fyxmll IKnweMitg Jitotg
THE GIFT OF
Q.ai/-yt^iji--^j-T.^ .-j,^]A'^
.Aj^...TT^f
.MMA^'
The dateshows w^en this volume was taken, j
All books not in use]
for instruction or re-^
, search are limited to'|
all borrowers. -l
Volumes of periodi- '
cals and of pamphlets
comprise so many sub-
jects, that they are held
in the library as much
as possible. For spe-
cial purposes they are
given out for a limited
time.
Graduates and sen-
iors are allowed five
volumes for two weeks.
Other students may
have two vols, from the
circulating library for
two weeks.
Books not needed
during recess periods
should be returned to
the library, or.arrange-
hients made for their
return during borrow-
er's absence, if wanted.
Books needed by
more than one person
are held on the reserve
list.
Books of , special
value and gift books,
when the giver wishes
it, are riot allowed to
circulate.
Cornell University Library
QB 3.H64
V.1
The collected mathematical works of Geor
3 1924 020 314 757
XI
Cornell University
Library
The original of this book is in
the Cornell University Library.
There are no known copyright restrictions in
the United States on the use of the text.
http://www.archive.org/details/cu31924020314757
THE COLLECTED
MATHEMATICAL WORKS
OF
GEORGE WILLIAM HILL
VOLUME ONE
THE COLLECTED
MATHEMATICAL WORKS
OF
aEORGE WILLIAM HILL
VOLUME ONE
Published by the Carnegie Institution of Washington
JuifE, 1905
c
\
.'nH^J-^1
CAENEGIE INSTITUTION OF WASHINGTON
Publication No. 9 (Volume One)
THE FniEDENWALD COMPANY
BALTIMORE, MB., u. B. A.
EEEATA IN FIEST VOLUME.
(Lines counted from the bottom of the page are noted as negative.)
Page
Line
Page
Line
10
14
for T read T^,
252
6
for aj_ read ai_y
13
2
for 3^ read gy,
252
18
for -. + m read -, + m'
13
12
for ^q read ^q^
252
-3
for U_ read U_
13
15
for 2 F/' read 2 F,"
259
■-7
for m^ read m'
14
9
for-ae,-K)
264
4
for TT^j read, ttSJ ,
read +(|-e,-|eD
265
-1
for 4^/ read '4 5"
18
2
for m^ F„ read in' V„
266
11
for S read S
18
-5
V V
for — 5%r read ^
374
-1
for aP read aP^ v
nf F„ rre F„
278
13
for aP^ read aP^,
19
21
for - 1."3679 read + l."3679
293
13
add the exponent i to the right
60
-3
'"•S^-^'w)
member
293
17
add the exponent J to the
63
-9
add the term 6 (j!'' to the factor
denominator
of tan 8
for +^ read+^)
396
17
for {liny read {im)i
65
-9
304
23
for (2C) read (3C)I
69
14
for Ja read A^a
308
3
for (us) read (Ms)i
78
20
for A read X^
308
-3
for a(a_j__i read aia_i_j_i
79
21
for an read and
309
10
for aja_( i read aja_i_y_,
124
15
for cos e' read cos 6
310
-1
for + [3, 3] read + [3, - 3]
133
11
for cos
r"
and add the exponent i to the
193
-3
for 5— read 5—,
334
-6
last factor
for ao read aj
201
204
4
16
add V before(a'— 2aZcos^ + ?)
for / read / t
334
336
-1
23
for [y read [£,]'
for r sin read r sin v
210
19
for established read establishes
336
37'
forrcos, rsino readrcosuj^sino
210
20
for / read /
336
326
31
36
u it c( ii ti a
It tc 11 a 11 «
219
-3
for 3 E Els read 3 E^ El,
for ^ read ^,
for - read -r
338
6
224
12
r r^
1 2
331
-9
for^ffj read 9^ a*
and transpose an arm of paren-
340
-9
for — ^n read — f m'
thesis to one line lower
346
8
for (Z . e' read t^ . e"
225
10
for^£"''-read-e'"'-'
d CI
346
-5
for w' read n' '
,7"-l ,,/'*-"
349
-9
for J read J «
226
4
for'^f read*^f
3 3
230
2
for = read —
353
-7
add exponent i to last factor
230
3
for = read —
in denominator
for 9^ read 1^
y 3y
for wv + "wv' + -vrv"
359
-9
for A' read A„
238
19
360
36
for i V)* read J ij*
242
-7
361
15
for -^ 5; e' read -g- >? e'
read wi* + w'v' + w"/'
363
33
for r; read f,
251
9
for af read of
363
4
for
Z2 read ?2
[Insert in Piest Voujmb Mathematical Works of G. W. Hill.]
CONTENTS.
MBMOm p^Qj,
Introduction by H. Poincarfi vii-xvm
No. 1. On the Curve of a Drawbridge 1
2. Discussion of the Equations which Determine the Position of a Comet or
other Planetary Body from Three Observations 2-4
3. On the Conformation of the Earth 5-19
4. Ephemeris of the Great Comet of 1858 20-22
5. On the Reduction of the Rectangular Coordinates of the Sun Referred to the
True Equator and Equinox of Date to those Referred to the Mean Equa-
tor and Equinox of the Beginning of the Year 23-24
6. Discussion of the Observations of the Great Comet of 1858, with the Object
of Determining the Most Probable Orbit 25-58
7. On the Derivation and Reduction of Places of the Fixed Stars 59-76
8. Determination of the Elements of a Circular Orbit 77-84
9. New Method for Facilitating the Conversion of Longitudes and Latitudes of
Heavenly Bodies, near the Ecliptic, into Right Ascensions and Declina-
tions, and Vice Versa 85-88
10. Correction of the Elements of the Orbit of Venus 89-104
11. On the Derivation of the Mass of Jupiter from the Motion of Certain Aster-
oids 105-108
12. On the Inequality of Long Period in the Longitude of Saturn, whose Argu-
ment is Six Times the Mean Anomaly of Saturn Minus Twice that of
Jupiter Minus Three Times that of Uranus 109-112
13. Charts and Tables for Facilitating Predictions of the Several Phases of the
Transit of Venus in December, 1874 113-150
14. A Method of Computing Absolute Perturbations 151-166
15. On a Long Period Inequality in the Motion of Hestia Arising from the Action
of the Earth 167-168
16. Solution of a Problem In the Theory of Numbers 169
17. A Second Solution of the Problem of No. 8 170-172
18. Remarks on the Stability of Planetary Systems 173-180
19. Useful Formulas in the Calculus of Finite DifEerences 181-185
20. Elementary Treatment of the Problem of Two Bodies 186-191
21. The Differential Equations of Dynamics 192-194
22. On the Solution of Cubic and Biquadratic Equations 195-199
23. On the Equilibrium of a Bar Fixed at one End Half Way between Two
Centers of Force 200-202
24. The Deflection Produced in the Direction of Gravity at the Foot of a Conical
Mountain of Homogeneous Density 203-205
25. On the Development of the Perturbative Function in Periodic Series 206-226
YI CONTENTS.
MEMOIB PAGE
No. 26. Demonstration of the Differential Equations Employed by Delaunay in the
Lunar Theory 227-232
27. Solution of a Problem in the Motion of Rolling Spheres 233-235
28. Reduction of the Problem of Three Bodies 236-242
29. On the Part of the Motion of the Lunar Perigee which is a Function of the
Mean Motions of the Sun and Moon 243-270
30. Empirical Formula for the Volume of Atmospheric Air 271-281
31. On Dr. Weiler's Secular Acceleration of the Moon's Mean Motion 282-283
32. Researches in the Lunar Theory 284-335
33. On the Motion of the Center of Gravity of the Earth and Moon 336-341
34. The Secular Acceleration of the Moon 342-347
35. Note on Hansen's General Formulas for Perturbations 348-350
36. Notes on the Theories of Jupiter and Saturn 351-363
INTRODUCTION
Par M. H. Poincaeb
M. Hill est une des physionomies les plus originales du monde scienti-
fique am^ricain. Tout entier a ses travaux et a ses calculs, il reste 6tranger
a la vie fi^vreuse qui s'agite autour de lui, il recherche I'isolement, hier dans
son bureau du Nautical Almanac, aujourd'hui dans sa ferme tranquille de la
vallee de I'Hudson. Cette reserve, j'allais dire cette sauvagerie, a 6te une
circonstance heureuse pour la science, puisqu'elle lui a permis de mener
jusqu'au bout ses ingenieuses et patientes recherches, sans en etre distrait
par les incessants accidents du monde exterieur. Mais elle a empech6 que
sa reputation se repandit rapidement au dehors; des annees se sont ^coulees
avant qu'il eut, dans I'opinion du public savant, la place a laquelle il avait
droit. Sa modestie ne s'en chagrinait pas trop et il ne demandait qu'une
chose, le moyen de travailler en paix.
M. Hill est ne a New York le 3 mars 1838. Son p^re, d'origine an-
glaise, etait venu en Amerique en 18 20 a I'age de 8 ans; sa mere, d'une vieille
famille huguenote, lui apportait les traditions des premiers colons de la terre
am6ricaine.
Quoique n6 dans une grande ville, M. Hill est un campagnard ; peu de
temps apres sa naissance, son pere quitta New York et vint s'6tablir a "West
Nyack, N. Y. ; c'etait une ferme, pres de la riviere Hudson, a 25 milles en-
viron dela grande Cit6. C'est la que M. Hill passa son enfance; il aimatou-
jours cette residence; il y revenait toutes les fois qu'il le pouvait, et quand
il eut quittS le Nautical Almanac, c'est encore la qu'il s'etablit d6finitivement;
c'est la qu'il poursuit tranquillement ses travaux, 6vitant le plus qu'il peut
les voyages a New York.
Ses aptitudes exceptionnelles pour les math^matiques ne tarderent pas a
se manifester et on d6cida de I'envoyer au college. En octobre 1855, a I'age
de 17 ans, il entra au College Rutgers, New Brunswick, N. J. Son pro-
fesseur de mathematiques etait le Dr. Strong, ami de M. Bowditch, le traduc-
teur de la M6canique Celeste de Laplace.
Le Dr. Strong etait un homme de tradition, un laudator temporis acti ;
pour lui Buler etait le Dieu des Mathematiques, et apres lui la decadence
avait commenc6 ; il est vrai que c'est la un dieu que I'on peut adorer avec
yill INTRODUCTION
profit. De rares exceptions pr&s, la bibliotheque du Dr. Strong 6tait im-
pitoyablement fermSe a tons les livres posterieurs a 1840. Heureusement
on a 6crit d'excellentes choses sur la Mecanique Celeste avant 1840; on
trouvait la Laplace, Lagrange, Poisson, Pontecoulant. Tels furent les maitres
par lesquels Hill fut initie au rudiment.
En juillet 1859 il reput ses degres au College Rutgers et se rendit
a Cambridge, Mass., dans I'espoir d'accroitre ses connaissances matli6matiques,
mais il n'y resta pas longtemps, car au printemps de 1861 il obtint un poste
d'assistant aux bureaux du Nautical Almanac a Washington. II resta au
service de cette 6phemeride pendant trente annees de sa vie, les plus
fructueuses au point de vue de la production scientifique.
Les bureaux du Nautical Almanac etaient a cette epoque a Cambridge
(Massachusetts), oil ils pouvaient profiter des ressources scientifiques de I'Uni-
versite Harvard et ils etaient diriges par M. Runkle. Ce savant avaitfonde
un journal de mathematiques elementaires. The Mathematical Monthly, dans
le but de favoriser les etudes mathematiques en Amerique en facilitant la
publication de courts articles et en proposant des prix pour la solution de
problemes mathematiques. L'un des premiers articles publics rev61ait la main
d'un maitre, et gagna ais6ment le prix. II s'agissait des fonctions de La-
place et de la figure de la Terre. L'auteur etait M. Hill, qui venait de sortir
du college.
C'est ainsi que 1' attention de M. Runkle fut attiree sur ce jeune homme et
qu'il songea a utiliser ses services pour les calculs de I'ephemeride ameri-
caine.
On I'autorisa n6anmoins a continuer sa residence dans sa maison fami-
liale de West Nyack (village qui s'appelait alors Nyack Turnpike). II y
resta encore quand en 1886 les bureaux du Nautical Almanac furent trans-
fer's a Washington.
Mais en 1877 M. Simon Newcomb prit la direction de r6phemeride.
II voulut entreprendre une tache colossale, la reconstruction des tables de
toutes les planfetes ; la part de M. Hill etait la plus difficile ; c'etait la theorie
de Jupiter et de Saturne, dont il avait commence a s'occuper depuis 1872.
II ne pouvait la mener a bien qu'aupr&s de son chef et de ses coll&gues. II
fallut done se resigner a I'exil; I'importance de I'oeuvre a accomplir lui fit
facilement accepter ce sacrifice.
Ses services furent hautement apprecies; en 1874 il fut eiu membre de
I'Academie Nationale des Sciences. En 1887 la Societe Royale Astrono-
mique de Londres lui accorda sa medaille d'or pour ses recherches sur la
theorie de la Lune. II fut president de la Societe Mathematique Americaine
pendant les annees 1894 et 1895. L'universite de Cambridge (Angleterre)
INTRODUCTION jj-
lui confera des degres honoraires, et il en fut de meme de plusieurs
universites americaines.
En 1892 il prit sa retraite et quitta les bureaux du Nautical Almanac;
il eut hate de s'installer pour ses dernieres annees dans cette ch&re maison
ou il avait pass6 son enfance ; au debut, il la quittait encore plusieurs fois
par semaine pour venir professer a I'Universite Columbia a New York; mais
il ne tarda pas a se lasser de cet enseignement et depuis il y vit seul avec
ses livres et ses souvenirs.
Le travail quotidien du Nautical Almanac, qui est fort absorbant, lui lais-
sait cependant assez de temps pour ses recherches originales, dont quelques-
unes portent sur des objets etrangers a ses etudes habituelles. Dans les
premieres ann6es surtout, on trouve frequemment son nom dans ces recueils
periodiques, ou les amateurs de mathematiques pures se proposent de petits
problemes et se complaisent dans I'^legance des solutions, par exemple, dans
" The Analyst."
Mais il ne tarda pas a se sp^cialiser. Non seulement ses fonctions I'y
contraignaient, mais ses gouts I'y portaient. Le travail courant, n6cessaire
pour la preparation de l'ephem6ride, lui foumissait deja des occasions de se
distinguer. Nous citerons des tables pour faciliter le calcul des positions
des 6toiles fixes et qui sont pr^c^d^es d'une note de M. Hill ou la theorie de
cette reduction est exposee d'une fapon simple et claire.
A cette epoque le prochain passage de Venus preoccupait tons les astro-
nomes. En vue des expeditions projetees, le bureau de I'ephemeride dut se
livrer a de longs travaux preliminaires. M. Hill fut ainsi conduit a refaire
les tables de Venus. C'etait son premier ouvrage de longue haleine, et on
pent y voir deja le germe des qualit^s que I'on admirera plus tard dans tous
ses Merits. Dans cette premiere p6riode de sa vie scientifique, il revint a
plusieurs reprises sur le calcul des orbites. C'est la un probleme qui se
pr^sente constamment au calculateur astronomique et qui devait naturelle-
ment retenir I'attention d'un praticien constamment aux prises avec les diflB-
cultes qu'il fait naitre. Citons une el6gante discussion de I'equation fonda-
mentale de Gauss et diverses notes relatives au meme sujet. Les progres de
I'astronomie d'observation avaient d'ailleurs fait entrer la question dans une
phase nouvelle; les decouvertes de petites planfetes se multiplient et devien-
nent de plus en plus fr6quentes. EUes se succ^dent avec une telle rapidite
que les calculateurs sont distances par les observateurs. Ceux-ci fournissent
aux premiers plus de besogne qu'ils n'en peuvent faire, et ils veulent etre
servis promptement, parce que d^s qu'une nouvelle planete est d^couverte
ils craignent de la perdre. La question aujourd'hui est done avant tout de
faire vite ; il faut des m^thodes rapides, qui n'exigent pas de trop longe cal-
INTRODUCTION
culs et permettent d'utiliser les premieres observations. On a et6 ainsi con-
duit a n^gliger d'abord l'excentricit6 des ellipses et a calculer des orbites
circulaires. Tel est le point de vue ou s'est place M. Hill dans une s^rie de
notes qui ont paru dans divers recueils entre 1870 et 1874.
Mais j'ai hate d'arriver a son oeuvre capitale, a celle ou s'est d6voil6e
toute I'originalite de son esprit, a sa theorie de la Lune. Pour en bien faire
comprendre la portee, il faut d'abord rappeler quel 6tait I'etat de cette the-
orie au moment oii M. Hill commenpa a s'en occuper.
Deux oeuvres de haute sagacit6 et de longue patience venaient d'etre
raenees a bonne fin ; je veux parler de celle de Hansen et de celle de Delau-
nay. Le premier, par une voie inutilement detourn^e, etait arrive le premier
au but, devanpant de beaucoup ceux qui avaient pris la bonne route. Ce ph6-
nom^ne, au premier abord inexplicable, n'etonnera pas beaucoup les psycho-
logues. Si sa methode, qui nous parait si rebarbative, ne I'efifrayait pas, c'est
precis6ment parce qu'il etait infiniment patient, et c'est pour cela aussi qu'il
est alle jusqu'au bout. Et c'est aussi parce qu'elle etait etrange qu'elle lui
semblait avoir un cachet d'originalite, et c'est dans le sentiment de cette
originalit6 qu'il a puise la foi solide qui I'a soutenu dans son entreprise. Une
autre raison de son succes, c'est qu'il n'a cherche que des valeurs purement
numeriques des coefficients sans se preoccuper d'en trouver I'expression
analytique ; ce qui chez les autres representaient de longues formules, se
reduisait pour lui a un chiffre, et cela d&s le debut du calcul.
Quoi qu'il en soit, c'est encore sur les tables de Hansen que nous vivons
et il est probable que les nouvelles theories plus savantes, plus satisfaisantes
pour I'esprit, ne donneront pas des chiffres tres difif6rents.
Delaunay est a I'extreme oppose ; ses inegalites se presentent sous la
forme de formules alg^briques; dans ces formules ne figurent que deslettres
et des coefficients numeriques formes par le quotient de deux nombres en-
tiers exactement calculus. II n'a done pas fait seulement la theorie de la
Lune, mais la theorie de tout satellite qui tournerait ou pourrait tourner
autour de n'importe quelle planMe. A ce point de vue il laisse Hansen loin
derriere lui. La methode qui I'avait conduit a ce resultat constituait le
progres le plus important qu'eut fait la Mecanique Celeste depuis Laplace.
Perfectionnee aujourd'hui et alleg^e, elle est devenue un instrument que
chacun pent manier et qui a rendu deja bien des services dans toutes les
parties de I'Astronomie. Telle que Delaunay I'avait d'abord conpue, elle
etait d'un emploi plus penible. Peut-etre aurait-il abrege considerablement
son travail s'il en avait fait un usage moins exclusif, mais il faut beaucoup
pardonner aux inventeurs.
INTRODUCTION XI
II mena a bonne fin sa tache d'algebriste, mais les formules demandaient
a etre r^duites en chifFres; quand un accident imprevu I'enleva a ses admi-
rateurs, il etait sur le point de commencer ces nouveaux calculs. Sa mort
arreta ce travail, et ce n'est que dans ces derniers temps qu'il put etre repris
et termini.
Malheureusement les series de Delaunay ne convergent qu'avec une de-
sesp6rante lenteur. Elles precedent suivant les puissances des excentricites
de I'inclinaison, de la parallaxe du soleil, et de la quantite que Ton appelle m et
qui est le rapport des moyens mouvements. Cette quantite est de tw environ,
et si les coefficients numeriques allaient en d^croissant, la convergence serait
suffisante. Malheureusement il n'en est pas ainsi, ces coefficients croissent,
au contraire, tres rapidement par suite de la presence de petits diviseurs.
Aussi desesperant de pousser assez loin le calcul des series, Delaunay fut-il
oblig6 d'ajouter aujuge des termes complementaires.
M. Hill s'assimila promptement la m6thode de Delaunay, et en a fait
I'objet de plusieurs de ses ecrits, mais celle qu'il proposa etait tout a fait
dififerente et tres originale. C'est dans un memoire de 1' American Journal
of Mathematics, tome 1, que nous en voyons les premiers germes.
Les series de Delaunay, nous I'avons dit, dependent de cinq constantes,
qui sont les excentricites, I'inclinaison, la parallaxe du soleil et enfin la quan-
tite m. Si nous supposons que les quatre premieres sont nulles, nous aurons
une solution particuliere de nos equations difFerentielles. Cette solution par-
ticuliere sera beaucoup plus simple que la solution generale, puisque la plu-
part des inegalites auront disparu, et qu'une seule d'entre elles subsistera,
celle qui est connue sous le nom de variation. D'autre part cette solution
particuliere ne represente pas exactement la trajectoire de la Lune, mais elle
pent servir de premiere approximation, puisque les excentricites, I'inclinaison
et la parallaxe sont eflfectivement tres petites. Le choix de cette premiere
approximation est beaucoup plus avantageux que celui de I'ellipse Keple-
rienne, puisque pour cette ellipse le perigee est fixe, tandis que pour I'orbite
reelle il est mobile.
Les equations difierentielles sont d'ailleurs elles-memes plus simples,
puisque I'excentricite et la parallaxe etant nulles, le Soleil est suppose decrire
une circonference de rayon tres grand. M. Hill simplifie encore ces equations
par un choix judicieux des variables. II prend non pas les coordonnees
polaires, mais les coordonnees rectangulaires, et c'est la un grand progres.
Que ces dernieres soient plus simples a tout egard, c'est de toute evidence,
et cependant les astronomes repugnent a les adopter. Je comprends a la
rigueur cette repugnance pour la Lune, puisque ce que nous observons, ce
que nous avons besoin de calculer c'est la longitude, mais j'avoue que je me
XII INTRODUCTION
I'explique difficilement en ce qui concerne les plan&tes, puisque ce n'est pas
la longitude hSliocentrique, mais la longitude geocentrique qu'on observe.
En tous cas, pour la Lune, elle-meme, M. Hill a juge que les avantages
I'emportent sur les inconvenients, et qu'on peut bien se resigner a faire a la
fin du calcul un petit changement de coordonnees, pour ne pas trainer pen-
dant toute une th6orie, un encombrant bagage de variables incommodes.
Les variables de M. Hill ne sont pas d'ailleurs des coordonnees rectan-
gulaires par rapport a des axes fixes, mais par rapport a des axes mobiles
animes d'une rotation uniforme, 6gale a la vitesse angulaire moyenne du
Soleil. D'ou une simplification nouvelle, car le temps ne figure plus explicite-
ment dans les equations. Mais I'avantage le plus important est le suivant.
Pour un observateur lie a ces axes mobiles, la Lune paraitrait decrire
une courbe ferm^e, si les excentricit6s, I'inclinaison et la parallaxe etaient
nulles. Comme les equations differentielles sont d'ailleurs rigoureuses,
c'etait Ih le premier exemple d'une solution periodique du prohleme des 3 corps,
dont I'existence 6tait rigoureusement d^montree. Depuis ces solutions peri-
odiques ont pris une importance tout a fait capitale en M6canique Celeste.
Mais I'auteur ne se borna pas a demontrer cette existence, il ^tudia dans le
detail cette orbite (ou plutot ces orbites periodiques, car il fit varier le seul
parametre qui figurat dans ces Equations, le param&tre m); il determina
point par point ces trajectoires fermees et calcula les coordonnees de ces
points avec de nombreuses decimales. Les developpements de Delaunay
furent remplaces par d'autres plus convergents et pour de grandes valeurs de
m, quand les series nouvelles elles-meme ne suffirent plus, M. Hill eut re-
cours aux quadratures m6caniques. II arrive finalement au cas, on, pour I'ob-
servateur mobile dont nous parlions, I'orbite apparente aurait un point de
rebroussement.
Une derni&re remarque ; M. Hill, dans le memoire que nous analysons,
trausforme ses equations de fapon a les rendre homogenes et il tire de ces
equations homogenes un remarquable parti; il serait aise de faire quelque
chose d'analogue dans le cas general du probleme des trois corps; il sufiirait
d'eliminer les masses entre les equations du mouvement; I'ordre de ces equa-
tions se trouverait ainsi augmente, mais on arriverait a n'avoir plus dans
les deux membres que des polynomes entiers par rapport aux coordonnees
rectangulaires et a leurs derivees. Les equations ainsi obtenues ne pourraient
servir a I'integration, mais elles pourraient rendre de pr6cieux services
comme formules de verification.
Par ce memoire les termes qui ne dependent que de m se trouveraient
enti&rement determines avec une precision infiniment plus grande que dans
aucune des theories anterieures ; les termes les plus importants ensuite sont
INTRODUCTION XIII
ceux qui sent proportionnels a I'excentricite de la Lune et ne dependent
d'ailleurs que de m.. Ces termes dependent des memes equations differen-
tielles; mais comme on connait deja une solution de ces equations et que
celle que I'on cherche en differe infiniment pen, tout se ramene a la conside-
ration des " equations aux variations." Or ces Equations sont lin^aires,
elles sont a coefficients p6riodiques; elles sont du 4eme ordre, mais laconnais-
sance de I'integrale de Jacobi permet de les ramener ais6ment au 2eme ordre.
La theorie g6n6rale des equations lin6aires a coefficients periodiques nous
apprend que ces Equations admettent deux solutions particuli^res suscepti-
bles d'etre representees par une fonction periodique multipliee par une expo-
nentielle. C'est I'exposant de cette exponentielle qu'il s'agit d'abord de
determiner et cet exposant a une signification physique tres simple et tres
importante, puisqu'il represente le moyen mouvement du perigee.
La solution adoptee par M. Hill est aussi originale que bardie. Notre
Equation diflferentielle doit etre resolue par une serie. En y substituant une
s6rie S a coefficients indetermines, on obtiendra une autre s6rie 2 qui devra
etre identiquement nulle. En egalant a z^ro les difFerents coefficients de
cette s6rie X, on obtiendra des equations lineaires ou les inconnues seront
les coefficients indetermin6s de la serie S. Seulement ces equations de meme
que les inconnues etaient en nombre infini. Avait-on le droit d'egaler a
zero le determinant de ces equations ? M. Hill I'a ose et c'^tait la une grande
hardiesse; on n'avait jamais j usque-la considere des equations lineaires en
nombre infini; on n'avait jamais etudie les determinants d' ordre infini; on ne
savait meme pas les definir et on n'etait pas certain qu'il fut possible de donner
a cette notion un sens precis. Je dois dire cependant, pour etre complet,
que M. Kotteritzsch avait dans les Poggendorf s Annales aborde le sujet.
Mais son memoire n'6tait guere connu dans le monde scientifique et en tout
cas ne I'^tait pas de M. Hill. Sa methode n'a d'ailleurs rien de commun
avec celle du geometre americain.
Mais il ne suffit pas d'etre hardi, il faut que la hardiesse soit justifi6e
par le succ^s. M. Hill 6vita heureusement tous les pieges dont il etait envi-
ronne, et qu'on ne dise pas qu'en operant de la sorte il s'exposait aux erreurs
les plus grossieres; non, si la methode n'avait pas ete legitime, il en aurait
ete tout de suite averti, car il serait arrive a un rSsultat numerique absolu-
ment different de ce que donnent les observations. La meme methode donne
les coefficients des diverses inegalites proportionnelles a I'excentricite et dont
les plus importantes sont I'equation du centre et I'evection. Comparons ce
calcul avec celui de Delaunay; la methode de Hill avec deux ou trois appro-
ximations donne un grand nombre de decimales ; Delaunay pour en avoir
moitie moins devait prendre huit termes dans sa serie, et ce n'etait pas
^jy INTRODUCTION
assez, il fallait evaluer par des precedes approches le reste de la s6rie ; s'il
avait fallu attendre qu'on arrive a des termes negligeables, la plus robuste
patience se serait lassee. A quoi tient cette difference? Le mouvement
g du perigee nous est donn6 par la formule
cos gTf=
tV 1 this value of tan 6 is imaginary, and the left member of
the equation differentiated is not susceptible of a maximum or minimum
value, and the equation in sin 6 has only two real roots, which are among
those rejected. Hence we conclude, that when the data are taken from
observation, the q uantity — ; will always be contained between the
' ^ •' 1 -\- m cos c
4 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
limits ± I . If we substitute for 6 in the equation sin* 6 — A sin {6 — (3)
= , the result is A sin ^ , and for 6 = 7t , the result is — A sin ^ ; showing
the existence of and odd number of roots between the limits 6 = and Q-=7t,
which odd number is three, since c and the root which the problem demands
are within these limits. If we make Q=^ c -\- dc, there results the quantity
or, smce
the quantity
[4 siv? c cos c — A cos (c — /S)] dc ;
J sin'c , a w! sin c
A = -. — J , tan /? = ;
sin (c — /?) 1 + m cos c
(3 cos c — m) sin' cdc.
And — A cos /? , the result, on putting 0=7t, is equal to — m sin* c.
Therefore, sin c being always positive, 6 has two real values, or only one
(between the limits c and n), and, consequently, the problem two or one
answer, according as m and m — 3 cos c have the same or opposite signs.
It is evident that A cos /3 must be positive, in order that the equation in
sin 6 may have three positive real roots ; so the quantity 1 + m cos c is always
positive, and tan (3 has the same sign as m . If |3 be taken between the
limits zfc -— , J. is always positive. Since the equation in sin 6 must have
no root greater than one, unity substituted for sin 6 in the first derived
function of its equation must render it positive ; that is, the expression
4 — 5 A cos (i + A^ ia positive, which gives J. <; 2 and A cos /? <^ f . Accord-
ing as m is positive or negative, the equation for finding d presents itself
under two shapes, sin* 0=. A sin {d — (3) and sin* 6 = A sin (6 + ^), in
which A is always positive and less than 2, and ^ never exceeds 36° 53'-
From the expression for p in terms of r, it is clear that r is less or
greater than E , according as m is positive or negative. Therefore, in the
first case, S is contained between the limits c and n — c ; and, in the second
case, if c is in the first quadrant, between n — c and n — (3 ; but if c be in the
second quadrant between c and n — ^. These remarks may be of use to
shorten the tentative process of finding 6 .
With regard to 6, it is clear it is the angle subtended at the comet by
its radius vector and the line joining it and the earth prolonged beyond the
comet.
CONFORMATION OF THE3 EARTH
MEMOIE No. 3.
On the Conformation of the Earth.*
(First Prize Essay, Eunkle's Mathematical Monthly, Vol. Ill, pp. 166-182, 1861.)
1. All the particles which compose the mass of the earth are animated
by the attraction of gravitation. The law of this force is, that the attrac-
tion of any atom for a spherical surface of material points, described about
it as a center, is constant. Hence, if the attraction of an atom for a mate-
rial point be represented by A, and r be the radius of the spherical sur-
face and N the number of material points in a unit of surface, the attrac-
tion of the central atom for the spherical surface is inNr^A = a constant
M
= — 4nNM. Whence A = ^ ; that is, the attraction varies inversely
as the distance squared. The constant M is called the mass of the attract-
ing atom. We have given A the negative sign because it represents a force
tending to decrease the line r .
2. Making A = -^- , then V= — . V is called the potential function,
and has this property: that if the partial derivative of it be taken with
respect to any of the three spaceal coordinates of which it is necessarily a
function, the result will be the partial force in the direction of that coordi-
nate axis.
M
3. If the attraction of a single atom give ^^^ -jj > D denoting the
distance, the attraction of an indefinite number or assemblage of atoms will
give F= S . -jY • If ^ . 4' > 4> represent any three lines at right angles with
each other, then -^^ , -=-t . o— ^.re the forces acting in each of these direc-
o^ 0-4' o(p
tions respectively. In a rectangular system
D^\(x'- xy + (y'-yy + (z'-zy }i,
*Thi3 memoir, written at the end of 1859 and beginning of 1860, was designed to show how all the
formulae connected with the figure of the earth could be derived from Laplace's and Poisson's equa-
tions, combined with the hydrostatic equilibrium of the surface, without any appeal to the definite
integrals belonging to the subject of attraction of spheroids. Some of the assumptions are quite
unwarranted, nevertheless I allow them to stand.
g COLLECTED MATHEMATICAL WORKS OP G. W. HILL
where the accented letters pertain to the attracting atom, and the unaccented
to the attracted. Consequently,
dV_„ M,._. dV_g ^(v'-y) ^=S.-^(z'-z). (1)
4. Differentiating again,
15=..^ {^.(.'-.,--1},
(2)
In this differentiation M has been regarded as independent of x^y, z;
but, in order to render equations (2) altogether general, the attracting mass
must be considered as extending into the point x ,y , z . Let p be the den-
sity of the atom occupying this point, which becomes zero when the attract-
ing mass does not reach the point. This atom may be regarded as spherical,
then for it ilf= t7tp2>'; substituting this value in equations (1), the results
are
Hence, we must add the term — f Ttp to the right members of equations (2),
and then we can regard D as having always a finite value. By adding
these equations, there results
The integration of this gives F; p is a function of x , y , z; in the case of
solid bodies, as the earth, a limited function.
5. Transform (3) to terms of polar coordinates ; put
X = r )^/ (1 — fi'') COS u) , r=^l {x* + y' + «') ,
z = r p., to = tan~' -^ .
X
CONFORMATION OF THE EARTH 7
Then
dx' ^ dy' "^ 82;' "■ 1 dx' ^ df dz' J dr' ^ \ dx' ^ df dz' i dr
+
a r^^Z a fl — a^-l^^ " "
dv
9(u
Hence (3) becomes
i?^ + M + ?^l 9'^, /9V , av , avi dv
I dx' "^ dy' dz' i dp' \ dx' dy' dz' ) 3/^
1 aar" "^ ay' a«' ; d'^' \ dx' "*" ay' a^' /
^rara^ ara/i ara^i ^^ •
I ax aa; a?/ dy dz dz) drdfi
jj f a/i So) 3^ aw a^ a(u 1 a'F
I dx dx dy dy dz dz J diJ^d>"
2 ( a<« ar au; ar ac> ar •) d'v
1 aa; a* ay ay az az J d<«dr
d'V )
+ 1 - fiO
dr + aA
a.r^ar a.(i-/.')9r ^^
-~aF-+ a^ + i^^, + 4-/'»" = o.
6. To show the application of (4), take the simple case when the sur-
faces of equal density are concentrically spherical. Placing the origin of
coordinates at the common centre, p becomes a function of r alone, either
continuous or discontinuous as the case demands ; and evidently ^ = ,
97
--— = ; therefore, (4) becomes
ou ^ '
a ^>9^
dr , - , „
—Q^ + i7:pr' = .
By integration
lf={G-i-fpr'dry-'.
Between the limits and r, in J pr^dr is equal to the mass contained
within the sphere whose radius is r ; denoting this by M, it is clear that
C= 0, since the expression must agree with that for the attraction of a
dV M
single atom. Thus ^— ■= ^ . Or the principle may be stated : The
force acting on any point, wherever sittiated, equals the mass of all the particles
nearer the center than the point attracted, divided hy the square of the distance
of the point from that center, taken with the negative sign.
g COLLECTED MATHEMATICAL WORKS OF G. W. HILL
7. The earth revolves about a constant axis ; hence, to remove our
problem from dynamics to statics, it is necessary to introduce the force of
pressure called the centrifugal force. Making the coordinal axis of z coin-
cide with the earth's axis, and T denoting the period of the earth's rotation,
the potential of the centrifugal force is
Since this force animates every particle, include its potential in V and make
F the potential of both gravitating and centrifugal force. It then becomes
g_,3
necessary to add to (4) the term =^ r^ , whence
8. To apply this equation to the solution of the earth's conformation,
we must combine it with some condition of equilibrium. From the manner
in which the atmosphere and ocean cover the earth, we may conjecture it
was once fluid, and in solidifying, preserved the form it had taken by the
laws of hydrostatics. In passing from the solid earth to the ocean, and
from the ocean to the atmosphere, there occur two faults in the continuity
of the earth's density ; hence p is strictly represented by a discontinuous
function. But, as the mass of the ocean and atmosphere is about rinnr of
the whole, its influence may be neglected, and p supposed continuous from
center to surface.
9. If^ is the pressure, then dp = pd V, and F+ 5 = is the equation
to surfaces of level, B having a different value for each surface. Let p be
a function of p, and thus of V. In (5) make
4.(,-^,)=/(F).
10. The centrifugal force being small compared with gravity, may be
regarded as a perturbing force. Supposing at first this force is zero, the
particles would arrange themselves symmetrically about a center, since
there is no reason why they should accumulate more in one place than in
another. Take the origin of coordinates at this center, then .5— =: 0, >r— = .
oix oa
Thus (5) becomes
CONFORMATION OF THE EARTH 9
Consequently F is a function of r alone, and the general equation of sur-
faces of level F+ 5 = 0, when solved gives r = a constant ; these surfaces
are then concentrically spherical.
11, The term -=2 r^ (1 — n^) , which the centrifugal force introduces
into F, and which causes a departure from the spherical form, does not con-
tain 6), and so whatever derangement it may produce, cannot introduce w
into F; that is, the earth is a solid of revolution. Consequently (5) becomes
1 2. From the form of this same term, it may be concluded that
V= F, + Yy +Y,u.^ + . . .^ S.Y,u?' , (7)
where Yi is a function of r alone, and a quantity of the order of the i^^ power
of the centrifugal force. Substitute this expression of F in (6) and put the
coeflBcient of /tt^' resulting equal to zero, and let the coefficient of y?* in
rV(^) be ?7,; then
J idYt
a . r -T-'
_5f!l J. r2i: + n (ii + 2^ f,,. - 2?;c2?: + 1^ f + rz = o. (^)
dr
+ (2t + 1) {%% + 2) F,+, - 2i (2^ + 1) F, + R = .
This equation has the inconvenience of introducing F^ + j; let us therefore
assume more generally F= 2 . YiM^, F being a function of r of the same
order as Y^ and Mi a function oi ^ . Making these substitutions in (6) ,
which may be written
d.r" ^ d.(l—/j.)-j— /Q-v
10 COLLECTED MATHEMATICAL, WORKS OP G. W. HILL
•In order that the left member may be arranged in a series of the same form
as 2 . ViMi, we must have
3-f = »«'•
in which n is independent of ^u. We may determine n from the considera-
tion that, Vi being of the same order as F* , Mi cannot contain any higher
power of (J. than (i^\ Making J^ = 2 . V^'- ^^^^ relation results :
'^•+'- (2s + l)(2s + 3)*'-
To make this series end at ki, n must equal — 2i (2i + 1) ; and
J. _ (2i - 2s)(2t + 2 s + 1) i. .
•+' ~ (2s + l)(2s + 2) "■ '
hence, putting A;o= 1, which is allowable,
^ _ 1 _ 2t (2t + 1) (2i - 2) 2» (2i + l)(2t + 3) ,
' 1.2 '^ ^ 1.2.3.4 '^ ' • • '■^"-'
^ 2. . .2t(2i + l). . .(4i + l) «
1.3.3...2i '^ •
For Fj we have from (9), by rejecting the sign 2 and dividing by Mi,
the equation
— 3^- 2z (3i + 1) F; + T = 0. <^^^)
(12)
From the expression (10) we easily deduce
Y - ± (^^ + 1)(^^' + 3) ■ • . (4t - 1) r ^ , (^• + l)(4t + 1) ^
1 . 3 . . . (2i - 1) I '^' + 1 . (2i + 1) '^'+^
(t + l)(^• + 2)(4t + 1)(U + 3) p. , -1
1.2.(2i + l)(2i+3) ''H2 + ••■]-.
The inversion of which is
v-± i.3...(2t-i) r J, (i + iX2i + i) y
"^'-^ {2i + l){2i + 3) . . . (4t-l) t ^' + 1 . (4* + 3) ^'+' (13)
(t + l)(t + 2)(2^^ + l)(2i + 3) ^ ,
1 . 2 . (4t + 3)(4i + 5) •+' "^
}■
* The complete integral of this equation when « = — 2i (3t + 1) , i being an integer, is
where K and JS"' are the arbitrary constants.
CONFORMATION OF THE EARTH
11
The upper sign is to be used when i is even, the lower when odd. From
(10) we obtain
Ml=\-i^M, + llM,. (14)
From this and the equation rV (2 . ViM,) = 2 . TiMi , pursuing the approxi-
mation to quantities of the second order, we get these expressions for Ti ,
,--y„=/(F.) + |yiV"(F.)+. . ., \
r-^T, = rj' ( F.) - f T^Y" ( F„) + . . . , I (15)
r-y, = VJ' ( r„) + f^ V,r' ( F„) + . . . J
If quantities of the second order are neglected, the two differential equa-
tions to be integrated are
-^-^_6F + r'F/'(Fo) = 0.
(16)
To pursue the analysis farther would require a knowledge of the form /(F).
13. However, when the point r, jw, o is without the surface of the
earth, (11) can be integrated. Supposing the point not to partake in the
motion of rotation, the centrifugal force must be neglected, and, since p = ,
generally 7'^ = ; consequently (11) becomes
-i^-2U2^+l)F=0. ^''^
The integral of this is
F=a,r-« + 5, + ^_|c,(25.c,-65. + If^ =0,
- he, + h + U<^ (2*oC. - 65, + ^^ = .
From the equation ^ ~ — I = 0, we also have
(33)
dr
F.'+|c.(Frc. + 2FO
-h+^, + fcJ- 66„c, +345, + ^'
37"
F,"c, + F;+f^ c,( F."'c, + 3 F") = 35oC, - 55, + f^ c, (- 65,c + 345, + ^ .
If we make
^^=.^[5,(1 -3c,) + 35,],
equations (22) and (23) can be reduced to the following simpler forms :
5.= - (1 + |y + i? -f ?c, + ^«-0[F.' + f c,(F."'c + 3F,")], 1
K = [(1 -4x^)0.- W - i? + iV?^] *o,
5, = (c,-A5'C + |4c,')5o,
K"ci + T7 - f c C Fo"'Ci+ 2 F") _ [(1 _ |i 5) c, + J^ c,^ - 1 g - ^5^ g^] F.' = ,
F."c, + F; + fl c, ( F."'c, + 3 F,") - (3c, - f | g'c, + || c,») F„' =
(23)
(34)
:::} f^')
15. In this article we shall neglect quantities of the order of g'^- Let
i w denote the quantity obtained by dividing the second member of the
second equation of (23) by the second member of the first equation ; then
w= 3
— 35, + ^,+ 35.c,
- = 6c, + 5-9^;
_ Q ^1
from the second equation of (24), 3ci=3~+ \q:=z e, the approximate
value of the earth's compression, as is clear from the equation
7-=l + C, (1— 3/^^)+ ....
Hence, by addition,
e + w=6c, + fg'-6-^=|^. (26)
The relation enunciated by this equation is known as Claieaut's Theorem.
14 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
16. The equation of the earth's surface is
= (1 + c, + c,-) {1 - [3ci (1 — cO + 10c,-] ix' + s^c,!J.* + . . .] I (27)
= a [1 + ei sin' +6^ sin' 6» + . . . J ; J
in which 6 is the geocentric latitude. If e represents the compression, or
part of its own length by which the equatorial radius exceeds the polar,
e = — ei — 62 = 3ci (1 — Ci) — | Cj . (28)
17. Denoting the normal or astronomical latitude by 6', we have
e' = 0- tan-^ = 0-(er + e, — ^e,') sin 20 - (ie, — i O sin i0 , (29)
the inversion of which is
0=e' + (e, + e,-^ 61') sin 20' + (i e^ + f e^') sin 4:0' . (30)
18. A line geodetically measured on the earth's surface is clearly the
shortest possible ; hence, if ds is the element of the curve, by the principles
of the Calculus of Variations
3 C ds + ISV=0,
/I being the indeterminate multiplier of ^F. Also,
ds' = dr" + 4^2 + J-" (1 — /»') dw' ■
1. fJ.
The coefficients of br , hfi, Sa each equal zero; retaining only that of ^o ,
as sufficient for our purpose, we have
That is, the sine of the angle made by the curve with the meridians varies
inversely as the distance from the earth's axis. Hence,
CONFORMATION OF THE EARTH
15
Taking account only of terms of the first order with respect to q ,
(1 — /J.') /^1—fi' - fi'~ {l — fi' - fj.')i'
which, integrated, gives
1 L±_^ .
1 — ^ — p/*
- i sin- iLj + ^e. l'^-' vi-A'- Vl-l'-/>^' ]
(31)
This, then, is the equation to the curve ; if O/ , W/^ are the extreme values of
0) , and jtfy , (III those of ^tf , h can be found from the expression for an — Oy . If e
is the angle made by the curve with the meridian at the commencement,
then h = cos 6j sin e{1 + e^ sin^0,) , and, as affording an approximate value
of e , we have
, tan d,, cos S, ■ „ , / V
cot e= -. — j^ ( — sm e^ cot (««„—<«,).
sin (u>i^ — u>i) ' ^ II u
For the length of the curve,
which, integrated, gives
If ^ = , this expression gives the length of any arc of the meridian, but
in this case
ds = i^W+¥W = r[l + I' sin^2^ \de,^a{l + e^ smH + e^ sin* e + ^ sin»2« \ d0,
which, integrated, gives
8= G + a
"( 1 + -| + ^^ + I .,) ^ - ^-L + ^ Sin 28 - ^^l^ sin 4/
(33)
19. All areas on the earth's surface, bounded by lines whose equations
are (31), can be divided into a finite number of parts, each contained by an arc
of a meridian, an arc of a parallel of latitude, and a line whose equation is
(31). Let A denote the area of this, then ds being the element of the meri-
dian, A = r Cr cos Q da ds , or, neglecting quantities of the second order,
A= r Ct^ d(i da. If this is integrated along a meridian, the result is
A=a' y [(1 + I eyj /.„-(l + I /x») ^ ] ^o,
16 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
The arbitrary constant G in all these formulas is determined by the condi-
tion that the length or area must vanish when the beginning and end of the
geodetic line coincide.*
20. Let F denote the force of gravity at the surface, then
^=lFVWiJ=IFti.i8*'(i-.-)].
If ^0. ^1. ^2 are the numerical values of the members of equations (23),
then
L Aa + A-i Ao J
= Foll + w,fi' + w,/x'+ . . .].
21. Thus far the general theory of the subject. We shall now assume
some particular law of density. Suppose that the matter of which the
earth is composed is compressible inversely as its density. This gives
<^P = ^ — , m being a constant. Substituting for dp its value pd V, and
• • tyi^
mtegratmg, p = _ F. No constant is added, because it may be supposed
contained in F. Then
/•(F) = m^F-^,
and from (15) generally
T, = m'r' Vr,
but
and thus (11) becomes
— ^^ — [2i (3i + 1) — m'r'-] F, = . (36)
In integrating, the part of Fj involving negative powers of r may be
neglected, since it belongs to F. If a^ is an arbitrary constant, and /^'
denotes the operation -j- — performed 2i times, the integral of (36) is
F; = a,r"-'/"(sinmr).
* Equation (34) in the original memoir is erroneous ; the correct form is given here.
GONFORMATION OF THE EARTH
17
It may also be obtained thus : put (36) under this form
d'rVt r , 2i(2i + l)~\ ^ „
-^ + [m'- ^^ ^ y F; = . (37)
Assume
rVi = P sinmr + P' coamr;
which, by substitution, gives
Make
P ± P'= /?o ± A»-' + Aj--" ± /?.»—+ . . . ;
then this equation results
± 3m(w + 1) /J„+, + [w(w + 1) - 3i"(2t + 1)] ^„ = ,
whence
(w-at)(n + 2t + i)
/J. + i-T 2(w + l)m '^"'
the upper sign being taken when n is even, the lower when it is odd.
Then making f}Q=±m^'a(, in order to agree with the expression
Vi =: ajr^'-y^* (sin mr), we have
(2i-n + l)...{2i + n) . ,
'*»-* 1.2...W.2" "* ''*•' ^ ''
the upper sign having place when 2i — n is of the forms 4v + 2 , 4v + 3 ,
the lower when it is of the forms 4r , Av + 1 .
V= J^, + a„r - y (sin mr) + a,rf' (sin mr) (1 - 3/.') (39)
+ a^^-T^Csin mr) (1 — 10/j?+ ^//i*) + • • ■ >
or, expanding/^' (sinmr) by using (38),
,^ 8tc' , am mr
3
)^-^^-^52i^^](l-3.') (40)
+ a. [(^-i^+ ..)^^-(i-^ -i^)Mr](l - W+ ¥.^) + . . .
22. Since Fcontains a constant, the term —3™ may be neglected except
in finding the value of the density. Moreover, for simplicity, let Oq = 1 and
%r = — E; then, from (36) and (40), we derive
Vo"=(2H-l)m'r,,
F„"'= [2 + (m'- 6)H-\ m'V,,
Fi =a,(3H—l)m'r,,
F/ = a, [3 + (m^- 9)5"] m' F„ ,
Fi" = Oi [m'— 12 - <5m''- 36)5'] m' F„ ,
F, = a, [m=— 35 - (lOm'- 105)ir] m' V, ,
F,' = a, [- lOm^ + 175 - (m*- 65m' + 525)5-] m' F„ . J
(41)
18 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
By substituting these expressions in (21) and (25), neglecting quantities of
the second order, they become, after removing the factor WgFo,
{%H-l)(h—Hc^ =0,
[3 + {m?- 9)fi] «!+ (3 J5" - 1) Ci = f qH.
Whence,
_s g' _s n{ZH- 1) ,,„.
«i - ¥ ^m^m- (_dB - 1) ' "' - ^^m'H'— (3R- 1) * '- -*
23. Represent the volume of the earth by v, its superficial density by
B, its mean density by B'. Then, neglecting quantities of the second order,
V = — ; and, if ( V) denote the value of F at the surface, B = — — ( V).
Since L is the mass of the earth, B' = —^ =^ — - . Hence — ^^ — ^ = ^^ , or,
putting for b^ its value from (24),
V' Sf
m?iV)— 3i2(l + |9;-
If, in the expression for F, we make Z^^=- 1, and, consequently, r-:=\
then
H R'
1 + 2qH-3R(l + |g)'
or
"^~-3R-2q(R'- Ry
R'
To find m we have, since H=
mW,'
i(i-cotm) = l?. (44)
24. In order to test the preceding theory by numerical calculation, we
adopt the following values for q, B, B', the best we can find:
q=^, R = 2.56, 5' = 5.67.
OONFOJIMATION OF THE EARTH
19
We shall mark with an accent the numbers of the formulas from which the
numerical values of the following quantities are obtained :*
(43)'
H =0.7432817,
[ F„"'=
2.39744m= F, ,
(44)'
\m = 146° 27' 56".2 ,
t =2.556307,
V, =
1.3398451m^F„ai,
F' =
1.1675917m'' F„«i ,
(42)'
Jo, =0.0006715666,
Ic, =0.0011111845,
(41)'
F/'=-
-2.99279wi'F„ai,
F =
1.00801m' F^a ,
(41)'
F„"=0.48656m=F„,
[ F' =
3.40342m' Fa^.
To obtain the values of aj, a^, Cj, c^ true to quantities of the order of 5^ by
substituting the preceding in (21) and (25), we have
(21)' 0.7432817ci— 1.3298451ai + 0.0000006695 = ,
(25)' 1.2398451ci + 1.1675917«,— 0.0021543100 = ,
(21)' 0.74328cj - 1.00801aj — 0.0000009039 = ,
(25)' 2.71641c, + 3.40342a2 - 0.0000054038 = .
The solution gives
(h = 0.0006730400 , c^ = 0.0011127213 ,
«2 = 0.0000002964 , c, = 0.0000016180 .
(34)' F = 5o [r~-'+ 0.0005328715r-' (1 - 3//) + 0.0000010445r-^ (l-iOf/+ ^/,*)] ,
(27)' r = a [1 - 0.003350630 sin' 8 + 0.000018877 sin* i?] ,
(28)' e = 0.003331753 = gQQ^ ,
(39)' d' = d + 688."3811 sin 20 - l."3679 sin id ,
(30)' = 6'— 688."3811 sin %0' + 3."6836 sin 4:0',
(33)' s=a [0.998334571 + 0.000832938 sin %0 — 0.000000112 sin 4e] ,
(35)' F= F, [1 + 0.005406990 sin' - 0.000041419 sin* e] .
The following table contains the values of p and of e, the compression
of the surfaces of level, for every tenth of the equatorial radius, calculated
from the equations
R sin mr
f
-^' —
rsinm
- , aui
r
a
f
e
0.0
11.800
0.1
11.673
1 —
3773'"
0.2
11.393
ISSO'"
0.3
10.677
1 —
1242'"
0.4
9.848
919'"
0.5
8.839
733"'
and e = ■
3F
F
7i=3ai
~3
_r'
m
«!
— — coi
mr
mr
r
a
P
s
0.6
7.688
1 - 587'"
0.7
6.437
1 — 489'"
0.8
5.133
1 - 413'"
0.9
3.832
1 - 351'"
1.
3.550
1 —
300'"
* The numbers following this In the original memoir are erroneous ; they are here rectified.
20
COLLECTED MATHEMATICAL WOKKS OF G. W. HILL
MEMOIR No. 4.
Ephemeris of the Great Comet of 1858.
(Astronomische Nachrichten, Vol. 64, pp. 181-190, 1865.)
The coordinates given in the following ephemeris are unaffected with
aberration ; the constant intended to be used is that of Struve. The columns
Aa, A8, contain the excess of the present ephemeris over that used for
comparison.'
f
Wash. Oh
True a
Aa
Trues
Ai
Logr
Log A
1858, June
6
141° 14 9.88
41
—2.66
+24
13
51.41
11
—8.47
0.33754
0.39511
9
141 15 38.45
2.56
24
33
33.59
7.65
0.32900
0.39669
12
141 20 1.24
2.51
24
52
32.08
6.95
0.32024
0.39795
15
141 27 11.39
2.52
25 10 52.97
6.22
0.31124
0.39887
18
141 37 2.40
2.45
25 28 42.01
5.34
0.30198
0.39942
21
141 49 28.00
2.30
25
46
4.62
4.60
0.29247
0.39959
24
142 4 22.65
2.39
26
3
5.98
3.82
0.28267
0.39936
27
142 21 41.87
2.34
26
19
51.10
3.29
0.27259
0.39871
30
142 41 22.12
2.31
26
36
25.07
2.48
0.26219
0.39762
July
3
143 3 20.57
2.29
26
52
52.10
1.93
0.25147
0.39609
6
143 27 35.01
2.21
27
9
17.53
1.32
0.24041
0.39407
9
143 54 3.55
2.15
27
25
46.19
0.72
0.22897
0.39155
12
144 22 44.48
2.22
27
42
23.55
—0.17
0.21716
0.38851
15
144 53 36.34
2.33
27
59
15.31
+0.48
0.20493
0.38492
18
145 26 38.65
2.45
28
16
27.20
1.23
0.19226
0.38074
21
146 1 52.22
2.55
28
34
4.66
1.90
0.17914
0.37595
24
146 39 19.33
2.65
28
52
13.27
2.52
0.16552
0.37052
27
147 19 4.27
2.87
29
10
58.54
3.07
0.15138
0.36440
30
148 1 13.43
3.12
29 30
26.24
3.69
0.13669
0.35756
Aug.
2
148 45 55.13
3.29
29
50
42.07
4.40
0.12142
0.34994
'5
149 33 19.76
3.58
30
11
52.17
5.08
0.10552
0.34149
8
150 23 40.13
4.02
30 34
2.98
5.75
0.08896
0.33214
11
151 17 12.15
4.45
30
57
21.02
6.35
0.07170
0.32181
14
152 14 15.79
5.03
31
21
52.34
6.97
0.05372
0.31041
17
153 15 17.36
5.59
31 47
41.92
7.72
0.03498
0.29785
20
154 20 51.06
6.08
32
14
52.80
8.41
0.01545
0.28402
23
155 31 41.08
6.75
32
43
25.42
9.06
9.99514
0.26878
26
156 48 45.37
7.59
33
13
15.90
9.74
9.97404
0.25196
29
158 13 19.50
8.16
33
44 14.36
10.59
9.95219
0.23337
Sept.
1
159 47 1.10
9.15
34
16
0.63
11.49
9.92968
0.21279
2
160 20 37.81
9.30
34 26
41.02
11.50
9.92205
0.20544
3
160 55 35.02
—9.55
+34 37 21.35
+11.61
9.91437
0.19783
* It sbould be stated that this ephemeria is constructed from the final theory of Memoir No. 6, pp. 35-58.
EPHEMERIS OF THE GREAT COMET OF 1858
21
Wash. Oh
True a
Aa
True S
AS
Logr
Log A
1, Sept. 4
161 31 58.87
— 9.83
+34 47 59!31
+11.79
9.90665
0.18994
5
162 9 56.20
10.14
34 58 32.25
11.90
9.89889
0.18177
6
162 49 34.55
10.44
35 8 57.37
12.12
9.89112
0.17330
7
163 31 2.11
10.75
35 19 11.02
12.34
9.88334
0.16451
8
164 14 27.99
11.01
35 29 8.94
12.57
9.87557
0.15540
9
165 2.13
11.25
35 38 46.22
12.76
9.86782
0.14595
10
165 47 55.45
11.53
35 47 57.10
12.95
9.86011
0.13614
11
166 38 19.94
11.79
35 56 34.79
13.14
9.85247
0.12595
12
167 31 28.74
12.04
36 4 31.28
13.38
9.84492
0.11537
13
168 27 36.18
12.33
36 11 37.43
13.60
9.83748
0.10439
14
169 26 57.89
12.55
36 17 42.34
13.64
9.83019
0.09299
15
170 29 50.86
12.82
36 22 33.31
13.72
9.82307
0.08114
16
171 36 33.54
13.08
36 25 55.69
13.86
9.81616
0.06884
17
172 47 25.71
13.29
36 27 32.35
13.98
9.80949
0.05607
18
174 2 48.55
13.40
36 27 3.32
14.15
9.80310
0.04281
19
175 23 4.32
13.56
36 24 • 5.57
14.25
9.79703
0.02906
20
176 48 36.37
13.68
36 18 12.45
14.34
9.79133
0.01480
21
178 19 48.74
13.74
36 8 53.06
14.42
9.78602
0.00003
22
179 57 5.84
13.83
35 55 32.06
14.56
9.78115
9.98476
23
181 40 51.78
13.72
35 37 28.78
14.63
9.77677
9.96898
24
183 31 29.48
13.64
35 13 57.05
14.74
9.77291
9.95272
25
185 29 19.90
13.49
34 44 4.69
14.74
9.76960
9.93601
26
187 34 40.58
13.27
34 6 53.07
14.74
9.76687
9.91889
27
189 47 44.44
12.94
33 21 17.77
14.79
9.76477
9.90143
28
192 8 38.03
12.56
32 26 8.32
14.73
9.76329
9.88372
29
194 37 19.97
12.03
31 20 10.26
14.76
9.76247
9.86587
30
197 13 39.26
11.52
30 2 6.42
14.61
9.76230
9.84804
Oct. 1
199 57 13.94
10.91
28 30 43.24
14.45
9.76280
9.83042
2
202 47 29.98
10.13
26 44 44.87
14.19
9.76395
9.81324
3
205 43 40.69
9.41
24 43 20.34
13.68
9.76574
9.79678
4
208 44 47.46
8.52
22 25 51.80
13.22
9.76816
9.78137
5
211 49 40.83
7.58
19 52 13.03
12.47
9.77118
9.76736
6
214 57 2.54
6.40
17 2 57.06
11.59
9.77477
9.75514
7
218 5 29.17
5.79
13 59 22.54
10.65
9.77890
9.74507
8
221 13 35.77
4.75
10 43 36.87
9.56
9.78352
9.73748
9
224 19 58.83
3.63
7 18 32.39
8.37
9.78862
9.73264
10
227 23 20.16
2.69
3 47 36.33
7.16
9.79413
9.73070
11
230 22 30.47
2.01
+0 14 34.17
5.45
9.80002
9.73171
12
233 16 30.95
1.28
—3 16 49.39
4.80
9.80625
9.73558
13
236 4 34.32
0.74
6 43 9.47
3.80
9.81279
9.74211
14
238 46 5.66
—0.31
10 1 35.21
2.94
9.81958
9.75101
15
241 20 41.21
+0.09
13 9 57.92
2.17
9.82660
9.76194
16
243 48 7.80
0.39
16 6 52.25
1.76
9.83381
9.77453
17
246 8 21.80
0.55
18 51 33.65
1.24
9.84118
9.78843
18
248 21 27.23
0.53
21 23 50.16
0.96
9.84868
9.80331
19
250 27 33.93
0.62
23 43 55.37
0.90
9.85628
9.81886
20
252 26 56.04
+0.46
25 52 21.11
0.85
9.86395
9.83485
21
254 19 51.19
—0.01
27 49 49.80
1.00
9.87168
9.85107
22
256 6 39.54
0.40
29 37 10.18
1.12
9.87944
9.86734
23
257 47 42.34
0.64
31 15 13.14
1.22
9.88722
9.88355
24
259 23 20.88
1.15
32 44 48.82
1.35
9.89500
9.89958
25
260 53 56.93
—1.58
—34 6 45.50
+1.54
9.90277
9.91538
22
COLLECTED MATHEMATICAL WORKS OF G. W. HILOL.
Wash. Oh
True a
Aa
Trues
AS
Logr
Log A
1858, Oct 26
262° 19' 51.45
—1.98
—35° 21 48.24
u
+1.68
9.91051
9.93087
27
263 41 24.71
2.57
36 30 38.26
1.87
9.91821
9.94602
28
264 58 56.27
3.21
37 33 53.30
2.15
9.92586
9.96081
29
266 12 44.50
3.86
38 32 7.34
2.35
9.93346
9.97521
30
267 23 6.42
4.47
39 25 50.85
2.44
9.94100
9.98923
31
268 30 18.13
5.16
40 15 30.68
2.68
9.94847
0.00285
Nov. 1
269 34 34.73
5.84
41 1 31.33
2.85
9.95587
0.01608
4
272 32 6.07
8.02
43 1 0.91
3.36
9.97760
0.05349
7
275 10 36.95
10.27
44 38 22.12
3.71
9.99857
0.08767
10
277 34 14.07
12.42
45 59 3.78
3.80
0.01876
0.11892
13
279 46 8.19
14.54
47 7 0.96
4.17
0.03815
0.14754
16
281 48 46.92
16.56
48 5 4.46
4.12
0.05676
0.17382
19
283 44 5.42
18.49
48 55 19.62
4.03
0.07462
0.19801
22
285 33 34.41
20.64
49 39 20.44
3.94
0.09176
0.22034
25
287 18 26.11
22.73
50 18 18.84
3.82
0.10821
0.24100
28
288 59 38.31
24.46
50 53 10.72
3.58
0.12400
0.26015
Dec. 1
290 37 56.18
26.31
51 24 39.95
3.41
0.13918
0.27794
4
292 13 55.21
27.98
51 53 22.34
3.24
0.15377
0.29450
7
293 48 3.40
29.38
52 19 46.72
2.98
0.16782
0.30992
10
295 20 42.09
31.00
52 44 16.96
2.51
0.18135
0.32431
13
296 52 8.76
32.41
53 7 12.21
2.35
0.19440
0.33775
16
298 22 37.88
33.86
53 28 48.88
1.92
0.20699
0.35031
19
299 52 22.11
35.15
53 49 20.28
1.67
0.21915
0.36205
22
301 21 32.66
36.28
54 8 67.88
1.09
0.23090
0.37305
25
302 50 19.37
37.35
54 27 51.54
0.68
0.24227
0.38334
28
304 18 50.80
38.34
54 46 10.16
+0.27
0.25328
0.39298
31
305 47 13.68
39.38
55 4 1.96
—0.23
0.26394
0.40200
1859, Jan. 3
307 15 33.18
40.37
55 21 34.49
0.78
0.27429
0.41045
6
308 43 53.24
41.23
55 38 54.59
1.51
0.28432
0.41835
9
310 12 17.04
41.91
55 56 8.15
2.13
0.29407
0.42573
12
311 40 47.05
42.66
56 13 20.59
2.77
0.30354
0.43264
15
313 9 26.42
43.33
56 30 36.49
3.42
0.31275
0.43908
18
314 38 18.98
43.70
56 48 0.05
4.09
0.32171
0.44510
21
316 7 28.53
44.10
57 5 35.13
4.78
0.33044
0.45071
24
317 36 59.54
44.34
57 23 25.61
5.56
0.33894
0.45594
27
319 6 55.45
44.53
57 41 35.30
6.36
0.34722
0.46079
30
320 37 19.46
44.73
58 8.20
7.16
0.35530
0.46530
Feb. 2
322 8 14.64
44.72
58 19 8.24
8.00
0.36319
0.46949
5
323 39 43.18
44.61
58 38 39.00
8.87
0.37088
0.47335
8
325 11 47.56
44.48
58 58 43.52
9.79
0.37840
0.47692
11
326 44 30.89
44.19
59 19 24.71
10.72
0.38575
0.48022
14
328 17 56.99
43.80
69 40 44.85
11.74
0.39294
0.48325
17
329 62 10.91
43.29
60 2 46.12
12.77
0.39996
0.48604
20
331 27 18.49
42.63
60 25 30.70
13.82
0.40684
0.48860
23
333 3 25.63
41.78
60 49 0.77
14.89
0.41357
0.49094
26
334 40 38.55
40.75
61 13 18.80
15.92
0.42017
0.49308
Mar. 1
336 19 3.23
39.57
61 38 27.12
17.08
0.42662
0.49503
4
337 58 45.67
38.18
62 4 27.82
18.24
0.43296
0.49681
7
339 39 52.15
—36.55
—62 31 22.53
—19.44
0.43916
0.49843
RECTANGULAR COORDINATES OF THE SUN
23
MBMOIE No. 5.
On the Reduction of the Rectangular Coordinates of the Sun Referred to
the True Equator and Equinox of Date to those Referred to the
Mean Equator and Equinox of the Beginning of the Tear.
(AstronomlBche Naciricliteii, Vol. 67, pp. 141-143, 1866.)
In computing an ephemeris of any planetary body, it is quite the
easiest plan to get the heliocentric rectangular coordinates referred to fixed
planes, as those defined by the mean equator and equinox of the beginning
of Bessel's fictitious year, either of the current year or of the nearest tenth
year. Then, by the addition of the sun's coordinates referred to the same
planes, to obtain the geocentric rectangular coordinates, and from thence to
proceed to the corresponding polar coordinates, which may be very readily
changed to the true equator and equinox of date by using the three star
constants f,ga.To.d G.
But the coordinates of the sun hitherto published in the various ephem-
erides have not been rigorously reduced to these planes.
The following method of reduction is offered as being quite simple,
since it involves only the star constants in addition to the coordinates them-
selves.
Let B denote the sun's radius vector and a, h its true right ascension
and declination referred to the mean planes of the beginning of the year,
and a', h' the same referred to the true planes of date, and let X, Y, Z,
X\ T', Z' be the corresponding rectangular coordinates.
Whence result these relations
X= R cos 5 cos a , X'—R cos &' cos a',
F= jR COS 3 sin a , F'= i? cosS' sin a',
Z-R&\-a.S, Z'=R sin d'.
Through subtraction, in which we can neglect all but quantities of the
first order with respect to the small differences a' — a and B' — S, since the
error which results in the values of X, Y and Z is less than half a unit in
the seventh decimal place, we get
X- X'= R cos S' sin a' («'- a) + R sin d' cos a' (d'— d) ,
Y— Y'=—R COSd' COSa'(a'—a) + R Sin 3' sina'(5'— 3),
Z - Z'=- RcosS' (5'— g) .
24 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
But, from the well known formulas for the reduction of the fixed stars, we
have
a'—a = aA-irbB-i-E and S'- 8 = a'A ^VB ,
in which
a = wi + « sin a' tan S', a'= n cos a',
b — cos a' tan 8', 5' = — sin a'.
Making these substitutions, we shall obtain
X-X'={mY'+nZ')A+ Y'E,
Y- Y'= — mX'A - Z'B — X'E,
Z—Z'=-nX'A+Y'B.
Since mA -|- ^ is usually denoted by /, and we may write A' instead
of nA = g cos G and B =:■ g sm G, our equations may be written
X- X' = fY'+ A'Z',
Y- Y'=:—fX'—BZ',
Z-Z'=-A'X'+ BY'.
In most of the ephemerides /, log B and log A are given ; then to the last
add log of n expressed in seconds of arc ; /, A' and B being thus expressed in
seconds of arc, it will be most convenient to add to their logs the constant
log 1.68557, whence the reductions above will be expressed in units of the
seventh decimal place.
If it is required to reduce the coordinates to the equator and equinox
of the beginning of a year previous to or following the current one, it is
only necessary to increase, in the first case or diminish in the second, the
value of A by the requisite number of units. This, however, must not be
too large, otherwise the quantities of the second order may become sensible.
In computing the ephemeris of a planet, if we have not the mean co-
ordinates but only the true coordinates of the sun, it will evidently be a
saving of labor, to employ the formulas above to reduce the heliocentric co-
ordinates of the planet from the mean to the true equinox and equator of
date, and not those of the sun in the opposite direction.
ORBIT OP THE GREAT COMET OF 1858 ' 25
MEMOIE No. 6.
Discussion of the Observations of the Great Comet of 1858, with the
Object of Determining the Most Probable Orbit.
(Memoirs of the American Academy of Arts and Sciences, Vol. IX, pp. 67-100, 1867.)
Communicated hy T. H. Bafford, April 12, 1864.
The interesting physical aspect of this comet attracted to it, in an unusual
degree, the attention of astronomers, a large part of whose energies were ex-
pended in obtaining observations for position. Consequently, we have a large
mass of material for determining its orbit, not a little of which is of very good
quality. Added to this, the long period of the apparition of the comet
(nine months), would enable us to obtain the elements with considerable pre-
cision. Moreover, hints were thrown out that some other force besides
gravity might affect its motion. Although these seem to have had no foun-
dation other than the fact that the orbits derived from three normals did not
well represent the intermediate observations, yet it is a matter of some
interest to clear up the suspicion.
As the first step in the work, I determined to reduce the observations
to uniformity, in respect to the places adopted for the comparison stars ;
which last I proposed to derive from all the material accessible to me. The
desirableness of this course is evident when we consider that the observers
at Bonn, Kremsmiinster, Ann Arbor, and the two observatories in the south-
ern hemisphere reobserved their comparison stars, in consequence of which
their observations agree much better among themselves ; while the rest
contented themselves with places from Lalande, Bessel's Zones, or the
British Association Catalogue, and their results exhibit larger probable
errors. And as the comet was observed nearly simultaneously in Europe,
the same comparison star was frequently used by a dozen observatories for
the same night's work ; and thus the stars of the latter class of observatories
mentioned above are often found among those reobserved by the former.
The result of this labor has convinced me that it has not been wasted ; the
good effect is apparent, particularly in the Liverpool and Gottingen obser-
vations.
4
26 COLLECTED MATHEMATICAL WOEKS OF G. W. HILL
A catalogue of all the stars used for comparison having been formed,
the following authorities were consulted for material :
Baily's Lalande, Piazzi, Bessel's Zones (Weisse's Reduction), Struve
Catalogus Generalis, Taylor, Riimker, Argelander's Southern Zones (Oelt-
zen), Robinson's Armagh Catalogue, Johnson's Radcliffe Catalogue, Green-
wich Twelve Year and Six Year Catalogues, Madler, Greenwich Observa-
tions, 1854-1860, Henderson Edinburgh Observations, Challis Cambridge
Observations, Leverrier Paris Observations, 1856-59.
Leverrier commenced, in 1856, to reobserve the stars of Lalande ;
hence quite a number of the stars the observers had taken from this source,
were found in the Paris Observations. The searching them out and reduc-
ing them entailed considerable labor. In addition to the material before
mentioned, that furnished by the observatories at which the comparison
stars were reobserved, was, of course, not omitted.
All this material was reduced to 1858.0, and to the standard of Wolfer's
Tabulae Reductionum, by applying the systematic corrections given by Au-
wers, in Astr. Nachr., No. 1300, with the modifications suggested by Mr.
Safford, in No. 1368. The systematic corrections for Robinson are found in
Astr. Nachr., No. 1408. Also, the following, kindly furnished by Mr. Saf-
ford, were employed :
B. A. DEC.
Greenwich Six Year Catalogue, . . . -|-0'.017
Greenwich Observations, 1854-60, . . .-(-0.027 -1-0. "70
Paris Observations, 1856-59, .... -1-0.056 +0. 19
In a few cases, mostly Piazzi stars, where the observations indicated
proper motion, it was taken into account. With regard to the stars used in
the southern observations, those common to the northern being excepted,
they were retained without change, or when the same star had been used at
both observatories, the observations were combined, allowing a weight of
3 to the Cape and of 2 to the Santiago observation. However, the place of
the Santiago star, No. 57, equivalent to Cape No. 95, is wrong, seemingly
an error of reduction ; hence the Cape place has been adopted. And San-
tiago, No. 49, differing 7". 5, in declination, from its equivalent. Cape No. 87,
the Cape declination appearing the better, has been retained.
ORBIT OF THE GREAT OOMET OF 1858
27
S 1858.0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
h. m s
9 11 35.277
9 23 19.992
9 25 52.434
9 29 26.949
+25 59.98
25 2 12.96
24 5 6.93
25 1 49.81
9 29 41.987
25 18 21.93
9 30 47.635
26 34 35.94
9 32 23.230
26 38 48.99
9 33 27.857
9 37 42.094
9 37 47.087
9 38 33.273
26 33 26.79
27 41 55.28
24 25 33.49
27 34 38.43
9 38 42.482
9 44 17.169
9 45 49.118
9 45 51.470
9 46 34.633
27 48 43.19
28 26 25.61
28 21 41.66
27 57 29.24
28 1 10.89
9 48 45.001
9 49 3.501
9 50 10.023
9 51 24.453
28 46 15.38
29 14 1.93
29 15 28.20
30 19 26.44
9 53 8.109
9 56 54.727
29 27 50.85
30 26 9.00
9 58 59.212
30 12 16.94
10 3 36.641
10 6 0.703
30 50 50.42
32 7 41.13
10 6 56.647
32 10 17.05
10 8 9.965
10 9 27.359
30 58.15
31 35 38.88
10 9 50
10 10 27.200
31 8 36
31 19 36.16
10 12 33.258
10 12 45.450
32 8 25.35
31 22 26.94
10 14 12.375
10 14 47.246
32 15 26.28
31 2 47.57
10 14 56.648
10 16 57.222
31 33 9.66
31 5 41.78
10 23 37.058
31 46 9.62
10 23 47.154
10 25 56.094
10 26 29.210
33 6 25.56
33 14 35.93
32 24 43.22
10 27 27.290
32 30 36.72
10 29 41.545
33 28 13.95
10 29 46.127
33 25 30.26
10 30 43.132
32 42 45.11
10 34 4.401
10 34 13.347
33 53 25.44
32 26 21.00
10 35
34 10
10 35 11.612
10 36 27.312
10 37 50.279
34 6 20.68
33 21 49.84
33 20 33.31
10 38 50.569
34 18 20.60
10 39 45.817
10 44 6.512
34 20 17.23
+ 32 7 11.86
No.
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
a 1868.0
S 1868.0
h m s
10 44 8.277
10 45 21.594
+33°
34
47 57"50
58 45.71
10 47 3.944
34 47 31.18
10 47 51.964
34
15 49.07
10 52 35.910
35
13 36.25
10 56 35.054
35
7 11.70
10 59 36.820
35
36 32.64
11 45.884
35
29 1.63
11 1 29.939
37
4 43.39
11 1 58.610
35
40 37.52
11 2 24.790
36
6 12.73
11 4 16.855
35
46 40.87
11 4 37.860
35
33 27.80
11 10 16.610
36
13 5.54
11 10 48.100
33
52 6.00
11 11 4.961
36
15 52.34
11 13 48.264
36 25 24.60
11 14 24.766
36
6 48.63
11 17 49.011
35
56 46.68
11 19 30.334
36
32 58.19
11 20 16.048
36
9 7.63
11 22 8.300
36
25 12.10
11 27 39.248
36
11 24.50
11 28 14.746
36
42 40.78
11 29 52.400
36
23 30.10
11 30 28.154
36
23 31.78
11 31 6.811
36
23 1.60
11 33 33.925
35
12.15
11 38 8.137
36
40 53.08
11 41 22.161
35 37 17.78
11 42 18.698
35
43 13.14
11 48 39.684
36
7 52.28
11 48 57.507
36
14 16.51
11 54 23.109
36
50 12.34
11 55 23.490
36 31 5.11
11 57 25.064
36
21 29.55
11 59 22.626
36
7 52.04
12 8 41
36
2
12 9 21.473
33
51 20.73
12 14 5.054
35
28 35.44
12 18 0.818
35
33 5.54
12 23 36.015
34
32 7.24
12 24 3.679
34 40 32.25
12 24 38.593
34
42 4.90
12 26 38.907
34
1 58.92
12 30 5.468
33 48 31.59
12 40 14.318
33
20 42.67
12 44 8.808
32 15 8.42
12 48 56.000
32
46 19.64
12 49 22.827
39
5 10.26
12 53 28.505
31
33 8.05
12 53 38.459
32
32 45.83
12 55 34.619
+31
7 17.08
28
COLLECTED MATHEMATICAL WORKS OF G. W. HILL
No.
a. 1858.0
S 1858.0 1
No.
a 1858.0
: 1858.0
h
m
B
/
«
b
m
s
o / //
107
12
57
5.635
+31 31 16.16
160
14 59
35.593
+ 6 19 28.56
108
12
57 16
31 14
161
14 59
57.698
6 54 50.52
109
12
57
26.035
30 58
58.19
162
15
33.375
6 49 9.03
110
12
59
23.612
29 47
28.59
163
15
4
21.645
3 22 6.91
111
13
21.817
28 23
16.41
164
15
5
11.046
7 10 34.03
112
13
2
21.623
31
9.03
165
15
8
54.017
6 59 39.72
113
13
2
45.467
31 11
36.59
166
15
12
35.554
+ 3 51 2.73
114
13
7
53.393
30 9
19.48
167
15
17
4.230
— 2 17.19
115
13
9
5.077
30 5
55.53
168
15
20
28.516
— 6 57.69
116
13
10 14.299
29 47
44.00
169
15
20 44.108
+ 23 21.61
117
13
12
20.788
29 18
25.90
170
15
23
56.400
— 14 16.79
118
13 18 20.109
24 35
44.78
171
15
30
20.818
3 7 57.60
119
13
20
10.842
26 59
50.88
172
15 33
46.943
3 31 59.42
120
13
21
46.769
28 5
9.80
173
15
37
0.458
— 3 23 9.11
121
13 22
2.800
29 11
20.02
174
15
37 16.575
+ 6 52 30.92
122
13
23
8.620
28 24
36.87
175
15 41
30.788
— 3 22 46.67
123
13
23
45.303
28 23
16.90
176
15
43
44.425
+ 4 54 29.08
124
13
25
28 20
177
15
44 11.680
— 7 36 47.93
125
13
30
3.869
26 36
19.12
178
15
44 33
6 53
126
13
33
22.650
26 38
49.62
179
15
46
54.770
7 40 54.50
127
13
37
33.182
26
5.60
180
15
52
4.738
6 53 37.22
128
13
40
7.651
26 24 59.35
181
15
52
26.783
6 42 53.62
129
13
44
19.729
24 20
51.41
182
15
53
7.954
8 23.10
130
13
45
56.310
24 15
58.17
183
15
55
1.103
10 13 57.47
131
13
46
12.072
24 2
8.33
184
15
56
33.959
10 58 40.88
132
13
46 46.651
24 51
40.80
185
16
21.617
13 22 56.05
133
13
51
39.354
24 38
30.49
186
16
41.310
9 42 57.87
134
13
51 59.705
22 23
26.35
187
16
2
59.572
14 27.35
135
13
54 25.217
22 39
58.50
188
16
3
6.618
13 36 59.00
136
13
55
20.693
22 14 33.72
189
16
4
24.098
13 22 3.31
137
14
7
56.650
19 9
59.50
190
16
4
41.570
10 6 50.01
138
14
9
11.160
19 55
24.82
191
16
5
42.980
12 40 2.27
139
14
9
23.644
19 34 29.31
192
16
5
59.266
16 22 13.69
140
14 11
14.667
19 6
2.59
193
16
6
12.084
13 37 42.35
141
14
13
2.053
16 57
35.06
194
16
6
29.082
10 3 1.35
142
14 17
27.790
16 55
11.16
195
16
6
59.660
14 16 29.96
143
14 20
0.953
17 3
22.25
196
16
8
32.770
13 17 23.70
144
14
21
31.064
16 45 49.85
197
16
10
3.544
13 5 23.42
145
14
23
11.387
16 50 40.76
198
16
11
34.281
16 8 19.19
146
14 28
12.905
13 43
16.55
199
16
14
44.967
16 40 51.94
147
14 33
46.108
13 52 14.60
200
16
20
9.964
15 53 23.81
148
14 33
55.070
14 8
48.84
201
16
23
0.959
16 17 57.48
149
14
34
22.174
14 20 23.45
202
16
23
43.537
21 9 29.66
150
14 34
54.307
12 16
29.77
203
16
30
18.736
18 32 9.72
151
14 39
4.926
13 42
18.62
204
16
34 32.419
21 29 43.20
152
14
42
33.357
10 38 27.29
205
16 34 36.256
21 4 1.58
153
14
42
47.985
10 47
39.30
206
16
37
12.274
18 52 12.21
154
14
44 10.403
10 18 35.81
207
16 40
7.990
21 41 3.72
155
14
44
35.229
10 35 48.49
208
16
41
6.008
24 23 11.15
156
14
51 54.337
7 10
15.57
209
16
41
7.071
21 35 54.42
157
14
57
3.620
6 3
17.78
210
16 41
50.606
24 15 51.12
158
14
58
1.431
7 15 39.93
211
16
52
37.309
26 25 39.09
159
14
58
12.582
+ 6 51 17.26
212
16
53
3.366
—27 43 31.09
ORBIT OF THE GREAT OOMET OF 1858
29
No.
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
a 1863.0
S 1858.0
16 53 10.777
13 20 26.52
16 55 4.596
28 2 57.96
16 55 31.065
28 22 0.37
16 57 44.663
28 3 54.25
16 58 39.016
27 55 53.96
16 59 24.776
27 54 39.07
17 5 10.403
29 52 34.77
17 5 37.478
29 41 14.99
17 6 47.473
30 2 30.62
17 8 16.677
30 8.39
17 9 20.175
29 42 53.91
17 10 7.169
31 12 16.83
17 12 14.840
31 25 56.61
17 13 4.999
31 26 22.48
17 17 17.581
32 50 3.02
17 19 44.388
32 52 53.83
17 23 0.424
34 10 1.01
17 23 54.929
34 16 20.54
17 29 16.449
35 21 48.14
17 31 4.961
35 33 46.60
17 33 12.654
36 52 6.28
17 34 26.669
36 42 1.87
17 40 15.178
37 28 49.04
17 41 32.567
37 45 43.95
17 44 36.555
38 35 8.95
17 45 57.092
38 38 45.01
17 50 27.549
39 13 45.64
17 50 39.138
39 39 2.66
17 54 38.324
40 38 8.40
17 55 11.130
40 26 50.86
18 2 23.867
41 44 28.49
18 5 14.187
41 56 26.36
18 5 36.073
43 12 19.51
18 7 1.414
42 30 48.85
18 7 5.615
42 15 28.83
18 8 31.282
42 20 5.76
18 10 43.566
43 49 49.10
18 10 52.913
43 1 59.55
18 11 7.779
42 37 40.29
18 12 9.145
42 59 37.64
18 12 36.869
42 39
18 13 58.237
44 10 30.94
18 18 12.611
44 14 43.36
18 18 54.896
43 55 46.90
18 21 39.243
44 41 8.96
18 27 50.763
45 34 44.15
18 33 11.685
46 18 24.55
18 35 45.167
46 43 42.28
18 36 12.466
46 31 17.59
18 41 53.432
46 45 22.59
18 43 23.840
—47 26 22.74
No.
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
2S3
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
a 1868.0
18 44
10.419
18
44 28.573
18 46
26.978
18
46
31.226
18
48
13.570
18
49
54.520
18
52
54.066
18
53
55.641
18
54
10.142
18
56
41.428
18
59
22.395
19
3
52.330
19
5
52.858
19
6 44.688
19
12
11.637
19
14 33.567
19
19
10.963
19
19
18.450
19
23
22.183
19 23
34.615
19
26
53.202
19
29 44.020
19
30
15.827
19
30
35.665
19
33
0.830
19
33 15.841
19 34 26.520
19 38
11.319
19
39
33.763
19 40 57.485
19
42
1.927
19
45
4.527
19
45
36.756
19 50
33.215
19 50 43.777
19 56
47.461
19
57
18.613
20
15.512
20
2 33.653
20
5 17.695
20
6
48.770
20
9
15.916
20
11
41.803
20 15
45.780
20
16
25.769
20
17
13.505
20
18 46.916
20
19
7.030
20 21
57.156
20 22 58.968
S 1868.0
-47° 49' 47.56
47 47 17.12
47 45 18.57
47 34 3.31
48 9 23.70
48 28 21.45
48 54 32.28
48 36 18.11
48 51 11.31
49 14 21.17
49 32 0.29
49 46 24.01
50 13 41.25
49 42 18.51
50 30 20.14
50 46 57.51
51 3 5.09
51 16 7.09
50 51 50.34
51 34 45.35
51 45 , 7.15
51 51 59.82
51 50 49.35
52 5 43.17
52 8 8.42
52 16 22.64
52 21 40.38
52 25 21.16
52 35 4.72
52 47 37.75
52 40 19.22
53 10 20.41
53 4 53.37
53 21 50.75
53 12 39.53
53 30 37.25
52 58 52.81
53 45 3.25
54 1 31.30
54 11 0.36
54 14 53.47
54 29 49.18
54 42 28.61
54 16 41.37
54 39 2.12
54 45 46.37
55 33 8.81
55 2 3.04
54 59 26.98
-54 56 2.85
30
COLLECTED MATHEMATICAL WORKS OF G. W. HILL
No.
314
a 1859.0
li m 8
20 25 26.390
1 1859.0
o / //
—55 3 20.79
No.
339
a 1869.0
h m 8
21 23 0.430
» 1859.0
-68° o' 17.76
315
20 27 7.650
55 18 29.95
340
21 23 22.890
57 42 4.33
316
20 27 13.290
55 24 33.43
341
21 25 2.280
58 4.65
317
20 31 23.834
55 36 23.14
342
21 28 35.340
58 20 30.78
318
20 33 40.930
55 36 2.15
343
21 29 54.810
58 4 24.05
319
20 34 34.990
55 41 47.50
344
21 30 52.172
58 22 23.85
320
20 38 21.560
55 43 23.42
345
21 32 9.160
58 15 1.52
321
20 39 58.730
55 53 23.14
346
21 33 18.394
68 32 14.83
322
20 43 12.100
56 6 49.19
347
21 33 39.750
58 28.20
323
20 44 29.930
55 59 27.04
348
21 33 57.470
57 56 21.35
324
20 45 35.640
55 45 12.47
349
21 35 8.380
58 41 34.64
325
20 47 1.230
56 14 47.53
350
21 37 51.400
58 40
326
20 47 55.460
56 20 9.53
351
21 40 18.106
58 67 17.60
327
21 1 2.480
57 5 12.43
352
22 8 41.970
60 32 20.95
328
21 2 14.520
67 5 6.31
353
22 8 48.310
60 57 36.81
329
21 4 50.880
57 8 13.18
354
22 9 36.790
60 49 14.47
330
21 8 3.260
57 18 3.90
355
22 11 13
61 8 11.00
331
332
21 10 45.750
21 11 6.720
57 12 15.77
57 26 34.46
356
357
22 12 6.080
22 16 36.870
60 39 16.90
61 5 50.15
333
334
21 12 39.970
21 14 20.600
57 23 55.34
57 51 22.09
358
359
22 18 40.420
22 21 12.300
61 17 31.70
61 13 40.79
335
21 18 19.350
57 45 21.06
360
22 23 53.980
61 32 27.09
336
21 20 26.300
57 29 5.95
361
22 25 40.970
61 40 32.46
337
21 20 48.940
57 46 28.59
362
22 27 25.610
61 43 53.15
338
21 21 55.210
—57 55 14.83 r. _ .
363
22 30 54.250
—61 57 58.99
The following are the authorities for the observations and the places of
the comparison stars :
Altona. Astr. Nachr., L. 187.
Ann Arbor. Astr. Nachr., XLIX. 179. Brunnow's Astr. Notices, I. 6, 53.
Armagh. Monthly Notices, XIX. 305.
Batavia. Astr. Nachr., L. 107.
Berlin. Astr. Nachr., XL VIII. 333, LI. 65.
Bonn. Astr. Nachr., XLIX. 253, LI. 187.
Brbslau. Astr. Nachr., L. 37.
Cambridge, Eng. Astr. Nachr., L. 243.
Cambridge, U. S. Astr. Nachr., LI. 273. Briinnow's Astr. Notices, I. 71.
Cape of Good Hope. Mem. Astr. Soc, XXIX. 59-83. The observations were made with
two different instruments ; those made with the larger have been denoted in the list
of observations which follows by "Cape 1," and those made with the smaller by
"Cape 2."
Christiania. Astr. Nachr., LII. 277.
Copenhagen. Oversigt kgl. danske Videnskabernes Selskabs, 1858.
Dorpat. Beob. Kaiserl. Sternw. Dorpat, Vol. XV. These observations are published in
a crude form, and I was unable to reduce and use them, from a want of the instru-
mental constants.
Durham. Astr. Nachr., L. 11.
ORBIT OF THE GREAT COMET OF 1858
rLOEENOE. Astr. Nachr., XLVIII. 347, 355, XLIX. 57, L. 97. The observation of Octo-
ber 13 is erroneous as regards the comparison star, which it seems should be Piazzi
XV. 227.
Geneva. Astr. Nachr., XLIX, 115, L. 21.
GoTTiNQBN. Astr. Nachr., XLIX. 235, L. 11.
GEBBirwiCH. Greenwich Observations for 1858. Monthly Notices, XIX. 12.
KoNiGSBEEG. Astr. Nachr., L. 71, LIII. 289.
Kkemsmunster. Astr. Nachr., XLIX. 68, 79, 257, LI. 23.
Leyden". Astr. Nachr., L. 157. The observer is mistaken in the comparison star of his
last observation ; it should be Weisse, XV. 369.
Liverpool. Astr. Nachr., XLIX. 267. Monthly Notices, XIX. 54.
Markree. Observations on Donati's Comet, 1858, at Markree.
Padua. Astr. Nachr., XLVIIL 357.
Paris. Annales de I'Observatoire Imperial, Paris. Tome XIV. Observations.
PuLKOVA. Astr. Nachr. L. 307. Beobachbungen der Grossen Cometen 1858. Otto
Struve.
Santiago. Astr. Nachr., LIII. 131. Astr. Jour., VI. 100.
Vienna. Astr. Nachr., XLVIII. 349, XLIX. 43, 53, L. 227, LII. 57.
WiLLiAMSTOWN. Astr. Nachr., L. 7. As the latitude and longitude of the place are
uncertain, I have not reduced these observations.
Washington. Astr. Nachr., XLIX. 55, 113, 363. Astr. Jour., V. 150, 158, 166, 180.
The comparison star of October 1 is mistaken.
The typographical errors to be met with are so numerous I cannot
undertake to mention them. To render the reduction of the comparison
stars from mean to apparent place uniform, the elements of reduction in
the British Nautical Almanac for 1858 were adopted as the standard ; and
the same will be used in reducing our normals from apparent to mean places.
Consequently, it becomes necessary to add to the observations in which the
elements of the Berlin Jahrbuch were used, quantities easily obtained from
this small ephemeris.
June
15
+ 0.09
+ 0.18
Sept.
18
+ 0.08
+ 0.03
July
15
+ 0.02
+ 0.22
Oct.
3
+ 0.07
-0.04
Aug.
14
+ 0.03
+ 0.18
Oct.
18
+ 0.04
-0.19
Sept.
3
+ 0.05
+ 0.10
Nov.
2
+ 0.14
-0.23
For the reduction of the observations for parallax, and the computation
of the perturbations, and for comparison, an ephemeris was computed from
these elements published by Searle in the Astronomical Journal, V. 188,
Searle's own ephemeris not being sufficiently exact for the purpose of com-
parison.
T= Sept. 29.75230 1858 Washington Mean Time
^ — gj = 129 6 24.8]
g2 = 165 18 46.2 [ Mean Equinox and Ecliptic 1858.0
i = 116 57 46.1 J
9'= 85 21 21.2
log ^ = 9.7622362
32
COLLECTED MATHEMATICAL WORKS OF G. W. HILL
In the following list the observations of the comet are given reduced
for parallax, and are made to accord with the places of the comparison stars
given in the foregoing catalogue. Gould's list of Longitudes (in the Ameri-
can Ephemeris) has been used in getting the Paris M. T. of Observation.
The comparisons in the last two columns are Obs. — Cal. The declinations
of the southern observations have generally been reduced to the time of
observing the right ascension ; that observation of right ascension being
selected which was nearest in time and which had the same comparison star.
Paris M. T.
Place of
S
Number ot a „
Oomp. Star " °
Ad
of Observation
Observation
a
June 7.41071
Florence
141° 14 47.79
+24° 2l' 54.73
3
+21.69
+ 6.26
8.37659
"
141 15 36.99
24 27 52.30
10
+39.17
—15.66
9.42802
"
141 16 20.54
24 34 48.42
10
+27.71
— 7.33
10.39044
"
141 17 25.48
24 41 10.00
10
+23.44
+ 5.67
11.40973
"
141 19 3.43
24 47 35.12
1
+28.89
+ 5.05
12.37591
Padua
141 20 31.82
24 53 36.67
4
+11.55
+ 5.14
12.41803
Florence
141 20 21.71
24 53 56.68
1
— 3.35
+10.15
13.37729
Padua
141 22 34.98
24 59 27.99
4
+ 6.81
—14.10
13.40557
Florence
141 22 16.33
25 14.83
1
—15.72
+22.34
13.43268
Berlin
141 22 43.08
24 59 50.15
2
+ 7.30
—12.31
14.41069
"
141 24 58.54
25 5 52.65
2
— 0.89
— 7.69
14.41609
Vienna
141 25 15.40
25 5 55.69
2-5
+15.13
— 6.61
15.39007
Florence
141 28 20.08
25 11 23.30
5
+37.49
—31.86
15.40675
Vienna
141 27 58.18
2-5
+14.65
15.44201
Berlin
141 27 36.29
25 12 2.13
5
— 9.96
— 4.80
16.39944
Kremsmiinster
141 30 54.96
25 17 48.71
5
+10.41
— 8.58
16.41628
Berlin
141 30 39.41
25 17 49.11
5
— 8.36
—14.19
17.39261
Florence
141 34 31.47
25 23 26.85
5
+28.40
—23.47
19.37441
"
141 41 42.14
25 35 32.96
5
+12.05
+ 7.07
19.38451
Padua
141 42 8.91
25 35 39.64
5
+23.63
+10.24
28.38292
Florence
142 29 25.62
26 26 8.67
8
+26.28
— 6.86
28.61976
Cambridge, U. S.
142 30 24.43
26 27 36.77
6
— 7.25
+ 2.80
29.38224
Florence
142 35 56.97
26 31 43.74
8
+22.16
— 2.41
29.41947
Berlin
142 36 2.15
26 31 52.88
6
+13.31
— 5.57
30.37599
Florence
142 42 46.34
26 37 8.61
8
+22.98
— 5.44
30.38577
Vienna
142 42 24.72
26 37 20.26
8
— 2.72
+ 2.98
ruly 2.37816
Florence
142 56 57.17
26 48 14.56
7
+ 4.77
+ 1.86
8.38159
"
143 46 55.06
27 20 55.54
11
+34.65
— 9.74
9.38324
Vienna
143 55 34.04
27 26 44.86
12
+ 6.24
+ 6.85
9.60789
Washington
143 57 42.00
27 27 56.51
11
+ 9.40
+ 6.19
10.37333
Florence
144 5 0.20
27 32 2.33
11
+16.80
— 1.59
10.59343
Washington
144 6 50.62
27 33 16.24
9
+ 1.75
— 0.72
10.59343
"
144 6 59.45
27 33 18.41
11
+10.58
+ 1.45
11.59576
"
144 16 28.58
27 38 51.84
9
— 0.61
+ 1.42
12.37144
Florence
144 24 2.74
27 43 13.41
9
— 5.59
+ 3.93
13.37158
"
144 34 23.43
27 48 47.14
12
+10.19
+ 2.20
13.59089
Cambridge, U. S.
144 36 26.15
27 50 6.11
12
— 1.67
+ 7.56
14.36879
Florence
144 44 34.08
27 54 18.74
12
+ 3.24
— 2.50
14.58534
Washington
144 46 51.57
27 55 38.77
9
+ 4.72
+ 4.34
15.58781
Cambridge, U. S.
144 57 23.93
+28 1 20.09
15
— 1.44
+ 4.97
ORBIT OF THE GREAT COMET OF 1858
33
ParlB M. T.
ol Observation
1868
Place of
ObserTatlon
a
s Number of .
" Oomp. Star " "
A
To pass to any new system, we shall have the known equations
x' ^= ax -^ hy -^ Gz , "j
y'=a!x-\-Vy^dz, \ (2)
But in the case where we wish to obtain the differentials of a; , ?/ , z for an
infinitesimal time dt, a, V and c" are each unity, being the cosines of angles
infinitely small ; and all the other constants will contain dt as a factor.
Hence we may write
4§ = al'x + V'y .
(3)
The equation x^ + f + z^ = 1 gives 'isx^ + 2/^ + 2^ = 0- Sub-
stituting in this the above values of ~ , etc., there result these three equa-
tions of condition between the six remaining constants
b + a'=0, c + a"=0, c'+h"^0. (4)
60
Hence,
COLLECTED MATHEMATICAL WORKS OF G. W. HILL
dx _
by + cz,
dy
'di~
— bx + dz.
dz
dt~
— ex — c'y .
(5)
It belongs to Celestial Mechanics to deduce the values of the three
remaining coeflBcients of these equations. When precession alone is con-
sidered, c' = 0, and — b and — c are the quantities usually denoted by m
and n . Thus we have, the unit of t being one year,
dx
^^=-my-nz,
dy
dz
~dt
= nx.
(6)
If the values of x, y and z are now substituted in these equations, we find
that
da ■ J. »
~ = m + w sm a tan d ,
dS
dt
i.t)
m and n are functions of t which admit of being expressed by power series.
Differentiating (7) and always eliminating ^ and -=- by means of the
at €vt
primitive equations, we obtain
d'a _ dm n'
dt'-~dt'^
-g- sin 2a + (
dn .
dt
")
sin a + mn COB a] tan S + n' sin 2a tan'' S ,
dn
TTj- = — mn sin a + -Tj cos a.—vi? sin" a tan & ,
d?a mn'' , , , n . X <^^ ■ a
-j^ = -g- + f mw cos 2a + f « TT sm 2a
(2n^ — m'+ Sn' cos 2a) m sin a +I2m-^ + w-yrjcosa tan 5
+
3mn' cos 2a + 3w ^ sin 2a tan'' S
+ 2m' sin a (1 + 2 cos 2a) tan' S ,
dt
5 /„ dn dm . , , , • , ^
i= — I 2m-T7 + w-^sina — (m'+ w" svo.^a)n cos a
- f OTw' sin 2a + 3wtj sin' a tan S
— Sw' sin' a cos a tan' <5 .
(8)
DERIVATION AND REDUCTION OF PLACES OF THE FIXED STARS gj
In writing these equations, it has been assumed that -^ and -^ vanish.
The right ascension and declination of a star, as far as regards preces-
sion, are then found by the formulas
dtt\^^dti\<.^ 1
«=-+(a).' + i(S).'' + i(S).''+
\.
(9)
2. Let us next consider the effect of proper motion. If the values of
-5- and -5- for any star are obtained from observation for a certain epoch,
we may compute the functions m + re sin a tan h and n cos a, and subtract
them from these quantities, the remainders ^ and ^i' are the effect of proper
motion in right ascension and declination at that epoch. But, to deduce the
values of ^ and ^' for any time in general, we may adopt the assumption
that the proper motion is uniform on the arc of a great circle, and on this
supposition derive the rigorous values of the differential coefficients of a and
h with respect to the time.
Considering now the effect of proper motion only, let
p denote the velocity of the star's motion on the arc of a great circle,
j(^ the angle of position of this arc,
a' and h' the right ascension and declination of the star at the end of
the time t.
The consideration of the spherical triangle formed by the pole of the
equator and the two positions of the star will give these equations,
sin 5'= sin S cos {pt) + cos 8 sin {pt) cos/, -v
cos 8' cos (a' — a) = COS 8 COS {pt) — sin 8 sin {pt) cobx> \ (10)
cos 8' sin (a'— a) = sm{pt') sin;^ . 3
Eliminating p and x by means of the equations
;o sin;^ = iOi cos5, pcoax = p-',
we derive from the first and third of the preceding equations the following
values of a' and S' in series arranged according to the powers of t :
a'=a + fjLi + fip.' ta.nS.f-^\_/M' sm''8-p.p.'\l + 3tanU)-]f+ . . ., 1
8'=S + ,x't-\fi'sm28.t'-}tiy{l + 2sm'8)f+... J ^^^^
62
COLLECTED MATHEMATICAL WORKS OF G. W. HILL
(12)
3. In order to have the combined effect of precession and proper
motion, a' and 8' should be substituted for a and 8 in the series whicn give
the effect of precession. Hence, we obtain
da ■ , ,.
-jj z= m + n sm a tan S + /j. ,
dd
^ — n COS a + fi' ;
and, (I and fi' being considered as variable quantities,
dfi
di'-
dt^' ■ 1 2 ■ OS.
-TT = — n/i Sin a — ^ At'' SID 2ff .
It may be useful to note the rate of variation of the angle of position ;u
through the effects of precession and proper motion ; it is
nix cos a tan S + w/ sin a sec" S + 3^/ tan d ,
(13)
-^ =n sm a sec o + fj. am
(14)
By differentiating the values of -^ and -=- , and eliminating jt > ^ > -Jr
du'
and -^ by means of their values just given, we obtain
€tt
(fa dm n' . „ n / •
+ jT sin a + (m + 2/i) n cos a + %ixn' tan 8
+ 2w sin a (n cos a + fi') tan' d ,
d'S . dn „ . m" . .,
J ™ = — mm sin a + -^ cos a — 3w^ sin a — -^ sin is
— n' sin" a tan 5 ,
^ = '^' + 2/./.'"+ 3 §/ sin a + 3V (m + 2/i) cos a
+ f (wi + 2fj.) n" cos 2a + f w ^j sin 2a — "Up.' sin" 8
(2 w"— to"— 6m"— 3TO/i + 3>i" cos 2a) n sin a
(^ dn dm „» \ ,02^-0
"•"( "5^ ■*" " "ST "^ W^ j ^°^" "*" '' sin 2a
+
tand
+
6/x/"+ 3 ^Z sin a + (3m + 12/n) w/ cos a
dn
tan" 5,
+ 3 w ^ sin 2a + (3m + 6/i) w" cos 2a
+ [(2w"+ 6/i'")w sin a + Gw"/ sin 2a + 4w" sin a cos 2a] tan' 5,
^, = - Ai"/i' — (2m + 3/*) ^ sin a — (m" + 3m/i + 3/*") w cos a
— w -jx sin a — w' sin" a cos a — 3w"Ai' sin" a — 3^"/ sin" 8
at
V dn ~\
— 6w/iAt' sina + f (m + 2/i)w" sin 2a + 3«-^ sin" a tan5
— 3 (« cos a + ii) w" sin" « tan" 8 .
(15)
DERIVATION AND REDUCTION OF PLACES OP THE FIXED STARS
63
The values of a and S, computed by means of Maclaurin's Theorem, using
the above values of the differential coefficients, will give the mean place of
the star. The last term of -=-g- and also that of -j^ are nearly always insen-
Cut Gut
sible.
The expressions for -=-3 and ,-3 above are too complicated for use in
Civ a*
computation ; hence, if their values are wanted, it will be much easier to
compute the values of the second differential coefficients for 50 years before
and after the epoch, and divide the differences of those by 100 for the value
of the third differential coefficients at the epoch.
4. We have next to consider the effect of nutation. Resuming equa-
tions (5), putting for x, y and z their values from (1) in terms of a and h ,
and writing Aa and AS instead of -^ and t- , we obtain.
at at
Aa= — b — csina tan 8 + c' cos a tan S ,-t
AS =z — c cos a — c' sin a . / ^ ■'
Changing the notation so as to correspond with that usually employed in
this subject, we make
l=-mA'-E, c=-nA', c'=:B, (17)
where A' is the quantity usually denoted by A with the term r, the frac-
tion of the year, omitted. Then
Aa = (m + n sin a tan S) A'+ B cos a tan d + E ,\
Ad = A'n COB a — JB sina. J
These formulas give the effect of nutation when terms multiplied by the
squares and products of A', B and E are neglected.
The following formulas contain those which involve the squares and
products oi A' and B, still neglecting the square of E and its products with
A! and B as of no moment :
Aa-^^^nA + ^A'dB^ ^ ^ ^ dB" ^ '
(19)
We have from (18)
d.Aa . ^ ^ d.AS
-T-j7-=m + M sina tanff, -jjT- = n cos a,
d.Aa , , d.AS
n r, = COS a tan S , ,p == — sin a .
(30)
(54 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
Differentiating these again with respect to A! and B, and eliminating
da da
dl'' dB
^-ji , -Yn , etc., which are the same as j ^, , jp , etc., we obtain
dA' ' dB
d?Aa n'
■j-jn = "o sin 2a + mw cos a tan S + if sin 2a tan' 8 ,
J AijT, = w cos'' a + n cos 3a tan" 5 — m sin a tan S ,
-yw = — f sin 2a — sin 2a tan'' 8 ,
-T-jTz = — mn sm a — w sin'' a tan 5 ,
J AijT ) = — 2 Sin 2a tan S — m cos a ,
d''A8
els'
— cos" a tan 5 .
(21)
It will be sufiBcient to retain in A^a only the terms multiplied by tan* h,
and in A^5 those multiplied by tan ^, and to put A' =^ — 0.34236 sin SJ
= sin g2, and 5 = — 9". 2235 cos g^ = — m cos g2 , where Q, denotes
the longitude of the moon's ascending node. Thus we get
J''a =
A''8 — -
-o" COS 2a sin 2fJ r — sin2a cos 2SS tan' 8 ,
'uv . „ . „ ^ /m'— v' V? +
-^sin2a sm2 SJ +( — g 1 g-
— cos2aj C0S2SJ tan^.
(22)
Hence, if we put
a = ^ (m + m sin a tan 5) , h-=^ cos a tan 8 ,
a'= n cos a , J'= — sin a ,
(23)
the formulas for the whole effect of nutation will be
+ [O'.OOOOIOS cos 2a sin 2 SJ -O'.OOOOIO? sin 2a cos 3Q] tan' 5 ,
A8 = ciA'-vl'B
— [0".000077 sin 2a sin 2 S3 + (0".000023 + 0".000080 cos 3a) cos 2 ft ] tan 5 .
(24)
5. The effect of aberration is next to be considered. If a! and V denote
the right ascension and declination of the star as affected by aberration,
while a and h denote the same unaffected by aberration, and -t— , ^r- and
■^ dt dt
-=- denote the velocity of the earth projected on the three axes of coordi-
DERIVATION AND REDUCTION OF PLACES OF THE FIXED STARS
65
nates, and h denote the velocity of light, we have, R' being a fictitious dis-
tance to be eliminated.
1 liY
R' COS d' COS a'=: COS 5 COS a + i^ ,
K at
1 dY
R' COS 8' sin a = cos 5 sin a + ^ ^^ .
K dt '
R' sin S'
= sin
+
IdZ^
h dt ■
Whence are derived
R' cos 8' sin (a' — a) = — ^ (^^^° " ~ ^°°® ") »
R' cos 5' cos (a' — a) — COS 5 + j^f-^-COS a +
dY
dt
sin
•).
R' sin {8'— 8) == — ^ f^sin 5 cos a + ^sin 5 sin a - ^cos s]
k\dt dt dt j
1 , JdX . dY V
^' COS (5'- 3) = l+y^cos5cosa+^cos5sina+^sin3]
. 1 IdX . dY V
From which, to quantities of the second order, we have
1 » fdX
a' — u. = — ^ sec 5 -^^
k \dt
dY \
sm a ^ cos a j
1 , . IdX .
^ sec' 5-^ sin a
dt
dY„ „ ydX , dY . \
^COSaj^^COSa+^Sinaj,
3'— 5 = - i f~ sin 3 cos a + 4? sin 3 sin « - ^cos 8]
fc\ dt dt dt J
1 , , fdX . dY
-W^'"' 'U '''"'— dt'
-1
+ -Ti -^rr sin 5 cos a + -rr sin 5 siu a rr COS 5 )
^(dX . dY . . , dZ
X -jT cos 5 cos a + -rr- COS 5 sm a +
^^ dt
sin 5
(25)
(36)
(27)
\dt — • — - ' (ii; — "'" (^^
If /• is the radius vector of the earth, the sun's true longitude and o the
obliquity of the ecliptic,
X=— rcosO, Z= — r sin O cos w, Z= — rsinOsinw. (38)
And, if e denotes the eccentricity of the earth's orbit, T the longitude of the
solar perigee and n the mean sidereal motion of the sun,
dr an
dt V 1 - ,
Jo
an
dt
VI — e'
iesin(o — r),
[1 + e cos iO—r)^.
(39)
66 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
Whence we derive
tl. "y nvt.
■ [sin O + e sin r] ,
^ cos sin3asin3G]
In these formulas, terms multiplied by x^e have been neglected, as also the
terms in 6' — 8 multiplied by x^ which are not also multiplied by tan S .
Substituting for x Struve's value 20". 4451, these formulas become
a' — az=— 30".44:51 sec S [sin a sin O + cos o cos = — [1.31059] sin © ,
Aq= — [9.53457 + 0.4?;] sin Q, + [7.6128] sin 2Q, ,
Bq_^ - [0.96490] cos Q, + [8.9518] cos %Q, ,
IIq= — [7.4951 — 6.6;;] sin Q. .
(43)
The term Eq being neglected, we write
cO + dD + /IT , '
(44)
Jga = aAQ + IBq + cC + dD + /tr ,
Aq8 = u'Aq + I'Bq + c'G + d'D + ii't.
DERIVATION AND REDUCTION OF PLACES OF THE FIXED STARS
69
To Aqk and Aq8 should be added the terms of the second order in
aberration, and to A^a and Aq5 the terms of the second order in nutation
whenever they are sensible.
If we make
p Q^— [1.31059] d + [6.5942] a - [7.9609] b ,
q ©=—[1.27313] c + [7.4644] a — [7.2370] 5,
p ,o== — [8.4012 ] a + [5.7922] sec' d cos 2a ,
q ,0= — [9.7410 ] b — [5.7938] sec" 8 sin 2a ,
p' Q= - [1.31059] d' + [6.5942] a'— [7.9609] b',
?' ©= - [1.37313] c' + [7.4644] a'— [7.2370] b',
yjQ= — [8.4012 ] a'- [6.6673] tan d sin 3a ,
g',0= - [9.7410 ]b' — [6.6688] tan S cos 2a + [5.6042] tan d ,
we shall have, terms of the second order included,
:^ O sin o + g cos O + PiQ sin 2© + g^Q cos 5
, ■ ^ , , ■ c, t c,^ d8 ,d'^d ,
■■ p'q sm O + g-'o cos © + p'iQ sin 3© + ? 2© cos 2© + -^^t + | ^^ t .
A a =p Q sin Q + q qcos Q + p^Q sin 2© + q^Q cos 2© + ^^ t + 1 ^, r%
zf©5 =
Let us make
Pq=1cq cos Eq ,
qQ—lcQ sin Eq , q q— k q am
PiQ= he cos E^Q , p'2Q= Jc\q cos E'
q^Q= k,Q sin E^Q , q\Q= k\Q sin
p' Q— k' cos E' ©
q' Q— k' © sin X' © ,
q\Q= k\Q sin E\q .
Then equations (46) take the form
J©a = f?r + l^rH * © sin (© + ^©) + ^^O sln (2© + Z,©) ,
(XT " Cut
/I©^ = ^r + i^^'^- k'Q sin (© + E'q) + Td^Q sin (2© + ^,©) .
(45)
(46)
(47)
(48)
10. To compute the variations of Aga and Aq^ for a certain interval of
time as 10 years, we compute the variations ofp©, g©, etc., in that interval;
calling them 5^©, ^g-©, etc., and certain very small terms being neglected,
we have evidently these equations :
5 . Jga = 5po sin © + 8qQ cos © + 8p^Q sin 2© + Sq<,Q cos 2©
+ 10^"r+ kQ C0S(©+ ^©)5©,
5 . Aq8 = dp'Q sin © + Sq'Q cos © + 8p\Q sin 2© + Sq'^Q cos 2©
+ 102r + ^'GCOs(©+^'©)5©.
(49)
(50)
70 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
The value of 50 is [6.0057] sin (0 — 15°) ; substituting this, we have
a . Jo« = 10 ^ r - [5.7047] h Q sin {Xq+ 15°) + dpQ sin O + ^fe cos ©
+ [5p,0 + [5.7047] A ©COS (^0-15°)] sin 2©
+ [.Sq^Qi- [5.7047] ^©sin (_Kq- 15°)] cos 2© ,
S.AqS = 10^,t -[5.7047] ^q sin (K'q+ 15°) + S^'q sin © + Sq'q cos©
+ [.¥^Q + [5.7047] k'Q cos (E'q- 15°)] sin 2©
+ [VsO +[5.7047] ¥q sin (Z'©- 15°)] cos 2© .
As in the case of Aga and A©5 , these quantities can be made to take
the form
d . AQa = a + lz + TiQ sin (© + Hq) + A,© sin (2© + H,q) , -i ,..^-.
8.AQS = a' + Vt + A'© sin (© + H'q) + A',© sin (2© + ^',©) .] ^ ^
Except for stars near either pole, the first and last terms of these equations
may be neglected, and regard be had in computing S^q, Iq^, etc., only to
the variations o^ c , d, d and dJ in the formulas for 2>© , g'© , etc. Then
5./I©a = 10£r+ A© Sin (©+5-©),
d^
5./J©5 = 10^r+ A'o sin (© + ^'©) ,
(52)
11. In computing A©a and A©5, we may either suppose Tcq, Kq, Jc'q
and S^Q constant throughout the year, and afterwards add to A©a and
Aq5 thus obtained, the proper fractional part of Aq sin (0 + Sq) and
A'© sin (0 + H'q) for the fraction of the year ; or, we may make them vary
from date to date. For a star, whose declination is within the limits ±65°,
there is, however, no need to attend to this correction.
Having formed a table of for every 10 sidereal days, beginning with
the fictitious year, we can readily get for the time of the star's transit
„ 18*40'"
over the fictitious meridian with the constant interpolation factor . — .
^ 240* '
and thus form the arguments + ^q, 20 + K^q, + Eq, etc. Terms
with small coefficients can be most readily formed by means of a Traverse
Table.
1 2. We can reduce Aj^a and Aq5 to the forms, terms of the second
order included,
^fi« = ^n sin (Q + Eq) + h^ sin (_2Q, + K^^) , \
■A^d = k'a sin (SJ + Z'n) + Jc',a sin (2£J + K',,^) . / ^^^''
DERIVATION AND REDUCTION OF PLACES OP THE FIXED STARS
71
by making
k QCoaK Q = — [9.53457 + 0.4;;] a — O'.OOSl,
* n sin ^ n = - [0.96490] b ,
h ,n cos K^a= [7-6128 ] a + [5.0114] cos 3a tau' S ,
h 2Q sin ^j n = [8.9518 ] i - [5.0294] sin 2a tan'' $ ,
k' Q cos ^' n = - [9.53457 + 0.4^;] a',
y Q sin ^' j^ = — [0.96490] V,
*',Q cos ^',n = [7.6128 ] a'— [5.8865] sin 2a tan 3 ,
Tc\^ sin Z^'jfj = [8.9518 ] V— [5.9031] cos 2a tan 5-[5.3617] tan <
But perhaps it will be as well to adopt the formulas
(54)
or,
■}
(55)
(56)
^n«=/n+^nsin(G'n+a)tand, -,
^n^= 5'ncos((?n+ a). j
13. Tor stars near either pole, it will be well to construct tables giving,
with the arguments Q + Q, and © — Q,, the values of the small terms in
(33) and (34). These will be most readily computed with the aid of a Tra-
verse Table, when they have been reduced to the forms
^G+Q« = ^ o+a sin (O + £2 + iT Q+jj) ,
AQ+^d = k'Q+n sin (Q+Q,+ X's^.^) ,
^G-a« = *o-a sin (0—^ + ^0-n) ,
Aq_^S = k'Q_n sin (0- ^ + E'Q^a) ,
14. Tables for A^a and A^h may be computed in the same way.
by making
/fc d cos ^ + d) = a,,
r sin
(w + 0-) = Oi ,
r sin 11}
COS. -«xY-
r sin {u> + d) = a„,
r p.OH til
sin ^ ,
cos 0- sm 1 , there will be only two real roots .
We will now show how to arrive at a direct solution of the problem by the
employment of trigonometric formulas. If tan c is taken for the unknown
quantity, the equation, on which the solution of the problem depends, is
[c cos /J tan
— c' (1 + sin 2/3) , always negative;
— c" sin 2/3, negative or positive, according to the sign of sin 2/3;
+ c" (1 — sin 2/3) , always positive .
Moreover, it is plain that there is one real value of fi, which makes
sin 2(1 and sin 2/1? have like signs ; this value we shall adopt.
Making, according as c^ is greater or less than unity, c^ =: sec^ y or
c? = cos^ y', the above cubic is solved by these formulas (see Chauvenet's
Trigonometry, p. 96), it being necessary to make three different cases.
Case I.
. 2sinVtan)' j. , j. i 9> a 3, ,„,
tan «> = j^^ . — ^ , tan = tan » -— , sm 2/* = -7^ tan r cot 24> .
V 27 sin 2/3 2 V 3
Case 11.
2 sin / tan" /.,,,*' .„ 2.,
siny= /-g^g^^g^ . tani& = tani-|-, sin 2/t = --;^ sm / cosec 2^ .
Cflwe ZZ7.
'"^ ^' = 2W^' ' ""^ ^'^ = 73 ^^"^ '-' «i" (^ ± ««°) •
When ^ is impossible in Case TI, the formulas of Case III must be
used ; and the upper or lower member of the double sign in the second
equation must be taken according as sin 2^ is positive or negative, in order
that sin 2^ may have the same sign with sin 2/3. All the auxiliary angles
^, 1^ and fi may be taken between the limits ± 90°. Since sin 2/if sin 2^1
is always positive, tan /? tan (i and tan /? cot (i are so likewise, since they are
respectively equivalent to
sin 3/3 sin 2/t , sin 2/3 sin 2/*
and
4 cos" /3 cos" II 4 cos" ^ sin" ix '
11
82 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
Let US take two auxiliary angles 6 and B', determined by the equations
tan* /? tani fi cos /J cos 2fi
sm 29 = ■
sin25'=.
sin fi cos (1^ + fi) '
tan* j9 cot* /t cos /? cos 2fi
cos /^ sin (/3 — /t) '
or by the equations
. „^ co8 2fi /sin3/S
s^° ^^ = T cos (/? + ^) Visa;;:'
. „„, cos2/i /sin2/J
sm2g = T gi„(^_^) Vsm2:;r'
where the upper or the lower of the signs must be taken according as -. — ^
in the first and 1- in the second are positive or negative ; and 2d and 26'
cos jU
may also be taken within the limits ± 90°. The four values of x or tan a
are then
tan ? denoting the corresponding latitude. The formulas used are
given in Watson's Theoretical Astronom'jf, pp. 153-159.
In the following equations we have put
x=AL^ -'Jls.in'^AQ,', y = lQOAn', z=Ae', u = e' (a::' ..^2 ain'-^ AQ'V
all expressed in seconds of arc ; and x', y', z' and u' denote the corresponding
quantities in reference to the solar elements. In the computation of the
coefficients of the last, roughly approximate formulas have been used.
A mean of the Transits of 1761 and 1769 gives
+ 0.992a; - 0.839y + 1.61^ + 1.17w + 1.00a;'- 0.84/ + 0.832'- 1.82m'= + 1".745 .
THE ORBIT OF VENUS 95
The indeterminate correction of the Sun's semi-diameter nearly disappears
from this mean.
The following equations of condition are numbered with the same num-
bers as the normals from which they are derived. The last column contains
the residuals which remain after the elements have been corrected as shown
in the sequel.
Equatio;ns of Condition,
No.
Residuals.
X
— 0.40ar
+0.05J/
— 0.36«
—1.44m
+1.43i»'
—0.191/'
—0.21s'
— 3.O61*'
= +l'.()l
+o'.'97
2
—1.37
+0.18
—0.87
—2.97
+2.41
—0.32
—1.45
—4.69
= —0.95
—1.02
3
—2.05
+0.28
-0.87
—4.16
+3.08
—0.41
—2.17
—5.74
= —0.69
—0.74
4
—2.07
+0.28
—0.02
—4.28
+3.11
—0.41
—2.65
—5.57
= +3.37
+3.37
5
—0.80
+0.11
+0.31
—2.15
+1.80
—0.24
—2.22
—3.16
= +1.44
+1.43
6
—0.31
+0.04
—0.42
+1.41
+1.30
—0.16
+1.86
+2.32
= +1.27
+0.68
7
—0.98
+0.12
—0.93
+2.31
+1.98
—0.24
+3.56
+2.59
= —0.23
—1.25
8
—2.27
+0.27
—2.31
+4.06
+3.27
—0.39
+5.84
+3.12
= +1.48
—0.38
9
—2.44
+0.29
—3.04
+3.85
+3.40
—0.40
+6.26
+2.85
= +3.13
+1.22
10
—1.70
+0.20
—2.53
+2.69
+2.70
—0.32
+5.18
+2.00
= +1.13
—0.27
11
—0.90
+0.11
—1.76
+1.56
+1.91
—0.22
+3.95
+1.01
= +0.96
+0.08
12
—2.06
+0.21
+3.53
—2.38
+3.08
—0.32
—6.12
—0.14
= —1.66
—2.07
13
—2.51
+0.26
+4.64
—2.02
+3.51
—0.36
—7.01
+0.39
= +0.29
—0.34
14
—2.00
+0.17
—4.12
—0.54
+3.00
—0.26
+4.57
—3.99
= +0.22
—0.91
15
—2.09
+0.18
—4.05
—1.48
+3.10
—0.27
+4.28
—4.47
= +4.08
+3.09
16
—1.12
+0.10
—2.39
—1.18
+2.12
—0.14
+2.72
—3.49
= —0.59
—1.14
17
—2.69
+0.19
+3.74
+3.87
+3.69
—0.26
—1.82
+7.26
= —2.09
—4.20
18
—1.58
+0.11
+1.80
+2.98
+2.58
—0.18
—0.65
+5.38
= —0.63
—2.10
19
—0.27
+0.01
—0.63
—1.18
+1.27
—0.07
+0.21
—2.76
= —0.81
—0.95
20
—2.40
+0.13
—0.41
—4.82
+3.40
—0.18
—2.47
+6.18
= —0.57
—1.14
21
—0.47
+0.02
—0.47
+1.64
+1.47
—0.06
+2.19
+2.48
= —0.98
—1.79
22
—1.54
+0.06
—1.31
+3.16
+2.54
—0.10
+4.37
+3.00
= —1.19
—2.93
23
—1.95
+0.07
—2.61
+3.15
+2.95
—0.11
+5.79
+2.51
= +1.84
—0.20
24
—0.40
+0.01
+1.03
—1.15
+1.40
—0.03
—2.79
—1.18
= +0.42
+0.24
25
—2.29
+0.05
+3.87
—2.60
+3.28
—0.07
—6.58
—0.29
= —2.95
—3.92
26
—0.55
+0.01
+1.78
—0.38
+1.55
—0.03
—3.33
+1.04
= +3.15
+2.75
27
—2.22
+0.02
—4.51
—0.53
+3.22
—0.02
+4.94
—4.09
= +2.47
+0.72
28
—1.22
+X).01
—2.59
—1.14
+2.22
—0.01
+3.02
—3.46
= +0.04
—0.85
29
—1.55
—0.01
+2.84
+1.97
+2.55
+0.02
—2.07
+4.94
= —2.24
—3.81
30
—2.73
—0.03
+3.75
+3.93
+3.72
+0.04
—2.06
+7.28
= —0.41
—3.18
31
—0.88
—0.01
+1.17
+2.02
+1.88
+0.02
—1.96
+4.10
= +4.28
+3.11
32
—2.18
—0.05
—1.24
—4.31
+3.18
+0.08
—1.77
—6.06
= —0.38
—1.64
33
—1.26
—0.03
+0.04
—2.92
+2.26
+0.06
—2.00
—4.22
= +2.07
+1.48
34
—0.44
—0.01
+0.19
—1.60
+1.44
+0.04
—1.92
—2.50
= +1.59
+1.39
35
—0.68
—0.03
—0.51
+1.96
+1.68
+0.07
+2.58
+2.70
= +0.46
—0.68
36
—1.37
—0.06
—1.02
+2.96
+2.37
+0.10
+3.95
+3.09
= +0.71
—1.20
37
—2.43
—0.10
—2.15
+4.44
+3.42
+0.14
+5.83
+3.70
= +2.52
—0.59
38
—0.54
—0.03
+1.14
—1.33
+1.54
+0.09
—3.10
—1.20
= —0.19
—0.52
39
—2.27
—0.13
+3.71
—2.77
+3.27
+0.19
—6.49
—0.59
= +0.25
—1.20
40
—2.32
—0.13
+4.26
—2.04
+3.27
+0.19
—6.59
—0.07
= +1.34
—0.27
41
—0.46
—0.03
+1.64
—0.39
+1.46
+0.09
—3.12
+0.99
= +2.17
+1.73
42
-1-0.13
+0.01
—0.89
+0.28
+0.87
+0.06
+1.91
+0.52
= +1.01
+0.85
43
—0.25
—0.02
—1.34
+0.16
+1.24
+0.09
+2.71
—0.68
= —0.68
—1.18
44
—1.37
—0.10
—3.10
+0.07
+2.37
+0.17
+4.12
—2.66
= +0.01
—1.57
96
COLLECTED MATHEMATICAL WORl
SS OF (
G. W. HILL
No.
Beslduals.
45
—2.17a!
—0.162/
— 4.30Z
— 1.15W
+3.17a;'
+0.233/'
+4.76«'
—4.18k
.' = +i:'79
-d'.3l
46
—0.86
—0.06
—2.06
—0.91
+1.86
+0.14
+2.50
—3.03
= +0.55
—0.31
47
—0.41
—0.03
—1.38
—0.72
+1.41
+0.11
+1.57
—2.63
= +1.49
+1.01
48
+0.28
+0.02
+0.81
—0.23
+0.72
+0.06
—1.45
—0.59
= —2.32
—2.18
49
+0.13
+0.01
+0.92
+0.17
+0.87
+0.08
—1.91
+0.43
= —0.21
—0.26
50
—0.05
0.00
+1.06
+0.48
+1.04
+0.09
—1.96
+1.41
= —0.98
—1.26
51
—0.80
—0.07
+1.95
+1.04
+1.79
+0.16
—2.15
+3.30
= +0.98
—0.08
52
—2.13
—0.19
+3.57
+2.60
+3.10
+0.28
—2.37
+5.96
= +4.52
+1.97
53
—2.58
—0.23
+3.49
+3.83
+3.54
+0.32
—2.09
+6.93
= +3.16
—0.06
54
—1.28
—0.12
+1.70
+2.46
+2.26
+0.20
—0.80
+4.79
= +0.13
—1.67
55
—0.48
—0.04
+0.81
+1.49
+1.47
+0.13
+0.19
+3.29
= +1.94
+1.03
56
+0.09
+0.01
—0.55
—0.78
+0.92
+0.09
+1.02
—1.78
= +0.55
+0.58
57
—0.17
—0.02
—0.62
—1.07
+1.18
+0.12
+0.55
—2.81
= +0.44
+0.27
58
—1.03
—0.11
—1.08
—2.31
+2.05
+0.21
—0.57
—4.25
= +0.95
+0.14
59
—2.27
—0.24
—1.37
—4.41
+3.29
+0.35
—1.62
—6.32
= +1.02
—0.62
60
—0.52
—0.05
+0.10
—1.72
+1.52
+0.16
—1.83
—2.76
= +2.87
+2.49
61
—0.06
—0.01
+0.16
—1.09
+1.07
+0.11
-1.89
—1.47
= +2.61
+2.58
62
+0.07
+0.01
—0.02
+1.02
+0.92
+0.11
+0.26
+2.12
= —0.58
—0.88
63
—0.12
—0.01
—0.16
+1.21
+1.11
+0.13
+1.00
+2.37
= +0.07
—0.48
64
—0.62
—0.07
—0.36
+1.89
+1.61
+0.19
+2.35
+2.74
= +0.72
—0.48
65
—2.12
—0.26
—1.60
+4.07
+3.07
+0.37
+5.13
+3.73
= +1.87
—1.35
66
—2.11
—0.26
—2.45
+3.60
+3.07
+0.37
+5.44
+3.15
= +4.21
+1.04
67
-0.23
—0.03
—0.97
+0.86
+1.22
+0.15
+2.71
+0.18
= +0.87
+0.23
68
+0.03
0.00
—0.80
+0.55
+0.96
+0.12
+2.05
—0.56
= +0.23
—0.06
69
+0.09
+0.01
+0.47
—0.82
+0.92
+0.12
—1.17
—1.65
= —0.45
—0.54
70
—0.17
—0.02
+0.73
—1.00
+1.18
+0.16
—2.14
—1.49
= +0.89
+0.73
71
—0.41
—0.05
+0.98
—1.22
+1.41
+0.19
—2.79
—1.34
= +3.22
+2.87
72
—1.19
—0.16
+1.98
—2.02
+2.20
+0.30
—4.49
—1.05
= —1.61
—2.63
73
—2.29
—0.31
+3.67
—2.89
+3.29
+0.45
—6.57
—0.78
= +0.77
—1.19
74
—1.13
—0.16
+2.56
—0.97
+2.14
+0.29
—4.55
+0.41
= +3.09
+1.95
75
—0.31
—0.04
+1.41
—0.36
+1.31
+0.18
—2.77
+1.06
= +3.64
+3.24
76
+0.13
+0.02
—0.88
+0.33
+0.97
+0.15
+1.93
+0.85
= —1.06
—1.21
77
—0.48
—0.07
—1.66
+0.23
+1.64
+0.25
+3.39
—0.84
= +0.13
—0.74
78
—1.09
—0.17
—2.64
+0.21
+2.25
+0.34
+4.25
—1.90
= +1.03
—0.58
79
—2.05
—0.32
—4.24
—0.17
+3.16
+0.48
+5.41
—3.27
= +1.30
—1.37
80
—2.51
—0.39
—4.99
—0.64
+3.50
+0.54
+5.80
—3.80
= +3.15
+0.05
81
—1.84
—0.28
—3.75
—1.02
+2.67
+0.41
+4.44
—3.24
= +3.72
+1.53
82
—0.78
—0.12
—2.03
—0.71
+1.64
+0.25
+2.66
-2.47
= +2.94
+1.95
83
—0.45
—0.07
—1.47
—0.65
+1.27
+0.20
+1.89
—2.22
= +1.89
+1.30
84
—0.07
—0.01
—0.98
—0.51
+0.94
+0.15
+0.91
—2.01
= +1.51
+1.32
85
+0.04
+0.01
+1.02
+0.20
+1.08
+0.18
—2.25
+0.55
= +0.69
+0.53
86
—0.24
—0.04
+1.28
+0.45
+1.39
+0.23
—2.49
+1.68
= +0.95
+0.41
87
—0.51
—0.09
+1.62
+0.66
+1.67
+0.28
—2.61
+2.47
= —0.10
—1.00
88
—1.36
—0.23
+2.76
+1,48
+2.52
+0.43
—2.92
+4.38
= +2.78
+0.78
89
—2.36
—0.40
+3.97
+2.75
+3.45
+0.58
—3.24
+6.23
= +5.62
+2.27
90
—1.95
—0.33
+2.75
+3.06
+2.73
+0.46
—2.12
+5.43
= +4.99
+2.07
91
—0.20
—0.03
+0.64
+1.14
+1.02
+0.17
+0.16
+2.49
= +2.58
+2.00
92
+0.24
+0.04
—0.01
+0.86
+0.66
+0.11
+1.23
+0.92
= +0.53
+0.47
93
+0.04
+0.01
—0.58
—0.82
+0.97
+0.18
+0.98
—1.93
= —1.45
—1.45
94
—0.14
—0.02
—0.64
—1.01
+1.15
+0.21
+0.65
—2.46
= +0.68
+0.49
95
—0.35
—0.06
—0.76
—1.27
+1.36
+0.25
+0.34
—2.95
= +1.31
+0.90
96
—0.81
—0.15
—1.05
—1.93
+1.83
+0.34
—0.20
—3.87
= +0.68
—0.18
97
—1.75
—0.32
—1.47
—3.45
+2.78
+0.51
—1.00
—5.52
= +2.86
+1.13
THE ORBIT OF VENUS
97
No.
BesldualB.
98
— 2.47i»
—0.46.1/
— 1.39.«
— 4.75W
+3.490!'
+0.652/'
—1.61s'
— 6.68it'
■ = +2'.'36
+0"02
99
—2.10
—0.39
—0.73
—4.28
+3.12
+0.58
—1.72
—5.98
= +2.60
+0.64
100
—1.05
—0.20
—0.11
—2.57
+2.06
+0.38
—1.67
—3.99
= +0.79
—0.19
101
—0.60
—0.11
+0.02
—1.85
+1.61
+0.30
—1.70
—3.02
= +1.48
+0.92
102
—0.35
—0.06
+0.06
—1.47
+1.35
+0.25
—1.75
—2.40
= —0.92
—1.26
103
—0.09
—0.02
+0.11
—1.13
+1.09
+0.20
—1.82
—1.62
= +0.20
+0.12
104
+0.12
+0.02
+0.09
+0.96
+0.87
+0.17
—0.09
+2.01
= +0.46
+0.23
105
—0.09
—0.02
—0.10
+1.21
+1.10
+0.22
+0.87
+2.40
= +0.79
+0.26
106
—0.32
—0.06
—0,20
+1.48
+1.31
+0.26
+1.49
+2.60
= +1.84
+0.95
107
—1.24
—0.25
—0.70
+2.84
+2.22
+0.44
+3.45
+3.30
= +5.70
+3.37
108
—2.63
—0.53
—2.15
+4.76
+3.58
+0.72
+5.86
+4.25
= +4.49
—0.01
109
—1.44
—0.29
—1.91
+2.58
+2.42
+0.49
+4.44
+2.42
= +3.11
+0.51
110
—0.63
—0.13
—1.27
+1.40
+1.62
+0.33
+3.34
+1.13
= +2.10
+0.78
111
—0.42
—0.09
—1.09
+1.13
+1.41
+0.29
+3.02
+0.72
= +4.67
+3.68
112
—0.20
—0.04
—0.92
+0.87
+1.20
+0.24
+2.65
+0.23
= +1.74
+1.08
113
+0.14
+0.03
—0.76
+0.46
+0.86
+0.17
+1.70
—0.82
= +0.76
+0.63
114
+0.24
+0.05
—0.80
+0.21
+0.75
+0.15
+1.09
—1.20
= +1.98
+2.03
115
+0.39
+0.08
—0.49
—0.22
+0.68
+0.14
+0.24
—1.39
= +0.96
+1.27
116
+0.37
+0.08
—0.55
—0.61
+0.64
+0.13
—0.42
—1.24
= —0.50
—0.12
117
+0.38
+0.08
—0.29
—0.77
+0.62
+0.13
—0.77
—1.00
= —1.55
—1.11
118
+0.40
+0.08
+0.15
—0.82
+0.60
+0.12
—1.07
—0.59
= —1.13
—0.63
119
+0.41
+0.08
+0.33
—0.77
+0.60
+0.12
—1.16
—0.34
= —0.66
—0.17
120
+0.41
+0.09
+0.60
—0.59
+0.59
+0.12
—1.19
+0.28
= —2.75
—2.27
121
+0.42
+0.09
+0.81
—0.23
+0.58
+0.12
—1.08
+0.48
= —2.23
—1.81
122
+0.42
+0.09
+0.84
+0.12
+0.58
+0.12
—0.85
+0.80
= —0.45
—0.09
123
+0.42
+0.09
+0.43
+0.73
+0.57
+0.12
—0.12
+1.16
= —0.44
—0.21
124
+0.42
+0.09
+0.22
+0.82
+0.57
+0.12
+0.14
+1.16
= —0.25
—0.05
125
—0.62
—0.08
+1.24
—1.44
+1.63
+0.22
—3.30
—1.23
= +0.50
—0.03
126
—1.88
—0.26
+2.98
—3.12
+2.88
+0.39
—5.80
—0.87
= +0.49
—1.02
127
—1.65
—0.24
+3.29
—1.47
+2.65
+0.37
—5.52
+0.04
= +3.61
+2.02
128
—0.43
—0.06
+1.57
—0.43
+1.43
+0.20
—3.07
+0.93
= +0.82
+0.31
129
+0^8
+0.03
—0.85
+0.36
+0.81
+0.12
+1.70
+0.76
= +0.51
+0.43
130
+0.06
+0.01
—1.00
+0.24
+0.96
+0.14
+2.18
+0.27
= —0.35
—0.56
131
—0.15
—0.02
—1.22
+0.19
+1.15
+0.17
+2.58
—0.38
= +0.88
+0.42
132
—0.57
—0.09
—1.81
+0.19
+1.56
+0.24
+3.19
—1.29
= —1.10
—2.05
133
—1.62
—0.25
—3.52
+0.03
+2.67
+0.41
+4.51
—2.92
= +0.68
—1.50
134
—2.41
—0.37
—4.84
—0.46
+3.57
+0.55
+5.46
—4.03
= +3.34
+0.29
135
—2.21
—0.34
—4.40
—1.02
+3.22
+0.49
+4.97
—4.07
= +1.74
—0.93
136
—1.18
—0.18
—2.60
—0.95
+2.18
+0.34
+3.20
—3.21
= +1.44
0.00
137
—0.35
—0.05
—1.32
—0.63
+1.36
+0.21
+1.51
—2.54
= +1.09
+0.56
138
+0.02
0.00
—0.83
+0.52
+0.97
+0.15
+0.35
—2.12
= +1.32
+1.23
139
+0.15
+0.03
+0.92
+0.09
+0.84
+0.14
—1.89
+0.22
= —0.60
—0.59
140
—0.07
—0.01
+1.10
+0.35
+1.07
+0.18
—2.09
+1.26
= —0.20
—0.50
141
—0.30
—0.05
+1.35
+0.52
+1.29
+0.22
—2.13
+1.99
= +1.23
+0.64
142
—1.01
—0.17
+2.28
+1.15
+2.00
+0.34
—2.30
+3.70
= +2.00
+0.49
143
—2.22
—0.38
+3.80
+2.57
+3.21
+0.54
—2.66
+6.05
= +5.49
+2.34
144
—2.54
—0.43
+3.66
+3.57
+3.51
+0.60
—2.36
+6.78
= +7.55
+3.84
145
—0.61
—0.10
+1.02
+1.61
+1.60
+0.27
—1.65
+3.56
= +3.16
+1.98
146
+0.42
+0.03
+0.30
+0.80
+0.58
+0.05
+0.38
+1.13
= 0.00
—0.06
147
+0.41
+0.03
—0.80
—0.24
+0.59
+0.05
+1.02
-0.58
= —0.11
+0.11
148
+0.39
+0.03
—0.04
—0.83
+0.62
+0.05
+0.06
—1.26
= +1.08
+1.40
149
+0.34
+0.03
+0.56
—0.61
+0.66
+0.06
—0.81
—1.12
= —0.51
—0.25
150
—0.02
13
0.00
+0.38
+1.00
+1.04
+0.10
+1.03
+2.11
= +0.79
+0.40
98 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
No. Beslduals.
151 +0.16® -fO.Oli/ +0.12a +0.89m +0.83af' +OMy' +1.432:' +1.18w' = +3'.'21 +3'.'00
152 +0.38 +0.04 —0.83 —0.03 +0.62 +0.06 +0.61 —1.11 =+0.34 +0.49
153 +0.40 +0.04 —0.58 —0.59 +0.60 +0.06 —0.13 —1.19 =—0.29 0.00
154 +0.42 +0.04 +0.06 +0.83 +0.59 +0.06 —0.83 —0.83 =—0.08 —0.03
155 +0.42 +0.04 +0.72 +0.43 +0.58 +0.06 —0.81 +0.85 =+1.04 +1.21
156 +0.41 +0.04 —0.25 —0.81 +0.59 +0.06 —0.10 +1.21 =+0.38 +0.75
157 +0.39 +0.04 —0.44 +0.72 +0.60 +0.06 +0.65 +1.08 =—0.61 —0.57
158 +0.35 +0.04 —0.85 +0.04 +0.65 +0.07 +1.38 +0.24 =—1.16 —1.04
159 +0.19 +0.02 —0.61 —0.65 +0.81 +0.08 +1.26 —1.29 =—0.19 —0.07
160 +0.21 +0.02 +0.35 —0.80 +0.77 +0.08 —1.76 —0.15 =+1.50 +1.69
161 +0.34 +0.04 +0.81 —0.24 +0.66 +0.07 —0.94 +1.06 =—0.53 —0.30
162 +0.34 +0.06 —0.63 +0.52 +0.66 +0.11 +1.25 —0.55 =+1.72 +1.80
163 +0.38 +0.07 —0.82 —0.05 +0.62 +0.11 +0.60 —1.10 =+0.98 +1.22
164 +0.42 +0.07 —0.02 —0.85 +0.59 +0.10 —0.75 —0.90 =—0.97 —0.50
165 +0.42 +0.07 +0.83 —0.13 +0.58 +0.10 —1.13 +0.28 =+0.52 +0.89
166 +0.41 +0.07 +0.64 +0.55 +0.58 +0.10 —0.70 +0.95 =+0.01 +0.24
167 +0.17 +0.03 +0.29 —0.85 +0.83 +0.16 —1.81 —0.43 =+1.38 +1.57
168 +0.35 +0.07 +0.82 —0.10 +0.65 +0.12 —1.16 +0.75 =+0.61 +0.90
169 +0.38 +0.07 +0.75 +0.38 +0.61 +0.12 —0.07 +1.29 =+0.85 +1.09
170 +0.42 +0.08 —0.79 +0.24 +0.58 +0.11 +1.10 —0.35 =—0.73 —0.48
171 +0.42 +0.08 —0.35 —0.76 +0.59 +0.11 +0.18 —1.15 =—0.27 +0.20
172 +0.42 +0.08 +0.27 —0.81 +0.58 +0.11 —0.50 —1.05 =—3.06 —2.55
173 +0.38 +0.07 +0.79 —0.27 +0.62 +0.12 —1.27 —0.50 =—0.26 +0.10
174 +0.31 +0.06 +0.81 +0.20 +0.64 +0.13 —1.35 +0.18 =—2.86 —2.65
175 +0.32 +0.06 +0.65 +0.55 +0.68 +0.13 —1.25 +0.80 =—0.73 —0.60
The equations derived from the latitudes y; contain two more unknown
quantities,
v= M', w = Bmi'.^Q,',
but, in them, the variation of the solar elements will be neglected.
The mean of the Transits of 1761 and 1769 gives
- 0.059a; + 0.050^ — 0.0952 - 0.069m + O.OOOi; + l.OOOw = - 1".165 .
From this mean the indeterminate correction of the Sun's semi-diameter is
nearly eliminated.
No.
1
Uquations
of Condition.
— O.OliB
+0.002/
— 0.01»
+0.00tt
+0.61«
+1.24W
= +0"82
2
—0.10
+0.01
—0.21
—0.08
—0.36
+1.95
= +0.41
3
—0.12
+0.02
—0.31
—0.11
—1.09
+2.04
= —0.49
4
+0.17
—0.02
—0.41
+0.25
—2.13
+0.88
= —0.14
5
+0.20
—0.03
—0.37
+0.17
—1.60
—0.40
= -1.51
6
+0.09
—0.01
—0.14
—0.10
+0.12
—1.35
= +0.02
7
+0.20
—0.02
—0.23
—0.35
+1.17
—1.42
= +5.62
8
+0.19
—0.02
—0.30
—0.49
+2.32
—0.77
= +1.54
9
—0.14
+0.02
—0.54
—0.16
+2.42
+0.46
= +0.64
10
—0.23
+0.03
—0.54
—0.07
+1.88
+1.10
= —1.70
11
—0.18
+0.02
—0.36
—0.10
+1.05
+1.38
= — L48
THE ORBIT OF VENUS 99
No.
12
— 0.22i»
+0.02J/
—0.012!
—0.58m
—2.341;
—0.09m;
= -1:49
13
+0.11
—0.01
—0.33
—0.36
—2.06
—1.55
=: +1.04
14
+0.12
—0.01
+0.21
—0.24
+1.57
+1.68
= +0.34
15
—0.03
0.00
—0.02
—0.09
+0.06
+2.34
= +0.63
16
+0.02
0.00
+0.01
+0.05
—0.75
+1.69
= —0.77
17
+0.01
0.00
—0.07
+0.04
+0.27
—2.68
= +2.21
18
—0.15
+0.01
—0.12
+0.32
+1.60
—1.45
= +0.21
19
+0.01
0.00
+0.02
0.00
+0.78
+0.97
= +1.13
20
+0.10
0.00
—0.38
+0.18
—2.05
+1.36
= —0.77
21
+0.11
0.00
—0.17
—0.11
+0.33
—1.45
= +0.53
22
+0.23
—0.01
—0.28
—0.45
+1.63
—1.35
= +0.73
23
—0.23
+0.01
—0.57
—0.06
+2.13
+0.86
= —4.07
24
—0.13
0.00
—0.12
—0.25
—0.86
+1.09
= +0.95
25
—0.17
0.00
—0.07
—0.56
—2.43
—0.35
= —0.67
26
+0.07
0.00
—0.09
—0.11
+0.09
—1.54
= +0.53
27
+0.10
0.00
+0.18
—0.24
+1.52
+1.83
= —0.65
28
+0.01
0.00
0.00
+0.02
—0.62
+1.79
= +3.82
29
—0.06
0.00
+0.14
—0.05
—1.03
—1.91
= —2.56
30
0.00
0.00
—0.07
+0.07
+0.49
—2.67
= —0.52
31
—0.15
0.00
—0.15
+0.29
+1.60
—0.73
= —0.13
32
—0.10
0.00
—0.30
—0.06
—1.04
+2.13
= +0.55
33
+0.21
+0.01
—0.38
+0.27
—1.92
+0.18
= —0.52
34
+0.16
0.00
—0.30
+0.10
—1.27
—0.63
= —0.79
35
+0.14
+0.01
—0.21
—0.20
+0.59
—1.54
= —0.38
36
+0.22
+0.01
—0.27
—0.39
+1.41
—1.47
= +1.29
37
+0.16
+0.01
—0.37
—0.44
+2.39
—0.81
= —0.72
38
—0.16
—0.01
—0.12
—0.30
—0.98
+1.12
= +0.33
39
—0.18
—0.01
—0.09
—0.56
—2.43
—0.21
= +2.84
40
+0.17
+0.01
—0.42
—0.29
—1.88
—1.59
= —1.00
41
+0.06
0.00
—0.08
—0.10
+0.20
—1.44
= +0.10
42
+0.06
0.00
—0.06
—0.11
+0.31
—0.86
= +0.03
43
+0.13
+0.01
+0.04
—0.25
+1.18
—0.46
= +0.64
44
+0.18
+0.01
+0.22
—0.32
+1.78
+0.86
= —0.92
45
—0.05
0.00
—0.03
—0.14
+8.33
+2.36
= —0.04
46
+0.02
0.00
+0.01
+0.05
—0.80
+1.48
= +0.66
47
+0.05
0.00
—0.01
+0.10
—1.04
+0.90
= +0.64
4S
—0.03
0.00
—0.05
—0.05
+0.03
+0.70
= +0.07
49
—0.07
—0.01
—0.02
—0.13
—0.68
+0.61
= —1.75
50
—0.09
—0.01
+0.09
—0.16
—1.15
+0.14
= —0.90
51
—0.11
—0.01
+0.19
—0.12
—1.37
—1.06
= —3.16
52
—0.03
0.00
+0.09
—0.03
—0.71
—2.35
= —4.41
53
—0.03
0.00
—0.07
+0.13
+0.84
—2.49
= —1.78
54
—0.14
—0.01
—0.10
+0.29
+1.54
—1.26
= +3.67
55
—0.12
—0.01
—0.14
+0.19
+1.45
—0.21
= +0.35
56
+0.04
0.00
+0.06
—0.05
+0.93
+0.14
= +1.39
57
+0.03
0.00
+0.05
—0.02
+0.90
+.079
= +1.57
58
—0.05
0.00
—0.10
—0.02
+0.18
+1.80
= +2.67
59
—0.08
—0.01
—0.30
—0.03
—1.11
+2.13
= +2.72
60
+0.16
+0.02
—0.31
+0.13
—1.37
—0.50
= +0.58
61
+0.10
+0.01
—0.19
—0.01
—0.63
—0.90
= —0.01
62
0.00
0.00
—0.01
0.00
—0.70
—0.73
= —1.35
63
+0.04
0.00
—0.07
—0.02
—0.37
+1.15
= —1.50
64
+0.13
+0.01
—0.20
—0.17
+0.44
—1.54
= —0.74
65
+0.22
+0.03
—0.34
—0.47
+2.04
—1.19
= —3.38
100 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
No.
66
— 0.21iB
—0.031/
—0.592
— 0.02tt
+2.26!;
+0.54W
= +2.46
67
—0.08
—0.01
—0.14
—0.09
+0.16
+1.25
= —0.29
68
—0.03
0.00
—0.03
—0.05
—0.40
+0.89
= +3.00
69
—0.04
0.00
—0.07
—0.03
+0.26
+0.90
= +0.62
70
—0.07
—0.01
—0.09
—0.11
—0.36
+1.15
= —2.16
71
—0.14
—0.02
—0.13
—0.25
—0.80
+1.15
= —3.21
72
—0.23
—0.03
—0.07
—0.49
—1.74
+0.76
= —0.32
73
—0.17
—0.02
—0.12
—0.56
—2.41
—0.21
= +2.40
74
+0.16
+0.02
—0.28
—0.18
—0.74
—1.75
= —2.59
75
+0.04
+0.01
—0.05
—0.07
+0.33
—1.30
= —0.70
76
+0.06
+0.01
—0.07
—0.10
+0.26
—0.86
= —0.80
77
+0.16
+0.02
+0.09
—0.30
+1.43
—0.21
= +0.69
78
+0.19
+0.03
+0.21
—0.34
+1.75
+0.50
= +0.68
79
+0.13
+0.02
+0.20
—0.30
+1.72
+1.54
= —0.50
80
+0.01
0.00
+0.03
—0.17
+1.22
+2.20
= +0.45
81
—0.04
—0.01
—0.03
—0.09
+0.07
+2.22
= +0.26
82
+0.02
0.00
+0.01
+0.04
—0.76
+1.50
= —0.82
83
+0.04
+0.01
0.00
+0.08
—0.99
+1.02
= —0.72
84
+0.05
+0.01
—0.04
+0.10
—1.06
+0.30
= —0.07
85
—0.08
—0.01
+0.01
—0.16
—0.88
+0.49
= +0.83
86
-0.11
—0.02
+0.12
—0.17
—1.29
—0.09
= —1.35
87
—0.12
—0.02
+0.17
—0.16
—1.41
—0.56
= +0.45
88
—0.10
—0.02
+0.20
—0.09
—1.26
—1.63
= —0.07
89
—0.03
0.00
+0.07
—0.04
—0.61
—2.49
= +0.58
90
—0.09
—0.02
—0.05
+0.23
+1.23
—2.00
= +1.14
91
—0.11
—0.02
—0.17
+0.15
+1.24
+0.17
= +0.70
92
—0.04
—0.01
—0.07
—0.03
+0.01
+0.75
= —0.49
93
+0.04
+0.01
+0.06
—0.05
+0.96
+0.26
= +0.83
94
+0.03
+0.01
+0.06
—0.02
+0.95
+0.69
= +0.99
95
+0.02
0.00
+0.04
—0.01
+0.82
+1.06
= +0.89
96
—0.02
0.00
—0.04
0.00
+0.44
+1.61
= +0.92
97
—0.09
—0.02
—0.21
—0.05
—0.44
+2.14
= +0.44
98
—0.03
0.00
—0.31
+0.05
—1.44
+2.02
= +1.56
99
+0.14
+0.03
—0.33
+0.27
—1.96
+1.25
= —1.15
100
+0.20
+0.04
—0.34
+0.26
—1.79
+0.12
= —0.91
101
+0.17
+0.03
—0.31
+0.17
—1.48
—0.35
= +0.19
102
+0.14
+0.03
—0.27
+0.09
—1.18
—0.62
= —0.52
103
+0.10
+0.02
—0.20
0.00
—0.72
—0.86
= —0.60
104
—0.01
0.00
+0.01
0.00
—0.79
—0.52
= +0.02
105
+0.03
+0.01
—0.06
—0.01
—0.43
—1.12
= +0.74
106
+0.07
+0.01
—0.13
—0.06
—0.08
—1.38
= +1.18
107
+0.20
+0.04
—0.28
—0.33
+1.14
—1.61
= +0.54
108
+0.05
+0.01
—0.46
—0.30
+2.47
—0.62
= +0.96
109
—0.24
—0.05
—0.53
0.00
+1.77
+0.94
= +0.22
110
—0.16
—0.03
—0.31
—0.07
+0.88
+1.27
= +1.94
111
—0.12
—0.02
—0.23
—0.09
+0.56
+1.28
= +1.03
112
—0.08
—0.02
—0.14
—0.09
+0.16
+1.20
= +1.81
113
—0.01
0.00
—0.01
—0.03
—0.57
+0.65
= +0.34
125
—0.17
—0.02
—0.12
—0.33
—1.12
+1.08
= +1.69
126
—0.23
—0.03
—0.07
—0.49
—2.24
+0.23
= +1.42
127
+0.19
+0.03
—0.39
—0.22
—1.30
—1.73
= +0.59
128
+0.06
+0.01
—0.08
—0.10
+0.16
—1.43
= +0.76
129
+0.05
+0.01
—0.07
—0.08
+0.11
—0.85
= —0.11
130
+0.08
+0.01
—0.05
—0.16
+0.56
—0.85
= +0.64
THE ORBIT OF VENUS
101
131
+0.12a!
+0.02J/
+0.01«
—0.24m
+1.03t?
— O.eito
= +0'.'74
132
+0.17
+0.02
+0.12
—0.31
+1.51
—0.08
= +1.60
133
+0.18
+0.03
+0.24
—0.33
+1.81
+1.08
= +1.68
134
+0.07
+0.01
+0.13
—0.29
+1.47
+1.98
= —0.50
135
—0.06
—0.01
—0.05
—0.18
+0.46
+2.34
= +0.90
136
—0.01
0.00
0.00
—0.01
—0.51
+1.80
= —0.11
137
+0.05
+0.01
—0.01
+0.09
—1.03
+0.84
= -0.35
138
+0.05
+0.01
—0.06
+0.09
—0.99
+0.03
= +0.88
139
—0.06
—0.01
—0.02
—0.12
—0.56
+0.66
= —0.91
140
—0.09
—0.02
+0.07
—0.17
—1.13
+0.23
= —0.84
141
—0.11
—0.02
+0.14
—0.17
—1.34
—0.22
= —1.90
142
—0.11
—0.02
+0.21
—0.11
—1.37
—1.26
= —1.37
143
—0.04
—0.01
+0.11
—0.05
—0.77
—2.38
= —0.65
144
—0.03
—0.01
—0.05
+0.12
+0.77
—2.51
= +3.52
145
—0.14
—0.02
—0.14
+0.24
+1.48
—0.50
= +0.28
To apply to these equations the rigorous method of least squares would
be very laborious; hence the method of "Equivalent Factors" has been
used ; the equations have been multiplied either by whole numbers or by
fractions which are ready multipliers. In this way the following Normal
Equations were derived from the equations of condition which have cos j? .A0
for their absolute terms :
+195.84ic
— 44.809J/
+127.71«
+ 73.19W
-251.9037
+43.027^
— 85.48a
+119.25U
= — 8.77
— 44.78
+47.099
— 83.68
— 62.84
+ 41.04
—48.460
+ 41.17
— 96.06
= —113.43
+120.94
—83.889
+427.28
+133.17
—136.59
+82.936
—410.76
+400.15
= +162.30
+ 70.03
—62.965
+135.64
+365.81
— 73.13
+63.350
+114.76
+508.04
= +197.06
—255.15
+42.172
—138.12
— 80.06
+425.64
—27.182
+ 91.22
—132.67
= + 92.63
+ 40.68
—48.373
+ 82.84
+ 61.99
— 26.27
+51.815
— 41.45
+ 94.13
= +121.18
— 83.42
+41.537
—422.53
+119.76
+102.83
—40.091
+644.06
—111.82
= — 23.87
+112.81
—95.792
+406.68
+505.65
—126.69
+94.621
—120.34
+902.21
= +264.18
If u is eliminated from these equations, the result is
+181.830!
— 32.213J/
+100.57^
-237.27®
+30.3522/
-108.442
+ 17.60it
= — 48.20
— 32.75
+36.284
— 60.38
+ 28.48
—37.577
+ 60.88
— 8.78
= — 79.58
+ 95.45
—60.971
+377.90
—109.97
+59.874
—452.54
+215.20
= + 90.56
—239.82
+28.394
—108.43
+409.63
—13.317
+116.34
— 21.48
= +135.76
+ 28.81
—37.705
+ 59.85
— 13.S8
+41.080
— 60.90
+ 8.04
= + 87.79
—106.35
+62.147
—466.94
+126.77
—60.831
+606.49
—278.15
= — 88.38
+ 16.01
— 8.770
+219.18
— 25.60
+ 7.053
—278.97
+199.94
= — 8.21
and if from these a is eliminated, the result is
+156.43a;
-15.9872/
— 208.00a>'
+14.418J/'
+11.99i2:'
—39.67m'
= — 7^30
— 17.50
+26.542
+ 10.91
—28.055
—11.42
+25.60
= — 65.11
—212.43
+10.900
+378.08
+ 3.863
—13.51
+40.27
= +161.74
+ 13.69
—28.049
+ 3.54
+31.598
+10.77
—26.04
= + 73.45
+ 11.59
—13.190
— 9.11
+13.151
+47.33
—12.25
= + 23.52
— 39.35
+26.593
+ 38.18
—27.674
—16.60
+75.13
= — 61.46
10 2 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
It is evident now, that since the principal coefficients of z' and u' have
fallen from 644.06 and 902.21 to 47.33 and 75.13, no very reliable values
of these quantities can be obtained from these equations. The elimination
of y gives
+145.89» —201.43a;' — 2.480j/' + S.llis' — 24.25-m' =— 111.'52
—205.24 +373.60 +15.384 — 8.82 +29.76 = +188.48
— 4.80 — 15.07 + 1.950 — 1.30 — 1.01 = + 4.64
+ 2.89 — 3.69 — 0.791 +41.65 + 0.47 =— 8.84
— 21.82 + 27.25 + 0.435 — 5.06 +49.48 = + 3.78
The elimination of x from these gives
+90.233?' +11.895J/' — 1.632' — 4.35«' = +3l'.'63
+ 8.44 + 1.868 — 1.13 + 0.21 = + 0.97
+ 0.30 — 0.742 +41.55 + 0.95 =— 6.63
— 2.88 + 0.064 — 4.30 +45.85 =—12.89
The elimination of x' from these gives
+0.7552/' — 0.983 + 0.62«' =— l^^g
—0.782 +41.56 + 0.96 =— 6.74
+0.444 — 4.35 +45.71 =—11.88
The only condition, relative to the solar elements, which can be obtained
with any weight from these equations is
a/ +0.132/= +0".335.
That is, the mean longitude of the Sun of Hansen and Olufsen's Tables
ought to be increased by a third of a second at the epoch 1863. As, how-
ever, these Tables will, probably, be used for a long time to come in com-
puting the solar coordinates of the American Ephemeris, y\ z' and u' will be
put severally equal to zero ; and, as it has been decided to use the Pulkova
constant of aberration, x' will be put equal to +0".19. With these assump-
tions, the values of aj, y , z and ware
a; = — 0".502, ?/ = — 2".863, z = — 0".040, m=+0".195.
The equation of condition derived from the Transits of 1 76 1 and 1769
being excluded, the normal equations, determining the corrections of the
inclination and the longitude of the ascending node, are
+2.51®
+0.3902/
+1.843
—0.67m
+163.26);
— 0.42W
= +2^.'02
—4.46
—0.105
—0.29
—1.06
— 5.86
+188.58
= +24.11
From these are obtained the following values of v and w :
V- + 0".18, w = + 0".13 or Agi'= + 2".o .
THB ORBIT OF VENUS 103
But, from the equation furnished by the Transits in 1761 and 1769,
If the first result is supposed to belong to 1855.0, and the second to 1765.4
the proper value of the correction is
JQ,'= +0".9 + 0".222t.
The origin of the pretty large correction — 0".02863, of the mean
motion of Venus, is easily shown. In his investigation, Leverrier {Annales,
Vol. VI, p. 72) found the following value of An' :
An'= + 0".00035 + 0".0689>/ + 0".0959k'+ 0".1207/';
but the value of this quantity used in forming his Tables is the first term
only. If the values of r , v', v" corresponding to the change from Leverrier's
values of the masses to those here adopted, be substituted in this expres-
sion, the correction of Leverrier's mean motion, from this cause, is found
to be
z)w'= — 0".01588.
Moreover, a comparison of the values of the Sun's mean longitude in the
Tables of Hansen and Olufsen and of Leverrier gives
Han.-Lev. = — 0".93 — 0".010r4^ .
From the way in which An' and An" are involved in the equations of con-
dition, it may be concluded, that if An" were left indeterminate in the solu-
tion, the value of An', obtained, would be roughly
M'= {An') + l.Szlw",
(An') denoting the value of An' on the supposition of An" = . Thus, on
making An" = — 0".01074, the correction of the mean motion of Venus
from this cause is An' = — 0".01289. The sum of these two corrections is
An' = — 0". 02877, which is almost identical with that derived from the
equations of condition.
The increment of the motion of the node, 0".222, requires that the mass
of Venus should be reduced from tu-stsz to Tjrmi- This agrees with Lever-
rier's result: setting out with the mass 0.0000024885, he found that it
should be multiplied by the factor 0.948, which would make the mass
]
The corrections to be added to the elements, with which we set out, to
obtain the elements, from which the Tables are constructed, are
AL'= — 0".502, /l7r'= 4- 28".46, AQ,'= + 0".90 + 0".222t,
Ai'= +0".18, Je'= — 0.000000196, An' = — 0". 02863.
104 eOLLBGTED MATHEMATICAL WORKS OF G. W. HILL
The Tables have been compared with the occultation of Mercury by
Venus, observed at Greenwich, May 28, 1737. The observations made are
Qreenwloli M. T.
9" 40°" 3'.9. Mercury distant from Venus not more than a tenth part
of the diameter of Venus.
9 48 10.2. Mercury wholly occulted by Venus.
The position of Mercury being derived from Prof. Winlock's Tables, the
apparent position of the two planets, as seen from Greenwich, and in longi-
tude and latitude, are
Greenwich M. T. I b V b' I' — li' — b
May 28- 8 89° 24' 23"o5 +2°9'l2'.'90 89°3l' 49'.'97 +2°10' 9'.'98 -)-446.'92 +57'.'08
9 89 27 56.68 +2 9 5.67 89 31 14.38 +2 9 42.02 +197.70 +36.35
10 89 31 30.35 +2 8 58.43 89 30 39.63 +2 9 14.28 — 50.72 +15.85
and, interpolating,
Greenwloh M. T. I' — l b'—b Dlst. of Centers,
h m s ,/ „ /,
9 40 3.9 +31.73 +22.64 38.96
9 48 10.2 — 1.79 +19.87 19.95
With the addition of 0".57 for irradiation, the semi-diameters of Mercury and
Venus are respectively 3". 98 and 26".97 ; hence, at the first observation, the
distance of the limbs of the planets is 8".01, 2".6 more than a tenth part of
the diameter of Venus ; at the second observation, the distance of the cen-
ters is less than the difference of the semi-diameters ; hence, the Tables are
verified by the statement of the observer. Venus being, at the time, a thin
crescent, and about half of Mercury's disc being illuminated, it is plain that
it would be diflScult for the observer to estimate the distance in fractional
parts of the apparent diameter of Venus.
Leverrier's remarks on this occultation are impaired by a mistake
made in the last line of his computation.
DERIVATION OF THE MASS OF JUPITER
105
MBMOIE No. 11.
On the Derivation of the Mass of Jupiter from the Motion of certain
Asteroids.
(Memoirs of the American Academy of Arts and Sciences, Vol. IX, New Series, pp. 417-420, 1873.)
The object of the present note is to show that the discussion of the
observations of certain asteroids, provided they extend over a suflBcient
period of time, will furnish a far more accurate value of the mass of Jupiter
than can be obtained from measurements of the elongation of the satellites,
or from the Jupiter perturbations of Saturn. It is to be hoped that observers
will hereafter pay particular attention to those asteroids which are best
adapted for the end in question.
The magnitude of the Jupiter perturbations of an asteroid depends at
once on the magnitude of the least distance of the two bodies, and the
greater or less degree of approach to commensurability of the ratio of their
mean motions, and also on the magnitude of the eccentricity of the asteroid's
orbit.
Those asteroids which lie on the outer edge of the group, and whose
mean motions are nearly double that of Jupiter, will best fulfil the two first
conditions named above. For they will have inequalities of long period
whose coefficients will be of the order of the first power only of the eccen-
tricities, while all other classes of long-period inequalities are necessarily of
higher orders, and hence demand longer periods in order to have their coeffi-
cients brought up to an equal magnitude.
In order to exhibit the relative value of these asteroids for the purpose
in view, I have computed the terms of the lowest order in the coefficients
of these inequalities of long period for all the asteroids, yet discovered,
whose daily mean motion lies between the limits 550" and 650" ; and have
appended herewith tables, by which the value of these terms can be readily
computed for any which may hereafter be discovered between these limits.
The formulas for computing these terms are found in the Mecanique
Cileste, Tom. I, pp. 279-281. Here i must be put equal to 2, in the terms
which involve the simple power of the eccentricities. We will employ the
usual notation for the designation of the elements of orbits, and make some
reductions in Laplace's formulas for the sake of ready computation.
14
106
COLLECTED MATHEMATICAL WORKS OF G. W. HILL
If we put y = ^ ^ or in Laplace's notation — . and recollect
(I ^ n
that we need the formulas only for the case of an inferior perturbed by a
superior planet ; and moreover make
ri?'"" = -^, and rG^'^=J,
F^^^ and G^^'> being Laplace's symbols, we shall have
H
If, in the next place, K and /? are derived from the equations
-S'cos(/3 — tt) = jy sin y> — / sinv"' C0S(7r'— ?:) ,
-ff'sill(^— 7r)=: — Jsinp' Sill(7r' — tt),
the inequality in longitude we are computing is
m
~KamlL—2L' + ^].
H and J may be regarded as functions of a , and are positive between
the limits corresponding to ju = 550" and yi = 650". The common loga-
rithms of these quantities are here tabulated for every 0.001 of a between
the limits above mentioned ; the values of if' and hf and their differentials
were obtained from Kunkle's Tables of the Goefficients of the Perturbative
Function.
a
logH
log J
u.
logfl-
log J
0.595
0.3153369
9.871828
0.610
0.3323864
9.889836
.596
.3165277
.873131
.611
.3334562
.890910
.597
.3177113
.874420
.612
.3345169
.891967
.598
.3188875
.875695
.613
.3355683
.893007
.599
.3200561
.876956
.614
.3366103
.894030
.eoo
.3212173
.878202
.615
.3376427
.895036
.601
.3223707
.879434
.616
.3386652
.896022
.602
.3235163
.880652
.617
.3396777
.896990
.603
.3246540
.881855
.618
.3406801
.897939
.604
.3257838
.883043
.619
.3416723
.898869
.605
.3269054
.884214
.620
.3426539
.899780
.606
.3280187
.885370
.621
.3436248
.900671
.607
.3291236
.886511
.622
.3445848
.901542
.608
.3302199
.887636
.623
.3455337
.902392
0.609
0.3313075
9.888745
0.624
0.3464714
9.903221
DERIVATION OF THE MASS OF JUPITEE
107
a
logff
log J'
a
log H
log J
0.625
0.3473975
9.904028
0.643
0.3618323
9.914446
.626
.3483119
.904814
.644
.3624928
.914764
.627
.3492144
.905578
.628
.3501047
.906320
.645
.3631366
.915051
.629
.3509827
.907040
.646
.3637632
.915306
.647
.3643722
.915528
.630
.3518480
.907736
.648
.3649632
.915717
.631
.3527005
.908408
.649
.3655358
.915871
.632
.633
.634
.3535399
.3543659
.3551782
.909056
.909679
.910277
.650
.651
.652
.3660897
.3666246
.3671400
.915991
.916076
.916125
.635
.636
.3559767
.3567612
.910850
.911396
.653
.654
.3676354
.3681103
.916136
.916108
.637
.638
.639
.3575313
.3582866
.3590269
.911916
.912409
.912874
.655
.656
.657
.3685644
.3689972
.3694082
.916040
.915933
.915785
.640
.641
.3597519
. .3604614
.913310
.913718
.658
.659
.3697969
.3701628
.915595
.915362
0.642
0.3611550
9.914097
0.660
0.3705053
9.915085
The values of the elements of Jupiter's orbit for the epoch 1850.0 which
we shall use are
: 299.1286,
loga'=: 0.7162372,
¥.' = 2°45'54".55,
n' = 11° 55' 2".
The values of the corresponding elements of as many of the asteroids
as lie between the limits above mentioned are contained in the following
table. The longitudes of the perihelia are referred to the mean equinox of
1850.0.
fi log a (^ TT
Hygea
634;'3118
0.498 4692
5° 4^ 56'.4
234° 58 40:6
Themis
636.7634
0.497 3523
6 42 52.9
139 56 11.2
Euphrosyne
633.8508
0.498 8680
12 44 10.3
93 27 51.5
Doris
647.1295
0.492 6769
4 23 42.9
74 10 11.3
Pales
655.6209
0.488 9025
13 43 18.3
32 3 13.1
Europa
650.0877
0.491 3564
5 49 14.3
101 45 37.6
Mnemosyne
632.6897
0.499 2106
5 58 17.1
52 58 47.8
Erato
640.8591
0.495 4961
9 46 4.3
33 55 38.0
Cybele
560.8775
0.534 0920
6 54 36.4
258 11 24.3
Freia
569.0505
0.529 9038
10 49 12.0
93 2 36.6
Semele
652.9848
0.490 0690
11 49 36.5
28 25 39.1
Sylvia
543.5800
0.543 1620
4 39 22.6
337 8 6.1
Antiope
632.3591
0.499 3618
11 39 2.7
293 49 3.5
108
COLLECTED MATHEMATICAL WORKS OF G. W. HILL
The expression of
which result from the
formulas, are
the inequalities, and the length of their periods
substitution of these values of the elements in the
Hygea
14676.2 sin [L-
-2L' + 228 58 1.4L
97.96 years
Themis
14606.2 sin [L-
-2L' + 146 4 4.5],
91.72 "
Euphrosyne
28996.5 sin IL-
-21,'+ 97 58 58.4],
99.23 "
Doris
5086.7 sin [L-
-2L'+ 85 41 49.4],
72.27 "
Pales
11639.2 sin [L-
-21,'+ 33 36 12.6],
61.57 "
Europa
6584.4 sin [L-
-2Zy' + lll 29 19.2],
68.14 "
Mnemosyne
12956.0 sin [L-
-2L'+ 60 9 1.9],
102.58 "
Erato
13654.9 sin [L-
-2L'+ 36 21 16.9],
82.91 "
Cybele
13145.4 sin [L-
-2L' + 251 13 31.6],
94.49 "
Freia
32243.5 sin [L-
-2L'+ 98 15 25.5],
120.93 "
Semele
10860.7 sin [L-
-21,'+ 29 55 45.1],
64.54 "
Antiope
28567.8 sin [L-
-2L' + 288 44 3L6],
103.57 "
These expressions can be regarded as rough approximations only to the
actual values of these inequalities, since all terms of the third and higher
orders with respect to the eccentricities and inclinations, and of the second
and higher orders with respect to the disturbing masses, have been neglected.
Yet they are suflBciently exact to show the order of magnitude of the Jupi-
ter perturbations of the asteroids in question.
The effect of these inequalities at the time of opposition will be magni-
fied in the proportion roughly of a to a — 1 . Thus in the case of Freia, the
determination of the mass of Jupiter will depend on the observation of an
arcof 12°.7.
INEQUALITY OF LONG PERIOD IN THE LONGITUDE OF SATURN 109
MEMOIE No. 13.
On the Inequality of Long Period in the Longitude of Saturn, whose
Argument is Six Times the Mean Anomaly of Saturn Minus Twice
that of Jupiter Minus Three Times that of Uranus.
(AstronomiBche Naohrichten, Vol. 83, pp. 83-88, 1873.)
This inequality is proportional to the product of the masses of Jupiter
and Uranus. In its coefficient we shall have regard only to the part which
is divided by the square of the motion of the argument.
Employing the notation in general use, the quantities having no accent,
or one, or two, according as they belong to Jupiter, Saturn or Uranus, p
designating / ndt, and putting
P _ m r 1 r' cos (p~\ m" f 1 r' cos g'—'2g and g'-3g"
4g'-2g " 2g'-3g"
3g'- 2g " 3g'- 3g"
belonging to terms of the several factors involved in the expression.
The values of the factors proportional to Jupiter's action on Saturn
have been derived from Hansen's Untersuchung uber die gegenseitigen Storung-
en des Jupiters und Saturns ; the values of those proportional to the action
of Uranus have been specially computed. The values of the masses adopted
are m = T~stT!, m'-=-^iT5^, m" ^= YrhiF ■ In the following expressions the
common logarithms are written in place of the coeflScients, and the values
of n'Sz' and Sir' are in seconds of arc.
"^'•^^'"^' I^ =+ 8.00004 cos (5ff'- 23 + 66=51/.6) («'(J3')2= + I.47638 3in( s''-3ff'/ + 316°12'.2)
*" ^^ + 8.80451 cos (V-23 + 97 39.5) +1.40046 sin (33'-3sf'/+ 60 7.5)
+ 9.50716 cos (3^'- 2^ +137 52.9) +0.17010 ain (3sr'-3sf'/+ 112 48.0)
+ 0.00653 COB (2s'-2sr +158 3.7) +8.6777 sin (4s/-3ff'' + 129 )
+ 9.79703 cos ( ff'- 2^ +258 50.7), +7.1505 sin (5s'-3s'/ + 135 ),
112
COLLECTED MATHEMATICAL WORKS OF G. W. HILL
a'a + mn \ Br' I
m 3g ■
a'-B,
= + 7.96411
+ 8.80930
+ 9.59413
+ 0.33413
+ 9.82355
" " +94.72630
+ 95.25666
+ 94.36848
+ 93.1858
ag"
= +93.98809
+ 94.85117
+ 95.39557
+ 94.1683
+ 92.9164
sin [Sg'-
sin {ig'-
sin &g'-
sin (2g'-
3in( g'-
cos ( g'-
cos {2g'-
cos (3gr'-
cos l,ig'-
cos (5^'-
sin ( g'-
sinCSjr'-
sin {.ig'-
sin (.ig'—
sin (5,0'—
-2g +228°48/.3) {Slr'\--
-2g +264 16 .2)
^gr +300 16.3)
-2g +335 43.0)
•3^ +256 31 .4),
-3g"+ 35° 16'. 9) {n'Sz%--
-3s'"+238 47.0)
-3/' + 125 24.6)
-3ji" + 146 30.5)
-3ff" + 148 34 ),
-3g''+ 40° 2'.0) ((!«»•')„=
■3s'" + 350 37.4)
■3sr" + 134 50 .1)
■35" + 153 36 )
3sr"+130 43 ),
= +0.1370 cos( sr'-3s'" + S20°58' )
+ 1.08430 cos (Sg''-3sr" + 240 30.9)
+ 0.05350 cos (3fir'-3ff"+ 289 25 )
+ 8.7413 cos(4sr'-3s" + 303 36 )
+ 7.4637 C03(5sr'— 3sf" + 301 )^
= + 3.46348 sin (5ff'-3^ +346°53'.9)
+ 3.83450 sin (4sr'-3sr +277 10.8)
+ 1.41347 sin (3sr'-3sf +135 15.1)
+ 1.50550 sin (3^'- 3^ +156 17.9)
+ 0.4363 8in( 5r'-2^ +350 30 ),
= + 1.73766 cos (Sjr'-S^ + 63° 43'.0)
+ 3.52156 cos (49''-2^ +277 2.4)
+ 1.30103 cos (3ff'—2sr +141 32.0)
+ 1.48499 cos (25r'-23 +156 1.7)
+ 0.4146 cos( fl''-3ir + 97 54 )■
In the next place
3 mnn'
log
2il + m')(Qn' - 2n - 3n"y
= 1.01570, log
3n'n"
3(6w'— 2w — 3n"y
3.18681 .
Thus we get, the coefiScients still replaced by their logarithms.
+ 0.50212 sin (e^r' -
-2g-
-3^" + 103°
3'.8) + 0.25546sin(6i?'-
-2g-
-3y + 102°10'.8)
+ 1.22067 sin (
CC
+ 337
47.0) + 0.74761 sin(
(C
+ 335 57.8)
+ 0.69296 sin (
iC
+ 60
41.0) + 9.8569 sin(
i(
+ 80 40 )
+ 9.6999 sin(
a
+ 107
) + 8.9608 sin(
a
+ 123 )
+ 7.9632 6in(
ti
+ 214
) + 6.8089 sin (
«
+ 219 )
+ 9.1068 sin(
a
+ 269
46 ) + 8.9026 sin(
((
+ 282 45 )
+ 0.90920 sin(
it
+ 324
37.1) + 0.55954 sin (
«
+ 347 29.8)
+ 0.66333 sin(
Cl
+ 49
41 .3) + 9.7834 sin (
a
+ 86 22 )
+ 9.9910 sin(
C(
+ 98
19 ) + 8.8401 sin(
11
+ 129 28 )
+ 8.3019 sin(
u
+ 17
30 ) + 6.5178 sin(
(1
+ 48 36 )
By the addition of these terms is obtained
Sp'= + 34".752 sin (6^'- 2g — Bg"') + 1".312 cos (6/- 2g - Sg")
= + 34".776 sin (6/— 2g — 3g" + 2° 9' 43") .
The inequality in the mean longitude of Uranus, having the same argu-
ment, has been calculated by Leverrier (Additions aux Connaissance des
Temps, 1849, p. 85). He found
dp"= + 32".74 sin (6/- 2g — 3g" + 181° 1' 58") .
Thus, contrary to what might be expected, the inequality in the case of
Saturn is larger than in the case of Uranus.
CHARTS AND TABLES OF THE TRANSIT OF VENUS
113
MEMOIE No. 13.
Charts and Tables for Facilitating Predictions of the Several Phases of
the Transit of Venus in December, 1874.
(Papers relating to the Transit of Venus in 1874, Part II, 1873.)
CONTENTS.
PAGE.
Constants and elements employed, 114
Hourly ephemerides of the snn and Venus, 115
Axis, diameters, &c., of the enveloping cones, 116
Curves represented on the charts, 116
Problem I. To find the point of the earth's surface at which contact takes place
at a given time and altitude, 117
Peoblem II. To find the point on the earth's surface where the contact takes
place at a given point of the sun's limb and at a given altitude, 121
Values of quantities required in the computation of the curves, 134
Times, &c., of the beginning and end of each contact, and of its occurrence in the
zenith, 127
Approximation of the curves to circles, 128
Tables of positions of points used in drawing the curves, 135
Explanation of the charts, and their use, 139
Tables and formulas for computing more accurate values of the times of contact,
Example, 142
Corrections to be applied for determinate changes in the elements, 145
Tables and formulas for finding the position of the planet on the sun's disc, . . . 147
Localities favorable for the determination of parallax, 150
CHAETS.
No. I. Ingeess, exteeioe contact. No. III. Egeess, interioe contact.
II. Ingeess, inteeioe contact. IV. Egeess, exteeioe contact.
All the constants and elements which have been used in the computa-
tions on the transit are given below. The quantities having no terms mul-
tiplied by t are either constant or may be regarded as such for the duration
of the transit ; and the quantities which vary may be regarded as varying
uniformly. The unit of t is an hour.
15
114
COLLECTED MATHEMATICAL WORKS OF G. W. HILL
Epoch: 1874, Decemher 8*^ 11", Washington Mean Time.
VENUS.
Orbit longitude, referred to the mean equinox of date,
Longitude of the ascending node,
Log sine of inclination, .
Periodic perturbations of the latitude,
Log radius-vector, ....
Semi-diameter at mean distance,
76° 58' 13". 84 + 243". 332i(
75° 33' 24". 1
8.7722486
+ 0". 11
9.8575310 — 27.6(5
8". 546
THE SUN.
True longitude, referred to the mean equinox of date.
Latitude,
Log radius-vector, ....
Semi-diameter at mean distance,
True obliquity of the ecliptic,
Equation of the equinoxes in longitude,
Sidereal time, at Washington, in arc, .
Constant of solar parallax, .
Constant of aberration.
Eccentricity of the earth's meridians, .
Horizontal refraction, ....
256° 58' 41". 62 -1- 152". 533i!
— 0". 41
9.9932845 — 21.3<
959". 788
23° 27' 27". 67
— 7". 43
63° 44' 9". 6 + (15° 2' 27". 84)^
8". 848
20". 4451
0.0816967
35'
The elements of the heliocentric position of Venus are from the new
Tables of Venus,* and may be readily deduced from the first example given
in pages 16-19 of the introduction.
The apparent position of the sun which results from the above elements
coincides with that derived from the tables of Hansen and Olufsen, but the
true longitude is 0".19 greater, owing to the adoption of Struve's value of
the constant of aberration, 20". 445, instead of the value 20". 255.
The value of the sun's semi-diameter is adopted from Bessel. (See
Astronomische Nachrichten, No. 228, and Astrunomische Untersuchungen, Vol.
II, p. 114.) This value is used in the cotoputation of eclipses for the Ameri-
can Bphemeris. Hansen has also used it in his disquisition on the transit of
Venus. In the British Nautical Almanac the value 961".82 is used, and is
the same as that given for the reduction of meridian observations of the
sun. Leverrier states {Annates, Vol. VI, p. 40) that the value, deduced
from the previous transits of Venus, is 958". 424. Hence, it is probable that
predictions from the elements of the British Nautical Almanac will be found
to be considerably in error from this cause.
* Tables of Venus, prepared for the use of tlie American Ephemeris and Nautical Almanac, by
George W. Hill, Washington, 1872.
CHARTS AND TABLES OF THE TRANSIT OF VENUS
115
From the data given above are obtained the following hourly ephemer-
ides. For the sake of completeness they are expressed in terms of longi-
tude and latitude, as well as in right ascension and declination.
Wash. M.
T.
a =
App. R. A.
TENUS.
App. dec.
App. geocentric
longitude.
App. geocentric
latitude.
Logr =
log distance
from the
earth.
1874.
Dec. 8a
8h
255° 58' 56'.'03
—22° 38' 9'.'96
257° 4 53'.'34
+11 40'.'84
9.4221505
9
57 21.96
37 22.29
3 22.30
12 19.91
482
10
55 47.90
36 34.60
1 51.27
12 58.99
467
11
54 13.86
35 46.90
257 20.24
13 38.06
460
12
52 39.84
34 59.18
256 58 49.21
14 17.13
461
13
51 5.83
34 11.44
57 18.18
14 56.21
470
14
255 49 31.85
—22 33 23.67
256 55 47.16
+15 35.28
9.4221488
Wash. M.
T.
App. B. A.
THE SUN.
App, dec.
App.
longitude.
App.
latitude.
Log r'=
log distance
Irom the
earth.
1874.
Dec. 8a
81
255°42'l6"80
— 22°48'24'.'39
256° 50' 35'.'86
-6'ax
9.9932909
9
45 1.47
48 39.36
53 8.39
0.41
888
10
47 46.15
48 54.28
55 40.93
0.41
867
11
50 30.84
49 9.15
256 58 13.44
0.41
845
12
53 15.54
49 23.98
257 45.99
0.41
824
13
56 0.25
49 38.77
3 18.52
0.41
802
14
255 58 44.98
—22 49 53.51
257 5 51.05
—0.41
9.9932781
From these quantities the position of the center of the sun, as seen from
the center of Venus, is derived.
Wash. M. T.
0=E.
A.
i = dec.
Log G —
log distance.
1874.
Dec. 8a Sn
255° 36'
9'.'50
—22° 52' 9'.'48
9.8575394
11
255 49
8.82
54 3.58
309
14
256 2
8.52
55 56.63
227
In the next place are obtained the following quantities, which are desig-
nated by the eclipse notation* of Chauvenet's Spherical and Practical As-
tronomy, which, for the most part, is identical with that of Bessel's Analyse
der Finsternisse. It must be remembered that Venus here takes the place
of the moon.
* The plane of reference passes through the center of the earth perpendicular to the axis of the
enyel oping cones ; a and cJ are the right ascension and declination of the vanishing point of the axis;
^1, the hour-angle of that point at the first meridian ; (?, the distance of the sun and planet ; x, y, the
coordinates of the axis in the plane of reference, y being taken positive toward the north, x positive
dx
toward that point whose right ascension is 90° + a; ■— and _i. are the hourly changes of x and y ; / is
the angle of the cone ; I, the radius of the cone in the plane of reference ; i = tan /.
116
COLLECTED MATHEMATICAL WORKS OF G. W. HILL
Wash. M. T.
1874.
Dec. 8d 8h
11
14
+37.6744
+ 8.4134
—20.8759
dz
dt
-9.74895
9.75838
-9.76782
+25.0318
32.7602
+40.4042
dt
+2.59020
2.56207
+2.53393
122 36.6
166 55 0.8
211 49 24.6
Wash. M. T.
1874.
Dec. 8a 8h
11
14
Exterior contacts.
/ I log ! log i
22'24'.'272 41.1254 1.614110 7.8141
.299 62 19 41
.324 68 24 41
Interior contacts.
I log I
.570
.595
38.4845
54
59
1.585286
296
301
lost"
7.8063
63
63
CURVES REPRESENTED ON THE CHARTS.
Having novr the necessary data, I proceed to explain the computations
which have been made for the purpose of drawing the charts. These charts
are designed to give the principal circumstances attending each of the four
contacts at any point of the earth's surface where it is visible. These circum-
stances may be taken to be the time at which the contact occurs, and the
position of the point of contact on the sun's limb. Hence, two classes of
curves have been plotted on the charts — first, curves upon which contact
occurs at the same instant ; and, secondly, curves upon ivhich contact takes
place at the same point on the sun^ s limb. These curves are evidently limited,
in both directions, by the curve upon which contact takes place in the hori-
zon. The readiest method of drawing them will be to compute a suflBcient
number of positions conveniently distributed on these curves, and through
these positions, plotted on the chart, draw the curves.
As convenient formulas for the purpose are not found in the treatises
on practical astronomy, I will develop them here.
It will be amply sufficient to determine the position of these curves to
within a minute of arc. Hence, as the horizontal parallax of Venus is
only 33", the effect of parallax on the right ascension and declination of the
point of contact may be neglected. Then the position of this point can be
found by the equations,
the upper sign being used for the exterior contacts, and the lower for the
interior. With sufficient approximation, these equations may be written
£^'=5±^('V-5).
CHARTS AND TABLES OF THE TRANSIT OF VENUS 117
The exterior contacts last about 21 minutes on the earth's surface, and
the interior contacts about 25 minutes. The quantities a' and d' vary so
slowly that they may be computed for the middle of the duration of each
contact on the earth's surface, and then supposed constant for this duration.
In this way the following values have been obtained :
Wash. M. T. a' d'
b m „ ; „ ,
For exterior contact at ingress 8 40 255 57 —22 38
For interior contact at ingress 9 10 255 57 22 37
For interior contact at egress 12 48 255 51 22 34
For exterior contact at egress 13 18 255 51 — 22 34
The investigation to be made is conveniently divided into two problems
Problem I. — Tv find the point 6f the earth's surface at which contact takes
•place at a given time and at a given altitude.
Let
0) = the longitude of the required point west from the first meridian ;
y = its latitude ;
/I = the sidereal time at the first meridian ;
h = the given altitude ;
6 =: the parallactic angle at the point of contact ;
if— fj. — a'— (o = the hour-angle of the point of contact.
The general formulas of spherical trigonometry, applied to the triangle
formed by the zenith, the pole, and the point of contact, give these equa-
tions :
cos ^ sin S' = cos hsind,
cos f cos !?'= cos d' sin h — sin d' cos h cosO ,
sin ?> = sin d' sin h + cos d' cos h cos9.
As soon as 6 is known, these three equations, together with the equa-
tion,
oj = /J. — a'— y= jUi — ^'
give the position of the required point. To obtain B, resort must be had to
the equation defining the condition of contact, viz. :
= x'+y' — 2ixS + yyj) + p' — C^
In place of x and y make the usual substitutions,
a; = m sin M,
y = m cos M,
then
S Bin M + rj COS M= ^ 2^^ .
118 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
The numerical value of each member of this equation is always less
than unity, and it will be determined, to a suflScient degree of precision, with
four decimals. The average value of the denominator 2m is about 80 ;
hence, in the numerator it will be sufficiently accurate to put p" =: 1 , and
2U^ = 2mi^, and neglect the term — ^*f^; and if terms multiplied by
i and e^ are neglected, it is plain that ^ = sin A. Thus simplified, the equa-
tion becomes
c • j# , Tir '"^' — ^^ + ^ . ■ ■ I 1-21.
^ 8\nM + T] cos M = 5 h t sin h — q— sm^ h .
The right hand member of this equation is a known quantity, and it
only remains to discover the expressions of £ and y; in terms of 6 to have the
equation determining 6.
The known expressions for ^ and t^ are
$ = P cos {»' sin I? ,
7]^p cos d sin ^' — p sin d cos y' cos i? .
But if terms of the order of e* are neglected,
, cos f
p COBp — —— ,
, (1 — e^) sin
' = sin S ,
cos
= Ml — ^'.
It is worthy of remark that the equation determining 6 remains the
same if h, instead of being exactly zero, is a small positive or negative
angle ; for sec h will be sensibly unity, and, B and G being small, the terms
B sin h and C sin^ h may be neglected. Hence, in taking into account the
effect of refraction on the position of points, where contact takes place in
the horizon, 6 may still be derived from the equation,
cos (0 — y)=:Ap,
but it will be necessary to make A =: — (the horizontal refraction) in the
equations determining ^ and S-'.
The particular case where h = 90°, or contact in the zenith, requires
notice. Here the equation determining 6 reduces to
A + B + C = 0.
This determines the time at which the phenomenon takes place ; and the
equations for the position of the point reduce to
? cos iV— OTo cos (ilfo — -A'^) + (^ — i sin h) cos (Q — iV^) .
The values of ^ and >7 found in the first problem must be substituted in
these equations. The first member of the first of these equations is obtained
simply by writing iV-f- 90° for M in the first member of the corresponding
equation of the first problem. Hence, making
L'= 1— e'coa' d' sin' JSr,
X' = — \e^ cos^ d' sin 2N + v sin d',
yf ^N+k' + 90°,
these quantities are constant for the duration of each contact on the earth's
surface, and there is obtained the equation
p\^ cosN—Tj siniV] = i^'cosAcos(» — /) + ^' cos(iV+'<') sinA.
Consequently, if
A' = '^ sin {M, -N)-^, sin {Q -N),
B' = ^ sin iM„ -N)—^ cos {N + x') ,
where Q has been put equal to M^ in the term multiplied by i, the equation
determining 6 in this problem becomes
cos {e — r')=P sec h [A' + B' sin A] .
CHAKTS AND TABLES OF THE TRANSIT OF VENUS 123
The equation giving the value of nt is only needed for the purpose of
obtaining ii[, which it is necessary to have in order to get co from S''. In
this it will be sufficiently accurate to put for £ and vj their approximate
values,
f = cos A sin e ,
rj = cos hsinO,
and neglect the term multiplied by i ; then
nf = cos h cos (0 — W) — nio cos (J/"„ — W) + I cos (Q — N).
If 1^0 denote the value of ^( at the epoch from which t is counted, fi' the
motion of /u{ in a unit of time, and
the expression for iii[ is
M
f4 = A" + '=- cos h cos (5 — iV") .
71
After 6 and ^[ have been determined from the equations just given, the
position of the point on the earth's surface is found by means of the same
equations as in the first problem. Thus it appears that the solutions of the
two problems are quite similar, the only differences being that the term cor-
responding to O sin^ h, in the factor of the equation which determines 6, is
wanting, and that a separate computation must be made for n[ ; and the
remarks to be made regarding the solution of the equation determining 6 ,
and the limits between which Q and h must be assumed, in order that solu-
tion may be possible, are quite similar to those made in the first problem.
While B' and y' are constant for the duration of each contact on the earth's
surface. A' and A" involve the variable Q, and may be tabulated with Q as
the argument within its limiting values. The equation determining 6 gives
two values for this quantity, corresponding to the two points on the earth's
surface, which satisfy the conditions of the problem ; and p must be deter-
mined separately for each.
The condition of contact taking place at a given point on the sun's
limb, and at the maximum altitude, is
cos A = ± p \_A' + B' sin A] ,
and the equations
A'p=±\,
124 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
give the limiting values of Q. In finding the points on the curves of the
second class, which are common to the curve of contact on the horizon, 6 is
derived from the equation
cos (e — /) = A'p ,
but h=^ — (the horizontal refraction) in the equations which determine ^
and S''. In computing the value of [I'l for each of the two solutions of the
problem, it will be noticed that, with sufficient approximation, the second
terra has the same numerical value but opposite signs in the two solutions ;
and, in the case of maximum altitude for a given value of Q, the equation
becomes simply
K = A".
In this case also, it will be advantageous to compute four auxiliary
quantities from the equations,
^ cos £ = cos d', y sin e' = sin d',
^ sin £ = sin d' cos d , p' cos e' = cos d' cos 8',
by means of which the equations determining (p and ^ take the form,
cos y sin &' = cos h sind,
cos ^ cos &' =p sin (h — e) ,
sin 9> =p' cos (h — e') .
As & is constant in this case, p,p', e, e', are so likewise, provided that
after the point of maximum altitude has passed the zenith, h be supposed
to increase from 90° to 180°, or, in other words, that 180° — h be used
instead of h.
VALUES OF THE QUANTITIES EMPLOYED.
Denoting the four contacts in their order by the symbols I, II, III, and
IV, the values of the various quantities employed in the foregoing discus-
sion are :
I II III IV
Epochfrom which; is counted, 8" 40°" 9" 10°" 13" iS" 13" 18""
V +18' +16' -6' -8'
logX', 7.9187 7.9158 7.9359 7.9417
/, 54° 20' 58° 33' 100° 46' 104° 14'
logZ, 9.9989 9.9986 9.9977 9.9977
^ +3' +4' -3' -1'
logB, W7.1880 7.2238 m7.3475 M7.3411
N, 284° 50' 30".5 284° 48' 49".5 284° 36' 36".5 284° 34' 65".6
CHARTS AND TABLES OF THE TRANSIT OF VENUS
125
L',
m,
-ji-siniM.-N),
log[-^], .
r
logB'. .^
^'cos(if,-iV),
log
log
"V.
in minutes of arc
— in minutes
n
logi?,
logi>' ,
e', .
of arc ,
I
II
III
IV
9.9977
9.9977
9.9977
9.9977
-2'
-1'
+ 7'
+ 8'
-34.0289
+ 34.0187
+ 34.0188
+ 34.0290
1.616412
Ml.587591
wl.587600
1.616423
14° 49'
14° 48'
14° 44'
14° 43'
7.3788
7.3517
7.3349
m7.3605
166° 23'
166° 26'
166° 43'
166° 45'
3.5657
3.5369
3.5368
3.5656
1.9516
1.9516
1.9515
1.9515
9.99758
9.9978
. 9.9979
9.9979
31° 7'
21° 56'
-21° 54'
-21° 54'
9.9876
9.9876
9.9877
9.9877
-156° 40'
-156° 41'
-23° 15'
-23° 15'
The quantities which vary with the time and with Q are given in the
following tables.
I. — For Exterior Contact at Ingress.
Wash.
M.T.
A
log c
7
/*'!
Wash.
M. T.
A
log C y
/^'i
h m
V
/
b m
8 29
+1.0339
•n 8.0752
51 29
128 56
8 40 -
-0.0313
n 8.0864 49° 25
131 41
30
0.9360
.0762
51 18
129 11
41
0.1268
.0874 49 13
131 56
31
0.8383
.0772
51 7
129 26
42
0.2220
.0884 49 1
132 11
32
0.7408
.0782
50 56
129 41
43
0.3170
.0894 48 50
132 27
33
0.6435
.0793
50 44
129 56
44
0.4117
.0904 48 38
132 42
34
0.5464
.0803
50 33
130 11
45
0.5061
.0914 48 26
132 5T
35
0.4495
.0813
50 22
130 26
46
0.6003
.0924 48 14
133 12
36
0.3529
.0823
50 11
130 41
47
0.6942
.0934 48 2
133 27
37
0.2565
.0833
49 59
130 56
48
0.7878
.0943 47 50
133 42
38
0.1603
.0844
49 48
131 11
49
0.8811
.0953 47 38
133 57
39
+0.0644
.0854
49 36
131 26
50
0.9742
.0963 47 26
134 12
8 40
—0.0313
n 8.0864
49 25
131 41
8 51 -
-1.0670
n 8.0973 47 14
134 27
e
A'
A"
1
A'
A"
« A'
A"
46°50'
—1.0360
133° 54'
48° 30'
—0.3841
132° 26
50° 10' +0.2969
130° 58
47
0.9721
133 45
40
0.3173
132 17
20 0.3666
130 49
10
0.9079
133 36
50
0.2502
132 8
30 0.4366
130 40
20
0.8435
133 27
49
0.1828
131 59
40 0.5069
130 31
30
0.7788
133 19
10
0.1152
131 51
50 0.5774
130 23
40
0.7138
133 10
20
—0.0472
131 42
51 0.6482
130 14
50
0.6484
133 1
30
+0.0211
131 33
10 0.7193
130 5
48
0.5827
132 52
40
0.0897
131 24
20 0.7907
129 56
10
0.5168
132 43
50
0.1585
131 15
30 0.8624
129 47
20
0.4506
132 35
50
0.2276
131 7
40 0.9343
129 39
48 30
—0.3841
132 26
50 10
+0.5
!96g
130 58
51 50 +1.0065
129 30
126
COLLECTED MATHEMATICAL WORKS OP G. W. HILL
II For Interior Contact at Ingress.
Wash.
M.T.
A
log c
7
l^\
Wash.
M.T.
A
log C
y
Z''.
h m
^
1 o i
h m
• /
a /
8 57
+1.0501
n 8.1034
46
1 135 57
9 10 -
-0.0238
n 8.1155
43 13
139 13
58
0.9651
.1044
45 48 136 12
11
0.1037
.1164
43
139 28
59
0.8807
.1053
45 36 136 27
12
0.1832
.1173
42 47
139 43
9
0.7967
.1063
45 23 136 42
13
0.2623
.1181
42 33
139 58
1
0.7130
.1072
45 10 136 57
14
0.3410
.1190
42 20
140 13
2
0.6296
.1081
44 57 137 12
15
0.4193
.1199
42 7
140 28
3
0.5466
.1091
44 45 137 27
16
0.4972
.1208
41 53
140 43
4
0.4640
.1100
44 32 137 42
17
0.5746
.1217
41 39
140 58
5
0.3817
.1109
44 19 137 57
18
0.6515
.1225
41 26
141 13
6
0.2998
.1118
44
6 138 12
19
0.7280
.1234
41 12
141 28
7
0.2183
.1127
43 53 138 28
20
0.8040
.1243
40 58
141 43
8
0.1372
.1137
43 39 138 43
21
0.8795
.1251
40 44
141 58
9
+0.0565
.1146
43 26 138 58
22
0.9546
.1260
40 30
142 13
9 10
—0.0238
n 8.1155
43 13 139 13
9 23 -
-1.0292
n 8.1268
40 17
142 28
e
A'
A"
Q.
A'
A"
<2
A'
A'l
39° 50'
—1.0401
142° lO'
42° O'
—0.3965
140° 13
44° 10 +0.2964
138°19'
40
0.9924
142 1
10
0.3449
140 4
20
0.3517
138 10
10
0.9444
141 52
20
0.2930
139 55
30
0.4073
138 2
20
0.8961
141 43
30
0.2409
139 47
40
0.4632
137 53
30
0.8474
141 34
40
0.1885
139 38
50
0.5194
137 44
40
0.7985
141 25
50
0.1358
139 29
45
0.5758
137 35
50
0.7493
141 16
43
0.0827
139 20
10
0.6325
137 26
41
0.6997
141 7
10
—0.0294
139 11
20
0.6895
137 18
10
0.6499
140 58
20
+0.0242
139 3
30
0.7468
137 9
20
0.5998
140 49
30
0.0781
138 54
40
0.8044
137 1
30
0.5494
140 40
40
0.1322
138 45
50
0.8623
136 52
40
0.4987
140 31
50
0.1866
138 37
46
0.9204
136 44
50
0.4477
140 22
44
0.2414
138 28
10
0.9788
136 35
42
—0.3965
140 13
44 10
+0.2
964
138 19
46 20 +1.0375
136 27
III. — For Interior Contact at Egress.
Wash.
M.T.
A
log c
7
V-'^
Wash.
M.T.
A
log c
7
y-'^
h m
• 1
o ;
h
m
/
a /
12 35
—1.0188
n 8.1276
—10 53
190 41
12 48
—0.0100
■n 8.1162
—13 50
193 57
36
0.9440
.1268
11 7
190 56
49
+0.0706
.1153
14 3
194 12
37
0.8687
.1259
11 21
191 11
50
0.1515
.1144
14 16
194 27
38
0.7929
.1251
11 34
191 26
51
0.2329
.1135
14 29
194 42
39
0.7166
.1242
11 48
191 41
52
0.3147
.1126
14 42
194 57
40
0.6399
.1233
12 2
191 56
53
0.3968
.1116
14 55
195 12
41
0.5627
.1224
12 16
192 11
54
0.4793
.1107
15 8
195 27
42
0.4850
.1216
12 29
192 26
55
0.5622
.1098
15 21
195 42
43
0.4069
.1207
12 43
192 41
56
0.6454
.1089
15 34
195 57
44
0.3284
.1198
12 56
192 56
57
0.7290
.1079
15 46
196 12
45
0.2494
.1189
13 9
193 11
58
0.8129
.1070
15 59
196 27
46
0.1700
.1180
13 23
193 27
12
59
0.8972
.1060
16 11
196 42
47
0.0902
.1171
13 36
193 42
13
0.9819
.1051
16 24
196 57
12 48
—0.0100
71 8.1162
—13 50
193 57
13
1
+1.0669
n 8.1041
—16 36
197 12
CHAKTS AND TABLES OF THE TRANSIT OF VENUS
127
10 30
—1.0148
191 4
40
0.9669
191 13
60
0.9187
191 22
11
0.8702
191 31
10
0.8215
191 40
20
0.7724
191 49
30
0.7230
191 58
40
0.6733
192 7
50
0.6233
192 16
12
0.5731
192 25
10
0.5226
192 34
20
0.4717
192 43
30
0.4205
192 51
-12 40
—0.3690
193
-12 40
—0.3690
193
50
0.3173
193 9
13
0.2653
193 18
10
0.2131
193 27
20
0.1605
193 36
30
0.1076
193 44
40
0.0545
193 53
50
—0.0010
194 2
14
+0.0528
194 11
10
0.1069
194 20
20
0.1612
194 29
30
0.2158
194 37
40
0.2706
194 46
-14 50
+0.3258
194 55
-14 50
15
10
20
30
40
50
16
10
20
30
40
-16 50
+0.3258
0.3813
0.4371
0.4931
0.5494
0.6059
0.6628
0.7200
0.7775
0.8353
0.8933
0.9515
+1.0101
194 55
195 .4
195 13
195 21
195 30
195 39
195 47
195 56
196 5
196 13
196 22
196 30
196 39
IV. — For Exterior Contact at Egress.
Wash. .
M. T. -*
log a y
H-\
Wash.
M.T.
A
log C
7
/''i
h m
13 7 —1.0536 n
8.0983 —17° 48
198* 43
h m
1318 —0.0144
n 8.0874
—19° 59
201° 29
8 0.9605
.0974 18 1
198 58
19 +0.0817
.0864
20 10
201 44
9 0.8671
.0964 18 13
199 13
20 0.1780
.0854
20 22
201 59
10 0.7735
.0954 18 25
199 28
21 0.2745
.0844
20 33
202 14
11 0.6795
.0944 18 37
199 44
22 0.3713
.0834
20 45
202 29
12 0.5853
.0934 18 49
199 59
23 0.4683
.0823
20 56
202 44
13 0.4908
.0924 19
200 14
24 0.5655
.0813
21 8
202 59
14 0.3960
.0914 19 12
200 29
25 0.6629
.0803
21 19
203 14
15 0.3010
.0904 19 24
200 44
26 0.7605
.0793
21 30
203 29
16 0.2057
.0894 19 36
200 59
27 0.8583
.0783
21 41
203 44
17 0.1102
.0884 19 47
201 14
28 0.9563
.0772
21 53
203 59
1318 —0.0144 n
8.0874 —19 59
201 29
13 29 +1.0545
n 8.0762
—22 4
204 14
Q A'
A"
e
A'
A"
«
A'
A"
—17° 30 —1.0021
199° 18
o ;
—19 10
—0.3487
200° 48
— 20°5o'
+0.3341
202° 16
40 0.9380
199 27
20
0.2817
200 57
21
0.4039
202 25
50 0.8737
199 36
30
0.2144
201 6
10
0.4739
202 34
18 0.8091
199 45
40
0.1468
201 15
20
0.5443
202 42
10 0.7443
199 54
50
0.0790
201 24
30
0.6150
202 51
20 0.6791
200 3
20
—0.0109
201 33
40
0.6859
203
30 0.6136
200 12
10
+0.0575
201 41
50
0.7572
203 8
40 0.5478
200 21
20
0.1262
201 50
22
0.8288
203 17
50 0.4817
200 30
30
0.1952
201 59
10
0.9006
203 26
19 0.4153
200 39
40
0.2645
202 8
20
0.9727
203 34
—19 10 —0.3487
200 48
—20 50
+0
3341
202 16
—22 30
+1.0451
203 43
BEGINNING, ETC., OF EACH CONTACT.
From the foregoing data are readily derived the times, and position of
the places, at which the following phenomena occur.
128 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
Wash. M. T. Longitude. Latitude.
h m p , o /
Contact I begins on the earth 8 29.335 55 27 +35 24
occurs in the zenith 8 39.530 131 34 —22 38
ends on the earth 8 50.292 244 25 —38 24
Contact II begins on the earth 8 57.572 65 53 +40 15
occurs in the zenith 9 9.520 139 6 —22 37
ends on the earth 9 22.630 257 24 —44 22
Contact III begins on the earth 12 35.216 36 40 —64 33
occurs in the zenith 12 48.314 194 2 —22 34
ends on the earth 13 0.244 235 18 +62 48
Contact IV begins on the earth 13 7.548 58 15 —61
occurs in the zenith 13 18.300 201 33 —22 34
ends on the earth 13 28.471 251 17 +59 20
APPROXIMATION OF THE CURVES TO CIRCLES.
The curves to be drawn on the charts approximate so closely to circles
of the sphere that it has been deemed sufficient to compute the positions of
three points on each curve, namely, the two at which contact occurs on the
horizon, and the one for which the altitude is a maximum, and then regard
the curve as a circle of the sphere passing through these points ; and, as the
stereographic projection has been chosen for the delineation of the charts,
the projected curves will also be circles.
But it will be of interest to determine beforehand how great an error
can be produced by this assumption. And first, in the case of the time-
curves, let a be the radius of the circle of the sphere passing through the
three points, and adopt the subscripts (0), (l), (2), (3), for the quantities
which refer respectively to the pole of this circle, the points of contact on
the horizon, and the point of maximum altitude. Then (T and the position of
the pole of this circle are determined by the equations,
sin fi sin ^o + cos fi cos f^ cos {■ft[ — '9j) = cos „ = sin y^ ,
sin y„ = cos >-„ cos {d' — r) ,
<"a = All — ''o •
CHARTS AND TABLES OF THE TRANSIT OP VENUS 13 j
If the distance of any point on the time-curve from this pole be denoted
by a', then a' — a may be taken as a sufficiently exact measure of the error
committed by our method of drawing the curve.
But
cos a' = xXo + yyo + zz^ ,
X = cos h sin e ,
y = cos d' sin h — sin d' cos h cos d ,
z = sin d' sin h + cos d' cos h cos ,
whence
cos 0-' = cos h sin sin y^ + cos h cos cos y^ cos t + sin A cos ^„ sin t .
or, as cos r may be put equal to unity,
cos 0-' = (A3 — ;3) sin h + cos h cos (0 — ;-„) .
The quantity a' — a is composed of two parts independent of each
other ; the first depending on the curvature of the cone enveloping the sun
and Venus, and proportional to the quantity we have denoted by G ; the
second due to the non-sphericity of the earth and proportional to e'. These
parts can then be determined separately.
First, from the equations,
cos A3 = ±{A + Bsmh, + G sin' h,) ,
cos ^ = ± A ,
is obtained, with sufficient exactness,
h,-p = :f (5 + C sin /?) .
But
cos h cos (6 — Ya) = ± (_A + B sin h + C sin7t) ,
cos a — ± A ,
thus
cos a' — cos (T = ± C sin A (sin h — sin /J) .
Secondly, from the equations,
cos 7*3 = ± Ap^ = ± ^ Pi — -^ (sin d' sin /3 ± cos d' cos /S cos yf ,
cos /5 = ± A\l—\^ cos' d' {A"" cos %r + sinV)] ,
we find that the part of A3 — ^ proportional to ^ is
A3 — /? = I e' cos /9 [sin' d' sin /3 ± sin M' cos /3 cos >- — cos' d' sin ji sin' •,-] .
132 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
Also
COS h cos (0 — j-„) = ± Ap ^: ^e'' cos'' d' sin 2/ cos' /3 cos h sin (0 —)■) ,
= ± ^ 1 — -^ (sin d' sin h + cos d' cos h cos ^)' ,
q: ~ cos" (^' sin 2r cos= /? V (sin' /? - sin' k) ,
cos A cos e = ^ COS >- — V (sin' /9 — sin' h) sin ;- ,
where the sign of sin (0 — y) must be attributed to the radical
is/{s\v? ^ — sin^A).
After some reductions it will be found that
e'
cos a' — cos ff = -„ (sin' d' — cos' d' sin' y) cos /3 sin h (sin ^ — sin h)
') cos /J q= C (sin ^ — sin A) sin h
e'
+ ^ sin M' sin >- cos /J sin A V(sin' /? — sin' h) .
Al
It will be seen that this expression vanishes when ^ =: and when h = ^ ,
as it should. Differentiating it with respect to the variable sin h , in order
to obtain its maximum value, we arrive at an equation of the fourth degree
in sin h. Hence we are obliged to content ourselves with a superior limit
to the maximum value, which, however, for practical purposes, may be
regarded as identical with it. The first term of the expression has its
maximum value when sin h^= i sin j3 , and the second when sin h =
sin /3. Substituting these values in their respective terms, we obtain
"' — 0- = TTT (cos' d' sin' y ± 2 sin 2d' sin r — sin' d') sin 2/J ± -^ sin /J ,
Id 4
where the ambiguous signs, in both cases, must be so taken that the largest
numerical value of the expression will be obtained. Replacing e" and C by
their values, and taking for the factor which involves d' and y the greatest
value it can have, it results that a' — a cannot exceed
11' sin ^ + 2' sin 2^ ,
and the maximum value of this with regard to the variable /3 is less than 1 2.'
Having regard to the scale on which the charts have been constructed, this
CHARTS AND TABLES OP THE TRANSIT OP VENUS 5^33
quantity may be considered as within the unavoidable errors produced by
imperfection of drawing.
It is worthy of remark that, in our method of drawing the curves, the
error is only a fourth part of that which results from neglecting altogether
the curvature of the cones enveloping the sun and the planet, as has gene-
rally been done in treatises on practical astronomy.
The investigation of the error in the case of the second class of curves
differs somewhat from that of the first class, on account of [i[ not being con-
stant for all points on the curve. The equations determining a and the
position of the pole are
sia ,, cos (wj — w,,) = cos (t ,
sin w. sin «>„ + cos v^ cos «>„ cos (w, — <«„') = cos «■ .
where
sin ^2 sm ^0 + cos y^ cos „ cos (w^ — <«„) = cos «■ ,
, = A"+'Lsin(d,-r')-^',
If we put
0,3 = A"- K
9
= ^sin(e,-/)^±^V(l-^'^),
= A"-K,
g is 2u small angle, whose square may be neglected, and the equations, using
the notation given in the case of the first class of curves, take the shape
(^1 + 9yi) a^o + iVi - c/^i') % + 2, 2o = cos cos
= (^3 — i3) sin A + cos A cos (5 — ^o)
+ '^ V (sin" /J — sin" h) [cos <^' sin y^ sin A + sin d' cos A sin {6 — ;-„)] ,
— {]h — /5) sin A + cos A cos (ff — ^-j) ± — sin d' (sin" /3 — sin" A)
± — cos d' sin / sin A V (sin" /3 — sin" h) ,
where the upper or lower sign is taken according as A' is positive or nega-
tive, and the sign of sin (0 — y') is assigned to the radical
V (sin" /3 — sin' h) .
The part of cos a' — cos a which involves the factor ^ will be found to be
^ n
± — sin d' sin h (sin B — sin K) ,
± — cos d' sin / sin h V (sin" /J — sin" h) .
The part proportional to e^ is obtained from the analogous expression
in the case of the first class of curves, simply by changing y into y', and
thus is
-^ (sin" d' — cos" d' sin" /) cos /? sin A (sin /J — sin h)
Z
e"
+ sin SfZ' sin / cos /J sin h V (sin" /5 — sin" A) .
CHARTS AND TABLES OF THE TRANSIT OP VENUS
135
Combining these two parts, we have
cos 0-' — COS = the latitude of the place, positive when north,
01 = its longitude from Washington, positive when west,
loge = 8.9123, log (1 -«'') = 9.99709, sin;f = esin^,
h = secxcosp, ^= (1 — e') sec/ siny>,
a = A — h sin (fi — w),
1 = B— Eh + Gh cos(fi — (u),
c— — C+ Fk — Hh cos(At — w),
m=^ i\l be, (usually with the same sign as a).
CHARTS AND TABLES OP THE TRANSIT OP VENUS ^45
If m = a, the time Tq is correctly chosen. If m differs from a, a cor-
rection of the assumed time may be obtained in seconds, by the formulas,
log/ =9.8617,
a' = A' — ij.'h cos (/i — (u),
V = B' - ii'Ohsin{[i — a>),
, _ 10000 (m — a)
a' + b' cot Q '
and the actual Washington time of contact will be
and the local mean time of the phenomenon will be
T, + t-w.
Q must be taken of the same sign with a, and is a suflSciently near
approximation to the angular distance of the point of contact, reckoned from
the north point of the sun's limb toward the east.
To find F, the angular distance of the point of contact from the vertex
of the sun's limb, positive toward the left, we have the formulas,
^ sin P = sin = + 21° 18' 12"
w
==80=
' 51' 45"
(1)
loge = 8.91220
(3)
log sin = 9.96926
(4) + (5)
logA = 9.96945
Prom chart No. 2 the Washington mean time of contact is found to be
nearly S^ 58™ 24^ which will be taken as the value of Tq.
19
146
COLLECTED MATHEMATICAL WORKS OF G. W. HILL
Computation of t , tfie correction of T^ .
fi = 136°32'.8 (9)
f^—a>= 55°4r.O
(1) log sin (m — )- 9.31684?j
(7) + (8) \ogHhco&{ix—u)) - 9.30367m
(5) + (7) log t^'h cos {ii-w)- 9.5823
(4) + (5) + (6)log/ffAsin(M - <«) = 9.3444w
(30) log 5 =1.8163035
(21) log c = 1.0596051
(33) = i [(30) + (31)] log m = 1.4379043
(32) — (30) log tan i e = 9.6317108
Angle from north point, Q = 45°35'10"
(10)
(11)
(9) + (10)
(10) + (11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
\ogE= 9.96333
log^= 9.55755
logi?'= 9.96558
log Ek= 9.53078
log Fk = 9.52313
A= + 28.1840
- Asin(/i — «<) = — 0.7698
B= + 66.0334
-Ek = — 0.3317
Gh cos {iJ. — w)=:— 0.2074
- C = + 10.9364
Fk- + 0.3335
- Hh cos {ti—m)= + 0.3012
(13) + (13) a = + 37.4143
(14) + (15) + (16) 5 = + 65.4943
(17) + (18) + (19) c = + 11.4711
OT = + 27.4097
m—a = — 0.0045
(29)
log cot Q = 9.9936
(33)
A' = —
27.08
(30)
log V = 0.8686
(34)
— fi-'h cos (/i — m) = —
0.38
(29) + (30) log V cot Q = 0.8623
(35)
^'= +
7.17
(26)
— ix'Gh s\n{ix — w) = +
(35) + (36) h'= +
0.32
7.39
(31)
\og{m—a) + 4 = 1.6533W
(27) =
= (23) + (24) a' = -
27.46
(32)
log (a' + V cot Q) = 1.3050W
(28)
V cot G = +
7.28
(31) -(33) log if = 0.3483
(27) + (38) a' + 5' cot § = —
20.18
Assumed time, ....
, ,
h m B
. To = 8 58 24.0
Correction of the assumed time, .
.
;;= +2.3
Washington time of the contact, .
,
8 58 36.2
Honolulu time of the contact.
.
3 34 59.2
We have
F= — 9° 36.0.
G = 55° 1'.2, and the angle from the vertex,
CHARTS AND TABLES OF THE TRANSIT OP VENUS 147
The corrections which should be applied to the times of the four con-
tacts for determinate changes in the elements, exclusive of the effect of a
change in the constant of solar parallax, are given by the following formulas.
In these
Sq =the correction of the sun's longitude,
dL=the correction of the orbit longitude of Venus,
5g2=the correction of the longitude of the node of Venus,
SB=the correction of the sun's latitude,
5s=the correction of the semi-diameter of Venus at the mean distance,
3s'=the correction of the semi-diameter of the sun at the mean distance.
All these quantities being expressed in seconds of arc, the corrections
of the times of the four contacts, in their order, are
8 S B S
dT,= + 48.4 (5© - dL) + 3.00 {$L -dQ,+ 16.2SB') - 97.4 Ss - 26.1 ds',
ST, = + 50.9 (5© - SL)+ 3.94 (5i -dQ,+ U.9SB) + 116.3 ds - 31.3 Ss',
ST, = + 30.3 (SQ-SL)- 4.68 {8L -8Q,+ 16.955) - 116.3 Ss + 31.3 Ss',
ST^ - + 30.1 (5© - SL)— 3.75 {dZ -d5i+ W.9SB) + 97.4 Ss + 36.1 Ss'.
These expressions have been computed for the center of the earth, but
they may be taken as approximately exact for any point on the surface.
An approximate value of the co-efficient of the correction of the con-
stant of solar parallax, for any place, may be found by subtracting from the
ascertained Washington mean time of contact at the place, the Washington
mean time of the same contact occurring in the zenith, given on page 128.
Thus in the example for Honolulu, given above, one finds that
dT,= (8''58""26'.3 — 9''9"'.530)^,
= — 665=.0— %
To
where Tto denotes the constant of solar parallax. It must be understood,
however, that this method gives quite rude approximations.
POSITION OF THE PLANET ON THE SUN S DISC.
All that precedes relates to the contacts; but it may be desired to find
the position of the planet, when on the sun's disc, relative to the center of
this body. For this purpose the following tables of data are appended.
148
COLLECTED MATHEMATICAL WORKS OF G. W. HILL
Waeli. M. 1
h m
8 30
D.
Change of x
in 1 minute.
y
Change of y
in 1 minute.
V-
d
+32.7995
—0.16251
+26.3257
+0
.04309
129°29'.7
— 22°52'.5
40
31.1744
16252
26.7565
4306
131 59.4
52.6
50
29.5492
16252
27.1870
4304
134 29.0
52.7
9
27.9239
16253
27.6173
4301
136 58.7
52.8
10
26.2985
16254
28.0473
4298
139 28.4
52.9
20
24.6730
16255
28.4770
4296
141 58.1
53.0
30
23.0474
16256
28.9065
4293
144 27.8
53.1
40
21.4217
16257
29.3357
4291
146 57.5
53.2
50
19.7960
16258
29.7647
4288
149 27.1
53.3
10
18.1702
16259
30.1934
4286
151 56.8
53.4
10
16.5443
16260
30.6219
4283
154 26.5
53.5
20
14.9183
16260
31.0501
4281
156 56.2
53.7
30
13.2922
16261
31.4780
4278
159 25.9
53.8
40
11.6660
16262
31.9057
4275
161 55.6
53.9
50
10.0397
16263
32.3331
4273
164 25.3
54.0
11
8.4134
16264
32.7602
4270
166 55.0
54.1
10
6.7870
16265
33.1871
4267
169 24.7
54.2
20
5.1605
16266
33.6137
4265
171 54.3
54.3
30
3.5338
16267
34.0401
4262
174 24.0
54.4
40
1.9071
16268
34.4662
4259
176 53.7
54.5
50
+ 0.2803
16268
34.8920
4257
179 23.4
54.6
12
— 1.3465
16269
35.3176
4254
181 53.1
54.7
10
2.9734
16270
35.7429
4252
184 22.8
54.8
20
4.6004
16271
36.1680
4249
186 52.5
54.9
30
6.2276
16272
36.5928
4247
189 22.2
55.0
40
7.8549
16273
37.0173
4244
191 51.9
55.1
50
9.4822
16274
37.4416
4242
194 21.6
55.2
13
11.1096
16275
37.8656
4239
196 51.3
55.3
10
12.7371
16276
38.2894
4236
199 21.0
55.4
20
14.3647
16276
38.7129
4234
201 50.6
55.5
13 30
—15.9924
—0.16277
+39.1361
+0.
04231
204 20.3
—22 55.6
The distance B in seconds of arc of the center of Venus from the center
of the sun, and the angle of position Q, of this distance, counted from the
north point toward the east, are obtained by the formulas,
J sin § = « — p cos 55' sin * ,
id cos Q = y — ;o sin
= + 13° 4'.3, w = 203° 43'.6,
whence for this place
AsinQ =x— [9.9886] sin ■» ,
AcosQ = y — 0.2070 - [9.5787] cos ^ ,
. ^_ [8.0589JsinO» — 34°1')
^■^ 1 + [8.3896] cos (^ — 5° 50') "
Assume ll'' 4". 6 as an approximate value of the time; for which
M= 168°3'.8, 1? = - 34° 38'.8 , a; = + 7.6653,
y-+ 33.9566 , JsmQ-+ 8.2190 , Aco8Q= + 33.4378 ,
Q= 14°13'5".6, ^=-36'0".6, Q — E = 14° 49' 6".2 .
150 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
The error is, then, — 6' 22". 9, and the correction to the assumed time,
— 383".9
1025"
X 1"°=: + 0^3?'34.
If the computation be repeated for the time ll*" 4™.9734, the error of
the value of Q — E will be found to be only 13". Regarding this result as
suflBciently accurate, we compute, for this time, Q and D, and find
Q = 14° 6' 33", D = 819".42 = 13' 39".42 .
These distances and angles of position are, it must be remembered,
actual, not apparent. To obtain the last, the effect of refraction would have
to be considered.
LOCALITIES FAVORABLE FOR THE DETERMINATION OF PARALLAX.
A list of localities favorably situated for observations of the contacts,
with a view to the determination of the parallax, may be given in a few
words.
For the ingress accelerated by parallax, we have, in the first place, the
Hawaiian Islands; next, the most southerly and westerly of the Aleutian
Islands, the southern part of Kamchatka, and Japan, especially the northern
islands; also the Marquesas Islands, and, if more stations are desired, per-
haps in the long series of islands stretching west-northwest from the Ha-
waiian Islands some might be found available. We may mention the small
islands lying between the Hawaiian and Marquesas Islands.
For the ingress retarded by parallax, we have the islands of Saint Paul,
New Amsterdam, Kerguelen, Bourbon, Mauritius, Diego Rodriguez, Crozet,
Prince Edward, and Madagascar, where, however, only the interior contact
will be visible, and on the eastern coast at an altitude from 5° to 6°.
For the egress accelerated by parallax, we have New Zealand and the
small islands to the southward and eastward. With respect to the latter,
we may note that on some maps may be found a group of small islands,
called the Nimrod Islands, and placed in longitude 80° west from Washing-
ton and in latitude 57° south. Here the interior contact occurs at an alti-
tude of 9°, and if these islands are of a suflBcient size for the establishment
of an observing station on them, it would be a tolerably good one, as far as
geographical position is concerned. To these we may add Norfolk Island,
New Caledonia, the Fiji Islands, Van Diemen's Land, and the southeastern
part of Australia.
For the egress retarded by parallax, Southwestern Siberia, the region
immediately east of the Caspian Sea, Persia, the Caucasus, Asia Minor,
Syria, Arabia, and Egypt contain the best stations.
CARNEGIE INSTITUTION OF WASHINGTON
CHART No.
TRANSIT OF \a^:NUS. DEC. 8.1871
CtlARTXO.l. INGRESS, EXTKJ^rOR CONTACT
Scale for filtitudeH
is° 2o° 25" :w
LEG END
The broken lines in blue are for synchronism of runiaci.
The broken lines in red ai-e for contact at Uie same point
oftlie sular disU.
CARNEGIE INSTITUTION OF WASHINGTON
CHART NO. 2.
TRANSIT OF VKXl^S, DEC. 8. 1874
CHART NO. 2. INGKESS, INTERIOR COXTACT
Scale for altitudes
5" 10° 1G° 20° 2S°
LEGEN D
The broken Lines in blue are for syuchronismof tonlact.
The broken lines in red Qi-e for contiujt altlie some poinl
of the solijr disk.
CARNEGIE INSTITUTION OF WASHINGTON
CHART NO. 3.
TRANSIT OF VKX US, DEC. 8. 1874
CHART XO. 3. TiGRESS.rNTERlOR COXTACT
Scale for altitudes
B* iO" 15°
LEGE N D
The broken lines in blue fire for synchronism of euntat't.
TKo broken, lines in red are for contact at tlie same poini
of lUe soloi- disU.
A METHOD OF COMPUTING ABSOLUTE PERTURBATIONS
151
MEMOIE No. 14.
A Method of Computing Absolute Perturbations.
(ABtronomisohe Nachrlchten, Vol. 83, pp. 309-324, 1874.)
The object of this article is to call the attention of astronomers to the
notable abbreviations which are produced in some parts of the formulas for
perturbations by the introduction of the true anomaly as the variable
according to which the integrations are to be executed. Prof. Hansen, in
his later disquisitions, has substituted the eccentric anomaly as the indepen-
dent variable in place of the mean anomaly, or what is the same thing, the
time; and he regards this step as constituting a remarkable amelioration of
the method. The method here explained will, as far as coordinates are con-
cerned, be the same as that of Laplace, but the same use will be made of
the true anomaly in the elliptic orbit, as independent variable, as that which
Hansen has made of the eccentric anomaly.
The following notation and equations are so familiar that they seem to
need no explanation:
E = m'[^-^^)+m"[^-^-^) +
cPx II _ dR
dt-
d''z fi
(1)
Let us now suppose that each coordinate of the disturbed planet is sep-
arated into two parts, such that
x = x, + Sx, y — yo +
Z =:Zo + Sz,
the first of which, Xq, y^, Zq satisfy the differential equations
where rl=^XQ + yl +zl, and the second, 8x, 8y, Sz are of the order of the
disturbing forces.
152 COLLECTED MATHEMATICAL WORKS OP G. W. HILL
It is evident that this separation is, to a certain extent, arbitrary, as
certain functions of t might be added to Xq, «/o, Zq without their ceasing to
satisfy the differential equations determining them, and then hx, hy , hz
would necessarily be diminished by the same functions. This indetermina-
tion is eliminated in different ways according to the circumstances attending
the computation of the perturbations.
If ccq, ^oi 2o are derived from the elements osculating for a certain epoch,
it is plain that hx, 8y, 3z ought to vanish at this epoch, as also their first dif-
ferentials with respect to the time. This will be accomplished by taking
all the integrations, which 8x, Sy, Sz involve, between the limits ^ = and
t^t. If the perturbations are computed from so called mean elements, the
six arbitrary constants which 8x, 8y, Sz involve, must be determined in
accordance with the suppositions upon which the mean elements have been
derived.
We will now write
r = r„ + dr,
or ox ^ oy 9z
dR is then the differential of jB when the coordinates of the disturbed planet
alone vary. The last equation is evidently correct, when, in the first mem-
ber, we suppose B to be expressed in terms of r and two other coordinates
which make — , — , — independent of r.
r r r
By multiplying the equations which determine x, y, z, severally by
2dx, 2dy, 2dz, adding the products and integrating,
df raj' ^ '
where ^ is the constant added to complete the integral, and we suppose
that it is such that the equation
dx\ + dy\ + dzl ^+ A =
W r„ a
is satisfied; if there is any residual constant part, it must be supposed con-
tained in the term 2 / dR. By multiplying the differential equations deter-
A METHOD OP COMPUTING ABSOLUTE PERTURBATIONS
153
mining x, y,z, severally by these quantities and adding the products to
equation (2), we get
dr r a J Qr
By using the equation r =.ra + Sr, this can be readily transformed into
tSrfil + lirJr = 2rdB + r^^-iiP-\'^.
dv r, J dr dt' rlr
In like manner equations (1) can be transformed into
df rl dx \ri r'J
d
_ dR
+
dt
■ ri oz \r% r I
For the sake of brevity put
J dr df rir
Then our differential equations take the form
(3)
d\r,br)
df
+ -^roSr=Qr,
dt' ^ rl
Q'.
(4)
The problem of elliptic motion being supposed completely solved, -4-
is a known function of t, and
df ^ ii^-^'
(5)
is a linear differential equation. According to the theory of this class of
differential equations, the value of q has the form
q = K^q^ + K,q, ,
20
154
COLLECTED MATHEMATICAL WORKS OF Gk W. HILL
K^ and E^ being the arbitrary constants and q^ and q^ any particular solu-
tions independent of each other. Then there must necessarily exist the two
equations
^ + Aff =0 ^ + ^ , = 0. (6)
By the elimination of -^ from these is obtained
This is an exact differential and integrating
g.'^g.-gAi^ a constant. (7)
dt
This constant is arbitrary and may be taken at will ; for the sake of sim-
plicity, we assume it equal to unity.
Taking now the more general equation
^'1 + JLa-0
let us eliminate ^ from this and equations (6). We get
r%
dt' ^^' '
and, taking the integrals.
iJSL=J^=-K,+fq,Qdt.
Whence is obtained, regard being had to equation (7),
q = E^q^ + K£, + q^fq^Qdt — q^fq^Qdt.
The constants K^ and K^ may be regarded as contained in the integrals
/ q^ Qdt and / q^ Qdt. Applying these results to equations (4), there
result
A METHOD OP COMPUTING ABSOLUTE PERTURBATIONS
156
^« = 9'2 y ^iQ-dt — lif q^Q'<^i ■
(8)
These equations must satisfy the relation
r^Sr = x^Sx + y^Sy + z^Sz + \ [Sn^ + dy'' + 5«' — dr'] . (9)
It is, however, necessary to employ all of equations (8), since, in pro-
ceeding by successive approximations, as we are obliged to, we cannot get
the values of Q^, Qy, Q^, until hr is known. These equations contain, in
the general case, nine arbitrary constants, viz., the one added to the term
2 / dR in Q^, and the eight introduced by the eight integrals of equations
(8). But the last will be reduced to six, independent of each other, by the
condition (9), and the constant annexed to / dB will be determined in
function of these six, by the condition
In the case, mentioned above, of osculating elements, all the constants
are determined by making each integral expression vanish with t .
There is a remarkable procedure for reducing the right members of
equations (8) to contain a single integral expression, which is due to Prof.
Hansen. The factors qi, q^, outside the signs of integration, may evidently
be removed within, if it is agreed to regard the t they contain, as constant
in the integration. As it is necessary to keep this t distinct from the t of
the quantities already under the sign of integration, we may write r for it,
and, to denote that any quantity, which is a function of t, has its t changed
into t , we will write (~) above it. Thus making
N=Mx-M,, (10)
we have the very simple expressions
r,Sr= r NQ4t, te = C NQ.dr, Sy =, C NQ.dr , dz^ f NQ.dr . (11)
156 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
After the integration is finished, t will be replaced by t. Since t is
regarded as constant, an arbitrary function of t must be added to each of
these expressions, which, after t is changed into t, becomes an arbitrary
function of t. These must, in each case, be so determined that the expres-
sions (11) may coincide with (8). Any consideration of these arbitrary
functions will be rendered unnecessary, by agreeing to take the integra-
tions between limits, the upper of which is t itself, and the lower may be
any constant. In the general case, then, an arbitrary expression of the form
jE'i^i + E^q^ must be added to each equation. In the case of osculating
elements, mentioned above, if the lower limit is taken at zero, this arbitrary
expression vanishes.
Equations (11) may be exhibited in the form of definite integrals, thus
r,8r^- r NQ4r, Sx = — f NQ^dr, Sy = - f JSrQ,dT, Sz=- f" NQdr.
»/y t/o t/o «/o
A^may be regarded as an integrating factor whose value is virtually zero,
but a part of the time, involved in its expression, is regarded as constant in
the integration.
The values of ji and g'g must now be determined. If
- \ a"
nt + c=:Z^u — e sin u ,
where c and e are constants, and ^ and u respectively the mean and eccen-
tric anomalies in the elliptic orbit of the disturbed planet, then
-?i = 1 — e cos w , dl^ = -^du.
Equation (5) becomes then
dZ'^ rl ^
If u is made the independent variable, it becomes
(l-ecosM)^-esinw^-l-5' = 0. (12)
Differentiating this and removing the useless factor 1 — e cos u, we get
dvJ^ du '
the integral of which is
g' = ^i COB M -I- E, sin u + E^.
A METHOD OP COMPUTING ABSOLUTE PERTURBATIONS
157
Determining K^ so that equation (12) maybe satisfied, K^=. — K^e.
Hence the complete integral of (12) is
q = E^ (cos u ~ e) + K^smu .
It is evident now that we may take
q^ = h (cos u — e), q^ = ksmu.
If these values are substituted in equation (7), it is found that P = — •
thus
if V is the true anomaly of the disturbed planet in its elliptic orbit. Thus
■N'= — [sin (m — m) — « sin M + e sin w]
an
Wl — e"
r„r„ sin (v — v).
We now change the independent variable t for the variable v- We
have
whence
Ndt =
■ r„rl sin {v — v) dv .
fia{l-e')
Thus the expressions for the perturbations become
Sr = ^^^^_^,^ fQ.nsinCv-v^dv,
^''^ ij.a {1- e'-) f^'^° ^^^ ^^ -v)dv,
^^ = — /7° ^^ / C/o sin (v — v)dv.
iia {! — (?) J
(13)
It will be perceived that, by this transformation, we have been enabled
to get rid of the factor Tq before h, with a simplification of the right mem-
ber of the equation.
158 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
These equations, although very symmetrical, present the inconvenience
of being one too many. Hence, for the second and third, we substitute a
single one. From the differential equations of motion.
^^^ir^^-^-^/C^lf-'ll]*'
where h is such a constant that
and i denoting the inclination of the plane of the elliptic orbit to the plane
ofxy,
h= >J iia{l — e') cos i .
Denoting by /I the longitude measured in the plane xy, so that
tan ;i = -^ , and putting
X
^^ dy ^dx-d^'
we shall have
Supposing ^=:2,o + 6l^, where tan ;\,„ = ^ , the following equation is
obtained for the determination of S^ :
Or, if V is made the independent variable, and for brevity we put
j9 = a (1 — e") , the expressions for the perturbations are
dr= — I Q^rl sin (v — v)dv ,
dz= ^ CQA sin (v — v)dv.
(U)
These formulas are absolutely rigorous, since no terms have been
neglected, and also perfectly general, as no restriction has been put upon
the position of the plane ofajyfrom which the coordinate z is measured. By
adopting the plane of the elliptic orbit of the disturbed planet as the plane
A METHOD OF COMPUTING ABSOLUTE PERTURBATIONS 159
of 032/, the ^*s^ equation is somewhat simplified. For then i = 0, and Zo= 0,
and z = Sz; thus
"=/[/i^*- '^"--ir~"' ]f^- w
Perturbations of the first order with respect to the disturbing forces.
Since, in this case, elliptic values are to be substituted for the coordi-
nates in the functions Qr,Qg, Qk> there is no need any further of making a
distinction between ro and r; hence the (o) will be omitted from the former.
If we put
and h^ is the latitude of the disturbed planet measured from the plane of its
elliptic orbit, and S/l the perturbation of the longitude measured in this
plane, our formulas, in this case, reduce to
Sr= r TBin{v -v)dv, di3 = C Z sin (v — v)dv, SX= Tf C Ydv — 2—~\dv.
Put now
fip dr '
then it will easily be found that
±dB = r-'[tmix+v\.
up L p J
Thus the shape, in which we shall employ our equations, is
dr= r[x+%r^ r r-'(?-^^X+ v) dv~] sin (v — v) dv ,
dX:^ rr r rdv — % ^~\ dv,
[Z sin (v — vy] dv .
/'
The chief thing now to be done is to expand X, Y and Z in periodic
series as functions of w. The elliptic values of the coordinates of the dis-
turbed planet are readily expressed in terms of this variable, but the coor-
dinates of the disturbing bodies will naturally be expressed in terms of
their mean anomalies t,', ^". etc. These last variables must be eliminated
by means of the identities
n n n
n n n
160 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
Let us then put
so that
n n n n
M' n' diV n" .
-5- = — , -5- = — , etc.
av n dv n
Then ^', f", etc., will be replaced by the following values :
r = *'-i^(i;-C), :"=.*"— ^(w-O, etc.
In the development of X, Y and Z in periodic series from particular
values of these quantities, it will be better to make the differences of S'', ^",
etc., from v, the variables to be employed. Thus we shall put w' = ^' — v,
w"^^'i — v, etc.
The formulas, to be written now, will be confined to the case of the
action of one planet. The expressions for X, Y and Z are
^=^^*[i-^3]/co«^'cos(.-.)-^^,
where h^ = iia{l — e'), and
A' = r' + r" — 2rr' cos /J' cos (/ — A).
If the inclinations of the orbits of the two planets to some fixed plane,
as the ecliptic, are denoted by i, i', and the longitudes of their ascending
nodes by Q,, Q,', and the longitudes of their perihelia by 7t, 7^, we compute
/, 0, 0', n and n' from
cos I = cos i cos i' + sin i sin i' cos (^2' — Q,) ,
sin / cos (^ — Q) = — sin i cos i' + cos * sin i' cos (Q' — Q) ,
sin / sin (6» - Q,) = sin i' sin (Q; - Q)
sin 7 cos (8' — Q') = cos i sin i' — sin i cos i' cos (Q,' — Q) ,
sin / sin {ff — Q,') = sin i sin {Q' — Q,) ,
n = 7: -9, n' =n' -8' .
The circumference being divided into A; equal parts with reference tow,
pute for each of
following quantities :
compute for each of the h values oi v, 0,-=-7t, ^n .... — L^ 1 ji ^ the
A METHOD OP COMPUTING ABSOLUTE PERTURBATIONS Jgl
tan
u
-Vl + .t^"
V
C
=:M —
e sinu,
V
= V-
n' .
P
■0,
1 + e cos V '
Kcos(n'—J)= cos(v + Z7), ^'cos(/7'-^') = cos/cos(«+ //),
^8iii(/7'-^) = cos/sm(i; + /y), K' sin (^n' - A') = sm(v + n),
a = 2Kr, G' = ^r'E, G" = ^r', G"'=~r'K', G"" = "^ sin I . r\
Several of these quantities, as «, ^, F, r, will need to be computed only
k . . . . Jc ■ ■
— times, if ^ is a multiple of 2 ; and K, K', A , A! only — times in the
same case.
The circumference being divided into h' equal parts with reference to
the variable w', compute for each of the hh' values of v and w' the following
2 4 2 (Id 1^
quantities, w' taking in succession the values , -^ 7t, -=j Tt .... '■ .^ — 'n:
Z' — Vi- w', u'- e' sin u' = Z',
Vr' cos-^ = Va'(l — e') cos-s-, V*"' sin-s-= V' (1 + «') sin-s-.
If we have tables of the disturbing planet giving the true anomaly or
the equation of the center and the radius vector or its logarithm with the
argument mean anomaly, we can derive log r' and v' by means of their aid,
and thus dispense with computing the last three equations. We now com-
pute kh' times
A' — r'+ r"— Gr' cos (v'+ A) ,
X= G' [4r - ^] r' cos ^v'+A)-^,
Y^ G'" [^ - ^t] r' sin (v' + A') ,
Z= G"" [^ - ^] r' sin {v' + 77') .
From these M' special values of each of the quantities X, Fand Z,
we deduce their developments in periodic series of the form
^u, \K\^ cos {iv - i'w') + K\tl sin {iv - i'lu'y] .
21
162 COLLECTED MATHEMATICAL WORKS OP G. W. HILL
This process is so well known that we need not here insert the formulas
required for it ; they will be found in Hansen's Auseinandersetzung, Part I,
p. 159. A double application of these formulas will be necessary, the first
relative to v, the second relative to w'. After these series are obtained, w'
can be replaced by S'' — v.
The series X is now to be multiplied by — sin v, which, for every
periodic term in X, will give two periodic terms, which will be added to T.
This result is next to be multiplied by
/I + e cos vV l + U' 2e e'
= r 1- —2- COS V + -^r-r COS 2v .
\ P / p^ p Ip^
There is now an integration to be effected. A table of logarithms of
the integrating factors
1
li, i'-\ = ■
n
will now be made for all combinations of i and i' which occur in the periodic
series. If the last result contains a term
sin "^ ^^
the corresponding term of the integrated result will be
A multiplication by 2r^ is now to be made. We have
-^ =^Eo — Eicoav + E^ cos 2v — E^co&Sv + . . . ,
where the rigorous value of the coefficients is given by the equation
This multiplication accomplished, the product is to be added to X. If
this result has a term
^sTn(*"^-^''»')'
then 8r has its corresponding term
- [t- 1, t'][t + 1, i'] Kl?^ (iv-W) ,
A METHOD OF COMPUTING ABSOLUTE PERTURBATIONS jgs
except in the case where i = 1, i' = 0, when we have, instead of this,
2 cos
Having thus obtained Sr, we multiply it by
1 1 e
— — 1 — cos V .
r P p
The result, which is the perturbation of the natural logarithm of r,
must be doubled and then subtracted from / Tdv. Another integration
being executed on this result, we have h'k the perturbation of the longitude
measured in the plane of the fixed elliptic orbit.
Finally, ^^ will be obtained by treating Z to the same kind of integra-
tion as that last used in obtaining hr ; that is, in general, each coefficient of
Z will be multiplied by the proper value of — \i — 1, i'][i + 1, i'] which
corresponds to it.
Perturbations of the second order with respect to the disturbing forces.
Calling the parts of the perturbations of r, ^, ^, which are of two
dimensions with respect to the planetary masses, 5V, 8^(3, 6^^, so that we
have, with errors of the third order,
where hr, 5/3, h7u are the perturbations which have just been determined,
we shall have
dv ,
Sr
S'r = J -^ SQ, sin (v - v)
8'^ = J" ^SQ, sin (v-v)
pr P r' 3V /drV Sr d.SX~\
dv.
where, as before, there is no need of any distinction between rg and r. The
following are the expressions for 6Qr and 8Q^,
164 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
Bearing in mind that X, Y, Z are homogeneous functions of r and r',
it will be easy to deduce the following equations :
r' ( dR\ I dX ^^\Sr I dX „^\5r'
(5A _ SX') + r(r 1^- 3^) 5/? + |^5/S'.
,(|?U^.F
(4f-^)^-(^i?-^) ?-!?(---) - w^^ - w^^''
, 5r
-a
„ »•' r^*K. o,r ,rJ^ »• ^d.dX ^d.S^ e . r' J dR\ r-^-l,
n Sr
where the differential coefficients — L, — ^ — , ' " are complete with
dv dv dv
respect to the independent variable v.
In computing the values of these functions, — , 5^' and ^(3' must be
expressed as functions of «. Hence, if they are at first expressed in terms
oft, it must be eliminated by means of the equation
nt + c = v — Bi sinv + ^E, sin 2v — ^^3 sin3t> + . . .,
where the rigorous value of ^j is
We may have given only the perturbation of the orbit longitude and
the latitude above the elliptic orbit of the disturbing planet ; in this case,
calling the latter Sri', the values of 6X and 5/3' will be given by the equations
^^, cos J , , ain I COS (v' + n') ^ ,
' cos/S ' cos /J
We see that, in order to obtain the perturbations of the second order
it will be necessary to have, expressed in periodic series in terms of v, the
following nine quantities :
dX dZdZdXdYdZdXdYdZ
^ dr ' '^dr' 3/J' dX ' dX ' QX > dfi" d[i" W
A METHOD OF COMPUTING ABSOLUTE PERTURBATIONS 155
For six of these whose expressions are
*■ dr
dZ m' ^/3r''sin»/3' 1
[ 4J= 2/)" +4^/'
„_ 3 m' //' — r' 1\ , . ^,
dl3~ h' ' \ A
dX m! ^f3r' — r" 1 1
8/9'~ A' ^\2
— r'^ 11 1 \
ji — + 2^+ r^j'" sm/3'cos(A' — ;),
9F m' ,/3r» + /^ 11 15\ , . . . ^„ ,^
9Z m' r/1 1\ , „, 3rr''sin'/3'cosa'— /I)"!
9^ = IF ^ Ll^~^ J "■ ''°' '^ ^^ ^^ J '
the same method must be used as that which has been given for X, Y, Z.
The remaining three, X, Y and Z being considered as functions of the two
variables v and S'', can be obtained from the equations
9X_9X w^/ r' \gX e_ . ,aX
aA - Sv + w V~a'^'T^^I a*' i? sin?;, r ^^ ,
8F_aF «;_/ r- ^ar e_ . ,dY
a^_az jj[^/ r' \dz _e_ . a^
3^ -at; + « V aV"T^^V9*'~i'^''''' S*"*
The factor
where
1 „ ,_ = E, cos V — E, cos 2v + E, cos 3v -
a V 1 — «
Moreover, we have the relation
,a^_3^ o„Tr
The factor r is given by the equation
-— = \E^ — EiCosv + Ei cos 2i; — E^ cos 3« +
where
166 COLLECTED MATHEMATICAL WORKS OP G. W. HILL
The values of ^' — ^ and /?', necessary for the computation of the first
six quantities, can be obtained from the equations
cos /3' cos (A' — X) = K cos (i;' + A) ,
cos/3' sin (>(' — X) = K' sin (v' + A'^ ,
sin /9' = sin Iam(v'+ 11').
The terms to be integrated in the second approximation have the gen-
eral form
(iG+G'v)f^^Civ-i'&'-i"n.
If these terms are integrated with respect to v, we have
T Li, i', i"](C + G'v) ^?f {iv — i'»' — i"»") + [i, i', i"T G' ^|f„ {iv — i'»' — i"»") ,
SlU COS
where
[i, l', *"] = - —7 — TT -
I l' 1
n n
If they are integrated after having been multiplied by the factor
sin (w — «) , the result is
— \i—\, i', i"Ji + 1, i', i"XG + G'v) ^"^ (iv — i'»' — i"»")
COS
except in the case where i z= 1 , i' = , i" =: , when we shall have
±(lG'-^Gv-l G'v') 2 v + i(G + G'v) f^^ v .
The labor of computing perturbations of the second order is, in some
sort, measured by the number of multiplications to be made of two periodic
series, each involving double arguments. In this method, in the case of one
disturbing planet, there are 22, or 25 if one thinks that the multiplications
involving 8X' ought to be considered as distinct from those involving 5/1 . If
all the terms involving sin /as a factor be neglected, the number of these
multiplications is diminished by 1 2.
It is my intention to illustrate this method by applying it to the com-
putation of the perturbations of the first order of Ceres by Jupiter.
LONG PERIOD INEQUALITY IN THE MOTION OF HBSTIA 157
MEMOIE No. 15.
On a Long Period Inequality in the Motion of Hestia Arising from the
Action of the Earth.
(Astronomische Nachrichten, Vol. LXXXIV, pp. 41-44, 1874.)
While the attention of all is directed to the more exact determination
of the constant of solar parallax from the approaching transit of Venus, it
may be of interest to notice another source from which, at least in the
future, can be obtained the value of this constant.
Several of the asteroids have periods of revolution approximating quite
closely to four years ; hence, in their longitudes are long period equations
of the form
Ic sin [45* — g' ^-K'\,
g and g' being the mean anomalies of the asteroid and the earth. Should
It, be quite large, after the inequality has run through a considerable portion
of its period, we can, from this source, determine a pretty exact value of
the earth's mass, and thence, by the known formula, the corresponding value
of the constant of solar parallax.
In order to see what may be expected in this direction, I have com-
puted this inequality, as far as the first power of the disturbing force is con-
cerned, for Hestia. This asteroid has been selected on account of its large
eccentricity and the near approach of its period to four years. The ele-
ments employed (as many as we have need of), from the Berlmer Jahrbuch
for 1875, and from Leverrier's Annales de V Ohservatoire, Tome IV, are as
follows :
Hestia. The Baeth.
Osculating, 1865, July 26. Mean Elements for the same epoch.
■K = 354° 14' 18".7
95 = 9 36 55.8
gi = 181 30 35.3
i— 2 17 30.0
fi = 883".56391
log a = 0.4035124
M. B. 1870.0
7:' = 100° 41' 35".0
p' = 57 38.1
/ = 3548".19386
*" — 422800
These elements give fi' — 4(1 = 13". 937 22, whence the period of the in-
equality, in this case, is 254.6 years.
168 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
By a quite rigorous process, similar to that employed in Hansen's Aus-
einandersetzung, the terms of — depending on the argument Ag — g' have
been found to be
— 0.001 74933 cos (4^ — g') + 0.011041 88 sin (4^ - g') .
And, in like manner, the second part of the disturbing function ^ cos i^
contains the terms
+ 0.00257586 cos (ig—g') — 0.00872291 sin (ig —g') .
Thus aB contains the terms
+ 0.00082663 cos {ig—g') + 0.00331897 sin ij^g—g') .
Multiplying these by the factor
■^^''"'"-U X 306364".8 ,
(4/.-/^')"
we have the inequality sought,
^ndt = 75".869 sin {^g—g'+ 109° 37' 10") ,
/»
The eflfect of this inequality on the geocentric position of Hestia at
opposition is got, somewhat roughly, by multiplying the preceding expres-
sion by , and hence, at a maximum, may amount to about 125".
It must be confessed that the determination of the earth's mass from
this source is attended with the inconvenience of having to compute very
accurately the perturbations of Hestia by Jupiter; and among these is a
very large inequality having the argument g — Zg", whose period is nearly
the same as that of the inequality just determined. Hence it will be neces-
sary to proceed with a very accurate value of Jupiter's mass obtained from
other sources.
It will be noticed from the expressions given above that the portions
of the inequality, contained in the two parts of the disturbing function, have
a strong tendency to cancel each other. This is always the case where
either one of the mean anomalies is involved in the argument only to the
simple multiple. This tendency does not occur in the inequalities having
arguments of the form Ig — 2g', and perhaps quite large coeflBcients might
be obtained for these in some of the asteroids whose periods approach 3^
years, especially if their eccentricities are large. Melpomene would seem to
afford the best chance, and the period of the inequality would have the
recommendation of being much shorter than that of the one here computed,
namely about 80 years.
PROBLEM IN THE THEORY OP NUMBERS
169
MEMOIE N"o. 16.
Solution of a Problem in the Theory of Numbers.
(The Analyst, Vol. I, pp. 37-38, 1874.)
The following problem appeared in the Mathematical MontJdy, Vol, I,
p. 29, and no solution was published in that periodical :
" Show that the product of six entire consecutive numbers cannot be
the square of a commensurable number."
Since the square root of every integer, not an exact square, is a surd, it
will be sufficient to show that the product cannot be the square of an inte-
ger. Let the six numbers be denoted, n being an odd integer, by
n — 5 n — 3 n — 1 w + 1 w + 3 w + 5
Then it is required to prove the impossibility of —^ — . — -— . — - — = n.
Let us put — - — := X, where x is integral since it is the product of two
integers. Then it will suffice to prove the impossibility of a; (a; + 2)(a; — 4)=n.
Let us suppose x =■ l_! = r sin (;^o — v) + R-^ sin L_i,
^0 cos X„ = r cos Xo + -^0 cos A ,
/!„ sin A„ —r sin /„ + R„ sin i, ,
J, cos X^ = r cos {xa+ v) + Ri cos Li ,
Ai sin X-i zzzr sin (/« + i?) + R\ sin L^ .
These equations contain the six unknowns A_i, Aq, A], r, %, >?. If we
eliminate A_i, Aq, Aj from them, we shall have the three equations of the
first solution. But by retaining Aq as the unknown, we shall arrive at an
elegant solution. Let us first eliminate A_ j and Aj ; this we do by putting
P = ;i,_ 1 for the first two equations, and P = ;ii for the last two. The equa-
tions for determining the four remaining unknowns, are
= r sin (;^o — 5; — A_,) + R_, sin (i_i — A_,) ,
Jo cos A„ = r cos Xt + Ri cos i„ ,
J„ sin Ao = r sin/„ +i2„sinZo,
0=:»-sin(/o+ 1? — Ai) + i?i sin (A — ■*!) .
If, in the second and third of these equations we put successively
P = >7 -f ;i_i and P = — )7 + ;ii, we get
A SECOND SOLUTION OP THE PROBLEM OF NO. 8. jyj
AsinC^o — v — -i_i) = j-sin(;ifo— 5?— -»_i) + i?„ sin (i, — ij — >l_ i) ,
^0 sin (>lo + 1? — -^i) = »• sin (/„ + ly — ;,) + i?„ sin (Z„ + 5; — Ai) .
If, from these equations we subtract the first and last of the preceding
four, we get
/)„ sin(A„ — r, — A_,) = ^0 sin (£, — ■>] — X_^) + E_, sin (Z_ 1 — /i_ ,
Jo sin (/i„ + t; — Ai) = R, sin (/,„ + ^ — X,) — ij^ sin (A — X,) .
Two equations with two unknowns are thus arrived at without complicating
the form of the original equations.
It is very easy to eliminate Aq from these, and we get
[^„ sin (i„ - , — A_ ,) — i?_ , ain (Z_ 1 — A_ 1)] sin (/l„ + ^ — X^)
— [i?„ sin (i„ + )j - Xi) — El sin (A — ■*!)] sin (-)» — )j — X_ ,) .
But we prefer to keep Ao as our final unknown. Let us put for the
sake of brevity
^-1 = ^-1— A_i, 4\ = Li — K-
All these are known quantities with the exception of cr, which will take the
place of >7 as an unknown. Our two equations can now be written
A sin (d — s) = Ba sin (5' — a) + B_i sin (p_ ^ ,
/(„ sin (3 + (t) = E„ sin (d' + a) + E^ sin ^1.
Or, by taking in succession half the sum and half the difierence
/)„ sin5 cos -...).
184 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
The expressions given for D^" and i)^" + ■* are equally applicable when n
is negative ; they then give the formulas to be used in mechanical quadra-
tures, thus :
If these expressions are expanded in powers of A, we obtain
n-i_J^-i ±a,1La^ 191 2497 14797 ,,
■^ -'*r 12 TOO ~ 60480"^ + 3628800^ ~95800320'^
92427157 ,„ \
4- ,
3615348736000 --J'
~ '* r 13 240 + 60480 3628800 + 22809600 " " j"
These are the expressions to be used in computing the values of the inte-
grals / ydx and / / ydaf. It must be noticed that A~^ virtually contains
an arbitrary constant G, and A~^ an arbitrary expression Cx + C In
fact, the quantities in the columns to the left of that of the function y can-
not be written until we know one quantity in each column. These constants
G and G' are usually determined from the given values of / ydx and
/ / yd'^^ for x = a. If we denote them by D^'^ and D^^, and if, in gene-
ral, the subscript („) denote values which obtain when a; = a, it will be seen
that
^0 - h + 12 ^« 720 '^» + • ■ ■ '
D~' 1 1
~ h' 12 " + 240 "
Having thus the sum and difference of the quantities A~^y_^ and A~^3/j, it
will be easy to get the quantities themselves.
The preceding formulas give the values of the integrals for the series
of values of cc, .. a — h, a, a + h^ . . It is generally preferable to
compute them for the values, .... a — ^h, a-\-\h, a -\- ^h, .... Formu-
las for this purpose can be obtained by the simple consideration, that in
the scheme, given at the beginning of this article, it is allowable to treat
the odd orders of differences as if they were even, and the even as if they
were odd.
OAliCULUS OF FINITE DIFFERENCES jgs
In this way all the quantities obtained will correspond to the middle of
the intervals of the former supposition. Thus, calling D"^ and D~^ in this
case Dj ^ and D^ ^, it is evident we must have
= - Vi + i^nj Vi + i^V '
or, expanded in powers of A ,
iJi -n\^^ +24^~5760 +967680 464486400'^ + •• 7'
^4 -A ^^ 24+1920 193536^+66355200'^ •
The diflferences of the first formula, although they are of odd orders,
are to be taken as equivalent to the simple numbers standing in the original
scheme, while the diiFerences of the second, although of even orders, are all
the averages of two adjacent numbers of the same scheme.
It is plain we have
d .Df
dA
In using the method of mechanical quadratures, it is usual to multiply
the values of y by A , if the single integral only is wanted, but by h^ if the
double is also to be obtained ; in the last case then it is necessary to divide
the results obtained by Ji in order to have the single integral.
These formulas appear to have been first obtained by Gauss ( WerTce,
Vol. Ill, p. 328). Bncke has given them in the Berlin Jahrhuch for 1838.
For use they are much superior to the formula given by Laplace {Mecanique
Celeste, Vol. IV, p. 207).
186 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
MEMOIR No. 20.
Elementary Treatment of the Problem of Two Bodies.
(The Analyst, Vol. I, pp. 165-170, 1874.)
The deduction of the motion of the planets, in accordance with the laws
of Kepler, from the principle of universal gravitation, is important, not only
on account of the extensive role this theory plays in Astronomy, but also
for its interest, in a historical point of view, as Newton's principal discovery.
Hence it is desirable that the demonstration should be made as elementary
and as brief as possible, in order that it may be brought within the compre-
hension of the largest number of persons.
The polar equation of the conic section, referred to a focus as pole
r = . =^ —
1 + e cos (A — o) '
is well known ; a denotes half the greater axis, e the eccentricity and o the
angle made by the axis with the line from which ^ is measured. It will
be advantageous to replace a(l — e^) by ^, p being the semi-parameter, also
to put
o = e cos 10 , /9 = e sin (u .
Thus the equation becomes
r + ar cos A + /?r sin X =p .
Hence it is plain that the equation, in terms of rectangular coordinates, the
origin being at a focus, but the axes of coordinates having any direction we
please, is
^x'+f+ax + ^y=p. (1)
"We take for granted the following theorems, since they are demon-
strated in the most elementary treatises on mechanics :
In determining the relative motion of one body about another, it suf-
fices to regard the latter as fixed, and to attribute to it a mass equal to the
sum of the masses, and then to suppose the moving body without mass.
When a body describes a plane curve, and the radius vector, drawn
from a fixed point in the plane of the curve, passes over equal areas in
PKOBI/EM OF TWO BODIES
187
equal times (which we shall express by saying that the areolar velocity
about the fixed point is constant), the force acts always in the direction
of the radius ; and the converse.
Now let a body describe a conic section about another occupying a focus,
the areolar velocity about this focus being constant ; it is required to deter-
mine the force acting.
In the figure, let PP"T be an arc of the
conic section so described, S being the focus.
Let P and P" be any two points on the curve
at an indeterminate but small distance from
each other. Draw 8P, and PP a tangent at
P, P"P' parallel to, and' P'A and P"B per-
pendicular to SP- Let SP be taken as the
axis of cc, and SY perpendicular to it, as the b ^
axis of y. The coordinates of P are then
33 = SP = ro, 2/ = ; substituting these in the equation of the curve, we get
(1 + a)r„=^.
(2)
Since the ordinate y can here be supposed always very small, the term
Va;^ + y^ in (1) can be expanded, by the binomial theorem, in a series of
ascending powers of y . Neglecting y^ and higher powers, we get
or, as X differs from r^ only by a quantity of the order of y, by neglecting if
Or, by (2),
il!
p-^y-i^
' n
1 + a
x = r„
1 + a^~^
^ p
This is the value of a; from (1) expanded in a series of ascending powers of
y, the cube and higher powers being omitted. The equation
x==r„
1 + a-
belongs to a right line, which can be nothing else than the tangent PP.
Hence it is plain, from the figure, that taking P"B = P'A = y,
188 COLLECTED MATHEMATICAL WORKS OP G. W. HILL
tan PF'A ^ ^4- ' (3)
1 + a
PA =
1 + a'
P'P"^AB = \^, (4)
the last equation being only approximate, but more and more nearly true
as P"B or y becomes smaller.
Let F denote the force acting on the moving body, and t the small inter-
val of time in which the latter passes from P to P". Then we have
P'P"-^i^=\Ft\
If we denote double the areolar velocity by h, since P"B = y is very small,
we have
SP.P"B = r,y = M.
Eliminating t from these equations, we get
Since there is no limit to the supposed smallness of ^ and t, this equa-
tion is rigorously exact. The force is then inversely as the square of the
radius-vector, and its intensity at the unit of distance is found simply by
dividing the square of double the areolar velocity by the semi-parameter.
It is evidently attractive except when, the motion being in a hyperbola, the
focus, about which the areolar velocity is constant, is the exterior, in which
case it is repulsive.
Taking up the inverse problem, let a body start from P towards P' with
a velocity ■«, which would carry it to the latter point in the time t, and let
it be subjected to the action of a force varying inversely as the square of its
distance from a second body supposed fixed sX S: it is required to find the
curve described.
Let the masses of the bodies, measured by the velocities they are able
to communicate by their action, in the unit of time and at the unit of dis-
tance, be denoted severally by m and M. The force acting at P is then
M -\- m _M + m
SP' - ~~P~'
and, if at the end of the time t , the body is at P" instead of P', we must
have
p, p„ _ J M+ m
PROBLEM OF TWO BODIES jgg
But, as before, the constancy of the areolar velocity gives rx-=-}it. Whence
This equation coincides with (4) if we suppose
K
ni
V = M^- W
Let now a conic section, having this value for its semi-parameter, be
described with 8 as focus and touching PP' at P. That this is possible is
evident from the general equation (1) ; here are only two unknowns, a and
/3, to be determined, and they are given by equations (2) and (3), whence
we see the solution is always unique. A body, moving upon this conic sec-
tion, would have, at the point P, the same velocity, and the same direction
of motion, and be subjected to the action of an equal force having the same
law of variation, as the moving body in the problem. Hence, if the path of
the latter is thoroughly determinate, and it would be absurd to suppose
otherwise, the conic section just described must be the curve sought.
We can easily find the elements of this conic section. Thus, let the
angle P'P>S' be denoted by i^, then evidently,
h^rv mi
I 1/ ai-LA. y
M + m
whence we derive
„ rv^ sin' (h r^if sin' ^ 1 3
M + m '^ (^M+my a r M + m'
Consequently the greater axis, and the species of conic section described,
are independent of 4^. We have an ellipse, a parabola, or a hyperbola,
according as v^ is less, equal to, or greater than 2 .
From the last equation
v^ = (M+m)[-^-^), (6)
190 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
which may evidently be taken as a general expression for the square of the
velocity, if r denote the general radius vector.
Also from (5),
Thus, in different orbits, the areolar velocities are as the square roots of the
parameters, and as the square roots of the sums of the masses. In an ellip-
tic orbit, if T denote the time of revolution, the double of the area of the
whole ellipse
Whence
i\/ M + m'
Thus the theorem that, provided the sum of the masses remains the same,
the squares of the periods in different orbits are as the cubes of the
greater axes.
The mean angular velocity is usually denoted by n ; thus
2;r /,
2-7: I M + m
n =
It is customary with astronomers to assume the earth's mean distance
from the sun as the linear unit. If M and m are the masses severally of the
sun and earth, and m/, a' and n' belonging to another planet are introduced,
the mean distance of the last is given by the equation
m'
^^ M
To complete the subject, it is necessary to notice a particular case of
the problem, viz., when 4' = . Here the motion is in a right line, and
from (6) it appears the velocity is infinite when the body arrives at 8. As
the existence of another body here ought not to be considered, at least in a
mathematical sense, as an obstacle to its further motion, it is plain the body
will pass beyond and move in the same right line until its velocity is
reduced to zero, when it will return on its path, which will thus be a por-
tion of a right line of which S is the middle point. This cannot be consid-
ered as a degenerate form of a conic section of which S is the focus. For
when an ellipse is varied by augmenting the eccentricity but maintaining
the greater axis constant, at the point the first has attained the limit unity,
the ellipse has degenerated into two equal portions of right lines overlapping
PROBLEM OF TWO BODIES jgj
each other and having their extremities on. one side in the point S. Hence
this case must be regarded as a singular solution. However, most of the
properties of motion can be deduced from those of elliptic motion. Thus, if
the length of the whole path denoted by 4a, the duration of an oscillation
will be
n/ M + fn'
Whence we gather that the time, in which a planet, at rest at its mean dis-
tance, would fall to the sun, is found by dividing its periodic time by 4\/2.
192 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
MEMOIE ISTo. 31.
The Differential Equations of Dynamics.
(The Analyst, Vol. I, pp. 200-303, 1874.)
The general formula of dynamics is
In the usual treatment of this equation, we have been asked to attribute
to the symbols 8x, Sy, Sz, . . . . the signification they have in the calculus of
variations. This, however, is unnecessary, except when we wish to deduce
from it the principle of least action ; and the student unacquainted with
this calculus may regard these symbols as multipliers, which, when all the
points of the system are free, have any finite values we please, but when
the coordinates are restricted to satisfy an equation U= 0, are subject to
the condition
9a; cy dm
an equation which, for brevity, we shall write 8U= 0.
We shall confine our attention to those cases in which the equations of
condition and the accelerating forces are functions of the coordinates and
the time only, and in which the latter are equivalent to the partial differen-
tial coefficients of a single function li taken with respect to the coordinates
whose acceleration they express.
Whenever a function as U involves, in addition to x, y, z, .... their
differential coefficients with respect to the time, quantities which we shall
denote by x', y', z' ,...., we shall suppose that 5 U involves, besides the terms
written above, the following
3a; 92/ oz
Moreover, as we shall have to differentiate such functions as 5 Z7 with
respect to t, we shall meet with such quantities as -j— , and shall suppose
that the order of the symbols d and S may be inverted, that is, we shall
have equations such as
ddx ^dx , ,
^=5^= to'.
DIFFERENTIAL EQUATIONS OF DYNAMICS 193
The reader will see in this only a notational assumption, without quantita-
tive significance, serving merely as machinery of demonstration. It will be
noted that tis a variable not subject to the operation S.
We have
liXdx + Y8y +Zdz) = 8a ,
and for convenience may put
Then it will readily be perceived that the general formula can be written
thus
The coordinates x, y, z, . . . . , can be expressed as functions of the time and
certain variables qi, independent of each other and whose number is equal
to that of the variables x, y, z, . . ■ ■ , diminished by the number of equa-
tions of condition. Substituting for x, y, z, . . ■ ■ , their values in terms of
the new variables qt, it is plain that the last equation will take the follow-
ing form:
We can find the value of pi without actually making the substitution,
from this consideration j since the original equation contains only the varia-
tions 8x, 8y, Bz, ■ ■ ■ ■ , without the variations 8 -^, S ~, S -^, ■ ■ ■ ■ , it fol-
Ctt Cvt (tt
lows that, in its transformed state, it should contain only the variations 8qt
without the variations 8 -~ .
at
Then writing ql for -^, the coefficient of ^q^ should vanish in the
equation
That is, since 11 does not contain ql,
dT
dqi
Thus the general formula becomes
23
194 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
Because in this equation the variables qi are independent, we may
equate the coefficient of each hqi to zero. Thus
dt • dq'i dqt
This is Lagrange's canonical form of the differential equations of motion.
A simpler form may be obtained by substituting the variables ^^ for qi.
By adding to and subtracting from the general formula, the term S . 2^ {pi, q't) ,
and writing
it becomes
Equating the coefficients of each variation Sqi and 8pi to zero gives the
equations
dpt^__dH_ dq, _ dH
dt ~ ~ dqt ' dt ~~ dPi '
which are known as Hamilton's canonical form.
The expression for H can take a simpler shape. From the value of T,
it is evident that a certain part of it is independent of the variables ql ,
which may be denoted by Tq, another part Tj, involves the first powers,
and a third Tg involves the squares and products of the same ; then
T= To + 2\ + T^. By the theory of homogeneous functions
I.(p,q,') = S,(^^q/)=T,+ 2T,.
Hence, if we write
we shall have
CUBIC AND BIQUADRATIC EQUATIONS 195
MBMOIK No. 22.
On the Solution of Cubic and Biquadratic Equations.
(The Analyst, Vol. II, pp. i-8, 1875.)
In nearly all treatises on algebra, the solution of these equations is pre-
sented as accomplished by the aid of analytical artifices, which one seems,
by some happy chance, to have stumbled upon. No doubt the processes
were found in this manner by the original discoverers, Tartaglia, Cardan
and Ferrari. But, for many reasons, it would be better to treat the sub-
ject as one demanding invention rather than artifice. The equations can,
as it were, be interrogated and compelled to yield up their secrets, if they
have any.
To say that an equation is solvable algebraically, is to say that an alge-
braic expression can be found equivalent to the general root, that is, one
involving a finite number of the operations of addition, subtraction, multi-
plication, division and the extraction of roots of prime degree. If the
expression does not involve the last mentioned operation, it is called
rational, and if free from the two last, integral.
However complex an algebraic expression involving radicals may be,-
it is evident that there must be at least one radical which is involved
in it rationally. Supposing this to be denoted by i2", n being a prime inte-
ger, it is not difficult to convince one's self that, by the proper reductions,
the expression can be exhibited thus :
p, + PiR' + p^B" + . . . +p„-iB " ,
where ^01 Pii ■■ • ■ > ^^ 'lot involve the radical B". With no loss of gene-
rality, we can suppose pi = l ; for if p^ is not zero, we can multiply the
quantity under the radical sign by pi, and then take (pi i^)" as the radical ;
and in the contrary case, if p,c is one of the quantities p which is not zero ;
the simplification can be accomplished by putting R^ = plR''. Then
1_ 2_ « — 1
Po+ Ii^ + p,R-+ . . . + p„--,R "
may be regarded as the most general form of an algebraic expression.
196 COLLECTED MATHEMATICAL WORKS OP G. W. HILL
Here may be enunciated a general proposition, which, although I am
not aware that it has ever been proved, is doubtless true and may be used
for purposes of discovery. If an algebraic expression exists, equivalent to
the general root of the equation
a;'"4- aa;'"~'4- bx'"-''+ . . . + g = 0,
it can be exhibited in the form given above, n being one of the prime fac-
tors of m. Thus the algebraic expression of the root of the general equation
of the S"* degree, if it existed, could be presented in the form
p,+ m + p^R^ + PiRi + p^R^ ,
and that of the 6'*' degree in either of the two forms
p,-ir Ri + p^Rl , p, + Ri.
Solution of Cubic Equations.
According to the foregoing proposition, the root of the general cubic
equation
sf+ ax'+ bx + c = 0,
if it has an algebraic expression, must be presented in the form
x=p + Ri +p'Ri.
But, since we suppose that this is an irreducible expression involving radi-
cals, it follows that it must satisfy the given equation, whichever of its three
values is attributed to the radical \/B. Thus, calling either of the imagi-
nary cube roots of unity a, the three roots of the cubic equation must be
Xi=p+ Ri + p'Ri,
Xi=p + aRi + a'p'Ri,
X,=p + a'Ri+a*p'Ri.
The first method that suggests itself for obtaining equations which shall
give the values of p, p' and B, is to substitute these expressions in the sym-
metric functions which are equivalent to the several coefficients a, b, c, viz.,
Xi+ Xi+ x,= — a, XiXi+ XiXi+ x^x^ = b, x^x^x^ =z — c.
But a simpler proceeding is to employ the three symmetric functions 2. a;,
2 . a' and 2 . x^. Since any cube root, &s \/ B is a root of a;^ — B:=0, in
which the coefficients denoted above by a and b are each zero, it follows
that the sum of the three cube roots of any quantity, as well as the sum of
CUBIC AND BIQUADRATIC EQUATIONS 197
their squares, is zero. Now, it is plain that if the value of x is raised to
the n^^ power,
a? = A + ^i2i+ ORl,
where A, B and Care free from the radical \/-B, and are consequently the
same whichever of the three roots x denotes. Thus, since 2.'v'i2 = 0,
2.^J?^ = 0, we have
2:. a;" = 3^.
Thus, for computing the value of 2 . a;", we need only the part A which
is free from the radical \^Ii. In this way we obtain and equate to their
known values in terms of the coefficients a,h^ c,
2 .X =3p = — a,
S.x^ = Z{p^+ 2p'R) = a'- 2b ,
i:.x'' = 3(p'+ R + Qpp'R + p'^R") = — a'+ 3aS - 3c .
These equations afford the values of^, j?' and R; from the first two
a , „ a^— 35
P = 3", p'R= 9 .
and by substitution of these values in the last,
a quadratic equation in B] thus the general cubic admits solution by
radicals.
For the sake of brevity, putting
. a"- 35 _ 2a'— 9a5 + 27c
we have
R = B±i^B'-A\
and, as we may take at our option either of the two roots, we have choice of
the two expressions for x,
x= - :^a + IB + /^ B'— A'y + A[Ji + 1^ B'- A'}-K
x=-^a + iB — ii/B'-A'-]i+AlB — >/B'-A']-i.
The three values of x are obtained by attributing in succession to the
single cube root appearing in either of these expressions its three values.
I do not know why almost all algebraists prefer to put the root in the
form
x = —^a + y\.B + ^B'—A'}+''^\_B — >^/^—A'].
198 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
It is certainly easier in practice to make a division than an extraction of a
cube root ; moreover, we are troubled, in the latter form, with the selection
of the proper three values out of the nine of which it is susceptible, a diflS-
culty which does not occur in the two former expressions.
Solution of Biquadratic Eqtiations.
An algebraic expression for the root of the general equation of the
fourth degree
ai'+ aa^+ bx''+ ex + d = ,
if it exists, can be presented in the form P + V Q. And if this denotes one
of the roots, another will be P — V Qj but since x has four values, it is
plain that P and Q must receive each two values. This condition will be
fulfilled if we suppose that these quantities, in their turn, similarly to x, are
rational functions of a second radical \/B. Thus we put
P=pJr\IB, Q^q + q's/B.
Then we have
x=p + i^R + i^q + q'/^R.
The four values of x are obtained by giving in succession to the radicals
\/ Q and V R all the values they are, in combination, susceptible of. Thus
Xi=p + ^/E + \'q + q'i^R,
X2=p — i^R + ^q — q'^R,
X3 =p + ^ R — ij q + q' si R,
Xi-=p — nj R— >J q — q' hj R.
By substituting these in the four symmetric functions 2 . a; , 2 . x^ X .7?
and 2 . a;*, equations will be found determining p, q, §' and R. Here again,
in computing 2 . a;", the radicals all disappear ; for, whenever a radical is
present with one sign in any root, there is always another root in which it is
present with the opposite sign ; thus these expressions in pairs cancel each
other. Then, in deriving 2 . a;", it is necessary to preserve only the terms
which are free from radicals. In this way we get
J. a; =4p =—a,
S.0(? = 4:ip^+mq + R) + 3g'^] --a?+^ab-Zc,
S.y*z=4: f (/ +q + Rf + (4/ + l^pq' + q") R + ^q (/ + i?)] = a*- ^a?i + iac + W- id .
CUBIC AND BIQUADRATIC EQUATIONS 199
From which we derive
a _ 3a'— 8& , _ a'— 4aJ + 8c
i'=— X' 1-^^ = — 16 — > §'-«= 32 ,
PS 3a'- 85 ™ 3a*- ICa'S + 16ac + 165'- ^U _ /3a'- 4a5 + 8c\' „
^ 16-^+- — ^BT ^-t 64 J = «-
The last is a cubic in i2, which, by the foregoing, is solvable by radi-
cals; hence the general equation of the fourth degree is so solvable. In
forming the value of £c, we may attribute to R as its value any one of the
three roots of this equation. When a = , the case usually treated, the
equations are simpler, viz.,
^ = 0, q^R=-\l, q'R = —\c,
If we should attempt to treat the general equation of the fifth degree in
the preceding manner, we would be led to equations of higher degrees than
the fifth, which must be regarded as a strong argument for the non-existence
of an algebraic expression equivalent to the root of the general equation of
this degree.
200 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
MEMOIE No. 33.
On the Equilibrium of a Bar Fixed at One End Half Way between
Two Centers of Force.
(The Analyst, Vol. II, pp. 57-59, 1875.)
"A very small bar of matter is movable about one extremity which is
fixed half way between two centers of force attracting inversely as the
square of the distance ; if Z be the length of the bar, and 2a the distance
between the centers of force, prove that there will be two positions of equi-
librium for the bar, or four, according as the ratio of the absolute intensity
of the more powerful force to that of the less powerful is or is not greater
than (a -f 21) -^ (a — 2l) : and distinguish between the stable and unstable
positions."*
Solution.
Assume the fixed extremity of the bar as the origin of coordinates and
the direction of the line joining the two centers of force as that of the axis
of a;. Then x and y being the coordinates of a material point of the bar,
and X and Y the forces acting on it, we have from the well-known equa-
tions for the motion of a rigid body
S '^'^-y^''' dm ^SixY-ylT).
If ilf and M' denote the intensities of the forces at the unit of distance, we
have
_ M{a — x) dm M' (a + x) dm
^- ^^a-xf+f-\i-i{a + xf+f]V
_ M ydm M'ydm
Introduce polar coordinates, and put
a; = rco8^j y = r&va.O,
and since the mass of the bar may be supposed evenly distributed along its
length, put dm^=dr, and take the integration with respect to r between the
* Cambridge Problems for 1845.
EQUILIBRIUM OF A BAR BETWEEN TWO CENTERS OP FORCE 201
limits and I. These substitutions made in the equations of motion, we get
V cfe . „ PT Mrdr M'rdr
3" ^ = « «'^
/*r Mrdr M'rdr ~|
i/o L [«'— 3a?- cos e + r'li + [a' + %ar cos d + r'YA '
Or, the integration performed ,
Z d^__ Msine
3 df ~ [a - ; cos ^ + V (a'- 3a? cos + P) a/ ](a^ ^ 2al cos « + ?')
M' sin
+
[a + i cos (? + V (a'+ 3aZ cos + ?')] V (a'+ 2a? cos + P)'
This differential equation determines d and thus the position of the bar
at any moment. For equilibrium the right member must vanish; thus
= 0, = 7t are two positions of equilibrium. If there are any others, the
equation
[g — ? cos g + V (a"- 2al cos d + P)'] V (a'- 2al cos g + P) _ M
[_a + lcoB0 + n/ la'+ %al cos 6 + ?^)] V (a'+ aa? cos (? + ?')"" M'
must be satisfied. But the numerator of the left member of this equation
evidently has its minimum value when := , and constantly increases from
this point until 6 = 7t when the maximum value is attained. On the other
hand, the denominator has its maximum value when = 0, and constantly
diminishes from this point until 0:=7t, when the minimum is attained.
From this it is plain that the minimum value of the left member is ( ^ ) ,
^ \a + Z/ '
the maximum value { J , and that the member continually augments in
going from first to second. Hence if -=j^ lie between f^^jj and C^^j) ,
there will be two additional positions of equilibrium, one between =
and = 7t, and the other between 6=: n and = 27t ; in the contrary case
there will be none.
When we have nearly 6=0, the differential equation reduces sen-
sibly to
73 Ma
±.^ = 1- M{a-l)-'+ M'{a + ?)-'] sin ^
and when nearly B=-n, to
^^ = l-M{a + l)-'+M'{a- I)-'-] sin d .
Thus the position of equilibrium when =: is stable or unstable according
as -Tjy- is greater or less than ( — -j— ^) i and when = 7t, the equilibrium is
stable or unstable according as j^ is less or greater than ( — ^^J .
202 COLLECTED MATHEMATICAL WORKS OP G. W. HILL
The two remaining positions of equilibrium, when they exist, are always
unstable, as will be plain from considering the mode of increase of the func-
tion of which is equivalent to -^ .
NOTE.
The foregoing solution agrees with the statement of the problem, if we
suppose that I is so small that its square may be neglected. It may be
added that the preceding expression for -^ is complex only because it is
necessary to make sin Q appear as a factor. If i^ and •^' denote the angles
at the base of the triangle formed by the two centers of force and the ex-
tremity of the bar, the difierential equation can be written thus
I ^ = - 2M&m' |- + %M' sin' |^ •
GRAVITY AT THE FOOT OF A CONICAL MOUNTAIN 203
MEMOIR No. 34.
The Deflection Produced in the Direction of Gravity at the Foot of a
Conical Mountain of Homogeneous Density.
(The Analyst, Vol. II, pp. 119-130, 1875.)
Assume the station as the origin of coordinates, the axis of x being
directed toward the center of the base of the mountain, and that of z vertical.
Let a be the radius of the base and c the altitude of the mountain. The
equation of the mountain's surface is then
a^ (c - zf - & [(« — xf + fl .
The equation in terms of polar coordinates is obtained by putting
a; z=: r cos S cos a>, y = r cos d ainu), z = r ain0,
and thus is
. c cos ff cos o) — a sin 9
r — 2ac
& cos' e — a^ sin' 6
The element of volume of the mountain may be regarded as a rectan-
gular solid whose sides are cZr, r cos 0(^cj, rcZ0, and p being its density, the
element of mass is p r^ cos ddrddda. Its attraction on the unit of mass at
the station is p cos 6 drdd da. From the symmetry of the cone it is plain
that the component of the mountain's attraction in the direction of the
axis of y is zero ; and the vertical component which diminishes the intensity
of gravity at the station may be neglected. The component in the direc-
tion of the axis of x is
-///
cos' cos at dr de d .
Integrating with respect to r, the limits are r = and r =. the value
given by the equation of the surface. Thus
^ /* /*c cos (? cos o — « sin (9 _ . ,
X=2acpJ J c'cos'g-a'sin'g cios^ cos o> d0 da, .
Next we integrate with respect to o. As r must be always positive, the
limiting values of a are the two roots of the equation c cos o = a tan $ .
Hence
204 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
/.r ccos^gcos-^[-^taiig] sing cos g 1
V L c' coo' e - a' Bin' e ~ ^c'coa'd-a'Bm'ej^^-
The limits of integration are now from 6 =^0 to 6= the value given by the
equation a tan = c. The second term within the brackets is integrable,
and between the limits is „ " . . To simplify the first term, revert to
a^ + (^ "^
the variable a , that is, put a tan d = c cos a . Then
r /*rr (odio a' "1
L*'° sin o 1 + -^ cos' w J
The expression within the brackets is a function of — , calling it
F f — j , we have
Now p' being the mean density and B the radius of the earth, the force
of gravity is
and S the deflection of the plumb-line is given by the equation
tan 5 = •— =
\a J P c
g — ^t: y R
The definite integral
/* ludm
it appears, must be computed by mechanical quadratures.
As an example in illustration, suppose a = 5 miles, c = 2 miles,
R = 3956 miles, p = 2.75 and p' = 5.67. For evaluating the definite inte-
gral, divide the interval between and -— into 9 equal parts ; then h = 10°
= 0.1745241. Compute the value of the function to be integrated mul-
tiplied by h for the middle of each of these parts, that is, for
6) = 5°, 15° 115°. The three values beyond 90° are for the sake of
the differences. We get
GRAVITY AT THE FOOT OF A CONICAL, MOUNTAIN
205
tii.
Ao.
w.
Ao.
Ci>.
Aq.
5°
0.1400956
45°
0.1737216
85"
0.3594408
15
0.1432880
55
0.1893800
95
0.3899633
35
0.1497300
65
0.3094393
105
0.3358781
35
0.1595134
75
0.3337701
115
0.3705385
As the function integrated remains the same when the sign of o is changed,
all the odd orders of differences vanish for the argument w = 0. Then
making A"^ = 0, for the argument 0=0, by summing and differencing, we
get for the argument a =: 90°,
J-^ = 1.6563687, A^= + 0.0305234, J' = + 0.0015408 , A' ^ + 0.0007833 .
Thus the value of the definite integral is
1.6563687 + ^ (0.0305224) - ^^fj (0.0015408) + ^^^(0.0007833) = 1.6576363.
Consequently i^(0.4) = 0.7955673, and the deflection
5 = 19".21174,
206 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
MBMOIE Ko. 25.
On the Development of the Perturbative Function in Periodic Series.
(The Analyst, Vol. 11, pp. 161-180. 1875.)
1. There are two modes of developing this function. In one, the numer-
ical values of the elements involved are employed from the outset, and the
results obtained belong only to the special case treated. This mode has
been, almost exclusively, followed by Hansen, and is, perhaps, to be recom-
mended when numerical results are chiefly desired. In the other, all the
elements are left indeterminate, and thus is obtained a literal development
possessing as much generality as possible. Certain investigations, arising
from Jacobi's treatment of dynamical equations and Delaunay's method in
the lunar theory, have invested the latter mode of development with addi-
tional interest, and with it we shall be exclusively engaged in this article.
In Liouville's Journal for 1860, M. Puiseux has given us two memoirs
on this subject, in which appears the general term of this function, but his
formulas seem susceptible of modifications which would render them much
simpler. More recently, in the volume of the same journal for 18 73, M.
Bourget has presented the development in a more concise form by employ-
ing the Besselian functions, but as he discards the use of the functions 6^'\
his formulas on this account are more complex. It is hoped, that, even if
the expressions, given hereafter, are deemed too cumbrous for practical use,
they may still possess some interest from a theoretical point of view.
2. It is known that if we have a function yS' of a variable ^ , which is
never infinite, and such that the relation
function (C + 2in) — function (C)
is satisfied for all integral values of i both positive and negative, it can be
developed in a series of the form
^i. (Z;'"' cosiC -t- -ff;'" siniQ ,
in which i denotes a positive integer ; and that, in the cases where this
series is infinite, it is convergent.
THE PERTURBATIVE FUNCTION 207
In general, the handling of periodic series is easier if we introduce
imaginary exponentials in the place of the circular functions. Thus, e denot-
ing the base of natural logarithms, we shall put z = e^^~'^, whence
3 cos C = 2 + 2-\ 2 COS iZ = z'+ e-\
^VC— 1) 8iDC = «-z-S 2V(— 1) sin *■£: = «'- 2-',
« = cosC + V(— 1) sine, 2* = cosi? + V(— 1) siniC.
The above theorem then comes to the same thing as to say that 8 is
developable in a series of the form
where the summation is extended to negative as well as positive values of i.
The coeflBcients ^are given in terms of the coeflBcients (7 by the equations
z;.w = c.H- c_,, ^/" = (C— c,,) V - 1 ,
except the case where i = 0, when ^'"^ =^G^. It will be seen that when
8 is real, (7; is a complex number a + &\/ — 1, and C_i, its conjugate
a — hs/ — 1, which renders the coefficients ^real, as they should be.
The integral
A'fi?C - I (cos iC + V (— 1) sin iCj dZ,
taken between the limits and 27t, vanishes in all cases except when i^=0,
when its value is 271. Hence any function, capable of expansion in a series
of positive and negative integral powers of z, integrated with respect to ^
between these limits, gives, as the result, 2n times the coefficient of z° in its
expansion. And as the coefficient of z" in the function &~' is evidently (?<,
we have
1 /*2ir
G,^-=- I Sz-'dZ.
This equation holds for all values of i, negative as well as positive, zero
included.
3. Let us now suppose that t, denotes the mean anomaly of a planet,
and let u be the eccentric anomaly, connected with the former by the equa-
tion, e being the eccentricity,
w — e sin M — C.
In like manner as for ^, we introduce the imaginary exponential s = e"*'~^
Then the last equation can be written
£(M— eslntt))/— 1 — eSy— 1,
208 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
and, by the introduction of the variables s and z, this becomes
which is the transcendental equation connecting s and z . "We have
dZ = (1 — e cosu)du=\ 1— -n[s + —] du.
Substituting these values in the equation giving the value of Gi , and
noticing that, as ^ and u both take the values and 27t together, the limits
of integration, when u is the independent variable, are the same as for ^,
we get
^.=ir*-*'-"[-i(-T)]
du.
But, from what precedes, we conclude that the coefficient of s* in the expan-
sion of any function W, according to positive and negative powers of s , is
1 P^"
-ij- / Ws~'du.
Thus, from the foregoing expression for G^, we derive the following propo-
sition :
i being a positive or negative integer or zero, the coefficient of z', in the
development of S, according to the powers of z, is equal to that of s^ in the devel-
opment of
*«-''[-l(-4)].
according to the powers of s .
As most of the functions S, which are presented by astronomy for
development in powers of z, are quite readily expanded in powers of s, this
theorem is of much use. Another form can be given to it. For we have,
integrating by parts
Csz-'dZ = — V (— 1) fsz-^' + ^^dz
Taking the integrals between the limits ^ = and ^ ^ 27t, we get
^'- 2w J ds^ "-^
Mt! Ja dS
Whence we conclude this proposition :
THE PERTURBATIVE FUNCTION 209
The coefficient of z' in the development of S according to the powers of z is
equal to that of s'~^ in the development of
i ds
according to the powers of s .
This theorem however is not applicable when i = .
4. We shall often have occasion for the expansion of the function
in powers of s ; let us, for simplicity, put ;i = — , and
We have
whence we conclude that
■^^ — 1 . 2 . . . I L 1.(1 + 1)^ 1. 2(i + l){i + 2) — • • J •
This series is not applicable when i is negative ; but if, in the function
-- . 1
e"'. e ', we substitute — for a, and change the sign of /l, the function
remains unchanged, hence
and, consequently,
by which the values of these functions for negative values of i can be derived
from those in which i is positive. These functions are known as the Bes-
selian. By putting
^< — 1 1 A' J. 1 ^ +
i.(i + i)'^i.a(i + i)(i + 2) •••
one will have no difficulty in deducing the equation
^'-''=^'~i(i + 1) ^' + 1*
34
210 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
5. We come now to the more complex function S of two variables ^
and ^' ; it is known that when this is never in6nite and is such that
function (C + 2^, C'+ 2i'7r) = function (C, C)
it can be developed in a series of the form
2,,„ [^1,1, cos (tC + i'C) + El:\, sin (iC + i'Z')] ,
where to one of the quantities i and i', we need assign only positive integral
values, but to the other both positive and negative values. If we adopt
another imaginary exponential z' =s^'^~^, this is the same as saying that
where the summation is extended to all integral values positive and negative
for i and i'. Since we have
z'z''' = (COS iZ + V (— 1) sin iC)(cos i'Z' + V (— t) sin i'Z')
= cos (iZ + CC) + V (— 1) sin (iC + i'C) ,
the relations, which connect the coefficients K with the coefficients G, are
jrifi. = ((7,„-c_,,_,0V-i,
unless i and i' are both zero, when
-0-0,0 — ^0, •
A course of reasoning, similar to that in the case of one variable, established
that
which holds for all integral values of i and i', positive, negative and zero.
6. Supposing that t,' denotes the mean anomaly of a second planet,
whose eccentricity and eccentric anomaly are respectively e' and u', we have
u' — e' sin u' = C',
and by the adoption of the imaginary exponential s' = e"'''"^, this is trans-
formed into
It is not difficult to see that we have the following theorem :
The coefficient of z^^' in the development of S , according to the powers of z
and z', is equal to that of s's'*' in the development of
according to the powers of s and s'.
THE PEBTURBATIYE FUNCTION 211
7. After these preliminaries relative to the general development of func-
tions in periodic series, we come to the matter more immediately engaging
our attention. The perturbative function for the action of a planet, whose
mass is m! , on another, whose mass is wi , is usually written
T> r r 1 r cos ^n
and that for the action of m. on ir^
n r 1 r' cos (l>~\
where A denotes their mutual distance, '^ their angular distance as seen from
the sun, and r and r' their radii vectors. The problem proposed is then
to develop these two functions in series whose general term is of the form
(7j, i'zV*'. To this end it seems better to discuss the two portions of the gene-
ral perturbative function, — and — — -j^ , separately, and not, as most
investigators, attempt, by a particular notation, to combine, in a whole,
these two parts. Thus, in developing — , we shall have the term common
to both fimctions, and may suppose that r' denotes the radius vector which
belongs to the planet more distant from the sun. But, in treating the sec-
ond part, we shall suppose that r' belongs to the disturbing planet. The
following equations are well known :
j» — /2 — 2rr' cos > + r',
cos ip = cos {v + n) cos (v' + B') + cos / sin (v + 11) sin («' + H') ,
= cos(v — ^'4- 77 — 77') — 2 sin' I J sin (« + 77) sin(t)'+ 77'),
where v and v' are the true anomalies, and n and 11' are the angular dis-
tances of the perihelia from either point of intersection of the planes of the
orbits, and /is their mutual inclination.
8. Attending then, in the first place, to the development of -r- , we have
to notice what are the conditions under which this quantity can be devel-
oped in powers of a and z!. In the case of two elliptic orbits, the only one
we shall consider here, it is plain that -r- is always finite and continuous,
provided the orbits have no point in common. Here we must make two
cases according as the value of sin I is not or is zero. In the first case it is
evident that the orbits can meet only on the line of intersection of their
planes. Hence, p and p' denoting their semi-parameters, there will be
212 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
two, one or no points in common, according as two, one or none of the
equations,
/ (1 + e' cos /7')-' =^ (1 + e cos n)-\
p' (1 — e' cos /?')"' =p (1 — e cos /7)-\
are satisfied. In the second case, where the orbits lie in the same plane,
there will be two intersections or none, according as the equation
p' [1 + e' cos (A — w')]- ' = ^ [1 + e cos (A — w)]- ',
/I being the unknown quantity and a and o' the longitudes of the perihelia,
admits real or imaginary roots. If we put
pe' cos u>' — p'e cos ui^A cos a ,
pe' sin lu' — p'e sin u) =^A sin a ,
this equation takes the form
A cos (A — 0-) =p' — p ■
The roots of this are imaginary when
(p' — pY > ^V — 2pp'ee' cos (to — u>') + ^"e^
9. If we put
we have
P — r'' — 2rr' cos {v — v'+n—- 11') + r\
Q =4: sin' |/. r sin (v + ll) . r' sin (v' + II')
a series we shall denote thus
1 J^ ,_ ^., 1 . 3.. .(2^-1) p-H±i
10. In order that this development of —r- in a series of ascending pow-
ers of Q, or, if one likes, of s,vd?\I, may be legitimate, it is necessary that
the elements of the orbits should be such that the numerical value of-^
should be always less than unity. P is the square of the distance of the
two planets after the plane of the orbit of one has been brought into
coincidence with the plane of the other by revolving it about the line of
THE PERTURBATrVE FUNCTION 213
intersection of the two planes. Taking then a system of rectangular axes
passing through the center of the sun, and directing the axis of x along the
line of intersection, it is plain the equations of the orbits may be written
V(a;' + f) + ax + ^y =p,
V(a;''+^") + a'x'+^'y' =p',
a, /?, a', /3' being constants. And the variables x , y , x', y' satisfying these
equations, the question depends on the finding of the values of them which
render the expression
D., yl_
(x-x'f+iy-y'y
a maximum or a minimum. According to the known theory of maxima and
minima, the equations, which, in combination with the equations of the
orbits, give these values, are
- 2i? C=« - ^') + Z' [t^s^^) + « ] = ,
%D {x-x') + / [^TC^^F) + «'] = 0,
y'-2I){y-y')+^[jj/^rf)+P'\==^,
2/ + 2i)(y-2/')+M'[y(^l!p^) + /3']=0,
where ^ and yl are the multipliers of the partial derivatives of the two equa-
tions of condition. A complete investigation of this question would be con-
ducted in the following manner. Eliminate from the seven equations last
given the six quantities x, y, x', y', (i, fi' ; the result will be an algebraical
equation determining the unknown D. Having derived the Sturmian
functions of this, one will ascertain by the substitution of the values
D = , — ■„.,, , Z) = + 00, and again of i) = — — . ,. ., ^ , Z) = — oo, whether
any roots lie between these limits ; if none, -— can be expanded in a series
of ascending powers of sin^ \ I, in the contrary case not. In this way we
shall arrive at the condition or conditions necessary and sufficient for the
legitimacy of this expansion.
11. This procedure would doubtless lead to very complicated formulas,
hence we are obliged to pass over it. However, equations can be readily
got, which, by a tentative process, afford the maximum and minimum values
of D. Multiply the four equations last given respectively hy x,x',y, y'
214 COLLECTED MATHEMATICAL WOEKS OF G. W. HILL
and add the resulting equations, having regard to the equations of the orbits
and the value of D; we thus arrive at the simple relation
p^i +p'/ji' = 0.
Putting, for simplicity, x = r coad, x' =7^ cos6', the addition of the first
and second of the same group of four equations gives
/i (cos d + a) + ti' (cos e'+ a') = 0.
By combining this with the preceding is obtained
cos 0' + a' COS ^^ + a
J' "" P "
Again the addition of the same equations, multiplied severally by x,
— x', 1/, — y', gives the equation
2I)(r" — r'')=p'fi'—p/i.
Dividing the left member of this by 2D [x' — x) , and the terms of the right
member by its equivalents derived from the first and second equations, we
get
r" — r^ p' p
x' — X ~ cos &' + 0' "*" COS (? + a '
or
r' COS 6' — r COS cos + a
r'^ — r' 2p •
This and the equation
cos 0' + a' COS 6 + a
p p
determine the values of the variables Q and 0' which render D a maximum
or minimum. When the orbits are nearly circular these values are in the
neighborhood of 4 7t or !«. When both orbits are circles the solution is
very simple, and we find, in order that the development may be legitimate,
we must have
. / a' — a
a and a' being the mean distances of the planets from the sun.
12. Assuming that this development is legitimate, we have to develop
_2S+_1
P 2 g* in terms of s and s'. We have
THE PEE.TURBAT1VE FUNCTION 215
r cosv = a{cosu — e) = -g-fs-l 2ej,
r sin z; = a V (l-e") smu= ^f_^ V (1 — e') (« — t) '
whence
and by putting
l + V(l-e') e
2 -''' 1 +VC1 — eO"*"'
we get
revV-l = arisil — —]•
And the value of re""''"^ is evidently obtained by substituting in this -— for
s, hence *
s
From these two equations may be derived
r = ari(l — S
Writing y for n — IT', we have
(r'-'P)- ' =[l-2yC08iv-v'+r) + ^j-~-
The right member of this is developable in a series of integral powers of the
exponential gC-^'+v)!^-! when —is always less than unity. This con-
dition is fulfilled when we have a(l+e) . ^»+i BJie-v'+y )V- 1
j^ — CO
Bok + i is the same function of -r that Laplace's b^u+i is of ~= a . The
-y- / -2- a'
approximate value of — j- being a, any function of —j- can be expanded in a
series of ascending powers of -^ — a by Taylor's Theorem. And as we
have
216
consequently,
COLLECTED MATHEMATICAL WORKS OE G. W. HILL
[V(i--V)(i--f) J
■ V(l — a>V)|^l-^j
n being an integer, and n ! denoting the product of all integers up to n inclu-
sive, it being understood that ! = 1 . Expanding the last factor of this
expression by the binomial theorem, and employing the notation [t,y] for
the coefficient of a;'' in the expansion of (l + ^c)', we have, p being an inte-
ger,
13. In the next place the development of Q in terms of s and s' must
be formed. We have
r sin (w + n) — 2J _i C*"^*" + ""'- 1 - rs- c + ^)V- 1] ,
r' sin {v' + /7') = ^ ,_^ [/£(«'+ n')V- 1 _ /e- («'+ nW- 1] ,
n + n' = o, h = e»v-i,
and putting
we find
^[''(^~t-^-t(i-'"'*')v]-
Raising this expression to the k*-^ power, and multiplying by
r'-'^'+" = [aV (1 - «''s')(l - -T-j]
we find that the part of r'- '^'^ + " Q«^ which has h'"' as a factor is
J_ „Y)?'- '* + " sin'' 4 2 (— l)""[^j w'][*, * — i'"— w']
(„, \ai"' + aii'
1 — — )
C„/ \-2»'-l
1 ^j gl'"-l,+ l,'J^ill,^
THE PEETUEBATIVE FUNCTION 217
14. We are now in the possession of all the developments necessary for
exhibiting the function -r- in terms of s and s'. In order to obtain the part
of this function which has g^"h^"' for a factor, we must put, in the formulas
of §12,
j = i" — i"'+k — 2n',
and the chief operation here is the addition of the exponents of the quan-
tities s, 1 — as, 1 , and the similar functions of s' which are found
s
U + l (J)
in the three formulas for (r'^^P) 2 , £?^and r'-'^* + "g\ For brevity
we will write
_ i.3...(a/fc-i)
L*-"- 2.4. ...2^ •
Then the part of — , which has g*"h^"' for a factor, is
_1_2 ""5;"''' "2 V (- i)^-^"'+»-> W[^> ^'][^. ^ -i"'-n'Jn,p]
j„7(i"— iV'+s_2,i')
Cm' \—j!— 31— 1 — <" + <"'
XV
/-i-
We observe that in this expression the summation with respect to n'
affects only the integral coefficients \h, «'] , \h,'k — i"' — n'] and the upper
index of the quantity 6, hence if a new function of a is assumed, which is a
linear function of the 6's, and such that
it will take the following simpler form :
%a' jLl Li ^^ I n\ d^~ 2
k = i"' n = p =
1 — —]
/ B,' \-l-J.-l-4" + «"'
X,'-«-P-V-'" + ""(l_a,V)-«:-p-i + <"-Wl _^j /'r".
15. In order to get the coefficient of zV*'- in the expansion of —^ ,
according to the foregoing investigation, we must multiply the preceding
expression by
218 COLLECTED MATHEMATICAL WOKKS OF G. W. HILL
Hence if for brevity we adopt the functional notation
the coefficient of z'z'' g'"h^"' in —^ will be equal to the coefficient of s's''' in
k — CP n = co p —n
j_ y y V(_]y-<"+.-p W[w>j?]
/vd ^^ ^J ^m fll
k — i'" n = p=0
da" 2 \v y \ ^,' /
If then the coefficient of s' in Lhe expansion of S is denoted by E fol-
lowed by the same indices, and the coefficient of s'*' in the expansion of S^
by E' in like manner, E will be a function of e only, and E' a function of e'
only ; and, it being understood that each argument is taken but once, that
is, the negative of the argument is not considered, the coefficient of
in the expansion of — is expressed thus
a^n + l • T /ic-rp\ ,— \^K-rp + i-)^
As in this formula, h ought to be a positive integer, it will prevent
embarrassment, if the arguments are so taken that i'" may not be negative.
In the case where i, i', i" and i'" are all zero, the expression must be divided
by 2.'
16. Thus we have arrived at an expression for the general coefficient
involving only three signs of summation ; and it may be remarked that all
the coefficients are exhibited in precisely similar forms. Thus, to pass from
one argument to another, we have only to make the suitable changes in the
two lower indices of the functions E and E' and in the upper indices of B ,
and commence the summation with reference to k with the new value of i'"
instead of the old. Hence, from this expression, we can write out a scheme
or blank form, which, when the indices proper to the argument are filled
in, will be the coefficient of the cosine of it in the expansion of -r- . Such
THE PERTURB ATIVE FUNCTION 219
a blank form is written below ; the indices i" and i'" are omitted from B ,
and the two lower indices from E and E', and the upper indices of these
quantities, for the sake of facility in writing, are placed to the right and at
the foot. The factor —j , common to the whole expression, is also omitted,
so that the formula gives the coeflBcient in the expansion of — r- . In making
use of it, one must commence at the portion which has sin^'"' h I for a factor,
all the preceding parts being supposed to be suppressed. It is hoped that a
sufficient number of terms have been written to render the law evident, so
that they may be continued as far as desired.
+ 5^a"-§[E„E'_,-3E,E'_,+ E,E'_3]
- i;^a'||[E.E'_,- 3E,E'_,+ 3E,E'_3- EsE'.,]
+
+ ^sin4{ "B^E^E'-
_ya^4!-^tE.E'_,-E,E'_3]
+ j^a'^[EiE'_,-2E,E'_,+ E3E'_,]
— -12 3 " da? [EjE— J — 3E2C_3+ ot3t_4 — t4L_5j
+
+K-4{ ''^^^^^'-
-4-'f'[E.E'.3-E3E'_J
+ l^a*^[E,E'_3-2E3E'_,+ E,E'_,]
1 .cm-.
a
1.2.3" da'
+ . . . .
[E,E'_3- 3E3E'_,+ 3E,E'_,- E,E'_e]
+ M^6'^'y{ «'B5E3E'_,
_Ji.„4^rEE' — E E' 1
+ j^«^^'[E3E'_,-2E,E'_,+ E,E'_c]
1 ,^'Bs |-E3E'_,_ 3E,E'_,+ 3E E'_e- £«£'_,]
~ 1.2.3 da"
+ . . . .
220
COLLECTED MATHEMATICAL WORKS OF G. W. HILL
For illustration, let it be desired to obtain the coefficient of
cos(2C-5r+2r),
from which arises the larger part of the great inequality of Jupiter and
Saturn ; we have only to imagine that the lower indices ( 2 I are everywhere
applied to E, and the indices I _2 ) to E', the indices (2, 0) to B ; and as
we have *'" = , we suppress nothing.
17. The quantities B are very simply expressed in terms of the h's.
The following are all that are needed when terms of the eighth order with
respect to the inclination of the orbits are neglected.
B|''°
) 7,0
^■'
=*r'+*r''
Br
7,(0
-0,,
B<'-»
= Jf^>+45«+J|-»',
B«-^
= 2J|'+"+25|*-»,
Bf
^6f,
Bi''»
= *!'+"+ 9*1'+''+ 9i|*-"+ 5<'-",
^r
= 3J»+" + 95|« + 35i'-»,
Bp
= 35i'+"+3M<-'>,
Bi''"
= iT.
18. In computing the factors of the preceding formula which depend
on E and E', the following abbreviation can be used. M^ denoting the fac-
tor which multiplies
1 a" Bm -t- 1
«! da" '
and A being the symbol of finite differences with respect to «, it is plain
that
J"ilf„ = (-l)"E,+„E'_„+„+„.
Hence, if the products £* + „ E'_(i + „ + i) are computed for the various values
of n, and are taken alternately with the positive and negative sign, and are
written as if they were the successive differences of a function, we shall get
the values of the factors ilf„ by filling out the scheme of differences. This
abbreviation is applicable equally whether we are making a numerical
THE PEETUEBATIVE FUNCTION 221
computation of the coeflBcient or a literal one. In the latter case the
abbreviation can be applied separately to each term of the form CeV'' in
the products £*£'_(», + i).
19. We proceed now to discuss the functions E. From their definition
we have
whence
(^Jcosy. = i 5 [e (i) + E(-y)]cos^c,
(-^Jsiny. = i 5 [e (|) - <-|)] '''''''■
From which we gather that the functions E can be computed by definite
integrals, thus
Let us now suppose that the coefficient of s*, in the expansion of
/S 'v^" /■'"*" ^N
in powers of s , is denoted by -E" | y i , then evidently
By writing in the expression 1-^s for s and changing the sign ofy, it
remains unaltered ; hence the relation
By developing the factors of the expression
by the binomial theorem, we get
E0^ = i-iy-^ii -y, Jc -y] v<"'-^
V fi -^ (i±M=l) ,,., (^•+y)(^•+y-l)(^•-/fc)(^-/fc-l) , 1
222 COLLECTED MATHEMATICAL WORKS OP G. W. HILL
This equation, as written, is correct only when h — j is not negative, but by
the relation given above we can reduce the case of h — j negative to that
where it is positive. The factor in the brackets is a case of the series
treated by Gauss in a memoir entitled " Disquisiiiones generales circa seriem
infinitam, cfcc." (See Gauss' Werke, Vol. Ill, p. 1 23, and especially the
" Nachlass,'' p. 207.) According to Gauss' notation
^ (j) = (- 1)'-' [i - j, k -j-] iiJ--' F{- i -j,k — i, k-j + 1, o,^) .
Whenever, of i -{- j and i — j, one is not negative, this series terminates
after a certain number of terms, thus aflfording a finite expression for the
function. But when these integers are both negative, the series is infinite.
However, it can be easily transformed into another which like the former is
finite. From Gauss' investigation of these series (see the volume just
quoted, p. 209, equation [8 2]), we have
F{a, /S, r, x) = (1 — a;)v-»-P F{y—a,r — P,r,x).
Applying this to our expression, we get
e(j\ = i—lf-'ii—j, k—j] ri'io'-^l — my + 'FCi + k+l, i—j + 1, k—j + 1, w").
This expression is evidently finite when i — j and i -\-j are negative.
20. The developments of the functions E in powers of e as far as e' have
been tabulated by Prof. Cayley in the Memoirs of the Royal Astronomical
Society, Vol. XXVII. It would conduce to the ready employment of the
preceding formulas if we had the function E ( y i explicitly expanded in
\h)
ascending powers of e, but the attempts I have made to write such a series
lead to extremely complex forms of the coefficients. Hence I shall give here
only the coefficients of the lowest power of e in this function, which suffices
for obtaining all the terms of the lowest order in any coefficient of the expan-
sion of 1 -^ A. We have, when/ — h is positive,
E (]" ) = [[t +j,j-k-] + ii +y,i-/fc-l]^ + [i +j,j-k-%] ^
THE PEETUKBATIVE FUNCTION
223
and when Tc — j is positive
^ (4 ) = [f *' -^ ' ^ -■^■] - 1^' -i ' -^ -i - 1] T + 1« -y > * -j - 2] o
-... + [^-i,o]^^](-J
21. Thus in the example alluded to above, of the coefficient of
cos (2^ — 5^' + 2y), we find that the terms of the lowest order in E and E'
(omitting here, as in the scheme, the two lower indices), are
E„=E,= E,= E3 = 1,
E'-i= - [[-3,3]-[-3,2]| + [-3, 1]^ -[-^'0^li](T)=^^'"■
E'_,= - [[- 3, 3] _ [- 3, 3]f + [- 3, 1]^^ - [- 3, 0] jJi
_ 5 9 -,'3
_8 4 5 p'i
'-3=-[[-4,3]-[-4,3]f + [-4,l]f2-[-4,0]^3
'-*= - [[-5, 3] - [- 5,3]A + [_5,1]^^_[_5, 0]if3](4J=HF^"
Bringing into use our method of abbreviation, we multiply each of the
preceding numerical coefficients by 48 in order to avoid fractions, and then
write them alternately with the positive and negative signs in a diagonal
line, and from these, as successive orders of differences, derive the numbers
standing in the vertical columns, thus :
+ 389
— 590
— 201 + 845
+ 355 —1160
+ 54 —315
— 60 + 381
— 6 +66
+ 6 — 73
— 6
+6
and dividing the numbers of the first column respectively by 1, — 1, 1.2
— 1 . 2 . 3, we get the following as the terms of the lowest order in the coeffi-
cient of cos (2^ — 5^' + 2y) in a' -i- A ,
18" L * + ^"^ "1^ + ^^" ^S^ + " do? Y '
which agrees with that found in the books. The following additional terms
224 COLLECTED MATHEMATICAL WOBKS OF G. W. HILL
of the same coefficient can be written from the second, third, &c., columns,
viz., those which are multiplied by e'^ and the various powers of sin^ ^ /,
-Us [S90a Br>+ 255a^%- + 30a3 %- + a^^] e'^ sin4
+ gi[845«^Br"+ 315a3^ + 33a^^' + a^^j.^sin^
— &c
22. When we wish to obtain only the terms independent of ^ and ^',
that is, those on which the secular perturbations depend, i =; and i' = ,
and the Besselian function / disappears from the expressions giving the
values of E and E', and the coefficient of cos {i"'y + i"'d) in the expansion of
-r- can be written
A
k =00 n = ca p'=~n
v„* + " ~
d¥- «1^
J- .k+p + U .— {{h+p) .
23. In leaving the subject of the development of 1 -^A, it may be well
to note that two other forms can be given to the expression of the general
coefficient, by employing, instead of the expression given above, either of
the following :
— 7 ^— a ^^ OL 1 — z ^ — r
^-^V + 7)
,'(l-.Y)(l-4)
But as they do not possess as much symmetry and brevity as the form given
above, we will pass over them.
24. The second part of the Perturbative Function, omitting the factor
m', is
r r r I I ~1
— ^ji-coB4' = :;ji-\ cos'' Y COS (v — v'+r) + sin'-g cos (v + v'+ e)
^ COS ¥' = — -p?
THE PEKTUEBATIVE FUNCTION 225
According to the first theorem of §3, the coefficient of 2" in — e'"^~^ is
equal to that of s" in
or it is equal to
And, according to the second theorem, the coefficient of 2* in the same func-
tion is equal to that of s* in
ii['('-Tr]-«-"'-''=i('-^)-*<--*-
Hence we have
.■=+00
a .^^ * L ^ 2 J
And by simply writing 1 -f- a for z,
r .
— e
a
I = + 00
The well-known differential equations of elliptic motion
^ J^ -n
J + TT^/^o,
supposing the axis of x to be directed towards the perihelion, give us the
equation
fi. f(i£i:).
r"^ — dC
and consequently these two
* = + {»
4- e-v-i = ^ ^'J f'"''^'""- '^■■ + "1 «'■
By substituting these values in the expression given above for ^^ cos 4'.
it is not difficult to see that, in it, the coefficient of
cos(iC + i'C'-l-r)
IS
25
a 4i I \_ -2 Xj L2 aj
226 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
and the coefl&cient of
cos(iC + i'C'+fl)
is
— 4- sin»4 -^ U?-' - '"'^?+"l . i'-n' r^Sl'-"- "'V/i'.+ "l .
In the special case of i = the middle factors of these expressions take the
indeterminate form 0-^0, but then, in accordance with what has been shown
above we should read — |e. Thus, by means of the Besselian functions,
these coe£Scients take finite forms.
DIFFERENTIAL EQUATIONS EMPLOYED BY DELAUNAY 227
MBMOIE Fo. 36
Demonstration of the Differential Equations Employed by Delaunay
in the Lunar Theory.
(The Analyst, Vol. Ill, pp. 65-70, 1876.)
The method of treating the luaar theory adopted by Delaunay is so
elegant that it cannot fail to become in the future the classic method of
treating all the problems of celestial mechanics. The canonical system of
equations employed by Delaunay is not demonstrated by him in his work,
but he refers to a memoir of Binet inserted in the Journal de TEcole Poly-
technique, Cahier XXVIII. Among the innumerable sets of canonical ele-
ments it does not appear that a better can be selected. These equations can
be established in a very elegant manner by using the properties of Lagrange's
and Poisson's quantities (a, h) and [a, &]. But a demonstration founded
on more direct and elementary considerations, is, on some accounts, to be
preferred.
Let a denote the mean distance, e the eccentricity, i the inclination of
the orbit to a fixed plane, I the mean anomaly, g the angular distance of the
lower apsis from the ascending node, h the longitude of the ascending
node measured from a fixed line in the fixed plane, ^i the sura of the masses
of the bodies whose relative motion is considered, and R the ordinary per-
turbative function augmented by the term |p . Then if we put L = V^a,
6r=\/ [jLLa{l — e^)], H^s/ [}ta{l — e')] cosi, Delaunay's equations are
m_ dR dH_ dR
dt dg ' dt dh'
dg _ _dR dh dR
irt~ dO' dt dH'
In terms of rectangular coordinates
j._ fj? m[^ w! {xx' -\- yy' + z/)
2Z^ + iix' - xy + {y'- yj + iz'- zf]^ r"
In this expression, for x,y,z, ought to be substituted their values deduced
from the formulas of elliptic motion, and expressed in terms of L, G, H,
2
I, g, h. It should be noted that the term ^3= |^' of ^^^ ^^^^ ^^^^^
with respect to the disturbing force, has been added to B only to preserve
dL
dR
dt
er
dl
dR
dt
~dL'
228 COLLECTED MATHEMATICAL WORKS OP G. W. HILL
in the equations the canonical form : it is only by amplifying the significa-
tion of the word that I can be called an element, as it is not constant in ellip-
dl u?
tic motion, but augments proportionally to the time and -5- = w := tr_ .
dit Li
It is chosen as a variable in preference to the element attached to it by addi-
tion simply to prevent t from appearing in derivatives of B, outside of the
functional signs sine and cosine.
The equations
d^x iLX _ 9J? d^y ny _ dR d^z ij.z _ dR
d¥ '^7 -Qx' W ^ 7 -dy ' W^ 1^~ dz '
are well known ; here, however, B does not contain the term -^^ . By
multiplying them severally by dx, dy, dz, adding and integrating, is
obtained
dx^+dy''Jrdz^ 11 m- PfdR . 9^, dR -, \
¥dt^ T + W ^ J (d^'^'' + dy "^y + W^V-
When the elements are made variable, this gives
d I iJ. \ _ _ IdR dx dRdy dR dz\
dt U« / \dx dt dy dt '^ dz dtj '
dx _ dx_ dy _ dy dz dz
But we have
and hence
dt -^^ dl ' di -^ di ■ dt -'^ dl '
d I iJ.\ IdR dx dR dy dR dz\ dR
r)
dt\2a) — ~"'\dx dl ^ dy dl '^ dz dl)~ ~ " dl
Dividing both members of this equation by — n=- — \/fia~^, the left
member is seen to be the differential of V^ = L. Consequently,
dL _dR
dt ~ dl'
Denoting the true anomaly by v, the orthogonal projection of the
radius vector on the line of nodes is r cos {v-\-g), and on a line perpendicular
to it and in the plane of the orbit r sin {v ■\- g). And the latter projected
on the plane of reference is r sin(« + g) cosi, and on a line perpendicular
to this plane r sin {v + g) sin i. If the two projections lying in the plane of
reference are again each projected on the axis of a;, their sum will be the value
of the coordinate x, and the sum of their projections on the axis of y, the
value of the coordinate y. Hence
x = r cob{v + g) cos h — rsm{v + g) cos i sin ^,
y = ?• cos (« + g) sin A + r sin {v + g) cos i cos h ,
z = r aiii(v + g) sin i ,
DIFFERENTIAL EQUATIONS EMPLOYED BY DELAUNAY 229
or, substituting for i its value in terms of G and H,
x = r cos(v + g) coah — ^r sin (v + g) sin h ,
TT
y = r coa (v + g) sin h + -^r sin (v + g) cos h,
z = ^ r sin {v +g).
As r and v are functions o£ L, G and I only, the preceding equations show
the manner in which H, g and h are involved in JR.
S' denotes double the areal velocity projected on the plane ay, or
„ xdy — ydx
^= dt
Consequently
dH _ dB BR
dt -^ dy ~y dx'
But the foregoing values of », y, z show that we have
and thus
dx _ dy_ 9^ _n
dff_dRdx dRdy dE dz _ dR
dt -dx 9A + dy dh + dz dh- dh '
G denotes double the areal velocity, and evidently, if for the moment
we suppose x and y to be drawn in the plane of the orbit, the axis of x
towards the lower apsis,
dG_ dR_ dR_dR
dt ~ dy y dx ~ dv '
where, in the last B, for x, y, z must be substituted their values given above
in terms o{r, v, G, H, g, h. Now, as the only way in which g is involved
in these values, is by addition to «, it follows that
dR _dB,
dv ~ dg'
and this equation is not affected when, for r and vin R, are substituted their
values in terms of i, Gandl. Consequently
dG_dR
dt ~ dg '
In the elliptic theory
'^''-^''^^ = V Q'- H' cos A,
ydz — zdy
dt
:V'— -ff'sinA.
230 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
Whence we deduce
dt dz dx '
dliJ6'-JI'sinh] _ .dR_ dR
dt ~^ dz ~^dy '
Eliminating d^ CF — H^ from these equations, we obtain
dh _ e sinh dR z cosh dR x sin h — y cos h dR
~df-i{W~TPdx~i^W^np'dy~ i^G'—H^ dz'
Comparing the coefficients of the three derivatives of B in the right member
of this equation with the values oi x, y and z in terms of r, v, G, H, g,h,
we recognize that they are severally equivalent to the negatives of the partial
derivatives of these quantities with respect to H. So that
dh ___' d^dx_ dR^^dy_ , dR dz\_ __ dR
dt ~~ [ dx dJI dy dH dz dHl~ dH '
It is a well-known principle in the theory of varying elements, that if
we diiferentiate any function, which is a function of the coordinates and t
only, but expressed in terms of t and the elements, with respect to t only inas-
much as it is explicitly involved, we obtain the correct value. Hence, if the
differentiation is performed on the supposition that the elements are alone
variable, the result should be zero. Applying this to the function r we get
or
or again
drdL drdG^ Sr(dl \_
dL dt '^ d& dt + dl\dt -«j-0'
dr^dR^drdR^drldl^ \_.
dLdl ^d^dg'^ dl\dt -^j-v.
d
r^ldRdr dRdv\ dr^dR drldl_ \_
dL\dr dl ■*■ dv dl)^ dO dv'^ dl\dt -'^j-^-
Whence we derive
dl _ dr dR (dry^rdrdy dr~\dR
dt -^~ dL dr~\dl) [_dLdl '^ dGJdV
From the expression for r we can eliminate I and introduce v in its place by
means of the expression for v in terms of L, G and I; the result is the well-
known equation
y^ a(l-0 G"
1 +ecosw n JL'-G' l*
M 1 + - — J cos V
DIFFERENTIAL EQUATIONS EMPLOYED BY DELAUNAY 231
And we have
dr _drdv dr _ (dr\ dr dv
'Wdvdl' dZ~\dL)'^ dvdL'
the parentheses denoting the derivative with respect to L only insomuch as it
enters the preceding equation for r. By making these substitutions, the co-
efficient of ■^- in the expression for -t— becomes
dv (dr\-^r(dr\dv dr~]
-dL-\dlJ iKdLJdl^dGJ
From the preceding equation for r, we derive
{dr\ _ fir' cos v
\dL) -~ m '
also the following is a well-known equation in the elliptic theory
dl ~ nr' '
For obtaining the value of -^ , u being the eccentric anomaly, we have
the equations
r = a(l — e cosm), i = w — esinw.
Their dififerentials give
= (1 — e cos w) du — sin « .
dr , . 9m
~- = — a coau + 06 amu^- i
06 oe
Whence
And
dr . cos w — e
^ = — [« i — rr:::^. = — acQav.
06 '1 — 6 cos U
_ »jL'-G' de_ G^
^- X ' dG~~L'e'
dr dr de Q cos v
dG ~ d6dQ~ p-6
By substituting the values, it is found that
In consequence
(dr\dv dr _
dl _ dR dr dRdv __ dR
'dt-'^~'dr dL~ dvdL-^ dL'
232 COLLECTED MATHEMATICAL WORKS OP G. W. HILL
As i2 is a function of the coordinates and the time only, we can treat it
as we have done r. Then
dBdL dR(dl _ ^^"^ ^dRcLQ _^ dR dg ^ dB dH ^dRdh
\ dRdG
V + Wdi
dL dt "^ dl\dt ~") '^ dG dt ^ dg dt '^ dH dt ^ dh dt
On substituting in this the values of the differentials of the elements which
have already been determined, it is seen that all the terms but two, mutually
37?
cancel each other. And, on dividing the result by -=- , we get
dg^_ dR
dt — dG'
By adding to R the term -^ = i^ , its partial derivative with respect
to L is augmented by the term c_ = — n, but all the other derivatives
are unchanged. In consequence of this addition, the value of the differential
of I becomes
dl__dR
dt - WL'
An objection may be made against the preceding method of obtaining
the differentials of I and g, that the quantities ^ and ;=— , which both
ai dg
periodically vanish, have been employed as divisors. But this objection has
force only when it is admitted that the differentials of I and g, or the corre-
sponding derivatives of B, may be discontinuous. For, having proved the
truth of the equation for all times, except when the divisors, just mentioned,
vanish, it follows, that if both members are continuous, the equations must
still hold even for the moments of time when ^ = or ^^r— = 0.
dt dg
SOLUTION OP A PROBLEM IN THE MOTION OF ROLLING SPHERES.
233
MEMOIR Fo. 37.
Solution ot a Froblem in the Motion of Soiling Spheres.
(The Analyst, Vol. Ill, pp. 92-93, 1876.)
A sphere, of radius r, rolls down the surface of another sphere, of the
same material, of radius It , placed on a horizontal plane. The surfaces of
both spheres and plane are rough enough to secure perfect rolling. It is pro-
posed to determine the motion of the sphere, the point of separation, and
the equation of the curve described by the centre of the upper sphere.
Let X and be the coordinates of the center of the lower sphere,
x' and y' those of the center of the upper, Q and Q' the amounts of rotation,
and ^ the angle the line joining their centers makes with the horizon, and
for brevity put h-= R-\- r.
The expression for the living force is
^- 2 L
dt'
+ iB'
df
+
m^rd^ d^ 2 2^1
2 ^dt^ + dt^ +^^ dt^y
and the potential is H = — 'm'gy'.
According to the frictional conditions, the variables x, a/, y', and 6'
satisfy the following equations :
Rd-x
= 0,
= 0,
>^ioi-xf+y"^-h =0.*
re'+x + h tan-' ^73-^ = ,
(1)
With Lagrange's method of multipliers, if we denote these equations
respectively by i = 0, ilf = 0, N-=0, and the multipliers of their difieren-
tials by ^, j« , V, and take ^ to represent any one of the five variables x, x',
y\ 6, 6', the general equation of the problem is
dt ^d^~ d^ ~ d? d? '^ '^ d? '^'' dS
* These equations subsist only as long as the spheres are in contact.
234
COLLECTED MATHEMATICAL WORKS OP G. W. HILL
Applying this in succession to each of the five variables, and writing for
simplicity ^ for tan'
-1 y
a/ — X
, we get
d'x
in -jii = — A + ;u (1 + sin ?>) — p cos + i- sm y ,
^ml?^= XB,
(2)
Adding the first and second of (2),
d' (mz + m'af)
dt'
= ti-X.
The two first of (1) and the two last of (2) give
■-m
d^x
df
„ , d'e' , ,rd'x , ,«^V1
Substituting these values for % and ^ in the last equation,
d'imx + m'od) „, , ,.d^x „ ,,d^v
Integrating once and eliminating a/,
^ (wi + m') -^ + m'h (I - sin f) -J- = ,
(3)
where the constant is zero because the spheres are supposed to set out
together from a state of rest. As -^ , in general, is negative (4» can always
be supposed in the first quadrant), it is evident from this equation, that if
sin <^> I, the lower sphere will move horizontally towards the side on which
the upper sphere is; but if sin^<|, in the opposite direction.
Integrating (3) twice
(7j» + %m') X + fim'x' + 3»i'Ay = a constant .
Eliminating x and 4> from this by substituting their values in terms of x' and
SOLUTION OF A PROBLEM IN THE MOTION OF ROLLING SPHERES. 235
y, we get as the equation of the path of the center of the upper sphere
7 (m + m')[a/- i^h^-y"'] + m' [^2h sin"' -|- + 5 = is equivalent to
^ cos ¥> + ( -^ + ^f 1 sin f = .
And if we eliminate »' and y' from this by means of their values in terms of
X and ^, we get
cPx , . , dp' .
■^coa ^ + g amy — h ^ = ,
By eliminating second derivatives this becomes
49m \_j^- sin ?>] + 10m' (1 + sin vf [y ^' - l] = '
which, by substituting the value of-^, becomes {(3 is the initial value of ^)
70 (m + OT')[49m + 10m' + 20m' sin ][49m + 45m' + 20m' sin y — 25m' sin' ?>] = .
236 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
MEMOIE No. 28.
Reduction of the Problem of Three Bodies.
(The Analyst, Vol. Ill, pp. 179-185, 1876.)
The object of this article is to find the three differential equations which
virtually determine the sides of the triangle formed by the three bodies,
bringing to our aid all the known finite integrals of the problem.
Lagrange was the first to treat this question in his Essai sv/r le Problhme
des Trois Corps (Oeuvres, Tome VI, p. 227) ; but the formulas lacking sym-
metry, his editor, Serret, has, in a note, supplied this and pointed out an
important error into which Otto Hesse, who had investigated this subject
{Journal fur die Mathematik, Band LXXIV) had fallen.
By adopting an orthogonal substitution, at the outset, for reducing the
number of coordinates from nine to six, we can prevent the masses from
entering the equations except through the potential function or its deriva-
tives. In this way symmetry, indeed, appears to be lost, but there is so
great a gain in condensation of the formulas, that we can carry out some
of the eliminations which previous writers have been content only to indi-
cate.
Let I, »7, ^ ; ^', J?', ^'; ^", >/", ^" be the rectangular coordinates of the
masses m , m', m", the expression for the living force will be
d^'+dri'+dZ' ,d^"+dri"+di:" „d^"'+d7i"'+di:"'
%df ^ "- Uf "^ "' 2df
and A, A', A" being given by the equations
J" = (r- 1")"+ W- >?"/+ (f'- <^")\
A" = (I"- ^y + (ri"- rif + (C"- C)^
the potential function
+ -77- +
"■ — J ^ A' ^ A"
Without lessening the generality, the origin of coordinates can be put
REDUCTION OF THE PROBLEM OF THREE BODIES.
237
at the center of gravity, when the principle of the conservation of this cen-
ter will furnish the equations
mf + m'f ' + m"e" = ,
mt] + m'fj' + m">?" = ,
m? + m'C' + m"C" = ,
(1)
By means of these relations three of the variables can be eliminated and the
number thus reduced from nine to six. This transformation is most ele-
gantly accomplished by putting
^ — ax + ^x', rj = ay + ^y' , i: — aZ + jSz',
S' = a'x + /3V, rj' = a'y + ^'y', Z' = a'z + ^'z',
^"= a"x + ^"x', 7i"= a"y + [i"y', :"= a"z + /3"z',
where a , a', a", /? , P', /3" are six constants which may be so taken that they
satisfy the five equations
ma + m'a' -^ in"a" = ,
mj5 -I- m'/J' + m"/3" = ,
map + m'a'p' + m"a"p" = ,
m,a'+m'a"+m"a"' =1,
mP'+m'p''+m"p"' =1.
(2)
The first two are necessary in order that equations (l) may be satisfied ; the
third is adopted in order that nothing but squares of diflferential coefiicients
may occur in the transformed T; and, evidently, the last two may be adopted
without thereby diminishing the generality of the transformation.
These equations may be solved elegantly in the following manner: Put
sj m^=Tc sin x cos £ , tj m' ^Tc sin y sin e , iij m" = Jc cos y ;
and adopt the four quantities ^ , ^', u , a', such that
sj ma — sin
' -I- sin ;- sin ^' cos ' = 2^ +
rfw' + dyf" + dw'"
df
■2(wv + wV + w'V),
(5)
an equation which is symmetrical.
REDUCTION OF THE PROBLEM OF THREE BODIES.
241
It is evident now that, since the values of v, v', v" are known from the
first three equations of (3), we shall have, as the equations determining
w, w' and w", (4), (5) and the last of (3), provided we can find a relation
connecting p with w, w', w", v, v', v" and the differentials of the first three.
Such a relation can be found in the following manner : Assume the four
indeterminates X, X', X", X'" so that the equations
dx
IF
xX+ x'X'+ ^X"+ ^ X"'= 0,
d£
dt
§]L Y"^^ Y'"-
yX+y'X'+^X"+--I^X"'=0,
dz
~dt
zX + z'X' + -^ X" + ^ X"'= ,
d£
dt
are satisfied ; and treat the last as if they were equations of condition in the
method of least squares, that is, multiply the first by x, the second by y,
and the third by z, and take the sum for a first equation; and so on. In
this way the normal equations formed from them are
vX+v"X+^^X"+[^% + p)x"'=0,
v"X+v'X+i[^-p)x"+i%X"' = 0,
(i^ + p]x+i% X'+u"X"+ u'X"'=0.
As the number of these equations exceeds that of those from which
they are derived, they are not independent, and the determinant, formed
from the coeflBcients, vanishes ; which is the condition determining p . This
equation is
dvdv'-dv"''-\\ . , „„, , „,.
+ 2
P'+
v'p-' +
idt'
v'dv"— v"dv' v'dv"' — 2v"dv'dv" + vdv'^
dt
■P +
v"dv — vdtf' vdv"^— %v"dvdv"+ v'dv'
V+ — ^, p + —
dt
idf
V"p' +
dt
v'dv — vdv' . (v'dv -f vdv') dv"— v" (dvdv'+ dv'") -} „_r.
T+ 4 (w-^ _ w'» - w"^)(v» - /» - /'»)
-P + 5S?
or, expressed in terms of the new variables,
-4
+ 8
-4
—4
-4
P' +
dvr'- d^ff"- dvf"'T
dt'
~'w'd-w"—yf"dvf' vr"d'w — wd-w" , w%+ f m'a_.-.]g'^
252 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
Let the last term of the second member of this equation be denoted by the
series
since r is a series of cosines, we must have, in consequence of the equations
of condition which the a^ satisfy B^i= Bi, and the equations, which
determine these coeflBcients, can be obtained from the formula.
i^,. a,_ Bj = U(i + 1 + m) a< + f m^'a-,.!,
when we attribute to i, in succession, all integral values from i^ to *= <»,
or which is preferable, from i = to i = — oo. The following are all the
equations and terms which need be retained when it is proposed to neglect
quantities of the same order of smallness as m^" ;
a„i2„ + (ai + a_i) B^ -I- (aj + a_2) R^ = f m'a_i ,
a_ii?o + (a„ + a_2) A + a^Ri = — 4ma_i + f m'^a„ ,
a_272„ + (a_i + a_3) Bi + aoiJj + a^Rs = 8 (1 — m) a_2 + | m^a,,
a_,jBi + a_ii?2 + S'oRs = 1^ (2 — m) a_3 + f 111%,
a_3^i + a_A + a_ii23 + HoRi = 16 (3 — m) a_4 + f m'a3 .
For the purpose of illustrating the present method, we content ourselves
with giving the following approximate formula: —
-^ + m' = l + 2m + fm — f m^aj + 4ma_i (&i + a_ 1)
+ [f m' — 4ma_0 (C' + r') + [8 (1 — m) a_, + f m^ (a, - a_.) + 4maL0 (^* + ^"O ,
where, for convenience in writing, it has been assumed that a^ ^ 1 , and
consequently that a^ denotes here the ratio to ao, which, as has been mentioned
above, is a function of m. The absolute term and the coefficient oi ^* + ^~*
are affected with errors of the eighth order, while the coefficient of ^^+^~'
is affected with one of the sixth order.
We attend now to the remaining terms of 0. If we put
Du ~ S,. (2i + 1) a,^" ~ ' • '^ '
it is plain that we shall have
D's_ . 2-,.(2t + l)'a,C-" _ y ^._„
Ds ~ 2',. (ai + 1) a,e-" ~ *• •^•^ »
and in consequence,
ON THE MOTION OF THE LUNAR PERIGEE.
253
From this it will be seen that the development of -=^ will suffice for obtain-
ing all the remaining terms of©. Let us put
h = (2i + 1) a^ .
The equations which determine the coefficients Ui are given by the formula
but, in order to exhibit some of their properties, I write a few, m extenso, thus :
+ hU_, + h_^U_, + h_,{U„ — l) + h_,U^ + h_,U, + . .
+ hU^, + ho CLi + A_i(C/"o — 1) + h-,U, + h^sU, + . .
+ hU_, + h U_i + ?i, {Uo — V) + h U^ + h Ui + --
(19)
When the subscripts of both the h and U in these equations are nega-
tived, and the signs of the right-hand members reversed, the system of
equations is the same as before. Hence, if we have found the value of Ui,
which is a function of the A, the value of U_i will be got from it by simply
negativing the subscripts of all the h involved in it and reversing the sign of
the whole expression. When this operation is applied to the particular
unknown Uq — 1 , we get the condition
27„-l = -(f7.-l);
whence we have, rigorously,
U,:
This result can also be established by the aid of a definite integral,
absolute term, in
the definite integral
The absolute term, in the development of -^^ — in powers of ^, is given by
Z)"M
'27tJo D'u ^ ^ 2tz»/ — iJo d'u
d'u
dz'
The indefinite integral of the expression under the sign of integration is
, d'u 1 rd'x , d'"ii -. — ^"1
254 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
and if, for the moment, we take p and ^ such that
this integral takes the shape
logp + fV — 1-
The first term of this has the same value for -r = and t = 2n, and
consequently contributes nothing to the value of the definite integral. Thus
we have
1
1 /»2''D-+% , 1 r IT
When T := , let ^ be assumed between and 27t : it will be found that
^ has the value or - or 7t or |7t according as v is of the form 4^ or 4|tf + 1
or 4(ti+ 2 or 4^ + 3. Moreover, when t augments, <^ also augments, and when
■r has passed over one circumference, 4> has also augmented by a circum-
ference. Hence
'Sir X).'+1m
1 ni^ D-+
^. ... ■^. = 1.
It follows, therefore, that v denoting zero or a positive integer, the
absolute term of the development of —jy^ — in integral powers of ij' is 1.
And, in like manner, the absolute term of _^ - is — 1 .
ix S
Equations (19) are readily solved by successive approximations, and
when terms of the tenth order are neglected, we can write
^ = 1 + 3 [ Ai — A_A + h h A_,]£;'
+ 2 [3^, — h h — 3A_A + 4Ai A_A —2Jh h h A_i] C*
— 2[2A_j— A_iA_i — 2^1 A_, + 4A_,Ai h^., — 'ih_-Ji_.Ji_.,\ ]£;—
+ 2 [3A, — 3Ai h + h K h ] C'
— 2 [3^._, — 3A_iA_, + h_Ji_Ji_i] Z-'
+ 2 [47*4 —iJ^k, + 4Ai Ai As — 2^, Aj — Ai Ai Aj A, ] C
— 2 [4A_4 — 4:h_ih_3 + 4Ji_Ji_Ji_i — 2h_Ji_, — h_-Ji_.Ji_^h_^'\ ?-«,
where we have supposed again that ^q = a^ = 1 .
ON THE MOTION OF THE LUNAR PERIGEE. 255
With the same degree of approximation we have used for -y + m", ©
can be written
e = l + 2m — |m» + fmX + 54a? + (12 — 4m)aia_i + (6 — 4m)ai.,
+ [(6 + 12m) ai + (6 + 8m)a_i — f mT(:^ + C"')
+ [20maj + (16 + 20m) a_, — (9 + 40m) a? + 6aia_. + (7 + 4m) aL^
— fmVax — a_0](:' + r*).
In the determination of the terms of the lunar coordinates which
depend only on the parameter m, it has been found that, with errors of the
sixth order,
_ 3 6 + lam +^9m^ „
^'-T^ 6-4m + m^ "" '
_ 3 38 + 2 8m + 9m'' ,
*-'- ^ 6-4m + m' °''
and, with errors of the eighth order,
07 2 + 4m + 3m' Fooo , An , n 2 00 !<59 — 35m -\^,
Sj = -T^ ra A , 2iro» a i n ^38 + 40m + Qra' — 32 ^ j-— 5 m',
'ISO [6 — 4m + m''][30 — 4m + m'] L 6 — 4m + m'J
a_, =
87 2 + 4m + 3m' F „„ ^^ _,_ 9^1 7 - m ~| ^. ,
"[6-4m + m'][30-4m + m'] [" ^^ " ^"^ + ^% - 4m + m'J "^ '
No use will be made of these formulas in the sequel of this memoir :
they are given only that we may at need easily deduce an approximate
literal expansion for the important function 0.
III.
In the preceding discussion it has been established that the determina-
tion of the lunar inequalities, which have the simple power of the eccentricity
as factor, depends on the integration of the linear differential equation
to the treatment of which we accordingly proceed. We assume that the
development of 0, in a series of the form
has been obtained. Here we have the condition ©_< = ©£. If ©i, ©a, &c.,
are, to a considerable degree, smaller than ©o, an approximate statement of
the equation is
* These expressions will be established in another memoir.
256
COLLECTED MATHEMATICAL WORKS OF G. W. HILL
the complete integral of which is
K and K' being the arbitrary constants and c being written for \/©o- When
the additional terms of are considered, the effect is to modify this value
of c, and also to add to w new terms of the general form J.^±°+^'. It is
plain, therefore, that we may suppose
w = ^f(C,c) + ^'f(£:,-c),
and may take, as a particular integral,
w = 2'..b,:<'+»',
hi being a constant coefl&cient. If this equivalent of w is substituted in the
differential equation, we get the equation
[c + a;Tb,-i'..^,-.b<=0, (20)
which holds for all integral values for j, positive and negative. These con-
ditions determine the ratios of all the coefficients b^ to one of them, as bo,
which may then be regarded as the arbitrary constant. They also determine
c, which is the ratio of the synodic to the anomalistic month. For the
purpose of exhibiting more clearly the properties of the equations repre-
sented generally by (20), I write a few of them in extenso: for convenience let
[i} = ic + 2iy-e,;
then
(21)
If, from this group of equations, infinite in number, and the number of
terms in each equation also infinite, we eliminate all the b except one, we
get a symmetrical determinant involving c, which, equated to zero, deter-
mines this quantity. This equation we will denote thus: —
S (c) = . (22)
If, in (20), we put — c for c, — j for j, and suppose that b,- is now
denoted by b_j, the equation is the same as at first; hence the determinant
... + [-2]b_
8 - e.b_, - OA - ffA — ^M — . .
= 0,
. . . - e.b_,
+ [-l]b-,-«A -0A -SA -..
= 0,
. . . - e,h_.
- , - (F + 1) ^or.i-f'' Jo\
V »o (»o - l)(»o - ^)[^. - (* + !/][»„ - (A - 1)^]
By attributing, in these equations, special integral values to Jc, will be
obtained the values of all the single summations appearing in the preceding
expression for n (0). With regard to the double summations, we may pro-
ceed as follows : Substitute i + h for *', then resolve the expression under
consideration into partial fractions with respect to i as variable, and sum
between the limits — oo and -f <» ; the fractions occurring in the result thus
obtained are next resolved into partial fractions with reference to k, and
the summations, with reference to this integer, are taken between the limits
2 and + «> ; or, which is the same thing, between the limits and -\- oo •
ON THE MOTION OF THE LUNAR PERIGEE.
267
and the terms corresponding to ^ = and k=l subtracted from the result,
The single triple summation may be treated in an analogous manner. Thus
we get
■^t, i'
mi +
1 _ "^°K'i"^^°)pcoK^Vg.) 1,^1 9 1
TIFIF+T} 33V».(i-«./L V»o ^0 1-^0 a (4- 0o) J'
^Ui' 77
{i}{i+l}{i'}{i' + 2}
_ -'^"Ha ^'°) pcot(W^o) __ 1,3,2,51
^i.i'
■' {i}{i + l}{i'}{i' + l}{i' + 3}
3:rCOt(|-V0„)
■^i, i', i"
64V0„(l-6'„y(4-0„)
1_^
' 7rCOt(7rVg„) _ J^ , 2 2 20 ~|
. V«o 00 1-^0 4-^0 3(9- »„)J
_ 2 ' V / f— J- + ^ , 9 -1 7rCOt(7rVg,) _ _25_
138V«„(l-y„)' iL 00 1-^0 2C4-0„)J V^o 80,
4 9 9 4 _ tt" -I
l-0„ ' (l-0„)^ 8(4-0„) + (4^::0J^ 9-«„ 30„r
From which it follows that
+
+
4_« ' 9_«
d
+
i^L V^o 00 1 — 00 2(4
32V0o(l-0o>
37rCOtf-|- ^00)
^]''
+ ;:r-T7r
8V0.(l-0.)(4-0.)
0;0.
^cot(^V0.) ^r_ 1 2 9 1.
+ 128V"0;(l-0o)'tL 00 ^l-0o^2(4-0„)J '
+.^+ ^
-00 2(4-
cot (tt V g„) _ 25 _ 1
+
+
+
V0O
80,
TTCOt^l-Vgoj p „cot(;r^g„) 1 2 2 20 -\^g
9;O--0,y(^-0,-)l V0. 0,^l-e„ + 4-0„^3(9-0„)J ^'
cot^^yg.j p ^cot(7rVg,) 1 2 2 10 1^,^,
9-„(i_«„)(4-0„)L V0O «o l-0o'^4-e,^9-»„J ^ ^
.(24)
(7 - 3e„) TT cot/-^ V 0.) 5'^ cot (-|- V 0o)
4V0.(l-0.X4-0oX9-0o)
010A +
16V0.(l-0o)(4-0o)(9-0o)
0^0.
268 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
This is the same result as would be obtained if, setting out with the
equation 2) (c) = 0, and assuming that c = \/0o is an approximate value,
we should expand the function sin^T— c) in ascending powers and pro-
ducts of the coefficients ©j, Q^, &c.
IV.
In order to obtain a numerical result from the preceding investigation,
we assume
n = 17325594:".06085, n' = 1295977".41516,
whence
m = 0.08084 89338 08311.6.
From an investigation (to be published hereafter) of the corresponding
values of the a^, we have
2h, = + 0.00909 42448 77375.5 - 2h_,=- 0.01739 14939 23079.4
4A, = + 0.00011 75731 31569.1 - 4.h_, = + 0.00000 19654 85829.2
6A3 = + 0.00000 13613 28523.8 — 6A_a = + 0.00000 00738 11780.8
8h, = + 0.00000 00126 19314.9 - 8h_, = + 0.00000 00006 87885.7
lOAe = + 0.00000 00001 21722.9 -lQh_, = + 0.00000 00000 05777.1
12^6 = + 0.00000 00000 01147.9 -12A_e = + 0.00000 00000 00047.5
14A, = + 0.00000 00000 00010.6 - 14A_ , = + 0.00000 00000 00000.4
The values of the Ui derived from these are
U^=z+ 0.00909 40932 76038.2 U_-,=- 01739 21860 78260.6
Z7, = + 0.00007 62192 02104.5 CL^ = + 0.00015 32094 08075.6
^73 = + 0.00000 06474 24638.8 [!_,=- 0.00000 12670 56302.6
V^ = + 0.00000 00055 23086.8 U_, = + 0.00000 00115 67648.9
U, = + 0.00000 00000 47309.0 U_,=- 0.00000 00000 95049.5
Ue = + 0.00000 00000 00403.9 n_, = + 0.00000 00000 00867.3
Z7, = + 0.00000 00000 00003.4 CL, = - 0.00000 00000 00007.2
In combination with the values of Ei, which will be given elsewhere,
these afford the following periodic series for :
e = 1.15884 39395 96583
- 0.11408 80374 93807 cos 3t
+ 0.00076 64759 95109 cos 4t
- 0.00001 83465 77790 cos 6r
+ 0.00000 01088 95009 cos 8r
- 0.00000 00020 98671 cos IOt
+ 0.00000 00000 12103 cos 13t
- 0.00000 00000 00211 cos 14r
ON THE MOTION OF THE LUNAR PERIGEE.
269
The values of the coefficients ©p. ©i. ©a. &c., are the halves of these
coefficients, except O^, which is equal to the first coefficient.
On substituting the numerical values of these quantities in (24), and
separating the sum of the terms into groups according to their order, for the
sake of exhibiting the degree of convergence, we get
Term of the zero order, 1.00000 00000 00000
Term of the 4*1 order, + 0.00180 46110 93432 7
Sum of the terms of the 8*'' order, + 0.00000 01808 63109 9
Sum of the terms of the 12tii order, + 0.00000 00000 64478 6
D (0) = 1,00180 47930 21011 2
As far as we can judge from induction, the value of n (O) would be
affected, only in the 14*'' decimal, by the neglected remainder of the series,
which is of the 16*'' order An error in n (0) is multiplied by 2.8
nearly in c.
The value, which is derived thence for c , is
= 1.07158 3277416016.
In order that nothing may be wanting in the exact determination of
this quantity, we will employ the value just obtained as an approximate
value in the elimination between equations (21). The coefficients [i], as
many of them as we have need for, have the following values :
[- 4] = 46.8 , [1] = 8.27577 98905 1 ,
[- 3] = 23.13045 , [2] = 24.56211 3 ,
[-2]= 7.41678 05615 1, [3] = 48.85.
[- 1] =-0.29688 63288 2300,
If the quantities b^ are eliminated from equations (21) in the order b_i, b^,
b_2, bj, b_3, bs, and b_4, it will be found that the coefficient of bp, in the
principal equation, undergoes the following successive depressions :
[0] = — 0.01055 32191 58933,
[0](-i) = + 0.00040 72723 11650,
[0](-i.i) = + 0.00001 50888 08423,
[0](-2.-i.i) = + 0.00000 00253 21700,
[0](-2. -1. 1. 8) = + 0.00000 00009 20430,
[0](-3' -2' -1. !• 2) = + 0.00000 00000 03941 ,
[0](-3. -3, -1, 1, 3, 3) = + 0.00000 00000 00155 ,
[0](-i, -3, -2. -1, 1, 2, 3) = + 0.00000 00000 00008 .
270 COLLECTED MATHEMATICAL WORKS OF G. "W. HILL
The last number is not sensibly changed by the elimination of any of
the bj beyond h_^ on the one side, or bs on the other. This residual is so
small that it will not be necessary to repeat the computation with another
value of c : it will suflBce to subtract half of it from the assumed value of c.
Thus we have as the final result :
and, consequently,
c =: 1.07158 33774 16012;
1 da
dt
= 0.00857 35730 04864.
Let us compare this value with that obtained from Delaunay's literal
expression,*
1 dw 3 , , 335 8 , 4071 « , 365493 ^ , 12833631 s
^-^ = T'"+ 32- *"+ W'"+ -204r"^+ -34576-"^
, 1373935965 , , 71038685589 , , 33145883707741 ,
^ 589834 ^ 7077888 ^ 679477348 '
where m denotes the ratio of the mean motions of the sun and moon. On
the substitution of the numerical values we have employed for these quan-
tities, this series gives, term by term,
-1-^ = 0.00419 6439 + 0.00394 3798 + 0.00099 5700 + 0.00030 3577
n dt
+ 0.00009 1395 + 0.00003 8300 + 0.00000 9836 + 0.00000 3468 = 0.00857 1503.
From the comparison, it appears that the sum of the remainder of
Delaunay's series is 0.00000 1070, somewhat less than would be inferred by
induction from the terms of the series itself. And, although Delaunay has
been at the gi'eat pains of computing 8 terms of this series, they do not
suffice to give correctly the first 4 significant figures of the quantity sought.
On the other hand, the terms of the highest order, computed in the expres-
sion for n (0), were of the 12**^ order only ; and yet, as we have seen, they
have sufficed for giving c exact nearly to the IS*'' decimal. As well as can
be judged frotn induction, it would be necessary to prolong the series, in
powers of »i, as far as m^', in order to obtain an equally precise result.
Allowing that the two last figures of the foregoing value of - t- may be
Th at
vitiated by the accumulation of error arising from the very numerous opera-
tions, we may, I think, assert that 13 decimals correctly correspond to the
assumed value of m. It may be stated that all the computations have been
made twice, and no inconsiderable portion of them three times.
* Compte) Eendus de VAcademie des Sciences de Paris, Tom. LXXIV, p. 19.
EMPIRICAL FORMULA FOR THE VOLUME OF ATMOSPHERIC AIR. 271
MEMOIK No. 30.
Empirical Formula for the Volume of Atmospheric Air.
(Analyst, Vol. IV, pp. 97-107, 1877.)
The formula of Mariotte and Gay-Lussac is generally employed, in the
laboratory, to reduce volumes, observed under one tension and temperature,
to those which would have place under other tensions and temperatures.
But Regnault, about 1845, made several series of experiments, which, if
they may be relied upon, establish marked deviations from this formula.
These experiments are detailed in the Memoires de I'Aeademie des Sciences de
Paris, Tom. XXI. I propose to investigate a modification of the formula,
the introduction of which makes it possible to satisfy nearly these experi-
ments
Let y denote the temperature, here always expressed in degrees of the
centigrade scale ; P the tension or pressure, measured by the altitude, in
meters, of a column of mercury, it is capable of supporting, the mercury
being at the temperature O" and under the action of gravity which obtains
at Regnault's laboratory ; and let V denote the volume. Then, for any
given mass of air, these three quantities are so connected that, if any two
of them are assigned, the remaining third is immediately determined. That
is, we must have
function (V, P, T) = 0,
or, solved with respect to V,
F= function (P, T).
But the mode, in which T is to be measured, is arbitrary, and we may take
atmospheric air as the thermometric substance, and assume that T increases,
in direct proportion, as the volume, under constant pressure, increases.
This gives
V=F(F) +/(P). T.
It is here taken for granted that, whatever may be the density of the
air inclosed in the thermometer, its indications will be the same. It is true
that the usual custom of experimenters has been to measure temperatures
272 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
by the augmentation of tensions under constant volume; but, when Ma-
riotte's law holds, this gives results identical with those obtained by the
former method. In this case we should have to write the equation
F = Fcr)+f(r).T,
but the first equation seems preferable.
Now since, for any given constant temperature, the volume ought to be
a function of the tension similar to what it is at any other temperature, it
follows that, if ^^(P) is supposed to consist of a series of terms, each of the
form KP'', where iT and k are constants, so that we may write
then we ought to have
f(P) = S.K,P'',
where K^ denotes a constant, in general, different from K. Thus we should
have
The formula of Mariotte and Gay-Lussac assumes that F(P) and f{P)
contain each only one term, in which ^^ — 1. But Regnault's experi-
ments having shown the insufficiency of this, it is in order to see whether
agreement between theory and observation cannot be brought about by
annexing to V an additional term, for which k has a value different from
— 1. Thus let us suppose that
F= [K + E^T\ P-^ + {E' + K[T^ P-'-i
_ E+K'P^ , K, + EiP^ rp
- p i- p
As F contains a factor, which is directly proportional to the mass of air
considered, and inversely as the unit assumed for the measurement of
volumes, we prefer to write the preceding equations thus :
When the temperature is constant, the volumes are represented by the
formula
EMPIRICAL FORMULA FOR THE VOLUME OF ATMOSPHERIC AIR 273
that is, the result from Mariotte's law must be multiplied by the factor
1 + aP^, which differs but little from unity ; a is a small constant which
measures the amplitude of the deviations from this law ; while |5 is a con-
stant exponent so chosen that the more or less rapid variation of the devia-
tions, in passing from one tension to another, may be represented as well as
possible. It is evident that, in this manner, we get the utmost advantage
that can be derived from the addition of a single term to V.
The experiments of Regnault may be divided into two classes ; first,
those where, the temperature remaining nearly constant, the volumes of the
same mass of air, under different pressures, were observed ; second, those
where, the volumes remaining nearly the same, the tensions were observed
at the temperature of freezing and boiling water. It is obvious that experi-
ments of these two kinds, extended over a suflEicient range of tension, would
afford the data requisite for obtaining the values of the four constants
a, a', a" and (i which enter into our adopted formula.
The experiments of the first class are enumerated at pp. 374-379 of the
volume quoted above. As the temperature is nearly the same for all, we
assume that they have been made at the average of all the noted tempera-
tures which is 4°. 747.
To save labor, we may take the average of the observed volumes and
tensions when they are nearly alike. In this way Regnault's 66 experi-
ments are reduced to the 23 given in the following table. It may be noted that
Fis here expressed by the number of grammes of mercury required to fill
the volume. The column containing log (PF) exhibits the deviation from
Mariotte's law ; did this law exactly hold, the numbers in this column would
be identical for each series. It will be noted that, in general, they dimin-
ish with increasing pressures. The volumes being supposed to be repre-
sented by the equation
a preliminary investigation has given the approximate values
a=- 0.0024337 , /3 = 0.645 .
With these have been computed the values of the expressions which
stand at the head of the two last columns of the table, and which serve to
obtain the coefficients of the equations of condition to be given presently.
As the mass of air operated on was different in each series of experi-
ments, K will have 9 different values ; it can, however, be eliminated.
28
274
Taking the
COLLECTED MATHEMATICAL WORKS OF G. W. HILL
common logarithms of each member of the equation last given,
log E + log (1 + aPP) = log (P F) .
Series.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
{
Y .
jr*
i.^U, \JUQ,
lug v-r r ).
\ + aP?
1939.76
0.73899
4
3.156387
0.8244
969.65
1.47630
4
3.155790
1.2897
1939.37
2.11228
3
3.612412
1.6262
970.40
4.21020
3
3.611254
2.5430
642.82
6.35034
2
3.610886
3.3213
1939.72
2.06887
3
3.603472
1.6045
969.78
4.12663
6
3.602268
2.5102
1940.65
4.14235
2
3.905194
2.5164
979.78
8.17850
3
3.903803
3.9155
1939.85
4.21910
4
3.912988
2.5465
970.29
8.40648
4
3.911516
3.9863
626.91
12.98195
1
3.910545
5.2926
1940.23
6.77001
3
4.118444
3.4623
970.32
13.47353
4
4.116396
5.4226
685.11
19.00213
1
4.114562
6.7913
675.15
19.30191
2
4.115000
6.8612
1941.23
6.39003
2
4.093580
3.3347
969.98
12.72859
2
4.091543
5.2248
633.82
19.39954
1
4.089757
6.8842
1940.44
9.33401
3
4.257968
4.2676
970.53
18.54702
5
4.255283
6.6842
1945.06
11.47357
2
4.348632
4.8824
1053.78
21.05700
2
4.346146
7.2643
—0.1083
+0.2182
0.5281
1.5876
2.6664
0.5066
1.5452
1.5532
3.5737
1.5921
3.6857
5.8925
2.8758
6.1247
8.6846
8.8206
2.6861
5.7721
8.8654
4.1398
8.4774
5.1740
9.6137
To reduce the matter within the treatment of the method of least
squares, it will be necessary to make some assumption regarding the prob-
able errors of the observed P and F. We will, for convenience, suppose
that they are such that the function log (P Y) has a probable error equal for
all the observations ; an assumption somewhat precarious, it is true, but it
seems that we cannot easily do better.
Let the small corrections, which it is necessary to apply to the approx-
imate values of log K, a and /3, be denoted by h log K, ha and ^/3, and let
us put
8 log K= X , MSa = y, ad^ = z,
where M denotes the modulus of common logarithms. 5 log (PF) being
the excess of observed over calculated log {PV), we shall have the equation'
of condition :
P^ .. . P»
x +
1 + aP
y +
l + aPf
logP.z = a(PF).
EMPIRICAL FORMULA FOR THE VOLUME OF ATMOSPHERIC AIR
275
A little consideration will show that x will be eliminated by taking the
diflference of every two equations of condition arising from the same series,
WW
and attributing the weight - — to the resulting equation, w and w' denoting
the weights of the equations whose difference is taken, and 2 . w the sura of
the weights of all the equations in the series. Since the coeflBcients ofy
and z, in the equations, are all positive and nearly proportional, it will be
advantageous to adopt a new unknown u, such that
Then the equations, with the weights that ought to be attributed to
them, are
Series.
Weight.
I.
0.4653i<
, — 0.4490«
= —0.000106
2
- 0.9168
—0.4685
= —0.000193
f
II. i
1.6951
—0.6869
= +0.000255
%
1
0.7783
—0.2184
= +0.000448
%
III.
0.9057
—0.4709
= —0.000252
2
IV.
1.3991
—0.3113
= +0.000076
1.2
r 1.4398
—0.3061
= +0.000038
Y
V.
2.7461
—0.2764
= +0.000432
i
1.3063
+0.0296
= +0.000394
i
1.9603
—0.0183
= +0.000002
1.2
3.3290
+0.2605
= —0.000407
0.3
VI.
3.3989
+0.2800
= +0.000104
0.6
1.3687
+0.2787
= —0.000409
0.4
1.4386
+0.2982
= +0.000102
0.8
0.0699
+0.0195
= +0.000511
0.2
1.8901
-0.0642
= —0.000059
0.8
VII. ■
3.5495
+0.2635
= —0.000117
0.4
1.6594
+0.3276
= —0.000058
0.4
VIII.
2.4166
+0.3099
= —0.000164
V
IX.
2.3819
+0.4699
= —0.000005
1
The derived normal equations are
58.672 M — 0.0790 z = — 0.0005352,
- 0.079 u + 2.4453 z = — 0.0000157.
Whence
M = - 0.000008962 , z = — 0.000006707 , y= + 0.000002216 ,
da= + 0.0000051, Sfi= + 0.00276.
get
Applying these corrections to the approximate values of a and (3 , we
a = — 0.0034286, /3 = + 0.64776.
276
COLLECTED MATHEMATICAL WORKS OF G. W. HILL
How well the experiments are represented by the formula, with these
values of the constants, will best be seen from the following comparison of
V P
the values of ° ° given by Regnault and those computed from the for-
ala:
'' p;p^6—
vjy o-uc
Obs.
Gal.
Diff.
1.001414
1.001133
+281
1.001448
1.001132
+316
1.001224
1.001133
+ 91
1.001421
1.001133
+288
1.002765
1.002233
+532
1.002759
1.002234
+525
1.002503
1.002236
+267
1.003539
1.004134
—595
1.003452
1.004133
—681
1.003309
1.004133
—824
1.002709
1.002209
+500
1.002724
1.002207
+517
1.002713
1.002206
+507
1.002528
1.002211
+317
1.002898
1.002203
+695
1.002762
1.002203
+559
1.003253
1.003417
—164
1.003090
1.003411
—321
1.003302
1.003407
—105
1.003336
1.003506
—170
1.003495
1.003508
— 13
1.003335
1.003508
—173
1.003448
1.003509
— 61
Obs.
Cal.
Difif.
1.005437
1.006694
—1257
1.005703
1.006694
— 991
1.004286
1.004777
— 491
1.004512
1.004770
— 258
1.004599
1.004779
— 180
1.004580
1.004771
— 191
1.008536
1.008106
+ 430
1.008813
1.008108
+ 705
1.008016
1.008286
— 270
1.008064
1.008269
— 205
1.007980
1.008288
— 308
1.004611
1.004601
+ 10
1.004752
1.004601
+ 151
1.008930
1.008648
+ 282
1.008755
1.008642
+ 113
1.006366
1.005876
+ 490
1.006132
1.005880
+ 252
1.006010
1.005869
+ 141
1.006346
1.005878
+ 468
1.005619
1.005738
— 121
1.005622
1.005736
— 114
1.005902
1.005832
+ 70
It will be seen that the diflFerences, in the extreme cases, amount to a
fourth part of the observed deviation from the law of Mariotte. Moreover,
it is plain that some cause, which, varied from series to series, has operated
to vitiate these experiments, since it is possible to determine a and /? so that
any two series are well represented, but not possible when all the series are
included in the investigation. It may be noted also that the experiments,
in which the original volume was reduced to one-third, are not, in general,
concordant with those where the reduction was to one-half.
That these discrepancies are unavoidable will be evident from the fol-
lowing exposition : Let us put
com. log (PFJ = 2? (f).
EMPIRICAL FORMULA FOR THE VOLUME OF ATMOSPHERIC AIR 277
The observations of Regnault may be condensed into the following nine
results, all formed by combining tolerably concordant data :
1. F( L476)— F( 0.739)= 0.000598
2. F( 4.168)— F( 2.091) = 0.001181
3. F( 6.350)— F( 2.112)= 0.001526
4. F( 8.292)— F( 4.182) = 0.001437
5. F(12.982)— F( 4.219)= 0.002443
6. F(13.101)— F( 6.580) = 0.002042
7. F(19.276) — F( 6.580)= 0.003743
8. F(18.547)— F( 9.334) = 0.002685
9. P(21.057)—F(11.474)= 0.002486
These are the data actually furnished by Regnault for the determina-
tion of the function F{P). Employing the graphical method, we endeavor
to construct the curve whose equation is y=:F{x). One of the special
values of F{x), as i^(0.739), may be taken arbitrarily, and then the value
of -^'(1.476) becomes known. This premised, we see that each of the nine
equations furnishes the length, direction and abscissae of the extremities of a
chord, of the sought curve. Placing the chord, corresponding to the first
equation, arbitrarily, and drawing the others on any part of the paper, but
with the proper direction and abscissae of their extremities, we endeavor, by
imparting a motion to all their points parallel to the axis of ?/, to make
them fall into line as chords of the same continuous curve. We find that if
we take 1, 2, 4, 6 and 7, they can be made to indicate a tolerably continuous
curve ; but then 3, 5, 8 and 9 are not satisfied.
Again, from this graphical process, we see that there cannot be much
variation of curvature between the extremities of each chord, and hence the
tangent to the curve, corresponding to the abscissa, which is the mean of
the abscissae of the extremities, ought to be, very approximately, parallel to
the chord ; or, in other terms,
dx \ 2 j x^ — Xg
This gives the following values of -^ :
■£ .
dy
dx ■
1.
1.108
+0.0008113
2.
3.130
0.0005686
3.
4.231
0.0003770
4.
6.237
0.0003497
5.
8.600
0.0002788
6.
9.840
0.0003131
7.
12.428
0.0002948
8.
13.940
0.0002914
9.
16.265
+0.0002594
278 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
From the general course of these values of -^ , it may be gathered
that this function, at first, diminishes rapidly, afterwards more slowly, and
then tends, with higher values of cc, to become nearly constant. But while
this is the conclusion from the tout ensemble, a comparison of some of the
values contradicts it. Thus, from 1, 2 and 3, while -^ diminishes 0.0002427
ax
in an interval 2.0 in x, it afterwards diminishes 0.0001916 in an interval
1.1 o{ X. All attempts then to represent these data by a curve, without
singular points, must, evidently, show large errors.
For the discussion of the second class of experiments, let us assume that
a has the signification we have given it in the general formula for V. Then
the volume remaining the same, if Pq ^^d P^ denote the tensions observed
respectively at 0° and 100°, we have
P, _ 1 + 100 «' + (« + 100 g") P?
Po 1 + «PP '
p
-^ is the quantity Regnault has designated by 1-f 100a, let us denote it
by A ; then if, for convenience, we put
r = 1 + 100a', / = a + 100a",
each determined value of A will give the equation of condition
r + Pl.r' = A + API.a.
The following are Regnault's determinations of A augmented, in gene-
ral, by 0.00018, for the reason we adopt the mean coeflScient 0.00018153
for the expansion of mercury between 0° and 100°, found by this experi-
menter, instead of the value ^^Vir used by him (see Note, p. 31 of the
volume) ; the last column contains the page of the volume, where the experi-
ments may be found.
f""
/.■
A.
No. Obs.
Ol)B.-Cal.
Page.
o!iio
0?149
1.36500
10
—0.00012
99
0.174
0.237
1.36531
3
—0.00004
99
0.266
0.362
1.36560
2
—0.00003
99
0.375
0.510
1.36598
4
+0.00005
99
0.548
0.746
1.36673
3
+0.00038
57
0.756
0.7535
1.36724
4
+0.00035
66
0.557
0.754
1.36651
18
+0.00014
43
0.656
0.757
1.36641
14
—0.00022
33
0.747
1.016
1.36663
3
—0.00014
68
0.771
1.049
1.36696
11
+0.00014
51
1.678
2.286
1.36778
2
—0.00059
109
1.693
2.306
1.36818
4
—0.00021
109
2.526
2.517
1.36962
2
—0.00018
114
2.622
2.614
1.36982
2
—0.00011
114
2.144
2.924
1.36912
2
+0.00007
109
3.656
4.992
1.37109
4
+0.00031
109
EMPIRICAL FORMULA FOR THE VOLUME OF ATMOSPHERIC AIR 279
Adopting, for convenience, as an unknown in the place of y ,
a; = ^ + / — 1.367,
we have the following equations, to each of which we attribute a weight
equal to a tenth of the number of experiments it is founded upon :
z - 0.7086/ - 0.3268a = - 0.00300
X — 0.6064/ — 0.4398a = - 0.00169
X — O.482I7/ - 0.5790a = - 0.00140
X — 0.3534y' - 0.7a37a = - 0.00103
X - 0.1728/ - 0.93580 = — 0.00027
X — O.I6747' - 1.1410a = + 0.00034
X — 0.1670/ - 0.9354O = - 0.00049
X — 0.1650/ — 1.040 a = - 0.00059
X + 0.0105/ - 1.131 a = - 0.00037
X + 0.0317y' — 1.155 a = - 0.00004
X + 0.708 / — 1.913 a = + 0.00078
X + 0.718 7' - 1.934 a = + 0.00118
X + 0.818 7/ - 3.496 a = + 0.00263
X + 0.863 y/— 2.558 a = + 0.00383
X + 1.004 y' — 2.343 a = + 0.00212
X + 1.834 /- 3.175 a = +.0.00409
The derived normal equations, for determining x and y', are
X - 0.0047/ - 1.162a = - 0.000144,
- 0.0415* + 2.9547/ — 3.373a = + 0.007074,
whence
and
x=— 0.000133 + 1.168a, / = + 0.002392 + 1.158a,
a' = + 0.00364475 + 0.00010a, a" = + 0.00002392 + 0.00158a.
The equation which determines a has already been obtained from the
discussion of the first class of experiments ; it is
°. "*" t'ly > = — 0.0024286 .
1 + 4.747a'
The last three equations being solved, we gather that the volume of
any mass of air is represented by the formula
F= ^ [1 + aPP + (a' + a"P^) T] ,
in which
a = - 0.002565, a'= + 0.0036445,
a" =+ 0.00001987, /J= 0.64776.
280 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
How well the second class of experiments is satisfied by this formula
may be seen from the numbers in the column headed Obs.-Cal.
If we have T = ^-^^^^^^ = i29°.l, F takes the form
0.00001987
F
Hence we have the noteworthy result that :
About the temperature 130°, air follows quite exactly the law of Mariotte.
For the following temperatures and pressures the volume vanishes :
T.
0°
P.
9995"49
— 50
4420.13
—100
2048.00
—150
896.26
—200
314.23
These numbers may be regarded as indications of the magnitude of
pressure necessary for the condensation of air. The table is in accordance
with the well-known fact that reduction of temperature facilitates conden-
sation.
A table is given below which will be found useful in the application of
a! A- a."P^
of the formula. It contains the functions log (1 + aP^) and — — — =^ ,
* ^ ^ 1 + aP^ '
the latter being the coefficient of expansion under a constant pressure.
As an example, let us suppose that the volume of a mass of air has
been observed under the pressure 2". 5 and the temperature 20° ; it is
required to find the factor necessary for reducing it to the pressure O^.TS
and temperature 0°. From the table we get 3.07064. By employing the
ordinary formula with the coefficient 0.003665 of expansion, there is
obtained 3.06482, which differs from the preceding by about a 525**' part.
Rigorously, observations of pressure made in localities having an inten-
sity of gravity different from that which prevails at Regnault's laboratory
ought to be multiplied by the ratio of the former to the latter. The latitude
of Regnault's laboratory is stated at 48° 50' 14", the altitude above sea level
at SO"", and the intensity of gravity at Q^.SOQe.
EMPIRICAL FORMULA FOR THE VOLUME OF ATMOSPHERIC AIR
281
P.
O^.O
.1
.3
,3
A
.5
.6
.7
.8
.9
1 .0
1 .1
1 .2
1 .3
1 .4
1 .5
1 .6
1 .7
1 .8
1 .9
2 .0
3 .5
3 .0
3 .5
4 .0
4 .5
5 .0
5 .5
6 .0
6 .5
log(l + aPO.
0.000000
9.999749
9.999607
9.999489
9.999384
9.999288
9.999199
9.999115
9.999035
9.998958
9.998885
9.998814
9.998745
9.998678
9.998613
9.998549
9.998487
9.998436
9.998367
9.998308
9.998351
9.997979
9.997725
9.997485
9.997357
9.997039
9.996839
9.996636
9.996430
9.996339
351
143
118
105
96
89
84
80
77
73
71
69
67
65
64
63
61
59
59
57
273
354
340
328
318
310
203
196
191
186
CoefE.ofEip.
0.0036445
36511
36548
36579
36607
36683
36656
36678
36698
36719
36738
36756
36775
36793
36809
36836
36843
36858
36874
36889
36904
36977
37044
37106
37166
37334
37280
37333
37385
37436
66
37
31
28
35
34
33
30
21
19
18
19
18
16
17
16
16
16
15
15
73
67
63
60
58
56
53
52
51
49
P.
7"'.0
7 .5
8 .0
8 .5
9 .0
9 .5
10 .0
10 .5
11 .0
11 .5
12 .0
12 .5
13 .0
13 .5
14 .0
14 .5
15 .0
15 .5
16 .0
16 .5
17 .0
17 .5
18 .0
18 .5
19 .0
19 .5
20 .0
20 .5
21 .0
21 .5
log(l + aPP).
9.996053
9.995872
9.995695
9.995521
9.995352
9.995185
9.995021
9.994860
9.994702
9.994546
9.994393
9.994242
9.994093
9.993946
9.993800
9.993657
9.993515
9.993374
9.993336
9.993098
9.992962
9.992828
9.992695
9.993563
9.993433
9.993303
9.992174
9.993047
9.991931
9.991795
181
177
174
169
167
164
161
158
156
153
151
149
147
146
143
143
141
138
138
136
134
133
132
131
129
129
127
136
126
Coeff.ofExp.
0.0037485
37533
37580
37626
37671
37715
37758
37801
37843
37885
37935
37965
38005
38044
38083
38121
38159
38196
38233
38269
38306
38343
38377
38413
38447
38482
38516
38550
38584
38618
48
47
46
45
44
43
48
43
42
40
40
40
39
39
38
38
37
37
36
87
36
35
85
35
35
34
34
84
34
282 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
MEMOIE NO. 31.
On Dr. Weiler's Secular Acceleration of the Moon's mean Motion.
(Aatronomisehe Nachrichten, Vol. 91, pp. 251.254, 1878.)
Dr. Weiler's conclusions are, in general, not admissible because the
expressions he gives for the forces X, Y and Z* are incorrect. It is well
known that the attraction of a body, whatever may be its bounding surface
and law of interior density, always admits a potential function TT, such that
"3F' 'W "35""
But if we form the expression
Xdx + Ydy + Zdz
from Dr. Weiler's values of X, Y and Z, it is found to be not an exact dif-
ferential : hence these values are erroneous.
They appear to have been derived by some illegitimate transformations
from the formulas in the Micaniqiie Celeste, Tom. II, p. 22. After changing
to Dr, Weiler's notation, Laplace's expressions for the attraction of a homo-
geneous ellipsoid of revolution become
i'du
yr_ 3hx P"^ u*du yr_3hy P^^i^du__ ^_ 3Az /*' m'
where k' is given by the equation
2Icf' = r' + X- y (r' - /)' + 4:Xz\
But Dr. Weiler seems to have put y = r. This cannot be done for the Jtf
which is outside of the sign of integration, without losing some part of the
attraction which is of the order of the small quantity A.
Hansen {Fundamenta Nova, pp. 1-16) has elaborated this matter with
great generality and much elegance. From this source we learn that the
proper expression for the potential function of the action between the earth
and moon is
" r L ^ Mr' ^ "Mr' "J '
* Astronomlsche Nachrichten, Vol. 90, pp. 372-373.
ON DR. WEILER'S SECULAR ACCELERATION OF MOON'S MEAN MOTION 283
where A , B and C are the moments about the axes oi x, y and a , supposed
to coincide with the principal axes of rotation. In getting this expression,
no assumption respecting the bounding surface or law of density of the earth
is necessary ; it is only assumed that terms of the third and higher orders
with respect to the ratio of the dimensions of the earth to the radius-vector
of the moon may be neglected.
Very nearly we have B = A, and, if this assumption is adopted, W
takes the simpler form
Q j^
If we put h=: X {M-\- m) , and a = f „ ^ — , a will be a constant inde-
pendent of the linear and time units ; and measurements of arcs of the meri-
dian, of the length of the second's pendulum, and the data afforded by the
phenomena of precession and nutation, show that its value is very approxi-
mately a= 0.0016395.
The expressions of the forces, which ought to be substituted for those
given by Dr. Weiler, are then
X=-
284 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
MEMOIE N"o. 32.
Besearches in the Lunar Theory.*
(American Journal of Mathematics, Vol. I, pp. 5-26, 129-147, 245-260, 1878.)
When we consider how we may best contribute to the advancement of
this much-treated subject, we cannot fail to notice that the great majority
of writers on it have had before them, as their ultimate aim, the construction
of Tables ; that is, they have viewed the problem from the stand-point of
practical astronomy rather than of mathematics. It is on this account that
we find such a restricted choice of variables to express the position of the
moon, and of parameters, in terms of which to express the coeflScients of the
periodic terms. Again, their object compelling them to go over the whole
field, they have neglected to notice many minor points of great interest to
the mathematician, simply because the knowledge of them was unnecessary
for the formation of Tables. But the developments having now been carried
extremely far, without completely satisfying all desires, one is led to ask
whether such modifications cannot be made in the processes of integration,
and such coordinates and parameters adopted, that a much nearer approach
may be had to the law of the series, and, at the same time, their convergence
augmented.
Now, as to choice of coordinates, it is known that, in the elliptic theory,
the rectangular coordinates of a planet, relative to the central body, the axes
being parallel to the axes of the ellipse described, can be developed, in
terms of the time, in the following series :
a; = a ^ — Ji.'"'' coeig,
i ^ — oo
i=+c
^=^ S 4^-^"''^^^^'
t = 03
a and h being the semi-axes of the ellipse, e the eccentricity, g the mean
anomaly and, for positive values of i , the Besselian function (in Hansen's
notation)
'^'^ ~ 1.2...il l.(i + l)^ 1.2.{i + l)ii + 2) J'
» Communicated to the National Academy of Sciences at the session of April, 1877.
RESEARCHES IN THE LUNAR THEORY 28 5
while, for negative values of i, we have
and, for the special case of i = 0, we put the indeterminate
1 r(-i) — 8 /,
Here the law of the series is manifest, and the approximation can easily
be carried as far as we wish. But the longitude and latitude, variables
employed by nearly all the lunar-theorists, are far from having such simple
expressions ; in fact, their coefficients cannot be expressed finitely in terms
of Besselian functions. And if this is true in the elliptic theory, how much
more likely is a similar thing to be true when the complexity of the problem
is increased by the consideration of disturbing forces ? We are then justified
in thinking that the coefficients of the periodic terms in the development of
rectangular or quasi-rectangular coordinates are less complex functions of
their parameters than those of polar coordinates. There is also another
advantage in employing coordinates of the former kind ; the differential
equations are expressed in purely algebraic forms ; while, with the latter,
circular functions immediately present themselves. For these reasons I have
not hesitated to substitute rectangular for polar coordinates.
Again, as to parameters, all those who have given literal developments,
Laplace setting the example, have used the parameter m , the ratio of the
sidereal month to the sidereal year. But a slight examination, even, of the
results obtained, ought to convince any one that this is a most unfortunate
selection in regard to convergence. Yet nothing seems to render this para-
meter at all desirable, indeed, the ratio of the synodic month to the sidereal
year would appear to be more naturally suggested as a parameter. Some
instances of slow convergence with the parameter m may be given from
Delaunay's Lunar Theory ; the development of the principal part of the
15
coefficient of the evection in longitude begins with the term — me, and ends
.^, ^, ^ 413277465931033 , • • +u • • i ^ ^ *v,
with the term — 15288238080 — "* ' ^Sam, in the principal part of the
coefficient of the inequality whose argument is the difference of the mean
21
anomalies of the sun and moon, we find, at the beginning, the term — mee',
n . .X. J iu ^ 1207454026843 , , ,, . u i.i +t, ^. i. .v
and, at the end, the term — rrvee. It is probable that, by the
adoption of some function of m as a parameter in place of this quantity,
whose numerical value, in the case of our moon, should not greatly exceed
286 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
that of m, the foregoing large numerical coeflBcienta might be very much
diminished. And nothing compels us to use the same parameter throughout ;
one may be used in one class of inequalities, another in another, as may
prove most advantageous. It is known what rapid convergence has been
obtained in the series giving the values of logarithms, circular and elliptic
functions, by simply adopting new parameters. Similar transformations,
with like effects, are, perhaps, possible in the coeflBcients of the lunar ine-
qualities. However, as far as my experience goes, no useful results are
obtained by experimenting with the present known developments ; in every
case it seems the proper parameter must be deduced from a priori consider-
ations furnished in the course of the integration.
With regard to the form of the differential equations to be employed,
although Delaunay's method is very elegant, and, perhaps, as short as any,
when one wishes to go over the whole ground of the lunar theory, it is sub-
ject to some disadvantages when the attention is restricted to a certain class
of lunar inequalities. Thus, do we wish to get only the inequalities whose
coeflBcients depend solely on »i , we are yet compelled to develop the disturb-
ing function B to all powers of e. Again, the method of integrating by
undetermined coeflBcients is most likely to give us the nearest approach to
the law of the series ; and, in this method, it is as easy to integrate a diflfer-
ential equation of the second order as one of the first, while the labor is
increased by augmenting the number of variables and equations. But
Delaunay's method doubles the number of variables in order that the differ-
ential equations may be all of the first order. Hence, in this disquisition, I
have preferred to use the equations expressed in terms of the coordinates,
rather than those in terms of the elements ; and, in general, always to dimin-
ish the number of unknown quantities and equations by augmenting the
order of the latter. In this way the labor of making a preliminary develop-
ment of B in terms of the elliptic elements is avoided.
In the present memoir I propose, dividing the periodic developments of
the lunar coordinates into classes of terms, after the manner of Euler in his
last Lunar Theory,* to treat the following five classes of inequalities:
1. Those which depend ouly on the ratio of the mean motions of the
sun and moon.
2. Those which are proportional to the lunar eccentricity.
3. Those which are proportional to the sine of the lunar inclination.
4. Those which are proportional to the solar eccentricity.
5. Those which are proportional to the solar parallax.
* Theoria Motuum Luna, nova methodo pertractaia. PetropoU, 1773.
RESEARCHES IN THE LUNAR THEORY 287
A general method will also be given by which these investigations may
be extended so as to cover the whole ground of the lunar theory. My
methods have the advantage, which is not possessed by Delaunay's that
they adapt themselves equally to a special numerical computation of the
coeflQcients, or to a general literal development. The application of both
procedures will be given in each of the just mentioned five classes of
inequalities, so that it will be possible to compare them.
I regret that, on account of the difficulty of the subject and the length
of the investigation it seems to require, I have been obliged to pass over
the important questions of the limits between which the series are con-
vergent, and of the determination of superior limits to the errors committed
in stopping short at definite points. There cannot be a reasonable doubt
that, in all cases, where we are compelled to employ infinite series in the
solution of a problem, analysis is capable of being prefected to the point of
showing us within what limits our solution is legitimate, and also of giving
us a limit which its error cannot surpass. When the coordinates are devel-
oped in ascending powers of the time, or in ascending powers of a parameter
attached as a multiplier to the disturbing forces, certain investigations of
Cauchy afford us the means of replying to these questions. But when, for
powers of the time, are substituted circular functions of it, and the coefficients
of these are expanded in powers and products of certain parameters produced
from the combination of the masses with certain of the arbitrary constants
introduced by integration, it does not appear that anything in the writings
of Cauchy will help us to the conditions of convergence.
CHAPTER I.
Differential Equations. — Properties of motion derived from Jacohi's integral.
We set aside the action of the planets and the influence of the figures
of the sun, earth and moon, together with the action of the last upon the
sun, as also the product of the solar disturbing force on the moon by the
small fraction obtained from dividing the mass of the earth by the mass of
the sun. These are the same restrictions as those which Delaunay has
imposed on his Lunar Theory contained in Vols, XXVIII and XXIX of the
Memoirs of the Paris Academy of Sciences, The motion of the sun, about
the earth, is then in accordance with the elliptic theory, and the ecliptic is
a fixed plane.
Let us take a system of rectangular axes, having its origin at the centre
of gravity of the earth, the axis of x being constantly directed toward the
centre of the sun, the axis of y toward a point in the ecliptic 90° in advance
288 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
of the sun, and the axis of z perpendicular to the ecliptic. In addition, we
adopt the following notation :
r' = the distance of the sun from the earth ;
A' = the sun's longitude;
fi = the sum of the masses of the earth and moon, measured by the Telocity these masses
produce by their action, in a unit of time, and at the unit of distance;
m' = the mass of the sun, measured in the same way;
n' = the mean angular velocity of the sun about the earth;
a' = the sun's mean distance from the earth .
In accordance with one of the above-mentioned restrictions we have the
equation:
m' — n'^a'^
The axes of x and y having a velocity of rotation in their plane, equal
to— -=— , it is evident that the square of the velocity of the moon, relative to
cit
the earth's centre, has for expression, in terms of the adopted coordinates,
^rp _ r dx _ dX' n" r dy dX' y d^
^^-lir y~dr]^l~df^''~drj^^F
dx^ + dy'^ -\r dz^ n dX' xdy--ydx dX''' ,, .,
w '^^^t di — ^-^^^'^ +2/;-
The potential function, in terms of the same coordinates, is
V (a;' + «/' + 2") V[(»^ -«)' + «/' + 2'] ~7^**
If the second radical in this expression is expanded in a series proceed-
ing according to descending powers of r' and the first term — j- omitted,
since it disappears in all differentiations with respect to the moon's coordi-
nates, the following expression is obtained :
a =
V {x' + y' + z')
+ n"^lx'-^(f + z'-)-]
+ -^ -^ [^* - 3^ cf + «') + ! (f + m
+
RESEARCHES IN THE LUNAR THEORY 289
Since the diflferential equations of motion are of the form
d dT _dT _ da
dt ' -, d
'
There does not seem to be any function of x, y and 2, which, adopted as a
new variable to accompany u and s, would reduce this to a very simple form.
However, when we are engaged in determining the inequalities independent
of the inclination of the lunar orbit, this transformation will be useful to us.
For, in this case, z = 0, and the values of u and s become
u = x + y nj — 1,
s =x-y>j—l,
and T is given by the equation
c.rp_duds dX' uds — sdu dX'^
^^-~dF ~~di dt +~dF"*-
Although il is expressed most simply by the systems of coordinates we
have just employed, the integration of the differential equations will be
easier, if we suppose that the axes of x and y have a constant instead of a
variable velocity of rotation, the axis of x being made to pass through the
RESEARCHES IN THE LUNAR THEORY 291
mean position of the sun instead of the true. To obtain the expression for
T correspondent to this supposition, it is necessary only to write n' for — 5—
in the former values. As for £i , it can be written thus
"— r ^ [r'" — 2r'8 + r'Ji r" '
where
r' = x' + y' + z' = the square of the moon's radius vector ;
S = xcosu + y sino;
o = the solar equation of the centre.
This function being expanded in a series of descending powers of r', as
before, we have
S'
f+i-'
''{7? + f)
+ w"
a"
lis^-
■i*^]
a
■ i^S^-
-fr-'^]
a"
/5
-[^„' dy -
dy \^l^ + 3^-w-p^Y
The problem is then reduced to the integration of two differential equations
of the first order. Were this accomplished, and p eliminated from the two
integral equations, we should have the equation of the orbit. If we put
W=2x+ fA + Sa;^ — 2(7— /)»T,
the differential equations can be written in the canonical form,
dx __dW
dy dp '
dp _ dW
dy dx
294 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
It may be worth while to notice also the single partial differential
equation, to the integration of which our problem can be reduced. Return-
ing to the arbitrary linear and temporal units, and for convenience, reversing
the sign of G, if a function of x and y can be found satisfying the partial
differential equation
and involving a single arbitrary constant h , distinct from that which can be
joined to it by addition, the intermediate integrals of the problem will be
dz dV , , dy dV ,
dt dx " dt dy '
and the final integrals
a and c being two additional arbitrary constants. The truth of this will be
evident if we differentiate the four integral equations with respect to t and
compare severally the results with the partial differential coefiBcients of the
partial differential equation with respect io x, y,h and G.
Although, in this manner, the problem seems reduced to its briefest
terms, yet, when we essay to solve it, setting out with this partial differen-
tial equation, we are led to more complex expressions than would be expected.
It would be advisable, in this method of proceeding, to substitute polar for
rectangular coordinates, or to put
a; = rcosv, y^raml denoting the longitude of the moon, we have
IdX J
dt
dt
^ being a constant. Thus, after the longitude is determined in terms of t,
the radius vector is obtained by a quadrature. But it can also be found,
without the necessity of an integration, by dividing the integral by r^ and
then eliminating the term —^ ^g- by means of its value derived from the
second differential equation ; in this way we get
*i3 M^ a
-^ + |w'»8ia25.
dt
+ iiS-fw''cosV.
df
As we desire to make constant numerical application of the general
theory, established in what follows, to the particular case of the moon, we
delay here, for a moment, to obtain the numerical values of the three con-
stants /[/, n' and G. The value of |U may be derived either from the observed
value of the constant of lunar parallax combined with the mean angular
motion of the moon, or from the intensity of gravity at the earth's surface
and the ratio of the moon's mass to that of the earth. We will adopt the
latter procedure. The value of gravity at the equator, g'= 9.779741 metres,
the unit of time being the mean solar second. We propose, however, to take
the mean solar day as the unit of time, and the equatorial radius of the earth
as the linear unit. This number must then be multiplied by w^ififooYT^ '
(6377397.15 metres is Bessel's value of the equatorial radius.) Moreover,
296 COLLECTED MATHEMATICAL WOEKS OF G. W. HILL
the theory of the earth's figure shows that, in order to obtain the attractive
force of the earth's mass, considered as concentrated at its centre of gravity,
a second multiplication must be made by the factor 1.001818356. "With our
units then this force is represented by the number 11468.338: and the
moon's mass being taken at ^. ^^^,„ of the earth's, her attractive force is
° 81.52277
represented by the number 140.676. Consequently
II = 11609.014.
The sidereal mean motion of the sun in a Julian year is 1295977".41516,
whence
n' = 0.017202124.
The value of G might be obtained from the observed values of the moon's
coordinates and their rates of variation at any time. However, as the eccen-
tricity of the earth's orbit is not zero, C obtained in this manner would be
found to undergo slight variations. The mean of all the values obtained in
a long series of observations might be adopted as the proper value of this
quantity when regarded as constant. But it is much easier to derive it
approximately from the series
which will be established in the following chapter. Here n denotes the
n!
moon's sidereal mean motion, and m is put for , . In this formula the
n — n'
terms which involve the squares of the lunar eccentricity and inclination and
of the solar parallax are neglected; this, however, is not of great moment,
as, being multipled by at least m^ they are of the fourth order with respect
to smallness. The observations give n = 0.22997085, hence
C= 111.18883.
If it is proposed to assume the units of time and length so that ^ and
n' may both be unity, it will be found that the first is equal to 58.13236
mean solar days, and the second to 339.7898 equatorial radii of the earth.
The corresponding value of Cis 3.254440.
Let us now notice some of the properties of motion which can be
derived from Jacobi's integral. This integral gives the square of the velocity
relative to the moving axes of coordinates ; and, as this square is necessarily
positive, the putting it equal to zero gives the equation of the surface which
RESEARCHES IN THE LUNAR THEORY 297
separates those portions of space, in which the velocity is real, from those in
which it is imaginary. This equation is, in its most general form,
+ ,.,„> :^^...^.n = ^ + \^'^'-^> [(«' - ^y + 2/1 ,
V (ar' + 2/' + «') V [(«' - «)' + 2/' + 2'] " ' 2 " "■ 3
which is seen to be of the 16th degree. As y and z enter it only in even
powers, the surface is symmetrically situated with respect to the planes of
xy and xz. The left member is necessarily positive, (the folds of the surface,
for which either or both the radicals receive negative values, are excluded
from consideration), hence the surface is inclosed within the cylinder whose
axis passes through the centre of the sun perpendicularly to the ecliptic, and
whose trace on this plane is a circle of the radius
As, in general, the second term of the quantity, under the radical sign, is
much smaller than the first, this radius is, quite approximately VSa'. Thus,
in the case of our moon, assuming — j =sin8".848, we have this radius =
V 3.00138 3a'. It is evident that, for all points without this cylinder, the
velocity is real ; and as, for large values of 2, whether positive or negative,
the left member of the equation becomes very small, it is plain that the
cylinder is asymptotic to the surface. Every right line, perpendicular to
the ecliptic, intersects the surface not more than twice, at equal distances
from this plane, once above and once below.
Let us now find the trace of the surface on the plane of xy. Putting p
for the distance of a point on this trace from the centre of the sun,
and it is evident that the cubic equation,
will give the limits between which the values of p can oscillate. If C is
negative, this equation has but one real root which is negative; consequently,
in this case, the surface has no intersection with the plane of xy. But, in
all the satellite systems we know, G is positive, and this condition is prob-
ably necessary to insure stability. Hence we shall restrict our attention to
the case where G is positive. Then all the roots of the equation are real,
298 COLLECTED MATHEMATICAL WORKS OP G. W. HILL
and two are positive. It is between the latter roots that p must always be
found. To compute them, we derive the auxiliary angle 6 from the formula
or, since 6 differs but little from 90°, with more readiness from
C08= =
[^+*^?V^]
or, as -72-72 is a small quantity, with suflBcient approximation from
_ V^^^'
COSS =
1 +
The two roots are then
The trace of the surface on the plane of xy is then wholly comprised in the
annular space between the two circles described from the centre of the sun
as centre with the radii pj and p^- Moreover, as in most satellite systems we
^*^® J^' ®*^"*^ *° * ^®^^ ^°^*^^ fraction, (for our moon -^3 = 322930.2 ) '
it is plain that, for points whose distance from the earth is comparable with
their distance from the sun, the trace is approximately coincident with these
circles. For the term -£^, in the equation, may then be neglected in com-
parison with the other terms.
In the case of our moon there is foumd
e = 87° 52' 11".53 ,
and hence
p^ =z 22815.15 , p, = 23816.09 ,
and, if r and p are regarded as the variables defining the position of a point
in the plane xy, the following table gives some corresponding values of these
RESEARCHES IN THE LUNAR THEORY
299
quantities, for each of the two branches of the trace approximating severally
to the two circles.
r.
P-
'/■.
P-
433.3257
22878.69
439.7922
23751.81
450
22876.17
450
23753.37
500
22869.68
500
23760.04
600
22860.13
600
23769.85
1000
22841.59
1000
23788.87
10000
22817.70
10000
23813.43
46127.70
22815.68
47127.55
23815.53
The first and last values correspond to the four points where the curves
intersect the axis of x on the hither and thither side of the sun. It will be
seen that the approximation of the branches to the circles is quite close, except
in the vicinity of the earth, where there is a slight protruding away from
them.
In addition to these two branches of the trace, there is, in the case
where C exceeds a certain limit, a third closed one about the origin much
smaller than the former. As the coordinates of points in this branch are
small fractions of a', its equation may be written, quite approximately,
r ^
It intersects the axis of j/ at a distance from the origin very nearly
« — ^
y"- G '
and the axis of a; at points whose coordinates are the smallest (without
regard to sign) roots of the equations
jx_ _^ _nV^ =G + I w'V - \n" {a' — x)\
-JL+ -^^ = G + in"a" - i n" (a' - x)'.
For the moon these quantities have the values
2/0 = 104.408, a^ = — 109.655, a;, = + 109.694 .
This branch then does not differ much from a circle having its centre at the
origin, more closely it approximates to the ellipse whose major axis = 0:2 — Xi,
and minor axis ■=■ 2^o.
300 COLLECTED MATHEMATICAL WORKS OP G. W. HILL
The value of the coordinate a, for the single intersection of the surface
with the axis of 2 above the plane ofxy, is given by the single positive root
of the equation
z V (a" + z')
For the moon the numerical value of this root is
00 = 102.956 .
The intersection of the surface with the perpendicular to the plane of xy
passing through the centre of the sun is, in like manner, given by the equation
having but a single positive root, which is nearly
2 „i
1 + 1-^'
or, with less exactitude,
*.-l'''
2„ = la-.
From this investigation it is possible to get a tolerably clear idea of the
form of this surface. When G exceeds a certain limit, it consists of three
separate folds. The first being quite small, relatively to the other two, is
close, surrounds the earth and somewhat resembles an ellipsoid whose axes
have been given above. The second is also closed, but surrounds the sun,
and has approximately the form of an ellipsoid of revolution, the semiaxis
in the plane of the ecliptic being somewhat less than a', and the semiaxis of
revolution perpendicular to the ecliptic and passing through the sun being
about two-thirds of this. This fold has a protuberance in the portion neigh-
boring the earth. The third fold is not closed, but is asymptotic to the
cylinder mentioned at the beginning of the investigation of the surface.
Like the second, it also is nearly of revolution about an axis passing through
the centre of the sun and perpendicular to the ecliptic. The radius of its
trace on the ecliptic is about as much greater than a', as the radius of the
trace of the second fold falls short of that quantity. The fold has a protuber-
ance in the portion neighboring the earth, and which projects towards this
RESEARCHES IN THE LUNAR THEORY 391
body. The whole fold resembles a cylinder bent inwards in a zone neigh-
boring the ecliptic.
What modifications take place in these folds when the constants involved
in the equation of the surface are made to vary, will be clearly seen from
the following exposition. Let us, for brevity, put
and, for the moment, adopt a', the distance of the earth from the sun, as the
linear unit, and transfer the origin to the centre of the sun, and moreover put
At
n"a"
Then the intersections of the surface, with the axis of aj, will be given by the
two roots of the equation
x*-x'' — h3? + (A + 2 — 2r)a; — 3 = 0,
which lie between the limits and 1 ; by the two roots of
«* — ai'-Aa^ + (A + 2 + 2;')a; — 2 = 0,
which lie between 1 and V^; and by the two roots of
x^ — a^-'h3? + (h — 2 — 2r)x + 2 = 0,
which lie between and — \/h.
Hence, if G diminishes so much that the first of these three equations
has the two roots, lying between the mentioned limits, equal, the first fold
will have a contact with the second fold ; and if G fall below this limit, the
roots become imaginary, and the two folds become one. Again, if G is
diminished to the limit where the second equation has the mentioned pair of
roots equal, the first fold will have a contact with the third; and when G is
less than this, these two folds form but one. And when G is less than both
these limits, there will be but one fold to the surface.
In the spaces inclosed by the first and second folds the velocity, relative
to the moving axes of coordinates, is real ; but, in the space lying between
these folds and the third fold, it is imaginary; without the third fold it is
again real. Thus, in those cases, where G and y have such values that the
three folds exist, if the body, whose motion is considered, is found at any
time within the first fold, it must forever remain within it, and its radius
302 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
vector will have a superior limit. If it be found within the second fold, the
same thing is true, but the radius vector will have an inferior as well as a
superior limit. And if it be found without the third fold, it must forever
remain without, and its radius vector will have an inferior limit.
Applying this theory to our satellite, we see that it is actually within
the first fold, and consequently must always remain there, and its distance
from the earth can never exceed 109.694 equatorial radii. Thus, the
eccentricity of the earth^s orbit being neglected, we have a rigorous demon-
stration of a superior limit to the radius vector of the moon.
In the cases, where G and y have such values that the surface forms but
one fold, Jacobi's integral does not afford any limits to the radius vector.
When we neglect the solar parallax and the lunar inclination, the pre-
ceding investigation is reduced to much simpler terms. The surface then
degenerates into a plane curve, whose equation, of the sixth degree, is
r ^
It is evidently symmetrical with respect to both axes of coordinates, and is
contained between the two right lines, whose equations are
, rw
and which are asymptotic to it. It intersects the axis of y, at two points,
whose coordinates are
The cubic equation.
gives the values of r, for which the curves intersect the axis of a;. If
(2C)3 > 9fin',
[20
this equation has two real roots between the limits and -|- y -^-fz' If
(2(7)8 = Q^n',
these roots become equal. And if
(2(7)»<9/iw',
RESEARCHES IN THE LUNAR THEORY
303
there are no real roots between these limits, and the curve has no intersec-
tion with the axis of x. The figures below exhibit the three varieties of this
curve.
Fig. 1 represents the form of the curve in the case of our moon. In Fig. 2
we see that the small oval of Fig. 1 has enlarged and elongated itself so as
to touch the two infinite branches; while, in Fig. 3, it has disappeared, the
portions of the curve, lying on either side of the axis of x, having lifted
themselves away from it, and the angles having become rounded off. In
Fig. 1, the velocity is real within the oval, and also without the infinite
branches, but it is imaginary in the portion of the plane lying between the oval
and these branches. Hence, if the body be found, at any time, within the oval,
it cannot escape thence, and its radius vector will have a superior limit; and,
if it be found in one of the spaces on the concave side of the infinite branches,
it cannot remove to the other, and its radius vector will have an inferior limit.
In the case represented in Fig. 2, the same things are true, but it seems
as if the body might escape from the oval to the infinite spaces, or vice versa,
at the points where the curve intersects the axis of x. However, at these
points, the force, no less than the velocity, is reduced to zero. For the
distance of these points from the origin is the positive root of the equation
3^-^ = «'
or
V2g _ V9Atw'
~W~ 3n'
304 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
and this value is the same as that given by the equation
4- — 3w"= 0.
r
In consequence the forces vanish at these two points, and thus we have two
particular solutions of our differential equations.*
In the case represented in Pig. 3, the integral does not afford any
superior or inferior limit to the radius vector.
The surface, or, in the more simple case, the plane curve, we have
discussed, is the locus of zero velocity ; and the surface or plane curve, upon
which the velocity has a definite value, is precisely of the same character
and has a similar equation. It is only necessary to suppose that the G of
the preceding formulae is augmented by half the square of the value attrib-
uted to the velocity. Thus, in the case of our moon, it is plain the curves
of equal velocity will form a series of ovals surrounding the origin, and
approaching it, and becoming more nearly circular as the velocity increases.
Applying the simple formulae, where the solar parallax is neglected, to
the moon, we find that the distance of the asymptotic lines, from the origin, is
V
|^, = 500.4992.
The distance of the points on the axis of a;, at which the moon would remain
stationary with respect to the sun, is
^3^, = 235.5971.
If the auxiliary angle Q is derived from the equation
we get
e = 32° 49' 6".63 ;
and the distances from the origin, at which the curve of zero velocity inter-
sects the axis of as, are given by the two expressions
y fll
3w'
s:n(60°-|-),
and the numbers are 109.6772 and 435.5623. These values differ but little
from the previous more general determinations.
* The corresponding Bolution, in the more general problem of three bodies, may be seen in the
Mecanigue Celeste, Tom. IV, p. 310.
RESEARCHES IN THE LUNAR THEORY 395
CHAPTER II.
Determination of the inequalities which depend only on the ratio of the mean
motions of the sun and moon.
If the path of a body, whose motioa satisfies the difierential equations
^"'^-t-[f-3^']^ = «'
dp
intersect the axis of x at right angles, the circumstances of motion, before
and after the intersection, are identical, but in reverse order with respect to
the time. That is, if t be counted from the epoch when the body is on the
axis of 33 , we shall have
X — function (f ) , y = t. function (t') .
For if, in the differential equations, the signs of y and t are reversed, but
that of X left unchanged, the equations are the same as at first.
A similar thing is true if the path intersect the axis of y at right angles;
for if the signs of x and t are reversed, while that of y is not altered, the
equations undergo no change.
Now it is evident that the body may start from a given point on, and
at right angles to, the axis of x, with different velocities; and that, within
certain limits, it may reach the axis of y, and cross the same at correspond-
ingly different angles. If the right angle lie between some of these, we
judge, from the principle of continuity, that there is some intermediate
velocity with which the body would arrive at and cross the axis of y at
right angles.
The difiiculty of this question does not permit its being treated by a
literal analysis; but the tracing of the path of the body, in numerous special
cases, by the application of mechanical quadratures to the differential equa-
tions, enables us to state the following circumstances:
If the body be projected at right angles to, and from a point on, the
axis of a;, whose distances from the origin is less than 0.33 .... y -^, there
is at least one (near the limit there are two) value of the initial velocity,
with which the body, in arriving at the axis of y, will cross it at right
angles. Beyond this limit it appears no initial velocity will serve to make
the body reach the axis of y under the stated condition.
If the body move from one axis to the other and cross both of them
perpendicularly, it is plain, from the preceding developments, that its orbit
30
306 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
will be a closed curve symmetrical with respect to both axes. Thus is
obtained a particular solution of the differential equations. While the general
integrals involve four arbitrary constants, this solution, it is plain, has but
two, which may be taken to be the distance from the origin at which the
body crosses the the axis of x and the time of crossing.
Certain considerations, connected with the employment of Fourier's
Theorem and the possibility of developing functions in infinite series of
periodic terms, show that, in this solution, the coordinates of the body can
be represented, in a convergent manner, by series of the following form :
x = A^GQi%\y(f, — t^'\ + A^ cos3[i'(^ — ^o)] + At cos5[i'(i — ^„)] + . . . ,
t/ = 5„Bin[.'(^ — O] + J?, sin 3 [v' (^ - g] + 5, sin 5 [^ (^ — i!,)] + . . .,
where U denotes the time the body crosses the axis of as, and is the time
V
of a complete revolution of the body about the origin. We may regard v
and the series, given above, may be written
X = 2,. a.- cos (2i + 1) t ,
y = Sf.&t sin (2i + 1) r ,
the summation being extended to all integral values positive and negative
zero included, for i. By adopting polar coordinates such that
X =z r cos ¥> , y ^1" sin . We may
also remind the reader that they determine rigorously all the parts of the
lunar coordinates which depend only on the ratio of the mean motions of
the sun and moon and on the lunar eccentricity. The Jacobian integral, in
the present notation, is
Du.Ds + J^. +im'(u + sy=G.
(us)* * ^ ''
The most ready method of getting the values of the coefficients a^, is that
of undetermined coefficients; the values of u and s, expressed by the pre-
ceding summations with reference to i, being substituted in the differential
equations, the resulting coefficient of each power of ^, in the left members, is
equated to zero, which furnishes a series of equations of condition sufficient
to determine all the quantities a;. For this purpose we may evidently
employ any two independent combinations of the three equations last written,
and it will be advisable to form these combinations in such a manner that
the process of deriving the equations of condition may be facilitated in the
largest degree. Now it will be recognized that the presence of the term
7-^, in one of the factors of the differential equations, is a hindrance to
(lis)*
their ready integration, being the single thing which prevents them from
being linear with constant coefficients. Hence we avail ourselves of the
possibility of eliminating it. Multiplying the first differential equation by
308 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
s, and the second by u, and taking in succession, the sum and difference,
uD's + sD'u — 2m (uBs-sDu)— J— + iia'(u + sy =0,
uD's — sD'u - 2m (uDs + sDu) + | m» (m' - s") = ,
then, adding to the first of these the integral equation, and retaining the
second as it is, we have, as the final differential equations to be employed,
D» {us) —Du.Ds — 2m {uDs — sDu) + | m= (w + s)' = C ,
D (uDs — sDu - 2mus) + f m^ (u' - s') = .
It must be pointed out, however, that these equations are not, in all
respects, a complete substitute for the original equations. It will be seen
that [i ov X, an essential element in the problem, has disappeared from them,
and that, in integration, an arbitrary constant, in excess of those admissible,
will present itself This will be eliminated by substituting the integrals
found in one of the original differential equations, in which fi or x is present ;
the result being an equation of condition by which the superfluous constant
can be expressed in terms oi (i and the remaining constants.
We remark that the left members of our differential equations are
homogeneous and of two dimensions with respect to u and s. If the first
were differentiated, the constant C would disappear, and both equations
would be homogeneous in all their terms. This property renders them
exceedingly useful when equations of condition are to be obtained between
the coefficients of the different periodic terms of the lunar coordinates, and
it is for this purpose that we have given them their present form.
From the signification of the symbol D ,
Bu =2,. (2i + 1) a,!:" + \ Ds =2,. (2t + 1) ■a_,_^C' + \
D'u = I,.(%i + lY3,j:"+\ Z)»s = 2', . (2t + 1)' a_ , _ iC" + ' ;
also
Du.Ds=- S, . [Z, . (2t + l)(2i — 2j + 1) a,a,_J C*,
uDs — sDu=- 2Sj . [2, . {2i -j + 1) a 7; - ay 2 (4/— 1) - 4m + m' '
r,-|_(_/)^ 27 ^, 3 16/-3;-5-(37+ll)m
L-^J *- ^' 8/ 2/ 2(4;-' — l)-4m + m'
RESEARCHES IN THE LUNAR THEORY
311
In making a first approximation to the values of the coefiScients, one of
the terms of the equation may be omitted; for, when / is positive, the term
2i . U) aia_4_^_i is a quantity four orders higher than that of the terms of
the lowest order contained in the equation; and, when j is negative, the
same thing is true of 2^. [/] aia_<+j_i. Hence, with this limitation, the
equation may be written in the two forms
^i • [[;■ . *■] a.a *-, + if\ a,a_ , + , _ J = ,
^i • [[— y. i'] aA+, + (— y) a,a_,+^_,] = .
where / takes only positive values.
From these two equations, by omitting all terms but those of the lowest
order, we derive the following series of equations, determining the coefficients
to the first degree of approximation :
aoE^ ^ |_1 J Eoao ,
aoa_i = (— 1) a„a„ ,
a„aj = [3][aoai + aia„] + [2, 1] aia_i,
aoa_2 = ( — 2) [ai,ai + aia„] + [— 2, — 1] aia_, ,
aoa, = [3][a„aj + &i&i + aja,,] +[3,1] aia_ j + [3,2] aja.i,
ai,a_3 = (— 3) [a„aj + a^aj + ajao] + [— 3, — 1] a_ia2 + [— 3 , — 2] a_ jBi ,
aoa^ = [4][aoa3 + a^a., + aja, + a^ao] + [4,1] aia_s + [4,3] a^a. j + [4,3] aaa_i ,
a,a_4 = ( — 4) [a„a, + aiaj + a,ai + aaa,] + [— 4, — 1] a_ias + [ — 4,-2] a_2a3
+ [— 4,— 3]a_3a,,
The law of these equations is quite apparent, and they can easily be extended
as far as desired. The first two give the values of aj and a_i, the following
two the values of ag and a_2 by means of the values of aj and a._i already
obtained, and so on, every two equations of the series giving the values of
two coefficients by means of the values of all those which precede in the
order of enumeration. A glance at the composition of these equations must
convince us that all attempts to write explicity, even this approximate value
of aj, would be unsuccessful on account of the excessive multiplicity of the
terms. However, they may be regarded, in some sense, as giving the law
of this approximate solution, since they exhibit clearly the mode in which
each coefficient depends on all those which precede it. As to the degree of
approximation aiforded by these equations, when the values are expanded
in series of ascending powers of m, the first four terms are obtained correctly
in the case of each coefficient. Thus a^ and a_i are afiected with errors of
the 6th order, ag and a_2 with errors of the 8th order, ag and a_3 with errors
of the 10th order, and so on.
312 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
The values of these quantities can be determined either in the literal
form, where the parameter m is left indeterminate, as has been done by Plana
and Delaunay, or as numbers, which mode has been followed by all the
earlier lunar-theorists and Hansen. In the latter case, one will begin by
computing the numerical values of the quantities [j, t], [y] and {J), corre-
sponding to the assumed value of m, for all necessary values of the integers
i andy.
The great advantage of our equations consists in this, that we are able
to extend the approximation as far as we wish, simply by writing explicitly
the terms, our symbols giving the law of the coefficients. How rapid is the
approximation in the terms of these equations will be apparent, when we
say, that, after a certain number of terms are written, in order to carry this
four orders higher, it is necessary to add to each of them only four new
terms; and thereafter, every addition of four terms enables us to carry the
approximation four orders farther.
The process which may be followed to obtain the values of the a< with
any desired degree of accuracy, is this: — the first approximate values will
be got from the preceding group of equations until the a^ become of orders
intended to be neglected; then one will recommence at the beginning, using
the equations each augmented by the terms necessary to carry the approxi-
mation four orders higher; substituting in the new terms the values obtained
from the first approximation, and, in the old, ascertaining what changes are
produced by employing the more exact values instead of the first approxi-
mations. A second return to the beginning of the work will in like manner,
push the degree of exactitude four orders higher. In this way any required
degree of approximation may be attained.
Whatever advantage the present process may have over those previously
employed is plainly due to the use of the indeterminate integers i and j,
which, although much used in the planetary theories, no one seems to have
thought of introducing into the lunar theory. This enables us to perform a
large mass of operations once for all.
For the purpose of making evident the preceding assertions, and because
we shall have occasion to use them, we write below the equations determin-
ing the coefficients a; correct to quantities of the 13th order inclusive.
aoai = [l][a5 + 3a_iai + Sa.ja^] + (1) [aLi + 3aoa_2 -|- 2aria_3]
+ [l,-2]a_2a_.-l-[l, — l]a_,a_j-|- [l,3]aja, + [l,3]a3a,,
aoa_i = [— l][a!-i + 3aoa_j 4- Saja.s] -f- (— 1) [aj + 2a_ia, -l- 2a_ja,]
+ [— 1, — 3]a_3a_,-F [- 1, — 2]a_2a_, + [—1, 1] a^a, -h [— l,2]asa3,
aoaj = [2][2a„ai + 2a_,aj + 2a_ja3] -t- (2) [2a_,a_j + 2aoa_3 + 2aia_,]
+ [2, 2] a_ja_. + [3, - 1] a_ia_, + [2, 1] a,a_x + [2, 3] a.a^ + [2, 4] a^,
RESEARCHES IN THE LUNAR THEORY 313
aoa_, = [ — 2][2a_ia_, + 2aoa_5 + 2aia_ J + ( — 2) [Saoa, + 3a_iaj + 2a_2a3]
+ [ — 2, — 4]a_.a_2 + [— 2, — 3]a_3a_i + [—2, — l]a_iai + [—2, Ijaja,.
+ [— 2,2]aA,
aoa, = [3] [a? + 2a„a2 + 2a_ia3j + (3) [aLj + 2a_ia_3 + 2a„a_4]
+ [3, — l]a_ia_,+ [3, l]aia_, + [3, 3]a2a_i + [3, 4]aA,
aoa_3 = [— 3][aL2 + 2a_ia_3 + 2aoa_ J + (— 3) [a^ + 2a„a2 + 2a_ia3]
+ [- 3, - 4] a_,a_, + [— 3, — 2] a_ ^aj + [— 3, - 1] a_ a + [— 3, 1] a^a, ,
aja^ = [4][3aia3 + 2a„a3 + 2a_iaJ + (4) [3a_2a_3 + 2a_ia_j + 2a„a_5]
+ [4, — l]a_ia_, + [4, l]aia_3 + [4, 2]a2a_2 + [4, 3]a3a_i + [4, SJa^ai,
aoa_j = [— 4][2a_2a_s + 2a_ia_i + 3aoa_s] + (— 4) [3aia2 + 2a„a3 + 2a_iaJ
+ [-4,- 5]a_5a_i + [— 4, — 3]a_3ai + [— 4, — 2]a_saj + [-4, — l]a_ia3
+ [-4, l]aia5,
a„a5 = [5][a^ + 3aia3 + 3a„aJ + [5, 1] aia_^ + [5, 2] aja_s + [5, 3] aaa., + [5, 4] a^a_i ,
aoa_j = (— 5) [a? + 3aia3 + 3a„aJ + [— 5, — 4] a_^ai
+ [ — 5j — 3] a_3a2 + [ — 5, — 2] a_ja3 + [ — 5, — 1] a_ia^,
aoae = [6][2a2a3 + 3aiaj + 2a„a5j + [6, 1] aia_B
+ [6, 2] aja_4 + [6, 3] a3a_3 + [6, 4] a "''*»'
and, after some reductions,
»» = TftV fs A ; jTT^ri A ; n *38 + 40m + 9nv — 32 5 j = m'a„,
2 6 6 [-g _ 4.m ^ m='][dO — 4m + m''] \_ 6 — 4m + m" J »'
07 3 + 4m + dm" f oc r/^ , oa 7 — m 1 ,
*-' = ^ [6-4m + m'][30-4m+m'] [" ^^ " ^°^ + ^* W^T^^-^^j ^\ ■
It is evident that, however far the approximation may be carried, the
only quantities, involved as divisors in the values of the a;, are the trino-
mials, whose general expression is
2 (if - 1) — 4m + m',
314 COLLECTED MATHEMATICAL WORKS OP G. W. HILL
or, particularizing, the series of divisors is
6 — 4nH- m',
30 — 4m + m.\
70 - 4m + m^
It will be remarked that they diflfer only in their first terms, which are inde-
pendent of m. Hence any expression, involving several divisors, can always
be separated into several parts, each involving only one divisor, without any
actual division by a trinomial in m. For instance,
1 _ 1 1 , 1
[6 - 4m + m'][30 — 4m + m=] ^ 6 — 4m + m' ^ 30 — 4m +
1 _ 1
[6 — 4m + m']'[30-4m + m'] ^[6 — 4m + m"]'
" '^ fi _ Am J- m2 + ^' I
6-4m + m' ' ^ 30 — 4m + m»"
Moreover when, after this transformation, any numerator contains more or
other powers of m than two consecutive powers, it is clear it may be reduced
so as to contain only these by eliminating the higher powers through sub-
tracting certain multiples of the divisor which appears in the denominator,
or, in other words, the fraction may be treated as if it were improper.
From this we gather that the value of a< can be expressed thus
ao ~ " ^ 6 — 4m + m» ^ [6 — 4m + m']' ^ [6 — 4m + m']' ^ ' " *
30-4m + m' [30-4m + m^]' [30-4m + m']'
P P P
■^ 70-4m'+m' "*" [70-4mV m^ "*" [70-4m + m^ "*"•••
+
where Mg, M^ . . . ■ Ni, N^ . . . ■ P^, P^. . ■ ■ are entire functions of m each of
the form
Am" + ^m*+'.
The advantage of this method of treatment consists in that nothing, which
is given by the successive approximations, would be lost, as must be the case
when the values are expanded in series of ascending powers of m . The
preceding expressions, when put into this form, become
a, -f- a_.i _ _ Q ^ + ra ,
-"-i;^ ^6-4m+m'°''
= 3
4- a_
\j 6-4m + mJ '
ViAA a. 323 + 109m _ q^ 23 — 11m 215 — 53m "1 ,
|_"TT^ ■'' 6 — 4m -i- m' [6 — 4m + m7 30 — 4m4-m'J '
_ « r,^« , 175 + 563m ,q 89 — 32m , ^ 361 - 10m l^,
- A|_^^ + 6-4m + m' [6 — 4m -h m"]' "^ 30 - 4m + m" J
Tf
RBSBAHCHES IN THE LUNAR THEORY 315
The evident objection to this form for the coeflScients is that it makes the
several terms very large, and of signs such that they nearly neutralize each
other, the sum being very much smaller than any of the component terms.
However it may be possible to remedy this imperfection by admitting three
terms into the numerators, but, in this way, the problem is indeterminate,
infinite variety being possible.
It is remarkable that none of our system of divisors can vanish for any
real value of m, since the quadratic equations, obtained by equating them
to zero, have all imaginary roots. In this they differ from the binomial
divisors met with when the integration is effected in approximations arranged
according to ascending powers of the disturbing force.
It is well known that the infinite series, obtained from the development,
in ascending powers of m, of any fraction whose numerator is an entire
function of m, and its denominator any integral power of a divisor of the
previously mentioned series, is convergent, provided that m lies between the
two square roots of the absolute term of the divisor. Hence any finite
expression in m, involving these divisors, can be developed in such a series,
provided that the numerical value of this parameter is less than \/6. The
same, however, cannot be asserted when the expression really forms an
infinite series, as it is in the equation just given for the value of -^ . Yet, on
account of the simplicity with which these quantities can be expressed in this
form, a,i and a_i containing each a single term, with an error of the sixth order
only, this limit is worthy of attention.
If the parameter m, hitherto employed by the lunar theorists, is taken
as the quantity in powers of which to expand the value of a^, we shall have
m =
m
And, substituting this value, the principal divisor 6 — 4m + m^
becomes 6 — 16m + llm*. Thus the limits, between which m must be con-
tained, in order that convergent series may be obtained where this divisor
intervenes, are ± Vtt- When we consider how little, in the case of our
moon, m exceeds m, it will be plain that the series, in terms of m, are likely
to be much more convergent than those in terms of m .
If we inquire what function of m , of the form — , the quantity
M
[6 — 4m + m'']"
can be expanded in powers of, with the greatest convergency, it is easily
found that a should be — i. Then putting
1 + itn
316 COLLECTED MATHEMATICAL WOEKS OF G. W. HILL
the divisor 6 — 4m + mMs changed into
6 + ^m',
and there is introduced the additional divisor 1 + iwt. Here the series will
be convergent provided m is less than 3. It is true the terms involving the
succeeding divisors 30 — 4m + m^ &c., are not benefited by this change of
parameter, but as they play an inferior role in this matter, I have chosen in
as the parameter for the developments of the coefficients a; in series of
ascending powers.
To illustrate this matter, we have, in terms of the parameter nt, and
with errors of the sixth order,
»! + a-i _ _ r 2 + im _ 1 "1 J
ao ~ U + iVm' 1 + itnJ'"'
a„ U + iVm^ l + im^[l + imTj'"-
Expanding these expressions in powers of m, we get
?i4^^ = -[m' + im'-|m* + ^m^ + . . . .],
3i] ■""■ 3i
'1-^=1= i^m' + fm' + T^tn* — ■Hm' + ....
Let these series be compared with those which correspond to them in the
lunar theories of Plana or Delaunay, viz:
nf + ^m' + ^m' + 4^m^ + ,
i^m' + H «i» + ^ra* + ^■H-m^ +
The superiority of the former, in convergence and simplicity of numerial
coefficients, is manifest.
Much more might be said relative to possible modes of developing the
coefficients aj in series, but we content ourselves with giving their values
expanded in powers of m, the series being carried to terms of the ninth order
inclusive. The denominators of the numerical fractions are written as pro-
ducts of their prime factors, as, in this form, they can be more readily used,
the principal labor in performing operations on these series being the reduc-
RESEARCHES IN THE LUNAR THEORY 317
tion of the several fractional coeflBcients, to be added together, to a common
denominator.
a. _ 3 , , 1 3 , 7 . , 11 . 30749 , 1010531 ,
_ 18445871 8 _ 2114557853 ,
2".3\5' 2".3''.5'
a-.__i9.ni,_ 5 , ^Z , 14 . 7381 , 3574153 ,
55218889^8 , 13620153029
"^ 2''.3^5'
2".3'.5
m'
a. _ 35 . , 803 . , 6109 . , 897599 , 237203647 „. 44461407673 ,
"a^ W ^ W:3J '^ 2\3\5' 2\3'.5' ^ 2^3=.5* 2^3*.5\7
^-»-0m*4. ^3 m»+ ^^^ „e 1 56339 , 79400351 8 , 8085846833 ,
ii:;~ ^'Wj ^FXP +2^3W + 2".3i5* ^ 2".3^5^7
a» _ 833 . , 27943 ^, , 12275527 ^8 , 27409853579 »
■57 ~ 2^ 3".5.7 2"'.3^5l7" ^ 2".3*.5'.7'
a-3_ 1 . I ?1 ip, , 46951 8 , 14086643 ,
"a7 ~ "2^3" ^ 2'.3.5 ^ 2«.3\5^7 2'.3\5l7'
a, _3537., 111809667 o
a_.__23_., 1576553 „
17 ~ 2".3 "^ 2".3^7'' ■ ■ ■
These values being substituted in the equations
r cos u z= Sf.sif cos 2iT ,
r Bmo = 2 .a, sin 2ir ,
and the parameter changed to m, we get
rcos. = a, {l+[-m'-i-m» + |-m'--i-m»-gf^m« + g|||m'
25239037 ._ 732931 . "l 3^
+ 14929920'" 37324800 "^ • • • J "OS x!r
r 25 . , 311 ^, , 9349 . _ 5831 ,
L'Me" "* "^ 960 "^ "^ 28800 216000
164645363 8 11321875589 . 1 „,^^
552960000 ^ 19353600000 J
r299 , , 30193 , , 379549 . , 181908179 , ■lcos6r
[40% "^ +107520'" +1003520'" +1580544000"' •••J''°«^^
, r 1134^ ., 2350381 , 1poa8r+ I
+ L1966O8"' + 9031680"' •• -J ''°'^'^ + ---|'
+
+
318 COLLECTED MATHEMATICAL WOEKS OP G. W. HILL
"-a„| |^_m +-^m +^m -3^m-33j^tn -
r sin t> = a„ < \ ^ m' + -^m' + ^m* — ~ m" — ;^i;SS tn' - lUlH m'
269023019 ^3 _i5mm9^,;|^.^ 2^
74649600 93312000
+ [im^4-4|i-m» + Sm»-^^m'
256 480 28800 432000
3500287 ^8 _ 43885512859 j^,. _ 1 ^^^ ^^
11520000 58060800000
+ r ^69 e 24481 , 4419347 ^8 , 398314169
[_12288 107520 15052800 4741632000
, r 9875 8 , 32608451 , 1 • q , 1
+ L1M6O8"' + 144506880 "' •••J^^°^^ + ---}-
Our final differential equations are capable of furnishing only the ratios
of the coefficients a.^, hence we must have recourse to one of the original
equations if we wish to determine ao as a function of n and (i. By substi-
tuting the values
in the differential equation
[i)» + 2mZ> + f m' - ^J M + |m»s = 0,
we obtain
(^ = ^'-^^^^^ + 1+ my + im»]a, + |in'a_._.} C"+'.
Considering only the term of this, for which i^O, and supposing that the
coefficient of ^ in the expansion of , ", 3 is denoted by J, we shall have
^J=l + 2m+fm' + fm'^.
For brevity call the right member of this H; then, since
(TO TO )^ to' ^ '^
we shall have
The value of H is readily obtained from the value of -^ given above, and
J must be found by substituting the values
RESEARCHES IN THE LUNAR THEORY 319
in . °. 3 , and taking the coeflScient of ^. We get
\US) s
a«Q L Ho flifl J ^0 ^0
^3 aja_^ +45?:i?^ + 3?2%_\
a,) a^ dif
where the terms neglected are, at lowest, of the tenth order with respect to
m. And, explicitly in terms of this parameter,
r_i , ^Lm*--^m^ 53_ ,_ W7, j^, _ 4201313 . 14374939 ,
By means of which there is obtained
^'-LWAV- T'°+T°'+2304'^ 288"°' 41472 ""
8761 ^, __ 4967441 . , 14829273 ^^
6912 7962624 "^ 39813120
or, in terms of the parameter m ,
M'[
1 1 „, , 4 , 163 . 1147 , 79859 .
4811 „, , 9530295 . , 139240651
+ i§H. m' + 9530295 . , 139240651 . "l
"^ 10368 "^ 71663616 1074954240 " J "
The quantity I -^^ I is usually designated a by the lunar-theorists; and,
to make this appear as a factor of the expressions for r cos v and r sin i; , it
would be necessary to multiply all the coefficients by the second factor of
the preceding expression for ao- It seems simpler however to retain ao as
the factor of linear magnitude ; for the astronomers have preferred to derive
the constant of lunar parallax from direct observation of the moon, or, in
other words, they have preferred to consider j« as a seventh element of the
orbit; with this view of the matter, there is no incongruity in making ao
everywhere replace jx.
The expression for ao can be obtained in several other ways, which
lead to more symmetrical formulae, and which also serve for verification of
all the preceding developments. Tf, in the preceding equation giving the
value of T^^ in terms of ^, we attribute to t the value 0, or, which is equiv-
(usy
alent, make (^ = 1 , we shall have m = s = 2i . a^, and, consequently
320 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
X
, = r. . l(2i + 1 + m)' + 2mT a,.
And thus, mindful of the value of x given above, we get
^° ~ ^^-' [2, . [(3t + 1 + m)» + 2m^] ^ ■ [^. • -^j\ '
Again the differential equation
d-r dz r "
gives
^.^^^..[(ai + l + my — m']a = a [1 — 0.08331972 cos 2t
+ 0.00114564 cos 4t
+ 0.00007409 cos 6r
+ 0.00000404 cos 8t],
r sm o = a
[ 0.12709553 sin 2t
+ 0.00098090 sin 4t
+ 0.00006099 sin 6t
+ 0.00000342 sin 8r].
log a = 9.5318013.
For m = -
r cos o = a [1 — 0.1622330 cos 2t
r sin y = a [ 0.3542740 sia 2t
+ 0.0048920 cos 4t
+ 0.00059858 cos 6t
+ 0.000081198 cos 8t
+ 0.000011873 cos IOt
+ 0.000001849 cos 12t],
log a = 9.5955815.
+ 0.0039840 sin 4t
+ 0.00049306 sin 6t
+ 0.000070196 sin 8t
+ 0.000010611 sin IOt
+ 0.0000016902 sin 12t],
For moons of much longer lunations the methods hitherto used are not
practicable, and, in consequence, we resort to mechanical quadratures. Here
we shall have two cases. The satellite may be started at right angles to and
from a point on the line of syzygies, and the motion traced across the first
quadrant; or it may be started at right angles to and from a point on the
line of quadratures, and the motion traced across the second quadrant; the
prime object being to discover what value of the initial velocity will make
the satellite intersect perpendicularly the axis at the farther side of the
quadrant.
328 COLLECTED MATHEMATICAL WOKKS OP G. W. HILL
The diflFerential equations
give, as expressions of the values of the coordinates, in the first case,
x = x, + 2 C'ydt - C f'\^- ^'\xdt\
y = ^f\^« - ^) dt -££t'^^" '
and, in the second case,
^ = - ^£{y. -y)dt- ££ [p - ^ ] ^*" '
Here the subscript (o) denotes values which belong to the beginning of
motion, and d) will hereafter be used to denote those which belong to the
end.
Let V be the velocity, and a the angle, the direction of motion, relative
to the rotating axes, makes with the moving line of syzygies. In the first
case then Gq = 90°, and we wish to ascertain what value of Uq will make
(Ti = 180°. Generally, for small values of v^, Oi will come out but little less
than 270°; but, as Vo augments, a^ will be found to diminish, and, if xq does
not exceed a certain limit, a value of Vq can be found which will make
>7i i ^i ; fz -
'72 > ^2 ; ^3 > >73 > ^3 • -^iid let the rectangular coordinates of the moon relative
to the earth be denoted by x, y, z; those of the sun relative to the centre of
c gravity of the earth and moon by x', y', and 2'; and those
of the centre of gravity of the three bodies by X, F,
and Z. Then from an attentive consideration of the
subjoined figure, where S, E and M denote the posi-
tions of the sun, earth and moon, G the centre of grav-
ity of the last two bodies, and C the centre of gravity
of all three, it will be seen that, if we put
P- =
nh + m^ + m^
we shall have
^2 = (i"' — l)x' — iix + X,
f3 = (/-l)a;'+(l-,«)a;-(-X,
MOTION OF THE CENTRE OF GRAVITY OF THE EARTH AND MOON 337
with two groups, of three equations each, for the r; and ^, obtained from
these by writing, in the second members, for x and X, y and Y, and again
z and Z.
If we differentiate the equations just written, then square and add the
results, after having multiplied them severally by Wj, m^, and m^, we shall
get
niid^l + m^d^l + m^d^l = ntifi'dx''' + m^ndx' + MdX''.
Prom this equation it is evident that, if £l denote the potential func-
tion, the differential equations, determining the variables x, y, z, x', y', z', are
d'x SG d'y Sfl d'z Sfl
'^''^-W^dx' '^''■^^W "'''-M^^Tz'
Hence it may be gathered that the disturbing function for the motion of the
sun relative to the centre of gravity of the earth and moon differs from the
corresponding function for the motion of the moon relative to the earth only
by a constant factor which depends on the masses.
The expression for fl is
where the A'sare given by the equations
J?,, = (a;' + y.xf + (2/'+ tJ-yf + {%' + i>.%)\
^l3 = [^'- (l-M)a;T + ly' - (1 - /^)«/T + [2' -(1-/^)^]%
A\ 3 = x' + y^ + z^.
Let us put
r' = x' + y^ + z\ r" = x'^ + y'^ + z'\ rr' S = xx' + yy' + zz' .
Then
JJ ,, = r" + 2/ji.rr'8 + fi'V ,
Al ,^r"-20--fi) r/S + (1 - iJ-yr' .
r 1 1
Since the ratio — ris only about — — ■, and u about-— , it is convenient to
r' '' 400 "^ 80
expand, in D. , the reciprocals of Aj, 3 and Aj, 3 in infinite series proceeding
according to ascending powers of — ^ . This, in both cases, evidently depends
r
on the development of
32
338 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
(1 - 'Hax + a") -i
in powers of a. By the Theorem of Lagrange, in solving the equation
y — a,F (y) = X with respect to «/ , we get
, „. s , a" d.Fixf , , a" d''-\F(x)'' ,
whence
^ = 1 + a ^^JIS^ + «° d\F(xy ^ _ _ ^ al d'. Fjxy ^
da; c?a; 1.3 dx' ' ' ' n\ dx"
Let us suppose that we have here F{y) ^= ^{'if — 1); the equation, on
which y depends, becomes then?/ — ia(y — 1) = a^, and the resolution of
this quadratic in y gives 1 — ay=-\/\ — 2aa;^-a^ and, by differenti-
ation, -^ = (1 — 2 aa; + a^)~^. Consequently,
2 dx 2.4 dx' 2.. 2" dx"
= 1 + ag-a;
3r6.5.4 3_3 4.3.2^-|
4r 8.7.6.5 ,_4 6.5.4.3 4.3 4.3. 2.1 "]
"^" L2.4.6.8'^ 1-2.4.6.8^''^ 1.2'2. 4.6. Sj
The law of the numerical coefficients in this series is so plain that we
can set down as many terms as we have occasion for.
In making the application to the reciprocals of Ai,2 and Ajg we must
put, in the first case, a = — ^u — , , in the second, a — (1— /ct) -^ , and in both
x = S.
We obtain as the potential function proper for the relative motion of
the moon about the earth.
]_^^«z,+ m3^ rj_ 1 ^ 1 1 -1
+ r«, {[(1 -;.)-'• +M-']^
MOTION OP THE CENTEE OF GRAVITY OF THE EARTH AND MOON 339
.[a-,).,]^[t|.-|.fci]
,[a-.,.-„.]i^[|-i^-|.|-|.]
- }•
To get the similar function for the relative motion of the sun about the
centre of gravity of the earth and moon, it is necessary to multiply the pre-
ceding expression by
The term of the potential function for the moon, factored by -^ , gives
rise to inequalities in the lunar coordinates factored by — ^ . As this term
has 1 — 2fias a factor, we see the correctness of the rule which directs to mul-
tiply this class of inequalities by 1 — 2/a, in order to include the effect of the
disturbance of the relative motion of the sun about the earth by the lunar mass.
In treating the motion of the sun about the centre of gravity of the
earth and moon, it will suffice to take two terms of the preceding expression
and put
Let the longitudes of the sun and moon be denoted respectively by TJ
and /I, and neglect the latitudes; then
_L,fl = !^+ \M,j.O~-t,) 4[3cos2(A-A') + 1].
The differential equations, determining / and %\ are
where, it will be noticed, we have put r = a, and, after differentiation, in the
final small terms, r'=a', -;o- = ft'l and ;i — ;i' = 'r the mean angular dis-
a'
tance of the moon from the sun. The integration of the second equation
gives
-L — + 3 „"a/^ (!_;,) __.|^_____ cos 2r + lJ =0.
340 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
-jj- = -5-7^ I r M (1 — /i) -Ts COS 3t ,
Oo being the arbitrary constant. We can now eliminate -j- from the first
equation, and we get
Let us suppose that this equation is satisfied by
^ = 00 + «'«! cos 2t ,
ai being a coefficient to be determined. Substituting this value of r' in the
differential equation, we get the two equations of condition,
(in' - 8»«' + 3 m'O «! - |to'' ^n — n!^ ^^ (1 — z^) -^^ = .
Whence may be derived
_ 3 w' (3 — m) ^^ X a"
"•-3 (1— OTX4:-8m + 3m») '*'■ ''''^■
where, as is usually done in the lunar theory, we have put — = m. The
value of r', thus obtained, being substituted in the expression for -=- , we get
dV , , TO 4 — 3m + m" ,^ ^ a' „
a^ * 1— to4 — 8m + Snv ^ ' a"
Integrating
„ f , „// Q to'' 4 — 2m + to' ,, \ a' ■ o
* (1 — nCf 4 — Sto + 3m' ^ -' d'
The numerical values of the constant quantities, which enter into these
formulas, are
TO = 0.0748 , I. = g2^ , |-, = 0.002587 , n' = 1295977".4 .
They give us
r' = a' [1.00000 00200 + 0.00000 00003 cos 2r] ,
X' =e' + n't - 0". 0001 sin 2t .
MOTION OF THE CENTRE OF GRAVITY OF THE EARTH AND MOON 341
The periodic terms of these equations are too small for consideration,
r'
but the constant term of —, may be noticed. If we should obtain the value
of a' from measured values of / on the assumption that the value of the con-
stant term is unity, it would be too large by the 0.00000 002 part, And
I'm
this value substituted in the equation n' = \ —j^ would give n' too small by
the 0.00000 003 part, or n' would be too small by 0".03895 ; or the error in
the mean longitude of the sun would amount to nearly 4" in a century, a
quantity which could not, in the present state of astronomy, be neglected.
However, it is only fair to state that astronomers proceed in a way the re-
verse of this ; that is, they observe n' and thence deduce a', and in this case
the term 0.00000 002 is without significance, since the logarithms of the
radii vectores in the ephemerides are usually given to 7 decimals only.
342 COLLECTED MATHEMATICAL WORKS OP G. W. HILL
MEMOIR Fo. 34.
The Secular Acceleration of the Moon.
(The Analyst, Vol. V, pp. 105-110, 1878.)
In the Philosophical Transactions for 1853, Prof. J. C. Adams, of Cam-
bridge University, England, showed that the values of the secular acceleration
of the mean motion of the moon, obtained by Plana and Damoisean, were
erroneous, for the simple reason that these authors had, inadvertently, made
the solar eccentricity constant throughout a certain portion of the investiga-
tion. This statement of Prof. Adams gave rise to an animated and pro-
longed controversy, the history of which will, no doubt, always possess much
interest.
It is proposed to obtain here the coeflBcient of the term in the moon's
mean motion involving the square of the solar eccentricity , supposed variable,
to quantities of the order of the square of the sun's disturbing force, when
the lunar eccentricity and inclination of orbit are neglected. The method
employed has no novelty, having been used before by Mr. Donkin. But, at
the end of the investigation, I have found that it is possible to do without an
explicit development of iB in a periodic series, and thus the treatment is,
to a considerable degree, abbreviated.
Let ^ denote the mean longitude of the moon as affected by this secular
inequality, and Wq the mean motion at a given epoch taken as the origin of
time ; we propose to prove that, in the equation
the true value of J? is
3/«'Y_37yi/w'Y
^U.i 64 [nj-
^
Employing the method of variation of the elements, we have, for deter-
mining the four elements n, ^, e and u of the lunar orbit, these equations
dn 3_9^ de Tutd-R
dt ~ fia^ 9C ' dt~ fie 9-^ ^3 [1 + 3 cos 2 (/I— /)] ,
where il and X' are the true longitudes of the moon and sun. The constant
part of i2 is evidently the same as that of ^n'^ a^—j^ , when we rejected, that
is, it is equal to in'^a*(l + |e'^) .
Considering first those terms in B which are independent of e (we need
those multiplied by e only when taking account of the effects produced by
the variations Se and 5u), we see that the only terms in B which produce
terms in -=- , and, consequently, can give rise to terms independent of sines or
(tt
cosines of arguments in -^ , have arguments of the form 2^ + 4', where i^ de-
notes an angle depending on the sun's mean motion. Hence, denoting any
one of these terms of B by n'^a^ A cos (2 where A is independent of
the lunar elements, but will generally contain e'^ and regard being had to this
term alone, the equations determining the elements become
^ = 6n"A sin (2C + <*) , ^^ = n — i^ Acoa (3C + 0),
where [i has been eliminated by using the equation fi = r^c^. Integrating
these, and considering •^ as constant, since its variability affects only the
terms in H multiplied by — g , but regard being had to the variability of a
d e'^
through e', where we may consider—^ as constant, we obtain
= -r-LAsin C2C + 0)-|^|f..^cos (2: + ^),
This being only a first approximation in which we have had regard only
to quantities of the order of n'^, we proceed to a second approximation. And
first, in the expression for -=-, we substitute for ^, ^-\-S^ ; and we find, for
344 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
that part of the increment which is independent of the sines or cosines of
arguments, the expression
~di~ 1? d.e"~df
Integrating this and putting e'* — e'l =8. e^ ,
2 n' d.e
8n and 8^ ,
Again, in the expression for -^ , increasing n and ^ by their variations
dt ^ w' d.e'^ n n"
Now the constant part of this value of 5 . -^ goes to form part of the constant
Uq , hence, desiring to retain only the varying part, we may write ' ,g 8 . c'^
for A^ , and thus obtain
■ df- '^' n'd.e" ^^ • ^^^
In the next place let us consider the terms in B multiplied by e ; they
are all of the form
n" d'e A cos (w + kZ + (p') ,
where A and t^ possess the same quality as before, and k may be — 3 , — 1
or 1 . Representing a -\- Jc^ + '4' hy 6 , the equations determining the ele-
ments are, regard being had to this term alone ,
di dt n
dt ^ n dt n e
In the last equation we have written only the term divided by e, since this
alone can produce terms 5 . -^ of the kind we seek. Integrating the last two
as we integrated in the former case, we obtain
;> 1 «'" 4 „..a /, _L 1 n"d.A'd.e'' . ,
se= l^Aame + ^^'^-i^i^^coso
he n' ¥e n' d.e'^ dt
THE SECULAR ACCELERATION OF THE MOON 345
Augmenting, in the expression for -5- , e and by these quantities, we ob-
etc
tain, regard being had only to the terms which are independent of sines or
cosines of angles,
. dn_ 3 n'* d.A'd.e"
' dt U rf d.^' dt '
Increasing, in the expression for -^ , the elements n, e and B b)' their vari-
ations 5n, he and hQ , and preserving only the terms independent of the sines
or cosines of angles, we get
• dt~ U n' d.e" ' '^ 'Wl? '
In like manner as before, rejecting the constant part of this which coalesces
with Wfl > we obtain
, <^C 5 n"d.A'
■ dt ~ k n" d. e'"
S.e" . (2)
When formulas (1) and (2) are applied to all the terms of i2, to which
each is applicable, and the results added, we shall have the complete value
of 5 . ^ , since it is plain that the combination of two different terms in R
will always produce terms in 5 . -^ involving the sines or cosines of angles.
Denoting the mean anomalies of the moon and sun by ^ and ^', and the
mean angular distance of the bodies by r, the part of a B, which is indepen-
dent of e, may be written
aR = A^+ Ai cos 2r + A^ cos f + A^ cos (2r— f) -f- A^ cos (2t + ^') .
Formula (1) applied to this series gives
dt ^^ d. e"
We can obtain the terms in R multiplied by e from the series just given
by using the equation
dR_^ dRd.logr dR dX
9e 9r de d^ de
= -2i2cose-|-2^sinf ,
346 COLLECTED MATHEMATICAL WORKS OP G. W. HILL
Whence
?^^ = - ZA, cos f — 3^1 cos (2t - e) + Ji cos (2t + f)
— A, cos (e— r) - A^ cos (f + f)
— 3 J, cos (3r _ f _ ^) + ^3 cos (2t — f + f)
— 3^4 cos (2t + e' — f) + At cos (3t + ?' + ?).
Applying formula (2) to this series gives
5 . ^ = w j^[- 5 (4^; + Al +Al)-i (Al + Al + Al)+5 (9A\ + 9Al-{- 9^J)] e . e'
= ^ ^ [i|i {A\ + ^« + ^D - 10 (2^S + AX)-\ d.e'\
€b m 6
Adding this to the expression given by formula (1) ,
S.^= n^ [-V (^; + ^3 + Al) -10 i2Al + Al)} S . e'\
But, denoting the constant term of a^B^ by K, we have
K=Al + ^iAl+Al + Al + Al).
Or
^; + ^5 + Al = %K— (2^S + ^D ,
and
But we have
a^R = tV$p-I[¥ + 6cos 2(A - /) + • cos 4(A_A')] ,
and hence ^ is equal to the constant term of fj -j — ^ . In consequence,
a"
denoting the constant term of -jj by Z , we shall have
Also we evidently have
iir~ = ^. + ^.cose',
and thus
\:^L=%Al^-A\.
Substituting this value
THE SECULAR ACCELERATION OF THE MOON 347
But the constant term of — ^ is known to be 1 + Y*'^ 5 hence, in fine,
dt 8* if
To obtain -^ we must add to n both this term and that which arises, in
at
Inc^ dR
the first approximation, from the term -^— in the differential equa-
tion for -^ , which is therefore equal to the constant term of jo- , that
dt n r^
is, to — — (1 + |e'2) . Thus
n
at n
L(l+3e'2)_(B!^_3|Jl^;)5..".
We could have added to the first two terms of this equation a term
B^ e'l, where 5 is a numerical coeflScient, equal to the aggregate of the
constants we have virtually neglected whenever we wrote h. e'^ for e'" , but it
will be easily seen that this would not change the final result. We evidently
have
From which, to a suflBcient degree of approximation ,
Substituting this value of n, we get
348
COLLECTED MATHEMATICAL WORKS OF G. W. HILL
MEMOIE No. 35.
Note on Hansen's General Formulae for Perturbations.
(American Journal of Mathematlce, Vol. IV, pp. 256-359, 1881.)
The last form in which Hansen expressed the perturbations of the mean
anomaly and equated radius vector is exhibited by the following equations :
.( + ..+ /{f+-*^(-3^)"}«,«,
n„z:
(Equations 36 and 37, p. 97.)*
It will be perceived that the right-hand member of the first of these in-
volves three quantities, viz. W, v and -^ . But the last of these quantities
has no share in defining the position of the body, and it is desirable to get rid
of it, provided that can be done without complicating the equation. This is
readily accomplished by means of the equation (33, p. 95)
dz Aq
~dt~ h(l + v)' '
The result is
w„«
/W 4- if
Why Hansen has not put the equation in this form I cannot imagine ;
the advantage, not only as regards simplicity of expression, but also in point
of ease of computation, is evident.
Hansen develops W by Taylor's theorem, and, limiting ourselves to
the second power of the disturbing force, we have
W=W„ + [^)n,Sz=W,-2^Sz.
When this value is substituted for W in the equation for n^^, we have a
differential equation of the first order and degree for the determination of Sz,
*See Auseinandersetzung einer zweckmdsaigen Methode zur Berechnung der absoluten Storungen der kleinen
Planeten. Von P. A. Hansen. Erste Abhandlung. {Abhandlungen der Koniglichen Siichsischen Oesellsehaft der
Wissenschaften. Band III.) The numbering of the equations and the paging are from this volume.
NOTE ON HANSEN'S GENERAL FORMULAE FOR PERTURBATIONS. 349
the integral of which is well known. Terms of three dimensions with re-
spect to disturbing forces being neglected, this procedure furnishes the equa-
tion
n,H = (1-— 3.) C [(1 + j}.)-r„ + "'] n,dt ,
which, however, is without interest other than analytical, as its use involves
more labor than that of the equation given by Hansen.
Hansen's equation for the determination of v has the disadvantage of
not affording the constant term of this quantity, and is inconvenient in com-
puting the portion, of the form
At + Bt' + Gf + . . . .,
which is independent of the arguments g , g' , &c., as the values of A, B, &c.,
must be determined to a degree of accuracy much beyond what is necessary
in the case of the other terms. As all the arbitrary constants admissible have
been introduced by the integrations which give z , it is evident there must
exist an equation determining v without additional integrations. Hansen
has virtually employed this in the place where he shows how the constant
term of v is to be obtained, but has nowhere given it explicitly. This lacuna
I propose to fill here.
The equation 39, p. 97,
smw
may be employed to discover the value oi -j— . The known expressions for
-£- cos 0) and J- sin u are
i- cos ui
p •
-i- sin ■ ■' 1 _ X [r" — 'i/rrs +
r"]*'
If for fl are substituted only the first two terms of this expression, the
differential equations are easily integrated, and the variables x, y,z and x',
y' , z' represent the motion of two planets moving according to the laws of
elliptic motion, whose mean motions are
/ Mm J / Mm'
V 7^ ^""^ \-7^
In terms of symbols whose meaning is well known, we will put
L= sj \Mmii.d\ , L' — 1^ [Mm'/i'a'] ,
= V {.Mmfia (1 - e')-] , (?' = V [Mm'ii.'a' (1 - e")] ,
H= n/ [Mmfia (1 — e=)] coai, 5"' = V [Mm'fi'a' (1 — e'^)] cos i' ,
and denote the mean anomalies by I and V , the distances of the perihelia from
the nodes by g and g' and the longitudes of the nodes by h and h' , and
moreover, put
mm' 1
IZIT [r" - Zrr's + r'] '
+
We have then the following system of differential equations for deter-
mining the elements L , G , H , U , G' , H' ,1, g , h, V , g' , h' : —
dL_dR dl___dR dI/^_dR dV _ dR
dt ~~ dl' dt~ az ' dt ~ dV ' dt dL' '
dG_dR dg__dR dG[_dR dg' _ dR
dt~d9' dt~ dG' dt '~ dg' ' dt dG"
dH_dR dh _ _ dR dH^_dR ^'-_?^
di ~ dh ' dt~ dH' dt ~ dh' ' dt ~ dH' '
in which it is understood that B is expressed in terms of these elements.
33
354 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
As r is a function of the three elements L, G,l only , and r' of X' , & ,1'
only, it follows that the six elements H, g, h, E' , g' and h' enter in B only
through s ; hence we have the equations
dJR_dB_ds^ dR_dR ds^ dB_dRds
dS ds dg' dS~ ds dH' dh~ ds dh '
dR _dR ds^ dR _dR ds dR_dR ds
dg' ds dg' ' dS'~ ds dE" dh' ~ ds dh' '
The expression for « being given by
rr's = xaf + yy' + zz' ,
and V and v' denoting the true anomalies, the rectangular coordinates have
the equivalents
x = r [cos h cos (v + g) — cos i sin h sin {y + g)] ,
y =:r [sin h cos {v + g) + cos i cos h sin {v + g^] ,
z = r sin i sin (v + g) ,
a/ = / [cos h' cos («' + jf') — cos i' sin /t' sin (v' + g')},
y' = r' [sin h' cos (v' + g') + cos i' cos h' sin (u' + /)] ,
z' =r' sin t' sin (v' + g') .
Whence the following expression for s ,
s = cos (A — h') cos (y + g) cos («;' + gi') + cos i cos i' cos {h — h') sin {v + g) sin («' + g')
+ cos t sin {h — A') cos {v + 5^) sin («' + g')
— cos 1 sin (A — A') sin (v + 5^) cos (v' + g')
+ sin i sin i' sin (w + g) sin (?;' + 51') .
Remembering that v and v' contain only the same elements as r and r',
and that
cosi=^. sini^V^!^, cost'^f, sin^'=VE^,
it will be found that
-^ [ V »' — H' cos A + i^G" — H" cos A'] = ,
^ [V*?' — ^' sin A + V (^"' - i?'' sin A'] = ,
^[^+^'] =0.
Hence we have the following integrals of the differential equations.
)/G' — B'cosh+ V " — E" cos A' = a constant ,
V " — ^-^ sin A + V " — 5" sin A' = a constant ,
H + ff' =a. constant .
NOTES ON THE THEORIES OF JUPITER AND SATURN 355
These integrals may be employed to diminish the number of differ-
ential equations. Thus far the system of planes to which x, y ,z . . . are
referred has been left indeterminate ; let us now assume that the plane of
maximum areas, called by Laplace the invariable plane, is chosen for the
plane of xy. In this case it is well known that the constants of the first
two of the integrals, given above, become zero. Then we shall have
>f/G' — H'coah + i^G" — H"cosh' = 0,
ifCP^^R^ sin A + V »" — H" sin A' = ,
H + H' = c,
being an arbitrary constant. But , since i and i' are supposed contained
between 0° and 180°, the radicals in these expressions must be taken posi-
tively. Consequently the equations are equivalent to
h' = h + 180°, H + H'=c, H—H'= <3" — <3"\
c
These equations determine the values of the elements E, H' and h' in terms
of the rest, and they may be used to eliminate them from R. Then it is plain,
from the expression of s, given above, that h will also disappear from R,
and we shall have
^ = function {L,'Q, L' , 0' , l,g, V,g'),
and s takes the much simpler form
s=-co6(v-v' +g-g')+ ^^ +^^^' ~ ^ sin (v + g) sin (w' + (/') .
As to the partial derivatives of R with respect to L, L' , I, V , g , g' , they
are evidently unchanged by this elimination of the elements H, E' , h, h' .
But (^^^ and (^t^i) denoting the derivatives of R on the supposition of
its containing the elements E, E' ,h,h' , we have
(dR\_dR _dR dH dR dB'
\mi~'dG dH dG dH' dO '
(dR\_ dR _dR^ dH_dR^ dH^
\W)~ dO' dH dO' dH' dG''
But we also have
hence
dR dR _ d(h' -h) _^
dH dH'~ dt
ldR\_dR^_dR dCH+H') _ dR
['dG)~ dG dH dO dG
ldR\-.dR _dR d{H+ H') _ dR^
\W) ~ dG' dH dO' dG'
356 COLLECTED MATHEMATICAL WORKS OP G. W. HILL
Moreover
dB^dB dH as 3^^ as 9 (g + H') _ dR
dc dS dc dB! dc dH dc ~ dH'
Thus the system of differential equations still retains its canonical form,
and is
dL _ dR dIJ^_ dR dG^_ dR dff _ dR
dt ~ dl' dt ~ dV ' dt dg' dt dg' '
dl__dR dV___dR di^_dR dg' dR
dt ~ dL' dt 327' dt dG' dt dO''
After this system of eight differential equations is integrated, the value of
h is found by a quadrature from the equation
dh__dR
dt dc '
These integrations introduce nine arbitrary constants which, together
with c, make ten. The reference of the coordinates to any arbitrary planes
introduces three more, but one of these coalesces with the constant which
completes the value of h.
The time t does not explicitly enter R, hence the complete derivative
of it with respect to t is
dR_dR dL dRdl
dt dL dt dl dt
If, in this are substituted the values of -^ , -^- , . . . , from the equations
at at
just given, we shall find that it vanishes ; hence
^ = a constant
is an integral of the system of differential equations. This integral may be
employed to eliminate one of the elements, as L, from the equations. We
can also take one of the elements, as I, for the independent variable in
place of t. The system of equations, to be integrated, is then reduced to
the ix
dR
dR
dR
dL'
dl'
da
dg
dG'
W
dl ~
■ dR''
dl -
■ dR'
dl ~
dR '
dL
dL
dL
dR
dR
dR
dl'
dL'
dg _
d&
dg' _
dG'
dl -
dR '
dl
dR'
dl
dR'
dL
dL
dL
NOTES ON THE THEORIES OP JUPITER AND SATURN 357
A simpler form can be given to them. If the solution of -B = a constant
gives
Zi= function {L' , G, G',l',g,g', I),
and L is supposed to stand for the right member of this, the foregoing equa-
tions can be written
dl dl" dl ~ dg' dl ~ dg"
dr___dL^ dg _ _dL M--^
dl ~ dL' ' dl " dO' dl~ dO' ■
When the values of L', G, G',V , g and g' in terms of I have been de
rived from the integrals of these, they can be substituted in the equation
-J- = — -^Y , which will then give t in terms of Z , by a quadrature. By
inverting this we shall have I in terms of t ; and by substituting this in equa-
tions previously obtained we shall have the values of all the other elements
in terms of t .
It will be noticed that ^ is a homogeneous function of Z , Z' , G , G' and
c of the dimensions — 2 ; hence we shall have
and, as a consequence of this,
L ^ + L'^ + G% + Q' % + c^4= 2i2 = a constant.
dt dt dt dt dt
Thus, if the rate of motion of each angular element 1,1' . . . , be multiplied
by the linear element which is conjugate to it, the sum of the products is
invariable.
The sines of half the inclinations of the orbits on the plane of maximum
areas are
Thus, in the special case where the two planets move in the same plane,
we have
G+G' =c.
This equation may be employed to eliminate one of the elements G or
G' from R , In the same case, the expression for s is reduced to
s = — cos (« — v' + g — g').
358 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
Then, if we put
Q-Q' = r g-9' = r,
R will be a function of L , L' , T ,1 , I' , y , and we shall have, for determi-
ning these variables, the system of differential equations
dL dR
dt~ dl'
dL' dR
dt ~ dl"
dr dR
dt~ dr'
dl dR
dt~ dL'
dl' dR
dt ~ dL' '
dr dR
dt~ dr'
After these are integrated, the value of g + g' will be got by a quadra-
ture from the equation
d(g + g') _ dR
dt Qc ■
If the value of L is obtained from the solution of -B = a constant, and
we have
L = function {L', F ,1' ,y ,T) ,
and I is adopted as the independent variable in place of t, the solution of
this special case is reduced to the integration of the four equations
dL' dL
dl' dL
dr dL
dr dL
dl ~ dV '
dl - dL' '
dl - dr'
dl ~ di"
The angle between the planes of the orbits of Jupiter and Saturn is
about li°. This is small enough to make the terms, which are multiplied
by the square of the sine of half of it, and which are besides of two or more
dimensions with respect to disturbing forces, practically insignificant. Thus,
while we are engaged in developing those terms of the coordinates which
demand the highest degree of approximation relatively to disturbing forces,
we shall assume that the planes coincide ; the determination of the effect of
non-coincidence of these planes being reserved to the end, when it will be
always sufficient to limit ourselves to the first power of the disturbing force.
The coordinates usually preferred by astronomers are the logarithm of
the radius vector, the longitude and the latitude. We suppose that the two
last are referred to the plane of maximum areas. Let these coordinates be
denoted by the symbols log . p , ;i and ^ ; and let the subscript (q) be applied
to /I and /? when we wish to designate the similar coordinates corresponding
to the variables x,y,z, a;',y,z'. Then we have
P cos /J COS -i = r cos /S„ cos A„ + zr* cos /?'„ cos A'„ ,
P cos iS sin l = r cos /S, sin A„ + xr' cos /S',, sin A'„ ,
(0 sin /S = r sin /S, + z/ sin /S', .
NOTES ON THE THEORIES OP JUPITER AND SATURN 359
From the first two equations are readily obtained the following two : —
P cos (J cos (A — A„) = r cos /J„ + xr' cos /S',, cos (A'„ — /l„) ,
P cos /? sin (A — Ao) = zr' cos /S'o sin (A'o — Ao) .
In the developments in infinite series which follow, the eccentricities
of the orbits will be regarded as small quantities of the first order, the
squares of the inclinations of the orbits on the plane of maximum areas as
quantities of the third order, and x also as a quantity of the same order.
Then all terms, whose order is higher than the sixth, will be neglected.
This degree of approximation will be found amply sufficient for the most
refined investigations.
Under these conditions, we get
log p = logr + I log [ 1 + 2x -^ s + x^^'J .
= logr + .^s + ix^^' (l+2s'),
^ = ^0 + X ^ ^', - X -^ s;9„ .
We will write >? for sin i* . Then, to the sufficient degree of approxi-
mation,
^ ^ s = - X I- COS {v — if + g - g') + 2^ {y, + 7,'y ^ sin {I + g) sin (l' + g') .
In like manner
X £l?2L^o sin (A' _ A„) = X (1 + v'-v") 4r sill i^ - ^' +9-9')
r cos /Jo r
^xri" -^ Sin {Zl—V + ^g—g') + xV'^- sin (I + i' + g + g') .
The expressions for 7^^ and /3o in terms of elliptic elements are given by
Delaunay.* Log r , as well as the following expressions
+ 3e-CO^«(3?+^')+ie"S(*^'+^'>
±i«'^s?n(^'-^')±^«"S(^^'-^')'
* Theorie du Mouvement de la Lune. Tom. I, pp. 56-59.
360 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
are found in a memoir by Prof. Cayley* With these data we get
logp = loga + ^e' + ^\^e* + ^e' + x' ^^
_(e_|fl3_^e=)co8Z-(|e»-^e* + ^e«)cos2?
-(H^- t¥V ^) cos 3? - (II e^ - 129 e^) cos il
— iU^ cos 5Z— 899 e« cos 61
~'4{ [l-e'-ie"- (>? + v')l cos (?-?' + t/-^')
+ (|e - Je'-fee") cos (2i -V +g—g') + (_ |e' + |eV) cos (l + g—g')
+ (— 1 e — I e» + ^ ee") cos (J,'-g + g') + (^ e' _ | e"— i eV) cos Q -%l' + g —g')
+ ^e' cos (dl — l' +g-g'y- Iff" cos (I + V —g + g')
+ I ee' cos (5' — p') — » ee' cos (2i + ^r —g')
— \ ee' cos (3? - 5' + 5^) + I «e' cos (2Z - %l' +g—g')
+ |e" cos (1—31' + g—g') + ^e" cos (l + V + g-g')
+ ^e' cos {H -I' + g -g') — ^\ e' cos (%l^l'-g + g')
— 41 ^e' cos (3? + 5r - ^f') + ^eV cos (?— ^ + ')
+ II e'e' cos (3? - 21' + g - g') - ^-^ e'e' cos (l + ^V -g + g')
— T^ ee" cos (3? — fir + /) + ^^ ee" cos (2/ — dl' + g —g')
— -^ ee'" cos {V +g—g') + ^ ee" cos (21 + I' + g - g')
+ 1 e'^ cos (i - 4? + fif -5^') + 5^ e" cos (i + 2? +g-g')
+ (r,+ Ti'y COS G + ? + 5^ +5^') J + l^t'^Icos (2Z - 21' + 2g - 2g') ,
; = Z + fif + A + (3e - 1 e' + ^e") sin l + (^e'- l^e* + ^e") sin 21
+ (If «°-il«') sin 3^ + (W «' - iUeO sin il
+ J^^e^ sin U + J^e« sin 6Z
+ ( —v'—V* + ^V'e' ) sin (2^ + 2g) + ^ t sin (U + 4^)
+ ( — 2,»e + -2ji,V) sin (3? + 2fir) + (2,»e - JijV) sin (Z + 2g)
— ^>)V sin (4Z + 25f) — I vjV sin 2^
- 8| ,v sin (5? + 2(/) + ^ ,V sin (? - 2g-)
+ x^|(l-e'-ie" + ,'-,") sin (?-?+£r-5f')
+ (f e— ^e' — f ee") sin (21 - I' ^g—g")
+ (he+\e'-\ee")sm(l'-g+^) +( - f e' + |eV) sin (l + g-g')
* Tables of the Development of Functions in the Theory of Elliptic Motion . Mem. Roy. Astr.
Soc, Vol. XXIX, p. 191.
NOTES ON THE THEORIES OF JUPITER AND SATURN 35 j
+ (i «' - I e'^- i eV) sin (I - 21' +g~g') ^ vr. ^a sin (3Z — Z' + ^ — /)
+ -I e' sin (Z + r - (/ + /) + f ee' sin (^r - ^') - f «e' sin {%i +g — g')
+ I ee' sin (2Z' — ^r + 5'') + I ««' sin (2? - 2Z' + ^ _ /)
4 f e'^ sin (; _ sr + - ^') + ^e" sin (Z + Z' + ^ _ ^')
+ l{ e' sin (4Z — I'+g- g') + ^1^ e' gj^ (g; + ^' _ ^ ^ ^.^
- 4 J eV sin (3/ + .^^ - /) - f^ eV sin (Z — ^ + «;')
+ i^ eV sin (3Z - 3Z' + (7— ^r') + ^1^ eV sin (Z + 2Z' - i? + g')
+ tV ce'^ sin {^I'—g + /) + ^a^. ee"" sin (3? - 3Z' + ^ - /)
- tV ee" sin (?' + gr - g') + ^s^ ee" sin (3Z + Z' + ^ — ')
+ ^e"sin(Z— 4Z' + g - g') + ^^ e" &in {I + 21' +g-g')
- yf sin {Zl—V + ^g—g') + ,'» sin (I + V + g + g')\
^ = (2ij — 2r,e^ + ^'^ ije*) sin (Z + ^) _ 1 ,' gin (3Z + S^') + (2ije - | ije') sin (21 + g)
— 2rie sin g + (^ ,e' - Y ,e') sin (3Z + fi') + Ci ^e' - 5V >?«*) sin G - g)
+ I -qe^ sin (4Z + ^) + ^ 'ye' sin {21 — ^) + e 25 ^g* gjn (5; + ^)
+ /j r,e' sin (3Z —5-) - )?'e ein (U + 35^) + yf'e sin (3Z + 3^)
+ ;< -^ I ij sin (3?- Z' + 25- -/) + (ij + 2,') sin (?' + g')
+ ^-qe sin (8Z — Z' + 2g -g') - f ije sin (Z — Z' + 2g -g')
— f ije' sin (2Z + 2g —g') + ^ );e' sin (2Z — 2Z' + 2g-g')
+ ^(v +2V)esin (I + I' + g') - ^(.v + 2^') e sin (I — l' — g')
+ i ('? + ^v') e' sin (3Z' +g')—B(y, + 2r,') e' sin g' \ .
As written, these expressions give the coordinates of Jupiter. Those
of Saturn are obtained by removing the accent from all the accented sym-
bols, and applying it to those which are unaccented, x excepted, for which
we have x' = x. Also it is to be remembered that we have A' = ^ + 180°.
The coordinates of the two planets are obtained by employing in these
formulas, for the quantities involved in them, the values they actually have
at the time in question. The latter are determined by the differential equa-
tions previously given ; but, instead of integrating these equations in one
step, we may, as Delaunay has done in the lunar theory, divide the process
into a series of transformations of the variables involved ; each of which
must be made not only in the expressions for log p , /I , (3 , log p' , /I' , ^' ,
but also in R .
As the introduction of I as the independent variable does not appear
to be advantageous, we will suppose that the six variables L , L' ,T, I, I' , y
are employed and that t is the independent variable.
34
362 COLLECTED MATHEMATICAL WORKS OF G. W. HILL
Delaunay's method, somewhat amplified, amounts to this: — selecting
the argument = i/ -j- i'V + *"/' suppose, for the moment, that B is lim-
ited to the terms
— B — A^ cos (t7 + i'V + i"Y) — ^j cos 2 {il + i'V + i"r) + . . . ,
where B , A^ . . . , are functions of i , Z' and T only. Then if it is found
that the differential equations, corresponding to this limited R, are
satisfied by the infinite series
0=0,{t + c) + dj sin [0„(^t + c)] + e^ sin 3[eo(^ + c)] + . . . ,
l-0) + /„ (i + c) + I, sin [»„ {t + c)] + I, sin 2 [So (i! + c)] + . . . ,
V = (V) + V„{t + c) + K sin [0„(t + c)] + n sin 2 [S„(if + c)] + . . . ,
;-=:=(;-) + ^„ (if + C) + ^ sin [Oo (I + C)] + J-, sln 2 [So (< + c)] + . . . ,
i = jLo + ii cos [0„ (J, + c)] + Zj cos 2 [So (< f c)] + . . . ,
i' = i; + A' cos [e^{t + c)] + X; COS 2 [So (< + c)] + . . . ,
r = To + A COS [So (/ + c)] + r, COS 2 [S„(^ + c)] + . . . ,
where c, (?) , (Z') and {y) are arbitrary constants, the last three being equiv-
alent to two independent constants, as we have the relation
i{l)^i'{l') + i"{r) = o,
and all the other coefficients are known functions of three other constants,
a, a' and e , we can replace
L hy L„ + A COS {il + i'V + i"r) + A cos 2 {il + i'V + i'V) + . . . ,
L' hy LI + L[ cos {il + i'V + i"r) + A cos 2 {il + i'V + i" y) + . . .,
r by r„ + r, cos {U + i'V + i"r) + r; cos 2 {U + w + i"r) + . . .,
I by I + I, sin {il + i'V + i"r) + h sin 2 {il + i'V + i"y) + . . . ,
V by V f l[ sin {il + i'V + V'f) + Z, sin 2 {il + i!V -I- i'V) -I- . . . ,
rbyr + n sin (i7 -1- i'? -1- i"r) + y, sin 2 {il + I'Z' + t"^) + . . .,
and will have, for determining the new variables, I , V , y , a, a' . e, pre-
cisely the same differential equations as we started with, provided we make
all these substitutions in the function R , and regard the new variables L ,
L' , r as connected with a , a', e by the relations
L ^L, + ^{o,L, + 20,L, +...),
L' = L'„ +l{0,Li ^20,L[ + ...),
r= n + ^(s, r, + 2s,z', + ...).
It will be perceived that, as long as we are dealing with terms of R,
whose arguments involve I or I' or both, the second members of the three
equations, last written, have values which differ from the elliptic values of L,
NOTES ON THE THEORIES OP JUPITER AND SATURN 3(53
L' and r only by quantities of the second order with respect to disturbing
forces. Hence, if we propose to neglect third order terms, until we have
reduced ^ to a function of the argument y only, we can assume that L, L'
and r which are the elements conjugate to the arguments I, T and y, are
expressed throughout in terms of a, a' and e, in the same way as in the ellip-
tic theory. It may be added that these third order terms are found in
experience to be much smaller than those which arise in other ways.
1