399 °29| 35.97 00° Tie a6 CoN eye eye 1.01) 1.11 90.34 2 Z Eats 3 Sars gee oe aaa ieOs 0.01] 14.96 ii Ae : 199 | 91.24 °9° oe ce 35.86 Oe TOG poet eas FSS 101 Ae ie 1.10 Soll ae eee ein Salata 85 \7.69-2°!| 1.27 | 1. 203 eee 200 | 92.14 Meas \esciga Seen og| ) e:02 1.97 | 1.04} 1.04) 1-10 uu 93.02 oS, 19a to-34 36.) 4.0.54 BU ac | Seok 1.98 | 1.02 |—1.05|—1.09 a 3. ; 5.28 9.3 QF 0.52 ONG su fot Sie Ba ele 0.0 EO 4 Bosone 085| 5.65 — 2! See lee Aes 0-03] 1.98 | 0.97 1.08 aR 204 | 95.59 0.85 6.03 0.38 = 56 0.48 9.37 : 750 Sela Oa 1.07 oa Biss Ne ae 39.04 5.51 O.14/n 46 0.04 oe J Vi 1.06 Bel gett hes 39) ong a7), O43 BG are ee 0.92 | 1.08| 1.05 506 | 97.93t°°82| 6.81722 3-34 10-45 ote omre (ote eal) ee] OU _1,09|—1.04 at 98.02 ae ony Sea We 0.44 ae oS tae 1.94 | 0.87 | 1.09 1.04 2 agai 79) 4.64 o4 0.41 6 | -12|; 93 0:07) 1.93 | 0.85 | 1-09 2 py (0-7 0 oy 4028S, 6.01 %22\7.21 22% sei se) .09| 1.03 09 | 99.57 ae 8 06 oo to 042(6-12 en oe Bone 1.92 | 0.83 | 1.09] 1.02 210 | 100.32 Ang 43) OST Nene oa (foes eal] Tee T Ae 1.09] 1.01 911 |101.06 +? 74 8 gato 43 eg to 38 6 39 FO-1° 7.05 45 119) | 0079) [1 L000 912 101.738 2727) 9.36 0.44 4993 o34 RoR Oe) RGR ity || OM) Loe 0.99 a 102.48 OF 930 O44| 42 a 0.31 en oars: Siar ale lena Ua 1.09| 0.98 2 103.17 ae 10.24 ae 49.89 0.30| 6.58 Ries Bes ca rei} 0.75 1.09) 0.97 915 | 103.84 wares -45 0.29 0.07 CONS mrG Tal |) ste} 1.09| 0.97 y z Vie 3 3.68 216 104.49 +? 2) sitll +9.45 aor 40.26 : be +0.06 GO ORair 1.09 | 0.72 1 (S3} DENS prcmitonia 2 03| 1159 oP Bicceios: Uae acileaae Sel) LAU) eee 1.03| 0.96 ae feta. 2 mon 3.08 0.22 pk eres pitas rey ice OTTO 0.95 rn 99 .60 : Dae 82 2 99 v Bg: 19 |106.33 |, 12.50 oe rot 0.20 \6 $7 0.05 Nee — 1.02 | 0.70 | 1.06) 0 94 220 | 106.90 2 12.95 “45 0.18 -Oo4 te 0.12 1.00 0.70 1.05 0.93 4 S A 415) 9 = Salons 55 | is4ite Weta Balog (2) 2 0.93 | 0.70 |—1.04|—0.92 999 1107.99 ° 2+ 13.86 0.451 44. Mamie. Osee aol Meas 0.95 | 0.70 | 1.04 0.92 Beuoss0, O52 | 14.30 . ae sales erale He G2 |D- 16 Oe een Wen 103| 0.92 95 0.5 o@) TE 910.9 BOS TS (5 ae gue 224 | 109.00 ae see Pek oe Roos DUP pone 0.90 | 0.70 | 1.02 0.92 925 |109.48 5 19 es fu 0.10 0.00 5.53 oar 0.88 | 0.71 1.01) 0.92 oat eae 1S 9 6.96 - 996 |109.94+°°45| 15.63 TO48) 45 55 -£0-08 | ( ner Sele ere 0.72 |—1.00}—0.92 227 110.37 2:43) 16.07 0.44 45.08 0.06 6.97 OL 5.31 5 0.84 | 0.72 1.00} 0.92 Bae 110.78 °4%| 16.50 °°43 nase, ee 6.96 o-0F Bec) ae 0.81 | 0.73 | 1.00) 0.92 999 |111.18 °4°| 16.93 °-73 naa cue 6.94 0:0? Os ee (MO | OC | 0.99) 0.92 930 |111.55 0-37 17.35 0.42 ‘ BOLO E|| 03 4.98 ane Oe |) Oo 0.98| 0.92 : 5! aa 5.18 6.9 g 1 231 111.39 +°: 34 iG ures ie He 0.00 : 0.03 4.88 Out 0.76 | 0.78 —0.97 | —0.938 933 |112.93 °34| 18.17 3. 4- fate eo een lep cack Ooi Td | 00194), Cost On8e 933 |112.54 0.31 18.57 0.40 ae 0.04 6.85 ee 4.68 = 0.72 | 0.81 0.96| 0.94 on4 [119.83 °29| 18.96 O33 oe oo oe Sales IU Ae ee 0.95) 0.95 935 | 113.10 0.2 ae 0.38 ce 6.06 8 9.05 4.49 Stool 0.69 OD, 0.94 0.96 236 |113.35 +0" 19.49 ooo gins! pe 6 4.40 4.09 0.68 | 0.87 |—0.93\—0.97 py (iisist °27| 20.09 oF Anes 202 6.53 2:09 NEilem ae ue 0.89 | 0.93] 0.97 Seelisace Cool 44k oe. a CCU 423 onl Gor OFOIy | 40:83) 0-88 90.80 °35| 44.59 0.12] 6 45 0.06 ae 0.07| 0.65 0.93 | 0.93) 0 99 0.35 0.12| - 0.07 -09 0.65 | 0.96 | 0.93) 1.00 0. 258 THE ORBIT OF URANUS: TABLE X, Ara. 3.— Continued. (v.c.0) Diff. |(v.s.1) Diff. |(v.c.1) Diff. |(v.s.2) Diff.|(v.c.2) Diff. (v.s.3) (v.c.3)| (v.s.4) ” u " ” ” ” ” "” " " 114.11 44.47 38 4.02 ny anes 44,88 284 Ges Osim 114.36 0.12 oO. 0.0 7 S0i ee Tiny eee ye 28} Gy) 114.46. ° Moy SG) ALi, © fig Soe lipiag 25 uy oO. oO. oO. 5.07 .08 0.05 oO. 0.3 ° 1 re e588 jo8|oe 0.05 60 99 3.41 TGS aes 91 2:08 | 54] 0-04 114.65 T° B20" 12s) ase oy Rae EG aes 114.64 ~~ 3:06 oy 43. A mee CONST Gros OS 6 fe . ¢ 0.03 BESS 82 Oo,| 42. ee 114.55 24. 42. 114. ; 42. 114.% 42. 114. 41. 41. = S " " —0.93 0.93 0.93 0.94 0.95 —0.95 0.96 0.97 0.97 0.98 —0.98 0.98 0.99 1.00 -O1 O01 01 -O1 for) So > oD a or) rss Dwr oT FNC M-T COwwoeo or Or maT Se ee ce (te) NwNNN onn ocn~r I Oo Os OOO OOo WOeTaT aT tore) | co co 02 mo S 1 Ko oo} oe) T= o or Ho cocooo So COOH HH Re ee eS Re eRe Ree Lt cell cee ell eel — i — eee eee fa ee ee 4 ae Aa ae ei ae aeicaice y GR cae oy Say Ueda oo ee et ee et et et et tt et tt Feat es ttt [Sie cooceo SOeoeoe SSeS) Sao ece oes eS THE ORBIT OF URANUS. 259 TABLE X, Ara. 3.— Continued. Arg. |(v.c.0) Diff. |(v.s.1) Diff. ((v.e.1) Diff. |(v.s.2) Diff.|(v.c.2) Diff. (v.s.3) (v.c.8), (v.84) (v.c.4) A wt ur ” ” w ” n” wr ” ” “ur ut ” 300 | 92.77 27.34 | 29.40 3.76 5.4 1.35 | 0.82 |—0.97/—1.03 301 93. 1as-°-64 Desi DOU Ghas -allseiiatc ce 5.51 +004 1.34 | 0.80 aE fae s02)| 91.50 © (9) 27.26 °°") 28.89 es aloes. OI RAE Uae) MULES] (05 sis 90-87 2 ¢2| 27.21 © 32] 28.63 Oo 618.82 OOF 15.61 9°95! 1.381] 0.76 | 0.98] 1.04 304 | 90.24 5,3] 27.16 Soo] 28.87 02 /8.85 Sa 5.66 oe 1.30] 0.74 | 0.99| 1.04 305 | 89.60 _| 27.10 _| 98.11 _|3.87 5.71 1.28 | 0.72 |—0.99|—1.04 306 Sie Se B05 e ? | 2.86 7213.91 1 O41 5 75 FO°4) Toy | O71 || 1.00) 1,04 eNieipe3:39) -1¢,| 26-99 | ¢| 27.60 — 7218.94 2515.80 °°5! 1:25] 0.69] 1.00! 1.04 303) 87.72 5 ¢,| 26-93 5 5,| 27-85 3 2/8.97 o03|5.84 oe4| 1.93 | 0.68 | 1.01/ 1.04 309 | 87.10 5 ¢,| 26-86 o.7| 27.10 oe 4.01 oo | 2-88 ae 1.21 | 0.66 | 1.01] 1.04 310 | 86.49 26.79 26.85 4.05 5.92 Te198\ O<65) |==1.02|==1204 311 Scie Asim 26.59 2-20 097 2 22 15.962 Sol) ears |ONG4 |). 10S) eK B12 | 85:29 35, | 26.65 5 2/| 26.34 $33 /4.14 $°5|6.00 4! 1.15 | 0.63 | 1.03] 1.03 813 | 84.69 5 6, 26-58 5.45] 26.09 5153 /4.18 Aes 6.03 aa 1.12 | 0.62 | 1.03] 1.03 B14 | 84.09 52g] 26-49 64g] 25-84 5 92/422 515. |6.07 6.03] P10 | 0.62] 1-04 1.03 315 | 83.51 | 26.40 25.59 4.27 6.10 1.08 | 0.61 |—1.04|/1.03 316 | 82.93—0-5°{ 26.31—°-09] 95.34 —°-25 | 4.39 +°-°5 16 14+°-°4] 1.05 | 0.61 | 1.04] 1.02 317 | 82.37 2-59! 26.22 °-99] 95.19 24/437 2°5/6 17 93] 1.03 | 0.61 | 1.04) 1.02 318 | 81.81 °5°| 26.12 3°! 94.85 25/443 %°5/g 19 9-021 1.01 | 0.61] 1.04] 1.01 Suid) || SEL CPt) DRAW PMO byitaay SaeEIZIZIEs SNC SNTA Coy CSS pee (ai | EO) sli) 0.54 O.I1 0.25 0.06 0.03 320 | 80.173 25.91 24.35 4.54 6.25 0.96 | 0.61 |—1.04/--1.00 321 | 80.90—°-53| 25.80—°-21| 94119-2414 60 +906] 6.97 +9-921 09.94 | 0.61 | 1.04] 0.99 epomie(acs) 22° || 25.68 222) 93.87 ° 24) 4.66 °°8lg.9 °07) 6.91 | 0.69 | 1.03] 0.99 Pam coatge 2125.56 —22)| 93.63 22414 72 C006 39 2:07! 989) | 0169\|| 1.03] 0:98 Beaeiesesy 2 19! 95.44 © 72) 93'39 9F4 14 7g -2°5l¢35° 9-021 9.87 | 0.63 | 1.08] 0.98 49 0.13 0.23 0.07 0.02 325 | 78.19 25.3 23.16 4.85 634 0.85 | 0.64 |1.03|—0.97 PORE 2) |) 258ltmc 4 |) 99/99 0-24 | 4) 99/1 0"O7 16°95 GO-OF1 0183 | 0.65 | 1203) 097 397 | 77.96 24° 95.02 275! 99.69 °23/4.99 9716.35 °°] 0,81 | 0.66 | 1-02] 0.96 Boe mes2 4+) 2487 275) 99:46 °°231'5.06 27 16.35 2°) 0o.79 | O67 | 1:02) 0.96 Soommccsge |) 24072, 72) 20193 S°F3i5.18 227 16.35 22°) ort | 0.69 | 1-0L) 0195 0.42 0.16 0.23 0.07 0.00 : 330 | 15.97 24.56 22.00 “| 5.20 6.35 0.75 | 0.70 |—1.01|_0.95 MIS spe tO! 2439-021 91.78 -0°2215.97 FO-O7 16.35 2-091 0.43 | 0.42 |) 1.00) 0.95 Beans 1g 0-59) 9491 27°) of 5, O22 15.34 0°71 6 .34—2-°%| 9.72 | 0.74 | 1.00) 0195 Setet 237 24,903 O25! 91.35 022 15.41 2271633 2°") O70 || 0.76 | 0:99] 0195 334 | 74.45 oy 23.84 a 21.14 ie 5.48 ceiaGe82 Sie 0.69 | 0.78 oe 0.95 Bane 74.111 23.65 20.94 5.56 6.30 0.68 | 0.80 |_0.99]0.95 Bap G3.79 097 | 93.45 70 2° | 90.78 0 2 15.68 8 oo 16.98 202) 0.67 | 0.89 | 0.98) 0.95 Pema ioe C35) oon S227) op 5g C222 ln ng 2-7 1695, 9-931 OG || 0.84) 05971) 0195 Sesmescs 2.4) 23.02 "|| 20.8 oe 5.18 coy [O22 £03! 0.65 | 0.86 | 0-96) 0.95 x ¢ 0.2 90 @ Bas C Ep 5 ro P lp yc +03 y Ry Say 95 QF 339 | 72.91 ae 22.80 575,| 20-15 5119 |5-85 007/019 bon Oe He ue ee 2.66 22.5 19.96 5.92 6.15 0.64 | 0.91 |_0.95]_0.95 an ago? se 9:24) 19 vg—O-F7 | 5, 9g +9-28] 6 17 —9-4] 9 63 | 0.93 | 0-95] 0.96 Eionleyao9 22) 99:09 ~:241 19.69 2 1716.05 °°" 16.07 °°4) 0:63] 0:95 |- 0-94) 0:96 EeoaNTe00), | 21.84 —°2| 19.45 SUNS 113 ol 6.02 °°5| 0.63 | 0.98 | 0.94] 0.96 344 | 71.82 ae 21.58 ae 19.30 ae 6.19 eee 5.97 °°) 0.63 | 1.00 | 9-93) 0.97 345) 71.66 21.3 19.15 6 25 5.91 0.63 | 1.02 |0.93|_0.97 346 | 71.59—°-14| 91.0427] 19.002: 15 16.31 +909 | 5.86 2-95] 0.64 | 1.05 | 0-93) 0.98 347 | 71.39 13! 90.75 29] 18.87 °1316.37 2°!5.80 2°°! 0.64 | 1.07 | 0.94) 0.99 348 | 71.99 1°! 99.46 29) 18.74 °1316.43 °°|5.73 °°7] 0.65 | 1.09 | 9:94] 1.00 349 | 71.21 9:98) 90.16 23°] 18.62 °1216.48 °°5|5.66 °°7| 0.66] 1.12 | 9-94) 1.01 0.06 0. 31 0.10 0.05 0.07 : 350 | 71.15 19.85 18.52 6.53 15.59 0.67 | 1.14 |—0-94\1.02 B51 | 71 10—°-°5| 19.54—°-3"| 18.43 0-09 | 6.58 +O-°5 |5.51—0-08| 0.68 | 1.16 | 9-94] 1.09 352 | 71.08—°-22| 19.92 232] 19.34 °°9|6.62 %°4/5.43 °°8) 0.69 | 1.18 | 0.95] 1.03 Boaleniios, 22°) 18.90 °37| 18.96 2°°l6.66 °°4)5.35 °°2! o.71 | 1:20:' 0-95) TL04 354 | 71.10 +°-02| 13.56 °34] 18.19 %°716.70 %°4/5.97 298] 0.72 | 1.22 | 0-95] 1.05 0.05 0.34 0.05 0.03 | 0.09 | si 355 | 71.15 18.22 18.14 6.73 |5.18 0.74 | 1.23 |—0.95|—1.06 356 | 71.91 +°-0°| 17.88—°-34| 18.10 —0-0416.76+2-°3|5.09 °° 09| 0.76 | 1.25 | 0.96) 1. B57 | 41.29 °°! 17.53 235) 18.07 °°3l6.79 °°315.00 2°99] 0.78 | 1.26] 0.97] 1 Bee gi4o 272 \ 17.17 °3°| 18.05 °°? /6.81 °°2\4.91 2°9| 0.80 | 1.97 | 0.98) 1. 359 | 71.52 are 16.80 ae 18.05 oo; 6.83 Pane 4.81 © 10] 0.82 | 1.28 CES 1 360 | 71.67 16.43 18.06 6.84 al 0.84 | 1.29 |—0.99|—1. 260 TUE OR Bel Ok SUeRAUNTUES. TABLE X, Arc. 3.—Continued. (v.c.0) Diff. |(v.s.1) Diff.|(v.ce.1) Diff. |(v.s.2) Diff.|(v.¢.2) Diff. (v.s.3) (v.¢.3) ” ” ” " ” ” ” 71.67 16.43 .| 18.06 gach OA) Melber c So lmrelos mise ce 2.02 ° 167 (237) ase 2.94 15.28 ° Islip ee 2.48 14. 18.24 ae ae 18: 14. ; 18.42 13 : 18.55 1s 18.66 12. 18.8 18.¢ 19. 19.3: 19. 19.8 20. 20.3 20. 20. com e2 0 69 > bO bree ow NNN Se oO. sU(3) OO Go Cs Go Co bo Oo Go oo09 ty sj 2 ph Oy C1 bh 9 02 OO bo bo bo w wwn Ss ee ~~ S90 _ ran) = fe) bo N ad CO 00 Se He He He He He Cr Or on A nur NON NN OH bp wor to Ww wamTHDs | fe) aS © <2) a} aap ie) NAH oO wn aT aT eT a eT eT et eT et HT oe eT -T -T -T nn nN fo BD Oo 6 ON —T-1TH cto Or bo bo oo MH 13 & [ Co oe ah OC) WwwnN SS 6owSo OATS On on <3) — aI -T or Wm ow n Oo on on on on CO mwhb = oo b oO cob > n a] bo bh bo bo bo for) boat PRS) Th) bho bo bo bo bo WO HOI on ao BR ARO do ax ee ee ee ee eee or | | 999 99999 990900 65 <5 cS 6 8 8 8 8 8 bo bo bo bo bo 9 bo bo bo bo bo OUR He Co OS OO ROO oT 9 | LO bo bo © + fe) ons GQ oo bo bo bo bo wpoowo ORO as I He ee He eH OT PARR MNRAS NN NN N oo =I 00 ( | Memos ossss sess5 sssss ssese @Oomsos socpcosd os sscsied <5 69 co 65 cS ow ef Co Co Co Co tO SS Saat -T coo bo to mC eecss ese: fr) See en ee WUAAD DON FPD Dee SOS? SeelL Ses =) LO 10 NOLO LO Go G9 6 L9 O09 SD Oc 0 6 NONRrR oS coop emo SOO we GS © to & 02 He C2 Cv lO bo bo po Oo 09 co 09 Gs bo bo Se a D> a oO | or | =r) o © for) ~ Dore oA OU RWwW WOW HK Oo OCOSoO COCO FYE ww Dp BS bo bo bo tO oo OF OO co OO bo bo bo bo bo QD -T fds) Oe ee a ee ee OS NN NO y) i T4 39.47 +o 40.19 40.92 41.66 42.39 Tey 43.84 44.57 45,29 46.01 ise) 3 Oe - - £O bo bo tO LO wW 99900 al on | Po esscs cesses Seses ie) oO rn Se a Siete COD oSo CO SK Oooo Vater, W Non ty bom ID HH OD bo 9990 “sIrUL nN OH @2omoesod Oooh oH O00 W WH NNW NW G% S&H io’2) yom CO oo CO OO CD THE ORBIT OF URANUS. 261 TABLE X, Ara. 3.—Continued. Arg. |(v.c.0) Diff. | (v.s.1) Diff. |(w.c.1) Diff. |(v.s.2 ) Diff.|(v.c.2) Diff. (v.s.3) (v.c.8)|(v.s.4)| (v.c.4) wu ” ” a 7 ” ” : ” uw ” uw id ” F 7 z ” ; 420 |113.51 imcolaee 46/011 1.59 5.84, | 0.53|| 0.92 |—0,98|—1.19 421 | 114. Saree Mloays So hae oe Taliep et 6.04 F270) 9.52 | 0.95 | 0.99) 1.12 499 115.55 7 53| 4.73 331) 47.41 = puilees» 2 Silicia8 So a0. lbs | 02990 |e ete oa iGeot ioe] 5:07 32) 48.10 229 11-65 Sieg Ose = | Or00) | W028 | ApO2| edie 494 |117.59 7735] 5.42 9 °33/ 48.79 ee 169) e661 ae 0.49 | 1.06 | 1.03] 1.11 495 |118.59 5.80. | 49.48 1.75 6.80 O49) 110110) Sofas eset 496 }119.5917-0°/ 6.91 F°-4T/ 50.15 +9-67 11.81 +9-06/6 93 +0-18| 09.49 | 1.14] 1.06] 1.10 Mee e2059) 2) 6.63; 242) 50:80 28S igs OOF, Py 19] 0.49 | 11s | 1.08; 110 ROM AIeGOy Soo) Oy 9°44) 51 44 ae Te9Gh ei oa led oN 0) 150. 50n | 22> hi T0911 eaten AZIe2259) 2°92) 7.54 0-47) 52.06 oe, | 2:06 ee 1.52 pee 0.51 | 1.26 | 1.10] 1.09 0.¢ oO. 0.01 . o.1 430 |193.58 | 8.03 | 52.6 2.16 7.10 0253) | 1-30 | =I = 109 431 124.57 19-99 Ly ae pa. +e ae 2.26 T° = it be 7 0.54 | 1.34] 1.11/ 1.08 BBO mlb25 5c) 9108) 2:54) 53185 ee 2.38 7 [808 & g| 0-56 | 1.38) 111] 1.07 433 | 126.52 ep 9.62 eee 54.41 °-5919.52 Boe ao neroS Leal | ates) e106 434 |127.48 ee 10.18 Ber 54.95 ees 2.66 ane 8:3 Nae ORG) 145} | To: KOS ‘: oO. : . Bp t eiiesas | 10.76 | 65.48 ~~ (9.81 [8.48 *) 0.64 | 1.48 [1.131104 436 |129.38+°-94) 11.37 +°-61) 55.99 +05! 2.96 F°-F5|8.62 +14) 0.66 | 1.51] 1.18) 1.02 437 |130.31 °-93) 11.99 Bee) Do or Beet Sres | cme | NObUO) Web 4y)) eeteros e0y 238 /131.34 ©93) 19.62 ©73| 56.94 °-40|3.30 °F7/8.86 O17) 0.73 | 1.57 | 1.13] 0.99 | Baomiis2als 91 13.96 224) 57.38 24413 44 eBags O: 17 1259) |) aes 10098 eens lance ses. leur lea | een lei —0.97 440 | 133.06 : 7.8 3.65 07 0.82 -61 |—1.1% HEI |183.95 $229) 14.61 +208) 53.19 +°-39/3.g4 +919 9.16 ¥e-09) 0.86. | 1.63 | 1.12) 0.95 442 |134.83 Oo") 15.31 ae 58.56 a 4.04 ooo (ot Oe! 0.90 | 1.64 1.12 0.93 ao On |) Wo "| 58.00 7 38 |4 24 2 89.81 cog; 0:95 | 1-65 | 1.12) 0.92 444 | 136.55 ae MST arta Cee eed orza|" 8! e.ce| | oC: eal 00 445 1137.39 17.45 SOLS 8 aIASG5) a 5 | 9242 _| 1.04 | 1.67 |1.11|—0.90 446 138.21 +° aa Too TAN an eee VES ee 9.45 F°-°3) 1.09 | 1.67 | 1.09 0:80} Hai) 129102) 2-7) 18.08 277) 60.02 PF 5.07 Cos 9.48 6-63] 113) 1.67 | 1.07) 0.8% 448 | 139.81 nae 19.68 — />) 60.22 75 (5.28 - 7. [9-49 ee 1.18 1.66 | 1.06) 0.88 449 |140.59 27°) 20.45 SUNT (S02) ao, a 9.49 9°/ 1.23 | 1.65 | 1.05] 0.87 Bo /i4ia6, | 21.99, | 60.55, (5.71 9.48 1.28 | 1.64 |-1.04|0.86 Pipi 75 || 2199277) 60.67 F212 15.9222? 19 45-903) 1:82 | 1.62) 1.02|| (0.86 452 |142.83 O77) 29.77 Bue 60.06 22 6.13 or |942 By, 1.37 He HA es 453 |143.54 © 77) 23.55 OF 60.82 1 oo3 [O34 cori 288 Soe] 141] 1:58 | 0.91 8 454 [144.23 69) 24.33 es 60.85 75 55 { 6:55 6.20 /932 Sion] 1-45 | 1-55 | 0.98] 0.8 oO. 7 : “2 5 0.85 25 .49 | 1.52 |—0.97|—0.86 455 | 144. lhe 65 ss ‘4 OU ey: 6.15 Die on BO Bees ie eee a pee 456 | 145.56 er 25.89 O78 60.82 6 og 6.96 0.19 Ou Biers eee ee he nae 457 |146.19 © 3) 26.67 375) 60.76 S75/7-15 5.,5/9-07 Soe feb | Ho meas at Mee) eee | 21.45 C7 | 60.66 5717-85 5g |8-9T Sto} 1.6 en ee 459 |147.40 ©°53| 28.92 0°77) 60.54 516 |7-53 5.77/8-86 Cr] 1.63] 1.3 re as TT) ogg (7.10, [8.74 | 1.66 | 1.83 |-0.89|~0.89 460 [147.98 ,, | 28.99. .6| 60-38_..18/7-704 0 1 BS | AO] ES ae ee 461 |148.54 ">>| 29.75 S72) 60.207 524) 7-87 "5.1, /8:60° ou LSS Ps aN 280 | gD jilasio7 22 | 30-51 OVA) ae Oana Ono seats 172 | 119 | 0.86] 0.92 463 | 149.59 st 2 31.26 0.74 59.73 0.28 8.20 0.15 eee) 0.17 = in ae i o8 t 464 | 150.08 ale 32.00 97%) 59-45 5.3, (8-85 914/814 Sue] L783 | 1. 1851|| 0: a 13) so. 8.49 1.97 1.74 | 1.09 |—0.85|—0.94 | fies |isbiee to-44 5 t0-72| 58.81 —°°33 |g 62to%3|4 79-218) 175 | 1.03 | 0.85| 0.96 466 150.98" O71 33 5 oy1| 28-817 9.57 / 8-62 SIC ea) Sa ae asa) RORY edlisie, 24° a gs 069 aane ©-39 he 11/7745 0-19] 145 | 0.93 | 0.85! 1.00 468 | 151.80 85 58.05 543/885 gyo/7-42 $22) 1.75 | 0.93 | 0.85] 1.0 469 | 152.18 Se 35.52 0°87) 57.62 ogo {8-95 Gog [7-22 Sizc| 1-75 | 0.87 | 0.85] 1.03 3 5 1.73 | 0.82 |—0.85|—1.03 470 | 152.53 36.18 4 ge} 57-17 6,48 | 9-93 1.0.08|7-91_ 9 56 82 eee Papi oie 2 |) 26.8307. 5) 06.09" <7 /811 “ocg|G-81 oe 1.72 | 0.77 | 0.86) 1.05 472 |153.18 338) 37.46 Sgr] 56-18 o54/9-1T O05 /8-60 o 2, tH Cs 473 1153-46 , 46] 38-07 4.60] 59-64 0.55 9.22 0.05 BrS8 eeaall arcs "69 0.87 1.10 MA 53:72 | 38.67 BoiOGE alt) peveri| ClG ee pml| aieGu| ROso au) miOseur ea ar a : 9.30 : 5.93 _ | 1.63 | 0.58 |—0.88|—1.11 3 54.51 9.3( Dave ae .63 o8 |—0.88 —l. “16 154.1649 21| 39, na t2 55] 53.91 —2-6°| 9.31 +98 5.71 —°-22| 1.60 | 0.54 | 0.90| 1.12 | 3.34 0-18 2-53| 53.98 63)9.39+°°F|5 48 23! 1.56] 0.50 | 0:91] 1.13 #18 |isd.1 ©27| 49.20 5°| 52.62 25l9.31—°°2%/5.05 23) 1/53 | 0.46 | 092) 114 | Reiteee ot 10°82 SIG etal OCilncn Moco ent call etal ice Goal ilk Beers se=8) Atal 12 | 51.04 6519-29 Go|? o.an| 2] 0.93] 1. . | 51.95 | 9.25 4.82 1.44 | 0.39 |—0.94 —1.16 154.77 41.77 TE OURS Ty ORE SUMRZACNSUES!: 262 eee TABLE X, Ara. 3.—Continued. Arg. |(v.c.0) Diff. |(v.s.1) Diff.|(v.e.1) Diff. |(v.s.2) Diff.|(v.e.2) Diff.| (v.s.3) (v.¢.3)] (v.s.4)| (v.e.4) ” ” dA ” ” ” 3 ” i ” ” 4 ems 480 154.77 ALT 51.25 9.25 _| 4.82 1.44 | 0.39 | 0.94|_-1.16 481 |154.86 1°02! 49.90 1°43 59.53- 0-72 |9.90°25 | 4.60 --27 || 1.89 | 0.86 | 0196 me 482 1154.93 2°97) 49.61 24%! 49.99 SO 7319.15 29351439 2-27) 1.85.) 0.34 | OF9S eee 483 1754.93 °°5| 49.99 03°! goo, °7S19.08 COM ay 222) 4.30 |W ole 1.00| ia@ 484 |154.99+°-°1| 43.34 ©35) Ggiog 7°19 09 C°8l3igg 222) 1.95 | 0130 |) auom memes —o.OI 33 O. 0.10} 0.21 485 | 154.98 Si 2eeare ne aI eS) 8.90 3.15 ile 0.28 |—1.02/ eine 486° N15 4,952 2 9)| 43.08 2: || Celso oS 80mm o) a5 meee 0.27 | 1.04 its 281 754290 2) 20)|| PAae Obed ibeR ee Casale Omare or ada mean al 0.26 | 1.06] 1.15 488 1154.89 O°2| 44.49 274) 45.95 OOS lg.57 272 13.15 79) 1.04 | 0126 | 1208) iiiene 489 [154.72 27°) aaga 227) gan OO4)e4a °TSi9'93 227) 0199 | 0:26 | 1805) uae 0.13 0.19 Oo. o.14 0.17 490 | 154.59 44.90 43.37 8.3 2.81 0.27 (1.10/24 491 }154.45 724] 45.05°F°-15| 49.51 —9-80|8 140-2919 ¢4 —9-77 0.28 | 1.11) are £92 Ni ae98 oe! | ab s18) S223) G4 CoO Tio 8 am oOo Mag oats 0.29 | 1.12) 1.12 29301154008 ia |/ 45008 © Ooo AON eC odie e inno: alloy aumotS 0.30 | 1.13) aa 494 1958.8% | | 45.12 34.57 45 1250) ae 0.51 |—1.17|_0.99 DOP | ik id mo S 0 cad 98 ON) Toate game 9lll Gnd mec cio demons 0.55 | 1.17) 097 502) 1751,36 ! C:3°| geist: O27) g0%e9) C2e8llienjan nce 2iliag §O-cS 0.59 | 1.16) 0095 503° 1750.95 C4") aero O22" | Saigo 0:°9il5 ggmuec- 22) ean mcce 0.64 | 1.16] 0.93 5041150152» 2-43)! saaisg: 2-22) siuod | SONe bg O22 iedons coce 0.68 | 1.15] 0.91 eee: || 0.44 0.26 oO. 0.22 —=(9),(0)7 505 | 150.08 ee Nall Sans 38 1.40 37 | 0.73 |— 1,15 2oege 506 | 149,61 0-47 | 43.8425 | 99390895 150-22 | a0 2°21 (0.86) one el aay O07 1148.18. S34") 43159) 2°37) 98.46 O22 4.93 29221 do) 2c || 0PSbu10.83. |) eile 508 1148.68 S50) 43.18 ©34) ov6g OCtlaea oF tla 430223 | 0:34 | 0.87 || 1et0 aaa 509 148.11 °5?| 42.89 O35 26:80 Siga[4-51 Sar [1-46 O03] 0.84 | 0.92) 1.09] 0.85 53 82 2 oO LADS eee | AoeaD E | 25098.) Sulake ies } .84 | 0.97 |—1.08|—ong4 BET | 147.08 25> | 49-0072 47) 05 17 2-8 | 409 1 Siete 2) Osea incl 02 ae en 512 (146.46 2-37 | 4156 ott) cag O79 lgig9 FO l1.64 © 26| 0:85 | 1.06 | 1.00) SSC es Se carl) |e il e4T3.69 27 1.42 || 0-86.) adi! | | 1802) aes 514 | 145.97 22°! 40.60 9°49) 99.95 OF°ls 59 O89 lg e7 2-29)! O13 | die | © ilOmlenames | 0.61 0.51 0.74 0.15 0.09 O40 | WHA 66q Se tl) 40209 eel oer 3.32 {1.90 .89 | 1.20 |-1.00|—0.82 | 516 1144.04 °°)? 39155 0°54) 91.38—°°73] 3.14072] 9.91 FO 22) 0.41 | 1.94 | Ol9s/ ones 517 |143.49 254) 38.99 55] 90.68 P7°l9.98 °Flor9 277! 0.43 | 1.98} 0.96| 0.89 518 | 149.74. ee 38.41 °58| 19.99 °P9!9'99 27?) 9.95 ora | 0-46 | 1-82] 0.94] 0.88 ¢ : SUT oC ° ¢ > 2) : y 5 a ae 0.68; 37-81 o1g5| 19-82 S64 {2-66 S75|2-39 oq 0-49 | 1.85] 0.92] 0.88 20 /141.5 37.19 18.68 2.51 2.53 0.52 | 1.39 |0.90|—0.84 521 /140.702°09] 36.56 2°63] 18.05 2-63 |9.3g 2-13 ]o'¢g +275] 0.55 | 1.42 | 0.98/ 0.85 522 1140.00 272) 35.91 2-2) 14.45 7212.05 3 olga | 22 | 0158 || 14 0nd 523 l139.99 © 77 | 35.25 @ 3) 16.87 92.12.18 -2 2218.00. op] Ole: | deat | NGeag meen 524 | 138.57 ee 34.57 ee 1GVS2 9 2008 oe 3.16 Bee 0.66 | 1.49 | 0.95] 0.89 525 | 137.84 33:87 | 15.79 1.94 3.33 0.70 | 1.51 |—0.94|—0.90 Peele gO acer eS 3eb waged ad e528 ae 1.86 0-02 }3.51F°-22| 0.74 | 1:53 | oloeliigeae B27 1136.85 8°75) 39.48 2-72) 14.79 49) 1.78 983.69 18) olns | 1.54) (0 gs] ame 628" |135.69 ote| 81.70 O75) 14.34 S45) ng OOO ls eg O20) 0:82) 55) 0eg 2mm 529 1134.83 ° 7?) 30.96 °74) 13.91 ° 431.67 °°514.96- °-29! 0:86 || 1.55 | = Os9ammonem 53 0.76 0.74 O°. 0.04 0.19 30 1134.07 30.22 3.51 1.63 4.95 1.56 |—0.91/—0.99 BSE 1188.80 o 4A 09146 = 0 101g ig ae OSE Ose a0 ea ace 1.55 | 0.91) 1.01 532 1139.59 C79) 98.69 ©7711 yong ©-35\lq 5g 021 41¢9 0:80 1.55 | 0.92} 1.03 533 V131.73\ 2379) $Om. 08 “Sata oaGs “OUSCi eee lac Naomi gn 1.54 | 0.92] 1.05 534 130.94 ~24 91 on 15 © 277) oes oO Se) yagec: Sls CON c 1.54 | 0.938] 1.07 535 |1> 0.79 0.78 | 0.26 0.02 0.18 585) (180.15 | es) 26:37 11.90 1.60 5.18 1.53 |—0.93|—1.09 536° | 109.35 2-20] 9558-0 79)| ai eGo 22 1G othe Oo Sse meee 1.51 |. (0%95)| eng 537 1198.55 ase o4:ei OFT! alas 98-2 hehe °-CSiipanamemeae 1.50 | 0.96] 1.12 538° /107.75) 282i oaio3 S78) nae 220i eon c-Collaiammerne 1.48] 0.97| 1.13 539 |196.95 °° 89)| logins, 0°78) Sani 2o-* Oligo Coll amaqumcats 1.46 | 0.98| 1.14] 540 0.80 0.77 | 0.12 0.06 | 0.17 126.15 22.48 10.98 1.82 |6.07 1.43 |—0.99|—1.15 J THE ORBIT OF URANUS. 263 TABLE X, Ara. 3.—Continued. (v.c.0) Diff.) (v.s.1) Diff. | (v.e.1) Diff. |(v.8.2) Diff. |( .| (v.8.3)] (v.e.8)| (v.8.4)] - | ” n" u" "” ” uv ” " ” u" uv '126.15 92.48 10.98 82 .43 |—0.89 | 79| 91.70 10.88 go +2. oo 41 one 20.93 ° 3! 3 20.16 8 BH) ce oes 40 ey CLS "33 | 0.961 - ners 33 | 96 OOM, a8 3 97 | Rtas 26 | 0.99| 52k | 38} .02| 65 ey .20 .04 | 79 es ay .06 93 i ata 08 Denes! Aira eiero} 23 ae 08 12) 3! pao. 04 13 fz .16 ue OL ae 0 98 15 Benisome 95 | 1.15 02 0-16) ).92 16 ).90 slit 19 O17 0.16 87 .18 .85 0.16 51 67 10:16 | 83 g4 O17 81 00 als ).79 LGA ae oe 17 76 .16 sue | —o.10O om) LO tO ho bo Oo co co OAT aAWwowds ed ios) eo v2 bo bo bo bo bo 99900 PREG HHO nw ow il CO G9 C2 02 G2 LO Now OL Nn Sata sy a Dy .00 (rmenmeat A~I Oo OH CON oo WwW nh coun CrCl ee He Re OO OO -I “ICO NW aT 3-7 1 -1 otto : on 29 32 47°15 | SiG Sits oro on 1 i) TT Oilmeor a ee -14 05 ergs ay A163} 04 nD 44 3 h2 -12 on -T -T - bobo © OoRRR RRA “I WHO f 9900¢ -T i=) 99900 -T poy a a -T 9.8 9.3: 8. SEs te Tey Galle 6.5 5.8¢ 5. 4.9} 4, 4.9% SES ded one 2: N oN Oo Oo GW COD N Lo — to} er) 2 = fony r Bit -I0 -09 -09 -09 .08 .06 .06 .06 05 205) -03| 03 03 .O1 | 9 —~I 0 - Co Mm bo Orc ON 9900 [SS DMHwW So bo bo tS 27 TT 990900 nm N ho b bok eS) | ie) | oo Go or ray CO Ul YD H f=) WSTOAD FPWNWWO Re BR eee = Ge ¢ mann non Nm nN NANA OWN Bp hob NN bo bo bo bo bo OV me He Co 09900 9900 % Lal Lol Leni w | | tc) erikor) | 9 G0 Ge oa By oe Se STITT, “1D SUN > CO J oo aT a aa TOT aT et aT eT OAT DIOOD ~ ee oo OO bE 2 Oe nD ¢ Lo bo tO bo bo CROEORO) 90990 OC Tt -1 orn OW WW OW Sie neds moron on So woo ly ve) Ce ee ee ey Preror) oo oS Om d Dm mMmOmDnynm mMmwnmo _ rte DO bob Oo He oa © 0 bo LO bo ORG boo 0) OD 99000 Monn UMM ee NN YN — | 9° .02 .03 .O4 05 .04 0 CO CO Co bo 00 OO awaSHe =TOO, © 09 0) LO 999090 or Nn oS Hee ee He He OT Or Or Or Ot Or or nr? = 5 1 Oi Ss CO < 0990 i] 4 NAMM NADANA ADAAD CU MEIEIES COPIENCURS Lal Or Ou wo 09 oD ~~ 2900. [oyo} bo To -¥-7 -T -1 + me HR Ooo Ss 264 THE ORBIT OF URANUS. TABLE X, Are. 3.—Continued. Are (p.c.0) (p-s.1) (p.e.1), (p.s.2)] ( 0 524 | 180 89 49 1 525 | 178 90 50 2 525 | 176 90 50 3 526 | 174 91 51 4 527 | 172 91 52 5 528 | 170 92 53 6 529 168 92 54 il 531 | 165 92 55 8 533 | 163 93 55 9 535 | 161 93 56 10 538 | 158 94 57 11 540 | 156 94 aT 12 543 | 154 95 58 3 546 | 151 96 | 59 14 550 | 149 96) 59 15 553 | 146 97 60 16 557 | 144 98 C0 17 560 | 141 99 61 18 564 | 138 100 61 19 568 | 135 | 101 62 20 573 | 132 | 109 62 21 Hf || 128) 103 | 62 22 582 | 196 105 62 23 586 | 123 |-106 63 24 591 | 120 108 63 25 596 | 117 109 | 63 26 601 | 113 it || GB 27 606 | 110 | 113 63 | 28 Git | wo” |) Ws 63 29 Gillza | OSiean ature 63 30 622 | 100 | 119 62 31 627 96 121 62 32 633 | 93 | 193 62 | 33 638 | 90 126 62 84 643 | 86 129 61 35 648 | 82 132 61 36 654 | 179 135 60 | 37 659 |\ 76 | 139 60 | 38 COL TQ MeL 59 39 669 | 69 145 58 | 40 674 65 149 58 4] 680 62 152 5Y 42 685 ) 58 156 56 43 689 | 55 160 55 44 694 | 52 164 54 45 698 | 49 168 53 | 46 703 46 173 52 | 47 707 43 177 51 48 TL AOR a S28 50 49 715 | 38 186 49 50 719 | 35 191 44 51 729 | 32 | 1196 46 OP, 1P2AB || BX) 201 45 53 729 | 98 | 206 3 54 732 |. 26 | 911 49 55 735 | 24 | 216 40 | 56 137 | 29) | 999 39 57 740 21 221 38 58 742 19 | 933 36 59 744 18 | 238 35 60 746 244 | 33 17 me Lo ppmprmPpmwmrnrn nonrwowe AT nee bo OW He Arg. (p.c.0) (el) Gre) (p.8.2)) (p c.2) 60 746 Ly, 244 33 61 TAT Wi 249 By4y bi 62 748 16 255 350 63 749 16 260 29 64 749 16 266 27 65 750 16 271 26 66 750 16 277 24 67 750 IU 282 293 68 750 17 287 22 69 749 18 293 20 70 748 19 298 19 fel TAT 21 303 18 72 746 22 308 Uy 3 744 24 313 16 74 742 26 318 14 ne) 740 29 323 | 33 76 737 3 327 | 12 TT eee) 332 | 11 78 731 3 836 | 10 79 728 40 340 09 80 724 43 344 09 81 721 47 348 08 82 717 51 352 O07 83 US 55 355 07 84 708 59 359 06 85 703 63 362 06 86 698 68 364 05 8T 693 72 367 05 88 688 i 369 | 05 89 682 82 372 05 90 676 87 373 05 91 670 92 379 05 92 663 97 376 05 93 657 102 378 05 94 650 108 378 06 95 643 114 379 06 96 636 119 379 06 OT 629 195 3879 07 98 622 130 379 08 99 614 136 379 08 100 606 141 878 09 101 598 147 HLT 10 102 590 1538 376 11 103 582 159 374 12 104 574 164 873 13 105 505 170 371 14 106 557 176 368 15 107 548 181 366 16 108 539 186 363 17 109 530 19¥ 360 19 110 521 197 857 20 111 512 202 353 21 aT, 503 207 350 23 113 494 212 346 24 1a 484 217 342 25 Wily 475 299 338 27 116 465 2207 3338 28 Divi 456 231 329 30 118 446 235 324 31 119 436 239 319 32 120 427 243 314 34 21 22 24 THE ORBIT OF URANUS. TABLE X, Ara. 3.— Continued. Arg. | (p.c.0)| (p.s.1) (p.e.1)) (p.8.2) (p.¢.2)] Arg. | (p.c.0)| (p.8.1) | (p.c.1) (p.8.2) (p.c.2)] 120 | 427 243 | 314 3 66 1S Opel eel 172 66 26 33 121 | 417 247 | 309 35 66 181 10 168 66 25 33 122 | 407 250 | 303 37 66 182 9 165 67 25 34 1233 |) BO 254 | 298 38 65 183 9 162 67 24 35 124 | 387 957 | 292 39 65 184 8 159 68 24 36 125 | 378 260 | 286 41 | 64 185 8 156 69 24 258 126 | 368 263 | 280 42 | 63 186 8 153 71 24 | 38 127 359 266 | 274 By | 8} 187 8 150 72 23 39 128 | 349 268 | 268 44 | 62 188 9 147 4 93 40 129 | 339 270 | 262 45 61 189 9 144 15 ey | Zt 130 | 330 272 | 256 46 60 190 10 141 17 23 42 131320 274 | 250 47 59 191 11 138 79 24 43 132 | 310 275 | 244 48 58 192 13 136 81 24 44 133 | 301 277 | 238 49 5T 193 14 133 84 24 45 134 | 292 278 | 232 50 56 194 16 151 86 24 46 135 | 289 279 | 295 50 55 195 18 129 89 25 47 3604) 213 279 | 219 51 54 196 21 127 91 25 48 137 | 263 280 | 213 52 52 197 23 125 94 26 49 138 | 254 280 | 207 52 51 198 26 123 97 26 50 139 | 245 280 | 201 52 50 199 29 121 99 27 51 140 | 236 280 | 195 53 49 200 32 120 | 102 28 52 141 | 297 280 | 189 53 47 201 36 118 | 105 28 52 142° | 218 280 | 183 53 46 202 39 Mile |) TOS 29 53 143 | 209 279 | 177 53 45 203 43 116 | 112 30 54 144 | 201 978 | 171 53 44 204 47 14 | 15 31 54 145 | 193 Q77 | 165 53 49 205 51 Uy |] aT 32 55 146 | 184 275 | 160 53 41 206 5d 112 | 199 33 55 147 | 176 gy4 | 154 52 40) 207 60 112 | 195 34 56 148 168 272 | 149 52 3 208 65 111 129 35 56 149 | 160 | 271 | 144 | 52 | 38 209 | 69 110) | esoun| sch eon 150 | 152 269 | 139 51 36 210 15 110 | 135 37 51 151 | 145 967 | 134 51 35 211 80 Tl) |) Tigy: 38 57 152 | 13 265 | 129 50 35 212 85 110 | 148 3s 57 153 || 13 963 | 124 49 34 213 91 110 | 146 40 57 154 | 193 260 | 120 49 33 214 96 110 | 150 41 5T 155 | 116 sth |) lal 48 32 O15) 102 I) |), 31538} 42 57 156 | 109 955 | 111 47 31 216 | 108 110 | 157 43 5T 157 | 103 252 | 107 46 30 Die |) ale! 110 | 160 44 57 158 96 249 | 108 45 30 218 | 120 itl |) GA 45 57 159 90 246 | 100 45 2 219 | 126 112 | 168 46 57 160 84 243 | 96 44 29 220 | 133 113 | 171 47 57 161 78 240 | 93 3 28 221 | 139 114 | 175 48 56 162 13 236 | 90 42 28 222 | 145 15 | 1738 49 56 163 67 933 | 87 4] 27 Dee} |) G9) 116 | 182 50 59 164 62 929 | 84 40 27 224 | 159 117 | 185 50 55 165 57 226 | 81 39 27 225 | 166 118 | 188 51 54 HGOmNmeSSie = 2920/79) /\| 88) ||| 2 226 | 173 119 | 192 | 52 | 53 167 48 DON ie Ti 36 27 227 | 180 120 | 195 53 52 168 44 915 | 75 35 27 298 | 187 12a os 54 52 169 40 219) || 13 34 24 229 | 194 123 | 201 54 51 Heel SG. |e 208 |5 41 1-38 |). -97 230 | 201 125 | 204 | 55 | 51 171 32 204 | 70 32 OH 231 | 208 127 | 207 55 50 172 29 201 | 68 31 28 232 | 216 128 | 210 56 49 173 26 197 | 67 30 28 233 | 223 130 | 213 56 49 174 23 193 | 67 29 28 234 | 230 32 | 216 51 48 115 21 190 | 66 28 29 935 | 238 134 | 218 5T 47 176 18 186 | 66 28 30 936 | 245 136 | 291 57 46 Ti 16 182 65 27 30 237 253 138 294 58 45 178 14 179 | 65 27 31 238 | 260 140 | 226 58 45 179 12 175 | 65 26 32 239 | 268 142 | 298 58 44 180 11 172 | 66 26 33 240 | 275 144 | 231 58 43 34 August, 1873. 266 THE ORBIT OF URANUS. TABLE X, Ara. 3.— Continued. - ; l } | Arg. (p.€.0) (p-8.1)| (p.e.1)} (p-s.2)} (p-€.2) (p.c.0) (p.s.1)| (p-€.1)) (p.s.2)} (p.¢.2) 240 275 144 | 9231 | 58 43 562 | 208 | 250] 49 32 241 283 1460\) 233) 85S 42 562 | 208 | 250| 42 32 242 990 | 148 | 235 | 58 41 562 | 208-| 250 | 42 32 243 298 | 150 237 58 | 40 561 208 | 250) 42 32 244 | 305 | 152] 239 | 58 40 560 | 208 | 250| 41 32 245 | 313 1547 240] 57 | 39 559 | 208 | 250] 41 31 246 320 | 156 | 242] 57 38 558 | 208 | 250] 41 31 247 328 159 | 244| 57 | 38 557 | 208 | 250| 41 31 248 335 | 161 | 245] 57 3 555 | 908 | 251 | 41 31 249 342 163 | 246] 57 36 553 | 208 | 251] 40 30 250 350 | 165 | 248| 56 | 36 B55 |) 208) | 2251) ko alee 251 357 | 167 | 249 | 56 35 549 | 209 | 251] 40 30 252 364 169 | 250 | 55 3 547 | 209 | 251 | 39 3 253 371 171 | 951 | 55 3 544 | 209 | 951 | 39 29 254 378 173 | 952 | 54 By 542 | 210] 251 | 39 29 255 385 175 | 253) 54 33 539 | 210 | 951] 38 29 256 392 177 | 954) 54 33 536 | 210 | 252) 38 29 51 399 179 | 254] 53 32 532 | 211 | 252) 38 28 258 406 180 | 255 | 52 32 599 | 211 | 252) 37 28 259 412 182 | 9256 | 52 32 525 | 212 | 2952] 37 28 260 419 184 | 956 | 51 31 521 | 213 | 2521] 386 28 ~261 426 185 | 256 | 51 21 | 517 | 213) 252) 36 28 262 432 187 | 257 | 50 31 | 518 | 14 || 252 | 85 28 963 | 43! 189 | 257 | 50 31 | 509 | 215 | 952) 34 28 264 | 444 190) 25ii 50 31 504 216 952 | 34 28 265 450 192 | 257 | 49 31 500 | 217 | 252] 33 28 266 456 193 | 957 | 49 31 495 | 918 | 252) 38 28 267 462 194 | 957 | 48 31 490 | 219 | 952] 32 28 268 | 468 196 | 257 | 48 3 485 | 221 | 951] 31 28 969 | 473 | 197] 257 |. 47 31 | 480 | 222] 251] 31 28 270) || — 478" |) 198) 257) Ar 31 330 | 474 | 2293 | 251 | 83 28 71 | 484 199) |) 25ie| 46) 133 331 469 | 224 | 250| 30 28 279 | 489 200 | 257 | 46 3 332 463 | 225 | 249 | 29 29 273 | 494 201 | 257 | 45 31 333 457 | 927 | 949 | 99 29 274 | 499 202 | 256 | 45 31 334 451 | 228 | 948 | 28 29 245 | 503 203 | 956 | 45 3 335 44¢ | 299 | 948 | 27 3 976 | 509 203 | 256 | 45 31 336 439.| 931.| 247 | 27 8 217 512 204| 256 | 44 31 337 433 | 933 | 246! 26 31 278 517 205 | 255 | 44 3 338 497 | 934 | 9245 | ‘25 31 279 521 | 205 | 255 | 44 39 339 421 | 936 | 244| 925 31 280 524 906 | 255 | 44 39 340 414 | 988 | 243 | 24 32 231 528 206 | 254 3 32 341 407 | 239 | 249] 24 33 282 531 207 | 254 | 48 39 342 401 | 241 | 941 | 24 33 233 535 207 | 253 | 43 32 343 394} 243 | 240) 93 34 934 | 538 | 207 | 253 | 43 32 344 387 | 244 | 938] 23 35 285 541 | 208 | 253 3 32 345 380 | 246 | 936 | 23 35 236 544 | 208 | 252] 48 33 346 373 | 248 | 935 | 929 36 237 546 208 | 252] 438 33 347 366 | 249 | 233] 29 37 288 549 | 208 | 252] 43 33 348 359 | 251 | 931 | 22 38 239 551 208 | 251 | 42 33 349 352 | 253 | 929 | 929 38 290 553 208 | 251 | 42 33 350 344 | 955 | 997 | 21 39 291 555 | 208 | 251) 49 33 351 33 256 | 225 | 21 40 292 556 | 208 | 251) 49 33 352 330 | 258 | 993] 21 41 293 558 | 208 |. 951 |, 42 33 253 393 | 960 | 221] 21 42 294 | 559 | 208 | 251 | 42 33 354 315 | 261) 218| 21 42 295 | 560 | 208 | 250] 42 33 355 308 | 263 | 216] 922 43 296 561 | 208 | 250 | 42 33 356 | 300| 264] 213] 22 44 297 | 562 | 208 | 250 | 49 32 357 293 | 266 | 211] 22 45 298 562 | 908 | 250 | 49 32 358 285 | 267 | 208] 22 46 299 | 562 | 208 | 950) 42 32 359 278 | 269 | 205 | 93 47 300 | 562 | 208 | 250 | 42 32 360 270 | 270 | 203 | 23 47 SR ES SR A SA TE THE ORBIT OF URANUS. TABLE X, Ara. 3.— Continued. 267 | Arg. }(p.c.0)| (p.s.1) 360 270 270 361 | 263 272 362 | 255 273 363 | 248 274 8364 241 275 365 | 233 276 366 | 226 277 367 219 278 368 || 211 278 369 | 204 279 370 | 197 280 871 | 190 280 372) | 183 281 373 | 176 281 374 | 169 281 375 | 162 281 376 | 155 281 377 | 149 281 378 | 142 281 379 | 136 281 380 | 129 280 881 | 193 280 33) |) alaliy 279 883 | 111 278 384 | 105 278 8385 99 QTT 386 94 276 38T 89 274 388 83 273 389 78 272 390 73 270 391 68 268 392 63 266 393 59 265 394 54 263 395 50 261 396 46 258 397 42 256 398 39 254 399 35 251 400 32 248 401 29 246 402 QT 243 403 24 240 404 22 237 405 19 23 406 17 231 407 15 228 408 14 225 409 13 221 410 12 218 411 iil 215 412 iil 211 413 | 10 208 414 | 10 204 415 10 201 416 10 197 417 11 194 418 12 190 419 13 187 14 183 420 | (p.e.1 ) (p.8.2 203 200 197 194 190 187 184 181 66 68 69 (p.c.2)} Arg. | (p.c.0)} (p.s.1) (y-0.1)) (p.8.2) (p.0.2) 23 Ay 420 14 183 | 69 44 39 23 48 42] 16 180 10 44 32 24 49 429 18 176 | 72 43 29 8 50 423 20 173 | 174 42 32 25 50 424 22 TON ane 41 32 a au 425 24 166 | 78 41 32 26 52 426 27 163 8] 40 32 27 52 427 30 160 33 39 32 28 53 428 33 157 | 86 38 29 29 53 429 36 153 | 89 38 33 29 54 430 40 150 | 92 31 33 30 54 431 44 147 | 95 36 34 3 54 432 48 144 |] 99 35 34 32 55 33 53 142 | 103 35 35 33 55 434 BY 139 | 106 34 36 34 55 435 62 1360 La 34 36 35 55 436 67 134 | 115 33 3 36 55 437 12 132 | 119 3: 38 37 55 38 it 130 | 124 32 39 38 55 439 83 128 | 128 32 40 39 55 440 89 126 | 133 32 40 40 55 44] 95 124 | 138 | 32 41 41 55 442 | 102 129 | 148 | 3 42 42 55 443 | 108 121 | 148 3 3 42 54 444 | 115 120 | 153 31 44 43 54 445 | 121 118 | 159 3 45 44 54 446 | 128 118 | 164 3 46 45 53 447 | 135 TG Ih EO 3 47 46 53 448 | 149 116 | 176 32 48 47 52 449 | 150 116 | 181 32 49 47 52 450 | 157 115 | 187 32 50 48 51 451 | 165 115 | 198 33 51 48 50 452 | 173 115 | 199 3 52 49 50 453 | 181 116 | 205 34 58 49 49 454 | 189 117 | 211 34 54 50 48 455 | 194 Tal || Cally 35 55 50 47 456 | 206 1alg: | 223 3 56 51 47 457 | 215 120 | 229 36 57 51 46 458 | 223 121 | 236 31 58 51 45 459 | 232 ee 949 38 58 52 45 460 | 241 125 | 248 39 59 52 44 461 | 250 127 | 254 40 60 52 43 462 | 259 129 | 260 41 60 52 42 463 | 268 131 | 265 42 61 52 41 464 | 278 134 pee 43 61 52 40 465 | 287 137 | 277 44 62 52 39 466 | 296 140 | 283 45 62 it 38 467 306 143 | 288 47 62 51 38 468 | 315 147 | 294 48 62 51 31 469 | 324 150 | 299 49 62 50 | 36 aio | 334 | 154/305 | 51 | 62 50 36 AT1 | 3438 158 | 310 52 62 49 35 472 | 358 162 | 315 53 62 49 34 473 | 362 167 | 320 54 62 48 34 474 | 372 171 | 325 55 62 ABie | 33 ais | 382 | 1%6:| 329 | 57 | 62 4T 33 476 | 391 181 | 334 58 61 46 33 4iT | 401 186 | 338 59 61 46 32 ATS) | AIL 191 | 342 60 60 45 32 479 | 420 196 | 346 61 60 44 32 480 | 430 202 | 349 62 59 t THE ORBIT OF URANUS. TABLE X, Ara. 3.—Concluded. (p.c.0) | (p.8-1)| (p.c.1)] (p.8.2)| (p.c.2)f Arg. | (p.c.0)) (p.8.1)] (p.¢.1)] (p.s.2)) (p.¢.2) | 430 | 202 | 349 | 62 | 59 540 | 741 | 381 | 186 | 29 | 19 Be S0e || SES) GE | Be 541 39°| 378) 18a Osi een 449 | 213] 356] 64 | 5% 542, | S087 |/ e815 6 el eae ene 45809) 28 ss5 90 |) 6505 lan 6 543) 7 1385)!) Sse ur ee SOG aa mee 468 ‘| 223} 361 | 66 | 55 ad | 732) S68) | N67 25 aimee Ati. | 23000 364 (Gm Bhetn4 545 | 730 |) 9365} 116240) 204 eae 486 | 236 | 366 | 68 | 53 546° | 727 | 361)) 158; 20 Sos 496 | 242] 368] 69 | 52 547 | 724) 357 {153 | 23 | 26 505. | 248 | 370 | 69 } 51 548 || G21 | 853!) 149) || 2a aoe Bla P2548 a3 leTOn | 50 549 |) 717") 3505) 145 7) 2a ies 523 | 260 | 373 | 70 | 49 550 | 714] 346) 141, | ot | 9 532 | 266 | 374 | “71 | 48 551 || U0) 1342 | 13h i) 21 ao 540 | 272] 374] 71 | 46 552) |) 706) eoesa|isdq ol al emer 549° | 278/375 | 72 9) = 45 553 |) 102") 28801309 | S20 mime 558 | 284] 375'| 72 | 44 5b4 “) 698") 329°) 126) | 920 Vas 566 | 289 | 375 | 72 | 42 555 |) 1693)! #325) | sl23)|| ON aimee Bid 295 | 84 | a 556 || 689 | 321°] 1201 | 20 || as SE | EOI | ue eX 55 S684.) SiG ch tara i One ieee SO BU | Bee TO Ns Bie 558 | 679 | 812°) 114) | 19) 38 599.) 313) Sr W297) sr 559 | 674) 808 | 112) )||" Ponmieees 606.7) ssy 37a) 72) 35 560 | 669| 303} 109 | 20 | 40 614 | 324] 369] 71 | 34 561 | 664| 299] 107 | 20 | 41 621. | 329) 36% | 3 562 | 659] 295} 105' | 20 | 42 629) 3340) 365 |) lies 568 | 654 | 2901) 103; | 20 | ae 636 || 339) 363'|| 70 “|= 564 | 649 | 286} 101 | 20 | 44 643 | 344] 360] 70 | 29 565 | 644] 282} 99 | 20 | 45 649 | 349] 357] 69 | 28 566 | 6381\) 278) |" oian| melema mete 656 || 358) 354)" 68) |= oi 567 | 683) 274) 96) | O1 ay 662 | 358] 351| 68 | 25 568 | 628] 270] 94 | 22 | 48 668 | 3862} 347} 67 | 24 569 | 623] 266} 93 | 92 | 49 674 | 366] 344] 66 | 23 570! |" G18 |) 262)" 928) osama 680 | 870 | 340] 65 | 22 BTL | (6131) 25800 2s alan 686 | 374] 336] 64 | 21 572 | 608 | 255 |. 90) | 24° |an GOL | 87 |) 332))) 63.4 1-20 BIB |} 602) 25 00%) 25) aoe 696 | 380 | 327) 62 | 19 Bid + | 598°) 241s 8907) S85) |eaos 701 383 323" |, 61 1 18 515 | 598| 244] 88 | 26 | 54 706 | 3886] 318] 59 | 18 BiG | 588) 240 | 88) on mee TOS eS SS Sis | a5 Sane 5a | 588 | 237 | Sti || 28) ibe 714 | 390} 308| 57 | 16 Bis) | 5TS) 2e40 erst, | Osmallmaa 718 | 392 | 303) 56 | 16 B19 | 574 || 2s1N lsat | oun 721 | 394 || 298) 54 | 15 580 | 570) 2281)" 86) |) SO miaeag 25° | 396) 293) | 53a" 15 581 | 565 | 225] 86 | 31 | 57 728) 39% || 28%) 59°) 14 582 | 561] 222]: 86 | 32 | 58 731 | 398] 282] 50 | 14 583 | 557 | 219 | 86 | 32) 958 733 | 399] 276 | 49 | 14 584 | 554] 216} 86 | 33 | 58 736 | 400] 270) 48 | 14 585 | 551] 214] 986 | 84 | 58 738 | 400 265 | 46 |) 14 586 | 547 | 2 | 860) Ssomlmeoe 740 | 400} 259| 45 | 14 587 || 544 | 5209 |) 186) || sab anieag 742 | 400] 253 | 44 | 14 588 | 541] 206] 86 | 87 | 59 743 | 399 | 247] 42 | 14 589 | 539] 204] 86 | 38 | 59 744 | 399] 242] 4y | 44 590 | 586 | 202] 87 | 39 | 60 745 | 398] 236] 40 | 14 591 | 534] 199] 87 | 40 | 60 745. | 397] 230°) 88 | Td 592, || 682 | 9m || “Sie at) sian (46 || 396)| B25, By" i) 15 593 | 530 | 1957) 88 | 42, || 760 746 | 394] 219} 36 | 15 594 | 528] 193] 88 | 43 | 60 145 | 392] 213 | 34 | 16 595 | 62% | 190) 88 | 44 || ho 745 | 390] 208] 33 | 17 596 | 526| 188} 88 | 45 | 59 Wad ||) 8895), 08M) oad aly 597 | 595] 186} 88 | 46 | 59 743.9) (S86 OTA mis mimes 598 | 525 | 184] 89 | 47 59 742 | 383] 192 | .3 19 599 | 524] 182) 89 | 48 | 58 381 89 | 49 | 58 741 186 29 19 600 524 180 EOE OFR BID Ov ap eAUN TEs 269 3 a 9. | = — . 0) (v.s 1)) (v.c.1) J (v.¢.0)) (v.8.1)| (v.e.1) J (v.€.0) (v.84) (v.c.1) ” " " ” ” me ” ” ” ” uw w” 0.03 0.21 5 3 8 ) OF 1.12) 0.03) 0.22 Bele OTe eae baie aah (oe ees en was ocular cckouelqis ne 0.18) 0.19 0.09, 0.15/0.19}0.12/0.10 0.12 1.11/ 0.03) 0.26]0.07 0.16 0.17 ii one ae Fal eetal Gea ubotatl ekac tines : ° 10 6 i 18 ¢ 99 € fl A 1.10] 0.03] 0.281 0.081 0.16|0.16}o,10'0.18)0.19[ 0.301 0.101 0.87] 0.11) 0.09). 0.08 i meee ile leas $/0.19]0.30 0.10] 0.37]0.11) 0.09 0.09 ae =a reel EH Fa oa | op 0.18) 0.19}0.39| 0.10) 0.44 0.10'0.08 0.08 oi OGL Cae (NOE aie ae Fans Fala ie 0.50 0.11] 0.5240.09)0.08) 0.07 0.39 11001 0.06|0.34 paolonel ol aly a aa 0-61) 0.18 0 5970 09 0.08 0.06 0.8510.29]0.91 0.07|0.85| 0.091 0.181 0.12[ 0.09 0.161 0.17|0.e7| 0.1710. 7510.08|0.07| 0.08 100 | 0.31) 0.31f0.93) 0.08 0.3640 po lolislouatores|a eile set selena A TPM ea aerate oe] 2810-88) 0.0910. 18) 0.128 is 0.16/0.17] 1.00) 0.21/0.8310.07/0.07) 0.08 elaeclocdocio vlogs A ee eee 1.14 0.25) 0.919 0.06) 0.07, 0.02 12D CEE Ee aed ba reseed bore pon 0.15]1.28) 0.29/0.98] 0.06) 0.06) 0.02 140 |0.17/0.4310.74) 0.14:0.38] 0.09| 0.18 0.081 0.06 oe aE es a Fa Parlier /0.38]0.09 0.18 0.0840.06 0.15 0.13]1.56) 0.40) 1.1: 5/ 0.06! 150 |0.14/0.4610.68| 0.16 0.3840.09| 0.18 0.070.060.1210 6 Daler eg baal eh 160 |0.11|0.49}0.63/0.18| 0.380.091 0.18 0.0640.05 0.11/0.11 Fe eee aaa: Lprac lene tog Re reese |e (2 280-88 8. 09/0:18) 0:08 10:05) 0.10/0"1 1.83]0.52 1.25]0.04 0.06, 0.00 Se Oe OnH 0 3110.08) 0.17) 0-05) 0.08 0.10 10]1.96 0.58 1.31]0.04 0.06 0.00 190 |0.05/0.561 0.461 0.23| 0.360.081 0.16 0.0440.04 0.09 aoe ae Nee Fee ecealree eee 200 | 0.04) 0.59)0.41) 0.25 0.35 0.08'0 16 0 10.03 0.08|0.07 a pine ear sel 4 210 |0.03/0.61| 0.36) 0.26 0.34] 0.07 0.15 0.021 0.03 0.07 0.05 Evan aie lard acedcas 920 |0.02| 0.63) 0.391 0.28 0.32]0.07 0.14 0.0210.02 0.0%, 0.05 BO One Eee Coe We diacene 230 |0.02| 0.641 0.271 0.30 0.3110.07, 0.13 0.02} 0.02 eae latosl otee ates Pele pale sea 240 | 0.021 0.660.253] 0.31, 0.301 0.06) 0.12, 0.01] 0.02 Bos ttodl orci itts| entctasloroe Roe -VO) 0,12) U. -UZ) 0.00) VU. Le 05) 1.505 NRX) (.0£ 950 [0.031 0.6710.20| 0.32 0.28] 0.061 0.11/ 0.01] 0.01 0.041 0.03] 9.65 ra a peed ee 260 |0.04| 0.67] 0.16) 0.34| 0.2610.06 0.11 0.01]0.01 0.04 0.0319 ¢s Hee sea (Scape es 270 10.061 0.6310.14| 0.35| 0.241 0.05| 0.10, 0.01} 0.01/ 0.0310. Ee ee ee eeu aee 980 |0.08| 0.6810.19| 0.35|0.2210.05| 0.091 0.011 0.00 0.03 ace PN ae eae Baan ee 290 | 0.10) 0.6810.10) 0.36, 0.21] 0.04 0.08 0 Oe eu aes Haasan Beli : Ole 0.04, 0.08) 0.0 wee 0.01]2.68 1,29} 1.40]0.06) 0.10 0.06 13] 0.6740 08) 0.37/ 0.19 70.02) ale 310 |0.16|0.67 BE we 0 i Bioalgids iL an Bien eee Foe lpcrcioerlenie 320 | 0.19| 0.66] 0.08| 0.37 0.1610.03| 0.05, 0.0: loan lege he Bie fee ae ene aH ae 330 | 0.22] 0.6490.09| 0.37 0.1470 0310.0410.0340.00/0.021 0.0149. SU a Fa eal oe | 0.64] 0.09] 0.37 0.14] 0.03) 0.04] 0.039 0.00 0.02) 0.01]2.48)1 39/1.1940.09/0.11| 340 | 0.26) 0.6310.10) 0.37 0.12]0.02) 0 04! 0.044 pore Deen creed raa 2 "1010.37. 0.1240.02! 0.04, 0.0410.00 0.02| 0.01] 2.40) 1.- 3 350 0.30) 0.6140.12 0.36, 0.10 0.02) 0.03} 0.0 10.00/0 02)0 3 2 a ai _ es ce ce ao ee 0.5990.14' 0.36 0.0940.01 0.03) 0. 540.00 0.02/ 0.02 9.90 1.39 ioeloati ee ee 370 |0.37|0.5610.16 0.35 0.0810.01/ 0.02/ 0.0640.01 0. 310.0219.0911.3710.910.11/ 0.19) 0.1. BEM or] ojo: 350.08 Jo-01) 0.03) oh Or 0 08 Oe 9.09/1.37,0.91]0.11 0.12) 0.14 Bi Woe 4 -o4 é ; UZ 10.01 0.03) 0.03591.96)| 1 35/0.8310.12) j ee Oe a Sees be 2) 0.07 | -96| 1.35/ 0.830.192) 0.13| 0.15 i - oe us Gagne 0.01, 0.02 0.08} 0.01 0.04) 0.03]1.83] 1.82 0.75]0.12 0.13) 0.16 -49|0.4910,27 0.32 0.0440.01| 0.02 0.08]0.02 0.04) 0 5710.13! 0.1: 410 |0.53/ 0.4610.32 0.31 0.0340.01) 0.01/0.0910.02 elke oe ie oaly: Hee nee 429 |0.57|0.4310.36| 0.29. 0.0310.01| 0.01 0.10] 0.03) 0.06| 0.06 Tali oilomolons | jute 430 0.60] 0.40f0.41, 0.28, 0.02] 0.01) 0,02 0.11]0.03 lO ROE SLUG ced ae! one nee aa Be ~ eal : J J J. ° Ap 216) U. a C Le 0 | 0.63] 0.37]0.46 0.26) 0.0240.01) 0.02 0.12]0.04 0.07,0.07]1.14)1.10 0.87] 0.15) 0.14) 0.19 450 | 0.66/0.34 0.52 0.24/ 0.024 0.01] 0.02 0.1: 1 0.04/0.08/0.0841.00/1.05/0.31] 460 | 0.69] 0.3140.57| 0.22! 0.0270.01/ 0.02 0.1490.05 0.09, 0.09 0.87/0.98 0.25) 470 | 0.71] 0.29] 0.63 0.21/ 0.03 0.02) 0.03 0.15]0.05 0.10) 0.10 0.741 0.92 0.19 io 0.73) 0.2670.68 0.19} 0.03 0.02) 0.03, 0.1690.06) 0.10) 0.11 0.61 0.85 0.14] = a 0.2410.74 oie 0.02) 0.04 0. } 0.07] 0.11] 0.12}0.50 0.78)0.10]0.16 0.13] 0.20 -76| 0.2110.79| 0.15| 0.0510.02/ 0.04) 0.17] 0.07) 0.12) 0.13}0.39 0.72) 0.07 3] 0.5 510 |0:77|0.19f0.84 0.14| 0.060.031 0.05 | 0.18] 0.07| 0-13) 0.14 Fcalintee levee 0 te\0115 0.90 520 |0.78/0.17]0.88 0.12 0.03[0.03 0.06 0.18]0.08 0.14 0.14}0.22 0.58 0.0240.16 0.13] 0.19 230 [o-velo-tefo.¢sl 0.101 0.09] 0.031 0.0g| 0.18[ 0.08) 0.14|0.15]0.15|0.52|0.00]0.16)0.19| 0.1: 340 (0-781 0.141 0.97/ 0.09) 0.10] 0.04| 0.08] 0.199 0.08] 0.15| 0.16] 0.09] 0.45) 0.00} 0. Apseleo ts yb a : 9| 0. ote 8/0.15)0.16]0.09 0.45 0.00] 0.16 0.12) 0.18 50 |0.77| 0.16 0.08 0.12]0,04/ 0.09 0.1940.09 0.16 0.17 [0.05 0.40, 0.00 50.15 3 560 |0.76|0.1311.04 0.07 0.14]0.04| 0.09 0.1! tPpalo tele ie arloroa fons on oe 570 |0.74|0.1211.06| 0.06 0.16] 0.05] 0.10 0.19}0.09 0.17 0.18]0.00 9.2910.0440.15|0.11| 0.16 580 | 0.72|0.12]1.08 0.05 0.18 0.05 0.11 0.19}0.10 0.17 0.18}0.00 0.25 qonfoutelunows 590 | 0.70) 0.12 Lot OO 0.190.06) 9.12 0.19} 0.10) 0.18 0.1960.02 0.21 0.10} | 600 |0.67|0.13}1.11) 0.03 0.21] 0.06) 0.13 0.18}0.10/ 0.18) 0.19 ) Year. TABLE MW IELIS; (O) 18155 IL ah XVIla. OF URANUS. TABLE XVIIO0b. 5) (w.e.1) Year. (v.s.1) (v.e.1) (v.e.1) 1800 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 | 1814 1815 b 1816 1817 1818 1819 | 1820 1821 1822 1823 H 1g24 | f 1825 } 1826 1827 1828 | 1829 f 1830 1831 | 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846 f 1847 } 1848 p 1849 1850 ” —240.33 DES) s) 238.06 236.92 935.79 | 234. 65 233.51 232.38 931.24 230.11 —228.97 227.83 226.70 225.56 224.43 29 15 02 .88 15 61 3.47 Dyan 214.20 213.07 93 80 -66 53 39 26 13 388) 86 02 .o9 45 .32 197.18 196.05 —194.91 193.78 192.64 191.51 190.37 —189.24 188.10 186.97 185.84 184.70 i—183.57 } | bo bo bo bo bo ee BO > k= bO CD oc t | bo bo ” —162.11 162.61 163.12 163.63) 164.14 —164.65 165.16 165.68 166.20 166.72 —— (2! 167.76 168.29 168.82 169.35 —169.88 170.41} 170.95 | 171.49 172.03} OT] -11} .66 21) 16 | 5.31) 5.86 5. 42 | 97 | D3 8.09 8.65 y OF Jv.g | — —t TT =7 —T -I He He CO OD LO ae a ee 32.06 82.64 33.22, 3.80 .38 84.96 35.55 36.13 36.72 87.31 87.90 88.50 89.09 89.69 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 Weis 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 " —183.57 182.44 181.30 180.17 179.04 —177.91 176.78 175.65 174.51 173.38 —172.25 171.12 169.99 168.86 167.72 —166.59 165.46 164.33 163.20 162.07 —160.94 159.81 158.68 157.55 156.42 ——155.29 154.16 153.03 ipyleeht 150.78 | —149.65 148.52 147.39 146.27 145.14 —144.01 142.89 141.76 140.64 139.51 —138.39 137.27 136.14 135.02 133.90 22 11 131.65 130.53 129.40 128.28 — 127-16 ” —189.69 190.29 190.89 191.50 192.10} —192.71] 193.32] 193.93 195.155 bene 196.39} 197.63 198.25 —198.88} 199.514 200.14} 200.774 202.045 202.685 203.325 203.965 204.608 —205.24 205.89 206.544 207.194 207.845 —208.49} 209.158 209.81} 210.474 211.135 —211.79} 212.45 § 213.12) 213.794 214.46} —— lls) 1S} 215.80} 216.484 ONT. 217. =e 219.208 219.8% 220. 221. —221.$ 194.544 197.01f 201.408 woeeoye Oo om DUN fo) CPCOnWwWf NN Am WO G2 NH n” Soquoe 95.81 +o°43 95.54 tc ae 95 40th ite Oo. 5.75 — 99. 100.9% 102. 104. 106.02 —108.0§ 110.° 112.8 115: 118. ee 124.5% 127.9% US BE 135.32 518). 143. 147. 152.42 Lois SSS (viii 167.242 13 172.57 178.09 ee | 3.80 2 2T| 18 oe —189.69 195.77 —6:08 202.04 27 208.49 ee 215.13 2: YW 991.95 . | "998 § 236.11 243. 251. —258. 266. 274.6 282.9: 291.3 —299.95 Nore.—The values of (v.s.1) and (v.c.1) must be taken from only one of the two tables XVIltaand XVIIb: THE ORBIT OF URANUS. TABLE XVII 6.—Concluded. 271 (v.s. 2) Gado eso aean Ss. uw Ge 152.43 152. ipyle 151. —150.69 150.2 149.738 149.32 TASS, te 0.47 —148.39 147.92 +°-47 147.44 2-48 146.96 °48 146.48 cee —145.99 145.50 +°-49 145.00 °5° 144.50 144.00 —143.49 142 98 T° 149°47 2: 141.96 ° 141.44 ° fo} —140.93 anit Boles. < 139.36 138.83 138; 137.75 137.2 136. 136. 135.69 135.175 ° 134.65 134.13 133.61 133.09 139.57 +° 132.06 131.55 131.04 —130.54 130.04 +° 129.55 °45 129.06 128.57 —128.09 ° nuns HO NW NWN fol IS} dst toi Tal NOW NON oO. oO. mnnin WINN KANN On NN NNN NN to) 0)" ©) & ownor 1 1 Hee N ee 9 09 o9 CO Go to WwW OO CD co oO ws Ww ww C2 or) we 1 OU He He OO UD bow we OT =1 ae ww o.) 9 aa w u Wwe Oo | CB OR: C ite) NN NNN Qui & Ww WwW Nm N NAO G2 CT Go GW NUN wW os) No} N ° nmonwnw7 o1 nwo unr mya AN HWNH i) 2) nn eH Oo on = > co ee) -I TT -r-a -7 6 -T-I cr) O-Te<-! oo oe 9 9. 9. 9.6 OuGe OR: OLE F i). mMmmnmmn WOannnn ONMMWSS eo co Go bo bo I -1 oH coo GO 7 c— > OU He 09 8 }-1163/—1123 145 +1379\— 891/+-2361 1369) 885) 2258 1359} 881) 2154 1349) 8T8} 2047 1338) 877| 1944 +1328|— 878|+-1838 1318! 881) 1731 1307) 886} 1623 1297 893} 1513 1286 901; 1403 ELT 911/+-1292 1264 923) 1180 1253 937) 1067 1241) 954 954 1230 972 840 41219'— 9929/4 725 1208 1014) 610 1197; 1038] 495 1185] 1064] 379 1174; 1092] 262 1152) 1155 27 1141) 1189 92 1130) 1226 212 1119} 1264 333 +1108|—1304|— 454 1097, 1346) 575) 414 1086) 1391 696] 417 1074) .1437 817| 420 1063] 1486] 938] 424 1052'—1536|—1059|—427 1041/ 1588] 1180) 431 1030, 1642) 1301) 435 1019| 1699} 1422) 439 1008, 1758] 1544] 443 + 997'—1819|—1665|—447 986, 1881| 1786, 451 975! 1945) 1906 456 964 2012} 2097) 461 953, 2081| 2147) 466 942'—9151|—9967|—471 9386, 476 2505) 481 2623) 486 2741) 492 29)|—-2859' 4.97 2976) 503 3092 509 2780) 3207) 515 9867 3321) 521 | )|+ 838 —2956 —3434 —527 [566 272 THE OR Bw OF URANUS: TABLE XVIIJ.—Repvcrion To THE Eciipric. ARGUMENT w. U u R ° fe) ” ° (eo) 0|180| 10.00 | 22h 270 | f 14) Teil 9 Gi oo 226 271 | “33 BED Osis “PSs 227 | 272 | qe 317 3/183} 9.02 °33 | 228 273 | "33 318 ASE Su S82 929 274 | 532 319 0.33 538 Se (SS SBF 9: 275 320 6 |186| 8.05 93° 2: 276 Se 321 1) Msg “dees 2 Se 235 277 | oe 322 S.C | 10 a 233 278 & 323 9 |189| 7%. 37] 934 | 79| Se 2 9 | 189 7.10 0.30} 34 27 26 324 10 |190| 6.80 935| 1. 13. ie |) Nn) A= Ss2 236| 1. 13.5 “3T 12 |192) 6.19 °3°} 237] 1. 13.8 73° 13 | 105]| S88 Ces) 238| 1.5: 14. 3 VY || OS) BRO S22) 939| 1.7% 14. 229) 0.28% .28 15 | 195) 5.32 5 240! 1.8 14.65 e 16 | 196] 5.04—°-%— 241| 2. 14. Bs ri SHE Cet ee 2.2 iN? = 18 1198} 4.49 °27 2. 15. 2) 19 |199| 4.93 °26 2. i 26 ; 0.25 2 20 |200| 3.98 | 2. $6 91/901) §3.730: 22 5, 22 92 | 902! 3.49 ° 24) 3.5 he 23 |203| 3.26 273] 3. £6) 16 94 |204| 3.04 ° 727% 3. Cos 16 0.225 25 95 |205| 2.82 | coe 17 26 | 206) 2.62077) ee 17 97 |207| 9.42 27°F 2 997] 17 28 | 208| 2.23 col a 298) 17 99 | 99 905 cei ; 299 7 29 | 209) 2.05 oI ae 299] 17 30 | 210] 1.88 300| 18. SOU Aor. 287 j01/ 301) 18.2 32 |919| 1.58 oT] 29. 1302] 19. 33/213] 1.44 O14 ae | 303| BA onal) seal ess ON “3 304 0.11 53H 35 |215| 1.20 I Di) BS | 305 | Yee) OMall AOC 232 | 306 | Te POL Oe eee) 3? | 307| 33 |o1s| 0.91 °°8F “ST | 308 39/1 219| 0.840 977 “3? 309, fe) O7 32 | 40 | 920| 0.77 310. 355 AL W991 “O67 = 0-23 33 311| 356 49 |999| 0.68 ©°4 Soe 312 357 TEV OpR) (G8) SSS “33 313 358 | Bay OOM MONG ASCO soe 314 359 | .O1 -33 THE ORBIT OF URANUS 273 TABLE XIX.—Princrpat TERM OF THE LATITUDE. ARGUMENT w. U 8 u | " 10° a 1o} Mmro 62 fo) Ro} non =) 9 fo) No} © DOWD: 0 90 oOo [Soho} wemoDmpDmonMnnMmn Oonnnnnon (e} CO Cio} Ore Go OC) DOR 02 Or TOR GON HnNnwWwpom OH QA~T fe) \o HNNWWH ANNI AD woououwowuowo wuowuwvnovsd nonunion or or 01 01 1 OT fo) [e/a] fo) fe) co te} (e} 2.0 (oe) Co oOo DODD RRR eRe rRrRoco ocoeoooCoc.cCc lh Ams CO Hm He OO bO Fr oe mmmowvs om Oooo RRR HHA N&® Be MH to bo bo bo bk bo Hw ARAB AD 5 fo) ma naInnann Ge G0 92 90 99 OP 2.00.00 G0 G0 Go 90 M9. G0 GO Go ¢ aS Gs Ga MONODMDnOnnnDn DDnDnnme HN 0 OO OO HO me G& O11 ~7 1 0 [e) ~r Ko) NNWWWWR WWW WH Ww o (vo) mr > oO CON OO ee eR RR RRR CO Wt te HNP AAAS O NR NNN NN N N Or 09 CO bo H \S DAOW FD DONKF SPD OW « ° OD CO 0 bo bo i H ~I Ro SSH RT ECAR EES CRP Ny Op Cop SPSS TIONS CSTR US RS FST FCS FIG IS Cf iC I sass bo b = moe + aD 5 = to a Go 00 90 00 0.0 00 G0 G9 G0 Oo C0 Ca Go Ho 9. Go 0 GAG DG GO Go a as Oro CO bo he oS ~1 CO © 0 oO 0 ND os co ho (0/0) Lo wD. DO BHN&w po on rwmornrnnwmownwrn wwl wl on Te Ao A ART RS CE RGR RESON BRR SIR SPR STS, STS SPSS CPS RES CSTE RCS CORI REREAD TENOR CRIES IER IIOS (CDCR IES BAO (o} rae — ~ oS ee oe eS ay wownrwonwnwmwnp www r looney) e} Lo we) N DR 36 July,1873. 274 THE ORBIT OF URANUS. TABLE XIX, Ara. u.— Continued. uU B 330° ” 320° " 150° 2 140° 29299 50’ i ’ 50’ 35 35.18 40 Pag ls 2 35 40.35 30 £ ‘ 3 3 35 45.50 20 4 : : 35 50.63 10 3 : 30) D010 149° : 0.85 50’ 8 ; F Ne ei5 5s} 40 3 BY ; 2 36 11.00 30 30 42. " 3 F 16.05 2 [ 21.07 26.08 31.07 36.04 ) 40.99 45.93 6 50.84 byayr(s} 0.61 5.AT 10.31 15.13 19.93 24.72 29.48 34.22 38.95 43.66 48.3 3.01 Site Ph. 40) 3F 6. 3 g oiey LUE 20 10 134° 50’ 40 20 3 30 20 40 10 50 183° | 57° 50’ 10’ 40 20 3 3 20 40 10 50 182° | 58° 50! 10’ 40 20 30 30 20 40 10 50 181° | 59° 50! 10’ 40 20 30 30 20 40, 10 50 130° | 60° 810° | 240° u RaAUMMIM nNONnNWnN 00000 wR UL T Naw Bo DAOoO No) © 0 rh NP IkK HenbhOnN nO 7 0 MC poreate S bo bo bo bo be oO nnnnmo oo no MI COA W Ee DO Nm G& U1 CG CO AwWO O ‘) fo oe) mown ft U1 BAeost OH WU CO HW NBO sy BERR PHPADA PRARER RRRREE PRES aon PHRBRRU ANNUM ADH agnawsss aw 4 2 WWW WOO 2 0D OD tO OO NWWWWwWW wht PHHR HPHhMANKNNK NNN AD ADAA QW WaT WT TINT AMHMDMDO WMH WHO HWW WO CwWoWwWHW WWWWWW WWWWHH WWW WHE tO Or CVO OU CT ee He He He OE EER ODD Ww ON OOO NW Wwe Ko) eo DNAAAAD AAAAANH DAAAAAH DAADAAAN AAADAAAD AAADAAA AAHHAAAA AAAAAA AAANS NaHCO NN NN NW Co WWW Ww HOE AwO NR NN NN N O° aAmnnimnn nAMnnnin UANNNniN NMNNINNH MNMoINnnN MoUn wn ¢ c t OS 9 Go Co Go GO THE ORBIT OF URA NUS. TABLE XIX, Are. u.— Continued. B u B u B ro” 300° }250°,/ » 290°|260°;, » 280° 40 8.00 ;| 120° | 70° |43 32.84 , | 110° | 80° [45 38.29 al) ste? 40 12.03 nae 50’ | 10’ |43 35.60 Fae 50 | 0 - ant no Bo / . on 3 38 9¢ <- 9 F OF < 40 16.04 7°05) 40 | 20 |43 38.33 7-75) 40 J 20 oe i ae 40 20.04 ° 30 30 3 41.04 ee 30 30 3 2.4 1.33 3 40 24.01 397] 90 40 /43 43.73 7°69) 20 | 40 |45 Sus eo 20 40 27.96 ee BO oD 8 2620 eal Oa 50g 5) 20,03 al IG 40 31.89 5 ,| 119° | 71° |43 49.04 || 109° | 1° 45 46.30 | 6] 99° 40 35.80 33°) 50’ | 10’ |43 51.66 322) 50’ | 10" | 45 47.56 (| 50 40 39.69 38 UN ONT Seale |) IU ea) eee) 40 43.56 357) 30 | 30 |43 56.84 325) 8 80 |45 50.00 J 7g 2 40 47.41 355| 20 | 40 [43 59.40 23| 20 | 40 [45 S118 I 76) 20 40 51.24 3750] 10.) 50 [44 1.93 2 sr| 10 | 50 [45 52.34 177! 10 ee) ce 5 j ° ° 145 53.48 98° 40 55.05 ae 118° 72° | 44 4.44 2.49 | L08 aA ve a eS a siti oa 40 58.84 376 50’ | lo’ |44 6.93 2°72) 50’ 0 |45 54.59 10 g 41 2.60 37°) 40 | 20 44 9.35 Pies) ea ||P I OSE ae |e Megs) 327° |" 3 30 |44 11.83 5 | 3 30 | 45 56-75 5 o4 2 41 10.07 gal, 20 40 /44 14.95 720) 20 40/45 57.79 oo HY 41 13.78 > 68 10 50/44 16.65 5 38 10 50 |45 58.81 ain See _ 9 ° 4117.46 | 5 || 117° | 73° |44 19.03 2.35| LOT | 83° |45 59.80 | eT Ziptreis 251) 50° | 10! |44 21.33 735) 50" | 10" 146 “o.78| 29° 3-64 90 |44 93.71 2:33] 40 WT) 2G skye Sa Ap 41 24.77 40 | 20 Pte ae 41 98.39 3-62] 39 | 30 |44 96.09 35] 30 | 30 [46 266 293) 30 41 31.99 36° 90 | 40 |44 28.30 2-28) 90 | 40 |46 a aa) 29 41 35.56 357) 10 | 50 |44 30.57 2 a,| 10 | 50 ]46 4.44 Oe 41 39.19 °°’) a6 TNE RS oe | EO eo | ae 41 42.65 go8 50’ aS 200" 2.19 oy 0 16 6.95 Osis 40 41 46.17 35°) 40 | 20 |44 37.93 Aen tare Fie Peunere eS, Fy | Oe ee fae BO) Se ee ey 41 53.13 3°47) 90 | 40 |44 41.55 773} 20 | 40 e a ea ae 4156.58 $45| 10 | 50 |44 43.68 2 73| 10 | 50 |46 9.24 074 oe 7 : mo 3 ( 9 42 0.01 © ,,| 15° | 75° Br are) Oo A OO a oes calle 42 3.42 3%) 50° | 10 TPR ZOO eli ga nae Cen at 42 6.30 33°] 40 | 20 49.93 oa 4 eee tees ier 4910.17 331) 3 Bee UN Bs a ae lee tatsonee Calms 49 13.51 2°97}. 20 40 |44 53.98 ap||, 4 Some S coltuee 42 16.83 2 a 10 50 |44 55.97, Or z, one; ; 42 90.13 ~~ .|114° 1 76° |44 57.94 104° | 86° 46 13.76 |__| 94° 42 93.41 3-28 50’ 10’ -144 59.89 1:92) 50 10’ |46 14.31 - ae) 50 Bee: 26 99 |45 1.81 192 2 6 1Age ToS) 40 4226.67 377) 40 | 20 [45 1.81 19°) 40 ees Peer vias, 42 29.91 37%) 30 | 30 [45 3.71 1g3| 30 | 30 |46 15.35 593] 20 9 32949 3-2 9 5 5.59 20 40 (46 15.83 z 42 33.12 2 7,| 20 40 |45 5 Baier: 70 (46 leon 24°) To 42 36.32 3 7>| 10 | 50 [45 7.44 199 (a os reese ee ie (Sr To 920 ge OS | STE eae ae 49 42.64 375| 50’ | 10’ /45 11.08 J on 99 |46 17.52 239] 40 245.77 3:13 20 |45 12.86 17° 20 |46 17.52 “9 42 48.87 en ae - ieee relence 40 -|46 18.93 °-34 20 ) PR a 9 Q oO 00 ane av 4 Ss 22 5 Pe 3-08! 10 | 50 |45 18.08 777) 10 | 50 [46 18.55 ©3°) 10 Z : 3.04 1.09 5 6 46 18.8 72 92° 42 58.06 ee 112° | 78° 8 Heat aa oS oS ASEH KOT Maer 7s ee 3.00 un a Fs 93.09 EPS 40 20 |46 19.36 mee 40 0S 20 | 45 23.08 219.59 923! 30 One coal) eS 30 [45 24.71 7°?) 30 | 30 |46 19.59 (3) 30 23 10.01 2.96 20 40 3 26.3 see 20 | 40 |46 19.79 22. 0 : 2.93| 7 F — on oq 1-58 5 MS WOE ae 43 12.94 oo 10 50 [45 27.89 735 oy ee 2 er | eee oF [fs is r; ¢ » ZU.14 s Ey ool ey Lue lan scgas 53 yy | 10/46 3095 O83) 50 ASS (Ss 50! 5 30.98 | 51 09 646 20.36 OF! 40 . 6 R26 . 20 46 20.3¢ > 43 21.60 ¢ e a 20 ra Bae 1.48 5 30 46 20.44 Ore8 30 43 24.45 % >| 30 30 49 33.97 Tle 7 > 20.50 O08 20 43 97.97 2:82] 90 40 |45 35.48 20 Urea e tos 10 43 30.06 779| 10 | 50 [45 36.87 144) 10 | 50 |46 20.58 oo ee pea? Revane es 9 ° 46 20.54 43 32.84 | 110° | 80° |45 38.29 Ly 90 ‘ 4 ee B 290° | 260° B 280° } 270 B 276 THE ORBIT OF URANUS. TABLE EXEXe VOU XOXeIIE f ARG. iL 2 3 |(b.8.1)| (b.c.1)] (b-c.0) (6.8.1) (b-c.1) (b.8.2) (b-¢.2) (6.0.0) (b-8-1)) (b.c.1)| (b.8.2)] (b.€.2) Fa in lf u" ” iy ” ” wm | ” ” uy ” " ” 0 1.20) V2) |} Os04 | 5299) |) 5-42) 10221) OT A On0Gs| 18585] 1345 Ono elmOntin 10 Tet5 Mealy (083) j) SHB yes 0.19 0.18 0.06 1.63 1.14 0.08 0.23 20 1.10 1.21 0.03 | 6.25 | 5.04 } 0.18 | 0.18 | 0.05 | 1.67 0.93 | 0.11 0.29 30 1.04 1.25 0.02 6.34 ANS Aa Oleg 0.18 0.04 1.68 | 0.72 0.15 0.32 40 0.98 1.28 0.02 6.41 4.62 0.16 0.18 0.03 6 i0 0.51 0.21 0.33 50 OL92A lao 0.02 | 6.45 4.40 OLS OFL8) 40203) | 15607) 10233) 022% 0.32 60 0.86 1.33 0.02 6.45 4.18 0.15 0.18 0.02 1.48 | 0.18 0.32 0.28 70 0.79 | 1.34 | 0.02 | 6.40 | 3.94 | 0.16 | 0.18 | 0.01 | 1.382 | 0.09 | 0:35 | 0:93 80 Di |) Ise3s) 0202 | 62382) | 3569) | 20816) |) OFT 02009), Fels) | Ox07 | O236 Osta 90 0.66 1.35 0.03 6.19 3.43 0.17 0.16 0.00 0.94 0.13 0.35 0.13 100 0.59 | 1.34 OL03) (62025 sald l= Ola Or Lan oro 0.77 | 0.25 | 0.32 | 0.09 110 02525) 1232 0.04 | 5.81 2.87 | 0.18 | 0.14 | 0.02 | 0.66 | 0.41 | 0.29 | 0.07 120 OFZ65 SISSON OL0 4a Moe o Gu leeeebe 0.19 | 0.12 | 0.04 | 0.62 | 0.60 | 0.25 | 0.06 130 0.40 WONT 0.04 | 5.29 | 2.26 |} 0.20 | O.11 OLO6 7) 1064 | OSTT 0222) OL0s 140 0.34 1.24 } 0.05 4.99 1.96 | 0-20 | 0.10 9 0.08 |) 0272 | OL9X |) 0220) 0x0 ) 150 0.28 1.20 0.05 4.68 1.66 0.20 0.09 0.09 0.83 1.00 0.18 0.98 160 0.23 1.15 0.05 4.36 1.38 0.20 0.08 0.10 0.95 1.04 0.17 0.09 170 0.19 1.10 0.06 4.03 WoL 0.20 0.08 OFS le Os: 1.04 0.17 0.10 § 180 0.15 1.04 0.06 3.69 0.86 0.19 0.07 0.11 Tey: 1.01 0.16 0.10 § 190 ONOTOZ9Sa OF Gm lors Om lm OnOOmImO LG 0.07 | 0.11 1.14 } 0.97 0.14 | O11 R200) 0.09 | 0.92 | 0.06 } 3.03 ) 0.56 0.16 | 0.08 | 0.11 PVA’ f 10:93) 05139) OSE2E 210 0.07 0.86 0.06 2.71 0.32 0.14 0.08 0.11 ileal 0.91 0.11 0.13 | 220 0.06 | 0.79 0.06 | 2.39 | 0.20 | 0.12-)_0.09 ff -0.10 1.07 0.91 0.10 | 0.15 § 230 0.05 0.72 0.06 2.08 0.12 0.11 0.11 0.10 1.02 0.92 0.10 0.18 240 0.05 0.66 | 0.07 NT 0.08 | 0.09 | 0.13 | 0.09 | 0.98 | 0.96 0.10 | 0.21 250 0.06 0.59 | 0.07 1.48 } 0.08 | 0.07 0.15 | 0.09 | 0.96 1.00 | 0.12 | 0.23 260 0.08 | 0.52 | 0.07 1.21 ORO) OS05 |e Osr 0.08 | 0.95 } 1.05 } 0.14 | 0.25 270 0.10 0.46 0.08 1.05 0.16 0.03 0.20 0.08 0.94 1.09 0.16 0.26 280 0.13 | 0.40 | 0.08 | 0.72 | 0.26 | 0.02 | 0.22 | 0.07 0.95 1S 0519) | Oe2e 290 0.16 0.3 OOO MOso 2 IMORS 0.01 0.25 | 0.06 0.95 1.16 | 0.22 | 0:20 300 0.20 0.28 0.10 0.35 0.54 0.01 0.27 0.06 0.94 We Ass || 3283 0.23 1 310 0.25 0:23 f 0.11 On225 Onsen OsOn! 0:29] 0206 |) 0893") 12200)" 0294) | ozo F 320 OFSON me OMLOm ORS OMMZ! || 0894 BCBS!) TOA ayy | 3.60| 12.07 3.57 11.97 3.54 11.86 Sey tle G 3.48 11.65 3.46 11.54 3.44 11.43 3.43) 11.32 —l1. iL 10.93 10 10. G2 > O) Co CO He He ee coor lt E10 10.5 10.38 o2 99 09 oH ¢ go 99 92 99 G9 ” oo So So G2 CO OD Yo 0D -I oc -T ooocco oo 0 OO OO -T -1 -1 -T Go a ” == (ods == ORO iT 0.34] 0.96 0.34) 0.95 0.34| 0.94 0.34| 0.93 —0.34|—0.92 0.34| 0.90 0.34) 0.89 0.34| 0.88 0.34) 0.87 ==0.35)|—=0.86 0.35| 0.85 0.35) 0.84 0.35] 0.838 0.35| 0.82 277 (b.s.2) (b.c.2)] Year. |(b.c.0)| (6.8.1) | (b.c.1) (b.s.2)| (b.c.2) wr ” 1800 |-+0.15| —4.60 1810 | 0.14] 4.68 1820 | 0.13) 4.75 1830 | 0.12] 4.83 1840 | 0.11] 4.92 1850 |+0.10| —5.00 1860 | 0.09} 5.09 1870 | 0.08] 5.18 1880 | 0.07) 5.27 1890 | 0.06] 5.36 1900 |+.0.05] —5.45 1910 | 0.04) 5.55 1920 | 0.03] 5.64 1930 0.02 5.74 1940 |+0.01! 5.84 1950 | —5.94 1960 6.04 1970 6.14 1980 6.25 1990 6.36 2000 | 6.4, 2010 2020 2030 2040 .08 .80 92 for or or mor) 2050 2060 | 2070 2080 2090 03 14 26 OF ol 48 aT aT 1 -7 -7 2100 2110 2120 2130 2140 .60 TaD) (a .84 97 .09 | OO -T -T -1 -1 2150 2160 2170 2180 2190 21 .30 44 56 68 DTmOMmMMOD 2200 2210 : 2220 31 2230 .33 2240 | 0.34 .80 92 04 16 .28 | co 6 3 HO CO 9950 |—0.35| —9.40 2260 | 0.36] 9.52 9270 | 0.38] 9.64 9280 | 0.39] 9.76 2290 | 0.40) 9.88 2300 ee —10.00 .69 | 7 | ” | _6.00|—0.44 5.80| 0.45 5.61| 0.45 5.41] 0.45] 5.21] 0.46| —5.00 —0.46 4.79| 0.47 | 4.58] 0.47 | 4.36] 0.48} 4.14} 0.48 |—3.92/—0. 49 3.69| 0.49 3.46) 0.50) 3.22} 0.50 2.97| 0.50 | me ho bo Lo TOD eH H-T He oe OD co _-o.10-0.55 |-+0.18| 0.55| 0.47| 0.56| 0.76| 0.56 1.05 0.56 + bo bee ee Gp oD 0 On Or or oO So aaa bom < Sat te a Oo Orbo OH m OO De bo <> ew co to Oo GO how Be OI ~ TABLE FOR FORMING THE PRODUCTS OF GIVEN NUMBERS BY THE SINE OR COSINE OF A GIVEN ANGLE. Tus table is formed for the especial purpose of facilitating the formation of the products (v.s.3) sin 3g, (v.c.3) cos 37, etc., (p.s.1) sin g, (p.c.1) cos g, for entire degrees ef g. It is so arranged that the required products can be taken out at sight. Supposing the number to be given in seconds and decimal fractions of a second, we first seek the given angle at the top or bottom of the page, and then enter one of the first nine lines of the table with the fraction part of the second, interpolating for the hundredths. We then add the result mentally to the number corresponding to the entire seconds. ‘The algebraic signs at the sides of the angles are those of the sines or cosines corresponding to the angle and to the column above or below. If the number does not exceed 3” we can enter the table as if it were ten times greater, and remove the decimal point one place to the left in the result. For example, to find the value of 21”.67 sin 280° + 2”.25 cos 280° we find the angle 280° at the bottom of a pair of columns, the right hand one being the sine column. Entering this column with 0.67 as the argument, we find 0.66. Entering with 2.1, we find 20.68, to which adding 0.66, we have 21”.34 as the sine product. Entering the other column with 22.5, and moving the decimal point, we find 0’.39 for the cosine product. Noticing the algebraic signs on each side of 280°, we find the result to be — 21.34 + 07.89 = — 20".95. (279 ) 280 TAB TL EOP SP SRiO DU CLL Sa OO SeN GE Se cAuNeD ma OLO) Silene base + w+] + 24+] + B+] 4+ 4°4 + 5°+ +179 — +178 — +177 — +176 — +175 — —181 — —182 — —183 — —184 — —185 — —359 + —358 + —357 + —356 + —s855 + sin | cos sin cos sin cos sin cos sin cos 0.1 0.00 | 0.104 0.00 0.107 0.0L | 0.109 0.01 0.10 0.01 0.10 0.1 0.2 0.00 0.209 0.01 | 0.20] 0.01 0.204 0.01 0.20 6.02 0.20 0.2 0.3 0.01 0.30} 0.01 0.307 0.02 0.3 0.02 0.30 0.03 0.30 0.3 0.4 0.01 | 0.409 0.01 0.407 0.02 | 0.40] 0.03 0.40 0.03 0.40 0.4 0.5 0.01 | 0.50} 0.02 0.50% 0.03 0.509 0.03 0.50 0.04 0.50 0.5 0.6 0.01 | 0.60] 0.02 0.60 0.03 0.60 0.04 0.60 0.05 0.60 0.6 0.7 0.01 | O.70f 0.02 | 0.709 0.04 | 0.70] 0.05 0.70 0.06 0.70 0.7 0.8 0.01 | 0.809 0.03 | 0.804 0.04 | 0.809 0.06 0.80 0.07 0.80 0.8 0.9 0.02 0.90} 0.03 | 0.909 0.05 0.909% 0.06 0.90 0.08 0.90 0.9 1.0 0.02 | 1.00 | 0.03 1.009 0.05 1.00 0.07 1.00 0.09 1.00 1.0 2.0 0.03 | 2.00] 0.07 2.009% 0.10 2.009 0.14 2.00 0.17 1.99 2.0 3.0 0.05 3.00} 0.10 3.00% 0.16 3.00] 0.21 2.99 0.26 2.99 3.0 4.0 0.07 4.00} 0.14 4.009 0.21 3.99] 0.28 3.99 0.35 3.98 4.0 0.09 5.009 0.17 5.00 | 0.26 4.99] 0.35 4.99 0.44 4.98 5.0 0.10 6.00] 0.21 6.00} 0.31 5.99] 0.42 5.99 0.52 5.98 6.0 0.12 7.00] 0.24 7.00} 0.37 6.99} 0.49 6.98 0.61 6.97 7.0 0.14 8.00} 0.28 8.00] 0.42 (991 0256 7.98 0.70 7.97 8.0 0.16 9.00} 0.31 8.99] 0.47 8.998 0.63 8.98 0.78 8.97 9.0 0.17 | 10.00] 0.3: 9.997 0.52 | 9.997 0.70 9.98 0.87 9.96 } 10.0 0.19 | 11.00] 0.38 | 10.99} 0.58 | 10.98] 0.77 10.97 0.96 10.96 | 11.0 0.21 | 12.00} 0.42 | 11.99] 0.63 | 11.98] 0.84 11.97 1.05 11.95 | 12.0 0.23 | 13.00] 0.45 | 12.99} 0.68 | 12.98] 0.91 12.97 1S 12.95 | 13.0 0.24 | 14.00] 0.49 | 13.99] 0.73 3.98] 0.98 3.97 1.22 13.95 | 14.0 0.26 | 15.009 0.52 | 14.99] 0.79 | 14.98] 1.05 14.96 sail 14.94 | 15.0 0.28 | 16.007 0.56 | 15.99] 0.84 | 15.989 1.12 15.96 1.39 15.94 | 16.0 0.30 | 17.00] 0.59 | 16.99] 0.89 | 16.98] 1.19 16.96 1.48 16.94 | 17.0 0.31 | 18.00 0.63 | 17.99] 0.94 | 17.987 1.26 17.96 Iga ye yy Ube(ethe3 | | USD) 0.33 | 19.00] 0.66 | 18.999 0.99 | 18.97] 1.33 18.95 1.66 18.93. | 19.0 | | 0.35 | 20.007 0.70 | 19.99]. 1.05 | 19.97] 1.40 19.95 Lv4 19.92 | 20.0 0.37 | 21.00] 0.73 | 20.99] 1.10 | 20.97] 1.46 | 20.95 1.88 | 20.92 | 21.0 0 38 | 22.00} 0.77 | 21.999 1.15 | 21.979 1.53 | 21.95 1.92 | 21.92 | 22:0 0.40 | 23.00} 0.80 | 22.99] 1.20 | 22.97% 1.60 22.94 2.00 | 22.91) | 23°50 0.42 | 24.00] 0.84 | 23.99] 1.26 | 23.97% 1.67 23.94 2.09 | 23.91 | 24.0 { i} | 0.44 | 25.00] 0.87 | 24.98] 1.31 | 24.97] 1.74 24 94 2.18 24.90 | 25.0 0.45 | 26.007 0.91 | 25.987 1.86 | 25.96] 1.81 | 25.94 IAT | 25.90 | 26.0 0.47 | 27.00] 0.94 | 26.989 1.41 | 26.964 1.88 26.93 9.35 26.90 | 27.0 0.49 | 28.004 0.98 | 27.98] 1.47 | 27.96] 1.95 | 27.93 2.44 27.89 | 28.0 0.51 | 29.00] 1.01 | 28.98] 1.52 | 28.96] 2.02 | 28.93 2.53 98.89 | 29.0 | | | 0.52 | 30.00] 1.05 | 29.98] 1.57 | 29.969 2.09 29.93 2.61 29.89 | 20.0 cos | sin cos sin cos sin cos sin cos sin SE OF fit — +272 — +273 — +974 — +275 — —269 — —268 — —267 — — 266 — —265 — — §$1 + — 92 + — 93 + — 94 + — 95 + foes $2 |) ea | ay + 86+ + 85 + a 7 io yo TABLE OF PRODUCTS OF SINES AND COSINES. 981 Cree \eae ke ae) ae Care ere Oe +173 — + 172 — + 171 — +170 — — 187 — —188 — —189 — —190 — — 353 + —352 + — 351 + —350 + cos sin cos sin cos sin cos 0.1 0.10] 0.01 | 0.10% 0.02 0.10 0.02 0.10 0.1 0.2 0.20] 0.03 | 0.20] 0.03 0.20 | 0.03 0.20 0.2 0.3 0.30] 0.04 |] 0.3 0.05 0.30 | 0.05 0.30 0.3 0.4 0.40 f 0.06 | 0.40 0.06 0.40 0.07 0.39 0.4 0.5 0.50 | 0.07 0.50} 0.08 0.49 0.09 0.49 0.5 0.6 0.609 0.08 | 0.5949 0.09 0.59 } 0.10 0.59 0.6 0.7 0.69 f 0.10 0.69 0.11 0.69 0.12 0.69 0.7 0.8 ONT OSL Ono Oss 0.79 0.14 0.79 0.8 0.9 0.897 0.13 | 0.89} 0.14 0.89 | 0.16 0.89 0.9 1.0 0.99} 0.14 0.999 0.16 0.99 0.17 0.98 1.0 2.0 1.99] 0.28 | 1.98] 0.31 1.98 | 0.35 1.97 2.0 3.0 2.98} 0.42 2.97 0.47 2.96 0.52 2.95 3.0 4.0 3.971 0.56 | 3.969 0.63 3.95 | 0.69 3.94 4.0 5.0 4.964 0.7 4.95% 0.78 4.94 0.87 4.92 5.0 6.0 5.964 0.84 5.94 0.94 5.93 1.04 91 6.0 7.0 6.957 0.97 | 6.93] 1.10 6.91 22, 6.89 7.0 8.0 MaQ4h Vd) |) 92h 195 7.90 | 1.39 .88 8.0 9.0 8.931 1.25 | 8.91] 1.41 8.89 | 1.56 8.86 9.0 10.0 1.05 | 9.95} 1.22 | 9.939 1.39 | 9.90] 1.56 9.88 | 1.74 9.85 | 10.0 11.0 1.15 | 10.944 1.34 | 10.92} 1.53 | 10.89 1.72 | 10.86 1.91 | 10.83 11.0 12.0 EOS OS eon ee Ot LG Ess 1.88 11.85 2.08 11.82 12.0 13.0 1.36 | 12.939 1.58 | 12.90% 1.81 | 12.87 2.03 12.84 2.26 12.80 13.0 14.0 1.46 3.921 1.71 | 13.90} 1.95 3.86 2.19 13.83 2.43 Stet) 14.0 15.0 1.57 | 14.92] 1.83 | 14.89 2.09 | 14.85 9.35 | 14.82 OGY || WES ag 15.0 16.0 | 1.67 | 15.91] 1.95 | 15.83] 2.23 | 15.84] 2.50 | 15.80 | 2.78 | 15.76 | 16.0 ita) 1.78 | 16.91] 2.07 | 16.87 9.37 | 16.83 2.66 16.79 2.95 16.74 17.0 18.0 1.88 | 17.90} 2.19 | 17.87] 2.51 | 17.82 2.82 | 17.78 BIG 183 18.0 19.0 1.99 | 18.90] 2.32 | 18.86] 2.64 | 18.82] 2.97 18.77 3.30 18.71 19.0 20.0 2.09 | 19.89) 2.44 119.85] 2.78 | 19.819 3.13 19.75 3.47 19.70 20.0 21.0 9.90 | 20.881 2.56 | 20.84] 2.92 | 20.80] 3.29 | 20.74 | 3.65 | 20.68 | 21.0 22.0 2.30 | 21.88] 2.68 | 21.84] 3.06 | 21.79] 3.44 21.73 3.82 21.67 92.0 23.0 2.40 | 22.874 2.80 | 22.83] 3.20 | 22.7849 3.60 22.72 3.99 22.65 23.0 24.0 9.51 | 23.87] 2.92 | 23.82] 3.34 | 23.77] 3.75 | 23.70 J 4.17 | 23.64 | 24.0 { | 25.0 9.61 | 24.86} 3.05 | 24.81] 3.48 | 24.76] 3.91 | 24.69 | 4.34 | 24.62 | 95.0 26.0 9.72 | 25.86] 3.17 | 25.81] 3.62 | 25.75] 4.07 | 25.68 | 4.51 | 25.61 | 26.0 27.0 2.82 | 26.85) 3.29 | 26.80] 3.76 | 26.74 4.22 26.67 4.69 26.59 27.0 28.0 2.93 | 27.85] 3.41 | 27.79] 3.90 | 27.73 4.38 27.66 4.86 27.57 28.0 29.0 3.03 | 28.84] 3.53 | 28.7849 4.04 28.72] 4.54 28.64 5.04 28.56 29.0 30.0 3.14 | 29.84] 3.66 | 29.78] 4.18 | 29.71 4.69 29.63 5.21 29.54 30.0 cos sin cos sin cos sin cos sin cos sin +276 — +2717 — + 278 — + 279 — + 250 — —264 — — 263 — — 262 — — 261 — — 260 — OG le On at 98-1 a OO —100 + se as || Eee | eepeel|s eeeae || an eee 36 August, 1873. 282 TABLE OF PRODUCTS OF SINES AND COSINES. + mep | + 12°4 | + ast | + 14°4 69) 23) <4 WeSe= EG Ie= hoe —Jor =" | 199 f= gan Eig. = 94974 | — s46-— P2847 eaten sin | ¢ sin sin sin 0.02 | .02 0.02 0.02 04 | 04 , 0.04 0.05 0.06 ). .06 : 01 0.07 FOSmOr39 038 A 5 0.10 2S ©) He 0 BO on 10 AS 10 49 j 4G 0.12 1 DF 12 0.56 ald 0.587 0.15 3 : 15 ; : 68] 0.17 Op ils) |) “Os ; 7 : TS 0.19 J.17 : : : : : 0 cooseo for) SOS 2 2 Oo OTS © Mw -T 24 48 13 5165) 08 Rov oo to =) SS) soso He OD RO eH ooc co 21 45 69 94 18 95 14 34 03 12 eococo Dee ee 42 66 91 POV =) 29 1 1 1 1 ] | ell coal cell eee col OO ATH He os Sooco, ooo GD CO CI. COTS) SRS. US) GS) Go 02 05 to 0D bo bo bo bo bo PPR Roe corr ro 51 D> 46 44 AL BY o) v0 bo bo bo bo bo mm WD Re © ocooooo ee ee UO maamawon me o> bo ; 24.36 5.52 5 Ae . 8 25.3¢ .00 : : D: 53 49 8 og 6.8 27.28 47 203) | ae .02 | 28.26 for) S or ~) fer) Or bo po) S> > Te ar, bo bo bo bo 1 ocoocoo UO OU ie aT bo bo bo Ww bo (ore) -t oa bo 45 B 2 Ore Sif) |) 28} sin ‘ sin iSv) o Oo bo (Je) 981 — |) 969) a oegie= =-959 => |e o58' = O57. |) ong 55 JOT tf 100 reste) rose 22305 = =- WM) =. ap KS Sp ap) WU sp Ont. =p 1/2) Se TABLE OF PRODUCTS OF SINES AND COSINES. 283 + 16°+ | + 17°+ | + 18°+ | 4 19°4 so TG ae |) ane GE ag) elas) ee || aie sae — 196 — — 197 — — 198 — — 199 — —344 + — 343 + —342 + — 341 + sin sin 0.03 0.03 0.06 g 0.07 0. 0.297 0.10 0.15 38] 0.13 cose He CO bo Re 2 zs — cess Hm CO bore Or 0.16 0.20 0.23 0.26 0.29 cosse (Joe oer Mor m)! cooss ROR Ss at oppo pwn Sa seiceseeee CO Orb OomnTS oH .30 .65 0.98 .30 coooo S22 1 co or bo MSCS Me He C2 bo rt wm OO bo et 63 55) .28 60 93 (Talo cet Mor msi oooco ho — -T i or se) o-2) oe bo no bo t EO Te OOo oo COT oo 0 0 .0 0 0 go 99 69 G9 tO amr wa Hm oo 00 CO 9D mm eR co CO CO = oO en co 0. 1 2 3 4 oe RR ae oon fk bo bo bo bo bo wm OO bo Re © bo bo bo bo bo CONTA MN ooocco wo = o —108 + 423+ [4 72+] 4+ 7 284 TABLE OF PRODUCTS OF SINES AND COSINES. + me+ | + 29°4 J + 23°+ | + 24°4 | 4 25°+ + 159 — + 158 — + 157 — + 156 — +155 — — 201 — —202 — — 203 — —204 — —205 — — 339 + — 3388 + — 337 + — 386 + —335 + sin cos sin cos sin | cos sin cos sin cos 0.1 0.04; 0.097 0.04} 0.099 0.04 | 0.09 0.04 0.09 0.04 0.09 0.1 0.2 0.07 | 0.19% 0.07) 0.197 0.08; 0.18 0.08 0.18 0.08 0.18 0.2 0.3 0.11) 0.287 0.11] 0.28% 0.12] 0.28 0.12 0.27 0.13 0.27 0.3 0.4 0.14) 0.379 0.15] 0.379 0.16 | 0.37 0.16 0.37 0.17 0.36 0.4 0.5 0.18} 0.479 0.19] 0.469 0.20) 0.46 0.20 0.46 0.21 0.45 0.5 0.6 0.22'| 0.569 0.22) 0.56%) 0.23) 0:55 0.24 0.55 0.25 0.54 0.6 0.7 0.25 0.65 0.26 0.65 0.27 0.64 0.28 0.64 0.30 0.63 0.7 0.8 0.29) 0.75% 0.30] 0.7449 0.31 0.74 0.33 0.73 0.34 | 0.73 0.8 0.9 0.32} 0.849 0.34) 0.83] 0.85) 90.83 0.3 0.82 0.38 0.82 0.9 1.0 0.36] 0.937 0.37) 0.93] 0.39 | 0.924 0.41 0.91 0.42 0.91 1.0 2.0 0.72) 18T] 0.75) 1.85) 0.78) 1.84 0.81 1.83 0.85 1.81 2.0 3.0 1:08) 2580p PIES) 227s Lelie 2506 1.22 2.74 127 2.72 3.0 4.0 Wey] BGS] | MeO) Beat 1.56) 3.68 1.63 3.65 1.69 3.63 4.0 5.0 1.79) 4.679 1.87] 4.64] 1.95) 4.60 2.03 4.57 Dell 4.53 5.0 6.0 9.15. 5.60 9295 5.56 2.34 5.52 2.44 5.48 2.54 5.44 6.0 7.0 9.51 6.54 2.62 6.49 9.74) 6.44 2.85 6.39 2.96 6.34 7.0 8.0 2.87] 7.47§ 3.00) 7.42) 3.13) 7.36 3225 Uses 3.38 25 8.0 9.0 3.23 | 8.40] 3.37) 8.34] 3.52] 8.28 3.66 8.22 3.80 8.16 9.0 10.0 Betsy ee) Seri) | CER iel | Baa 9.21 4.07 9.14 4.93 9.06 10.0 11.0 3.94] 10.279 4.12] 10.20] 4.30 10.13 4.47 | 10.05 4.65 Sees 11.0 12.0 4.30 | 11.20] 4.50) 11.13] 4.69 |} 11.05 4.88 | 10.96 5.07 | 10.88 12.0 13.0 4.66 | 19.14] 4.87] 12.05] 5.08 | 11.97 5.29 | 11.88 5.49 | 11.78 13.0 14.0 5.02} 13.079 5.24|/19.98] 5.47 | 12.89 5.69 | 12.79 5.92 | 12.69 14.0 } 15.0 5.38 | 14.00] 5.62|13.91] 5.86 | 13.81 6.10 | 13.70 6.34 | 13.59 15.0 16.0 5.73 | 14.94 5.99 | 14.83 6.25 14.73 6.51 14.62 6.76 14.50 16.0 17.0 6.09 | 15.87] 6.37 | 15.769 6.64 | 15.65 6.91 5) 8: 7.18 | 15.41 17.0 18.0 6.45 | 16.80] 6.74 | 16.69] 7.03 | 16.57 7.32 | 16.44 eGo | lees: 18.0 19.0 6.81 | VEt4§ V.¥2 | 27.624) 7.42 4) 27.49 Teno |liisoo 8.03 | 17.22 19.0 20.0 Yall 18.67 7.49 | 18.54 7.81 | 18.41 Sals 18.27 8.45 18.13 20.0 21.0 7.53 | 19.61 7.87 | 19.47] 8.21 | 19.33 8.54 | 19.18 8.87 | 19.03 21.0 22.0 7.88 | 20.54 8.24 | 20.40 8.60 | 20.25 8.95 20.10 9.30 19.94 22.0 23.0 8.24 | 21.47] 8.62 | 21.337 8.99 | 21.17 9.35 | 21.01 9.72 | 20.85 23.0 24.0 8.60 | 22.41 J 8.99 | 22.2549 9.38 | 22.09 Sek |) PIRCEY || UOEZD Cal rds 24.0 | 25.0 8.96 | 93.34] 9.37 | 93.18 Dalit P23 10.17 | 29.84 | 10.57 | 29.66 25.0 26.0 9.32 | 24.97 8 9.74} 24.1190 10.116 |) 23: LOS58 | 23775 ff) LOs99) | 23°56 26.0 27.0 9.68 | 25.21 | 10.11 | 95.03 110.55 | 24. 10.98 24.67 11.41 94.47 27.0 28.0 10.03 | 26.14 } 10.49 | 25.96 1 10.94 | 25. 11.39 | 25.58 §/ 11.83 | 95.38 28.0 29.0 10.39 | 27.07 | 10.86 | 26.89 } 11.33 | 26. 11.80 | 26.49 | 12.26 | 26.28 29.0 30.0 10.75 | 28.01 | 11.24 | 27.82 411.'79 | 27.6 19.20 | 27.41 | 12.68 | 27.19 | 380.0 cos sin cos sin cos sin cos sin cos sin + 291 — -++ 292 — + 293 — +994 — +9295 — —249 — —248 — —247 — — 246 — —245 — — lll + — 112 + — 118 + —114 + — 115 + + 68+ | + 67+ ap 5 S5 + 69 + TABLE OF PRODUCTS OF SINES AND COSINES. 285 eesonals + 29°+ + 30°+ +154 — 7.0L +150 — — 206 — 2-909) — eo1Ove= OES Sp Hl |) eee ecco He Co bo RS oooos Oo WOeIoo eo 08 to lO bo Cc Oe a bo os wm Cobo coco i) oo 99 99 bo b9 He OO OO bO bo 4.5 4.§ De as 6.3 bo -T bo -T bo or fi me Oo He i): 9.53 | 9. 10.44 | 2 10.90 > = no HD og bo bo bo 1 0 Hm Le Oo ho bo bo po bo coccoo for) 2. 11.35 23.37 | 11.80 | 24.27 | 12.26 95.17] 12.71 26.07 | 13.17 SO OTTO oocoo bo ho bo bo 19 1 ooocoo bo bo bo bo LO w bo bo bo bo 19 bo po bo bo 8 Ore 9 LO 19 Om oo bo bo bo po bo 1 Om Co LD nw ¢ a oO (Jt) o o Lo or) ~) for | 26.96 | 13.62 | 26.73 | cos | sin FLO BS = —244 — eee 4+ 644 286 TABLE OF PRODUCTS OF SINES AND COSINES. 4+ 84°4 41146 — SOTA n= = 3965. cos 0.08 0.17 25 .33 coss wm Ow Loe 0.41 SO .08 cosse OO -T 0.75 Ltt coco 5.0 6.0 7.0 8.0 9.0 eee ee m Whe oS cooccso _ o (lll eel oe ed Oo O-TS MN oooco —) bo bo bo bo bo CO So) ocooc'oS bo bo bo bo bo He CO LO oooco bo bo = or) waa bo b 9 moo 1 bo bo bo 19 19 8 HO -3 S Or ococo bo bo bo bo bo bo bo bo bo LO rm Co WO bo <3) v=) cocoo oe) i —) bo oo S So 4304 — Osea s 19402 =p 818 = TABLE OF PRODUCTS OF SINES AND COSINES. + 39°+ 287 st OOo deat + 38°- + 40°+ 44143) — | - yao. doe, +140 — — 216 — — 217 — — 218 — — 219 — — 220 — — 324 + — 323 + — 322 + — 321 + — 320 + sin cos sin cos sin cos sin cos sin cos 0.1 0.06) 0.08} 0.06} 0.08} 0.06] 0.08} 0.06 0.08 0.06 0.08 0.1 0.2 0.12| 0.16] 0.12) 0.16] 0.12] 0.16] 0.13 0.16 0.13 0.15 0.2 0.3 | 0.18] 0.24] 0.18) 0.24] 0.18] 0.24] 0.19 | 0.23 | 0.19 | 0.23 | 0.3 0.4 0.24) 0.32] 0.24] 0.32} 0.25] 0.32} 0.25 0.3 0.26 0.31 0.4 0.5 0.29} 0.40] 0.30) 0.407 0.31] 0.39] 0.31 0.39 0.32 0.38 0.5 0.6 0.35| 0.499 0.36) 0.48] 0.37] 0.47] 0.38 0.47 0.39 0.46 0.6 0.7 0.41| 0.57] 0.42] 0.56] 0.43] 0554 0.44 0.54 0.45 0.54 0.7 0.8 0.47) 0.65] 0.48) 0.64] 0.49] 0.63] 0.50 0.62 0.51 0.61 0.8 0.9 Os53s | OMT ORD 4a ONT Ono) (Onn IOLar 0.70 0.58 0.69 0.9 1.0 0.59! 0.81} 0.60] 0.80} 0.62] 0.79} 0.63 0.78 0.64 0.77 1.0 2.0 GIES |) aLee}) | LEX || TG) | Ta aIGaxs}] = 1.55 1.29 1.53 2.0 3.0 1.76) 2:43) 1.81) 2.40] 1-85) 2:36) 1.89 2.33 1.93 2.30 3.0 4.0 2.35 | 3.249 2.41 Seo One onl eeonlo: 2.52, By 10 2.57 3.06 4.0 5.0 9.94| 4.05] 3.01] 3.99] 3.08] 3.94] 3.15 3.89 3.21 3.83 5.0 6.0 3.53| 4.85] 3.61] 4.79] 3.69] 473] 3.78 4.66 3.86 4.60 6.0 7.0 4.11| 5.66] 4.21| 5.59} 4.381] 5.52} 4.41 5.44 4.50 5.36 7.0 8.0 A.70| 6.47] 4.81] 6.89] 4.93| 6.30] 5.03 6.22 5.14 6.13 8.0 9.0 5.291) 7.289) 5.42) Wot 5:54) 7209) 5.66 6.99 5.79 6.89 9.0 10.0 5.88] 8.09] 6.02) 7.99] 6.16| 7.88] 6.29 Teri 6.43 7.66 | 10.0 11.0 6.47 | 8.90] 6.62] 8.78] 6.77) 8.67 6.92 8.55 Ten 8.43 | 11.0 12.0 7.05 | 9.71) 7.22] 9.58] 17.39) 9.46] 7.55 9.3: rare 9.19 | 12.0 13.0 7.64| 10.52] 7.82|10.38} 8.00] 10.24] 8.18 | 10.10 8.36 9.96 | 13.0 14.0 8.23 | 11.83] 8.43) 11.18] 8.62]11.03] 8.81 | 10.88 9.00 | 10.72 } 14.0 15. 8.89] 19.14] 9.03] 11.98] 9.23 )11.82] 9.44 | 11.66 9.64 | 11.49 | 15.0 16.0 9.40 | 12.941 9.63| 12.78) 9.85 | 12.61] 10.07 | 12.43 ] 10.28 | 12.26 | 16.0 17.0 9.99 | 13.75 | 10.23 | 13.58} 10.47 | 13.40] 10.70 | 13.21 | 10.93 3.02 | 17.0 18.0 10.58 | 14.56 # 10.83 | 14.38} 11.08 | 14.18} 11.3: 13.99 | 11.57 | 13.79 | 18.0 19.0 911.17 | 15.379 11.43 | 15.17] 11.70 | 14.97] 11.96 | 14.77 ] 12.21 | 14.55 | 19.0 20.0 11.76 | 16.18 | 19.04 | 15.97] 12.31 | 15.76] 12.59 | 15.54 12.86 | 15.32 } 20.0 91.0 112.34| 16.99 | 12.64 | 16.77 | 12.93 | 15.55 | 13.22 | 16.32 ] 13.50 | 16.09 } 21.0 92.0 112.93| 17.80) 13.24 | 17.57] 13.54 | 17.34] 13.85 | 17.10 J 14.14 | 16.85 | 22.0 23.0 3.521 18.611 13.84 | 18.37) 14.16 | 18.12] 14.47 | 17.87 | 14.78 | 17.62 | 23.0 94.0 $14.11 | 19.429 14.44 | 19.17] 14.78 | 18.91] 15.10 | 18.65 } 15.43 | 18.39 | 24.0 95.0 114.69 | 20.23} 15.05 | 19.97] 15.39 | 19.70] 15.73 | 19.43 ] 16.07 | 19.15 | 25.0 96.0 115.28 | 21.03] 15.65 | 20.76 | 16.01 | 20.49] 16.36 | 20.21 | 16.71 | 19.92 | 26.0 97.0 $15.87 | 21.841 16.25 | 21.56 | 16.62 | 21.28] 16.99 | 20.98 | 17.36 | 20.68 27.0 298.0 116.46 | 22.654 16.85 | 22.36 | 17.24 | 22.06 | 17.62 | 21.76 | 18.00 | 21.45 28.0 99.0 117.05 | 23.461 17.45 | 23.16 117.85 | 22.85 | 18.25 | 22.54 | 18.64 | 22.22 29.0 30.0 117.63 | 24.27] 18.05 | 23.96] 18.47 | 23.64] 18.88 | 23.31 | 19.28 22.98 | 30.0 cos sin cos sin cos sin cos sin cos sin +306 — | +307 — | +308 — £3097 Jk OO) = — 234 — — 233 — — OS as — 931 — — 930 — — 1296 + — 197 + — 1298 + — 129 + — 1380 + + 54+ at (80. + 53+ + 52+ + 51+ 988 TABLE OF PRODUCTS OF SINES AND COSINES. + 41°+ | + 42°4+ | + 48°+ | + 444 | + 45°+ + 1389 — + 1388 — + 187 — + 186 — + 185 — — 2921 — — 222, — — 293 — —2294 — —225 — — 3819 + — 318 + — 317 + — 316 + — 315 + sin cos sin cos sin cos sin cos sin cos 0.1 0.07! 0.08 0.07} 0.07% 0.07 | 0.07 ORO ORO 0.07 0.07 0.1 0.2 (ONS3 IP OTUs | | SUBS |) Wass | | EUS ess 0.14 0.14 0.14 0.14 0.2 OS 0.20} 0.23] 0.20] 0.227 0.20; 0.22% 0.21 0.22 0.21 0.21 0.3 0.4 0.26] 0.307 0.27) 0.80] 0.2 0.29} 0.28 0.29 0.28 0.28 0.4 0.5 0.33| 0.38} 0.383) 0.387] 0.3 0.37 0.35 0.36 0.35 0.35 0.5 0.6 0.39 0.45 0.40 0.45 0.41 0.44 0.42 0.43 0.42 0.42 0.6 0.7 0.46 0.53 0.47 0.52 0.48 0.51 0.49 | 0.50 0.49 0.49 0.7 0.8 0.52| 0.60] 0.54] 0.59] 0.55] 0.59 O25 6a en OFdS 0.57 0.57 0.8 0.9 0.59!) 0.684 0.60) 0.679 0.61) 0.66 0.63 0.65 0.64 0.64 0.9 1.0 0.66) O75] 0.67 | 0.74) 0.68) 0.737 0.69 | 0.72 0.71 0.71 1.0 2.0 1.31 1.51 4; Ifo S62 146 1.39 1.44 1.41 1.41 2.0 3.0 1.97) 9.969%. 2.01 | 2.23% 2.05) 2.19 2.08 2.16 2.12 2.12 3.0 4.0 9.62 | 3.02] 2.68) 2.97 — 2.73] 2.93 2.78 | 2.88 2.83 2.83 4.0 5.0 S98 HST 8.00) So T2iin 8.414) 73266 3.47 3.60 aye! 3.04 5.0 6.0 3.94] 4.53] 4.01] 4.469 4.09] 4.39 4.17 4.32 4.94 4.24 6.0 7.0 4.59 | 5.98] 4.68] 5.20 4.77| 5.12 4.86 5.04 4.95 4.95 7.0 8.0 D5) |) (G04 t bs) O.9o1le O-40 | o-S0 5.56 5.75 5.66 5.66 8.0 9.0 5.90! 6.79] 6.02) 6.694% 6.14) 6.58 6.25 6.47 6.36 6.36 9.0 10.0 6.56| 7.55] 6.69} 7.43] 6.82) 17.31 6.95 7.19 7.07 Cy 10.0 11.0 TOON SeSOM isso |) Seles) |iero 04. 7.64 7.91 7.78 7.78 OO 12.0 7.87 | 9.06] 8.03) 8.927 8.18] 8.78] 8.34 8.63 8.49 8.49 12.0 13.0 8.53)| 928I) 18270) 895664 “8287 || 9.5L 9.03 9.35 9.19 9.19 13.0 14.0 9.18 | 10.57] 9.37] 10.40] 9.55} 10.24 9.73 | 10.07 9.90 9.90 14.0 15.0 9.84 | 11.32} 10.04 } 11.15 | 10.23 | 10.97 | 10.42 | 10.79 7 10.61 | 10.61 15.0 16.0 10-50: 120088 LOT LISS P LOL9r | OF LEE | TS a1 3) ae3T 16.0 17.0 11.15 | 12.83 } 11.38 | 12.63 [11.59 | 12.43} 11.81 | 12.23 | 12.02 | 12.02 17.0 18.0 11-81 | 13.58 1 12-04 | 13.38 | 12.28 | 13.16 | 12.50 | 12:95 — 12.73 | 12°73 18.0 19.0 12.47 | 14.34 | 12.71 | 14.12 1 12.96 | 138.90} 13.20 | 13.67 | 13.44 | 13.44 19.0 20.0 13.12 | 15.09 | 13.38 | 14.86 f 13.64 | 14.63] 13.89 | 14.39 | 14.14 | 14.14 20.0 21.0 13 78 | 15.85 | 14.05 | 15.61 | 14.32 | 15.36 | 14.59 | 15.11 § 14.85 | 14.85 21.0 22.0 14.43 | 16.60 | 14.72 | 16.35 715.00 | 16.09 f 15.28 | 15.83 | 15.56 | 15.56 22.0 23.0 15.09 | 17.36] 15.39 | 17.09 | 15.69 | 16.82 | 15.98 | 16.54 | 16.26 | 16.26 23.0 24.0 15.75 | 18.11 } 16.06 | 17.84 7 16.37 | 17.55] 16.67 | 17.26 | 16.97 | 16.97 24.0 25.0 16.40 | 18.87 § 16.73 | 18.589 17.05 ; 18.289 17.37 | 17.98 | 17.68 | 17.68 25.0 26.0 17.06 | 19.62] 17.40 | 19.32 | 17.73 | 19.02 | 18.06 18.70 18.38 18.38 26.0 27.0 17.71 | 20.381 18.07 | 20.06 118.41 | 19.75] 18.76 | 19.42 } 19.09 | 19.09 27.0 28.0 18.37 | 21.13] 18.74 | 20.81 1 19.10 | 20.48 | 19.45 | 20.14 ff 19.80 | 19.80 28.0 29.0 19.03 | 21.89 } 19.40 | 21.55 919.78 | 21.21 7 20.15 | 20.86 | 20.51 | 20.51 29.0 30.0 19.68 | 22.64 | 20.07 | 22.29 | 20.46 | 21.94 | 20.84 | 21.58 | 21.21 | 21.21 30.0 cos sin cos sin cos sin cos sin cos sin + 311 — + 312 — + 3138 — + 314 — + 315 — — 2239 — — 228 — — 297 — — 226 — — 225 — — 1381 + — 182 + — 1383 + —1384 + —1385 + + 49+ | + 484 + 46+ | + 454 4a yy = x a e : i al eee. ee Sonne ononrerseeon ease | Ne eee SERSUEE SE wey | Ne Nea Ne i J C Se fist = < : L \ r | g : =) SoeSee SS =e Tru SS \ Bucuccses Jae y= ehh ei og te weveer JIS | SO da eat SS S SSS SS SOOO ew NA IS s= PL A)\A HI, | ot WS a |S |S BS SS Oe FS SS Se SS Ss g \) = \ vac Vessel Veet! Veal Nepal Vee? \ \ we | | yy S ITT Vseeus= Cece eeges ~~ | WA | a SS ead Aes HS IGJSFAG Sooo oeivaeeeyY Wee Worcs Boo ewere" ewe ZY keg aacecaeeY o J Go sve eveeeee Se AAAI OO ES De | Cee cuss ' zs —j\— = SA Ss = BA AAA Ag \ | \ \ , | i H — BW ee | j a tp ae, . ee / BS Soe Se ONS Agog eS sSereg NNR AG ‘here they od Se eS =—\— |= Sat Z| } h | Whey a il SL IS StS t dn , SMITHSONIAN INSTITUTION LIBRARIES "HOM 3 9088 01421