pe foo eer ee agar on™ joemamgnr arene eS s viah pommuapreser meses fakes rarer
seSS : ——— a a a ——S————————S—
Cote: enna, ; oni ch ss arene wee
errors Seatac
| YWusyyuEEY ae Se 2 hada Ze SISSIES ISS = ) GS 2 SSS vu SECS eee, eveeveeu vere
PPDENEN Se SSS
a é nd cue a ~ ps ver we
&
UNITED STATES PATENT OFFICE :
Croacawacanv
aaa Ra Ra Ba Ra Ra RaSPaSXo Po PoSPoSP PSPSPS
Vwwweweee | \- yar es) WS. ; gs Na a ea NII IN a wwe { Le)
Se ROO NE as SS | ha
=
—_ 3 ) A
| S ded
BYUSBeSSeess Ue. vee as he Na Appp~ fe
“Lda eeeceeeee et ed IIIS | IFHNavd vesee? APS GS SSSI SS SSG, | Sw SS
Pe Ne
: SAG = FSIS) S we! 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. This gentleman left his property in trust to the United States of America, to found, at Washington, an institution which should bear his own name, and have for its objects the “increase and diffusion of knowledge among men.” This trust was accepted by the Government of the United States, and an Act of Congress was passed August 10, 1846, constituting the President and the other principal executive officers of the general government, the Chief Justice of © the Supreme Court, the Mayor of Washington, and such other persons as they might elect honorary members, an establishment under the name of the “SmirHsonrAN INSTITUTION FOR THE INCREASE AND DIFFUSION OF KNOWLEDGE AMONG MEN.” The members and honorary members of this establishment are to hold stated and special meetings for the supervision of the affairs of the Institution, and for the advice and instruction of a Board of Regents, to whom the financial and other affairs are intrusted.
The Board of Regents consists of three members ex officio of the establishment, namely, the Vice-President of the United States, the Chief Justice of the Supreme Court, and the Mayor of Washington, together with twelve other members, three of whom are appointed by the Senate from its own body, three by the House of Representatives from its members, and six persons appointed by a joint resolution of both houses. To this Board is given the power of electing a Secretary and other officers, for conducting the active operations of the Institution.
To carry into effect the purposes of the testator, the plan of organization should evidently embrace two objects: one, the increase of knowledge by the addition of new truths to the existing stock; the other, the diffusion of knowledge, thus increased, among men: No restriction is made in favor of any kind of knowledge; and, hence, each branch is entitled to, and should receive, a share of attention.
iv ADVERTISEMENT.
ablishing the Institution, directs, as a part of the plan of ary, a Museum, and a Gallery of Art, together and popular lectures, while it leaves to the rganization as they may
The Act of Congress, est organization, the formation of a Libr with provisions for physical research Regents the power of adopting such other parts of an o
deem best suited to promote the objects of the bequest.
After much deliberation, the Regents resolved to divide the annual income into two parts—one part to be devoted to the increase and diffusion of emerges 2 by means of original research and publications—the other part of the income to be applied in accordance with the requirements of the Act of Congress, to the ce formation of a Library, a Museum, and a Gallery of Art.
The following are the details of the parts of the general plan of organization
provisionally adopted at the meeting of the Regents, Dec. 8, 1847.
DETAILS OF THE FIRST PART OF THE PLAN:
I. To rycreAse Know.epcr.—It is proposed to stimulate research, by offering
rewards for original memoirs on all subjects of investigation.
1. The memoirs thus obtained, to be published in a series of volumes, in a quarto form, and entitled “Smithsonian Contributions to Knowledge.”
2. No memoir, on subjects of physical science, to be accepted for publication, which does not furnish a positive addition to human knowledge, resting on original research; and all unverified speculations to be rejected.
3. Each memoir presented to the Institution, to be submitted for examination to a commission of persons of reputation for learning in the branch to which the memoir pertains; and to be accepted for publication only in case the report of this commission is favorable. |
4. The commission to be chosen by the officers of the Institution, and the name of the author, as far as practicable, concealed, unless a favorable decision be made.
5. The volumes of the memoirs to be exchanged for the Transactions of literary and scientific societies, and copies to be given to all the colleges, and principal libraries, in this country. One part of the remaining copies may be offered for sale; and the other carefully preserved, to form complete sets of the work, to supply the demand from new institutions.
6. An abstract, or popular account, of the contents of these memoirs to be given to the public, through the annual report of the Regents to Congress.
ADVERTISEMENT. Vv
II. To rncrEASE KNowLEDGE.—Zi is also proposed to appropriate a portion of the income, annually, to special objects of research, under the direction of suitable
persons.
1. The objects, and the amount appropriated, to be recommended by counsellors of the Institution.
2. Appropriations in different years to different objects; so that, in course of time, each branch of knowledge may receive a share.
3. The results obtained from these appropriations to be published, with the memoirs before mentioned, in the volumes of the Smithsonian Contributions to Knowledge.
4. Examples of objects for which appropriations may be made :—
(1.) System of extended meteorological observations for solving the problem of American storms.
(2.) Explorations in descriptive natural history, and geological, mathematical, and topographical surveys, to collect material for the formation of a Physical Atlas of the United States.
(5.) Solution of experimental problems, such as a new determination of the weight of the earth, of the velocity of electricity, and of light; chemical analyses of soils and plants; collection and publication of articles of science, accumulated in the offices of Government.
(4.) Institution of statistical inquiries with reference to physical, moral, and political subjects.
(5.) Historical researches, and accurate surveys of places celebrated in American history.
(6.) Ethnological researches, particularly with reference to the different races of men in North America; also explorations, and accurate surveys, of the mounds and other remains of the ancient people of our country.
I. To pirrusE KNowLepGE.—IFt is proposed to publish a series of reports, giving an account of the new discoveries in science, and of the changes made from year to year
in all branches of knowledge not strictly professional.
1. Some of these reports may be published annually, others at longer intervals, as the income of the Institution or the changes in the branches of knowledge may indicate.
2. The reports are to be prepared by collaborators, eminent in the different
branches of knowledge.
vi ADVERTISEMENT.
3. Each collaborator to be furnished with the journals and publications, domestie and foreign, necessary to the compilation of his report; to be paid a certain sum for
his labors, and to be named on the title-page of the report. 4. The reports to be published in separate parts, so that persons interested in a particular branch, can procure the parts relating to it, without purchasing the
whole. 5. These reports may be presented to Congress, for partial distribution, the
remaining copies to be given to literary and scientific institutions, and sold to indi-
viduals for a moderate price.
The following are some of the subjects which may be embraced in the reports :—
I. PHYSICAL CLASS.
. Physics, including astronomy, natural philosophy, chemistry, and meteorology. . Natural history, including botany, zoology, geology, &¢
oo bk =
. Agriculture.
—
. Application of science to arts.
I]. MORAL AND POLITICAL CLASS.
- Ethnology, including particular history, comparative philology, antiquities, &e. . Statistics and political economy.
“I o> Cr
- Mental and moral philosophy.
. A survey of the political events of the world; penal reform, &e.
oe)
UI. LITERATURE AND THE FINE ARTS.
9. Modern literature. 10. The fine arts, and their application to the useful arts. 11. Bibliography.
12. Obituary notices of distinguished individuals.
Il. To pirruse KNoWLEDGE.— Ji is proposed to publish occastonally separate treatises
on subjects of general interest.
1. These treatises may occasionally consist of valuable memoirs translated from
prepared under the direction of the Institution, or procured by offering premiums for the best exposition of ,
2. The treatises to be submitted to to their publication.
foreign languages, or of articles
a given subject. ® commission cf competent judges, previous
ADVERTISEMENT. vii
DETAILS OF THE SECOND PART OF THE PLAN OF ORGANIZATION.
This part contemplates the formation of a Library, a Museum, and a Gallery of Art.
1. To carry out the plan before described, a library will be required, consisting, Ist, of a complete collection of the transactions and proceedings of all the learned societies of the world; 2d, of the more important current periodical publications, and other works necessary in preparing the periodical reports.
2. The Institution should make special collections, particularly of objects to verify its own publications. Also a collection of instruments of research in all branches of experimental science.
3. With reference to the collection of books, other than those mentioned above, catalogues of all the different libraries in the United States should be procured, in order that the valuable books first purchased may be such as are not to be found elsewhere in the United States.
4. Also catalogues of memoirs, and of books in foreign libraries, and other materials, should be collected, for rendering the Institution a centre of bibliogra- .
‘phical knowledge, whence the student may be directed to any work which he may require.
5. It is believed that the collections in natural history will increase by donation, as rapidly as the income of the Institution can make provision for their reception ; and, therefore, it will seldom be necessary to purchase any article of this kind.
6. Attempts should be made to procure for the gallery of art, casts of the most celebrated articles of ancient and modern sculpture.
7. The arts may be encouraged by providing a room, free of expense, for the exhibition of the objects of the Art-Union, and other similar societies.
8. A small appropriation should annually be made for models of antiquity, such as those of the remains of ancient temples, &c.
9. The Secretary and his assistants, during the session of Congress, will be required to illustrate new discoveries in science, and to exhibit new objects of art; distinguished individuals should also be invited to give lectures on subjects of
general interest.
In accordance with the rules adopted in the programme of organization, each
memoir in this volume has been favorably reported cn by a Commission appointed
viil ADVERTISEMENT. for its examination. 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,