NRLF
4Dfl 3DE
ASTRONOMY
IIBBAIY
AN
INTRODUCTORY TREATISE
ON THE
LUNAE THEOEY
SonDon: C. J. CLAY AND SONS,
CAMBKIDGE UNIVEESITY PKESS WAEEHOUSE,
AVE MAKIA LANE.
263, ARGYLE STREET.
F. A. BROCKHAUS.
$efo gorfe: MACMILLAN AND CO.
Bombay: GEORGE BELL AND SONS.
AN
INTKODUCTOBY TEEATISE
ON THE
LUNAR THEORY
BY
ERNEST W. BROWN M.A.
PROFESSOR OF APPLIED MATHEMATICS IN HAVERFORD COLLEGE PA. U.S.A.
SOMETIME FELLOW OF CHRIST'S COLLEGE CAMBRIDGE
CAMBRIDGE
AT THE UNIVERSITY PRESS
1896
[All Rights reserved]
31
Astron. Dept,
Cambridge :
PRINTED BY J. & C. F. CLAY,
AT THE UNIVERSITY PRESS.
ft-
PEEFACE.
rriHE researches made during the last twenty years into the particular case
-*- of the Problem of Three Bodies, known as the Lunar Theory, have had
the effect of creating a wider interest in a subject which had been somewhat
neglected by the majority of mathematicians. Enquiry has been made, not
only into the value of the various methods from a practical point of view, but
also into questions which have an equal theoretical importance but which,
until just lately, have been almost entirely neglected. The existence of
integrals and of periodic solutions, and the representation of the solutions
by infinite series, may be cited as instances.
In order to understand the bearing of these investigations on the lunar
theory, some acquaintance with the older methods is desirable, if not necessary.
In the following pages, an attempt has been made to supply a want in this
direction, by giving the general principles underlying the methods of treat-
ment, together with an account of the manner in which they have been
applied in the theories of Laplace, de Pontecoulant, Hansen, Delaunay, and
in the new method with rectangular coordinates. The explanation of these
methods, and not the actual results obtained from them, having been my chief
aim, only those portions of the developments and expansions, required for the
fulfilment of this object, have been given.
The use of infinite series requires that investigations into their
convergency should take an important place in any treatise on the Lunar
Theory, and it is with regret that I have been obliged to leave it almost
entirely aside, owing to the lack of any certain knowledge on the subject.
The applications of the results to the formation of tables have shown that
the series are of practical use, but the right to represent the solutions by
VI PREFACE.
means of them has been discussed only for a few of the simpler cases, and
the radius of convergence, when the series are arranged according to powers
of any parameter, has been determined for elliptic motion only. It has also
been found necessary to omit many other theoretical investigations which, in
a more extended treatise, might have been included ; but it is hoped that
the references will cause the volume to be of service to those who desire
to proceed to the study of these higher branches, as well to those who wish
merely to obtain information concerning the older methods.
The difficulties of the subject are, perhaps, less inherent to it than due to
the manner in which it has been presented to a student approaching it for
the first time. The classical treatises are, almost invariably, original memoirs,
and, as such, do not contain the details which are essential for a clear
understanding of the scope and limitations of the problem in the form in
which it is usually considered. Moreover, the authors generally confine
themselves to their own methods, and the discovery of the relations which
exist between the various forms of expression for the same function, is often
troublesome. I have therefore given special attention to this point and
also to another, closely associated with it, namely, the definitions and
significations of the constants in disturbed motion.
A selection of one of the five methods, mentioned above, being necessary
as a basis for the elucidation of the properties common to all, I have had
no hesitation in adopting that of de Pontecoulant. Laplace takes the true
longitude, instead of the time, as the independent variable a method which
renders the interpretation of the results difficult, until the reversion of series
has been made while the theories of Hansen and Delaunay are not well
adapted to the end in view. De Pontecoulant's method of approximation
being similar to that of Laplace, it then seemed to be sufficient, for the
explanation of Laplace's treatment, to give the equations of motion, the first
approximation, and a brief account of the manner in which the second and
higher approximations are obtained.
In the Chapters on the methods of de Pontecoulant, Hansen and
Delaunay, I have made some alterations in the form of presentation and
the methods of proof, whenever these seemed to tend towards greater
simplicity ; where the differences from the original memoirs are important,
they are noted. In order to facilitate references to the latter, the original
notations are adhered to as far as possible, and this has necessitated the
PREFACE. VII
employment of three distinct notations. The tables given at the end of the
volume will show the Chapters in which they are severally employed, and will
enable the reader to find, without difficulty, the meaning of any frequently
recurring symbol.
The first four Chapters respectively contain investigations of the form
of the disturbing function, of the equations of motion, of the expansions
relating to elliptic motion, and of the methods adopted in order to approxi-
mate to the solution. The term 'intermediary' is used to signify any
orbit which may be adopted as a first approximation to the path actually
described a definition somewhat different from that given by Prof. Gylden.
In Chap. V. the equations for the variations of the elements in disturbed
motion are obtained in an elementary way and also by the more elegant
and symmetrical method of Jacobi. The properties and methods of de-
velopment of the disturbing function are collected in Chap. vi.
The details of the second and of parts of the third approximation to the
solution of de Ponte'coulant's equations will be found in Chap. vn. ; the
inequalities are divided into classes in order to show their origin more
clearly. Chap, vili., devoted to the arbitrary constants, is made intentionally
simple and explicit.
Chaps. IX. and x. contain the theories of Delaunay and Hansen,
respectively. A special effort has been made to free the methods of the
latter from the difficulties and obscurities which surround them in the
Fundamenta and the Darlegung. In Chap. XI. I have attempted to give
a complete method for the treatment of the solar inequalities in the Moon's
jnotion, based on that initiated by Dr Hill for those parts of them which
depend only on the ratios of the mean solar and lunar motions. The
infinite determinant, which arises in the calculation of the principal parts of
the mean motions of the perigee and the node, is considered at some length,
the conditions for its convergency and its development in series being
included. Chap. XII. contains an account, necessarily brief, of methods
other than those previously discussed.
In Chap. XIII. the inequalities arising from planetary action and from
the ellipticity of the Earth are considered. It_being impossible to give_an_
adequate account of these in the space at my disposal, they are treated by
Dr Hill's modification of Delaunay 's method only. An exception is made
TrT favour of the inequalities due to the motion of the ecliptic, and of
Vlll PKEFACE.
the inequality known as the secular acceleration, since the effects of these
appeared to be more simply explained by other methods.
The various memoirs and treatises of which I have made use are
referred to in the text. In particular, the excellent collection of methods,
contained in the first and third volumes of the Mdcanique Celeste of
M. F. Tisserand, has been frequently consulted.
I take this opportunity of acknowledging a deep debt of gratitude to
Professor G. H. Darwin and Dr E. W. Hobson, not only for their
valuable criticisms and suggestions made while reading the proof-sheets of
this work, but also for their advice and assistance rendered freely at all
times during the last eight years. My thanks are also due to Mr P. H.
Cowell, Fellow of Trinity College, Cambridge, for much help in the correc-
tion of all the proof-sheets and in the verification of the formulae and
results.
I may add that the cooperation of the officers of the University Press
has made it possible for me to see the printing almost completed during
my temporary residence in Cambridge, and thus to avoid the delays and
difficulties which would otherwise have arisen.
ERNEST W. BROWN.
HAVERFORD COLLEGE,
1895, December 13.
CONTENTS.
CHAPTER I.
FORCE-FUNCTIONS.
ARTS. PAGE
1. Units 1
2. Problem of three bodies 2
3, 4. (i) Forces relative to the Earth 2
5, 6. (ii) Forces on the Moon relative to the Earth, and those on the Sun
relative to the centre of mass of the Earth and Moon . . 4
7, 8. Force-function and Disturbing function usually used .... 6
9. Distinction between the lunar and the planetary theories ... 8
10. Force-function for p bodies 10
CHAPTER II.
THE EQUATIONS OF MOTION.
11. Methods of treatment .......... 12
12-15. (i) De Pontecoulant's equations 13
16, 17. (ii) Laplace's equations 17
18-21. (iii) Equations of motion referred to moving rectangular axes . . 19
22. Particular case : the solar parallax neglected .... 23
23. the solar parallax and eccentricity and the
lunar inclination neglected 24
24. The Jacobian integral 25
25-30. (iv) Equations in the general problem of three bodies. The ten
known integrals. The Invariable Plane. Special cases . . 25
CHAPTER III.
UNDISTURBED ELLIPTIC MOTION.
31. Method of procedure 29
32-47. (i) Formulae, expansions and theorems connected with the elliptic
curve . ... . 29
CONTENTS.
ARTS.
32.
33-36.
37-42,
43.
44-47.
48-53.
48-52.
53.
54.
PAGE
General formulae referring to an ellipse 29
Series connecting the radius vector, the true anomaly and
the mean anomaly 31
Similar expansions in terms of Bessel's Functions. . . 33
Theorem of Hansen ......... 36
Ellipse inclined to the plane of reference .... 38
(ii) Elliptic motion . 40
Undisturbed elliptic motion of the Moon .... 40
Sun 42
Convergence of elliptic series ......... 43
CHAPTER IV.
FORM OF SOLUTION. THE FIRST APPROXIMATION.
55. The two principal methods of approaching the solution . . . 44
56. Form to be given to the expressions for the coordinates ... 44
57-60. Intermediate orbits .......... 45
61. Solution by continued approximation . ... . . . 47
62. the method of the variation of arbitrary constants . . 47
63. The instantaneous ellipse . ';"'""'. . .> . . . . 48
64-66. Application of the method of solution by continued approximation to
de Pontecoulant's equations '. .../.. 49
67,68. Modification of the intermediary . . . . . . . . 51
69. Convergency of the series obtained for the coordinates ... 53
70. Modification of the intermediary for Laplace's equations ... 53
CHAPTER V.
VARIATION OF ARBITRARY CONSTANTS.
71. The two methods of development to be employed .... 54
72-92. (i) Elementary methods of treatment . .. ;>, ,. . , . ... _, . . 54
73, 74. Change of position due to changes in the elements . . 55
75. Expression of the derivatives of the disturbing function in
terms of the forces . . ..-;. . . . 56
76. Meanings to be attached to the symbols 8, 8 . . . . 58
77-82. Differential equations for the elements in terms of the
forces. Departure points . . . ;...,* . . 58
83. Differential equations for the elements in terms of the
derivatives of the disturbing function .... 63
84-86. Delaunay's canonical system of equations deduced ... 64
87-92. Observations on the previous results ..... 66
93-105. (ii) The methods of Jacobi and Lagrange . pf . . 67
94. The dynamical methods of Hamilton and Jacobi ... 68
95-97. Elliptic motion by Jacobi's method. f-v / . . . . 69
98. Variation of arbitrary constants by Jacobi's method . . 71
99, 100. Lagrange's method .... 73
CONTENTS.
XI
ARTS.
101, 102.
103.
104.
105.
Pseudo-elements and ideal coordinates
Lagrange's canonical system
Hansen's extension to the method of the variation of arbitrary
constants
References to memoirs and treatises
PAGE
74
76
76
77
CHAPTER VI.
THE DISTURBING FUNCTION.
106, 107. Development in powers of the ratio of the distances of the Sun and
the Moon 78
108-121. (i) Development of R necessary for de Pontecoulant's equations.
Properties of R . . . . , 79
108. Development in terms of polar coordinates .... 79
109, 110. the elliptic elements and the time 80
111, 112. Form of the development 81
113. Connection between arguments and coefficients ... 81
114. De Pontecoulant's expansion ....... 82
115, 116. Deduction of the disturbing forces ...... 83
117-120. Relations between the orders of the coefficients in the dis-
turbing function and those in the coordinates ... 84
121. The second approximation to the disturbing function and to
the forces 87
122, 123. (ii) Development for Delaunay's theory 88
124-126. (iii) Development for Hansen's theory 89
127. (iv) Development for Laplace's theory ....... 92
128. (v) Development for the method with moving rectangular coordinates 92
CHAPTER VII
DE PONT^COULANT'S METHOD.
129. Summary of the previous results required ...... 93
130-133. Preparation of the equations of motion. Order of procedure . . 93
134-138. (i) Variational inequalities 96
134-136. Second approximation 96
137. Third approximation 98
138. Results - 99
139-141. (ii) Elliptic inequalities. Motion of the perigee 100
139, 140. Second approximation 100
141. Third approximation to the motion of the perigee . . 102
142. (iii) Mean-period inequalities 103
143, 144. (iv) ParaUactic inequalities . .104
143. Second approximation 104
144. Third approximation and results 105
145-147. (v) Principal inequalities in latitude. Motion of the node ... 106
148. (vi) Inequalities of higher orders 109
Xll
CONTENTS.
ARTS.
149.
150.
151-153.
154.
Summary of results
Direct deduction of the terms in the disturbing function required for
special cases .......
De Pontecoulant's Systeme du Monde
Slow convergence of the series for the coefficients
PAGE
110
111
112
113
CHAPTER VIII.
THE CONSTANTS AND THEIR INTERPRETATION.
155. The questions to be considered 115
156-161. Signification of the constants present in the final expressions for the
coordinates . . . . . . . . . . .116
162-164. Determination of the numerical values of the constants by observa-
tion. The values of the solar constants . . . . . 121
165. Mean Period and Mean Distance , .; . . v . . , . . . 124
166. The Variation, variational inequalities and the variational curve ' . 124
167. The Parallactic Inequality, parallactic inequalities and the parallactic
curve ............ 125
168. Methods for the determination of the ratio of the masses of the Earth
and the Moon .127
169. The Principal Elliptic Term, elliptic inequalities, the Evection and
the motion of the perigee . . . . . . . .127
170. Kepresentation by means of variable arbitraries . .. , 128
171. The Annual Equation and mean-period inequalities . . . .129
172. Inequalities in latitude and the motion of the node '. , . . 130
173. Magnitudes of the principal inequalities. References to memoirs in
which the numerical values of the constants are obtained . . 131
174. Motions of the perigee and the node when the true longitude is the
independent variable ,.-?*' 131
CHAPTER IX.
THE THEORY OF DELAUNAY.
175. Method used. Limitations imposed on the problem . . . ... 133
176. Defect in the canonical equations previously obtained, due to the
presence of the time as a factor when the equations are in-
tegrated ij. . .. 134
177. Method used for transforming to a new set of variables . . . 134
178. Transformation to avoid the occurrence of terms containing the time
as a factor 135
179. Change of notation. Signification of the symbols . . . .136
180. Form of the development of the disturbing function. Relations be-
tween the two sets of elements used to represent the coefficients. 137
181. Elliptic expressions for the coordinates in Delaunay's notation . . 138
182. The method of integration 139
CONTENTS. xiii
ARTS. PAGE
183, 184. Integration when the disturbing function is limited to one periodic
term and to its non-periodic portion. The new constants of
integration 139
185. The omitted portion of R is included by considering the new con-
stants variable. The resulting equations are canonical . . 142
186. Nature of the solution obtained in Arts. 183, 184 . . . . 144
187. Form of the new disturbing function ....... 145
188. Lemma necessary for the next transformation 146
189. First transformation of the new equations to new variables, in order
to avoid the occurrence of terms containing the time as a factor. 147
190. Second transformation to new variables, so that, when the coefficient
of the periodic term previously considered is neglected, the new
equations shall reduce to the old ones 148
191. Kelations connecting the old and new variables, and the old and new
disturbing functions 149
192, 193. Application to the calculation of an operation 150
194-196. Particular classes of operations 153
197. Delaunay's general plan of procedure 155
198. Integration when the disturbing function is reduced to a non-periodic
term 156
199, 200. Final expressions for the coordinates. Change of arbitraries. The
meanings to be attached to the new arbitraries . . . 157
201. The results obtained by Delaunay .... ... 159
CHAPTER X.
THE METHOD OF HANSEN.
202, 203. Features of the method. History of its development . . . . 160
204. Change of notation 161
205, 206. The instantaneous ellipse. The equations for the functions of the
instantaneous elements required later 162
207. Keasons for the use of the method 164
208, 209. The auxiliary ellipse. Its relation to the actual orbit . . . .164
210. Method of procedure 166
211-213. The equations for 2, v 166
214, 215. The equation for W. Certain parts of the expression may be con-
sidered constant in the differentiations and integrations . . 169
216. The constants arising in the integrations 171
217. Motion of the plane of the orbit. Definitions. Mean motions of the
perigee and the node 172
218, 219. Equations for P, $, K in terms of the force perpendicular to the
plane of the orbit. First approximation to P, Q, K . . . 174
220, 221. Form of the development of the disturbing function . . . .177
222, 223. Expression of the derivatives of the disturbing function with respect
to P, Q, K, in terms of the force perpendicular to the plane of
the orbit 179
224. First approximation to the disturbing function and to the disturbing
forces. 181
XIV
CONTENTS.
ARTS.
225.
226-228.
229, 230.
231, 232.
233, 234.
235.
236.
237, 238.
239, 240.
241.
Expression of the first approximation to the disturbing forces in the
plane of the orbit, in terms of the derivatives of the disturbing
function in its developed form ........
First approximation to the equation for W. Method for the calcula-
tion of the equation ..........
Integration of the equation for W. Determination of y and of the
form of the arbitrary constant. First approximation to W .
Integration of the equation for z. Signification to be attached to the
new arbitraries and to the elements of the auxiliary ellipse .
Integration of the equation for v. Determination of the arbitrary
constant in terms of the other arbitraries . . . - .
Equations for P, $, K ..........
Effect of the motion of the ecliptic .......
The first and second approximations to P, Q, K. Determination of
a, 77 and of the new arbitraries .......
Method of procedure for the higher approximations ....
Reduction to the instantaneous ecliptic. . .- J . ; . . *
PAGE
182
182
185
187
188
190
191
191
192
194
CHAPTER XL
METHOD WITH RECTANGULAR COORDINATES.
242. General remarks on the method. Limitations imposed . . . 195
243. History of its introduction 196
244-246. The intermediate orbit. Equations for finding it 196
247-256. (i) Determination of the intermediary, or of the inequalities depend-
ing on m only 198
247, 248. Form of solution required. ! . .: . . '''.' . . . 198
249-252. Equations of condition for the coefficients ,V . ' . . 200
253, 254. Method of finding the coefficients from the equations of con-
dition. The parameter in terms of which the series
converge most quickly . . . ' ; . . . 203
255. Determination of the linear constant . > . . . . 205
256. Transformation to polar coordinates = '>; . ' '^ . 206
257-274. (ii) Determination of the terms whose coefficients depend only on m, e.
Complete solution of the equations (1). Motion of the perigee . 206
257, 258. Form of the general solution of equations (1). Equations of
condition between the coefficients . - * *;. . . 206
259-261. Coefficients depending on m and on the first power of e.
Equations of condition. Signification to be attached to
the new arbitrary constant of eccentricity . .. * . 209
262-264. Determination of the principal part of the motion of the
perigee. Equation for the normal displacement . . 211
265. Calculation of the known quantities present in this equation 215
266, 267. Determinantal equation for c. Properties of the infinite
determinant. Deduction of a simplified equation for c . 216
268. Convergency of an infinite determinant 219
269, 270. Development of an infinite determinant .... 220
CONTENTS.
XV
ARTS. PAGE
271, 272. Application to the determinant A(0) 222
273, 274. Determination of the coefficients depending on powers of e
higher than the first. New part of the motion of the
perigee. Further definitions of the linear constant and
of the constant of eccentricity 223
275, 276. (iii) Determination of the terms whose coefficients depend only on
m, e' 225
277. (iv) Determination of the terms whose coefficients depend only on
m, I/a' 227
278-286. (v) Determination of the terms whose coefficients depend only on
m,y 228
278. The equations of motion ........ 228
279, 280. Terms dependent on the first power of y only. Principal
part of the motion of the node 229
281. Signification to be attached to the constant of latitude . . 231
282. Terms of order y 2 231
283-285. Terms of order y 3 . Determination of the new part of the
motion of the node 231
286. Further definition of the constant of latitude .... 233
287. (vi) Terms of higher orders 234
288, 289. Connections of the higher parts of the motions of the perigee and the
node with the non-periodic part of the parallax. Adams' theorems . 234
CHAPTER XII.
THE PRINCIPAL METHODS.
290. Newton's works 237
291. Clairaut's theory 238
292. D'Alembert's theory 239
293. Euler's first theory 239
294. Euler's second theory 240
295. Laplace's theory 242
296. Discoveries of Laplace. The secular acceleration 243
297. Damoiseau's theory ........... 244
298. Plana's theory 244
299. Poisson's method 245
300. Lubbock's method. Other theories. Airy's method of verification . 245
301. References to tables of the Moon's motion 246
302. Remarks on the various methods . 246
CHAPTER XIII.
PLANETARY AND OTHER DISTURBING INFLUENCES.
303. The method to be adopted
304-307. General method of integration founded on Delaunay's theory
308-313. Method for obtaining the planetary inequalities .
248
248
252
XVI CONTENTS.
ARTS. PAGE
308. The disturbing functions 252
309. Separation of the terms in the disturbing functions. Their
expressions in polar coordinates ..... 253
310, 311. Development of the disturbing functions. Connections be-
tween the arguments and the coefficients . . . 255
312. Example of an inequality due to the direct action of Venus. 258
313. Case of the indirect action of a planet. Example of an in-
equality due to the indirect action of Venus . . . 259
314-316. Inequalities arising from the figure of the Earth. Determination of
the principal inequalities ......... 260
317, 318. Inequalities arising from the motion of the Ecliptic. Determination
of the principal inequality in latitude . 263
319-322. Secular acceleration of the Moon's mean motion. Determination of
the first approximation to its value. Effect of the variation of
the solar eccentricity on the motions of the perigee and the node 265
I. KEFERENCE TABLE OF NOTATION .' '.', ; V' .... 270
II. GENERAL SCHEME OF NOTATION ,;,.,.,. ^72
III. COMPARATIVE TABLE OF NOTATION '.:;*' * . .. .:. . .- 273
INDEX OF AUTHORS QUOTED . . . '. '" . V '. . . 274
GENERAL INDEX 275
ERRATA.
Page 5. Head-line. For MOON'S read SUN'S.
60. Line 1. a a 2 .
108. 35. 4ffm*- +ffm + .
109. 8. -ffn*+ -*fm 4 -.
CHAPTER I.
FORCE-FU NOTIONS.
1. THE Newtonian law of attraction states that either of two particles
exerts on the other a force which is proportional directly to the product of
their masses and inversely to the square of the distance between them. Let
g be the force between two particles of masses m, m' placed at a distance r
from one another, then
8 = *~,
where & is a constant depending only on the units employed It is known
as the Gaussian constant of attraction.
If we use any terrestrial unit of mass, k will vary directly as the unit of
mass and the square of the unit of time and inversely as the cube of the
unit of length.
The accelerative effect of the force exerted by m on m' is
The Astronomical unit of mass corresponding to given units of length and
time, is obtained by so choosing this unit that k=l. Since 51 is an accelera-
tion, when the units of mass, length and time vary, the astronomical unit of
mass varies directly as the cube of the unit of length and inversely as the
square of the unit of time. It is used very largely in theoretical investiga-
tions in astronomy and the frequent repetition of the constant k is thereby
avoided.
For further remarks on this unit see E. J. Routh, Analytical Statics, Vol. II. pp. 1, 2, 3.
For the sake of brevity the term ' force ' is often used to indicate ' accele-
rative effect of a force.' In general, no confusion will arise in this use of the
word. With an astronomical unit of mass, an acceleration may appear either
as a mass divided by the square of a length or, in its usual form, as a length
divided by the square of a time.
B. L. T. 1
2 FORCE-FUNCTIONS. [CHAP. I
2. The general problem of Celestial Mechanics consists in the determina-
tion of the relative motions of p bodies attracting one another according to
the Newtonian law. This problem is not able to be solved directly : in order
to deal with it, certain limitations must be made.
The first simplification which we shall introduce, is made by eliminating
from consideration the effects of the size, shape and internal distribution of
mass of the bodies. A well-known theorem in attractions states, 'that a
spherical body, of which the density is the same at the same distance from
its centre, attracts a similarly constituted spherical body as if the mass of
each were condensed into a particle situated at its centre of figure.' The
larger bodies of our solar system differ but little from spheres in shape
and their centres of mass are certainly but little distant from their centres
of figure if at all ; the shapes of very small bodies when under the attraction
of a large one play little part in their motions. We assume therefore that
the bodies are spherical and attract one another in the same way as particles
of equal mass situated at their centres of figure. With these limitations
the problem is known as that of p bodies.
Again, owing to the conditions under which the bodies of our solar system
move, we are further able to divide the problem of p bodies into several others,
each of which may be treated as a case of the problem of three particles, or, as
it is generally called, the Problem of Three Bodies.
The greater part of the Lunar Theory is a particular case of the Problem
of Three Bodies ; it involves the determination of the motion of the Moon
relative to the Earth, when the mutual attractions of the Earth, Moon and
Sun, considered as particles, are the only forces under consideration. When
this has been found, the effects produced by the actions of the planets, the
non-spherical forms of the bodies etc., can be exhibited as small corrections to
the coordinates.
3. We proceed to consider the impressed forces in the problem of three
particles. There are two methods of treating them, from the combination
of which a suitable form of force-function can be obtained. In the first
method, we find the accelerations due to the forces acting on the Sun and
Moon relatively to the Earth, and in the second method, those acting on the
Moon relatively to the Earth and on the Sun relatively to the centre of
mass of the Earth and Moon.
(i) The forces relative to the Earth.
In figure 1, let E t M, ra', G respectively denote the places of the Earth,
Moon, Sun and the centre of mass of the Earth and Moon.
Let the masses of the Earth, Moon and Sun measured in astronomical
2-3]
FORCES RELATIVE TO THE EARTH.
J/,
m'
E m/_ E
r>* Fig. 1. ^2-
units be E, M, m'. Let the distances, ME, m'E, m'M be r, r', A : these must
always be considered as positive quantities.
We shall only deal here with the accelerations due to the forces and, in
accordance with the remark made in Art. 1, speak of them as forces.
The forces acting on E are M/r* in the direction EM and ra'/r' 2 in the
direction Em'. Similar expressions hold for the forces acting on M and m', as
shown in the figure. Hence the forces acting, relatively to the Earth, are,
On the Moon,
i n the direction of ME,
On the Sun,
E+m' . f ,n
^ m the direction of mE,
in'
A*
Mm',
m'E-,
_
A 2
M
m'M,
ME.
Take the Earth as the origin of a set of rectangular axes. Let x, y, 2, and
x f , y', /, be the coordinates of M and ra', and , 2), 3, ', ?)', 3', the forces
acting on M and m' relatively to E, referred to these axes. We have then, .
v__ E+M x_mf_ x-x' _m_x^
r 2 r A 2 A r' z r' '
~, _ E + m' x' M x x MX
7^2 r' ~ A 2 A ~~ r 2 r '
with corresponding expressions for g), 3, 3)', 3'-
If we put
,, E + M , ra' , oaf -I- yy + zz
M
we shall have
The expressions denoted by F, F' are the force-functions for the motions
of the Moon and the Sun relative to the Earth.
12
4 FORCE-FUNCTIONS. [CHAP. I
Let the cosine of the angle MEm' be denoted by 8. Then
and therefore F, F' are independent of the particular set of axes fixed or
moving which may be used.
The expressions for F, F' are quite general, and when they shall have been
substituted in the six differential equations which represent the motion of M,
m' relative to E, we shall have the general equations of motion for the problem
of three particles. (It has been tacitly assumed that the system is free from
any external influence having a tendency to disturb the relative motion
of the three particles.) It is not possible to integrate these equations
except in special cases. In order to obtain F in the most suitable form,
certain observations must be made before proceeding further.
4. In finding the motion of the Moon, the first assumption usually made
consists in considering the motion of the Earth about the Sun or, what is the
same thing, of the Sun about the Earth as previously known and therefore
the coordinates x ', ?/, z' as known functions of the time. Now the function
F' which is used to find the motion of the Sun contains the unknown
coordinates of the Moon. It becomes necessary to see what effect those terms
in F' which have M as a factor produce in the relative motion of the Sun.
If we limit F' to its first term (E + m')/r, the resulting motion of m'
about E will be elliptic. Now in the lunar theory, r is very small compared
to r' or A while r f and A are of nearly equal magnitude. Hence it appears
that the third term in F' is that which will cause most deviation from elliptic
motion. This term is due to the force M/r\ which is one of the forces acting
on E. If we had referred the motion of m to G this term would not have
entered. We shall therefore find the motion of m' relative to G and discuss
the other terms in the new force-functions for m' and M.
(ii) The forces acting on m' relatively to G and on M relatively to E.
5. We have EG = -^-^EM,
/ + M
and therefore the accelerations of G relative to E are M/(E + M) times those
of M relative to E. Hence the force on m' relative to G, parallel to the
axis of #, is 3' - $M/(E + M) which, by Art. 3, is equal to
E+M+m'/E x' Mx' x\
E + M~ V 2 r' + A 2 ~A~J '
3-5] MOOK^S FORCE-FUNCTION RELATIVE TO G. 5
Let x l ,y l ,z l , be the coordinates of in referred to parallel axes through G:
let m'G = r 1 . Then
M M M
E
etbre x x = x l ^
Also
and therefore x x = x l ~ ^ x, etc.
/. M \*
= * 1
E
If therefore we put
ff J/m'/g
J
where r', A are now expressed in terms of x } y, z, x lt y ly z l} the differentials
'd~ L ' JT^" ' 3 l w *^ ^ e ^ e f rces acting on m' relatively to G.
Again replacing in Art. 3, #', y', z' by their values above, we have
E+Mx w'/ E \_S( M
The forces $, g), 3 can therefore be derived, by partial differentiation with
respect to x, y, z, from the force-function
E + M m' E+M m'E+M
l ~ ~i^~ *" A E *" / M '
where A, r are expressed as before in terms of x, y, z, as lt y lt z t .
It is not difficult to show by expanding FI in powers of r/r 1} that it
differs little from (E -f M + m)/i\ . We have, putting cos MGm' = 8 l ,
Jjj. n
1 1 M r / M V r 2
_
whence -, = - - ^-^ - S, +
Similarly
6 FORCE-FUNCTIONS. [CHAP. I
and therefore
m' + E + MV E.M ,* 1
^ ~ ~^~ + (E + M)*r^ ( ^ 1 "_]
Now rji\ differs little from ^ at any time and M/E is approximately
gL. Hence the order of the second term of FI relatively to the first is
roughly
80 400 2 ~ 12,800,000'
a quantity which may be neglected here. We can then neglect the second and
higher terms of FI and consequently assume that the motion of m about G is
an ellipse.
6. The Moon thus produces little effect on the motion, of the centre of mass of the
Earth and Moon and consequently we can consider this point as moving in an ellipse in
accordance with Kepler's laws. The actions of the planets, however, produce effects which
become very marked after long periods of time. These effects, being exhibited by terms
in the expressions of x^ y lt z lt are transmitted to the Moon through F. We ought
properly to have considered the problem of p bodies or even less generally, to have
included in the force-functions F. F' the forces produced by all the planets. But the
action, both direct and indirect, of the other planets on the Moon is so small that in most
cases it may be neglected : where it should be considered, it is always possible to do so in
the form of small corrections. The variations produced in the motion of the Ea,rth or of G
by the other planets belong properly to the planetary theory and need not be considered
here. All we require here concerning the motion of the Earth is, that it should be
supposed known. The only reason for considering its force-function at all, is to see how
the unknown coordinates of the Moon enter therein and to eliminate them as far as
possible.
It is a remarkable fact, and one which shows the extraordinary care required in the
treatment of the lunar and the planetary theories, that the direct actions of the planets on
the Moon are in general much less marked than their indirect actions as transmitted through
the Earth. These indirect effects, though sometimes too small to be detected in the
motion of the Earth, may become sufficiently large to be observed in the motion of the
Moon.
7. With the assumption that the motion of G is elliptic, we ought
properly to use the force-function F^ for finding the motion of the Moon.
But as it is generally found more convenient to use F, we shall expand both
functions in order to see in what way the latter must be corrected to give
the former.
The expansions of I// and I/A given in Art. 5, furnish
1 1 1 r 1 1 1 r 3
+ +
5-7] CORRECTIONS TO BE MADE TO F.
When this is substituted in F lf the term
will contribute nothing to the forces since it is independent of the coordinates
of the Moon : it may therefore be omitted. We shall then have
Again, we find by expanding F in powers of r/r' and omitting the useless
term m'/r',
Now x, y', z' ', r' refer to the motion of the Sun about the Earth, and
x i, y\> ZD TI to that about G, and these are contained in F, F l respectively in
the same way. If therefore in F we consider x', y', z, r' to refer to the motion
about G, which motion is supposed known in terms of the time, F, F will be
the same except as regards the ratio M/E.
Let now a, a' be the mean values of r, r x (or /). It will be found later
that expansions will be made in powers of a/a' and that the parts resulting
from the term in F,
do not contain a/a'. The parts resulting from the term in F,
r ' r '3
contain a/a in its first power. Comparing F, F! we see that if in the results
produced by using F l} in addition to the change noted above, we replace the
ratio a/a wherever it occurs by
E-M a
E + Ma"
we shall be able to use F instead of F lf The terms containing (a/a') 2 we
should multiply by (E - M)~l(E+Mf, and so on.
This does not quite account for all the differences between F, F^ The
terms of next highest order in F, F are respectively
and n
8 FORCE-FUNCTIONS. [CHAP. 1
The first expression must be multiplied by
Hence, after the changes previously noted, there will still remain to be added
to F, the term
Now the order of this remainder, compared to the first of those terms
in F which depend on the action of the Sun, is
Ma* 1
E a' 2 ~ 12,800,000
approximately. The largest periodic term produced by the Sun has a
coefficient in longitude of less than 5000", so that it is very improbable
that such a small quantity can produce any appreciable effect. The effect of
the differences in the higher terms will be still smaller. We may therefore
conclude that the replacing of a/a' by a (E - M)jaf (E -f- M) will sufficiently
account for the remaining differences.
8. The problem of the Moon's motion is therefore reduced to the deter-
mination of the motion of a particle of mass M, under the action of a
true force-function MF, where
in which #', y' , z' are the known functions of the time obtained from purely
elliptic motion. We shall now consider E instead of G as the origin.
Let E + M = p, and put F= + R.
The quantity R depends on the action of the Sun and is known as the
Disturbing Function.
There is now no further need to consider the functions F l} F', F^.
9. The distinction between the planetary and the lunar theories is one of analysis only,
based on certain facts deduced from observations on the nature of the motion of the bodies
forming the solar system. Both theories are particular cases of the problem of three
bodies which, owing to the deficiencies of our methods of analysis, is at present only
capable of being solved by tedious expansions, even when the bodies are so favourably
placed relatively to one another as those which come within the range of observation.
Almost nothing is at present known of the possible curves which bodies of any masses and
placed at any distances from one another may describe. In the case of the planetary
theory, where it is required to investigate the perturbations of one planet by another, or
more properly, the mutual perturbations of two planets, we can use the functions F lt F^
7-9] THE LUNAR AND THE PLANETARY THEORIES. 9
where E stands for the mass of the Sun and J/, m' for the masses of the planets. For in
this case the ratios (E-N}I(E+M), (Em')l(E+m!) are so near unity owing to the great
mass of the Sun compared to that of any of the planets and the actual perturbations are
so small, that the differences of these ratios from unity can in general be neglected.
Rough observations extending over a sufficient interval of time show very quickly that,
during that interval at least, the planets describe curves which approximate more or less
closely to circles of which the Sun occupies the centres. A more exact representation of
their motion is given by Kepler's well-known laws. The known satellites, and in
particular the Moon, also approximately satisfy these laws with reference to the planets
about which they move, but for a shorter time; they also exhibit larger deviations from
them. Observation too has shown that the eccentricities of their ellipses and the mutual
inclinations of the planes of motion of all the principal planets oscillate about mean values
which are in no case very great. The same is true of the Moon and of most of the
satellites with reference to the orbits of their primaries.
It is assumed then that expansions may be made in powers of the eccentricities and of
suitable functions of the inclinations. But when we are considering the perturbations of
one body produced by another, it has just been seen that expansion will also be made
in powers of the ratio of the distances of the disturbed and disturbing body from the
primary.
It is at this point that the first separation of the lunar and the planetary theories takes
place. In the lunar theory, the distance from the primary the Earth of the disturbing
body the Sun is very great compared to that of the disturbed body the Moon, and
we naturally expand first in powers of this ratio in order that we may start with as few
terms as possible. In the planetary theory, the distances of the disturbed and the disturbing
bodies two planets from the Sun which is the primary, may be a large fraction. For
example the mean distances of Venus and the Earth from the Sun are approximately in the
ratio 7 : 10, and in order to secure sufficient accuracy when expanding in powers of this
ratio, a very large number of terms would have to be taken. In the case of the planetary
theory then, we delay expansion in powers of the ratio of the distances as long as possible
and form the series first in powers of the eccentricities and inclinations.
Again, in the lunar theory the mass of the disturbing body is very great compared to
that of the primary, a ratio on which it is evident the magnitude of the perturbations
greatly depends. On the other hand, in the planetary theory the disturbing body has a
very small mass compared to that of the primary. From these facts we are led to expect
that large terms will be present in the expressions for the motion of the Moon due to the
action of the Sun and, from the remarks made above, that the later terms in the expansions
will decrease rapidly ; and in the planetary theory we expect large numbers of terms of
nearly the same magnitude, none of them being very great. This expectation however is
largely modified by some further remarks about to be made.
In the integrations performed in both theories the coefficients of the periodic terms by
means of which the coordinates are expressed, become frequently much greater than might
have been expected a priori. In the lunar theory, before this can happen in such a way
as to cause much trouble, the coefficients have previously become so small that it is not
necessary to consider such terms beyond a certain limit. Suppose in the planetary theory
that n, ri are the mean motions of the two planets round the primary. Then coefficients
will, for example, continually be having multipliers of the form ri/(ini'ri) and n" 2 !(ini'n') 2
produced by integration (i, i' positive integers). In general, the greater z, i' are, the
smaller will be the corresponding coefficient. But owing to the two facts that the ratio
10 FORCE-FUNCTIONS. [CHAP. I
a : a' may be nearly unity and that the ratio n : ri may very nearly approach that of two
small whole numbers, a coefficient may become very great. For example, five revolutions of
Jupiter are very nearly equal in time to two of Saturn, while the ratio of their mean
distances is roughly 6:11. One result is a periodic inequality which has a coefficient of
28' in the motion of Jupiter and 48' in that of Saturn. Such inequalities take a long
period to run through all their values, the one in question having a periodic time equal to
about 76 revolutions of Jupiter or 913 years, so that the variation due to this term in one
revolution is small. The periods of the principal terms in the motion of the Moon
are generally short but some of them have large coefficients, so that the deviations from
elliptic motion are well marked.
One of the greatest difficulties in the planetary theory, perhaps owing chiefly to our
methods of expression, is the presence, in the values of the coordinates (when the latter
are obtained as functions of the time), of terms which increase continually with the time,
and thus, after the lapse of a certain interval, render the expressions for the coordinates
useless as a representation of the motion. Whether such terms can be eliminated by the
use of suitable functions is not at present certain. Recently the work of Gylden* has
gone far in the direction of achieving this object. In the lunar theory, the difficulty also
occurs, but, as regards the perturbations produced by the Sun, is easily bridged by
means of a slight artifice.
It will readily be conceived, from the few statements made here, that in general,
different methods will be required for treating the two problems of a satellite disturbed by
the Sun and of a planet disturbed by another planet. When the disturbing planet is an
inferior one, we use a function corresponding to F t but we have then to develope in powers
of a'/a instead of a/a'. In the cases of some minor planets again special methods are
required owing to the great eccentricity of their orbits. All the problems are essentially
the same : the analytical difficulties alone compel us to treat them differently.
As concerns questions of purely mathematical interest, the planetary theory has in the
past opened out a larger field for the investigator than the lunar theory. In the last few
years however the researches of Hillt, Adams t, PoincareJ and others have brought the
latter problem forward again and given it a new stimulus.
10. Let (ffiyi-O, (#2^), (#8^8*3) be the coordinates, referred to any
origin and any rectangular axes, of three bodies of masses m ly ra 2 , m 3 which
attract one another according to the Newtonian law. Let r la , r 23 , r 13 be the
distances between them. The force acting on m i} resolved parallel to the axis
of x is
r
13
the forces acting on m^ are therefore derivable from the function
* Acta Mathematica, Vols. vn., XL, xv. etc., also Traite analijtiquc des Orbites absolues, etc.
Stockholm, 1893.
t See Chapter xi.
t Les Methodes nouvelles de la Mecanique Celeste, Paris, Vol. i., 1892, Vol. 11., 1893. These
researches are outside the scope of this book.
9-10] FORCE-FUNCTION FOR p BODIES. 11
or, since 7*23 does not contain the coordinates x^,y lt z ly from the function
The symmetry of this expression shows that it is also the force-function for
the motions of m. 2 and m 3 .
Generally, if there be p bodies, it is easily seen that
where i,j receive values 1, 2...j;, the terms for which ij being excluded.
CHAPTER II.
THE EQUATIONS OF MOTION.
11. THE methods used in the solution of the lunar problem may be
roughly divided into four classes. In the first class we may place those
methods in which the time is taken as the independent variable, the radius
vector (or its inverse), 'the true longitude measured on a fixed plane, and
the tangent of the latitude above this plane, as dependent variables; the
equations of motion are expressed in terms of these quantities and solved by
continued approximation with elliptic motion as a basis, so as to exhibit
these coordinates as functions of the time and the arbitrary constants intro-
duced by integration. Under this heading we may include the theories of
Lubbock and de Pontecoulant. In the methods of the second class four
similar variables are used, but the true longitude is taken to be the
independent variable and the other three variables are expressed in terms
of it. A reversion of series is finally necessary in order to obtain the
coordinates in terms of the time. Clairaut, dAlembert, Laplace, Damoiseau
and Plana followed this plan.
A third class will embrace those methods in which the Moon is supposed to
be moving at any time in an ellipse of variable size, shape and position ; this
is known as the method of the Variation of Arbitrary Constants and it was
used in different ways by Euler in his first theory, by Poisson and with great
success by Hansen and Delaunay. In the fourth class may be placed those
theories in which rectangular coordinates referred to moving axes are used,
with the time as independent variable. Euler's second theory and the general
methods resulting from the works of Hill and Adams may be included under
this heading.
We shall give here the equations of motion used by de Pontecoulant
and Laplace and the generalised form of Hill's equations, to illustrate the
methods of the first, second and fourth classes respectively: the principles
which form the basis of the methods of the third class will be given in
Chapter v. We shall also include here some considerations on the general
problem of three bodies with special reference to the known integrals.
11-12]
DE PONT^COULANT'S EQUATIONS.
13
The methods used by de Pontecoulant, Delaunay and Hansen will be found in
Chapters VIL, ix. and x. respectively; the methods of Hill and Adams with the
extensions to the complete problem are given in Chapter xi. A short summary of the
methods employed by other lunar theorists is made in Chapter xn.
(i) De Pontecoulant' s equations of motion.
12. Let x, y, z be the coordinates of the Moon referred to three
rectangular axes, fixed in direction and passing through the centre of the
Earth. The equations of motion of the Moon will be, according to Newton's
laws of motion, since = , -=- , =- are, by Art. 8, the forces parallel to the
ox oy oz
axes, v /v/
Mx =
or by the same article,
dF
where R is supposed expressed in terms of x t y, z, x', y', z'.
Let the plane of (xy) be the plane of the Sun's motion, supposed fixed,
and let the axis of x be a fixed line in this plane. Let x, y, z, M, M' be the
Fig. 2.
points where a sphere of unit radius cuts the axes, the radius vector and its
projection on the (xy) plane, respectively. Let
r* be the radius vector of the Moon,
T! its projection,
v the longitude of this projection reckoned from the axis of x, and
s the tangent of the latitude of the Moon above the plane of (xy).
14
Hence
and therefore
rcosv
THE EQUATIONS OF MOTION.
v = ocM', s = tan M'M,
[CHAP. II
rsnv
=
rs r
~~ ~
If we change the variables from x, y, z to r l} v, z, the equations of motion
become
ar, , dR \
dR
* >
dz
(i),
where i-^ I denotes partial differentiation of R when R is expressed as a
\dvj
7) 7?
function of n, v, z, in contradistinction to ^ which will denote partial differ-
entiation when R is expressed as a function of r. v, s; the two expressions
are easily seen to be equal though differently expressed.
Multiplying these equations by 2r 1? 2^4), 2z, respectively, and adding, we
obtain
where
/T1 - 7 , -, . - 7
Jt =.^- ar x H - r x dv + -5- w.
'
Whence, if a be a constant, we have on integration
(2),
the expression for the square of the velocity.
The integral on the right-hand side of this equation requires explanation.
From the way in which d'R was formed, it is evident that d'R/dt denotes
differentiation of R with respect to t, only in so far as t is present through the
variability of the coordinates r it v, z and not through its presence explicitly in
the coordinates of the Sun. We suppose then that d'R/dt has been formed
in this way and we can put
an equation which defines the meaning of fd'R.
In order that the integration may be actually performed, d'R/dt ought to be expressed
in general as a function of the time only. It will, however, contain the unknown coordinates
r lt v, z, and its value can only be obtained by a process of gradual approximation to the
values of these coordinates.
12-13] DE PONT COUL ANT'S EQUATIONS. 15
Multiplying the first of equations (1) by r lt the third by z and adding the
sum to equation (2), we obtain
But since we have r? + z CL = r i and therefore r-i'r-i + r? + zz + 2? =
the equation becomes
Also from the second of equations (1), if h be a constant, we obtain
whence
Finally, from the third of equations (1) we have, substituting the value of
z in terms of r, s,
d? l rs \ fju rs _dR ,_,
~*
10 w . ,
13. We must now express ^ , I = , ^ in terms of -= , - , ^- where,
9^ \9^ / 3* or ov os
in the former three R is considered a function of r^, v, z, and in the latter
three a function of r, v, s. Let Sr f &;, z be any independent variations
of r-j, v, 2 and 3r, &;, 3,s, SJ? the corresponding variations of r, v, s, R.
We have
dR ^ fvR\ * dR * *T* ^R * dR <?. . dR ^
^-Sr l + (-^)Bv+ ^ 8z = SR = ^- Sr-{- a- Si; + -- 65.
9^ \dv ) dz dr dv ds
Also, since r 2 = r x 2 + 2r 2 , 5 = ^/r^, and therefore
r&r = r a Srj + zSz, 8s = SZ/T! zbrjrf,
we obtain by substituting for Sr, 8s in the previous equation and equating the
corresponding coefficients of the independent variations 8r l} 8v, 8z,
dR = ^dR_z^dR (dR\ = dR ^ = i^,i^
8^ " r dr n 2 8s ' \dv ) " dv y dz r dr r, ds '
These furnish immediately
dR dR_ dR, dR_ _s dR ___
dr~ + 2 ~dz~ r dr' 9* ~" V^~~^ *~ - a -'
16 THE EQUATIONS OF MOTION. [CHAP. II
Substituting these results in equations (3), (4), (5), we obtain
2d^ v '' r^a
v
(A),
d? i rs
dt 2 WIT*'
the equations of motion sought.
14. The first of these three equations contains no differentials, with
respect to the time, of s, v, while the second has none of r, 5: the third
equation has differentials of both r, 5. But since it is found by observation
that the arbitrary constants introduced by integration are such that r differs
from a constant by a small quantity only and that s is itself always a small
quantity, we shall see that the three equations are respectively useful in
determining the radius vector, the longitude of the projection of the radius
vector on the fixed plane and the latitude above this plane. They will
therefore be referred to as the radius- , longitude- and latitude- equations
respectively. The equations (A) having been first used by de Ponte'coulant*
as a basis for the lunar theory, are referred to under his name. Equations
of an almost identical form were, however, obtained by Laplace in Chap. VI.,
Book ii. of the Mecanique Celeste.
15. One of the chief difficulties of the lunar theory is the interpretation
of the arbitrary constants arising from the integration of the differential
equations. It is necessary, in order that we may be able to find them
accurately from observation, to have them exhibited in such a way that their
physical signification can be exactly fixed. Special stress is laid on this
point. The same care is necessary also when it is desired to compare the
results of one mode of development with another, in order that the relations
between the constants used in the two sets of results may be determined.
It will be noticed that the equations of motion have been reduced to two
of the second order and one of the first ; the integration of them will there-
fore give rise to five arbitrary constants. These five constants with the two
h, a, already introduced, will make seven, while our original equations of
motion three of the second order only demand six. There must therefore
be some relation between these seven constants after the integrations have
been performed and, to determine it, use must be made of some other
combination of the original equations of motion. For this purpose we have
* Sy 'steme du Monde, vol. iv. No. 1.
13-16] LAPLACE'S EQUATIONS. 17
from the sum of the first and third of equations (1) multiplied respectively
by n, *,
ji dR dR
But since rf + z* = r 2 , s = tan u (where, for a moment, u denotes the
latitude) we obtain
and therefore
r^r, + zz = rr + r~ r^ z* = rr r 2 *7 2 = rr ^/(l + s 2 ) 2 .
. , 9-R dR dR
Also r _- + ^_ -= r - .
97*! 9-z 9r
Whence, after division by r 2 we obtain, since rf = r 2 /(l + s 2 ),
r v 2 s 2 p IdR
_i_ {_ _ . /f?\
r 1 + s 2 (1+s 2 ) 2 r 3 r dr '
When the motion is elliptic, that is, when we neglect the right-hand
sides of equations (A), the relation between the seven constants is very easily
found (see Chapter in.). When the general equations of motion are treated,
the equation (6) may be used to find the required relation. Six of the
constants are determined in the disturbed motion so as to simplify the
interpretation of the final results as much as possible (Chapter vill.). The
presence of a seventh constant greatly assists us in this respect.
(ii) Laplace's Equations of Motion.
16. Let ^!, Xi t 3i be the forces acting on the Moon, resolved parallel to
the direction of the projection of the radius vector on the fixed plane,
perpendicular to this direction in the fixed plane and perpendicular to the
fixed plane respectively. The equations of motion will be
Let MJ = 1/r-j and rfv = H. Transforming so as to make the independent
variable v, we obtain* from the first two equations, since dt r^dv/H,
/7\
* Tait and Steele, Dynamics of a Particle, 4th Ed., Art. 136.
B. L. T. 2
18 THE EQUATIONS OF MOTION. [CHAP. II
Also, since H= Xi^, we have
AH H X,
H -T- =H .
dv v Ui 2
Integrating, we obtain, if h be an arbitrary constant,
H* = # + 2 [ * (2v ;
J% 3
d* 1 1 /.. [ 1 , \~* , ft v
whence -j- = -^7 9 = j 9 [1 + 2 K^- cfo (8),
cfo Jit*!* i*\ MX /
and equation (7) may be written,
Since H = h when Xi is zero, /i will have the same meaning as in (i) when R = 0.
Again from the first and third equations of motion we obtain
^su! + si) 2
ds .. dJs vn
2 + -j- v + 2 -j- - + sv z
dv dv TI
ds (H
^fe
ds
7 . TT,
where the values v ; = ^ : (~i) > ^ = ^i r i nave ^ een substituted. Since
v 2 = irx 4 = #v (1 + 2 /a; dv/hv),
there results,
Also, since F (Art. 8) is a function of a, y, z and therefore of M I? v, 5, we
have, by the principle of virtual displacements,
and since S^ Buju^, Sz = Ss/X - sbujui*, we obtain by equating the
coefficients of the independent displacements Bu lt Sv, Ss,
8s
16-18] EQUATIONS REFERRED TO MOVING AXES.
Substituting for $ 1} X lf 3i m equations (9), (10), (8), we obtain,
19
_
fo A 2 V d^ 2
_
+ "
/
(
^_j_/ 2^ rap^\-i
dv~hu*( h*J ^%7
which are the equations found and used by Laplace*.
17. We have just obtained the equations of motion in the form of two
equations of the second order and one of the first, furnishing five arbitrary
constants on integration ; these with the constant h will form the six
constants necessary. The form of these equations renders them very useful
in certain departments of the lunar theory. For a complete development of
the perturbations produced by the Sun, with the accuracy demanded by
observation to-day, they are, nevertheless, almost excluded by the fact that,
after u^, s, t have been found in terms of v, a reversion of series is necessary
to get v the most important coordinate in terms of t. This last process
would probably demand as much labour as that necessary to find the other
coordinates and the time in terms of v.
(iii) Equations of Motion referred to moving rectangular
18. Take axes EX, EY moving in the fixed plane of (xy) with angular
velocity ri round the axis of z (Fig. 2, p. 13). Let X, Y, z be the coordinates
of the Moon and X', F, those of the Sun referred to these axes, so that, as
before, we assume the fixed plane of (XY) or (xy) to be the ecliptic. The
equations of motion of the Moon will be, according to the usual formula? for
accelerations referred to moving axes,
.(12).
If we define the function F', by the equation,
(13),
* Mecanique Celeste, Books n. 15; vn. 1.
t See remarks 011 Euler's second theory in Chapter xu.
22
20 THE EQUATIONS OF MOTION. [CHAP. II
the three equations of motion may be written,
Let now
We have then
, _ 9 .-i -9
A "' " z
Multiply the second equation of motion by V 1 : the first two equations of
motion become, by addition and subtraction,
Again put = e (n ~ n ' } <*-'> v=i ,
and change the independent variable from t to f; ?i, are two constants at
present not defined. Let
m
nnv
We have
according to the usual notation for operators. Substituting these values in
the equations for v, <r, they become,
9 OET/ O p)TT'
D 2 u + 2mDu = -- 2 ^ , DV-2mDo-=-- 9 ^- ..... (14).
v 2 oo- v* dv
\
19. We must now develope F'. From the expression of F given in
Art. 7, we have, since E + M ' p,
Let now n', 2a' be the mean motion and major axis of the ellipse
described by the Sun. It will be shown in Chapter in. that we can put
m' = ri*a' s . Let K = /A/V Z . With these substitutions, we obtain from equation
(13), since m = n'/v, X*+ F 2 = va; r z =
m
= + f m 2 (i; + o-) 2 - mV +
.(15),
18-20] EQUATIONS REFERRED TO MOVING AXES. 21
where
..................... (16).
In this last expression Q p stands for the terms of degree p in powers and
products of v, o-, 2, and therefore of degree p 4- 2 in powers of a' (r' is of the
same degree as a').
Substituting the value of F' given by the second of equations (15) in
equations (14), the latter become, since r 2 = v& + 2 2 ,
and the third equation of motion, after changing the independent variable, is
r\ ^ 1 /i Q\
The equations (17), (18) form the basis for a general treatment of the
lunar theory. We shall now give Hill's transformation in its most general
form.
20. Multiply the first of equations (17) by <r, the second by v and
subtract. We obtain
Again, adding equations (17) with the same multipliers to equation (18)
multiplied by 2z, we have
aD-v -f- vD 2 <r + 2zD 2 z - 2m (vDv - <rDv) + f m 2 (v + <r) 2 - 2m 2 2 2 - -
T
an an i
The last result is arrived at by applying Euler's theorem for homogeneous
functions to equation (16).
Further, multiplying equations (17) and (18) by Da, Dv, 2Dz respectively,
and adding, the result may be put into the form*
I> FU . !><r + (I^) 2 + fm 2 (u + ^
............... (21).
* The term Dv . Da- is the product of Du and D<r and must not be confounded with D (vDo).
Similarly (Dz) 2 = Dz x Dz.
22 THE EQUATIONS OF MOTION. [CHAP. II
But since O is expressible in terms of the coordinates of the Moon and
the Sun and since those of the latter are supposed known functions of the
time or of f, we may suppose O expressed as a function of v, <r, z, f We
therefore have
n _ an n an n an n an nj ,
Dfl = ^-Dv+ Da + ^- Ite + -~= -Rf
au ao- 9s a?
But ^ DZ; = ? || = AH
where A denotes the operation D performed on fl with reference to the
portions which contain t (or f ) explicitly, and D" 1 is the inverse operation
to D. Substituting these results in equation (21) and integrating, we have
J)v.D<r + (Dz)* + f m 2 (i; + <r) 2 - m 2 * 2 + = C - H + D~ l (AH). . .(22),
where C is a constant.
Adding this to equation (20), we obtain (since fl = fl 2 + fl 3 -f ...) an
equation which may be put into the form,
I> 2 (va- + z 2 ) -Dv.Da- (Dz)* - 2m (vDa -
(An) ...... (23).
The three equations (18), (19), (23) are the generalised form of Hill's
equations (see Chapter XL).
21. It will be noticed that these three differential equations are each of
the second order and therefore on integration will furnish six arbitrary
constants. A constant of integration C has already been introduced, while K
or fi has disappeared from (19), (23), the equations which furnish the motion
in the fixed plane. There is therefore a relation containing K, between these
seven arbitrary constants. This relation will be determined from one of
our original equations of motion. The constants n, t introduced into the
equations will be defined in Chapter XL as two of the arbitraries of the
solution.
The advantage possessed by the equations (19), (23), which are of principal
importance for the determination of v, cr, arises chiefly from the fact that
their left-hand members are homogeneous quadratic functions of u, <r, z.
When we neglect the parallax of the Sun, that is, when we consider the
Sun to be at an infinite distance, the right-hand members of the equations
are also of the same form except as regards the constant G. Even when
terms depending on the distance of the Sun are included, since it is not
generally necessary to take them beyond the order I/a' 2 , the terms thus
added will only be of the third and fourth degrees in v, cr, z. Equation (18)
20-22] PARTICULAR CASES. 23
has not this form, but it is not difficult to obtain an equation of a form similar
to (19), free from the divisor r 3 .
The remarks of this last paragraph apply equally to equations (19), (23),
when they are expressed in terms of the real variables X y Y, z, t. The use of
the conjugate complexes v, o- enables us however to put our solution in an
algebraic form. It will be seen later that X, Fare expressible respectively by
means of cosines and sines of the same multiples of t. As a consequence of
this, v, a- are expressible in series, with f as the variable and with real
coefficients. Also, <r can be derived from v by putting l/ for f, so that
it is only necessary to calculate either v or cr. The advantage of algebraic
over trigonometric series, when the multiplication of two series is in
question, will be easily understood.
Since Dv.D<r + (Dz}* = - (I 2 + F 2 + z 2 )/* 2 ,
the equation (22) is the Jacobian integral referred to moving axes. When the solar
eccentricity is neglected the term D~ l (Z^Q) vanishes. We may therefore look upon this
term as the variation of the constant of Energy due to the eccentricity of the Sun's
orbit.
Also since v<r + z 2 = r 2 , vD<r -oDv=-(YX-X Y)jv,
we can express immediately equations (19) and (23) in a real form.
Although, either of the equations (17) is, since v, <r are complex quantities, a complete
substitute for the first two of equations (12) the same cannot be said of equations (19), (23).
The reason of this is easily seen. If we give to v, er, their values in terms of A', Y, , each
of the equations (17) furnishes a real and an imaginary part. On the other hand when the
same substitutions are made in (19), (23), the former gives an imaginary part only and the
latter a real part only.
22. There are two particular cases of equations (12) which require notice
and, in order to treat them, we must know something further about the
disturbing function.
We have from Art. 7, putting w' = w'V 3 (see Act. 19),
E = *" Of-S' - if) + 'J ^ ft*S> - &*8) + . . . .
Let v' be the true longitude of the Sun supposed to move in an elliptic orbit,
and let the axis of X, which is rotating with the mean angular velocity of the
Sun, point towards the Sun's mean place. If e' be the angle, at time t = 0,
which this axis makes with the fixed line from which v is reckoned, we shall
have
X ' = r f cos (if - n't - e), Y' = r sin (v' - n't - e').
If now we neglect the solar eccentricity, these equations give
X' = r 1 = a', Y' = 0, rS = (XX' + YY')/r' = X,
and the first term of R becomes n- (%X- ^r 2 ).
24 THE EQUATIONS OF MOTION. [CHAP. II
If therefore we neglect the terms beyond the first in R, that is, if we
neglect terms which depend on the parallax of the Sun (retaining those
dependent on the solar eccentricity), we shall have
The second term of this expression then vanishes with the solar eccentricity.
Moreover since r 2 $ 2 is always a quadratic function of X, Y, we can put
E = ri* (f X 2 - -|r 2 ) - \A'X* -B'XY- |C"F 2 - \K'z\
in which A', B', C', K' are simple functions of the time depending on the
solar elliptic motion.
Substituting, equations (12) become
X - 2n'Y - 3ri*X + A'
Y + 2riX + B'X + C'Y = - -
z + n'*z + K'z = -
(24),
in which those terms depending on the distance of the Sun are the only ones
neglected. These equations form the basis of Adams' researches* into the
connection between certain parts of the motions of the perigee and node and
the constant part of 1/r.
23. A further simplification is introduced by supposing the solar eccen-
tricity and the latitude of the Moon neglected. Giving therefore A ', B', C r , K', z,
zero values, the equations are reduced to the two
Y+2riX =
These play an important part in Hill's method of treating the lunar theory f. They
are the equations of motion of a satellite disturbed by a body supposed to be of very great
mass m', at a very great distance a', such that m'/a' 3 = ri* is a finite quantity. The
disturbing body whose distance has just been supposed to be so great that I/a' is
negligible, is moving with uniform velocity in a circular orbit in the plane of motion of the
satellite and is placed on the positive half of the moving JT-axis.
The equations admit of a particular solution
Y= 0, X= r = const. =
The Moon is then always on the axis of X, or in other words, is constantly in
* Monthly Notices of the Eoyal Astronomical Society, vol. xxxvm., pp. 460472.
t Chapter xi.
22-25] GENERAL PROBLEM OF THREE BODIES. 25
junction with the Sun. The motion is however unstable, a fact which can easily be
obtained from the equations (25).
This is a particular case of a more general theorem mentioned below. (See Art. 30.)
24. It is not difficult to find expressions for the velocity when we neglect the
solar eccentricity oidy. Since in this case, rS=X t r f = a'^ we have, from Art. 22,
Hence R does not contain the time explicitly.
Multiply the equations (12) by X, 7, z and, after adding the results, integrate. We
obtain
X 2 +;n + 3 2 = ^ + n' 2 (2: 2 +r 2 ) + 2fl+const ...................... (26),
giving the velocity referred to moving axes. (If we include the solar eccentricity,
/p\ n
-~- dt must be added to the right-hand side of (26)).
But we have X* + Y* + z 2 = r t 2 + r^ (4 - n') 2 + z 2
Hence, since JT 2 + F 2 =:
we obtain * +y 2 +z 2 = -^ + 2ri (yx-xy)+2R+ const.,
giving the velocity in space, when the solar eccentricity is the only quantity neglected.
This expression was first obtained by Jacobi*.
%
(iv) The general Problem of Three Bodies.
25. The developments given above refer to the motion of one body about
a second when that of the third body about the second is supposed known.
The problem of three bodies or rather of three particles, considered without
any limitations, admits of a much more general treatment and moreover,
when looked at from this point of view, is seen to possess certain properties
in the form of first integrals which do not appear in the more restricted
problem.
Let TTij, T7i 2 , ms be the masses of the three bodies, r^, r 13 , r 12 their mutual
distances and (x^z^, (x. 2 y z z^), (x s y 3 2 3 ) their coordinates referred to rect-
angular axes, fixed in direction, through any origin. The nine equations of
motion are
.. _dF ,_<*? m "=^ (i-l 2 3}
9#i ' tyi ' 9^ '
where, according to Art. 10, Chap. I.,
m,w,
+
* Comptes Kendus, vol. in. pp. 59 61. Collected Works, vol. iv. pp. 37, 38.
26 THE EQUATIONS OF MOTION. [CHAP. II
Since rtf = (xi - xtf 4- (yi - ytf + (zi - ztf and therefore
we get immediately, by addition of the equations of motion,
2ra^ = 0, Zmij/i = 0, Zmi'zi = 0.
From these we obtain by integration
2m^ = a, Sm^i = 6, 2ra^ = c ............... (27),
where a, b, c are three constants. These constitute three first integrals of
the equations of motion. ^ '
On integrating again, we have three further equations of the form
Sm,;#{ = at 4- const.,
or, eliminating a, 6, c, three equations of the form *
2ra^ - t (2ra^) = const., ........................ (28),
which are also three first integrals of the equation of motion.
Again, since
a a ^ i _y i x j -x i y j _
'
we obtain from the equations of motion,
2 (^ii/i - y#t) = 0, S (ytl?i - *&) = 0, 2 (^^ - x^ = 0,
and therefore three more first integrals of the equations, of the form
^(^-2/^0 = const ......................... (29).
Finally, since F is a function of t only in so far as t occurs implicitly
through the presence of the nine coordinates in F, we shall obtain, after
multiplying the nine equations by their respective velocities, adding and
integrating,
wii (^i 2 + y\ + ^i 2 ) + m 2 (^ 2 2 + 2 2 + 4 2 ) 4- ra 3 (^ 3 2 + 2/ 3 2 + 4 2 ) = 2^+ const. . . . (30),
a tenth first integral of the equations of motion.
26. At first sight the general integral of Energy (30) seems inconsistent with the ex-
pressions obtained in Arts. 12, 20, 24 which, when the solar eccentricity was not neglected,
contained an unknown integral. It is to be remembered, however, that we have earlier
supposed the motion of the Sun round the Earth to be known and to be expressible
by known functions of the time. In so doing we have neglected a portion of the effect
produced by the Moon on the motion of the Earth, a portion which would produce an
unknown integral in the expression for the relative Energy of the Sun. We have then
divided the Energy, relative to the Earth, of the two bodies into two parts, one part being
25-28] THE TEN KNOWN INTEGRALS. 27
that of the Moon and the other that of the Sun. In the motion of the Sun, the portion
depending on the Moon is so small compared to its own great mass that we have
neglected it, while, in the motion of the Moon, the same portion, being great compared to
the mass of the Moon, cannot be neglected. In fact, if we denote, in the expression for
the square of the Moon's velocity, this portion by m'$, where is a function of the
coordinates and velocities, there will be, in the expression for the square of the Sun's
velocity, a term -M<p. When we add the Energy of the Sun to that of the Moon,
</> will vanish identically.
27. The ten integrals found above are the only known integrals for the
general problem of three bodies. It has been demonstrated further by
JVi. Bruns*, that no other algebraic uniform integral can exist for any values
of the masses. M. Poincaref has extended this result, from a practical
standpoint, by proving that if the ratios of two of the masses to the third
are sufficiently small quantities, there does not exist any other transcendental
or algebraic uniform integral. For the proofs of these theorems, which are
based on considerations altogether outside the scope of this book, the reader
is referred to the original memoirs.
It is evident that these ten integrals exist and are of the same form for
any number of bodies attracting one another according to the Newtonian
Law. The extension to this general case is made immediately, if we suppose
i,j to receive the values 1, 2,...p, there being p bodies under consideration.
The ten integrals might have been written down from purely dynamical
considerations. The first six integrals (27), (28) are known as those of the
Centre of Mass and they express the facts that the linear momentum in any
direction is constant and that the motion of the Centre of Mass is uniform
and rectilinear. The three equations (29) are known as the integrals of areas.
The dynamical equivalent is expressed by saying that the angular momentum
round any line, fixed in direction, is constant. Equation (30) is that of
Energy and expresses the fact that the sum of the Kinetic and Potential
Energies is constant.
28. Let the three constants of angular momentum be ^, h z , h 3 . The
straight line whose direction cosines are proportional to h lt hv,h z is invariable
in direction and consequently the plane perpendicular to it is so also. If we
consider all the bodies of the Solar System without any reference to those
outside, the plane determined in this way is known as the Invariable Plane
of the Solar System. Laplace suggested that this plane might be used as a
plane of reference to which the motions of the bodies might be referred.
There are however several difficulties in the way.
* Acta Mathematica, vol. xi. p. 59.
t Acta Mathenuitica, vol. xin. p. 264. Also Mecanique Celeste, vol. i. p. 253.
28 THE EQUATIONS OF MOTION. [CHAP. II
For further remarks on the Invariable Plane, see
E. J. Kouth, Rigid Dynamics, Vol. i. Arts. 301-305.
Laplace, Mec. Cel. Book vi. Nos. 45, 46.
De Pontecoulant, Systeme du Monde, Vols. i. p. 455 ; n. p. 501 ; in. p. 528, p. 555.
Other references are given by Tisserand, Mec. Cel. Vol. i. p. 158.
29. We may consider any one of the equations of motion as replaced by two others,
each of the first order. Let x be any coordinate, x its velocity. The equation
dx dF dx
may be replaced by ^ = ^ , -^ =x,
so that if there be p bodies we shall have 6p equations to determine Qp variables, namely,
the coordinates and the velocities. By means of the ten integrals, it is theoretically
possible to eliminate ten of the Qp variables and the resulting equations will contain
6p- 10 variables, or, in the case of three bodies, 8 variables. In general, it is found better
to eliminate only six by means of the equations (27), (28), leaving in the case of three
bodies, 12 variables between which four relations are known.
The literature on the general problem of three bodies dates chiefly from the researches
of Lagrange. An account of these is given by F. Tisserand, Me'canique Celeste, Vol. i.
Chap. vin. and by 0. Dziobek, Die mathematischen Tkeorien der Planeten-Bewegungen, pp.
80-82.
30. There are two special cases in which it is possible to integrate rigorously the
equations of motion. They are obtained by supposing that the mutual distances of the
three bodies remain constantly in the same ratio. In the first case, the three bodies are
constantly in a straight line and each describes an ellipse with the common Centre of Mass
as one focus. This motion is unstable. The result of Art. (23) is a special case of this.
In the second case, the bodies always remain at the corners of an equilateral triangle
of varying size.
These two problems are discussed by
F. Tisserand, Mdc. Cel. Vol. i. Chap. vm.
E. J. Routh, Rigid Dynamics, Vols. i. Art. 286, n. Arts. 108, 109.
In the former will be found further references to the papers on this subject. In the
latter the stability of the second case is considered.
CHAPTEE III.
UNDISTURBED ELLIPTIC MOTION.
31. THE subject of elliptic motion belongs properly to the problem of two
bodies, the solution of which presents no difficulties : it is treated in most of
the text-books on Dynamics. By the introduction of a certain angle (the
eccentric anomaly), the relations between the coordinates and the time can
be put into finite forms which, though useful for some purposes, are not
convenient when we proceed to the problem of three bodies, this latter problem
being generally treated, so far as the solar system is concerned, from the
point of view of disturbed elliptic motion. In this case it becomes necessary
to express most of the relations by means of series. These series, investigated
mainly by means of Bessel's functions, will be given briefly in this Chapter.
The subject will be divided into two parts, the first referring only to the
properties of the elliptic curve, and the second containing applications of
the results obtained in the first part, to elliptic motion about a centre
of force in the focus.
(i) Formulae, Expansions and Theorems connected with the
elliptic curve.
32. Let C be the centre of an ellipse, E one focus, A the apse nearer
to E, P any point on the ellipse, Q the corresponding point on the auxiliary
circle and QPN the ordinate drawn perpendicular to CA.
Let EP = r and let the angle ACQ = E, the angle A EP =/, and let w be
defined by the equation
w area AEP
2?r area of ellipse '
Then
UNDISTURBED ELLIPTIC MOTION.
/ is called the true anomaly,
E eccentric
w mean
[CHAP, in
Fig. 3.
Let the major axis of the ellipse be 2a, its eccentricity e and latus
rectum I From the well-known properties of corresponding points on an
ellipse and its auxiliary circle, and from fig. 3, it is evident that the
following relations hold :
r = a (1 - e cos E) = //(I + e cos/) = a (1 - e z )/(l + e cos/) (1),
(2),
r cos/= a (cos E e), r sin/= a V 1 e 2 sin E
w = E esmE.
tan = 6 tan
VI e
.(3).
Also, since a, e, Z remain constant and r, / E, w vary with the position of
P, we obtain easily
dE a
.(4).
There are two problems to be considered. The first consists in expressing
certain functions of w and r in series proceeding according to sines and
cosines of multiples of /and powers of e', the second consists in expressing
certain functions of r and / by similar series in terms of w and e. These series
will be investigated first in an elementary way and then by means of Bessel's
functions.
32-33] ELEMENTARY METHODS. 31
33. To obtain series for r, lu in terms off.
Define \ by the equation
e = 2X/(1 + X 2 ), or X =* (1 -
We then have by (1) I- <^ff
Putting for a moment 2 cos/= a; 1 -f a;" 1 , and therefore 2 cos if =%*-{- x~^ t we
easily obtain
r^l-X 2 / 1 Xar 1 \
a " 1 + X 2 U + Xtf 1 " + XarV '
and thence by expansion, since a? is a complex quantity whose modulus
is unity,
.} ......... (5).
From this equation, after expanding X in powers of e, we obtain the required
expansion of r.
Again, from (5) we have the identity
(1 + e cos/)- 1 = (1 - e z )-$ (1 - 2X cos/+ ... + (-!) 2X 1 ' cos ^/+ . . .}.
Since the series on the right is supposed to be convergent, we may differentiate
this equation with respect to e ; we then get, after multiplying by e,
cos
Adding this to the previous equation we obtain, since d\/de =
= (1 _ g2)-i i + 2 2 (- iy X* (1 + iVl - e 2 ) cos i/ .
But we have, by (4),
dw
cos/) 2 '
Substituting for (1 +ecos/)~ 2 its expansion just found and integrating, we
obtain, since w and / are zero together,
V^^ 2 )sm^/+ ...... (6).
32 UNDISTURBED ELLIPTIC MOTION. [CHAP. Ill
Putting for X its value and expanding in powers of e, we deduce the required
expansion of w. This is, as far as the order e 4 ,
34. To obtain f, r in terms of w.
Since we have supposed that convergent series are possible, we can obtain
/in terms of w by reversing the series (6') using /=w as a first approxima-
tion. By this proceeding we shall get, as far as the order e 4 ,
f=w + (2e - Je 8 ) sin w -f (f e 2 - \ef) sin 2w + jf e 3 sin 3w+ e 4 sinto +. . .(7).
Also as r = a(l e cos E\ w E e sin E, we can apply Lagrange's
theorem * to the expansion of cos E = cos (w + e sin E) and thus obtain
cos E = cos w 2 -3^ (sin p w sin w).
From this we have, after replacing powers of sin w by sines of multiples of w,
as far as the order e*,
** /?2
- = 1 + ^r (e f e 3 ) cos w (| e 2 Je 4 ) cos %w |e 3 cos 3w 4e 4 cos 4w ... (8).
(X Ii
Corollary. Since a-/r*= dfjdw, we can immediately obtain from (7) the
expansion of a 2 /r 2 .
35. From these expansions for/, r we can deduce those of any functions of/, r in terms
of w. As however all these expansions (except perhaps that of/) are obtained much more
easily by the use of other methods, those outlined in this Article will not be further
developed. For a fuller discussion of the methods of Arts. 33, 34, see Tait and Steele,
Dynamics of a Particle, Arts. 162-167.
36. An expansion for / in terms of w, exhibiting the general term of the series by
means of a simple expression, has been given by S. S. Greatheedf. The process may be
shortly sketched as follows. Let
then by Lagrange's theorem applied to equations (3) we have
... e*> el?- 1 /df .
/=/ +2 _ _ - . ( ^Q s
! p ! dw*>- 1 \dw
Also, differentiating the expression for/ we have
and*
* Williamson, Diff. Gal. Chap. vn.
t Camb. Math. Jour., 1st Ed. Vol. i. (1839), pp. 208211.
t E. W. Hobson, Trigonometry, Art. 52. The formulae there numbered (44), (45) can be
combined into this form.
33-38] BESSEL'S FUNCTIONS. 33
giving
-2q-i) w}.
Substituting this in the equation for / and finding the coefficient of sin/w in the
resulting expression, Greatheed finally arrives at the symbolic formula for/:
J=l
In order to obtain the coefficient of smjw, the expression for it given by this formula must
be first developed in powers of X as it stands, all negative powers of X rejected and the
terms of the order X divided by 2 *.
Cayley t has extended this result in a general manner to the expansion of any function
of r and /.
Expansions by means of Bessel's functions.
37. The following formulae will be found J in any treatise on Bessel's
functions, i being a real integer :
1 r*
J { (x) = - cos (i(f) xsm(f))d(f) (9),
(X) 4- J i+l (X)} = iJi (X\ | { /;_! (X) - J i+l (X)} = -j-Ji (X) (11),
where J { (x) is the Bessel's function of the first kind In the applications to
be made here, i is a positive integer and x a real quantity sufficiently small
for series in powers of x to be convergent. If i be negative we have imme-
diately from (9), Ji (x) = J_ t (- x).
38. To expand COSJE, sinJE in terms of w.
From (3) we may assume that cos JE, sinJE will be respectively expansible
in cosines and sines of multiples of w. Let
cos JE %i A i cos iw, sinJE = 2i Bi sin iw,
where j, i are positive integers. Then, by Fourier's theorem,
Ai ^ = I cos JE cos iwdw.
* Jo
* See note by Cayley on the expansion of this formula in Quart. Math. Jour. Vol. n.,
pp. 229232. Coll. Works, Vol. in. pp. 139142.
t Camb. Math. Jour. 1st Ed., Vol. m. pp. 162167. Coll. Works, Vol. i. pp. 19-24.
t E.g. Todhunter, Chapters xxx. xxxi. It will not here be necessary to suppose any
knowledge of Bessel's functions, beyond the assumptions that the series are possible and that they
converge. If we define Ji (x) by (9) the results (10), (11) can be found by a few simple operations.
B. L. T.
*r
34 UNDISTURBED ELLIPTIC MOTION. [CHAP. Ill
Integrating by parts,
2 fl ~\* 2? f 17 .
A< = - -cos JE sin iw\ + -^ I sin iw smiEdE
7T\_l J W =o WO
= o + J- I sin (IE ie sin .&) siuJEdE, by (3),
^'7^Jo
= J-T cos !(i -J)E- ie sin ^} dE-4-\ cos {(- i -./) ^ - (- te) sin E] dE
ITT J o 11T J o
= ?' J M (ie) + -i-. J-i-j (- ie), by (9),
1 v
except when i 0. For the determination of A we have
A Q TT = I cosJEdw =/ (1 e cos E) cosJEdE = - TT or 0,
Jo Jo
according as j is equal or unequal to unity.
If therefore we allow i to receive negative as well as positive values, we
obtain
* 3
COSJE= S ji-j(ie)cosiw ..................... (12),
i= -oo *
n O
in which -. JLj (0) = ^ or 0,
according as j is equal or unequal to unity.
In an exactly similar way we may find
Bi = - Ji-j (ie) -- -. J-i-j (- ie),
oo j
and sinJE = 2) -. Ji-j (ie) smiw .................... (13),
there being no constant term. From the results (12), (13) we can get most
of the expansions required.
39. To expand r, r cosf, rsinf, r~ l , r~ 2 in terms of w.
Putting j = l in the two results just obtained and substituting for cos E
and sin E in (1), (2), we deduce
r e
- = 1 2 - Ji-i (ie) cos iw (14),
a oo i
tp 00 1
- cosy = e + 2 J^i (ie) cos iw,
r oo ^
- siny = v 1 e* 2 - Ji-i (ie) si
(15).
sin iw
38-39] EXPANSIONS BY BESSEI/S FUNCTIONS. 35
Since JL$ ( ie) = Ji (ie) and since for i we have
we deduce, after the application of the formulae (11),
r t & 2edJi(i
a z
Ji(ie)
r cos/ =
[ 2 "1
2 Ji (ie) sin iw
Again, by (4), (3), (2), we have
a dE d , . , n d ( er sin/ \
- = -J = 1 + i (0 sin E) = 1 + 7- . - )
aw dw\ a *Ji e 2j '
r dw
and therefore from (15 7 ),
- = |+2S/i(e)oo8Mo (16).
Further, since r/a = l e cos E, we have
(fa; \a 2 / a d"w Va/ a dw
and since (15') gives the expansion of rsin/=a\/l e 2 sin ^, we obtain
cZ /T 2 \ 4
j ( - ) = 2 - Jj (ie) sin i'w.
rfw Vav i i
/*& QO ^
Integrating, = const. ^- z Ji (ie) cos iw.
Qi i 1
But since r^/a? = 1 2e cos ^ + Je 2 + J e 2 cos 2.E ;
and since it was shown in the last Article that the constant part of COSE,
when expanded in terms of w, is e/2 and that the similar part of cos %E is
0, the above equation shows that the constant part of r^/a 2 , when so expanded,
is 1 + 3e 2 /2. Hence
T* 2 4i
i Q o ^^ y/*\ * /i ^7\
Cb 1 1
Corollary i. From (1) and (4) we have
, 1 - e? a 1 ... \/l - e* dr
cos/= , sm/= - -.
ere ae dw
Whence, by using the developments (16), (14'), we can immediately deduce
those of cos/ sin/
The difference between the true and mean anomalies is called the
Equation of the Centre. Denote it by Eq.
32
36 UNDISTURBED ELLIPTIC MOTION. [CHAP. Ill
Corollary ii. If a be any angle, we have, since /= w + Eq.,
sin (a -f Eq.) = sin/ cos (a - w) + cos/sin (a - w).
By means of Cor. i. we can then get the development of sin (a + Eq.).
40. It will be noticed that the coefficient of sin iw or cos iw (i positive)
is always of the form a^ + a l e i+2 + 2 e* +4 + ..., ( , i, 2 numerical quanti-
ties). That this must be so in the expansions of all functions of r, f of the
forms treated here, is sufficiently evident from Art. 32. Hence if we are
considering any term whose argument is iw, we know immediately that the
lowest power of e contained in the coefficient is not less than e i . This fact
has an important bearing when we come to develope the disturbing function.
41. In the development of the disturbing function it is important to obtain
expansions for r*> cos qf, r? sin qf (p being any positive or negative integer and q any
integer including zero) in terms of w. These could be obtained from the expansions given
in Art. 39 by multiplication of series. Such a process would be somewhat tedious when
many terms are required. On pp. 163-179 of the Fundamenta*, Hansen obtains the
expansions by finding the finite expressions corresponding to each value of p and q
required for r? cos qf in terms of positive or negative powers of r, and for r*> sin qf in
terms of the differentials of the same with respect to w. That this is possible is evident
from the expressions for cos/ and sin/ given in Cor. i. of Art. 39. He then obtains a
general formula giving the coefficients of the development of r*> in terms of those of the
developments of r 2 and r~ 2 . The coefficients of r 2 are obtained as in Art. 39 and those of
r~ 2 as in the Cor. of Art. 34.
In a later workt, he has considered them in a much more general manner and has
obtained expressions for the coefficients of the development of r p exp. qf*J - 1 in powers of
exp. w\/ I, by means of Fourier's theorem. The definite integral corresponding to the
coefficient of exp. iw\/l is evaluated and a general expression which is the least
cumbrous to expand of any given up to the present time, is obtained for the coefficient.
This is even true of the case jo=-2, <? = 0, which by a simple integration gives the
development of/.
42. The literature on elliptic expansions up to 1862 has been collected by Cayley in a
report On the progress of the solution of certain problems in Dynamics J. Later developments
and references are to be found in Tisserand, Mec. Ce'L, Vol. i. chaps, xiv., xv.
43. The following Theorem and Corollary will be required later.
Let F, G, H be three functions of which F, G are developable in cosines (or
sines) and H in sines (or cosines) of a series of angles of the general form
(St + ft'. The function
a
* Fundamenta Nova Investigationis Orbitae verae quam Luna perlustrat. Auctore P. A.
Hansen. Gotha, 1838. This work will be referred to throughout as the Fundamenta.
t 'Entwicklung des Products einer Potenz des Eadius-Vectors mit dem Sinus oder Cosinus
eines Vielfachen der wahren Anomalie etc.' Abh. d. K. Sachs. Ges. zu Leipzig, Vol. n. pp. 183281.
Brit. Assoe. Reports, 1862. Coll. Works, Vol. iv. pp. 513593.
39-43] A THEOREM OF HANSEN. 37
can be developed into a series of the form
o r*rm
r = 2 2 . (iw
also, when the coefficients , & lt a_j have been found, all the other coefficients C/LI
can be obtained by a simple process.
Suppose that F, G are developable in cosines and H in sines of angles
of the form @t + ff. Let
- G = 2B cos (fit + ')> B. = 2# sin (fit + ff),
in which B, E' are the typical coefficients corresponding to the argument
fit + j3'. Let
The formulae (15') may be written
r cos/ " dRi r sin/ Vl-e 2 5 .
- ^ + 3 e = _2 ^ l cosiw, ^- = -- -Zt
a i de a e i
also from (17) we have
^ B =l + }c+22 J R i c08w ........................ (19).
a x
This last equation will be required in Chap. X.
Substituting in the expression for F we obtain
CO ///?. <
r - F = %ZB cos (Bt + ') . 2 - - cos iw - 22B' sin (ft + /3 7 ) . 2 t'JRj sin tw
=i e t -=i
= X I [f B ^ + ^tft) cos (iw + # + ^)
t=i l\ ae
- B'iR,] cos (- iw 4
de
If now we put -R_; = R i} R = 0, this may be written
r - jF= 2 2 Oj cos (iw + ft + P), (i = excluded)
t=-oo
where
If we put also
we obtain T = 2 Oicos(iw + j3t
t=-oo
for all values of i.
38 UNDISTURBED ELLIPTIC MOTION.
By the definition of 04 we have, since R$ = R-i,
l ~ de
[CHAP, in
whence B = (i
Substituting these values of B, B' in the expressions for a. iy o_, we obtain
de
de
de . Ri
dR l R^
de
de
~de~
de
de
When therefore a 1} OL_I have been found for all the different arguments
/S + /3', the coefficients a it a_; can be found without any trouble. This
simple method of obtaining the coefficients in the product of two series, saves
Hansen much labour in performing his developments.
Corollary. From the last two equations we deduce immediately
44. When the plane of the ellipse is inclined to the plane of reference,
expansions for the longitude in this latter plane and for the latitude above it
will be required.
Let M (fig. 4) be the position on the unit sphere corresponding to the
point P (fig. 3) whose true anomaly is / Let Ml be the position of the
Fig. 4.
plane of the orbit and draw MM' perpendicular to xM' the plane of reference.
Then, according to the notation of Chapter II., v = xM', s = tan M 'M. Let
a?n = 0, IM = L t ILflf' = z x , M'CIM = i* and therefore L = v-0.
* The letter i is frequently used, as in the previous articles, to denote any integer : the addi-
tional use of the letter to denote this angle will cause no confusion.
43-45] PLANE OF ELLIPSE INCLINED. 39
45. From the right-angled triangle MM'l we have
tan LI cos i tan L.
Putting i = V 1> we can write this
and therefore e? Lit = e 2 ^' ". . IV*
t - ..;*&*
Taking logarithms and expanding, we obtain
2z lt = ZPTTL + 2u - tan 2 \ (e* L <- - e~ 2Li ) 4- tan 4
Z J
where p is an integer. Since L^ = L when i is zero, we have p = 0. Hence
L = L tan 2 ^ sin 2z + 1 tan 4 ^ sin 4z ^ tan 6 ^ sin
Let now the angular distance from the apse to the node H be TX 6. We
have then
L =/+ ^ - ^ = w + CT - ^ -I- Eq. = 770 + Eq.,
where ^ = 10 + ^6. Substituting for this value and for ^ its value v 0,
we obtain
v =/+ or - tan 2 i sin (27; + 2 Eq.) + Jtan 4 \ sin (47/ -f 4 Eq.)- ...(20).
A '-
The terms involving i and constituting the difference between the longitude
in the orbit and that in the plane of reference are known as the Reduction.
We can expand
sin (277 + 2 Eq.) = sin 2rj cos 2 Eq. + cos 2rj sin 2 Eq.
by means of the formulae given in Cor. ii., Art. 39. Let
tan i = <y.
Then tan 2 | = (
In the case of our Moon, 7 is a small quantity of the same order as e : it
will, therefore, not be necessary to calculate a large number of terms in
the Reduction, when the latter is expanded in powers of e and 7.
40 UNDISTURBED ELLIPTIC MOTION. [CHAP. Ill
46. To obtain s, the tangent of the latitude, we have
sin M'M = sin i sin L,
sin i sin L 7 sin L
and therefore s = . = . ,
V 1 - sin 2 i sin 2 L V 1 + 7 2 cos 2 L
giving s = 7 sin L ^y s sin L cos 2 L + \<f sin z cos 4 z . . . ,
or s = 7 sin (970 + Eq.) - <f {sin (3^ + 3 Eq.) + sin (T/ O + Eq.)j + . . .(21).
Of these terms in s, the first is the most important and the method of
finding it has been given in Cor. ii., Art. 39. The other terms can be easily
calculated ; since they are multiplied by 7 3 at least, it will not be necessary
to take many of them.
47. It will be noticed that there is a connection between the index
of 7 and the multiple of rj similar to that between the index of e and
the multiple of w. In longitude we have even multiples of rj Q and even
powers of 7 ; in latitude, odd multiples of rj Q and odd powers of 7. In both
cases, the lowest power of 7 which occurs in the coefficient of sin irj Q or cos ir) ,
is 7*.
(ii) Elliptic Motion.
48. When we neglect the disturbing action of the Sun, the forces on the
Moon, relative to the Earth, are reduced to //,/r 2 acting inwards along the
radius vector. Such a force is known to produce motion in an ellipse of
which one focus is occupied by the Earth. We shall not here solve the
problem which is merely that of two bodies, but assume that the solution
has been completed ; all that then remains is to fix the constants to be used.
Let (fig. 3) Ey be a fixed line from which we may reckon angles. Let
yEA='&, and let e be the angle which a uniformly revolving radius vector
Efju makes with Ey at time t = 0. Let the time of a complete revolution
of this vector be 2,7r/n. Then yEp = nt + e and AEp ^nt + e-vr. But since
equal areas are described in equal times, we have, by the definition of w
in Art. 32, AEp = w. Hence
Mean anomaly = w nt + e t*r.
Let 2a be the major axis and e the eccentricity. Then we have the
following well-known results:
A6=7i 2 a 3 , (Velocity) 2 = y-^,
Twice the area described in a unit of time = na?\/l e?,
and n (or a), e, e, -cr may be taken as the four constants of integration.
46-51] MOTION IN AN ELLIPSE. 41
49. When the plane of motion is inclined to the plane of reference we
require two more constants. Let them be those defined in Art. 44, namely, i
the inclination and 6 the angular distance of the line of intersection of the
two planes from the fixed line Ex. In this case the line Ey is taken to
coincide with Ex, so that BT, e are reckoned from x along the fixed plane to
H (the ascending node), and then along the plane of the orbit. Let M
(fig. 4) be the position, on the unit sphere, of EJJ, (fig. 3). Then
and
The last angle is known as the mean argument of the latitude. The
constants introduced by the three equations which determine undisturbed
elliptic motion in space, are a (or n), e, e, -cr, 6, i (or 7).
These six constants are called the Elements of the ellipse. The meaning
to be attached to the word ' Element ' will be extended in Chapter v.
50. From the results of Arts. 34, 45, 46, we obtain the following values
of v, r, s in terms of the time, for the solution of the equations (A) of
Chapter II. when we neglect R, as far as the 3rd order of the small quantities
e,y:
v = nt + e + (2e ^e 3 ) sin w + ^e 2 sin
i7 2 sin 2*70 \e<f sin (w 2r) ) ^ey 2 sin (w + 2rj ) + . . .,
r/a= 1 + \& (e f e^cosw ^e 2 cos 2w % e 3 cos 3w ..., (22),
s = (1 - e 2 ^7 2 ) 7 sin rj +ey sin (w - T) O ) + ey sin (w + 770)
-f | e 2 y sin (2w rj ) + ftfty sin (2w + 170) ^y 3 sin 3^ + . . .
where w == nt -f e or, r] = nt + e 0.
51. It only remains to connect the constants a, h of Art. 13 with those
used here. We have found in equation (2), Chapter II., neglecting R,
Square of velocity in orbit = 2^/r //,/a.
This being the same expression as that given in Art. 48, a has the same
meaning in both cases, namely, the semi-major axis of the orbit. We have
also by Art. 12,
%h = \r^i) = rate of description of areas in the plane of reference,
whence
^ : = rate of description of areas in the plane of the orbit,
1 COS I
= ina 2 l-e 2 , by Art. 48.
Hence A = ^LE?.. ...(23).
Vl+7 2
This value refers to undisturbed motion only.
42 UNDISTURBED ELLIPTIC MOTION. [CHAP. Ill
52. The solution of equations (11) of Chapter n. when F=fj./r, may, since u^=u yT+s 2 ,
be put into the form
8=ymn(v-ff),
where tan (orj 6} = v 1 +-y 2 tan (or 0), e = ey\ +y 2 sin 2 (or - 6}.
For we have, by the figure of Art. 44, since M'M= u y
e cos {L (or 6)} . ^ cos L . . . sin
^ cos t7 cos u sin 17
/ /i\ / /i\ sm L \
e cos (tzr &) cos L t +e sin (w 6} . ;
cos i
whence, after substituting for e and 07 in terms of e 1 and w-^ and putting L I = V 0, we get
the required result. If AA 1J drawn perpendicular to the plane of the orbit, intersect xy
in A lt we easily obtain Tff 1 = xA 1 .
To obtain u in terms of v, we expand \/l+s 2 by the binomial theorem and, after
substituting for s its value y sin (v 6), express these terms in cosines of multiples of
2(v 0}. To obtain t in terms of v, we can expand l/^ 2 by means of the formulae of
Art. 33. For we have
which can be expanded in powers of ^(l+s 2 )"* and cosines of multiples of v-
Expanding next the various powers of \/l+s 2 in powers of s 2 and substituting for s its
value, we shall obtain dt/dv expressed by means of cosines of multiples of v w l and
2 (v - 6}. An integration will then give t in terms of v.
It will be noticed that i ar l , e differ respectively from sr, e by quantities of the order y 2 .
53. The expansions given in Art. 50 will apply equally to the motion of
the Sun, but become simpler since we suppose the plane of its orbit to be the
plane of reference. Only four constants will be required; these will be
called a', e', e, &'. The mean motion ri is denned properly by the equation
We have, in Arts. 19, 22, put m' = n' 2 a' 3 . The ratio //, : m is approximately
1 : 330,000, so that the error caused will be very small.
We should properly have put in the disturbing function,
The slight correction necessary is therefore obtained very easily after all the expansions,
giving the motion of the Moon as disturbed by the Sun, have been made. The correction
to be made to the largest coefficient in the expression of the longitude will not be so great
as 0"'02.
It is to be remembered that a', ri, e', c ', -or' are the constants of the ellipse which the
Sun describes about the Centre of Mass of the Earth and Moon, in accordance with the
principles laid down in Chapter I.
52-54] CONVERGENCE OF SERIES. 43
54. An important question in connection with the series given in this Chapter is their
convergence. Each coefficient of siniw or cosiw is represented by a convergent series,
and the series of coefficients thus arranged forms a convergent series as long as e is less
than unity. But if we arrange the series according to powers of the eccentricity, this
is no longer necessarily the case. We get in fact a double series, proceeding according
to powers of e and sines or cosines of multiples of w, the convergency of which, for a
certain range of values of e and W, depends on the manner of its arrangement. The
problem is to find the greatest value of e for which the series is absolutely convergent.
Laplace* has shown that if e be less than 0'6627432..., the series which have been
discussed, together with those of the form r p cos qf, i* sin g-/, will be absolutely convergent
for all values of w. Full references will be found in Cayley's Report already referred to,
and in Tisserand, Vol. I., Chap. xvi. In the latter are given some of the more important
theorems on the subject.
* Mem. de VInst. de France, Vol. vi. (1823), pp. 6180.
CHAPTER IV.
FORM OF SOLUTION. THE FIRST APPROXIMATION.
55. IT has already been pointed out in Chapter n. that there is no
known method of obtaining directly a general solution of the differential
equations which express the motion of the Moon as disturbed by the
Sun. In consequence, we are obliged to resort to indirect methods.
There are two well-recognized devices used, both of which depend on the
right to neglect certain parts of the equations of motion in the first
instance, so as to reduce them to forms which are capable of integration,
either by means of known functions or by the use of series, the coefficients
of which can be found according to a definite law.
The Form to be given to the expressions of the Coordinates.
56. In discussing these methods of obtaining a solution of the general
equations, it is necessary to keep certain physical considerations in view. It is
not sufficient to obtain mere expressions for the coordinates : they must
be put into such a form that practical applications may be possible and
sufficiently simple. Since infinite series will be used, this point becomes
of the greatest importance.
Now all records, ancient and modern, containing any mention of
lunar observations whether made in a scientific way or not go to
prove that, for a long period of time, the Moon has been circulating
round the Earth in an orbit which is confined between limits not very far
removed from one another. From this fact we infer that the motion is
of such a nature that, at any rate during a considerable interval, its
deviations from some mean state of motion (which, to fix ideas, we may
think of as circular) are never very great. We ought then to try and
express its coordinates in terms of the time, in a form which will give
the position . after any interval of time, whether it be short or long.
55-57] INTERMEDIATE ORBITS. 45
In order to be able to do this conveniently, the deviations from some
mean state of motion ought to be expressible by functions which oscillate
between finite and not very distant limits. The most convenient functions
of this nature are the real periodic functions* sine and cosine. Should
the variable, which is generally taken to be the time, occur, for example,
in the form of a real exponential in the expression of a coordinate, such a
term would cause the coordinate to increase indefinitely with t, for either
positive or negative values of t. Again, should a term of the form t 2 sinnt
be present, the same result would follow, provided the term has any indepen-
dent existence. It may happen that such a term is present as one of a series
which, in some other method of proceeding, would only have appeared through
the expansion of some periodic function an expansion only permissible for
small values of t. As it is desired to obtain expressions holding also for large
values of t, one object to be sought after is to try and obtain a solution in
which such terms are not present. If they should arise, they must, if possible,
be eliminated by some alteration in the form of the solution. In the case of
the Moon's motion as disturbed by the Sun only, when the latter is moving
in an elliptic orbit, it will be seen that the coordinates can be expressed by
periodic terms only. See Art. 69.
A discussion of the limits, upper and lower, of the Moon's radius vector is given by
G. W. Hill, Researches in the Lunar Theory, Amer. Journ. Math., Vol. i., pp. 5-26. See also
H. Gylden, Traite* analytique des Orbits absolus, etc. Vol. I., Chap. i.
Intermediate Orbits.
57. It has been stated in Art. 55, that the first step usually taken
towards the determination of the Moon's path, is a simplification of the equa-
tions of motion, made by neglecting certain portions of them, to forms which
can be readily integrated. A solution of the equations, thus limited, should
form an approximation to the true path of the Moon if it is to be of assistance
in obtaining the general solution of the complete equations of motion. This
approximate path is called the Intermediate^" Orbit, or more shortly the
Intermediary. The intermediary need not necessarily be a general solution
of the limited equations of motion ; it may not contain the full number of
arbitrary constants, but should be such that, when the general solution of the
complete equations is required, it forms in some way an approximation to the
path described by the Moon. It is not even necessary that the intermediary
shall exactly satisfy the limited equations of motion. We may, after having
obtained the exact solution of the latter, modify it in any way which experi-
* A periodic function is one which, after the addition of a definite and finite quantity to the
variable, returns to its previous value. We consider periodic functions of a real variable only.
t This term was introduced by Gylden. (German, intermediare ; Fr., intermddiaire.)
46 FORM OF SOLUTION. THE FIRST APPROXIMATION. [CHAP. IV
ence may suggest, provided that the modified solution differs from the exact
solution by quantities of an order not less than the lowest order of the
portions neglected in the original equations. The intermediary may then
be indefinite to a certain extent, until we have found the general solution
of the complete equations. Such indefiniteness will, however, be only
allowed for the purpose of facilitating the analysis and in order to put the
expressions for the coordinates into a suitable form, in accordance with the
remarks made in Art. 56.
58. The next consideration is the choice of an intermediary. In this
matter there is some freedom ; it must partly depend on the particular
method we intend to follow for the solution of the general equations. The
usual plan is to choose an intermediate orbit which is such that, after a
certain finite interval of time, the coordinates of any point on it, referred
to axes which may be fixed or moving, return to the values they had at the
beginning of the interval (an angular coordinate will have had its value
increased by STT). An orbit which possesses this property is called periodic*.
With reference to the axes used, the curve described will be closed.
59. In most of the older methods, the first limitation of the equations
of motion is made by neglecting the action of the Sun, so that the inter-
mediary is a fixed ellipse. A certain indefiniteness is then given by
supposing the apsidal line and the nodal line of its orbit on a fixed plane
to be moving with uniform angular velocities ; these motions are determined
on proceeding to the second and higher approximations. In other words,
the intermediary is periodic with respect to moving axes. By this choice
we begin by considering the problem of two bodies rather than the problem
of three bodies.
Dr Hill starts from a different standpoint. He begins by neglecting, in
the equations of motion, certain parts but not the whole, of the Sun's action,
and he is able to obtain for the intermediary a solution, periodic with reference
to axes moving in a definite manner ; this is really a particular case of the
problem of three bodies and the advantage of the orbit as a first approxima-
tion arises from this fact. This intermediary is not indeed a general solution
of his limited equations (which were given in Art. 23), in that it does not
possess the full number of arbitrary constants; nevertheless it serves as
a useful first approximation owing to the fact that one of the arbitrary
constants (the so-called ' eccentricity ' of the Moon's orbit), which has been
tacitly put equal to zero to get the intermediary, appears to be small enough
to permit of expansions in ascending power series.
* On periodic solutions, see Poincare', Mec. Gel. Vol. i. Chap. iv.
57-62] METHODS OF SOLUTION. 47
60. The subject of intermediate orbits has been treated by Gylden, Andoyer, Hill and
others. The usual plan is to express the disturbing function by powers of the ratio
of the distances of the Sun and the Moon and by cosines of multiples of their angular
distances ; the coefficients and the term independent of this angle are then functions of the
radii vectores of the Sun and Moon (or, in the planetary theory, of the two planets) only.
All the terms containing this angle are neglected, so that the disturbing function involves
only the radii vectores. A further simplification can be introduced by supposing the
motion of the disturbing body to be circular. The case treated in Art. 67 is a simple
illustration of the method followed. Gylden* uses a method akin to Hansen's to solve
the resulting equations ; Andoyerf follows Laplace in taking the true longitude as in-
dependent variable ; HillJ uses a direct method, by finding equations for the small
differences r-a, r'-a' (where a, a' are constants) and expanding in powers of them.
The principal part of the motion of the perigee is determined without much difficulty.
When the intermediary has been obtained, there are two methods of
proceeding to the solution of the general equations: (i) by continued
approximation, (ii) by allowing the arbitrary constants introduced into the
intermediary to vary.
(i) Solution by continued Approximation.
61. We suppose that, by means of the intermediary, the four variables,
namely, the three coordinates and the time, have been expressed in terms of
one of them ; in these expressions there will be a certain number of the
necessary six arbitrary constants present. With this solution or with a
modified form of it, we then proceed to find what small corrections must be
made to the variables when we include the omitted portions of the equations
of motion. If the motion be stable, these corrections should take the form of
small periodic terms. The method is then nothing else than that of small
oscillations about a state of steady motion that in the intermediate orbit.
In the case of the Moon we shall generally have to proceed to the third and
higher approximations in order to obtain the oscillations with sufficient
accuracy. It is necessary to consider the amplitude, the period and the phase
of each term.
(ii) Solution by the Variation of Arbitrary Constants.
62. The method is sufficiently well-known not to need explanation
here . In the case of the Moon we have three differential equations of the
second order and therefore six arbitrary constants in the solution. We assume
that an intermediate orbit has been found and that the resulting relations
* " Die intermediares Bahn des Mondes," Acta Math., Vol. vn. pp. 125-172 (1885).
t "Contribution a la Theorie des orbites interm&liaires, " Annales de la Fac. des Sc. de
Toul&use, Vol. i. M., pp. 1-72 (1887).
" On Intermediate Orbits," Annals of Math. (U. S. A.), Vol. vm. pp. 1-20 (1893).
See A. R. Forsyth, Differential Equations, Chapter iv.
48 FORM OF SOLUTION. THE FIRST APPROXIMATION. [CHAP. IV
between the coordinates and the time contain all the six arbitraries ; it is
required to find what variable values the arbitraries must have in order that
the same relations may satisfy the general equations of motion. The
coordinates expressed in terms of the arbitraries and the time will thus have
the same form for the intermediate orbit and the true orbit. There are three
relations, which may be chosen at will, between the first and second
differentials of the arbitraries. These are always taken such that the first
differentials of the coordinates have the same form whether the arbitraries be
constant or variable. Hence, the velocities, when expressed in terms of the
arbitraries and the time, have the same form whether the arbitraries be constant
or variable. This way of stating the relations enables us to change from
one system of coordinates to another without trouble. The six arbitraries
and any function of them not involving the coordinates, the velocities or the
time, are named elements. It is usual to take the undisturbed ellipse as the
intermediary. The method will be treated in the following chapter and an
important extension will be given to the meaning of the term ' element.'
The Instantaneous Ellipse.
63. We assume that the intermediary is an ellipse obtained when the
action of the Sun is neglected. It is evident that if at any instant during
the Moon's actual motion, the disturbing forces were to suddenly cease to
act and the Moon were to continue its motion from that point under the
mutual action of the Moon and the Earth only, it would describe an ellipse.
This orbit is called the Instantaneous Ellipse.
Now when a particle is describing an ellipse under the Newtonian Law,
if we are given the coordinates and the velocities* at any point, one ellipse
can be constructed which satisfies the given conditions, and its six elements
can be expressed uniquely in terms of the given coordinates and velocities.
Conversely, the coordinates and velocities of the point considered can be
determined uniquely in terms of the six elements. But since the coordinates
and velocities of this point on the Instantaneous Ellipse are the same as
those in the actual orbit, and since in the actual orbit the coordinates and
velocities, when expressed by means of the arbitraries and the time, have the
same form whether the arbitraries be constant or variable, the Instantaneous
Ellipse is the Intermediate Orbit at the time when, in the expressions for
the arbitraries, we have given to t the value which corresponds to the Moon's
position at that instant. Hence, the elements of the Instantaneous Ellipse
at any time ^ can be obtained, after the solution by the method of the
Variation of Arbitrary Constants has been carried out, by giving to t the
value tfj in the expressions which determine the arbitraries in terms of the
time.
* That is, the magnitude and direction of the velocity.
62-65] NATURE OF EQUATIONS (A). 49
Application of tJie Solution by continued Approximation.
64. Let us now return to the first method and see how it is to be applied
to the solution of equations (A), Chapter n. We may begin by neglecting
their right-hand members, that is, the terras dependent on the action of the
Sun. The equations so limited will give the intermediate orbit an ellipse
of period Zir/n; and we have seen in Chapter in. that, in this case, the
coordinates can be expressed by sums of periodic functions of the time*
which are sines and cosines of multiples of angles of the form nt + a,
where a is a constant.
To obtain the second approximation, we substitute these values of the
coordinates in the right-hand members. But the disturbing function also
depends on the coordinates of the Sun, which is supposed to move in an
elliptic orbit of period 27r//i', and these coordinates will be expressed by
sines and cosines of multiples of angles of the form n't 4- a'. Since the time
cannot enter into the right-hand members except through the coordinates, all
the portions which depend on the action of the Sun will be periodic functions
of the time and the arguments will be all of the form int + i'n't + A (i, i f
integers, positive, negative or zero and A a constant depending on the integers
i t i' and on the longitudes of perigee, node and epoch of the two orbits).
The equations being integrated, we obtain for our second approximation new
values of the coordinates which, when substituted in the right-hand sides of
equations (A), will, after a new integration, furnish a third approximation,
and so on. This process is repeated until the desired accuracy is obtained.
65. Let us consider the nature of the equations which we obtain for the
determination of the second approximation. The two integrals fd'R and
fdt dR/dv must first be treated. It will be seen in Chapter vi. Art. 116, that
the two expressions under the integral sign can contain no constant term when
elliptic values in terms of the time have been substituted for the coordinates ;
therefore, unless n, n are in the ratio of two whole numbers, no term directly
proportional to the time can be introduced by these integrals. W e assume
that n, n' are incommensurable.
Hence the right-hand sides of equations (A) consist entirely of periodic
terms, whose arguments are of the form int + i'n't + A. When therefore the
first of these equations has been prepared for the second approximation, we
may write it
i * ( r2 ) _ + = _ ^B cos (int + i'n't + A),
"
dV
where B is the constant coefficient corresponding to the argument int+i'nt+A.
* An angular coordinate will be considered to be periodic, if its rate of increase with respect
to the time can be expressed by periodic functions only.
B. L. T. 4
50 FORM OF SOLUTION. THE FIRST APPROXIMATION. [CHAP. IV
Let the elliptic value of r be r , and put
, *
- = h ou.
r r
Then 8u is a small quantity of the order of the disturbing forces. Since r*
satisfies the equation when the right-hand member is put zero, we obtain for
the left-hand member, by substituting the above value of r and neglecting
powers of Su above the first,
d?
For the purposes here, since the eccentricity e is a small quantity, we shall
neglect the product eSu and therefore put r<?Su = a 3 &u. Dividing by a 3 and
giving to fj, its value w 2 a 3 , the equation becomes
^- Su + n 2 8u = n^B cos (int + i'n't + A).
66. This is a linear differential equation of well-known form*. Its
solution consists of two parts the Complementary Function, containing two
arbitrary constants, and the Particular Integral. The former may be con-
sidered to be included in the first approximation, which already contains
a similar expression with the requisite number of arbitrary constants. We
are only concerned here with the Particular Integral. The latter is given by
n^B
Bu = 2 - T. -- y-/r, cos (int + i'n't + A).
n 2 - (in + in) 2
There are two classes of terms included under the sign of summation
(a) those in which n is different from in 4- i'ri, (b) those in which n = in + i'n'
for some values of i, i'.
Case (a) is simple. The resulting terms in 8u are of the same period as
those of R and they constitute forced vibrations of which the periods are the
same as those of the disturbing forces.
Case (b). Since n, n' are supposed incommensurable, this equality can
only hold when i' is zero and therefore when i=l. The corresponding
particular integral is then of the form
7)
S 2 nt sin (nt + A).
Proceeding to the third and higher approximations, it is evident that terms
involving t 2 , t 3 . . . in the coefficients will appear. As such forms are contrary
to the assumption of stable motion when only a finite number of them are
* A. B. Forsyth, Differential Equations, Chapter in. This particular form is given on
pp. 61, 62.
65-67] MODIFICATION OF INTERMEDIARY. 51
taken, the question arises as to whether all these powers of t are not in
reality the expansion of some periodic function an expansion which cannot
be convergent unless t be small and whether it is not possible, by including
certain portions of the Sun's action, to get a solution which shall consist
of periodic terms only.
Modification of the Intermediate Orbit.
67. For this purpose we shall examine more closely the first of equations
(A) and see how terms of period 2ir/n may arise through the Sun's action.
Neglect s the tangent of the latitude of the Moon, that is, suppose the
motion to be in one plane; neglect also the ratio of the distances of the
Moon and Sun. Putting m' = ?i' 2 a' 3 , the value of R given in Art. 7 will become
where, as before, v, v are the true longitudes of the Sun and the Moon. As
we substitute elliptic values in the first approximation, we may still further
limit the expression by neglecting e' the solar eccentricity. Then
and we have R = n' 2 r* [ + f cos 2 (v n't - e')].
Also, since r, v do not contain the angle n't + e', when we substitute their
elliptic values the second term of this expression will give no portion free
from the angle 2n't + 2e'. As only those terms which produce arguments of
the form nt + A are sought, we limit R to its first term. Hence
O D O D
i?iV + const., r
the equations (A) and (6) of Chap. II. now become
When the second and higher powers of n' 2 are neglected, the second and
third of these equations give the steady motion
by a suitable determination of the arbitrary constant in the first of the three
equations of motion, this value of r will also satisfy it. It is required to find
the small oscillations about this motion.
/ n' z \
Let r = ail ^ + a? ) . Neglecting powers and products of n' 2 /ri*, x
\ n /
42
52 FORM OF SOLUTION. THE FIRST APPROXIMATION. [CHAP. IV
beyond the first, we obtain from the substitution of this value in the equation
for r,
x + O 2 - f n' 2 ) as = 0.
The solution of this is given by
x = C cos (cnt + D\
where c 2 n 2 = n 2 ^ri 2 , and therefore
Hence the period of the oscillation differs from that of the original motion by
a small quantity of the same order as the small term introduced.
If in finding the oscillation about the state of circular motion we had
neglected ri, the solution would have been
x = C cos (nt 4- D),
which is nothing else than the second term of the elliptic expansion for r in
powers of the eccentricity. If we expand the previous value of x in powers
of w /2 /n 2 , we get
x = C cos (nt + D) + f (ri z /ri>) ntO sin (nt + D),
an expression which immediately shows how the occurrence of t in a coefficient
took place.
68. In order then to make the equations (22) of Art. 50 available as a
suitable first approximation we shall, in the terms dependent on the
eccentricity, put w = cnt + e w, where c is a definite constant which differs
from unity by quantities of the order of the disturbing forces and which is to
be determined in the process of finding the second and higher approximations.
Exactly the same difficulty occurs in the equation for s, which will
evidently give a form similar to that for 8u when we proceed to a second
approximation. The same artifice will serve. We put 77 instead of ij in the
expressions, where rj =gnt + e 0, g being a constant of the same nature as c.
The term nt+einv requires no modification since the difficulty does not
arise in the longitude equation.
Hence, the assumed first approximation to the solution of equations (A)
of Chapter II. wilt be obtained by giving to v } r, s the values (22) of Art. 50,
after we have substituted <f>, y for w } TJ O respectively, where
<j) = cnt + e r, ?; = gnt + e Q.
It is evident that the same change would have been effected if we
had substituted (l-c)nt + iff for w and (l-g)nt + for 0. A physical
67-70] MOTIONS OF APSE AND NODE. 53
meaning can therefore be given to these substitutions. Since -BT and 6 are the
longitudes of the apse and node, the action of the Sun not only produces
periodic oscillations about elliptic motion but also causes the apse and node
to revolve. (Fuller explanations of the physical interpretation will be given
in Chapter viii.) The intermediary chosen may therefore be considered to
be referred to moving axes.
For the subject of oscillations about a state of steady motion, E. J. Routh, Rigid
Dynamics, Vol. u. Chap. vn. may be consulted ; in particular, see Arts. 355-363 of the
same Chapter.
69. Although by this modification of the intermediary we have succeeded in avoiding
the occurrence of secular terms, there is no security that the expressions for the coordinates,
consisting as they do of sums of periodic terms, will actually represent the values of the coor-
dinates at any time. The periodic terms are infinite in number and, in order that they
may give the true values of the coordinates at any time, they must form converging series
for any value of t. At the present time little is known concerning the convergency of these
series. In Poincare's Mtcanique Celeste, certain groups of the terms are shown to converge
for sufficiently small values of the quantity in powers of which expansion is made, but no
definite numerical results have been obtained except in the case of purely elliptic motion
(Art. 54).
The comparison of theory with observation seems to indicate that we are justified in
assuming that these series will represent the motion. Nevertheless it must be stated that
Poincare's investigations just referred to, show that a limited number of terms of a divergent
series may, under particular circumstances, give with great accuracy the numerical values
of the function which the series was intended to represent.
70. The remarks of the previous articles apply also to equations (11) of Chapter n. in
which v is the independent variable. Before we can proceed to a second approximation it
is necessary to express the coordinates of the Sun, which are given in terms of t, in terms
of v ; this causes no difficulty since we have found the elliptic value of t in terms of v in
Chapter in. When this has been done, we substitute the elliptic values of u lt s (Art. 52)
in the terms depending on the action of the Sun. Putting u l (u l ) Q -\-8u lt s=s -f 8s, the
equations for 8wj, 8s immediately take the linear form obtained for 8u in Art. 65, with the
difference that v is now the independent variable. A device similar to that used for
equations (A) can be employed to avoid the presence of v in the coefficients of the periodic
terms. We substitute in the first approximation cv or for v ZJT, ge 6 for v - 6. It will
be seen in Chap. vni. that the constants c, g so defined are the same as those introduced in
Art. 68.
CHAPTER V.
VARIATION OF ARBITRARY CONSTANTS.
71. THERE are several ways of applying the method of the variation
of arbitrary constants (as outlined in Arts. 62, 63) to the problem of disturbed
motion. The assumption that the coordinates and velocities, when expressed
in terms of the arbitraries and the time, have the same form in the disturbed
and undisturbed orbits, lies at the basis of all these investigations. The
intermediate orbit is, in all cases, an ellipse obtained by neglecting the
action of the Sun, and the six elements of this ellipse or functions of
them are the arbitraries used.
The chapter is divided into two parts. The first part contains an elemen-
tary investigation of the differential equations which express the arbitraries in
terms of the time when the action of the Sun is taken into account. In
the second part, the equations for elliptic motion and for the arbitraries in
disturbed motion are treated by the more powerful method of Jacobi.
Certain results which will be required in later chapters follow.
(i) Elementary Methods.
72. We suppose that the equations for elliptic motion have been solved
and that the coordinates and the velocities have been expressed in terms of the
elements and of the time by means of the formulae given in Chapter ill. After
proving certain preliminary propositions, the equations which give the varia-
tions of the six elements a, e, -&, e, 0, i in terms of the resolved parts of the
disturbing forces in three directions, will be obtained. These equations will
be then expressed in terms of the partial differential coefficients of R with
respect to the elements. Finally we shall deduce the so-called ' canonical '
system of equations used by Delaunay.
71-73]
GEOMETRICAL RELATIONS.
55
To find the change of position due to small arbitrary variations
given to the elements*.
73. Consider a set of moving axes defined in fig. 5 by the points where
they cut the unit sphere, the axis of Y being along the radius vector, the axis
of X being 90 behind that of F in the plane of the orbit and the axis of Z
being perpendicular to this plane. Since the coordinates are supposed to be
expressed in terms of the time and of the elements (Chap, in.), small changes
in the latter will produce a change in the position of the Moon which may
be defined by 8r and by small rotations S0 lt S0 2 , S0 3 of the axes of X, Y, Z
about themselves. The point Y coincides with the point M of fig. 4, Art. 44.
Let zZ meet yx in C and let (as in Chap, in.) H be the node, a?fl = 0,
Fig. 5.
hence Cte = - a?0 = - (90 - 0).
geometrical equations -f-, we have then
S0j = sin LSi sin i cos z80 ^
>
J
By Euler's
(1).
Recurring to the notations of Art. 32, let Sf, BE, Siv, expressed in terms
of the elements and of the time, denote the changes in f, E, w due to the
variations a, Be, 8r, Se, Sn. These last are not all independent, owing
to the equation a*n 2 = p. But since n, e only occur in the coordinates
in the form nt + e (Art. 50), we can replace Sw, 8e by the single variation
$6! = tSn + 8e ; the four variations Sa, Se, 8vr t Se! are then independent.
* The assumption laid down in Art. 71 is not introduced until Art. 77.
t E. J. Ronth, Rigid Dynamics, Vol. I. Art. 256.
56 VARIATION OF ARBITRARY CONSTANTS. [CHAP. V
74. We have then the six arbitrary variations Sa, Se, SVT, Se 1} SO, Si and
these will produce changes rSO s , Sr, rStfj in the position of the Moon (whose
coordinates referred to the axes of X, Y, Z are 0, r, 0) towards the positive
directions of the axes. It is required to express the latter variations in terms
of the former.
From Art. 32, we have r = a(le cos E), w = E e sin E. Hence
Sr = - Sa (a cos E) Se + (ae sin E) SE,
a
- Sw + ( - sin E ) Se.
r \r J
Therefore
~ / a \ ~
cw + a I cos E + - e sin 2 E Se
\ r J
~ cos E + e ~
Sw + a -? Se,
- e 2 1 -ecosE
by equations (2), (1) of Art. 32. But w = nt + e-w and therefore
Sw = tSn + Se CT = Se l S-cr.
Hence, transforming the coefficient of Se by means of the relations of Art. 32,
Sr = -Sa+- t ^ L (S 1 -S^)-(acosf)Se (2).
a Vl - e 2
Again z = fW=arg. of lat. =/+ OT - 0; then SL = 8f+ S^ - SO. We
have (equation (3), Art. 32)
5/ $e SE ( 1 a\ , a *
sin7 = rz# + sin^ = (rr^ + ^J Se + ^i^^'
after the substitution of the value of SE given above. Hence, since
sin //sin E = a Vl - e*/r,
we obtain
and therefore, from (1),
S03 = sm/(^ + g^ + JVT^^^
(3).
Since L is immediately expressible in terms of /and of the elements, the
rotation S0j is immediately given by equations (1) ; S0 2 will not be required.
75. To express the partial differential coefficients of R with respect to
the elements in terms of the disturbing forces.
74-75]
VARIATIONS OF R IN TERMS OF THE FORCES.
57
We suppose that the values of the coordinates of the Moon, given in
Art. 50, have been substituted in R ; R will then be a function of a, e, nt + e,
or, 0, i and of the coordinates of the Sun : the latter, being expressed by
elliptic formulae, are considered known functions of the time and of definite
constants and they can therefore be left out of consideration.
The changes in the elements of the Moon denoted by the symbol 8, have
produced changes r&0 2 , Sr, r80 l in the position, towards the positive direc-
tions of the axes of X, Y, Z. Let the disturbing forces in these three directions
be X, $P, 3- Then 9j$ acts along the radius vector, X perpendicular to it in
the direction of motion and 3 perpendicular to the plane of the orbit.
The Virtual Work done by the forces is
Let the corresponding change in 21 be &R. Since the change in position is
produced by variations of the elements only, the Virtual Work is SR and
dR
dR
dR
dR*
/-v \S\M I ^ '-'t/ I /-* x , \ V *-l * O W I *"\/J vv I '"i *
dot- de o (nt -\- ) (TOT uu oi
Substitute in this equation the values of Sr, S0 3 , 8^ previously obtained;
since the variations of a, e, nt -f e, CT, 0, i are independent, we can equate to
zero their coefficients. The six resulting equations will give the values of
r) 7? r) 7?
... -~7 in terms of ^)3, X, 3- Before writing them down we notice that
da 01
since e never occurs except in the form nt -f e,
dR dR
~VJ = ~fc'
Also, dR/da is taken with reference to a, only as a occurs explicitly and not
as it occurs through n.
The resulting equations are easily found to be
ae
a*
<-\ T>
-^4
uu
sn
- $r sin i cos Z,
.(4).
58 VARIATION OF ARBITRARY CONSTANTS. [CHAP. V
Corollary. We deduce immediately
dR dR ~
+ - = Xr,
otz
dR dR dR ~
= h -5- 4- ^0 = 2> cos ^ - 3^ sin i cos L.
CTS oe ou
If r-j be the projected radius vector and v the longitude in the fixed plane,
dR/r^v is the disturbing force perpendicular to the projection of the radius
vector in the fixed plane. Hence, resolving in this direction, we have by
fig. 5, if MM' be perpendicular to xy,
|? = n (X sin SIMM' - 3 cos SIMM')
= r(X cos M. M ' sin SIMM' - 3 cos MM' cos SIMM')
= r( ( cos i 3 sin i cos z),
dR dR dR dR
whence -++-=_-.
0-or oe ou ov
The expression dR/dv implicitly supposes that R is expressed in terms of
r lt v, z. (See Art. 13.)
76. Let ^ be any function of the elements and of the time. The symbol
&<f> denotes the change in </> arising from the changes in the elements only
and therefore SQ/dt denotes differentiation of < with respect to t, only in
so far as t occurs through the variability of the elements and not through its
presence explicitly in <f>. If <f> is a function of the elements only,
rpi u^fn/T-ey u/t 06 dn
T """" v 7 ~T~ ' -f " ^~ v ^ "T" '"
Also, we denote by 9^/9$ the differential coefficient of < with respect to t,
only in so far as t occurs explicitly in <f>. Then
(*<p dd) o(b
As 3r/3^ occurs frequently in the following articles we shall denote it by f.
To find the differential equations required in order to express the elements
in terms of the time in disturbed motion.
77. According to the principles laid down for forming these equations,
the coordinates and velocities, when expressed in terms of the elements and
of the time, are to have the same form whether the motion be undisturbed or
75-77]
THE DIFFERENTIAL EQUATIONS.
59
disturbed. Hence the part of the change in position, due to the variability of
the elements alone, is zero.
Let the variations Sa, ... Si of the elements be now the changes which
actually take place in time dt, owing to the disturbing forces (see Art. 91).
Then - rSft, Sr, rSft, become the changes in position in time dt, due to
the variability of the elements only. We therefore have
j> 5J/5 SJ/)
OY Of 3 /% Ot/j _
dt~ ' T ~dt~ ' r <&~
Similarly Sv/dt = 0.
The three equations of motion of the Moon may be replaced by
.(5).
r_ r __p
dt* dt*~ r* +
M-5
dtr
~dtl~r^v
I A.2
rdt (
I d
(6).
For, by definition, dft is the angle, reckoned in the plane of the orbit,
between two consecutive positions of the radius vector; instead of the
equation for the motion perpendicular to the plane of the orbit, we use
the third of the above equations which, by the Corollary to Art. 75, introduces
the force 3- The first two equations may also be deduced from the general
formulae* for the motion of a point whose coordinates are 0, r, 0, referred to
the moving axes used here, by putting 8ft = = dft.
When the motion is undisturbed, we have
VL
r 2 '
rdt
But since dOJdt = d0 3 /dt + BBJdt and since by (5)
8ft__cZft . _d^r_d/r dv_dv
~di~~dt ) T ~di~dt' dt~dt'
(7).
0, etc. we have
From the second of these we get
Sr
d
* E. J. Routh, Rigid Dynamics, Vol. i. Art. 238.
60 VARIATION OF ARBITRARY CONSTANTS. [CHAP. V
g*^
Let r 2 ^ 3 = h = waVl^?, ry> |? = & = h. cos * (Art. 51).
dc C/&
We have then, by the subtraction of equations (7) from (6),
We shall deduce the required formulae from (5), (8).
78. If we refer to fig. 5, we see that since 8^, 80 3 are, by Art. 77, both zero, the moving
axes, as far as their rotation is due to the disturbing forces only, have the single rotation
8# 2 > th e instantaneous axis is therefore the radius vector. Hence, to get from one point to a
consecutive point in the actual orbit when the values of the elements at the moment under
consideration are given, we calculate the displacement in the plane of the orbit by means of
the elliptic formulae and then give the orbit a rotation 80% about the radius vector. The
effect of this rotation on the position will be of the second order of small quantities.
To obtain the rate of rotation of the orbit, we have from equations (1),
Hence, since 80, = 0,
79. Any line in the plane of the orbit which has no rotation about the axis of Z, is said to
be fixed in the plane of the orbit. Such a line will be absolutely fixed when the motion is
undisturbed ; when the motion is disturbed it will move only with the plane of the orbit.
The point where such a line cuts the unit sphere is also fixed in the plane of the orbit and
has been termed by Cayley * a Departure Point. When the plane of the orbit is in motion,
the line joining any two consecutive positions of a departure point is perpendicular to the
intersection of the orbit with the unit sphere. Hence the curves described by departure
points cut the plane of the orbit at any time orthogonally.
Since the variation in the position of the radius vector, due to the disturbing forces only,
is zero, longitudes and angular velocities reckoned in the plane of the orbit from the depar-
ture point have the same form whether the motion be disturbed or undisturbed.
80. We shall first find the equations for the variations of the elements in
terms of the forces *p, X, 3 : these will be required in Chap. x. It will
be then easy to deduce their values in terms of the partial differential
coefficients of R with respect to the elements by means of equations (4).
The Inclination and the Longitude of the Node.
We have, from equations (8) since h is a function of the elements only,
dh _~ d(h cos i) _ dR
~dt~ ~dt~ ~dv'
* On Hansen's Lunar Theory. Quart. Journ. Math. Vol. i. pp. 112-125, Coll. Works, Vol. in.
p. 19.
77-80] EQUATIONS FOR VARIABLE ELEMENTS. 61
dR dh . j_ . . di ~ , .di
Hence 5 = -j- cos i h sin i -j- = 2> cos i h sm i -=- .
ov at at at
Substituting for dR/dv in terms of X, 3 (Cor. Art. 75) we obtain, after
division by sin i,
., *~ 5-3
Whence, from the first of equations (1), since SOJdt = 0,
. dO ~ r .
sm i -j- = ^ 7- sin L ........................... (10).
The Major Axis.
We have in the ellipse,
V _ 2/^ _ ^
% 2 ~r a'
Submitting this to the operation 8/dt, we get, since &f/dt = ^, 8?-/cft = 0,
Whence, inserting the value of r obtained by putting dr/dw = r/n in Art. 32,
da 2na 3 e sin f^ 2/* & 2 o-
- ^ *P + X
I%e Eccentricity.
We have & 2 = /ia (1 e 2 ).
Differentiating and putting Xr for dho/dt, we obtain
^ $ sin/+
by equation (11). Whence, since na z \/l e 2 = h ,
As a (1 - e*)/r = l + e cos/, r/a = l e cos ., this may be also put into
the form
^ = -$sin/+-(cos/+cos20 .................. (12').
62 VARIATION OF ARBITRARY CONSTANTS. [CHAP. V
The Longitude of Perigee.
Since h<? = pi, we have r/sin/= pe/h 0) and therefore
Applying to this the operation &/dt and substituting as before, we obtain
$ cot/- *=(! + ),..
y sirffdt \r hf)
Whence, since h<? = /*, na = fj,/na?, we have, after inserting the value of r,
/- 2 vnp (i + 7) a; sin/
fj> \ M
e = cos
at
But since S# 3 /cfa = 0, the third of equations (1) gives
dO %L dO . 8/ dv dO
= -77 COS I + -T7 = -,- COSl+^ + -TT -^7
rf^ dt dt dt dt dt
Therefore, substituting for
Epoch.
We have, from the equations (4) and (2) of Art. 32,
nae . - na 2 e .
r = , sin / = - sin E,
Vl e* r
and therefore, since //, = ft 2 a 3 ,
r- = f e sin E sin/
Taking logarithms and applying the operation &/dt, we obtain
-FT + . TT + C0t/-r: .
r e
But, from equations (3) of Art. 32, we deduce
) & Sw 1 de
__
(ft ~ d dt ' 1 - e 2 d sinJ5; ^ ~ sin/eft '
Substituting for 8 (e sin .#)/cfa and then for 8jE/dt in the previous equation, we
shall find that de/dt disappears and that the equation becomes
r esinj^de V^sin/ sinf/dt ersinfdt'
80-83]
EQUATIONS FOR VARIABLE ELEMENTS.
63
Putting rr = ?ia 2 e sin , r sin/= a Vl - e 2 sin i? (Art. 32), this equation
reduces to
r
$w
_ $tt + _ _ VI _^2X/
-13 T 7. J. t/ 7 .
na 2 1 c?i rf^
= Vl^e 2 (^2 sin 2 2* -IT - -^) , by equation (13).
Finally, since Bw/dt = dejdt - dvrjdt (Art. 74), we obtain
(15),
which, by the help of equations (10), (14), gives the value of Sejdt.
81. We might of course immediately deduce the value of df/dt from this by obtaining
the value of dn/dt from that of da/dt in (11). But its value introduces the time in the
form t dn/dt, which possesses the inconvenience mentioned in Art.. 66.
From the definition of cj, we have
fl = f + ftdn = e +nt - jndt,
or nt + f = $ndt + f l .
So that by substituting fndt for nt, we change into fl . Since n only occurs explicitly in
the form nt + e, we shall consider the substitution to have been made. With this understand-
ing the suffix of e t is very generally omitted. The integral \ndt is called the mean motion
in the disturbed orbit.
82. The results obtained may be written, after a few small changes :
da
ae
no, v 1 e 2
Xr,
e 2
- $a cos/4- Xa (l + r ) sin/i + 2 sin 2 J* ^ ,
\ ft(J. 6 )/ Ctt
...(16).
83. Finally, we desire to express the terms on the right hand of
equations (16) by means of the partial differentials of R with respect to the
64
VARIATION OF ARBITRARY CONSTANTS.
[CHAP, v
elements and, if possible, with coefficients which are functions of the elements
only. This can be done by means of the equations (4). None of the substitu-
tions present any difficulty. The results are as follows :
da _ %na? dR
di~ ~fji~ de'
de na (1 -
dt ~ lie
- e* /dR dR
8e
d^ ?iaVl - e* dR
na
dt
dt
dO
3*
di
dt
fj>e
de
na
~da +
fie
1 dR
a /I 2\ I
V 1 e ) -o~ +
7 de
na
& sin i di '
dR
na (1
fji^/l _g2|sini
~ Q + tan U 1 + 5
dO 2 \de d'sr
...(17).
These equations are obtained without the intervention of ^>, X, 3 by C. H. H. Cheyne,
Planetary Theory ', Chap. n.
The ordinary Canonical System of Equations.
84. The system of equations just obtained is by no means the simplest
in form ; by taking certain functions of the elements used above, to form a new
set of elements, we can reduce the equations to a very convenient form. Let
! = (e r)/W,
2 = " ~
3 =
The equations for the new elements a, j3 take the form, known as canonical,
(19),
d@ 3 _ dR
~dt~da 3
dt
where ^2 is now supposed expressed in terms of a lt a 3 , 3 , ft, ft, ft,
85. To prove these we first notice that since the second three equations in (18) do not
contain e, or or 0, we can immediately deduce from the first three*
dd da 2
dR
1 dR dR
n
9a
9e
* It is to be noticed that the expressions 6.R/50... suppose that R is expressed in terms of
a, e...i and dR/Ba^.. that .R is expressed in terms of Oj, o 2 ...j8 3 .
83-86] DEDUCTION OF A CANONICAL SYSTEM. 65
For, since 6 is contained only in 03, a 3 , we have
9/2 9/2 9d2 9/2 9a 3 _ 9/2 9/2
^ = 9^ M + 9^ 3 90 = ~9^ + fo 3 '
and so on. Hence, by the Cor., Art. 75,
9/2 9/2 9/2 9/2 9/2 ~ 9/2 9/2 9/2 9/2 9/2
9^ =n 97' 9^ = 9r + 9^ = * r ' 9T 3 = 97+9^ + 90 = 9^
We therefore have, by equations (17), (8),
dpi_jj^da_ 9/2 _ 9/2
~dt~2tf~di~ n aT~cV
d&_^ _<jv_?^
dt~ dt~ 9o2 J
dgs _ <(A cosz) _ 9/2 _ 9^
eft ~ ~dt~ ~ 90~9a 3 '
giving the first three of equations (19).
86. Again, by the fifth of equations (17), we have
daz_d6_ 1 9/2 = 9/2 _9/2
dt ~ dt ~ A sin z 9i ~ A 9cosi 9/3 3 '
for i only enters into R through the element /3 3 .
Also, by the third and fifth of the same equations,
da^ _ d-aj dO _ A 9/2 cos i 9/2
dt ~ dt dt ~ pea de A sini 9i
A 9/2 .9/2
= - ^5 hcosi .
pea de^ 9^
But since e enters into R only through /3 2 j $3, we have
9/2 _ 9/2 9/3 2 9/2 90 3 _ /9/2
"" + " ~
since h Q =na \fl-e 2 . Substituting this value of 9^/9e in the previous equation we obtain,
after putting fj.n z a 3 , the equation da z jdt= -9/2/9# 2 .
Finally, since brjdt, dtijdt, 80 3 /cfo, are all zero and since the variations 5 are now those
which actually take place, we have from Art. 75,
Therefore
9/2 doi dRdaz .dRdas dRdh dR d& , M^? =
9 fll ^ 9o2 ^ + 9a 3 dt + dfa dt + 9/3 2 ^ + 9/3 3 cfe
Substituting for 2 , , i, 2 , the values just found, the second and fifth
cit dt dt dt dt
and the third and sixth terms respectively cancel one another ; after division by 9/2/9a x the
equation becomes dajdt= -9/2/9/Sj.
B. L. T. 5
66 VARIATION OF ARBITRARY CONSTANTS. [CHAP. V
87. All systems of elements which satisfy equations of the form (19) are said to be
canonical.
Other canonical systems of elements and the conditions which must be satisfied in
order to transform from one canonical system to another, will be found in the works of
Jacobi, Dziobek and Poincare referred to below. The general form of this transformation
is that known as tangential (Beriihrungstransformation).
88. The method of treatment given in this Chapter that of causing the elements to
vary in order to include the disturbing forces is more generally useful in its applications to
the planetary than to the lunar theory. The equations for the variations do not admit, any
more than the equations of motion examined in Chap, n., of a direct solution and we are
obliged here also to use some method of approximation. This proceeds according to the
plan explained in Chap. iv. We first find the values of dR/da... so that the right-hand
sides of the equations (17) become functions of the time and of the elements. To solve, in
general we may first consider the elements on the right-hand side to be constant or
we may combine the equations in any suitable manner to make them integrable ; we thus
obtain the values of the elements in terms of the time and of six new arbitraries. Using
these new values in the terms on the right-hand sides, we again get the latter expressed as
functions of the time and of absolute constants and we can proceed in this way until the
desired accuracy is obtained ; the new arbitraries introduced at each step can be determined
so as to simplify the final expressions as much as possible.
In the lunar theory, the necessity for a large number of terms and for many
approximations causes the process to become very tedious. Delaunay's theory (Chap, ix.)
the only one worked out on these lines is very fully expanded, but the labour of
obtaining the expressions was enormous and the results leave much to be desired. It
is also to be remembered that we cannot start by giving the constants their numerical
values a literal development is usually essential. Hansen's theory (Chap, x.) is not really
treated after this method. He uses the variable arbitrary constants in order to obtain
certain functions for the motion in the instantaneous plane but, having done so, he is able
to use numerical values for his constants from the outset.
In the planetary theory, secular terms that is, terms increasing in proportion with the
time appear, and also terms with large coefficients and of long period : these are very
much more easily managed by considering them as attached to the elements than by con-
sidering them as corrections to the coordinates.
89. One of the most important properties of the equations and of the corresponding
equations for all sets of elements which may be used is the fact that the coefficients of
the partials dR/d\ (where X is any element) are independent of the time explicitly, that
is, they are functions of the elements alone. The time only occurs explicitly on the right-
hand sides through the presence of the coordinates of the Sun in R. See Art. 99.
It will be noticed that the method practically replaces three differential equations of the
second order by six of the first order. For obtaining literal developments of the coordinates
this is of doubtful advantage, but for theoretical investigations it is of the highest importance.
Canonical systems of elements, as used by Poincare and others, have been shewn to be of
great value in this respect.
90. It is necessary to notice very carefully the meaning attached to dR/da in equations
(17). By means of the equations of Art. 50, R is expressed in terms of a, n, e, e, t<r, 6, i and
87-93] OBSERVATIONS ON THE PREVIOUS RESULTS. 67
there exists between a, n the relation 7i 2 a 3 =/n. It will be noticed that a only occurs as a
coefficient and that n only occurs in the form nt + (. Hence we must not use the relation
n 2 a 3 = p before forming dR/da but differentiate with respect to a only as it occurs in R ex-
plicitly. In the canonical system of equations (19) this difficulty is not present.
The replacing of nt + c by ^ndt+f l does not cause any trouble, since 9/2/3=9/2/9e 1 .
91. Attention must be drawn to the meaning of the symbol 5 as used in Arts. 73 75
and as used later. In the first case the variations for each element were quite arbitral**/
and it was therefore permissible to equate the coefficients of each of them to zero. Later
they were the variations actually taking place, owing to the disturbing forces. Thus, when
the variations were arbitrary, 8R had a certain value depending on the arbitrary variations
of the elements only ; when the variations were the actual ones it was seen (Art. 86) that
5\ /?
8R = -j-dt=Q. This last equation is merely a direct consequence of the fact that R is a
function of the coordinates only and not of the velocities and therefore that
8R _ dR &c 3R ty dfi 82
-di-fadi + dydt + Tzdi>
this expression is zero, since the velocities have the same form in disturbed and in un-
disturbed motion. This fact is used in Art. 86 to obtain the sixth equation when the other
five have been found. It might equally have been used in Art. 80 to obtain dfjdt. The
process would however have been somewhat longer.
92. The canonical system (19) is much more easily found by the method of Jacobi. In
fact the natural way is to obtain these equations first and then to deduce the results of
Arts. 83, 82. With the transformations given in Arts. 85, 86, it will be quite simple to
reverse the process.
The equations (19) have been obtained by R. B. Hayward* in a direct manner.
The constants used may be defined geometrically and dynamically as follows :
- a t = Time of passage through the nearer apse,
a. 2 = Distance from node to perigee,
a 3 = Longitude of node;
0! = Constant of Energy,
/3 2 = Twice area described in a unit of time in orbit,
plane.
(ii) The methods of Jacobi and Lagrange.
93. We shall now give a short account of the applications of the general
dynamical methods of Hamilton, Jacobi and Lagrange to the problem of
disturbed elliptic motion. In chronological order those of Lagrange should
come first; their application to the discovery of equations (17) is however
long and therefore his results will be stated only in so far as they are necessary
* " A direct demonstration of Jacobi's Canonical Formula," etc., Quart. Jour. Math. Vol. in.
pp. 2236.
52
68 VARIATION OF ARBITRARY CONSTANTS. [CHAP. V
for the explanation of Hansen's methods. The results of Jacobi's dynamical
methods, which were based on those of Hamilton, will also be merely stated ;
references will be given to the more advanced treatises on Mechanics in which
the proofs may be found.
94. The Methods of Hamilton and Jacobi.
Let T be the kinetic energy and F the force-function of a dynamical
system. Suppose that there are n degrees of freedom and let q lt q z ... q n be
the coordinates defining the position at time t. We suppose that there is no
geometrical equation connecting the coordinates, that F is expressible in
terms of the coordinates and of the time only, and that T does not contain
the time explicitly.
Let the velocities be q l} q z . . . q n ; then T is a function of q i} qi (i = 1, 2 . . . n).
Let
the quantities p^ are called the generalised component momenta of the
system or, more simply, the momenta.
Since T is a quadratic function of the velocities it can be also expressed
as a quadratic function of the momenta in the form
where A^ is a function of the coordinates only.
Theorem I. The equations of motion may be put into the form
. = dH _dH
where H=T-F*.
The principal function 8 is defined by the equation
S=j t (T+F)dt
Suppose that the dynamical equations have been solved and that 8 has
been expressed in terms of the coordinates, of the 2n necessary arbitrary
constants (exclusive of the constant to be added to 8 by definition) and
of the time. We then have
Theorem II.
* E. J. Routh, Rigid Dynamics, Vol. i. Art. 414.
93-95] ELLIPTIC MOTION BY JACOBl's METHOD. 69
Theorem III. S satisfies the partial differential equation
ds /asfy as as
ar+^iiU- + 2^1 ]2 5- + ...-^=0.
dt \dqj 9gr, 3g 2
(This follows immediately from Theorem n. by the use of equation (21).)
Theorem IV. If, knowing only F and the coefficients Ay, we can discover
any integral of this partial differential equation, involving n independent
arbitrary constants &, &...& (exclusive of that additive to S), of the form
S = (f> (q lt q,, ... q n , &, &, ..., *);
the n complete integrals of the dynamical system will be given by the equations
the ctj being n new independent arbitrary constants*.
Solution of the Equations for Elliptic Motion by Jacobi's method.
95. We shall first apply these theorems to the problem of simple elliptic
motion. There being three degrees of freedom, choose as coordinates the
radius vector r, the longitude v reckoned on the fixed plane and the latitude
u above this plane. We take the mass of the Moon for simplicity to be
unity, so that F=fi/r. The velocities fa are r, v, u and
2T=r 2 + (r 2 cos 2 u) v 2 + r 2 u 2 .
Hence from equation (20) the momenta will be
'= (r*C08 3 Zrtv = 7^^=
dr ' dv ' du'
and therefore
The partial differential equation satisfied by S is then (Theorems II., ill.)
__
a^"" 2 LVar/ r 2 cos 2 u \dv r 2 \du
We first require some integral of this involving three independent arbitrary
constants (Theorem IV.).
* Id., Vol. ii. Chap. x. Theorems n., in. are given fully. The result similar to Theorem
iv. for the characteristic function only is proved, but the proof for the principal function
is almost identical and may be easily reproduced.
70 VARIATION OF ARBITRARY CONSTANTS. [CHAP. V
96. To find one, assume
where S l contains neither v nor t and ft, /3 3 are arbitrary constants. Sub-
stituting, we obtain
A* i/afly 2^_
+ *W/' r ~ tt '
2 being a constant, so that
and assume S 1 = 8 2 + 8 3 ,
where S 3 is independent of r. Then, from the equation for S 1} we have
ru _
whence S 3 = I du Vy3 2 2 - /3 3 2 /cos 2 j7.
Substituting the values of $ 2 , ^ 3 , 8 1 in the assumed expression for S, an
integral of the equation for 8 is therefore given by
This contains three independent arbitraries ft, /3 2 , /3 3 . The constant additive
to $ may be fixed by inserting any lower limits to the integrals. Let that of
the second integral be and that of the first r a> where r a is the smaller root
of the equation
l .
r r 2
By Theorem IV. the integrals of the equations of motion are given by
. Whence
A- \-
3 ~
96-98] VARIATION OF ARBITRARIES BY JACOBl'S METHOD. 71
The parts of a lt 2 due to the differentiation with respect to the limit r a , are
.
and they therefore vanish by the definition of r a .
97. It only remains now to connect the six constants a t , <%, a 3 , ft, ft, ft
with those ordinarily used in elliptic motion.
Let the two roots of the equation defining r a be the greatest and least
distances in the ellipse, that is a (1 + e), a (1 e). We then have
*<!-*> g. * |,
whence ft = -^/2a, ft = V/m (1 - e 2 ) = h Q .
Again, as ft 2 - ft 2 /cos 2 u must be always a positive quantity, we give to u
its greatest value i and to ft a value such that the expression is then zero.
Hence
ft = ft cos i = h cos i.
Further, a x is the value of t when r = r a , that is, at perigee where the
mean anomaly nt + e is zero. Hence ! = ( w)/n.
Also, 3 is the value of v when tr= 0, that is, at the node. Hence 3 = 6.
Finally, we have
* " COS
Let sin r=sinisinz. Then (Fig. 5, Art. 73) since u = M'M, L is the
angular distance from the node to the radius vector as in Art. 44 ; the value
of the integral becomes L by the substitution. Hence 2 is the value of L
when r = r a , that is, at perigee : therefore a^ = TS 6.
The system of constants is then the same as that given by equations (18).
It is now very easy to obtain the canonical system (19).
98. Variation of Arbitrary Constants by Jacob? s n^ethod.
We have by Theorems IL, iv.
as as
* = W ^Wi'
Since S is a function of the independent quantities t -, <?;, these may be
written in one equation,
(22),
72 VARIATION OF ARBITRARY CONSTANTS. [CHAP. V
where 'ZpSq = pi$qi + ... +p n &ln> e tc. ; the operator S may denote any varia-
tion whatever.
The Hamiltonian equations are
Suppose we put F= p/r + R. Let the values of q i} p i} already obtained
for the case R = 0, be made to satisfy the equations when R is not zero, by
considering c^, ft variable. (This is merely another example of coordinates
and velocities having the same form in two problems when they shall have
been expressed in terms of the arbitraries and of the time.)
Let A<#, Ajpf be the small increments to be added to q i} p t in time dt,
due to the presence of R. Then from the Hamiltonian equations we have,
since T is unaltered,
dR ^, A dR 7
A = -^*' * pi = d^ dt -
These, expressed in one equation, give*
(23).
Again, as S denotes any increment, it may have the value A so that, from
equation (22),
AS = S (p&q + A/3).
Whence 2S (p&q + A/3) = SA# = ASS = 2 A (p$q + aS/3).
Therefore, as 8 (p&q) = Sp&q + p&Sq, etc., this equation gives
2 (SaA/3 - S/3Aa) = 2 (SgAp - SpA^) = ^8J?,
by (23).
Finally, as A denotes the actual increment due to the presence of R,
we have
'
Therefore, substituting in the equation just obtained,
dcn~dt' 8ft~ dt 1
where R is now supposed expressed in terms of t, OL I} ct 2 , ... A, $> ....
* Here Ap5j denotes the product of Ap and 8q, and so elsewhere.
98-99] LAGRANGE'S METHOD. 73
From this result it is evident that Jacobi's method of solution produces a
system of canonical constants. In the case of disturbed elliptic motion, we
shall therefore have as one system the values of cti, /3; given in Art. 97. From
the equations just found we can deduce (17) by reversing the processes of Arts.
8486.
Lagrange's Method.
99. Suppose for the sake of simplicity that in a dynamical problem there
are three degrees of freedom and that the complete integrals are
qi = <f>i (71 > 72> 7s> 74. 75, 7e, *)> Pi = & (7i> 7a, 7e, 0> (*' = 1. 2 > 3)
where q, p are defined as before and the system of constants of solution ^ . . . y s
is quite arbitrary. Since the constants are independent, we may suppose
them determined in terms of the coordinates and momenta by equations of
the form
Let now R be added to the force function and let the solution be made to
retain the same form by considering the arbitraries 7; as variable. Lagrange
has shown that the six equations which determine the yi are
dy t .dR .dR .dR
^ ('Yi 'Ya/ 1~ \7i ^Vs) ~f~ ~f~ (*Yi ( YR)
at ^72 ^7s ^7s
dy 2 _ .dR dR dR
- fa *%-***?%} + +
~
where (V
(7l
It is evident that after the differentiations have been carried out, the
coefficients (7^, jj) can be expressed in terms of the arbitraries and of the time.
Lagrange has shown however that, when so expressed, t is not present ex-
plicitly in any of these coefficients, so that the equations (24) only contain the
time explicitly through its presence in dR/dyi*. The equations (17) are a
particular case of these results and were so obtained by Lagrange f; the
original problem is that of undisturbed elliptic motion, R is the disturbing
function and to the arbitraries 7^, are given the values a, e...0.
It is easy to see that any function \ of the 7$ which does not contain
t explicitly, may replace one of the 7$, say y lt and that the equations
* Proofs of these results are given by Eouth, Rigid Dynamics, Vol. n. Arts. 477, 478 and
by Cheyne, Planetary Theory, Appendix.
t Mec. Anal., Pt. n. Section vn. Chap. ii. See also Tisserand, Mec. C61. Vol. i. Chap. x.
and 0. Dziobek, Math. The. der Planeten-Bewegungen, 10, 11.
74 VARIATION OF ARBITRARY CONSTANTS. [CHAP. V
(24) will be still true if in them we replace ^ by X: for the system of
constants % was arbitrary. The disturbing function is then supposed to be
expressed in terms of X, 72, 7s 7e> t.
100. The transformation of the equations for 7^ % ... to those for X, 72 ...
might also have been made directly by means of the assumed relation between
X and the 7^. This way of looking at the problem enables us to give an
extension to the meaning of X. If we define X by equations of the form
X = ^Ad or d\ = ^Ad
where the AI are functions of the 7^ only although the expression SAidji
may not be a perfect differential, the equations corresponding to (24) for
X, 7 2 , ... will still hold because the direct transformation only involves the
differentials of the arbitraries.
When R is not expressible explicitly in terms of X, 7 2 , ..., the expression
dR/d\ may be defined by the equation
dR , dR
aT^a*'
that is, dR/d\ has the same meaning as if R had been expressible in terms
of the new arbitraries. With this convention it will be unnecessary to make
a direct transformation.
101. Elements defined in this latter way have been called by Jacob! pseudo-elements*.
Hansen defines ideal coordinates to be such that they and their first differentials with
respect to the time have the same form whether the motion be disturbed or undisturbed*.
Such are r, r ly v, #, y, z, etc., these being all referred to fixed axes. We can however have
ideal coordinates referred to moving axes.
Consider a set of rectangular axes of which those of JT, T are in the plane of the orbit
and that of Z is perpendicular to it. Let the axis of X be placed at a departure point. Let
Fig. 6.
See the letters of Hansen and Jacobi referred to at the end of Art. 102.
99-102] PSEUDO-ELEMENTS AND IDEAL COORDINATES. 75
v l = XM=the longitude of the Moon reckoned from the departure point. Then (Art. 79)
v and dv l have the same form whether the motion be disturbed or undisturbed ; v is there-
fore an ideal coordinate. And yet v l is not expressible in terms of the elements and of
the time unless one of these be a pseudo-element. For if XQ = cr, we have from the figure,
since the line joining two consecutive positions of X is perpendicular to Q.X,
If then v be one of the coordinates in terms of which R is expressed, there will be present
in R the pseudo-element a.
102. Let us see how R will be expressed when these axes are used. Euler's formulae
of transformation of the coordinates of a point (xyz} to (XYZ) are*
l Y-\- c^Z,
where (o^Ci), ( 2 ^2 C 2)> (^3^3) are *^ e direction cosines of the axes of X, Y t Z, referred to
those of x,y,z: they are trigonometrical functions of o-, 0, i. If (XYZ] be the coordinates
of the Moon, we have Z=0. Hence R is expressible in terms of X, F, o-, 0, i, or in terms
of r, v lt o-, 0, i.
Here the differentials of R with respect to v lt o- have a meaning without further
definition. For dv^ is the angle between two consecutive positions of the radius vector
reckoned in the plane of the orbit and therefore dRjrdv l = Xj dRldr = ty ; the force perpen-
dicular to the plane of the orbit will now depend on the differentials of .ft with respect
to <r, 0) i (see Chap. x.).
The use of the pseudo-element a introduces another arbitrary constant, namely, the
value of a- at the origin of time.
The general condition that X, Y, Z may be ideal coordinates when the new rectangular
axes are any whatever, is
x da l +y da 2 + z da 3 = 0,
for then dX, dY, dZ will have the same form whether a^ ^ ... be constant or variable.
These three conditions, involving only the differentials of a lt a 2) ... are available whether
the elements be true elements or pseudo-elements.
On the subjects of Arts. 100-102, see
Lagrange, M4c. Anal. Pt. n. Sec. vn. No. 70.
Binet, "Sur la Variation des Constantes Arbitrages," Journal de Vficole Polytech-
nique, Vol. xvn. p. 76.
Hansen, " Auszug eines Schreibens," etc. ; Jacobi, " Auszug zweier Schreiben," etc.
Crelle, Vol. XLII. pp. 1-31.
Hansen, " Auseinandersetzung einer zweckmassigen Methode zur Berechnung der
absoluten Storungen der kleinen Planeten," Abh. d. K. Sachs. Oes. d. Wissensch.
Vol. v. pp. 41218.
* P. Frost, Solid Geometry (1875), Art. 146.
76 VARIATION OF ARBITRARY CONSTANTS. [CHAP. V
Cayley, " A Memoir on Disturbed Elliptic Motion," Mem. R. A. S. Vol. xxvu. pp.
1-29 ; Coll. Works, Vol. in. pp. 270-292.
Donkin, "On the Differential Equations of Dynamics," Phil. Trans. R. S. 1855,
Pt. II. pp. 352-354.
103. We can very quickly deduce a set of canonical equations from the formulae (24).
Let y lt 7 2 , 73 be defined as the values of q lt q 2 , q 3 at time t = r and 74,75, y 6 those of
P\i P2-> Ps a ^ ^ ne same time. Since, in the process of forming the partials *$ } _^ etc., it
"Pi **li
makes no difference whether t be constant or variable, and since t ultimately disappears
from the coefficients (7^, 7,-), we can give to t the value r, that is, we can put q l =y l etc.,
before forming these coefficients. We shall then have
Whence, since all the arbitraries are independent, we obtain
(7i> 74) = ! = - (74 7i)> (72 7s) = ! = - (75 72)* (7 3) 7e) = ! = - (7e> 7s) J
and all the other coefficients (y i} 7,-) will be zero.
Denote the values of the coordinates and momenta at time T by %, p if We have in
the present case 7 1 =Qi ..., y^ =p \ an d therefore equations (24) become
Hence the values of the coordinates and momenta at a given time form a system of
canonical constants ; p i} QJ are considered here as the arbitraries of the original solution
and R is supposed to be expressed in terms of them and of the time. This system of
canonical constants was first given by Lagrange.
104. The disturbed values of <&, P* are given by
the lower limits of the integrals giving new arbitraries which are absolute constants.
Suppose the integrations have been performed by any process so as to give the disturbed
values of p i5 <&; these latter will then be functions of t, r and of absolute constants.
But since the results must hold for every value of r, that is, at any point in the orbit, we
shall get the disturbed values of the coordinates and of the momenta at time t by putting
T = t in these equations. Whence
where the bar denotes that T has been changed into t after the integrations have been per-
formed.
This extension is due to Hansen*. It may also be written
* " Commentatio de corporum coelestium pertnrbationibus," Astr. Nach. Vol. xi. Col. 322.
102-105] REFERENCES. 77
where, under the integral sign, we suppose disturbed values substituted. From the former
of these results the general theorem, which lies at the basis of all Hansen's researches into
the lunar and the planetary theories, can be deduced. As however the form in which he
uses it can be exhibited as an elementary result of the integral calculus, it will not be
proved here.
The theorem in question is constructed to prove that any function of the elements and
of the time may be differentiated, the disturbed values of the elements substituted and the
result integrated, with the time as far as it occurred explicitly in the function constant
during the whole process*.
105. The earlier literature on the general dynamical principles of Lagrange, Hamilton
and Jacobi and on their applications to the subject of this chapter, is very large. It has
been collected and a summary of the results is given by
Cay ley, " Report on the recent Progress of Theoretical Dynamics," B. Ass. Rep. 1857 ;
Coll Works, Vol. in. pp. 156-204. See also "Report on the Progress of the
solution of certain problems in Dynamics," B. Ass. Rep. 1862 ; Coll. Works, Vol. iv.
pp. 514, 515.
The chief original memoirs are to be found as follows :
Lagrange, Me'c. Anal.
Poisson, " Mem. sur la variation," etc., Jour, de Vfic, Poly. Vol. vili. pp. 266-344.
Hamilton, "On a general method in Dynamics," etc., Phil. Trails. R. S. 1834,
pp. 247-308 ; 1835, pp. 95-144.
Jacobi, Vorlesungen iiber Dynamik.
The following treatises may also be consulted with great advantage :
Thomson and Tait, Natural Philosophy, Vol. I. Chap. n.
Tisserand, Mec. Ce'l. VoL I. Intro, and Chap. ix.
Dziobek, Math. The., etc., Abschnitt n.
Poincare', Mec. Cel. Vol. I. Chap. I.
* Fundamenta, pp. 22-25. A proof by Taylor's Theorem is given in the Commentatio, etc.
Cols. 323-326.
CHAPTER VI.
THE DISTURBING FUNCTION.
106. IN the equations of motion obtained in Chapter n. we have
expressed the forces in terms of the partial differential coefficients of R or F.
In order to obtain the forces in terms of the variables, R must be suitably
expressed. The object of this Chapter is to find expressions of such a
form that the labour of making the developments may be as small as
possible.
We have seen that with the methods of procedure usually adopted in
the lunar theory, the second approximation to the values of the coordinates
is obtained by substituting the results of the first approximation in the
terms previously neglected. In general, the first approximation being an
ellipse, this amounts to expressing the disturbing forces in terms of the
elliptic elements and of the time.
Now the determination of motion in space requires a knowledge of three
component forces. If we form these forces directly from the general expres-
sion of R in terms of the coordinates a process easily performed and then
develope the results in terms of the elliptic elements and of the time, there
will be three developments to be made. To save this labour we develope R
in terms of the elements and of the time ; the forces can then be deduced
by transforming their differentials with respect to the coordinates into differ-
entials with respect to certain functions of the elements and of the time
which occur explicitly in the development of R.
The principal object is then to develope R in terms of the time and
of the elliptic elements of the orbits of the Moon and the Sun. According to
the different functions of the elements used, there will be slightly different
forms of expression. They can, however, be all deduced from those given in
Section (iii) which contains Hansen's method. In connection with de Ponte-
eoulant's method, some general properties of the disturbing function will be
given. The variety of forms by which R can be expressed arises from the
fact that R depends only on r, r' and on the cosine of the angle between r, r'.
106-108] INITIAL EXPANSION OF R. 79
107. In the lunar theory, as already stated in Art. 9, we always begin
by expanding the disturbing function in powers of r\r'. We have (Art. 8),
T>_ _ ^ __ /a*
Let <S> be the cosine of the angle between the radii vectores of the Sun
and the Moon. Then xx + yy' + zz = rr'S,
Expand the first term of this expression in powers of rjr' by means of the
Binomial Theorem or by the use of Legendre's coefficients *. The first term,
which is ra'/r', may be omitted since it does not contain the coordinates
of the Moon ; the second term mrS/r'' 2 , will be cancelled by the term
- m'rS/r' 2 .
We therefore obtain
(i) Development of R necessary for the solution of Equation (A),
Chapter H. The Properties of R.
108. The first process is to develope R in terms of r, v, s, r, v'. Let m
be the place on xy (fig. 4, Art. 44) where the radius vector of the Sun cuts
the unit sphere. According to the notation previously used, we denote by
v, v' the true longitudes of the Moon and the Sun reckoned from x y and by s
the latitude of the Moon above the plane of xy.
From the right-angled triangle MM'm we have,
cos (v - v') = cos M 'm = cos.Mm/cos M'M = SJ(1 + s 2 ).
Hence S = C *^-^ = (I-^ + ^-...)cos(v-v') (2).
Substituting this value of S in (1) we have, neglecting s 4 , raV 4 // 5 and
* Todhunter, Functions of Laplace, Lame and Bessel, Chap. i.
80 THE DISTURBING FUNCTION. [CHAP. VI
higher powers of s 2 , r/r, and replacing powers of cosines by cosines of
multiples of (v v'),
E =-- (i + f (1 - s 2 ) cos 2 ( - v') - Is')
cos 0" - ^ + f(l- fs 2 ) cos 3 (
+ ................................................................... (3).
109. We have now to express R in terms of the time and of the elements
of the elliptic orbits of the Moon and the Sun, according to the principles laid
down in Chapter IV. Before doing so it is necessary to know something
further about the numerical values of e, e f , y, a/a' (in powers of which
the expansions will be made), in order that we may have some idea of
the number of terms necessary to secure a given degree of accuracy in
the results. We ought strictly to know the meanings to be attached to
these constants when the motion is disturbed ; but since in any of the
systems used to fix their meaning, the numerical values only vary to a slight
extent, for the purposes in view here it is sufficient to give a general idea of
their magnitude in the case of the Moon.
The most important ratio is that of the mean motions ri, n. It does not
occur directly in the expansion of jR; it will be seen, however, in Art. 114,
that ri z /n* is a factor of R. The numerical values are approximately,
n' , a
=1T3> e =20> e =SO> r Y = TI> ^ = ^T'
We consider n'jn to be a small quantity of the first order. Consequently
n'/n, e, e', y are small quantities of the first order and a/a' is one of the
second order.
On the basis that 1/13 is of the first order, e' 2 = -^Q would be of the third order,
e'a/a'= ^oo of the fourth order, and so on. But for simplicity we shall consider them of
the order denoted by the index. Hence (n'/n)f > ie p 2e' p 3y^(a/a')Ps will be said to be of the
order
110. The equations of Art. 50 in which w, TJ O are, by Art. 68, replaced by
cut + e w = <j>, gnt + e rj, give the values of r, v, s in terms of the ele-
ments a, n, e, e, r, 6, y and of the time. If, in the same equations, we accent
the letters and put c' = 1, g' = 1, they will give the values of r', v', s' in terms
of the time and of the elliptic elements a', n', e', e', TO-', 0', y. But since the
Sun's orbit is in the plane of reference, y' = 0, s f = : with these values 6'
disappears. Substituting in R, the disturbing function will be found ex-
pressed in terms of t and of the elements a, n, e, e, w, 6, y, a', n', e', e', TO-'.
108-113] PROPERTIES OF THE DISTURBING FUNCTION. 81
The form of the development of E.
111. Let %=(n-n')t + 6-e.
Since v t v' only occur in R in the form cosp(v v'\ (p any integer)
and as (Art. 50),
where A and A' consist only of periodic terms depending on the arguments
<f>, TJ and <f>'(= n't + e' CT') respectively, we have
cosp (v v') = cosp% cosp (A A') sin p% sin p (A - A').
Also, A, A' being small quantities of the first order at least, we suppose
that expansions in powers of A, A' are possible. Hence
Therefore, all the terms arising from v, v' can be expressed by means of
cosines of sums of multiples of the angles f, <j>, 2?/, <'.
Finally, r, r' and sr being expressible in terms of <, <' and of <f>, 2rj,
respectively, R can be expressed by a series of cosines of sums of multiples of
the four angles f , (/>, <', 2rj, with coefficients depending on m', a, e, rf, a', e'.
112. Owing to the introduction of c and g, the coefficients of t in these arguments
and in all the arguments which are present in R, will never vanish unless the argument itself
vanishes. For these coefficients of t will all be linear functions with integral coefficients,
of n - ri, en, ri, g?i, that is, of n, n', en, gn ; it will be seen later that c, g are not in general
commensurable with an integer or with one another, and n'/n was assumed to be an
incommensurable ratio. Hence no linear relation with integral coefficients will exist.
113. The connection between the arguments and the coefficients.
The constant y enters into R only through its presence in v, s. Since
only even powers of s are present in R, a glance at the equations of
Art. 50 will show that only even powers of y are present in R.
Also, if we leave aside the factor raV/a' 3 which arises from raV 2 // 3 ,
equation (1) shows that even powers of a/ a' in the coefficient of any term will
accompany even multiples of v v' and therefore of f in the argument of
that term ; similarly, odd powers of a/a' accompany odd multiples of f.
Combining these results with those of Arts. 40, 47, 111, we see
(a) that the arguments of all terms in R are of the form
jf p<t> p'<t>'
B. L. T.
82 THE DISTURBING FUNCTION. [CHAP. VI
(6) that the coefficient of the term having this argument is at least of
the order
ePe'P'yW or e*> e'?' y^ , ,
a
according as j is even or odd ;
(c) that any term in the coefficient is of the order
where p 1 , p/, q l are respectively equal to p, p', q or are greater than them by
even integers, and j, j^ are odd or even together.
The factor e p e fp 'y^ (or e p eVrfQa/a') which occurs in the coefficient of the
term with the above argument may be called the characteristic part of the
coefficient or, more simply, the characteristic.
De PonUcoulanCs expansion for R.
114. Since ra' = n' 2 a' 3 , p = n 2 a 3 , we have
ra'a 2 fin' 2 a , . ri
T =-- = - m 2 , where m .
a 3 a n z a n
It will be found convenient in de Pontecoulant's theory to choose the units
of mass, length and time so that /* = !. With these units, m' is the ratio
of the mass of the Sun to the sum of the masses of the Earth and the Moon.
We can now put n*a 3 = 1, and
m V m 2
( .
The development of R, complete as far as the first order in e, e', y z , a/a',
is given below ; for the sake of illustration, some terms of higher orders are
included. The shortest method of actually performing the expansions will be
explained in Arts. 124 126.
= ?[
- \e cos < - \e cos (2f -</>) + %e cos (2f + <)
+ f e' cos $ + Qe' cos (2f - </>') - e cos (2f + (//)
- f 7 2 - 7 2 cos 2? -f f 7 2 cos 2^ + fy 2 cos
+ f e 2 - -Je 2 cos 2</> -f . . . + f e /2 + |e /2 cos 2<' + - 5 F V 2 cos (2| - 2<//)
.(5).
113-116] PROPERTIES OF DERIVATIVES OF /?. 83
To deduce the disturbing forces.
OD O TO J} ~D
115. We have now to form -TT- , -- , -^- . Since the disturbing function
dr dv ds
has been expressed in terms of the elliptic elements, these partial differentials
must be transformed so that we can deduce the functions which they re-
present directly from (5). For the purposes of this and of the next article,
the factor m z /a must be supposed to be replaced in (5) by its value mV/a /3 .
In the first place, since a only enters into R explicitly through r, and
since r is of the form a (1 + p), where p is independent of a, we have
dR_ dR
dr da '
Here dR/da has the meaning assigned in Art. 90. Whence, if we consider
only the terms which have a 2 as a factor,
dR _ dR __ p , fi x
T dr ~ a da "
Similarly for those which have a 3 as a factor,
rg = 3 (7),
and so on.
Secondly, since v occurs only in the form v - v, and since f only arises in
R through the first term of the substitution of f 4- A - A' for v - v', we have
dR dR
-5- = -^
dv d
Thirdly, it is found to be simpler to deduce dR/ds directly from the
equation (3) and then to substitute elliptic values for the coordinates. No
transformation will therefore be necessary.
116. Some further properties may be noted. We have from the defini-
tion of d'R (Art. 12), since R is a function only of the coordinates of the Sun
and of the Moon,
== ___
dt = " dt dr' dt dv' dt
If we regard the first term of R only,
62
84 THE DISTURBING FUNCTION. [CHAP. VI
and generally,
dR dR _ dR , ft/ v
dv' = ~~fo = ~T% "
Hence jd'R = R + 3 JR^- + j^dv' (9),
a form which is frequently of value. If we are considering the term of R
which contains the factor mV >+2 / r/J>+3 > instead of 3 we must put p + 3.
By means of this result we only need to form the single differential dR/dj*
in the radius and longitude equations (A), Chap. IT., when R has been found.
It is easy to see the truth of the statement made in Art. 65, that dR/dv,
d'R/dt will contain no constant terms. For R contains only constant terms
and cosines and therefore dR/dv = dR/dJ; only sines of angles without any
constant term. Also in (9), R, r', dv are expressible by means of cosines and
constant terms while dr, dR/d% consist of sines only, whence d'R/dt contains
no constant term. All the functions we have to deal with are expressible
either by means of cosines and constant terms or by means of sines with or
without a term of the form t x const. 4- const.
The effect produced on the orders of the coefficients by the integration
of the equations (A).
117. The substitution of ra*/a for raV/a' 3 shows that the coefficient of
every term in R is at least of the second order of small quantities. It does
not however follow that the corresponding terms in the expressions for the
coordinates are of the same orders as the terms in R from which they arise.
The integrations will, in certain cases, cause small divisors to appear which
will lower the orders of the coefficients to which they are attached.
We have seen in Art. 66, that a term of the form
A cos (kt + a)
present in the right-hand members of any of the three equations (A) will
produce terms in r, s of the form
and it is evident that it will produce in v a term of the form
^
-JT sin (kt + a).
There are three cases to be considered, depending on the magnitude
of k.
116-117] ORDERS OF COEFFICIENTS AFTER INTEGRATION. 85
(a) If k be a small quantity of the first order, the terms in r, s will be of
the same order as A and the term in v will have its coefficient lowered one
order.
(6) If k 2 n? be a small quantity of the first order, the terms in r, s will
have the corresponding coefficients lowered one order, while the order of the
coefficient of the term in v remains unaltered.
ft) 7?
Further, in the longitude equation there occurs the integral I -^- dt, and
in the radius-vector equation the integral fd'R.
If a term of the form A sin (kt + a) is present in dR/dv and if k be of the
first order, the coefficient of the term will be lowered two orders by the
integration of the longitude equation. If the term occur in d'R/dt, its
coefficient will be lowered one order by the integration of the radius vector
equation.
(c) There is one term in R for which k = n, namely, the term with
argument f + </>'; its coefficient is of the order raVa/a'. Contrary to what
might have been expected from the remarks of Art. 66, this term does not
cause t to appear as a factor of the coefficient. The argument, expressed in
terms of the elements, is
nt + e n't e' + n't + e' - cr' = nt + e - **'.
To understand this, it is necessary to refer to Art. 67 where it was seen
that the first approximation could only be obtained in a suitable form by
supposing certain terms of the disturbing function (which should, by the
method of continued approximation, have been neglected) to be included. It
was seen that instead of the equation x + n z x = Q, the more correct equation
to deal with is x+ (n- -^-b 1 )x = Q, where 6 X and Q are small quantities arising
from the disturbing function. The first approximation (that is, the Comple-
mentary Function) then consisted of terms of period 2-ir/cn. If, with this first
approximation, Q be expressed in terms of the time and if a term A cos (nt + a)
arises from Q, we see that no modification is necessary, since its period is
2-7T/71 and not %7r/cn: further, no terms proportional to the time will arise.
Finally a term A' cos (cr?+ a') in Q will cause no difficulty owing to the defini-
tion of c.
The terms for which k is small are known as long-period inequalities.
Their effect is in general most marked on the longitude. The terms for
which k is numerically nearly equal to n, are those whose periods nearly
coincide with the mean period. They produce marked effects on the radius
vector and latitude and thence on the longitude.
86 THE DISTURBING FUNCTION. [CHAP. VI
118. Let us examine the case (c] of the last article a little more closely and see in what
way the ordinary method of approximation may be applied to a term of the form considered.
The equation for the second approximation to r can be put into the form (see Art. 130),
x+ n 2 x+ b^+ b^ 2 + byX?-{- ... = Q,
where 6 1 , b 2t ..., Q are the portions arising from the action of the Sun which, when the
results of the first approximation are substituted, consist entirely of known terms.
In dealing with the second approximation we neglect # 2 , ^ <3 ,... and substitute the results
of the first approximation in b^ Q, so that a term of the form A cos (nt -fa) in Q-b^x
appears to give an infinite value to the coefficient of the corresponding term in x. But
we have seen that this is not really so and that the coefficient can only be found by in-
cluding in the second approximation, terms of higher orders. It is the simplest plan, in
actual calculation, to leave this coefficient indeterminate until the third approximation is
reached : it can then be found because, in the third approximation, the results of the
second approximation, substituted in x 2 , ^ 3 ,... will produce terms of this form and these
can be equated to the corresponding terms in #, Q.
It is not difficult to see how a term with argument + <' and with a known coefficient
may arise in x 2 in the third approximation. In the next chapter we shall see that the
second approximation will produce in r or #, the terms A^w' cos<', A 2 ma/a'cosg,
(A ly A 2 numerical coefficients). On proceeding to a third approximation we should
substitute the results of the second approximation in, for instance, x 2 . "We thus get
amongst others a term of the form A 3 m 2 e r (a/a') cos (+<'), that is, in the equations for
finding the third approximation we have a term of the same order as that in the disturbing
function and therefore of the same order as that which would be used to find the second
approximation. Hence, as far as this term is concerned, it is necessary, not only for the
form of the solution to be correct, but also that the method of continued approximation
may be applicable, to include certain parts of the equations which, in the second approxi-
mation, would ordinarily be neglected.
There are other terms for which the third approximation appears to produce coefficients
of the same order as those given by the second approximation ; this peculiarity is chiefly
due to the direct and indirect effects of small divisors. De Pontecoulant* treats them
by leaving the coefficients indeterminate until the higher approximations have been
completed. Such terms illustrate the necessity, mentioned in Chap. iv. and insisted on
here, of continually bearing in mind the effects produced by the higher approximations and
the impossibility of obtaining the first and second approximations in the correct form,
without considering them.
119. There are certain results which we cannot stop to prove here but the statement
of which may perhaps prevent misconceptions. They are : (a) The coefficients resulting
from the action of the Sun in the coordinates are never of an order lower than the second ;
(/3) The coefficients can always be represented by series of positive powers of w, e, e', y, a/a',
with numerical coefficients ; (y) The characteristic of a term in radius vector or longitude
is the same as that of the term in R from which it arose : in latitude, it is always less by
one power of y ; (8) Nearly, but not quite, all the terms in the coordinates arising from the
action of the Sun, have the factor m in its first power at least ; () the constant portions of
the expansions of the functions considered contain only even powers of e, e', y, a/a'.
* Systeme du Monde, Vol. iv. pp. 103, 145, 151, etc. The terms in longitude and radius vector
of the form considered above, are those numbered 74.
118-121] COEFFICIENTS INDEPENDENT OF m. 87
With reference to the statement (y), it may be remarked that the divisors arising from
integration are, in de Pontecoulant's method, linear or quadratic functions, with integral
coefficients, of n, ri, en, gn. The constants 1 c, 1 g will be found to be represented by
infinite series in powers of m, e 2 , y 2 , e' 2 r (a/a') 2 . Their principal parts begin with the power
m 2 , so that the divisors involving c, g always contain powers of m. Hence none of these will
have e y y, e', a/a', as a factor.
The exceptions to (d) as given by Delaunay*, are the terms in radius vector and longitude
with arguments D + 1' -{-pi + 2qF, and those in latitude with arguments D + 1' +pl + (2g + 1 ) F t
or, with the notation used here, the terms with arguments
respectively (p, q = - oo . . . -f oo ).
120. Since the expression of R has the factor m 2 , when we put m=0 all terms
dependent on the action of the Sun should vanish in the expressions of the coordinates.
The apparent exception of the terms just mentioned has been explained by Gogouf. It
depends on the definitions of the constants in disturbed motion. When m is put zero the
motions of the perigee and node vanish and the arguments of those periodic terms which
remain, contain t only in the form pnt 4- const. After suitable changes of the arbitraries have
been made, Delaunay's expressions with m zero reduce to those for purely elliptic motion.
On the subjects of Arts. 117 118, see de Pontecoulant, Theorie du Systeme du Monde.
Vol. iv. Nos. 914, 90, 91, 100 ; Laplace, Mecanique Celeste, Book VIL, 5. On the short
period terms whose coefficients are large in comparison with their characteristics, see also
Part in. of a paper by the author, Investigations in the Lunar Theory $.
121. The Second Approximation to R.
It must be remembered that the substitution in R of elliptic values for the
coordinates of the Moon, is only a means of finding a first approximation to
R. Suppose that with these elliptic values for r, v, s substituted in the right-
hand members of the equations (A), Art. 13, we have solved the equations
and have found the new values r + 8r, v + $v, s + 8s, of the coordinates.
According to the principles of the method, these new values must be sub-
stituted in the right-hand members of the equations in order to find the
third approximation.
Let Q be a function of the coordinates of the Moon which may contain
also the time. Put
Q = F(r,v,s),
where Q is the new part of Q arising from the additions Sr, 8v, Bs to the
elliptic values of the coordinates. Expanding by Taylor's theorem,
* Mem. de VAcad. des Sc., Vol. xxix., Chap. xi.
t Ann. de VObs. de Paris, M6m., Vol. xvm. E. pp. 126.
J Amer. Journ. Math., Vol. xvn. pp. 318358.
88 THE DISTURBING FUNCTION. [CHAP. VI
By putting .ft, dR/dg . . . successively for Q we can find the new values of
these functions. In the partials dQ/dr, 3 2 Q/9r 2 , etc. we substitute the initial
values of the elements.
Care must be taken when we are proceeding to form such expressions as 8$dtdR/dv.
Omitting, for the sake of illustration, powers of fir, dv higher than the first and all terms
dependent on the latitude, we have
This, if we regard only the terms independent of a/a', gives
in which we substitute for 9 2 .S/3 2 , dR/dg the values obtained from (5).
(ii) Expansion of R for Delaunays Theory.
122. Let 3 , 2 be the angular distances #fl, 11-4 (fig. 4, Art. 44), so that
s = 0> 2 = 'C7 as in Art. 84. Suppose for a moment that the Sun's orbit
is inclined to the plane of xy and let /, 2 ' be the distances xl' , l'A f , where
A' is the solar perigee and fl' the intersection of the Sun's orbit with the plane
of xy. When the inclination of the Sun's orbit vanishes, IV will become
an indeterminate point on xy, but a 2 / + a 3 / = -c7' will be determinate. For
symmetry, we use a/, a 3 ' although they can only occur in the form / + a 3 '.
We have from the spherical triangle Mlm',
S = cos Mm cos H M cos Om'+ sin O M sin Hra'cos i ...... (12),
or, since XlJf =/+ 2 , Ilra'=/' + </ + a/- a 3 ,
s = (! ~ 7i 2 ) cos (/+ 2 4- 3 -/' - 2 ' - a s ') + 7i 2 cos (/+ 2 - a 3 +/' + / + ,')
......... (13),
in which we have put sin %i = <y l .
From this we may form S' 2 , S 3 ... and, after expressing them as sums of
cosines of multiples of angles, substitute them in (1).
The first part of R will consist of five separate terms of the form
77? V 2
K - cos
121-124] DELAUNAY'S DEVELOPMENT. 89
p, p taking the pairs of values 0, 0; 0, 1 ; 1,0; 1, 1 ; 1, 1 ; K being a
function of 7^, and a, a' depending on the angles etj, 3 , a/, a 3 '.
Delaunay proceeds by expanding
r 2 r 2 cos ... a' 3 a' 3 cos
(where a, a' may be any angles) in powers of e, e' and cosines or sines of
multiples of w, w'. These may be obtained by means of the formulae given
in Chap. in. above. By the direct multiplication of series he is then able to
form all the terms required. In a similar manner the rest of the terms in R
may be found.
The arguments of all the terms will evidently be composed of the four
angles,
w, w', w + ot 2 , 3 - (w' + a/ -f a 3 '),
or of w, w', w + 03, w + 2 + a 3 (w f -f a/ + a/)-
123. It is not difficult to see that, after one or two small changes, this method of
development will produce the same result as that obtained in Art. 114. In both cases we
shall have expanded in terms of the mean anomalies and elements of the two orbits.
In the former case the inclination of the Moon's orbit was introduced through v and s,
while in Delaunay's method it is introduced directly through S. For simplicity and
ease of calculation the latter method has a great advantage over the former, and more-
over, it admits of a much more general treatment.
Since y=tan i, 7^= sin Ji, if we put
and for w, w',
the symbols <, </>', ?/, , respectively, we shall immediately obtain the development (5).
Delaunay has performed the expansion so as to include in R all quantities up to the
8th order inclusive ; in addition certain terms are carried to the 9th, 10th and even higher
orders where it appears to be necessary for accuracy. His development of R consists of a
constant term and 320 periodic terms. See Mem. de VAcad. des Sc., Vol. xxvin.,
Chap. n.
(iii) Hansens development.
124. Hansen's method is a more general one than either of those out-
lined above since it is adapted to the case in which the Sun's orbit is in
motion. It will give, after a few small changes, the expressions both of
de Pontecoulant and Delaunay.
Let w, to' be the angular distances of the apses of the instantaneous
orbits of the Moon and the Sun from the line of intersection of the planes of
the orbits, that is, from their common node, and let J be the angle between
90 THE DISTURBING FUNCTION. [CHAP. VI
the planes (see Arts. 217, 220). The angular distances of the two bodies
from this node will be co +/, &/ +/', and therefore
S = cos (/+ &)) cos (/' + to') + sin (/+ ) sin (/' + ') cos J
= (1 - sin 2 %J) cos (/-/' + co - a/) + sin 2 %J cos (/+/' + o> + o>') . . .(14).
Let .#//* = Jfft 1 * + ^ 2 > + ..., where pPW is the term in (1) with coefficient
m ' r p+i/ r 'P+*. p u t m'a 3 /fjLa' s = m, 2 . Then
+ & cos (2/- 2/' + 2ft) - 2ft/) + & cos (2/4- 2ft))
4 cos (2/' + 2ft) 7 ) + &cos (2/+ 2/' + 2ft> + 2ft)'))... (15),
where ^ . . . /9 5 are definite functions of sin 2 J" which it is not necessary
to specify here. It is required to replace r/a, a'/r',f,f by series involving
the mean anomalies w, w' and the eccentricities e, e.
The symbol m 1 is here used instead of m because Hansen puts m' + n = ri 2 a' 3 and so
does not neglect the small ratio ^ : m'. We have then
,_w_(
/aa' 3
The ratio of the difference between m l and m to either, is the very small quantity 1/660,000.
See Art. 53.
r 2 a,' 3
125. Let = 2PjCosjw, 3 = S^T/ cos jV,
*
A \ ^jf i\ // A y
m'/ w 2 / \ m'/ / \ m'/
,
7*
cos * ^ cos ' cos v Gj ' C cos
v ) = 2
a? sin Qf sin r /3 sin * G f " sin
in which j, f receive all integral values from + 00 to oo *. The coefficients
P y Q will be functions of e only and K, G functions of e' only ; these may be
calculated after the methods explained in Chap. m. Since, in each case, the
coefficients for positive or for negative values of j, f are superfluous, it is
supposed that
P_ 7 -, K. f , Q_/, _/, Q_/> -/ = ^-> f. Of, #/> -Qi*> -/... (16),
respectively.
Consider trigonometrical series of the forms
COS (JV),
\J
* The letters c, a placed above the coefficients are simply marks to distinguish between the
coefficients of the cosine and sine.
124-126] HANSEN'S DEVELOPMENT.
91
and of the same nature as those just given. Their products may be expressed
in the forms,
j smjw x 2E f ' siuj'w' = - 22^- E/ cos (jw + jV).
Applying these results to the term of R with coefficient & we obtain,
Cr\ 2 /a\ 3
aJ(/) =2P
Also for the term with coefficient /8,,
cos (2oi - 2a>') g cos 2/. ^ cos 2/' + J sin 2/. ^ sin 2/')
+ sin (2a> - 2*/) cos 2/. sin 2/' - 2 sin 2/. cos 2/')
= cos (2w - 2ft) 7 ) 22 (Q// - Q//) cos (jw +jW)
+ sin (2ft) - 2ft/) 22 (Q// - QfG/) sin (> + JV)
= i22 (QfGf - QfG/ - QfG/ + QfGf) cos (> + JV + 2w - 2ft) x )
+ i22 (Q// - Q/ G/ + QfG/ - QfG/) cos (> +JV - 2w
Since J, / receive negative as well as positive values, we can in the second
line of this expression put j, j' for j, /. Whence from the relations (16)
the expansion becomes
22 (Qf + Qf) (G/ - Gj) cos (jw +fw t + 2w - 2ft)').
This form of the product of two series is a sufficiently simple one to
calculate, when we have obtained the values of Q in terms of e and of G in
terms of e. We can express by similar formulae the terms in R w whose coeffi-
cients are /3 3 , &, ft.
The terms in^ (2) are treated in like manner. The terms in R {3) , being at
least of the order m^tf/a' 2 , need not be calculated by means of the general
formulae. It is sufficient to choose out the various terms which will have
sensible coefficients and to find the latter directly.
126. Hansen's expansions are given in the Fundamenta, pp. 159179. His papers, Ent-
wicklungdes Products einer Potenz des Radius etc. wAEntwicldung der negativen und ungraden
Potenz&i der Quadratwurzel der Function r z +r"* 2r/(cos 7 cos U' + sin Usiu U'cosJ)*,
* Abhandl. d. K. Sdchs. Ges. d. Wissenech., Vol. iv. pp. 181281, 283376.
92 THE DISTURBING FUNCTION. [CHAP. VI
contain methods for the complete development of the disturbing function both in the
lunar and the planetary theories. A very clear and concise account of Hansen's method
and results has been given by Cayley in his first Memoir On the Development of the Dis-
turbing Function in the Lunar Theory*. Reference may also be made to two other papers
by the same writer on the development of the disturbing function f.
In order to deduce de Pontecoulant's developments from those of Hansen we put J=i
and 0, $', ;, for w, w', w + co, w+a-w'-a respectively. To deduce Delaunay's results
we put sin^e/=y 1 , also a 2 for a>, and a 2 '-t-a 3 ' a 3 for at.
(iv) Laplace's Equations.
127. In order to develope Ffor the purpose of treating Laplace's equations (Chap, n.),
we have by Arts. 8, 107,
And since
we obtain F =t+ { z co &(v-v')-%(l+s*}} + ...................... (17).
Laplace forms the forces dF/du^ dF/dv, dF/ds directly from this expression. Since the
independent variable is v, it is then necessary to expand the results in terms of v and of
the elements.
By means of the results of Art. 52 the coordinates u lt s are immediately put into the
required form. The coordinates u', v' being given as functions of t must be expressed by
means of the results of Arts. 52, 70 in terms of v. See Laplace, Me'c. Ce'L, Book vn.
Chap. i.
(v) Equations referred to Rectangular Coordinates.
128. The expansion of the disturbing function for the equations of Arts. 1820, has
been there performed as far as it is necessary. It is a feature of the method that we do
not substitute elliptic values for the coordinates of the Moon in the terms dependent on
the action of the Sun.
The part G 2 of equation (16), Chap. 11., is that portion of G which is independent of the
parallax of the Sun and which vanishes when e' is zero. As already pointed out in Art. 22,
we can put
(18),
in which A, B, C, K depend only on the motion of the Sun and are at least of the order
mV ; they can be easily expanded in powers of e' by the known elliptic formulae.
* Mem. ofR. Astr. Soc., Vol. xxvu. (1859). Coll. Works, Vol. in. pp. 293318.
t Mem. R. Astr. /Soc., Vols. xxvm., x*ix. Coll. Works, Vol. in. pp. 319343, 360474.
CHAPTER VII.
DE PONTECOULANT'S METHOD.
129. WE have, in Section (i), Chapter n., obtained the equations (A) on
which de Pontecoulant has based his method. In Chapter in. Art. 50, are
to be found the elliptic values of the coordinates which serve as a first
approximation after the modification, formulated in Chapter IV. Art. 68, has
been made. In Chapter vi. Art. 114, we have given a development of R
obtained by using these modified elliptic values ; and, in the same division of
Chapter VI., certain theorems which tend to simplify the algebraical processes
of the second and higher approximations, are proved. The object of this
chapter is to explain the manner of carrying out the various approximations,
by applying the principles already discussed to the discovery of some of the
larger inequalities in the Moon's motion. The arrangement of the inequalities
into classes, although a natural one, is not essential to the method (Art. 151).
It enables us, however, to explain the origin of the various periodic terms in
the expressions for the coordinates and to carry out portions of the third
approximation with greater ease and security.
Preparation of the equations (A) for the second and higher approximations.
130. Let r , U Q be the modified elliptic values of r, u(= 1/r). Let
and therefore r 2 = r 2 - 2r 3 Sw + 3 (r 2
Bu is then the part of 1/r depending directly on the disturbing action of the
Sun.
Substituting for r on the left-hand side of the first of equations (A),
Art. 13, we obtain, since /t is put equal to unity,
where P = r + 2fd'M .............................. (2).
94 I>E PONTECOULANT'S METHOD. [CHAP, vn
Since R contains only even powers of 7, the equations for radius vector
and longitude and therefore those coordinates contain only even powers of
7. We shall neglect powers of 7 beyond the first and consequently, in the
first two of equations (A), neglect 7, s entirely.
We can obtain other forms for P by means of the results of Arts. 115,
116. Neglecting s we have, in the second approximation,
(3),
where p = for the terms independent of the solar parallax and p = 1 for the
terms dependent on the first power of the ratio a/ a'.
Also from equations (9) of Art. 116, in the second approximation,
th/ ............ (4),
with the same definition of p. An arbitrary constant is considered to be
present in the expression for P. The value of SP, necessary for the approxi-
mations beyond the second, is found by Taylor's theorem as in Art. 121.
131. Let v be the elliptic* value of the longitude and let A be the
elliptic value of h. Put
Neglecting s 2 and substituting, the second of equations (A) may be
written,
We have also, since 7 is neglected and /* = 1,
h = Va (1 - e 2 ),
or, when e 2 is neglected,
fr = Va = na 2 ................................. (6).
Finally, the equation to be used to find the latitude is not the third of
equations (A) but
z dR 1 dR s dR
z + ^ = z- == --*- + - r ~r ........................ 00>
r 3 dz r ds r dr
from Art. 13, neglecting s 3 .
The equation (6) of Chapter II. for the determination of the constant part
of l/r is, when we neglect T 2 , a/a',
- .............................. (8).
r 77.2
* That is, modified elliptic. This abbreviation will be used throughout the chapter.
130-133] PRELIMINARIES. 95
When we are considering terms independent of e, we have
and not otherwise. For the introduction of c will cause terms of the order 1 c to appear
if we substitute the modified elliptic values of r , v in these equations. They are only
satisfied by the purely elliptic values of r , V Q .
The constant 8h and that considered to be present in P are theoretically superfluous,
but the presence of 8h is of great assistance in determining the meanings to be attached to
the arbitraries in disturbed motion.
132. We have seen in Chapter VI. that the characteristic of a term in R
is unaltered by the integration of the radius vector and longitude equations.
All terras in R depending on the latitude are at least of the order 7* : when
introduced into the latitude equation they will be at least of the order 7.
The order of the characteristic is not further lowered by the integration of
this equation.
Hence we can divide up the terms of the disturbing function and, instead
of finding the complete first approximation with all the terms of R, we can
separate them out according to the composition of their characteristics.
The order in which the terms will be taken is as follows: the terms
whose coefficients depend only on (i) m ; (ii) m, e ; (iii) m, e' ; (iv) m, a/a' ;
(v) m, 7 ; (vi) m and any combinations of e, e', 7, a/a' and of their powers.
In the second, third, fourth and fifth classes we shall here develope only the
terms depending on the first powers of e, e', a/ a, 7 respectively ; the terms in
the sixth class will not be developed.
The approximations, which will in certain cases be carried to the third
order*, are made according to powers of the disturbing forces, that is, of m 2 .
In the first approximation we neglected the disturbing forces ; in the second
approximation all the new parts added should be at least of the order m* ; in
the third approximation of the order m 4 , and so on. But, owing to the small
divisors introduced by integration, as already explained in the previous
chapter, some of the terms in the second and third approximations contain m
in its first power. It is this fact which causes the great labour necessary to
produce expressions for the coordinates with an accuracy comparable with
that of the best lunar observations of the present day.
133. It is necessary to make mention here of the two constants n, a. In
undisturbed motion we have
In disturbed motion, n will be defined (Art. 135) as the observed mean
motion ; n, a are two of the arbitraries of the solution. Since we cannot
have seven independent arbitraries, the relation w 2 a 3 = 1 will be supposed to
* The details of the third approximation are printed in small type.
96 DE PONT^COULANT'S METHOD. [CHAP, vn
hold between the symbols n, a in disturbed motion, whatever may be the meaning
attached to n. When n has been defined, a definite meaning is thereby given
to a.
The necessary changes in the meanings to be attached to n, e, 7 when the
motion is disturbed, are defined in the course of the chapter. Fuller expla-
nations will be given in Chapter vm.
(i) The terms whose coefficients depend only on m.
134. From equation (5), Art. 114 we have, since all terms dependent on
e, e', 7, a/a' are neglected,
/yvi2
# = (i + fco S 2) ........................... (9);
r = a } v = nt + e, r f = a', v' n't + e'.
Also, as 2 = 2 (n n') 1 4- 2e 2e', we see by the results of Art. 117, that
none of the terms here considered will have the orders of their coefficients
lowered by integration. Hence, in the results of the second approximation
all terms will be of the order m 2 at least, in those of the third approximation
of the order m 4 , and so on.
In the second approximation we neglect powers of u higher than the
first. The equation (1) therefore gives, since n^a 3 1,
And from equation (4), since we neglect a /a' and therefore have p = 0, we
obtain
8f
From the value of R given above,
_ o
_ = - f -- sin 2;
. , [dR ,. _ m 2 ri of .
therefore 2n I ^r dt = | - - 7 cos 2f .
J 9f 2 a n n
9f
The equation for Su then becomes, since n'/n = m,
1 rf 2 ,, ra 2 , OKV , m 2 m a
-- -Ji* u - u = (1 + 3 cos 2f ) + 1 n - cos 2f + -
n 2 dt 2 a ^ * a 1-m a
where a/a is the constant attached to the integral in P.
To find the particular integral assume
133-135] (i) VARIATIONAL INEQUALITIES. 97
Substituting and equating the constant term and the coefficient of cos 2f
to zero, we have
-& = ra 2 + a .............................. (10),
,
The second of these gives
6 2 {4 (1 - w) 2 - 1} = 3m 2 + f w 3 /(l - ra),
or, expanding in powers of m,
4 + .. ...................... (11).
The third approximation will produce terms of the order ra 4 and this
value of 6 2 is therefore only correct to ra 3 ; it is given here to the order m 4
in preparation for the next approximation. Since a is an arbitrary constant,
the constant term 6 is at present undetermined.
135. The equation (5) for the longitude is, with the same substitutions
and to the same degree of approximation, since h Q = no?,
d s , l [ dE * Sh
^ov = 2na&u + -^. at + .
dt a?J 3f a 2
/"/} 7?
This, from the values of 8u, I ^ dt found above, becomes as far as the
J 3%
order m 3 ,
Therefore integrating,
v -f B = (Znbo + ^]t + ^-^, (m 2 + J#m 3 + fm 2 ^- sn
2 + ff w3 +---)sm2^ ..................... (12),
where we have put n/(n ri) (1 m)" 1 = 1 + m + m 2 + . . ..
The longitude is v + Sv nt + e + Bv. This shows that the introduction
of .B is useless, for it merely adds an arbitrary part to e which was itself
arbitrary.
The coefficient of t is n + 2n6 + &h/a?. Since n was an arbitrary of the
original solution and since Sh is an arbitrary introduced into the second
approximation we can determine the latter at will. We shall always give
this arbitrary a value such that the coefficient of t in the expression for the
true longitude is always denoted by n ; this statement defines the meaning of
n in disturbed motion. Thus 2nb + 8h/a 2 = and
&; = (J g 1 -m 2 + ffra 3 )sin2f ........................ (13),
as far as the order m 3 .
B. L. T. 7
98 DE PONTCOULANT'S METHOD. [CHAP, vn
136. The constant term b in Bu is found by substituting the values
I/a -f Bu, nt + e + Bv for 1/r, v in (8). That is to say, we put
v = n + (n - n') (J^-m 2 + ^m 3 ) cos 2, - = 1 + 6 + (m 2 + -^ w s ) cos 2f ;
r
therefore, as 6 is of the order m 2 at least,
v* = n 2 + n (n - n') (^-m 3 + -^m 3 ) cos 2f , = 1 -f 36 + (3m 2 + -^m 3 ) cos 2f ;
and similarly for r/a, r/a.
The equation gives, since 272/a 2 can be put for 2R/r z ,
{I + b Q + (m 2 + J^m 3 ) cos 2f } 4 (n - n')* (m 2 + J^m 3 ) cos 2f
-ri*-n(n- n') (-^m 2 + ^m 3 ) cos 2f
I ,^,2
+ ^ {1 + 36 + (3m 2 + ^ 9 -m 3 ) cos 2?) = ^- ( J + f cos 2f ).
Put w ^^^(l m), I/a 3 = ri* and expand in powers of m, neglecting those
beyond m 3 . On equating the coefficient of cos 2 and the constant term to
zero, it will be found that the former vanishes identically, thus giving a
verification of the previous work, while the latter furnishes to the order m 3 ,
or, correctly to the order m 3 , there being no term of that order,
& = * 2 (14).
Thence Su = i {Jm 2 + (m? + *$-m?) cos 2f ) (15),
Cv
to the order m 3 .
137. In order to exhibit the method of finding the third approximation,
we shall calculate the coefficients of the periodic terms in 1/r up to the order
m 4 . They can be found to the order m 5 in this approximation, but for the
purpose of explanation it is not necessary to include the terms of this order
as they merely involve further expansions.
To find the third approximation we must include the first term of the right-hand
member of (1) ; this term being of the order m* at the lowest, we can use the result (15) of
the second approximation. We have therefore
torn 4 .
= 2^00821+12^00841,
ft Ct
since (n - ri) 2 m*=-n*m*=m*la 3 to the order wi 4 .
136-138] (i) VARIATION AL INEQUALITIES. 99
Also from (4),
i^dt
J ov
The terms being all of the order wi 4 at least, we can here put r a. Substituting the
values of R, dRfig, 3 2 /?/3 2 obtained from (9) and those of bv and 8r=-r*8u= -a?bu
from (13) and (15), we obtain to the order m 4 ,
ap= _ 8 (|+f cos 2) (m 2 +ro 2 cos 2) - 4 f sin
On integration, the second line of this expression will be seen to be of the order m 6 .
Hence
dP= - (V^+3 cos 2-f cos 4) to m 4 .
Let the new part of l/r be S^u. The equation for ^u therefore becomes
5 -TO &i u - 5, u = -- cos 2 + J4 5 cos 4 + const, part.
22 l a 'a
We do not put the constant terms into evidence since, as in the first approximation,
they are determined by means of (8).
Assuming ^u = - (8b Q + Sb 2 cos 2 + 6 4 cos 4),
we easily obtain ft^u = - (8b Q - ^ m 4 cos 2 + 1 m 4 cos 4).
Adding this to the value of 8u which in (11) was carried to the order m* for the coefficient
of cos 2, we have
138. In a similar manner the third approximation to the longitude may
be calculated. In doing this we can omit all the constant terms which
appear in the equation for dv/dt, since these merely add a known part and an
arbitrary part to the coefficient of t in the expression for v, and these new
parts, by the definition of n, always vanish. After the value of v has been
found we can obtain the constant part of l/r by means of (8) and at the
same time verify the results previously obtained.
When these calculations have been performed we shall get as far as m 4
for the terms whose coefficients depend on m only,
v = nt + e + (Jgi-m 2 + ffm 3 + ^ 3 -m 4 ) sin 2f + fm 4 sin 4
72
100 DE PONT^COULANT'S METHOD. [CHAP, vn
From the remarks just made, it is evident that in calculating the right-
hand member of the radius equation we can always omit the constant portions.
This evidently applies to the calculation of all inequalities.
Also, in calculating the terms in the equation for v, we always equate the
constant portions to zero. This rule also applies to all inequalities.
(ii) The terms whose coefficients depend on m and the first
power of e only.
139. The part of R required is*
A>l2
E = fi + J cos 2f-4* cos </>-fecos(2f-<) + fecos(2f + </>)}. ..(18).
a
Also, from Art. 50, since 7, e 2 are neglected,
r Q = a (1 e cos </>), v Q = nt + e + Ze sin </>, s = 0,
We first neglect all powers of m beyond m 2 . In forming P for the second
approximation, it is evident that we do not need the first two terms of R.
Hence by (4), remembering that = (n ri) t + e e', < = cnt + e OT,
P = ^ e {- 2 cos < - 9 cos (2f-<) + 3 cos(2f + <)}
a
2mV f Q 2cos(2g-j>) ? 2 CQS ( 2 ? +))
~ e ' *' '
Since 1 c is of the order m at least, the second line of this expression is
of the order m 3 and therefore, to the order m 2 , the value of P is given by the
first line.
The particular integral will be
a&u = Jm 2 + m 2 cos 2f + ec cos c + ec_j cos (2f 0) + ec^ cos (2f -t- 0).
The first two terms are those obtained in (i); c , c_i, ^ are the coefficients to
be found. We must substitute this expression for 8u in (1).
We have ^ (r 3 Bu) = ~ {a 3 (1 - Se cos </>) 8w}.
Whence, retaining only terms with the characteristic e,
d 2 d z
j- (r 5 &u) = a*e ^- {- f m 2 cos < - 3m 2 cos 2f cos + C cos <
+ cu cos (2f - <^) + G! cos (2f 4- <)}
= a?e j^ {(GO - m 2 ) cos </> + ( c _ x - f m 2 ) cos (2f -(/>) + (d - f m 2 ) cos (2? + 0)}.
* It is necessary to include the first two terms since, in combination with other terms of the
order e, they may produce coefficients of the order considered. There is the same necessity for
all inequalities.
138-140] (ii) ELLIPTIC INEQUALITIES. 101
Substituting this and the value (19) of P in equation (1) and neglecting
all terms but those which have the first power of e in their coefficients, we
obtain
P P
c 2 rt?a?e cos </> cos <f> (c cos <f> + c_i cos (2f </>) + c x cos (2 + <>)}
a a
+ tftfe {(c - im 2 ) c 2 cos $ + (c_ x - f m 2 ) (2 - 2m - c) 2 cos (2f - <)
+ ( Cl - f w 2 ) (2 - 2m + c) 2 cos (2{ -I- <)}
= e {- 2 cos - 9 cos (2f - <f>) + 3 cos (2 + <)}.
c&
Putting I/a for ?i 2 a 2 and equating the coefficients of cos<f>, cos(2f <),
cos (2f -I- <^) to zero we have, after multiplication by a/e, the three equations
of condition
c 2 - 1 -c + C 2 (c - Jm 2 ) = -2m 2 ,
- c_! + (c_! - f m 2 ) (2 - 2m - c) 2 = - 9m 2 ,
-GI +(G! - f m 2 ) (2 - 2m + c) 2 = + 3m 2 .
The first of these may be written
(c 2 -l)(l+c ) = -2m 2 + Jm 2 c 2 (20).
As c , c 1 are known to be of the order m at least, this shows that c 1 is
of the order m 2 at least. Hence, neglecting all powers of m beyond the
second,
c 2 - 1 = - 2m 2 -f Jm 2 = -f m 2 ,
or c = l fm 2
to the order m 2 . This value agrees with that found in Art. 67.
The other two equations of condition then give, neglecting all powers
of m but the lowest present,
c_j = ^g 5 -ra, d = ff m 2 ;
showing that the coefficient of cos (2f <f>) has been lowered one order by the
integration a fact which might have been predicted, since the coefficient of
t in 2f - <j) differs from n by a quantity of the first order.
It is evident that c is a new arbitrary constant, for the term c cos<
might be considered to be included in the elliptic value of a/r. We shall
not, however, neglect it here, but leave it arbitrary until the longitude
has been found.
140. To calculate the corresponding terms in longitude, the equation (5)
gives to the order required, namely to m 2 e,
dv h d 2/^ . Bh 1 CdR ,
102 DE PONTECOULANT'S METHOD. [CHAP, vn
Substituting the values of the various terms we have, since here h = na z ,
ff
dt
2 (c - 1 ) ne cos <j> + -j- Bv = 2n (1 + e cos </>) { m 2 + m 2 cos 2 + ec cos </>
+ - l me cos (2f -</>) + f f m z e cos (2f + </>)}
m2 is 2cos2f 9 2e cos (2f-) ^ 8 2ecos(2f +0)H
After expanding these expressions, we omit those periodic terms inde-
pendent of e ; we also put I/a. 3 = n z and c = 1 in those coefficients which are
of the orders em or em 2 . We then have
4 j^ m g cos (2f - </>) -f ^w a e cos (2f + </>),
- -U-vv\ 1 '^. Ctr^,/^ ^
in which terms of a higher order than those required are neglected.
The constant term is to be put zero. Hence
Sh = ^na?m 2 .
Since c is arbitrary we can determine it at will. Let it be such that the
coefficient of sin $ in longitude is the same as in elliptic motion. This gives
the definition of e in disturbed motion.
We have therefore,
i(-2 C + 2+2c + -im + 2^)=0 .................. (21),
or, giving to c, Bh their values,
c = - &m*.
Integrating the longitude equation, we finally obtain
- fim*e cos + Jme cos (2f - <) + f f m 2 e cos (2f + <), )
Jwe sin 72 " ' ( '
in which the terms with argument 2 </> are correct to the order me and
those with arguments <, 2f -f $ to the order m 2 e.
141. We can, by paying attention to the orders of the terms, obtain c to the order wi 3
without much further labour. Its value was obtained by equating the coefficient of cos
in the radius- vector equation to zero. The result (20) may be written
(c 2 -l) ........................... (23).
It was seen that c 2 - 1, c are of the order m 2 at least, so that c (c 2 - 1), m 2 (c 2 - 1) are
of the order m* at least. It is not then necessary to further approximate to c . The new
portions of the coefficient of cos< in (1) can therefore only arise from the terms
dt ............ (24).
140-142] (iii) MEAN PERIOD INEQUALITIES. 103
Now all terms in R are of the order m 2 at least, while only those terms in du, dv with
argument 2 < are of the order me. Hence a term of the order w 3 e, with the argument
<, can only arise in the part dP of (24) from the term in R of argument 2 combined
with the term in du or dv of argument 2-<. The last term of the expression (24)
furnishes only portions of the order m*e, and it can therefore be neglected. As (Art. 121)
we can, in this equation, put
,'2
r =a, = - cos 2, w= cos (2-
Whence 4d/2= -^ cos 2| cos (2 - 0) - 8 sin 2sin
of which the part depending on the argument </> gives
3P=4&ff=-if* cos0.
a
In the term -fcf 2 (r 2 8w) 2 /cfe 2 we can, by similar reasoning, put
r =a, adw = 7^ 2 c
whence (*o 2 ^) 2 contributes
and - f -T2 (V&O 2 contributes f im?a?e(?n 2 cos = ^f cos 0,
to the order required.
Adding this to the value of dP, the new term of argument in the right-hand member
of(l)is
The equation of condition (23) therefore becomes, neglecting quantities of the order m 4 ,
c-l=-fm J -W* i ;
or, taking the square root,
(25).
(iii) The terms whose coefficients depend on m and the first
power of e' only.
142. These are very easy to calculate and we shall only indicate the
steps. There are no indeterminate coefficients to find as in cases (i), (ii).
We put
R = { I + | cos 2f + y cos f + Qe' cos (2? - f ) - IB' cos (2 -f f )) >
a
r = a, V Q = nt + e, 5 = 0.
We use here the equation (3) to calculate P. The equation (1) becomes
_ J_ & $ u _ Su = e' (| cos # + ^f cos (2? - f ) - f cos (2? + f )},
_
71 CtC
giving, as far as the order raV,
= - f mV cos f + f mV cos (2f - f ) - JmV cos (2f + <') - -(26).
104 DE PONTECOULANT'S METHOD. [CHAP, vn
Thence we obtain
/ 2 cos
2 n-2n'-n' *2n-2' +
= nraV {- 3 cos <#>' + V- cos (2 - ') - -^ cos (2f + <'))>
and by integration
$v = - 3me' sin <' + f JmV sin (2 - <') - ftmV sin (2f + <')
One term in Sv has been lowered to the order me' by integration. This
will cause terms of the order mV to appear in the third approximation to
1/r and therefore, we should suppose, terms of the order raV in the third
approximation to v. But the only new terms in the radius-equation and in
fdt8(dR/dg) which are of the order mV will be easily seen to be those with
arguments 2f </>' and no terms are lowered in order by the integration of
the radius-equation; hence, the only new terms which are of order m s e' in
the longitude-equation are those with arguments 2 + <j>'. Therefore, as the
only terms in the longitude -equation which can have their orders lowered by
integration are those of argument <' and as such terms are at least of the
order raV, the values (26), (27) for &u, &v are correct to the order raV.
The third approximation may be carried out as in cases (i), (ii).
(iv) The terms whose coefficients depend on m and the first
power of a/a' only.
143. For these terms, we have
r = a, V = nt + e, s = 0, r = a', v' = n't + e,
and from (4), P = (p + 4) + 2*' dt ..................... (28).
In the second approximation to 1/r we shall not require the first two
terms of R : p = 1 for the third and fourth terms of R. Since the coefficient
of t in the argument of f differs from n by a quantity of the order mn,
the coefficient of cos f will be lowered one order by the integration of the
radius equation. We shall therefore, in preparation for the third approxima-
tion, carry this term to the order m 3 a/a' in the radius equation so that, after
the third approximation is complete, we may have it correct to the order
m?a/a f . The calculations are very similar to those necessary for case (iii).
We therefore have, to the order m?a/a' in the coefficient of cos and to
the order m*a/a' in that of cos 3f ,
p _ 2ri m? a ^r> m 8 a
P = 5R + - 7 | cos f = 5R -f J cos ;
n - n a * a * * a a fe
142-144] (iv) PARALLACTIC INEQUALITIES. 105
and equation (1) becomes,
m a , _ N m 2 a
The particular integral is
a /I* , ^ ,\ n 2 cosf a n 2 cos3f
(tf"* 1 +l^). + l'
= - (W + 2 ) cos f + |m 2 cos 3f ,
Cv (*
in which the coefficient of cos is only correct to the order ma/a'.
The equation for the longitude becomes, after neglecting superfluous
terms,
d , -A* 1 MR,.
-j- 8v= 2 - oti + |.-~jr at
dt a a 2 J df
2 ) cos f + f|m 2 cos 3f j
+ ^-, , f| cos f + | cos 3f}
a 3 a ri n (l
) cos f + ||m 2 cos 3f },
giving 5v = - (J^ra + fjm 2 ) 4 sin f + Jjm 2 ^ sin 3f ,
C& tt'
where the coefficient of sin f is only correct to the order ma/a'.
144. The third approximation. We are only going to find 8u, dv correctly to the order
m 2 a/a'. In order to do so, we require the new terms of order m 2 a/a' in 1/r, that is, the
new terms of order m 3 a/a' and of argument in equation (1). The only way in which such
terms can be produced is from the term in du or in 8v of argument , order ma/ a', combined
with terms of arguments 0, 2, order m 2 .
We have from (28), since the second term, being multiplied by n' t may be omitted,
8P=8R or 5dR,
O D O D
and SR = ^ 8v - 2Rr Q 8u or ^ 8v
according as we take terms in R independent of a/a' or dependent on its first power. The
only way in which we can get a term of argument , order m 3 a/a', from this expression is
by putting
= ^( + 1 cos 2), a&6=-iw-,cos, 8v= -^m^sinf
Whence, as the terms considered in R are independent of a/a',
106 DE PONTECOULANT'S METHOD. [CHAP, vn
and therefore the terms of argument in 8P are given by
^=W^|<*
For the first term on the right-hand side of (1) we take
adu = %m 2 + m 2 cos 2 - ^f m cos ;
therefore the term of argument | in (a8u) 2 is
- Am 3 , cos - |f w 3 cos = - |-m 3 -, cos ;
<z ct c&
so that f -7-2 (?'o 2 ^ w ) 2 contributes - ^ -- , cos .
Let the new part of l/r be djU. We then have
1 d 2 . , , x m 3 a ,. m 3 a
, . m 3 a w 2 cos , . m 2 a
Integrates ^=^ ^ a' (^^= ~ ^ -jooafr
Adding this to the value of dw previously found we obtain,
a$u = - (|f m + f^m 2 ) -, cos ^ + fjm 2 -, cos 3f , correct to m 2 -^ ... (29).
OE> a a
We shall now obtain the corresponding term in longitude. Since no coefficient has its
order lowered by the integration, the new portion is simply that arising from the term ^u
just found and it is of the order w 2 a/a'. Let it be denoted by d^v.
In the equation (5) the new portion 8jdtdR/dg is at least of the order m 3 a/a' and so
contributes nothing. The same is true of (d?*) 2 , dh x 8u, du x jdtdR/dg.
The only new portion of the order m?a/a' is therefore given by
^ * 2 ^ * i * * 9 a
d t V= V= - W nm * ~, c s &
whence di v = ^AP- - -. m 2 . sin ^ = *- m 2 , sin A
^ n-ri a' g a
Adding this to the value of 8v previously found, we have
Sv = - (if m + 93 m s) !L sin f + 1 5 m 2 ^ s i n 3^ correct to m 2 -^ . . .(30).
o d ft
(v) The terms whose coefficients depend on m and the first
power of 7.
145. We have s = tang, of lat. = z V(l + s 2 )/r, by Art. 12.
Since the inclination of the plane of the Moon's orbit is always a small
quantity oscillating about a mean value of 5, we can consider s and conse-
quently z as small quantities of the first order and, in the first instance,
neglect their squares and higher powers. Also, we have seen that the radius
144-145] (v) PRINCIPAL INEQUALITIES IN LATITUDE. 107
vector and the longitude will contain only even powers of the small quantity
7 (which is always a factor of s), and that the latitude will contain it in odd
powers only. Hence in the equation for s or in the equation for z we are
only neglecting quantities of the order s 3 or z 3 .
As stated in Art. 131, the equation (7) will be used to find z and thence
to obtain the latitude. In order to express its right-hand member in terms
of the coordinates, we use the development (3) of Chapter VL, neglecting
the parallax of the Sun, that is, neglecting the terms beyond the first. We
thus obtain,
m'rs mz
-- ;- to the order required*.
The equation of motion therefore takes the simple form,
+--'
or, neglecting e and dividing by ?i 2 , the form,
an equation which is sufficient to determine all terms in latitude whose
coefficients depend on m, e and the first power of 7 when those portions of r
which depend on m, e, are known. We shall, however, neglect e and therefore
give to a/r the part of its value dependent on m only that given by the
first of equations (17) as far as the order m 4 .
We are, in this method of finding the latitude, apparently departing from the principles
laid down in Chapter iv. concerning the method of solution by continued approximation.
That is to say, instead of considering the first or elliptic approximation to s as known and
then proceeding to find the new part due to the action of the Sun, we are including both in
the equation of motion, so that we shall find a portion of the first and second approximations
at one step. For the purposes here we do not need the value of * given in Art. 50.
Put a/r 1 4- a$u. We have
a/r = 1 + 3aSu + 3 (ofay + (aSu) s .
The value (17) of Bu gives (aSu) s = to the order ??i 4 , and to the same order
3 (abuy* = 3 (^m* + Jm 4 cos 2f + m 4 cos 2 2f )
= H ra 4 + m* cos 2f + f m 4 cos 4f .
The equation for z therefore becomes,
\ z + z jl + f m 2 - ^ra 4 + (3m a + m 3 + ^m 4 ) cos 2f + ^m 4 cos 4f } = 0.
* This result is nevertheless true when quantities of the orders y 3 , y 5 ... are taken into
account. The only quantity actually neglected, when r 2 = x 3 + y 2 + 2 3 , is a/a'. See Art. 150.
108 DE PONT^COULANT'S METHOD. [CHAP. VII
146. This equation is a case of the general form,
z + n*z{l+2f%costy't}=0, O'=l, 2, ...oo),
which frequently occurs in physical problems. It is of principal importance in celestial
mechanics for the determination of the mean motions of the perigee and the node.
The solution is of the form
z = "Sjq/ cos (gnt + 2jt + a), (j = - oo . . . + oo ),
the arbitrary constants being a and one of the qf. The chief part of the problem, which
we cannot stop here to investigate generally, is the determination of g. Hill and Adams,
as we shall see in Chapter XL, find g from the equation by means of an infinite determi-
nant. For the purposes of Case (v) we shall assume the solution to be of the form given
above and find the value of g to the required order by continued approximation.
An investigation of the differential equation and of its solutions will be found, together
with a large number of references to the literature on the subject, in Tisserand, Mtfc. Gel.
Vol. in. Chaps, i. ii.
147. Assume as the solution,
where yg is considered to be one arbitrary constant. The symbol tj stands
for gnt 4- e 0, where is the other arbitrary constant : g is a constant as yet
undetermined.
If we put m = the motion is undisturbed and we shall have g = 1 ; also
z ayg Q sin rj.
In undisturbed motion, 7 = tan i. If therefore we put g = 1, we shall be able
to define 7 in disturbed motion as being such that the coefficient of the
principal term in z is the same as in undisturbed motion. It is evident that,
in undisturbed motion, 0, e are the same as in Chapter ill.
We put then g Q = 1 and substitute the assumed value of z. It will
appear, when we write down the equations of condition, that since g differs
from unity by a quantity of the order m at least, #_!, g lt <7_ 2 , </ 2 are of the
orders m, m 2 , m 3 , m 4 respectively. Hence, omitting terms beyond the order
m 4 , the equations of condition become,
1 - f + f m2 - A m4 ~ (f m2 + ~f m3 ) ff-i + f m " #1 =
{1 - (2 - 2m - g) 2 } g^ + f m 2 0_j - (f m 2 + J^m 3 + ^m 4 ) =
(31).
- (4 - 4m - #) 2 } g_ 2 - f3 m 4 + ( 3
4 m 2 + m 3 =0
Neglecting powers of m beyond the second, we obtain from the first of
these,
l- 2 =-m 2 , or = l+?tt 2 .
146-148] (v) PRINCIPAL INEQUALITIES IN LATITUDE. 109
Therefore from the second and third,
#_! (4m - m 2 ) = f m 2 -f im* to m s ,
g l (- 8 + 12m - 7m 2 ) = - f m 2 - J^m 3 - ^m 4 to m 4 ;
or #_! = f m + $m 2 to m 2 ,
With these values we obtain from the fourth and fifth of the equations,
g- 2 (- 8 + 24m) = ff m 4 - (f m 2 + -^m 3 ) (f m + fm 2 ) to m 4 ,
g, (-24) =-^3
giving 9^ = Th m * + ffi m \ k 0. to 4 >
9* = <rft 4 torn 4 .
Substituting the values of g_ lt g l in the first equation we obtain for the
further approximation to g,
1 - f + f m 2 - ^m 4 - (f m 2 + J^m 3 ) (f m + fj m 2 ) + ^m 4 = 0,
or # 2 = 1 + f m 2 - ^m 3 - ^f m 4 to m 4 ,
or g =1 +|m 2 -^m 3 -f||m 4 torn 4 ............ (32).
With this value of g we can, from the second of equations (31), obtain a
more approximate value of g_ l} namely,
#_! (4m - m 2 - ^J m 3 ) = f m 2 + J^m 3 + -L^m 4 to m 4 ,
or </_! = f m + f|m 2 + ffff m 3 to m 3 .
The value of z is therefore obtained to the order 7m 4 in all coefficients
except in that of sin (2f 77) which is found to the order <ym 3 . As far as the
order ^m 2 , we have,
Z - = 7 sin 97 + (fm + f^m 2 ) 7 sin (2f - 17) + ^m 2 7 sin (2f + rj). . .(33).
Also to this order, from the first of equations (17),
s = z/r = z (1 + m 2 + m 2 cos 2f)/a.
Whence, to the order 7m 2 ,
s = (1 + fm 2 ) 7 sin 17 + (f m + ff m 2 ) 7 sin (2 f --/) + ^m 2 7 sin (2f + r)). . . (34).
(vi) The terms whose coefficients are dependent on m and the
products and higher powers of e t e', 7, a/a'.
148. It is not intended to develope the algebraical expressions of the
coordinates for terms other than those just given. To find the terms
included in this class the developments become of great length, but there is
110 DE PONTE"COULANT'S METHOD. [CHAP, vn
no change in the general method of finding them. When we wish to find
any particular inequality defined either by its argument or by the order of
its coefficient relative to e, e, y, a/a', it is in general sufficient for the first
approximation to the terms in l/r, to choose the corresponding terms present
in R ; to find the terms in longitude and latitude, we must consider those
of lower characteristics in l/r, s (these are supposed to have been previously
found) and in R which, by their combinations, may produce terms of the
required order. We shall then, as far as powers of m are concerned, have a
first approximation to the term. It is generally necessary to proceed to
further approximations.
If, after the third approximation to the coordinates has been completed, the order of the
new portion in the coefficient of any particular term is not higher than that of the portion
with the same characteristic obtained in the second approximation, we must, in computing
the second approximation, leave the corresponding coefficient indeterminate until the third
approximation is reached : the coefficient may then be found. In this case the second
approximation is not capable of giving the first term of the series for the coefficient in
powers of m. See Arts. 1 18-120.
Summary of the results.
149. We shall now collect all the results of the first and second
approximations and those portions of the third which have been determined
in the previous articles. The numerical coefficient of each term in the co-
efficients is correct to the order given. The elliptic parts of the values of
the coordinates will not be written down ; they are to be found in Art. 50.
The various portions of a/r are given by equations (17), (22), (26), (29).
Whence,
- = Elliptic value + m 2 - |f m 4 + (m 2 + -^m 3 + -^-m 4 ) cos 2f + f m 4 cos 4f
- -zm?e cos <f> + ^-me cos (2 -</>) + ff m 2 6 cos (2f + </>)
- f m 2 e' cos f + f m 2 e' cos (2? - <') - JmV cos (2f + </>')
- (H m + fi O -, cos ? + H^ 2 " cos 3f .
a a
The various portions of v given in equations (17), (22), (27), (30), furnish,
v = Elliptic value + (^m 2 + ff m 3 + ^m 4 ) sin 2f + f^m 4 sin 4
+ if-me sin (2f - </>) + ^-m^e sin (2 + <)
- (3m + Om 2 ) e' sin <' + {JmV sin (2f - <') -
2 ) -, sin f + Jf m 2 a ~ f sin 3f .
a a
148-150] SUMMARY OF THE RESULTS. Ill
The parts of these which were found by the third approximation are the
terms of order m 4 and those of order m 2 , in the coefficients of C ? S f ; all the
CL sin
other coefficients were found from the second approximation.
The value of s, as given by (34), furnishes,
s = Elliptic value + Jm 2 7 sin ij + (f m + ff m 2 ) 7 sin (2f rj)
We have also from equations (25), (32),
the former being only obtained correctly to the order m 3 while the latter is
found correctly to the order m 4 .
Finally, n is the coefficient of t in the non-periodic part of v ; e, 7 are
such that the coefficient of sin <f> in longitude and that of sin i) in z are the
same as in undisturbed elliptic motion.
150. For the cases (i) to (v) we have simply chosen out of the development of R,
given in Art. 114, the terms required. It is easy and often useful to deduce each particular
case directly from the disturbing function.
TIT i, o
WehOTe *
where S is the cosine of the angle between the radii to the Sun and the Moon.
Cases (i), (iv). Here, e=0, e' = 0, y=0; therefore r=a, / = ', v=nt + c, v' = n't+t'
S = cos ( v - -z/) = cos . Whence
Expanding as in Art. 107 and putting n' 2 a 2 = m?/a,
Case (v). Here r z =x 2 +y 2 +z 2 , S=xxf+yy' t and therefore
dR m'z
If we neglect the ratio a : a' or r : i* this gives
furnishing all inequalities in latitude independent of the solar parallax.
Cases (ii), (iii). Here y=0, a/a' = 0. We obtain by expanding R,
112 DE PONT^COULANT'S METHOD. [CHAP, vn
For case (ii), we have / = ', v' = n't + r and r, v take their elliptic values.
For case (iii), we have r=a, v=nt + c and /, v' take their elliptic values.
By proceeding in this way we can without much trouble obtain any class of inequality
denned by its characteristic.
De Pont&oulant's method as contained in his Systeme du Monde, Vol. iv.
151. We have already mentioned in Art. 129 that the plan used here of dividing up
the various terms into classes denned by the characteristic, is not essential for the
development of the method nor is it used by de Ponte"coulant. Further, if a complete
development of the expressions were required, it would hardly be a saving of labour to
proceed in this way. It will be readily seen that after R has been found in terms of the
time to the required degree of accuracy by using the elliptic values of the radius vector, the
longitude and the latitude, we should, in order to obtain the complete second approximation,
use the complete elliptic value of R in the equation (1) which serves to find 8u. Certain
coefficients would be indeterminate and they would be left so until, with this value of du,
we had found the value of 8v from equation (5), when they would be determined as in
cases (i), (ii). De Pontecoulant does not give full details of the method of procedure he
adopted, but it is not difficult to see his general plan which is somewhat as follows.
The first step is to neglect the latitude and with it all terms in the radius-vector
equation, the longitude equation and the disturbing function, which depend on y or s. In
order to obtain a second approximation to 1/r, v of the terms independent of y such terms
forming by far the greater part of R the parts on the right-hand side of (1) dependent on
(5w) 2 , (du) 3 ... are neglected, the value of P* is calculated from the expression for R and the
complete elliptic value of r is substituted on the left-hand side of the equation. In order
to solve the resulting equation for du, we assume a value for this quantity which consists
of a constant term and periodic terms of the same arguments as those occurring in the
equation all the coefficients being indeterminate. When this value for 8u is substituted,
by equating the coefficients of the various terms to zero we obtain definite values for all
the indeterminate coefficients except for that of cos$ and for the constant term. The
coefficients of cos <, however, give a first approximation to the value of c. Leaving these
two coefficients indeterminate we proceed with the value of 8u so obtained to find that of
fry from (5). We then determine the new arbitraries after the manner explained in cases
(i), (ii) and these, together with the coefficient of cos$, become definite. The constant
part of l/r is found from an equation corresponding to that numbered (8) above.
With these values of 8u, 8v we proceed to a third approximation by finding 8R
according to the method of Art. 121 and we thence obtain the whole of the new terms on
the right-hand side of (1). The resulting equations for the new parts of l/r, v are solved
as before and a third approximation to these coordinates is deduced. Proceeding in this
way by successive stages, all the coefficients are ultimately obtained accurately to quantities
of the fifth order inclusive. In addition, certain coefficients which are either expressed by
slowly converging series of powers of m, e, etc. or which, owing to the presence of small
divisors, have their orders raised, are calculated to higher orders by choosing out the
particular combinations required to obtain them.
The latitude equation is then treated. Neglecting powers of y (and therefore of z and s)
higher than the first and using the values of l/r, v already found, we can obtain z by means
* De Pontecoulant uses the letter P to denote the terms on the right-hand side of (1) which
depend on (3w) 2 , (Sw) 3 ....
150-154] THE SYSTEME DU MONDE. 113
of equation (7) ; in doing so, since all terms in z have the factor y, it will only be necessary,
except for certain coefficients, to use the previously found values of 1/r, v as far as the
fourth order, if we merely require z accurately to quantities of the fifth order. When z
has been obtained the value of * is easily deduced. The terms of the order y 2 in 1/r, v can
then be calculated by methods quite similar to those used before to approximate to these
coordinates. Returning to the latitude equation we find the portions of z and thence those
of * which are of the order y 3 ; and from the new parts of * so found we can obtain the parts
in 1/r, v which are of the order y 4 . De Pontecoulant, neglecting quantities above the fifth
order, stops at this point ; the terms of order y 5 in * are simply those given by the elliptic
formulae. In the course of the approximations, the value of g is found and the meaning of
y defined in disturbed motion according to the principles explained in (v). When all these
operations have been performed we shall have found complete expressions for the coordi-
nates, accurately to the fifth order at least, as far as the action of the Sun is concerned.
152. Two important differences between the expressions obtained above and those
found by de Pontecoulant, must be noted : they refer to the meanings of e, y in disturbed
motion. We have defined them to be such that the coefficient of sin in longitude and
that of sin 17 in z shall be the same as in undisturbed motion. De Pontecoulant, having
before him the earlier results obtained by those who followed Laplace's method and
desiring to compare his expressions with theirs, so determined these constants that, in the
final expressions of the coordinates in terms of the time, the results, if correctly worked
out, should agree. This point will be further explained in Art. 159 below.
153. The successive approximations are not given in detail by de Pontecoulant. He
merely states that the labour of performing them was very great, and then proceeds to
write out the complete value of R obtained by substituting the results of the various
approximations in the expression given in Art. 121 above ; he gives also the value of Bu
furnished by the previous approximations. With these expressions for M, du, he goes on
to find the complete value of 1/r and thence that of y, finally obtaining the value of *. The
labour of performing this last approximation is divided into several portions : first, we are
given those portions independent of y, whose coefficients include only the powers of e, e',
a/a' contained in their characteristics ; secondly, those terms in the latitude whose coeffi-
cients depend only on the first power of y and on those powers of e, ef, a/a' present in
their characteristics; thirdly, the omitted portions of the coefficients in all the three
coordinates are found.
He then proceeds to find the coefficients of certain long-period inequalities more
accurately; and also to obtain the inequalities due to the actions of the planets, the
non-spherical shape of the earth, etc. The results are worked out algebraically all through
and the final expressions for the parallax, the longitude and the latitude are given in
Chapter vn. of the volume. Numerical results are then obtained by substituting the
numerical values of w, e, e\ y, a/a', I/a, in the coefficients, for the particular case of the
Earth's Moon. These are followed by tables comparing his results with those obtained
by Plana and by Damoiseau and with those obtained directly from observation.
154. A brief examination of the literal expressions for the coordinates given by
de Pontecoulant in Chap. vn. of his volume will show that the series for certain
coefficients, when arranged according to powers of m, appear to converge very slowly owing
to the large numerical multipliers; it is doubtful whether all these series are really
convergent even for the small value which m has in the case of the Moon. If we arrange
them according to powers of e 2 , e* 2 , y 2 , (a/a') 2 , such slow convergence does not seem to take
place on account of the very small values of these four constants. It becomes important
P. L. T, 8
114 DE PONTCOULANT'S METHOD. [CHAP, vn
then to consider whether we can improve the slow convergence by substituting another
parameter instead of m and also how we may estimate the remainders of such series on
the assumption that they do really converge (see Art. 69).
One of the series which converges very slowly is that which represents the coefficient
of the term with argument 2 - + <' and characteristic ee'. The portion of this coefficient
in longitude, of the form ee'f(m}^ as given by de Pontecoulant, is*
The last term calculated has therefore a numerical multiplier of nearly 90,000 and its
ratio to the first term is about 24,000 m & = 1/18 approximately. The numerical value of
the terms given is about 30", so that six terms of the series are not sufficient to give the
number of seconds accurately, although the complete coefficient is small. It is such series
as these which make the literal development tedious and difficult.
It has been shewn by Hillf that if we expand in powers of m = m/(l m} instead of in
powers of m, most of the series will be rendered more convergent. Further, a careful
inspection will often enable us to estimate the remainders with some exactness, owing to a
certain regularity which these series appear to display J.
* Sys. du Monde, Vol. iv. p. 577. It must be remembered that the definition of e is that
used by de Pontecoulant ; this coefficient will therefore not be the same as that obtained by
Delaunay. See Art. 159 below.
t "Besearches in the Lunar Theory," Amer. Journ. Math. Vol. i. p. 141.
See two notes by the author in the Monthly Not. R. A. S. Vol. LII. pp. 71 80; LV. pp. 35.
In the latter, the complete numerical value of the above term is given.
CHAPTER VIII.
THE CONSTANTS AND THEIR INTERPRETATION.
155. ONE of the most important departments of the lunar theory is the
interpretation of the arbitrary constants which arise when the equations of
motion are integrated. It has already been mentioned that in obtaining
expressions to represent the coordinates at any time, we must keep in view
the necessity of putting the expressions into forms which enable us to readily
give a physical interpretation of the results. In one direction the elimina-
tion of all terms which increase in proportion with the time this object has
been achieved : the difficulty was only of an analytical nature. Another
question the convergency of the series obtained we have been obliged to
leave aside owing to the lack of any certain knowledge on the subject. Further,
assuming the convergency of the series which are obtained when the problem
has been formally solved and when the coordinates have been expressed in
terms of the time and of certain constants, it is necessary to so determine these
constants that the initial conditions of the problem may be satisfied. In many
physical problems it is sufficient to know the initial coordinates and velocities
in order to determine the constants easily. But the peculiar nature of the
problems of celestial mechanics 'makes it impossible to find them with any
approach to accuracy in this way; this is owing partly to the difficulty of
measuring distances, partly to the inaccuracy of a single observation and
partly to the complicated nature of the equations which it would be necessary
to solve. Consequently, indirect methods must be resorted to.
There are three questions to be considered in the interpretation of the
results. In the first place, we must give definite physical meanings to all the
constants involved in order that we may be able to apply the results to the
case of any satellite which moves under the general conditions initially
assumed. Secondly, we must be able to determine the numerical values
of the constants from observation, as accurately as the observations permit,
in the case of any such satellite and more particularly of the Moon so
8-2
116 THE CONSTANTS AND THEIR INTERPRETATION. [CHAP. VIII
that tables may be formed which will give the place of the satellite at any
time, with an error not exceeding that of a single observation. The discovery
of new causes of disturbance, and of their magnitude, is rendered far easier
when tables, calculated from theory alone and including the effects of all known
causes of disturbance, have been formed. It is the small differences between
the tables and the observations which will be most likely to lead to an
advance in our knowledge of the peculiarities in the motions of the bodies
forming the solar system provided these differences cannot be wholly
accounted for either by errors of observation or by inaccurate values of the
constants used in forming the tables. Thirdly, when the constants have been
determined, the magnitude, of the effects produced by the various terms
present in the expressions for the coordinates, will be inquired into.
The Signification of the Constants.
156. We have, in the previous Chapter, formed certain developments for
the coordinates and, in so doing, we have introduced new arbitrary constants
or defined those introduced into the first approximation in a new way. It is
immediately obvious that as soon as we begin to use the results of the second
approximation, the constants can no longer have their former significations.
They were specially defined for the case of elliptic motion, that is, motion
in a curve of known properties. When this orbit was modified by the
introduction of c and g, it was still possible to interpret the results
geometrically, namely, by the use of an elliptic orbit of fixed size and shape,
with its apse and its node moving with uniform velocities in given directions
(Art. 68). When, however, we go further and approximate to the path of the
Moon by the methods of Chap. VII., no such easy interpretation of the
results is possible : the curve described is not one with whose properties we
are familiar. In order to use the results, it will be necessary to consider
the constants separately; we must also give them such meanings that the
determination of them by observation shall be as easy as possible and that
the results of any other method, in which the arbitrary constants of the
solution may be introduced in a different manner, can be compared with
those just obtained.
Besides the arbitrary constants of solution there are present certain
constants which have been supposed known, namely, those referring to the
elliptic solar orbit. These are m', ri, a', e f , e', <sr', of which the first three are
connected by the relation m' = ri' 2 a' 3 . When the orbit of the Sun is supposed
not to be elliptic and not to be in one plane, two new constants depending on
the position of the plane of the orbit will be introduced, and all the
constants must be defined again. The problem of the determination of the
solar constants, although it does not differ much from that of the lunar
constants, belongs mainly to the planetary theory and we shall therefore
155-158] THE NEW ARBITRARIES INTRODUCED. 117
leave it aside. In the following we retain the former suppositions that the
orbit of the Sun is elliptic and that it lies in the fixed plane of reference.
In attaching meanings to the arbitrary constants used in disturbed
motion, the principal object which will be kept in view is the consideration of
definitions which depend in no way on the method of integration adopted.
We shall thus be able to compare the results obtained by any of the methods
used for treating the lunar theory.
157. In the first approximation we had seven arbitrary constants,
a, n, e, e, -cr, 0, i (or 7) and one constant /* present in the differential equations :
/JL was eliminated from the results by means of the necessary relation
fi ri*a 3 . This relation is not used to eliminate a or n because fj, is a
much more difficult constant to determine observationally than either a or n:
fi is, in fact, considered as an unknown, although definite, constant. When
we proceed to the second approximation which, in de Pontecoulant's method,
is really the discovery of particular integrals of the equations, it is theo-
retically unnecessary to introduce new arbitrary constants, the requisite
number being already present in the complementary function. The relation
fjL = n?a 3 would, however, have been replaced by one much more complicated
and certain important terms in the expressions for the coordinates would
have been much less simple. It is possible to make the required changes in
the arbitrary constants, after the expressions have been obtained, but we gain
much by defining them as we proceed in the way finally necessary. To do
so, the new constants must be retained and suitable meanings must be given
to them.
In order to simplify the explanations as much as possible, we shall first
neglect 7 (or i) and therefore 6 ; we then merely treat the constants present in
cases (i) (iv) of Chap. vii. In cases (iii), (iv) no new arbitraries were
introduced: we were only finding particular integrals. It is therefore only
necessary to pay attention to those present in (i), (ii). As already stated in
Art. 151, we might have taken the four cases at one step : the parts of 1/r, v t
due to the second approximation and collected in Art. 149, would then have
been found together. Had we done so, the terms considered in (iii), (iv)
would have appeared as parts of the solution which it is not necessary to
consider here. We therefore treat cases (i), (ii) only and suppose that the
second approximations, there separated, have been made together.
158. In these two cases there were five constants a, 6 > SA, -#, c which
were either new arbitraries introduced or parts of the assumed solutions
which were not directly determinable. Between a, 6 there was one relation,
namely, equation (10) of Art. 134. But since /u. was not present in the
solution, another relation involving /* existed between a, b , Bh. Hence only
118 THE CONSTANTS AND THEIR INTERPRETATION. [CHAP. VIII
three new independent arbitraries, which may be taken to be &h, B, c , were
introduced. It is required to find what connection there is between these
and the old arbitraries a, n } e, e, -cr.
The constant B arose exactly in the same way that 6 did, namely, as the
additive constant in the integration of the longitude equation. The constant
additive to the longitude therefore becomes B+ e instead of e. As both B, e
are arbitrary, the introduction of B is unnecessary and we can put it zero.
The longitude is then expressible in the form
t x const. + e + periodic terms.
Hence, in disturbed (or undisturbed) motion, e is the value of the mean*
longitude at time t = 0.
The constant Bh is determined in a similar way so that the meaning
of n may be fixed, but a complication arises owing to the way in
which Bh occurs. The expression for dv/dt consists of a constant term and
periodic terms. When the motion is undisturbed there is only one constant
term and it was denoted by n. (The arbitrary actually introduced was h ; n
is the constant term in the development of k/r~.) When the motion is
disturbed, definite constant terms are present as well as the arbitrary constant
&h. The expression for the mean longitude appears in the form
nt (1 + powers of m 2 etc.) + e.
The presence of Sh enables us to get rid of all these other terms, so that the
longitude is expressible in the form
nt + + periodic terms.
Hence n is the mean angular velocity of the Moon or, as it is generally
called, the mean motion, whether the orbit be disturbed or undisturbed.
Since the new definite terms which appear in the expression of dv/dt can
always be eliminated in any stage of the approximations, n will be the mean
motion at any stage and therefore in the final results.
It will be noticed that in case (i), dh was simply used to get rid of these new terms. In
case (ii), the result obtained by equating to zero the new terms multiplied by t, was required
in order to determine the new value of e. The introduction of 8k in cases (iii), (iv) is
unnecessary, for if we stop at the first powers of e', a/a', no definite constant terms arise in
the new part of dv/dt ; the value of 8A would therefore have been found to be zero and it
was neglected at the outset.
Next, to find 6 , we substituted in equation (8), Art. 131, assuming that
the relation /j, = n z a s still held. Had we simply used equation (1) without
introducing a, this would have been unnecessary for the purpose of finding
* The mean value of any quantity expressed in this form is defined, in celestial mechanics, as
its value when all the periodic terms are neglected.
158-159] THE CONSTANT OF ECCENTRICITY. 119
b Q which would have been determined by equation (10), Art. 134. But then
the new relation between ft, a, n and the other constants would have to be
determined from some equation of motion similar to (8) involving /z- ; a simple
definition for a in disturbed motion is also required. It is found best to
define a by the equation fi = n z a 3 , where n has the meaning just defined. This
we have done in case (i). The relation ytt = n z a 3 having been assumed to hold
and b having been found from equation (8), the introduction of a is un-
necessary as far as the discovery of the solution is concerned, but its presence
is necessary in order to make it evident that the equation for Su, p. 96, is
satisfied.
159. The constant ec in case (ii), was the coefficient of cos <f> in the value
assumed for aSu. Now the first approximation gave the value of a/r to be
1 + e cos </>,
neglecting powers of e higher than the first. Therefore, when we include in
the value of a/r the results of the second approximation, the coefficient of
cos <f> becomes
But e was an arbitrary of the first approximation to 1/r and ec is an
arbitrary introduced exactly in the same way in the second approximation to
1/r. We can therefore determine c at will. Instead of putting c zero, it is
found best to determine it so that the coefficient of sin <f> in longitude is the
same whether the motion be disturbed or undisturbed. Certain definite terms
will also occur with c in this coefficient (p. 102); these can always be
eliminated, by the proper use of C , at any stage of the approximations, in the
same way that Sh was used to cancel those occurring in the coefficient of t in
the expression for the longitude.
Other methods for fixing the meaning of e have been used. The older lunar theorists,
taking the true longitude as independent variable and expressing the time or the mean
longitude and the other coordinates in terms of it, fixed the meaning of e so that the
coefficient of the principal elliptic term, in the expression of the parallax in terms of the true
longitude, was the same in disturbed and undisturbed motion. After the true longitude
has been expressed in terms of the mean longitude (or of the time), the principal elliptic
term in longitude contains powers of m, e" 2 ,... in its coefficient. The characteristic is, in any
case, e. De Pontecoulant, when working out his theory, wished to compare his results with
those of the earlier investigators. He therefore determined c and e so that the coefficient
of sin< in longitude was the same as with them. We have not followed him in this
detail because the more complete theory of Delaunay has the definition of e used here and
because e so defined is obtained observationally with much greater ease.
In working out with rectangular coordinates the inequalities considered in case (ii), I
have defined the constant of eccentricity by the coefficient of sin </> in the expression of the
coordinate Y (Section iii, Chap. n.). This appeared to be the simplest plan, in view of
the later approximations necessary to form a complete development by this method. See
Section (ii), Chap. xi.
120 THE CONSTANTS AND THEIR INTERPRETATION. [CHAP, VIII
160. It is not necessary to introduce a new constant for the determination
of OT in disturbed motion. The reason for this will be seen more clearly when
we come to the method of the Variation of Arbitrary Constants as exemplified
by Delaunay's theory. It will then be seen that the variable longitude of
perigee, in disturbed motion, is expressed in the form
(1 c) nt + CT + periodic terms.
The constant or is therefore the value of the mean longitude of perigee at
time t = and it keeps this name in any theory ; also (1 c) n is the mean
motion of the perigee. As the longitude of perigee only occurs in the
elliptic expressions for the coordinates under the signs, sine and cosine, and
as the periodic terms which occur in the above expression for the longitude of
perigee have coefficients of the first order at least, we can, when substituting
in the elliptic expressions for the coordinates, expand the sines or cosines so
that no periodic terms appear in the arguments : the coordinates are therefore
ultimately expressed as in Chap. vn. Since or only occurs in the principal
elliptic term of de Pontecoulant's method in the form cut + e VT, and since e
may be supposed to be known, to- may be defined by means of the value of
this argument at time t 0.
It is usual to retain the term ' eccentricity ' for e in disturbed motion
whatever be the plan used to fix its meaning. The reason for giving to
(1 - c) nt + CT the term ' mean longitude of perigee ' will be evident from the
remarks just made. The constants e, -nr are called the ' epoch of the mean
longitude' and the 'epoch of the mean longitude of perigee/ respectively.
The constant a, defined by the equation //, ?i 2 a 3 , is often called the 'mean
distance ' or the ' semi-major axis of the orbit.'
The term * mean distance ' thus applied to a is inexact according to the usual definition
of the word 'mean' (see footnote, p. 118). With the above definition of a, we have
determined r in the form
a/r = 1+ ft + periodic terms,
which would give r/a = l+f ^-periodic terms (a);
0, ff being constants depending on m, e' 2 .... According to the definition of the word
1 mean,' the mean distance should be a (1 -f-/3'), while a/ (I +0) is the distance corresponding
to the mean value of the sine of the parallax. The latter is the quantity determined
observationally and therefore of most importance in this connection. The terms in ft are
small and are easily found when the values of the other constants have been determined by
observation.
161. Remarks quite similar to those made concerning e, OT, apply to 7, 0.
We have determined 7 in Art. 147 so that the coefficient of the principal
term in z is the same as in undisturbed motion. It is better to define it so
that the coefficient of the corresponding term in the latitude u, is the same
as in undisturbed motion. The transformation from the old constant to the
160-162] DETERMINATION BY OBSERVATION. 121
new one is easily made when u has been found. The coefficient of sin 77 in
u will be 7(1 + $"), where $' depends on m, e 2 , e"*,.... We replace 7 by
yj(l + ft") wherever the former constant occurs ; 7 is called ' the tangent
of the mean inclination.' The longitude of the node, when found as in
Delaunay's theory, will be expressed in the form
(1 g) nt + 4- periodic terms.
Hence (1 g)n is the mean motion of the Node, 6 is the epoch of the mean
longitude of the Node ; the latter is determined in de Pontecoulant's method
by finding the value of the angle gnt + 6 at time t = 0.
Determination of the Constants by Observation.
162. The three coordinates of the Moon which are observed directly,
are the longitude, the latitude and the parallax. Of these, an expression for
the longitude has already been obtained; the expressions for the parallax
and the latitude are deducible immediately.
Take the Earth's equatoreal radius as the unit of distance. Then 1/r
will be the ratio of the Earth's equatoreal radius to the distance of the Moon,
that is, the sine of the equatoreal horizontal parallax of the Moon. Let II
denote this parallax. We have approximately
The average value of sin II is about ^. The error caused by neglecting the
term sin 3 II is about 20t Q 00 of the whole, corresponding to an error of 0"'2 :
this is within the limits of error of a single observation. To this degree of
accuracy we can therefore put II = a/r.
To find the latitude we have, since tan u = s,
As s is a small quantity of the order 7, we can quickly find u when s is
known.
These three angular coordinates are therefore expressible in the form
Q = A + Bt + 2C*({it + /3') ..................... (1).
Mil
When Q denotes the longitude, the periodic terms are sines ; when Q denotes
the latitude, they are also sines and A, B are both zero ; when Q denotes the
parallax, we have B = and the periodic terms are cosines.
In all cases Q, and therefore A, C, are the circular measures of angles. To
express them in degrees we multiply by 180/Tr, or in seconds of arc by
180 x 60 x 60 + TT = 206,264'S.
122 THE CONSTANTS AND THEIR INTERPRETATION. [CHAP. VIII
The number of seconds of arc in any coefficient is therefore obtained by
inserting the numerical values of the constants and multiplying the result by
206,265.
163. Suppose that in the expressions for the coordinates, represented by
the general form (1), we stop at a given order; they will then be reduced to
a finite number of terms. If a number of values of Q, equal to the number
of constants A, B, C, 0, $' present, be given, each of these constants could be
determined independently. But our expressions have shown that only six or,
if we suppose /JL unknown, only seven of these constants are independent.
(We consider the solar constants known.) Hence, if the observations and the
theory were both correct, exact relations ought to exist between the various
constants thus found when the number of observed values of Q is greater
than seven. But these conditions are not quite fulfilled. In the first place,
each observation is only approximate and must be regarded as subject to
error. In the second place, the coefficients of the periodic series, being
each of them formally represented by an infinite series about the convergency
of which we have no information, can only be considered at the best as
approximate, apart from the question as to whether the infinite series is a
correct representation of the coefficient. Assuming that the infinite series
are possible and convergent, in order to determine the numerical values
of the seven arbitrages, it is still necessary to choose the particular terms
which are best adapted to our needs.
Now the methods used to find the constants present in an equation of the
form (1) enable us, in general, to obtain with a high degree of accuracy the
coefficient, period and argument of any term when the number of observed
values of Q is very great.
Suppose that it be required to determine the constant e. We naturally
choose out of one of the coordinates the term or terms in which a given
alteration to e will produce the greatest effect on the value of that coordinate.
This term is immediately recognised as being that with argument < in the
longitude. All the other terms in longitude containing e have either powers
of e higher than the first in their characteristics, or e is multiplied by some
small quantity such as m, 7 2 ; in parallax, all terms are multiplied by the
small quantity I/a. Again, the number of available trustworthy observations
of the longitude is far greater than those of the other coordinates. Finally,
since we have chosen that the coefficient of sin < in longitude shall be the
same as in elliptic motion, this coefficient can be obtained theoretically to
any degree of accuracy we desire. For all these reasons the determination
of e by observation from the term with argument <f> in the longitude is the
most suitable. The advantages of the definition of this constant, adopted in
Art. 159, now become very evident.
162-164] NUMERICAL VALUES. 123
It will easily be seen how the numerical values of all the seven arbi-
trary constants are determined. The values of n, e are obtained from
the non-periodic terms nt, e in v. The principal elliptic term then furnishes
e, CT and also en if we wish to find the period by observation. The principal
term in latitude that with argument rj (=gnt + e 0) gives 7, 6 and also gn.
Finally, the constant part of l/r furnishes the value of a; this constant part
contains also a few terms depending on m, e' 3 , etc. which are known, since
their numerical values were previously found.
164. At the present day the numerical values of most of the constants are known with
a very high degree of accuracy. Tables have thus been formed of the motion of the Moon
from theory alone. Notwithstanding the great care bestowed by various investigators in
including in them the results of all known causes, small differences between the tables and
the observations are continually to be found. Some of these can be put down to errors of
observation but many of them, especially when they exceed a certain limit and appear
to be either periodic or secular, are due to imperfections either in the theory or in the
numerical values of the constants used in the tables. Even when all the corrections due
to known causes have been made, certain empirical corrections, not indicated by theory,
have to be applied to these tables in order that they may agree with the observations.
The tables published in 1857 by Hansen, together with certain corrections investigated by
Xewcomb (see Art. 173), are still used to obtain the places of the Moon given in the
Nautical Almanac on pages iv to xii of each month.
A comparison of the values of en, gn, as determined from theory and directly from
observation, furnishes a valuable test of the sufficiency of the known causes to completely
account for the motion of the Moon. The recorded observations of eclipses have enabled
astronomers to obtain n, (l-c)n, (1 -g} n with a high degree of accuracy, and the values
of c, g deduced therefrom agree very closely with those calculated by theory. Nevertheless,
the small differences between theory and observation still leave something to be desired.
As far as may be judged, the results deduced from observation appear to be rather more
trustworthy than those deduced from theory.
In order to reduce the results of the preceding Chapter to numbers it is
necessary to know beforehand the numerical values of certain of the constants.
We take the units of time and length to be the sidereal day and the Earth's
equatoreal radius, respectively: the numerical values of the constants used
below are those by which Delaunay* reduced his theory to numbers (see
Art. 173). He takes
27T/7i' = 365-25637 days, l/a' = 8"-75, e' = 0'01677106 (2).
The object of the following articles being merely to find the extent by
which the principal inequalities affect the place of the Moon, we shall not
here require to know the values of e', tar' or of e, -cr, 6. The parts of chief
importance are the coefficients and periods. In other words, we consider
mainly the amplitudes and periods of the periodic oscillations and not their
phases.
* Mem. de VAcad. des Sc. Vol. xxix. Chap. xi.
124 THE CONSTANTS AND THEIR INTERPRETATION. [CHAP. VIII
The Mean Period and the Mean Distance.
165. The longitude is expressed in the form
V = nt + e + periodic terms.
The mean longitude is therefore nt + e and the Mean Period T is the
time in which the mean longitude increases its value by 2?r. Hence T = 2?r/n.
We find T directly from observation to be about 27 J days or more exactly
= T= 27-321661 (3).
With the value of n' given in equations (2) of the previous Article, we
deduce
m = ri/n= -07480133 = 1/13J, approximately ............ (4).
The parallax is given (Art. 138) by
-=-(l+4-ra 2 Mira 4 ) + periodic terms.
T a
The mean value of the sine of the equatoreal parallax is found directly
from observation to be 3422 //f 7. To obtain I/a we have therefore
which, with the value of m just found, gives
l/a = 3419 // '6 ................................. (5).
This value is very little altered by including the terms which have not been
calculated here for the constant part of 1/r.
The distance of the Moon corresponding to this value is 206,265/3419*6
equatoreal radii of the Earth or about 239,950 miles, taking the Earth's
equatoreal radius as 3,978 miles. The real mean distance, calculated by the
formula (a) of Art. 160, is 238,840 miles.
From the value (2) of a' and (5) of a, we deduce
-, = '002559 = 3^- approximately .................... (6).
The Variation.
166. The term with argument 2f in longitude or parallax is known
by the name of the Variation. Let T be the mean periodic time. The
Variation runs through all its values in time
27T/2 (/i - n') = T/2 (1 - m) = 14f days, by (3), (4),
or in half the mean synodic period of revolution.
165-167] THE VARIATIONAL CURVE. 125
The coefficient of this term in longitude was (Art. 138) seen to be
^-m 2 + f|m 3 + ^m 4 ,
which, when the value of m has been substituted and the result multiplied by
206,265, gives 34' 51". When the portions in the coefficient depending on
higher powers of m and on e 2 , e' z etc. are taken into consideration, the value
of the coefficient is found by Delaunay to be 39' 30".
The investigations of case (i) have shown that if the approximations
to the coefficients be continued, the result for the terms dependent on m
only, will be
cos m v - n't -<' = %+ 26' 2<? sin 2?f , (q = 0, 1, 2, . . .),
where b zq , b' 2q depend only on m and are each of the order m zq . The terms
considered in case (i) therefore constitute a curve, periodic with reference to
axes moving in their own plane with uniform angular velocity n. The curve,
relatively to these axes, will be closed and symmetrical ; the time of revo-
lution round it will be 2?r/(n - n'), or a mean synodic month. Since, in the
case of our Moon, all the coefficients b^ are positive, the maximum and
minimum values of a/r are given by f = and f = 7r/2 respectively. Hence
the shortest diameter of the oval is directed towards or away from the mean
place of the Sun. We shall call this line the X-axis of the oval.
All the inequalities with arguments 2q^ may be called 'Variational
Inequalities' and the curve just defined the 'Variational Curve.' This
curve has been calculated and drawn by Dr Hill for satellites of periods
differing from that of the Moon.
He has shown* that as the value of m is increased, the ratio between the
lengths of the two axes increases while the velocities at the ends of the
longer axes, that is, in quadratures, diminish. For the value m = 1/2*78 the
velocities in quadratures vanish and the curve has cusps at those two points.
M. Poincaref has shown that if the value of m be still further increased the
cusps are replaced by loops, so that a satellite whose mean period relative to
that of the Sun is greater than 1/278, would appear in quadrature six times
during one revolution.
The Parallactic Inequality.
167. The terms considered in case (iv) of the previous Chapter are
closely related to the Variational inequalities by their arguments, the latter
being simply odd instead of even multiples of f. The principal term
that of argument f is called the Parallactic Inequality.
* Amer. Jmir. Math. Vol. i. pp. 259, 260,
t Mec. Gel. Vol. i. p. 109,
126 THE CONSTANTS AND THEIR INTERPRETATION. [CHAP. VIII
The coefficient of this term in longitude was found in Art. 144 to be
by the values of m, a/of given in (4) and (6) respectively. Delaunay's value
of the whole coefficient is 2' 7". According to the results of Chap. i. these
numbers are to be multiplied by (E M)/(E + M) = 39/40 approximately
(Art. 168).
The period of the term is 27r/(n n'), that is, one mean synodic month.
Suppose that these inequalities be included with the Variational Inequali-
ties in the expressions for a/r and v. We shall have
a/r = %b q cos q% , v n't e' = j; + ^b' q sin q% ;
where the terms for which q is even are functions of m only and those for
which q is odd, besides being functions of m, have the factor
a(JE-M)/a'(E + M).
When we put f for f , r is unaltered and v n't e' changes sign. The
curve, referred to the same moving axes as before, is therefore symmetrical
about the line directed to the mean place of the Sun. In this case, however,
f = and ( = TT do not give the same values for r. Let r , r n denote the values
of r when f = 0, TT respectively. We then have
= a negative quantity
in the case of our Moon, for 6j is then greater than the sum of the quantities
6 3 , 6 5 ... and it is negative (see equation (29), Art. 144). Hence
The longer JT-axis is therefore directed towards the Sun and the shorter
Sun
Fig. 7.
167-169] PARALLACTIC TERMS. 127
X-axis away from it. The effect of the inequalities depending on a/a is
to slightly distort the Variational curve in the direction of the Sun. The
general character of the remarks just made, is not altered by the introduction
of the squares and higher powers of a/a'.
All the inequalities which have arguments of the form (2q + 1) f may be
called Parallactic Inequalities. The principal Parallactic Inequality has
been used to determine the parallax of the Sun, that of the Moon being
supposed known. It will be immediately seen that if we find the coefficient
by observation and also by theory, we can, knowing m with great accuracy,
deduce the value of I/a'. This method is, however, defective since it involves
the accurate knowledge of the ratio (E M)/(E + M).
The variation as well as the other principal inequalities in the Moon's motion and
the motions of the Apse and the Node, were first explained by Newton on the theory
of gravity only. The values of their coefficients were obtained by him more or less approxi-
mately. The oval curve, called above the Variational Curve, was also recognised by him
and the ratio of its two diameters was shown to be approximately as 69 : 70, corresponding
to a coefficient 35' 10" of sin 2 in longitude. The results of Newton's investigations
in the lunar theory are contained chiefly in Props, xxn. xxv. xxxv. Book m. of the
Principia.
168. The determination of the ratio MjE is a matter of some difficulty. There is no
direct way of obtaining it. Probably the best method consists in finding the inequality in
the motion of the Earth due to the action of the Moon. It will be readily seen from
Chap. i. that E will describe a circle of radius Ma/(E+M) about G, if we suppose that S
describes a circle about G and that M describes a circle of radius a about E. As the
Moon performs a synodic revolution, the apparent place of the Sun as seen from the Earth
will therefore oscillate to and fro about its mean position. By observing the extent of this
oscillation we can, knowing the other constants with considerable accuracy, deduce the value
of M/(E+M).
The constant may be determined by comparing the tides produced by the Moon
with those produced by the Sun and also by comparing the observed nutation of the
Earth's axis with the value deduced from theory. See de Pontecoulant, Systeme du Monde,
Vol. iv., p. 651. The differences between the values so obtained, indicate that the
numerical value of the ratio MjE is not certainly known within five per cent, of its true
value.
The Principal Elliptic term, the Evection and the
Mean Motion of the Perigee.
169. The Principal Elliptic term in longitude, having the same coefficient
as in undisturbed motion, is, as far as quantities of the third order (Art. 50),
(20- J^sin^.
The coefficient is found directly from observation to be
617'19"-06 =
128 THE CONSTANTS AND THEIR INTERPRETATION. [CHAP. VIII
Dividing this by 206,265 and equating the result to 2e \&, we obtain
e = '0548993.
The argument of the term is cnt + e w and its period is therefore T/c.
The value of c given by equation (25) of the last Chapter, is
Whence 1 - c = '0041964 + '0029428 = '0071392.
Delaunay finds the complete value of 1 c to be '00845. The period of
the term is therefore about 27^ days.
The term with argument 2f <f> is called the Evection. The value of the
coefficient in longitude was found (Art. 140) to be +-me which, with the
values of m, e deduced above, gives 52' 56". The complete value of this
coefficient is 116'26".
The period of the term is
27r/(27i - 2ri - en) = T/(l - 2m + 1 - c)
= 31f days approximately.
In Art. 160 we have seen that (1 c)n is the mean motion of the
Perigee. The Perigee will therefore make a complete mean revolution in
time T/(l - c) = 3827 days with the value of c found above. The period as
calculated by Delaunay* is 3232'38 days or about 9 years.
The class of inequalities defined by the arguments 2^f + p(j>, (p, q integers)
may be termed Elliptic Inequalities. The characteristic of the terms with
arguments 2<?f p4> is eP -
170. We can combine the principal elliptic term and the Evection in a
manner which enables us to illustrate the connection between the results
obtained by the method of Chapter vir. and those obtained by causing the
arbitrary constants to vary.
The principal elliptic terms in longitude and parallax are, neglecting
quantities of the orders e 2 and m 2 ,
.=... + 2e sin <f> -f- f-me sin (2f <) -f . . . ,
-=...+ e cos $ + Jg 5 - me cos (2f </>)+....
Let 20! sin < a = 2e sin <f> + g-me sin (2 </>),
e l cos </>! = e cos </> + if-me cos (2f <).
* " Note sur les mouvements du p6rig6e et du noeud de la Lune," Comptes Rendus, Vol. LXXIV.
pp. 1721.
169-171] THE ELLIPTIC INEQUALITIES. 129
From these we obtain, neglecting quantities of the order ra 2 ,
or e l = e + -me cos
We also deduce
e l sin (</>! - <) = J^me sin (2f -
Since e,, e are of the same order, </>j must be a small quantity of the order
m at least. To the order required we therefore have
- = m sn -
The transformations give
v = . . . + 2^ sin </h + ..., a/r = . . . + e l cos lf
where ^ = e + ^ra0 cos (2 - 2<),
we sin ( 2 ? - 2^) = w + - {(1 - c) nt + w - ig-m sin (2 -
The effect of the action of the Sun as far as it produces the Evection, is
to cause periodic variations of the eccentricity and of the longitude of perigee
of the Moon. Had we solved by causing the arbitrary constants to vary, the
variable values of the eccentricity and longitude of perigee would have been
found to contain terms of this form.
In a similar manner, the other terms due to the action of the Sun may be
included by assuming variable values for the constants. In order to perform
the process completely, it would be necessary to assume that the velocities
have the same form as in elliptic motion, in accordance with the principles
of Chapter v. To the order considered in the above example this is easily
seen to be true. We should of course include the terms depending on higher
powers of e, e l . This method of expressing the coordinates is not generally
useful in itself: it is merely given here as an illustration. In fact, when we
use the method of the variation of arbitrary constants, the reverse process
has to be gone through to achieve the object in view, namely, the expression
of each coordinate as a sum of periodic terms.
The Annual Equation.
171. The terms in longitude and parallax with argument <f>' = n't + e' TS'
are known as the Annual Equation ; their period is one year. The coefficient
in longitude was found (Art. 142) to be
- e' (3m + 0??i 2 ) = - 12' 56",
the complete value being II 7 10".
B. L. T. 9
e*AR y
x*
RSIT
_*^
130 THE CONSTANTS AND THEIR INTERPRETATION. [CHAP. VIII
The coefficient of the Annual Equation in parallax is, from the same
Article,
- - e' (f m 2 + Om 3 ) = 0"'43,
CL
which is the value correct to one hundredth of a second of arc.
The terms with arguments 2<?f p<j>' in longitude and parallax may be
called the Mean Period Inequalities. The variable parts of their arguments
depend only on the mean motions of the Sun and the Moon ; they are in-
dependent of the longitudes of the perigee and the node of the Moon's orbit.
The Latitude and the Mean Motion of the Node.
172. We have found (Arts. 147, 162) as far as the order 7m 2 ,
U = s = (1 -f ra 2 ) 7 sin 77 + (f m + ff m 2 ) 7 sin (2f - 77) + Jgi-ra 2 7 sin (2f + 77).
We shall replace the constant 7 by that used in Delaunay's theory. In
undisturbed motion let 7! be the sine of half the inclination ; therefore, to the
order considered here, 7 = 27^ In disturbed motion, y t is denned so that the
coefficient of the principal term in u is the same as in undisturbed motion.
Hence, in disturbed motion, we have to the order 7m 2 ,
7(l+im 2 )=27 1 , or, 7= 2 7l (1 -Jm).
The value of u expressed in terms of % is then given by
u= 2 7l sin 7) + (f m + ff m 2 ) 7i sin ( 2 ? - ?) + ir m2 7i sin (2f + 77).
The coefficient of sin 77 is found from observation to be
57'41"-26 = 18461"-26
corresponding to the value '04475136 of 7^ The correct value is
7l = -04488663 (7),
the difference being due to the omission of the elliptic terms of higher orders
in the coefficient of sin 77.
The period of the term is ^-n-jgn = T/g. The value of g was found to be
given (Art. 147) by
1 -# = _ (| m 2 _ _^ m s _ I7| m 4) = _ -0040119,
the complete value being* '0040212. The period is therefore about 27^
days.
The mean motion of the node is (1 g)n. This being negative, the node
moves backward, that is, in the direction opposite to that in which the Moon
moves. It completes a revolution in time
T/(g-l) = 6794-4 days or about 18J years.
* See footnote, p. 128.
171-174] INEQUALITIES IN LATITUDE. 131
It will be noticed that the first terms in the expressions for 1 - c, g - 1 are the same.
We have seen however that the apse moves forward twice as fast as the node moves back-
ward. The difference is principally due to the near equality of the first two terms in the
expression for 1 - c : the second term in the expression for g - I is quite small.
The method of Art. 170 may be applied to the remaining terms in the
expression for u, by assuming the inclination and the longitude of the node
to be variable. The two equations necessary for their determination are
furnished by the supposition that both u and du/dt have the same form as in
undisturbed motion.
173. The other terms present in the coordinates will not be examined here. Enough has
been said to show how the effect of any particular term on the place of the Moon may be
examined. The magnitudes of the coefficients can all be obtained in the manner explained
above. These are best seen in a paper by Newcomb, Transformation of Hanseris Lunar
Theory*. A reference to it will show that there are 2 coefficients in longitude (those of the
principal elliptic term and of the evection) which surpass 1, 11 coefficients lying between
1 and ]', 14 coefficients between 1' and 10" ; in latitude, 1 coefficient (that of the principal
term) greater than 1, 7 coefficients between 1 and 1' and 6 between 1' and 10" ; in
parallax, one term (the mean value) of nearly 1 in magnitude, one coefficient (that of the
principal elliptic term) of just over 3', and 7 coefficients lying between 35" and ]". The
number of large coefficients is therefore not great, as far as the solar perturbations are
concerned.
The methods used for deducing the numerical values of the constants from the recorded
observations will be found in the various memoirs which contain determinations of these
constants. The values of ri, ef which have been employed in the preceding articles were
obtained by Leverrier (Ann. de VObs. de Paris, Memoires, Vol. iv.), those of e, y by Airy
(Mem. of R. A. S. VoL xxix.) and that of I/a by Breen (Mem. of R. A. S. VoL xxxn.). The
values used by Hansen for the seven lunar constants will be found in the Darlegung (see
Art. 202 below) and the Tables de la Lune. Later determinations have been made by
Xewcomb (Papers published by the Commission on the Transit of Venus, Pt in. and various
memoirs in the first two volumes of the Papers published for the use of the Amer. Eph.}.
Further references will be found in the memoir mentioned in the previous paragraph, and
in the Nautical Almanac, the Connaissance des Temps, the American Ephemeris, etc.
174. It is not difficult to verify the statement made in Art. 70, that c, g, found by
Laplace's method with v as the independent variable, are the same as the values obtained
when t is the independent variable.
In Laplace's method we modify the first approximation by substituting for -si, & the
values (1 c)v + or, (l-<7) v+0, respectively. If v' be the true longitude of the Sun, the
disturbing function, which is expressed in terms of r, r 1 , s,v- 1/, will contain the angles
cv tzr, rit + f w', gv-6) v-v'.
In order to express v tf, rit+fw' in terms of v, we have
if =rit + +2e' sin (rit+f f -
= v Ze sin (cv w} + . . .
* Astr. Papers for the use of the Amer. Eph. Vol. i. pt. n. pp. 57107.
92
132 THE CONSTANTS AND THEIR INTERPRETATION. [CHAP. VIII
Hence v - #', rit + e' - or' can be expressed in terms of the four angles
v -- v '-\ e _1 ,y + e ' _ OT ' -- e ey-czr, gv 6,
n n n n
and therefore the disturbing function can be expressed in terms of v by means of the same
four angles.
After solving the equations used in Laplace's method, we ultimately obtain
where K consists of a series of sines of the sums of multiples of the four angles, the
coefficients being of the first order at least.
In order to get v in terms of t we must reverse this series. The first process in the
reversion is to put v = nt+e in the terms present in K. "We then get
where K v consists of a series of sines whose arguments are made up of the four angles
(n-n')t + <', n't + c' -ar', cnt + ce or,
The next process is to substitute this new value of v in K and to expand the sines of the
angles cv -us = cnt -f- ce tzr + cK l , etc. in powers of K . No new angles will be introduced. By
continuing the process, v is ultimately expressed in terms of the four angles which, after
putting for the arbitraries ce - or, gt 6 the new arbitraries e &, c-6, are the same as
those used in de Pontecoulant's theory. Also, v being expressed in the same form in
both cases, c, g must have the same values.
CHAPTER IX.
THE THEORY OF DELAUNAY.
175. THIS Chapter will be devoted to an explanation of the manner
in which Delaunay has applied the principles of the method of the Variation
of Arbitrary Constants to the discovery of expressions for the coordinates which
will represent the position of the Moon at any time. The principal object
which Delaunay had in view and which he fully carried out, is stated in the
preface to the two large volumes* containing his investigations, in the
following words ( :
' Determiner, sous forme analytique, toutes les inegalites du mouvement de
la Lune autour de la Terre, jusquaux quantites du septieme ordre inclusive-
nient, e)i regardant ces deux corps comme de simple points materiels, et tenant
compte uniquement de iaction perturbatrice du Soldi, dont le mouvement
apparent autour de la Terre est suppose sefaire suivant les lois du mouvement
elliptique!
The limitations imposed on the problem are therefore the same as those
made in the previous Chapters. The motion of the Sun is supposed to
be elliptic and in the fixed plane of reference, the disturbing function is the
same as that given in Art. 8, and the intermediate orbit is an ellipse
obtained by neglecting the action of the Sun. No modification of the inter-
mediate orbit, like that given in Chap. IV. and used in de Pontecoulant's
and Laplace's theories, is necessary here.
The use of canonical systems of elements being the basis of Delaunay 's
theory, we shall depart from the notation used above and, after Art. 179, adopt
that of Delaunay. The latter has the advantage of retaining a certain
symmetry in the formulae : it will also facilitate references to Delaunay 's
* Mem. de VAcad. des Sc. 4to Vols. xxvni. (I860) 883 pp., xxix. (1867) 931 pp. These will be
referred to in this Chapter as ' Delauuay, i., u.'
t Delaunay, i. p. xxvi.
134 THE THEORY OF DELAUNAY. [CHAP. IX
expressions and to the further developments (e.g. those in Chap, xin.) which
have been made according to his method by other investigators who have
generally adopted his notation in their memoirs.
The method by which the transformations contained in Arts. 178, 185,
189, 190 below, are carried out, is not the same as that of Delaunay; the
latter performed them by direct differentiation a process somewhat tedious.
Tisserand, in his account of the theory * , uses Jacobi's general dynamical
methods, stated in Art. 94 above, to perform the transformations. The
method used here is short and it has the advantage of showing immediately
the terms which are to be added to R (.
176. In Chapters IV., V. have been given the principles on which the
method of the variation of arbitrary constants is based. When the motion is
undisturbed it is elliptic, and the coordinates are expressible in terms of the
elements and of the time. When the action of the Sun is taken into
account by considering the elements variable, it has been shown (Arts. 84-86,
or 98) that the equations which express them in terms of the time are
dR . ( }
' 2 ' d
In these, c^, & have certain definite meanings with reference to the elliptic
orbit : they are explained in Art. 92.
The equations in this form possess a serious defect. It will be remembered
that R contains in its arguments, terms of the form nt + const. Now (Art. 84)
n = i$ar% .= fj,~ l ( 2ft)'l When, therefore, we form 9J?/3ft, the time t will
appear outside the signs sine and cosine and thus produce terms in the value
of tti, which increase continually with the time. It has been seen in
Chap. IV. that such terms are to be eliminated if possible. The artifice used
by Delaunay consists in simply replacing the variables ft, ^ by two others.
Before changing the variables to effect this object, some remarks must be
made on the method of performing this and similar transformations required
later.
177. Method for transforming from one set of variables to another.
Let any arbitrary variations S<^, Sft be given to ;, ft, and let $R be the
corresponding change in R. We have then
* M6c. Gel. Vol. in. Chaps. XL, xn. Also, Jour, de Liouville, Vol. xin. pp. 255 303.
t The method is used in a different way by Badau on pp. 336 340 of a paper "Bemarques
relatives a la Theorie des Orbites." Bulletin Astronomique, Vol. ix.
175-178] CHANGE OF VARIABLES. 135
The six canonical equations (1) can therefore be written, as in Art. 98,
or 2(dj3 i Soi i -d<x i $l3 i ) = dtSR, ..................... (I 7 ),
where R is supposed to be expressed in terms of 04, ft, t. The symbol d there-
fore denotes the^actual change taking place in time dt, while 8 denotes any
arbitrary variation of the elements. If we wish to transform from a;, ft to
another set of variables y i} 73, ... %> the process of finding the new equations
is rendered very easy. Suppose
i =/*(*, 7i>72>---76) Pi = fi(t, 7i> 72, -..7e);
we have, by the definitions of d, $,
and similarly for the variables ft.
The substitutions being made in the first member of (I'), we suppose R
expressed in terms of the new variables, so that
a r &R 5 , 9-R dR
oR**-*- 07! + 5- 672 + . . . + 3 67,5.
d7i 37-2 fye
Equating the coefficients of the independent arbitraries 87!, &y. 2 ,...Sy 6 to
zero, we obtain the new set of equations satisfied by y lt 72, ... 7 6 -
The transformations will in general be possible and definite if the Jacobian of o^ , ft
with respect to y l9 y. 2 , ...y 6 does not vanish. For transformations in which the system
remains canonical (Art. 87), see the works of Jacobi, Dziobek and Poincare referred to in
Art. 105. The formulae for these are not of great value here because, in Delaunay's
method, terms are added to R to keep the system canonical.
178. Transformation to the variables w, 02, s, L, ft, ft.
Let ft = -^ 2 / 2 M w^n^ + a,).
We have from Art. 84, ft = - /*/2a and ri* = par*. Hence
Z = \/a^ = M-2ft)-*, n=i*\L* ...................... (2).
The formulas of transformation are
ft = -
Whence 5ft = ^ 3 BL, B* = - 2 *w +
jL d p ft
136 THE THEORY OF DELAUNAY. [CHAP. IX
We have from these,
dft Set! - daJR = dLSw - dw$L + - dtSL ;
and therefore from (I 7 ), since the other variables remain unaltered,
/ u?
dLBw + dftS 2 + dfaSa* - dwSL - d 2 Sft - d* 3 S{3 3 = dt (W -
where R, = R+ i#ftl? = R - ft ........................ (3).
If we now suppose .R to be expressed in terms of the new variables,
the equations remain canonical. They are
dt dw ' dt 8 2 ' dt 8 3 ' . /^
dw 8-R doL. 2 ^RQ dct s
dt dL ' dt 8ft ' dt 8ft
It will be noticed that w is the mean anomaly in the undisturbed orbit
(Chap. in.).
179. Change of notation.
We now take up the notation adopted by Delaunay and replace
w > a z> &3> L, ft, ft, RO, 7!, w' y a 2 ', a. 3 '
by /, g, h, L, G, H, R, 7, I', </, h 1 ,
respectively.
The six canonical equations (4) may be written
dLSl + dGSg + dHSh - dUL - dgSG - dhBH = dtSR (4'),
where
u? n' 2 a'* .. xx 4- mi + zz'
It will not be necessary to change the signification of ri, a , e' the
constants referring to the solar orbit. Also n, a, e retain, for undisturbed
motion, the meanings previously given.
According to the definitions given in Art. 92, k, g, I are, in undistAirbed motion, the
angular distances xSl, QA, AM (fig. 4, p. 38) ; also H, G, L are double the rates of
description of areas by the projection of the radius vector in the plane of reference, by the
radius vector in the plane of the orbit, and by the uniformly revolving radius vector
supposed to be of length equal to the semi-major axis in the plane of the orbit,
178-180] THE DISTURBING FUNCTION. 137
respectively. The correlation of areas to angles will be readily noticed. We have also, in
undisturbed motion,
l+g+h=moa,n longitude of Moon,
g + h= longitude of perigee,
h = longitude of node,
=inean anomaly,
y=sine of half the inclination.
In the notation previously used, these were respectively denoted by
nt + fj or, 0, w=nt+c-*&, sin
It must be remembered that in de Pontecoulant's theory the letter y was used to denote
tani.
180. The form of the development of R.
The results of Art. 113 show that, after changing the notation, the
arguments of all terms in R are expressible by means of the four angles
I, V, l + g, h-l'-g'-h'.
The argument of any term is therefore of the form
where i, i', i", i'" may be any integers, positive, negative or zero, and
Since we suppose the perigee of the Sun's orbit fixed, q is a constant which
vanishes when i" = 0.
The coefficients of the development of R given in Art. 114 are functions
of e, 7, e, a/a with the factor m?/a ; since m = n'/n, n*a 3 = //-, n' 2 a' 3 = m *, they
may be expressed as functions of a, e, 7, a, e'. Hence if we put ra'a'/a' 3 ,
1 4- g + h I' g' h', I, I', l+g for m 2 , f , <, </>', 77, respectively and for y 2 the
expression 47 2 (1 7 2 )(1 27 2 )~ 2 we can, by expanding this latter quantity,
deduce the form of the development required for this Chapter. This is, after
adding the term //, 2 /2Z 2 =
a
in which only the principal periodic terms have been written. Delaunay's
development contains 320 periodic terms (.
* In Art. 114 we have put /* = ! and m' for the ratio of the mass of the Sun to the sum of the
masses of the Earth and the Moon,
t Delauuay, Vol. i. pp. 3354.
138 THE THEORY OF DELAUNAY. [CHAP. IX
If we neglect powers of j z above the second, it is sufficient (remembering
the change of notation) to put 4>y 2 for 7* in the result of Art. 114. In the
following, I' = n't + const, and a', e ', ri, /A, m', g', h' are absolute constants.
From the relations given in Art. 84 we have, in the new notation,
= no =
H = A cos i = (1 -
and therefore
(7),
by means of these relations the coefficients in R can be expressed in terms
of L, G, H. We shall see later that it is not necessary to actually perform
the substitutions.
Hence, as far as the elements of the Moon's orbit are concerned, the
arguments are functions of I, g, h only and the coefficients are expressible
as functions of L, G, H only.
It is to be noticed that R and the elliptic formula for v, 1/r, u, given in
the next Article, do not contain the time explicitly. It is only present
in the initial development of R through its presence in I, I'.
181. The elliptic expressions for the coordinates.
The longitude v is immediately obtained from the expression given
in Art. 50 if we replace therein, nt + e, w, rf Q by I + g + h, I, 1+ g, respectively,
and to the order given, 7 2 by 4><y\ The value of 1/r is obtained from equation
(16) of Art. 39 by expanding Ji(ie) and replacing w by L The latitude u
can be deduced from the expression of Art. 50 by means of the equation,
u = s-s s + ....
In the expression for s, just referred to, we replace v by 1 4- g, w by I and
2 7 (1 - 7 2 )* (1 - 27 2 )- 1 = 2 7 (1 + | 7 2 +...).
We thus obtain
>...(8).
Delaunay gives the values of v, u correct to the sixth order and that of
1/r correct to the fifth order*.
* Delaunay, Vol. i. pp. 55 59. He uses V to denote the longitude.
sn + e sn
- 7 2 sin (2(7 + 21) - 2j 2 e sin (2g + SI) + 2y z e sin (2
- = - jl + (e - e*) cos I + e 2 cos 21+ ... 1 ,
u= 27 (1 - e 2 ) sin (g+l)+ fyesin (g
180-183] INTEGRATION OF CANONICAL EQUATIONS. 139
Delaunays method of Integration.
182. The method of integration adopted by Delaunay consists, in
the first instance, in choosing out of R the constant term and one of the
periodic terms, neglecting the rest of R. It is then found that the canonical
equations can be integrated by means of series, and that the variables
L, G, H, I, g, h can be expressed in terms of the time and of six new
constants (7, (G), (H), c, (g), (h). By means of these values, R is expressed
in terms of the time and of the six new arbitraries. Having solved the
equations by neglecting a portion of the disturbing function, enquiry is
made in order to find what variable values these new arbitraries must have
when this omitted portion of R is included. It is shown that by adding
certain terms to the disturbing function, the equations which express C, (G),
(H), c, (g), (h) in terms of the time are canonical inform. The power of the
method arises from the fact that when the change in R from L, G, H, I, g,
h to C, (G), (H), c, (g), (h) has been performed, the periodic term considered
has disappeared.
The new arbitraries not being suitable for the purposes in view, two
transformations will be made. Finally, we shall have formulae similar to
those from which we started, that is to say, the equations which express the
new arbitraries in terms of the time are canonical, their relation to the new
disturbing function is the same as before and the new disturbing function is
of the same general form and has the same general properties as R.
Integration of the Canonical Equations when R is limited to one
periodic term and the non-periodic portion.
183. The canonical equations, given by (4/) Art. 179, are
dL_ dR dG = dR dH = dR
dt~ dl' dt~ d' dt = dh'
dl__dR dg__dR dh^_d
di~ dL> dt~ do* dt~
Let R = - B - A cos (il + i'g + i"h + i'"l' + q) + R l ,
or, putting il + i f g + i"h + i'"l' + q = 6 ........................ (9),
let R^-AcosO-B + R^
in which - B is the non-periodic part of R and - A cos is any one of the
periodic terms of R. We shall first integrate the equations (4") with R
neglected.
140 THE THEORY OF DELAUNAY. [CHAP. IX
Substituting in (4") we obtain, since A, B are independent of I, g, h and
6 is independent of Z, G, H,
dL dG dH
-dt =Alsme > rr$!gH dt =At sin ^
M-^A } o ^-M o-0 ^-^ A *0
The first three equations give
............ (11),
where (G), (H) are arbitrary constants and where
d/dt = Asin6 ........................... (11').
Differentiating (9) with regard to t, we obtain
d6 . dl ., do ... dh .... ,
-^ = l j*+' l -J7+' l ^ + l n
dt dt dt dt
which, by means of (10), becomes
d0
Now L, G, H and therefore A, B are, by (11), expressible as functions of
one variable @. Hence, putting
B + i'"ri = Bi .............................. (12),
there results, since i = dL/d, etc.,
dO dA dB,
We have therefore the two simultaneous equations (ll'X (13) for the
expression of , 6 in terms of t. Multiply (13) by dS and (11') by dB,
subtract and integrate. We obtain
AcosO + B^C .............................. (14),
where C is an arbitrary constant. Whence
sn
Substituting in (II 7 ) and integrating, we find, if c be an arbitrary constant,
Z + c =
The lower limit of the integral is taken to be the value of when = 0,
that is, when A G B^ Since A, B l are supposed to be known functions
of L, G, H, and therefore, by (11), of and of the constants (G), (H),
183-184] INTEGRATION OF CANONICAL EQUATIONS. 141
this integral gives the value of t + c in terms of and therefore of in
terms of t + c ; hence L, G, H can be expressed in terms of t -f c and of the
arbitraries (7, (G), (H). The equation (14) then gives 6 in terms of t + c.
184. The values of I, g, h are now to be found.
da dA a dB
Wehave ft = dG C se+ dG'
Substituting for dt from (15) and for cos 6 from (14), this becomes
dg (dAC-Bt am
~
Also, from the way in which (G), (H) were introduced,
dA_dA^ dB _ dB,
dG~d(G) y dG~d(G)'
Hence, integrating,
[dA C-B, + dB,
A
9 =
.(16).
('dA U - 1, dB,
d(3)~A~ + 8(^
^{A*-(C-B,)*}
Here (g), (h) are arbitrary constants ; the upper limit of each integral is
and the lower limit is, as before, that value of for which A = C B,.
Finally, as 6, g, h, I' are known in terms- of t, we can find I from (9).
The three equations (15), (16) can be put into a more convenient form by
assuming
.(17),
the upper and lower limits of the integral being the same as before. The
three equations then become
for K may be considered to be a function of , (7, (G\ (H).
The complete solution is contained in equations (9), (11), (14), (17), (18)
which, after the elimination of 6, 0, K, will give the values of Z, G, H, I, g, h
in terms of t and of the six new arbitraries C, (G), (H), c, (g), (h). The solu-
tion in this form is not, however, convenient for actual calculation ; the method
to be vised will be outlined in Arts. 192, 193.
142 THE THEORY OF DELAUNAY. [CHAP. IX
The Canonical Equations for C, (G), (H), c, (g), (h), when R lt the
portion of R previously omitted, is considered.
185. The canonical equations having been thus solved when jRj is neglect-
ed, it is required to find the solution when 1^ is included. The solution
just obtained contains six arbitrages (7, (G), (H), c, (g), (h). We are going
to inquire what variable values these six arbitraries must have when J? x is
not neglected. The method is then a further application of the principles of
Chapter v., for we suppose L, G, H, I, g, h to have the same forms whether
the new arbitraries be constant or variable. See Art. 98.
The canonical equations are given by
dLBl + dGBg + dHBh - dlBL - dgBG - dhSH
= dtBR = dtB(-Acos0-B + Ri) (19).
Into this equation we must substitute the values of L, G, H, I, g, h given
by equations (11), (14), (18), the arbitraries C, (G), (H), c, (g), (h) being now
considered variable. We suppose any arbitrary variation B given to the
latter and consider what changes are produced in R l} L, etc.
From (12), (14) we deduce
- A cos 6 - B = i'"n' - C.
The second member of (19) is therefore
i'"n'dtB - dt&C + dtBR,.
For the first member we have, from (11),
BL = iB, BG = i'S + 8(0), BH = i"S + B(H) ;
dL = id, dG = i'd + d(G), dH = i"d& + d(H).
Substituting these and remembering that
B0 = iBl + i'Bg + i"&h,
d0 = idl + i'dg + i"dh + i" f ndt
the left-hand member of (19) becomes
dB0 + d(G)Bg + d(H)Bh - d0B - dgB(G) - dhBH + i'"n'dtB.
But from the equations (18) we have
with corresponding expressions for dg, dh. Substituting these values of
Bg, Bh, dg, dh in the last expression for the left-hand member of (19), and
185] THE NEW CANONICAL EQUATIONS. 143
cancelling out the term fn'd&S common to both members, we reduce
equation (19) to
+ d(G)8(g)-}-d(H)8(h)-d(g)8(G)-d(h)8(H) =
Since the operators d, 8 are commutative, the part [...] may be written
(dx . m
- d fe s ^> + a(F
But by (17), .fiT is a function of , 0, (Q), (H) only, and each of these is
now considered to be a function of t, owing to the variability of the six
arbitraries. Hence
and a corresponding expression of dK. The portion [...] therefore becomes
w aer
~m d -dO
This expression, by reason of equations (17), (18), is equal to
or, since 8t = 0, 08d = 0d8, etc., to
- d80 + rf(7Sc + d08 - (dt + dc) 80.
If this be substituted in (20), the terms dS80, d08, dt80 disappear
and the equation reduces to
d08c + d(G)8(g) + d(H)8(h) - dc8C - d(g)8(G) - d(h)B(H) = dt8R l . . .(21).
Supposing now that, by means of the values of L, G, H, I, g, h, the function
E! has been expressed in terms of t and of (7, (G), (H), c, (g), (h), we obtain
from (21), in accordance with the remarks of Art. 177, the canonical
equations
dC_ dRi d(G)_ dE, d(H)^ dR, \
dt = fc> dt d(gy dt d(h)>[
dc__dRi d(g) dR, d(h) dR^
dt~ dC' dt d(G)' dt '' d(H)j
A glance at the results obtained in Arts. 183, 184 will show that the variables in (210
have meanings, with respect to the motion to which they refer, bearing a close analogy to
those of ai, ; (Art. 176) with respect to elliptic motion. We shall see in Art. 187 that a
transformation, similar to that of Art. 178, must be made.
144 THE THEORY OF DELAUNAY. [CHAP. IX
The nature of the solution obtained in Arts. 183, 184.
186. Let us examine the nature of the solution obtained in Arts. 183,
184 when the variables are expressed in terms of the time and of the
six arbitraries (7, (G), (H), c, (g), (h) : as R^ is there neglected, these arbi-
traries are still considered to be constant. Since A is the coefficient
of one of the periodic terms of the disturbing function, A, dA/dS are small
quantities of the first order at least. Hence, the equations
= A sin = cos - 1
show that, if we neglect quantities of the first order and remember that
A t B l are supposed to be expressed as functions of and of the arbitraries
(G), (H), a first approximation to the solution is given by
= const. = > = ft (t + c\
in which > c are arbitraries and is a definite function of , (G), (H).
A second approximation furnishes
= @ + j cos (t + c), = (t + c) + 0i sin (t 4 c).
Hence, assuming that developments in series are possible, the solution of
these equations is given by
@ = + ! cos (t + c) 4- B 2 cos 20 (t 4- c) + ...
Since 0=0 when t = c, the arbitrary c is the same as that defined by (15).
The arbitrary constant attached to is which, by (15), may be expressed
as a function of (7, (G), (H) ; hence , 1} 2 , ..., , B lf 2 ... are functions of
G, (G), (H).
For the sake of brevity, denote by cr c , a s the series
:'':' '' . ' >0'(t+c)+**w(t+)+~T ';:.< --^^.i -
The solutions may then be written
@ = O + <H) C , 0=0 (+ c ) + 0, (22').
With these values, the equations (18) give
9 = =(9) + Oo(t + c)-\-g S) k=(h) + ho(t + c) + h lt (23),
where g 9t g lt ..., h^ h^... are functions of C, (G), (H). Therefore, from
equation (9), we have
186-187] NATURE OF SOLUTION. 145
where l lt 1 2) ... are determined by
Finally, the value of & in (22'), substituted in equations (11), gives
L = L Q + l e , G = G Q + G e , H=H, + H e ,
where
Z = t , Zj = i ! , L 2 = i 2 , ... }
G t 1 = / e 1 , # 2 = i'e 2 , ...I. ..(24).
The coefficients ,, #,-, Aj, Zy, 6)-, fij being quantities of the order J at
least, the above investigation shows that the new values of the elements can
be formally expressed by series of cosines or sines of multiples of one angle
ft (t + c), with coefficients in descending order of magnitude. The values of
Z, G, H, I, g, h may then be substituted in the disturbing function and in the
elliptic expressions for the coordinates and the results expressed as sums of
cosines or sines; the arguments may be freed from periodic terms in the
manner explained in Art. 111.
187. The form of the disturbing function after the substitutions have
been made.
Let us now consider the effect of the substitution of these values of
L, G, H, I, g, h in R^ Previously, R^ consisted of periodic terms whose
arguments were multiples of I, g, h, I', g', h r and whose coefficients were
functions of L, G, H. After the substitutions have been made, R^ will
consist of periodic terms whose arguments, besides being multiples of
1' 9 g f , h', contain multiples of
ft (t + c), (g) +g Q (t + c), (h) + h (t + c) ;
ft) ffot h , and the coefficients of these periodic terms, are functions of C, (G), (H).
Hence, when we commence to solve the equations (21 X ) by differentiating ^
with respect to G, (G), (H), the time t -f c will appear as a factor of the
periodic terms. In order to avoid this, instead of G, (G), (H), c, (g\ (h),
a new set of variables can be chosen which are such that the equations
expressing them in terms of the time are still canonical ; when J^ has been
expressed in terms of the new variables, it will have a form similar to that
which R had, that is to say, ^ will consist of periodic terms whose arguments
contain multiples of three only of the variables and whose coefficients are
functions of the three conjugate variables only. In making the transformation,
the following Lemma will be required.
B. L. T. 10
146 THE THEORY OF DELAUNAY. [CHAP. IX
188. Lemma. Let
........................ (25);
Differentiating (22'), we have
d = - <ft [6! sin (* + c) + 2 2 sin 2<9 ( + c ) + ...],
and = (* + c) + 0i sin (t + c) + 2 sin 20 ( + c) + ... ;
whence 0d = - J0 eft [ A + 2 2 ^ 2 + 3 3 ft + . . .]
+ (t + c) dt x periodic terms -f dt x periodic terms.
Therefore, since K =/0d and since K,t+c are zero together, we obtain
K = J0 </> ( + c) + ( 4- c) x periodic terms + periodic terms. . .(26).
Now K was a function of , 0, (6r), (T) and is a function of (2+ c),
(7, (0), (H). Denote by , A fe?r the partial differentials o
with respect to 0, (G), (H) y after the value of has been substituted in K.
We then have
/d_K\ _dK dKd
(dc) ~ do + a aa f
and similarly for (0), (5"). Since dK/d = ^, we obtain from these equations,
a^_/a^\_ a
~ '
...... (27).
Now by (26),
+ periodic terms having (t + c), (tf + c) 1 , (* + c) 2 as factors.
Also, from (220,
O/3 -1
- (t + c) ^ {<H\ sin (9 ( + c) + 2 2 sin 2(9 (t + c) + . . . }
+ periodic terms having (t + c), (f -f c) 1 , (t + c) 2 as factors.
Further, by (18), dK/dC =-(t + c).
188-189] FIRST CHANGE OF VARIABLES IN THE NEW EQUATIONS. 147
/r) fT\ r)(Sl r) K
Substituting these values for ( ^ ) , -^ , ^ in the first of equations
(27) and equating those coefficients of (t + c) which are independent of
periodic terms, we find
In a similar manner, by (18), (23),
t>K/d(G) = c,-(g) =
and therefore, equating coefficients of (t + c),
1
9" = ~2
with a corresponding equation for A .
189. -Fir^ change of variables in Equations (21') to avoid the occurrence
of terms increasing in proportion with the time.
Put now
= =
"
and change the variables in (21') or (21) from (7, (G), (H), c, (gr), (h) to A,
(6r), (^T), X, K, T?. The Lemma just proved gives
We have also, from the relations (28),
Sc= -
B(h) = 8r ) -S\-\8, d(h)=dr)- d\-\d.
#0 C7 t7 t/
Substituting, the first member of (21) becomes
d (G) Sfc + d(H)Brj + dt&C- dS (G) -
102
148 THE THEORY OF DELAUNAY. [CHAP. IX
1 a, h,, 9A 9A 3A
Substitute everywhere for -^ , - , ^- their values -^ , . , ~. , g/g-y
The coefficient of SX becomes dA and that of dX becomes - SA. The
coefficient of +X is
= coefficient of X, since S, d are commutative.
The equation (21) therefore reduces to
d (G) S/c + d (H) Brj + dtSC - d/cB (G) - dr)B(H) + dASX -
(29).
Hence, putting R = R l -G (30),
and supposing R to be expressed in terms of A, (6r), (H), X, K, y, we obtain
JA 3 TV -j / /"r\ o D/ .7 / o~\ 3 IX \
aA cxt C& (CrJ 0xL a (.a j dti \
rfi 9X ' dt d/c ' dt drj ' I ^o/x
c?X 9J2' cZ/c dR' drj dR
!ti = ~dK' 'di = ~d(G) > dfe .d(Jff)'
The equations remain canonical and ^ x has a form similar to that which R
had, namely, it is expressed by cosines of sums of multiples of X, K, 77, I', g', h',
with coefficients depending on A, (6r), (H). The partials of R' with respect
to the new variables will no longer introduce terms proportional to the time.
Second change of variables.
190. It is advisable to make a further transformation in order that when
A (the coefficient of the periodic term previously considered) is put equal to
zero, the new canonical equations shall reduce to the old ones (4"), Art. 183.
When A vanishes, it is easily seen from (22), (28), (23) that
189-191]
Put therefore
SECOND CHANGE OF VARIABLES.
149
' = tA,
,
i i i i i '
and transform from A, (G), (H\ X, K, rj to A 7 , ', H', V, /c, rj.
We obtain by substitution in (29), after the elimination of G by means
of (30),
dtSR = dAB\ + d(G)Sie + d(H)Srj -
= * dM (i'5X' + i'Sic + i"Sri) + (dG f - \ dA') Stc + (dH f - ^ dA.'
i ^ i / \ i ,
i V / \ i
i"'
= dA*&\' + dG'&K + dH'&rj d\fSA? dfc&G' drjSII' n'dt&A'.
///
Hence, by putting R" = .R' + ^ n'A' (31'),
the equations can be written in the canonical form
dt
dt
__
dt ~ dG'
.(32),
and these new variables will reduce to the old ones when A is put zero. It is
evident that the remarks made at the end of the previous article will also
apply to these equations.
191. We will now see how the new variables are related to those found
in the solution given in Art. 186 ; also, how the new disturbing function R"
is related to R.
Let </>' = OJji + 2 L 2 +...=i',
by (11), (25). The equations (31), (28), (24) furnish
where, by (23'),
The new variables X', K, 77 are therefore nothing else than those non- periodic
parts of I, g, h which were obtained by the solution of the equations (4").
150 THE THEORY OF DELAUNAY. [CHAP. IX
Again, we have
R = - A cos 6 - B + R,
= -Acos0-B + C+R', by (30),
= - A cos 6 - B + C - C ro'A' + R", by (31').
But since A' = iA = i ( + i<),
we obtain R" = R l -C + i'"ri ( +
that is, JS" is the same as j^ except for some additional non-periodic terms.
The periodic term A cos 6 is therefore not present in the new disturbing
function. Since R^ does not contain A cos 0, the last equation also shows
that, when we make the change of variables in R by giving to L, G, H, I, g, h
their values as functions of A', G', H', A,', , 77, the periodic term A cos 6 in
R will identically vanish (See Art. 197). This furnishes a means of veri-
fying the calculations.
The application of the previous results to the calculation of an operation.
i
192. It is necessary to see how the results obtained in the previous
articles are applied in the actual calculations. In the first place, R was
expressed in terms of a, e, 7, I, g, h. Out of R the terms B A cos were
chosen : of these, B is the whole of the non-periodic part of R and A cos 6
is any one periodic term. In the canonical equations (4") for L, G, H, I, g, h,
we insert this value of R.
Equations (7) give the values of L, G, H in terms of a, e, 7. By means of
them, we find from the integrals
any two of the quantities a, e, 7 in terms of the third, say a, 7 in terms of e,
and substitute in the expression in (7) for L, which then becomes a function
of e, (G), (H) only. From this, dL/dt is deduced as a function of de/dt, e,
(G), (H).
But dL/dt = dR/dl, where R = - A cos - B. Whence, after expressing
A in terms of e, (G), (H), we find
de/dt = sin x function of e } (G), (H) (33).
Also
dO .dl .,dg .,,dh ., , .3R .,3R .,,3R
= l + l + l +* n *- l - % +
191-193] THE CALCULATION OF AN OPERATION. 151
,. T dR dR dR , , . f dR dR dR
Now az ; m ' m may be 13xpresi l m terms of as w &7 a - e ' 7 - b - v
means of the relations (7), (7'). We therefore obtain
7/1
-IT = function of a, e, 7 + cos x function of a, e, 7,
or, according to the remarks made above,
JS\
^7 = function of e, (), ( H) + cos x function ofe,(G),H... (34).
The equations (33), (34) involve only the two dependent variables e,
and the independent variable t. They are equivalent to (II 7 ), (13) and are
integrated as in Art. 186 by continued approximation or by some similar
method, giving
e 2 = e<? + cosines of multiples of (t + c), j
e = 0, (t + c) + sines of multiples of 6, (t + c) j " ''
where e , c are the two arbitrary constants and is a function of e Q , (G), (H).
In certain operations where e appears as a denominator in (34), it is found
to be more convenient to solve (33), (34) by finding
e cos 6 = const. + cosines of multiples of (t + c), ) /oc'\
e sin 6 = sines of multiples of (t + c) j "
Here the arbitrary e is the coefficient of sin (t + c) and it does not appear
in the right-hand members as a denominator. See Art. 196.
From these we can deduce the values of a, 7* (which were found in
terms of e, (G), (H)), expressed as functions of e , (G), (H) and cosines of
multiples of 6 (t + c). In all cases the coefficients of the sines and cosines
are functions of e , (G), (H) only. Let a , 7<> 2 be the non-periodic terms of a, 7*
and eliminate (G), (H). We have then
a = a + functions of a , e Q , % z multiplied by
cosines of multiples of (t + c), ,
e 2 = e 2 + similar terms,
ry2 = *y 2 + similar terms
When the form (35') is used, the expression for e 2 contains some other
non-periodic terms.
dl dg dh x f dR dR dR
193. Next, equate - _ , - |, - _ to the values of ^, ^, ^
152 THE THEORY OF DELAUNAY. [CHAP. IX
already found. Eliminating a, e, 7, from dR/dL, etc. by means of (35) or
(35'), (36) and integrating, we obtain
I = x' + functions of a , e , % multiplied by
sines of multiples of Q (t +
. .,
g = K + similar terms
h = 7j + similar terms
Here X', K, 17 are the non-periodic parts of I, g, h; also, by means of the
relation = il + i'g + i"h -f i'"l' + <?, we can express ( + c), which is simply
the non-periodic part of 0, in terms of X', K, 77, I', q.
The results (36), (37) contain the required solution of the equations.
In order to prepare for the next operation, we substitute these results in
R (and also in the expressions for the longitude, latitude and parallax,
previously obtained); R will then become a function of a , e Q , 70, V, K, rj.
Our new variables will be A', G' t H', X', K, 77,
where A', G', H' = functions of a , > % (38),
found by means of the relations given in Art. 191. For since a, e, 7 are
given by (36), the values of L, G, H can be deduced ; each of them will be
expressed by a constant term and by cosines of multiples of (t + c), the
former and the coefficients of the latter being functions of a , e , %. We then
have LO, GO, HO] the function (/>' is deduced from the series obtained for 6, L.
Finally, since (Art. 191)
R" = R + A cos 6 + B - C + C rc'A'
i'"
= R- i'"n'B + -T- n'Af, by (12), (14),
i
= E-Cn'(Z-A'), by (11),
we can obtain the new disturbing function. It will be found that, when R
and L have been expressed in terms of the new variables, the term - A cos 6
will not be present in R".
Taking the new variables and the new disturbing function the equations
for them are still canonical. Also, as (38) gives the values of A', G', H f in
terms of a , e , 7 , we are in a position to go through the whole process again
with another periodic term. Since the letters L, G } H, a, e, 7, I, g, h, R have
disappeared completely, there is no further need of the symbols A x , G', etc. :
as soon as the operation is finished, they are replaced for simplicity by the
letters L, G, etc. During the process, it is frequently more convenient to use
n instead of a : the relation between n, a is always defined by the equation
193-195] PARTICULAR CASES. 153
Particular cases.
194. It is evident that if any or all of the integers i', i", i'" are zero, the
same methods will hold ; the only difference will be a greater simplicity in
the results. When i' or i" is zero, we get
Q or =
respectively. When i'" is zero, the new part to be added to R vanishes.
When i is zero but either i' or i" not zero, the method requires a slight
modification. Suppose i' unequal to zero. We assume G = i f instead of
L = i ; the solution evidently proceeds on exactly the same lines since, in
the first instance, the equations were symmetrical with respect to L, G, H.
195. When i, i', i" are all zero, we can modify the solution in a way
which saves several operations.
The angle is reduced to i'"l', for q = when \" = 0. We take out of R,
instead of a single term, the terms
A! cos I' A 2 cos 21' A 3 cos 31' ... = - %A P cospl'.
When this is substituted in the canonical equations (4") we obtain
_ -
dt ' dt dt~
dl ~dA ,, da ~dA , dh
Hence L, G, H are constant and therefore
l-W + Z-^^smpl' ........................ (39),
pn oh
with similar expressions for g, h.
It is necessary to see what the new canonical equations become when R^
is not neglected. They are expressed by the equation
dtt (jRj - 2A P cospl') = dUl + dGBg + dHSh - dlSL - dgSG - dhSH. . .(40).
Since L, G, H remain unaltered, take L, G, H, (I), (g), (h) as new variables.
Substituting from (39) for I, g, h, the second member of (40) becomes
dGB(g) + dHS(h) - d(l)SL - d(g)BG -
154 THE THEORY OF DELAUNAY. [CHAP. IX
The second line of this expression can be shown to vanish in the same way as
a similar expression considered in Art. 189, p. 148 ; the third line is equal to
- 2,8 A p cos pl'dt = - dtB [2A P cos pl'] t
and it therefore disappears with the same term present in the first member
of (40). Hence the first line is equal to dtSR l} and when ^ has been
expressed in terms of L, G, H, (I), (g), (h), the equations for these new
variables will be canonical.
The rule for transformation is simple. We solve and find
l = (l) + 2l p sin pi', g = (g) + %g p sin pi', h = (h) + Zh p sin pi',
where l p , g p , h p are functions of a, e, 7, or of L, G, H. All that is necessary
is to replace I by I + 2l p siupl', with similar changes for g, h, in R, v, u, 1/r.
The new equations for L, G, H, I, g, h are canonical, they reduce to their
former values when all the coefficients A p are zero and, after the change
of variables has been made, the terms 2<A P cos pi' will not be present in
the new disturbing function.
It will be noticed that the constant term is not included here. Since it
is a function of L, G, H (which were shown to be constants) only, it can
produce no new parts depending on I, g, h and it need not, in this case, be
considered.
196. It has been mentioned in Art. 192 that, when the equations (33), (34) have been
prepared for integration, they sometimes take the forms
~
= !, 2,...);
here M is of the second order at least and N, J/i, JV 4 , P^ are of order zero \ these
coefficients are supposed to be independent of e, t, 6, so that, as far as the integration
of the equations is concerned, they are constants. If we integrate in series by continued
approximation, difficulties may arise owing to the presence of e as the denominator of a
fraction. This may be avoided as follows.
We deduce from the equations just given,
(e cos ff) = M (2 j^ea-a - SPi^- 2 ) (e sin 0) (e cos 6} -N(l + ^N^} (e sin <9),
(e sin 0) = Jf+ Jf[(e sm 0) 2 2Jf { e^
Since the least value of t is unity, e does not occur as a denominator in these expressions ;
also, the second members are expressible by integral powers of e cos 6, e sin 6.
To solve these equations assume
e cos 6=E Q + E l cos (t + c)+ E 2 cos20 (t + c) + ...,
EI sin (t+c) + E 2 f sin 20 (t + c] + ... .
195-197] DELAUNAY'S METHOD OF PROCEDURE. 155
If we put E^ = e^ it can be shown that E it E are of the order e * at least; also,
E 2 = E z ', E 3 = E 3 ', etc. The arbitrary constants are e 0t c : the quantity is a certain definite
function of J/, 3 T , M it ff it P it e Q 2 and it does not contain e as a denominator.
In the cases where e appears as a denominator, it is found, when we proceed to the
substitution of the values of 0, e in the expressions for R and for the coordinates, that we
only require to know e cos 0, e sin 0, e z and powers and products of these quantities ;
since their values do not contain e as a denominator, no difficulty will ensue. See
Delaunay, I. pp. 107, 108, 878882 and Tisserand, Mec. Cel. VoL in. pp. 216220.
The general plan of procedure.
197. A general view of the whole process will perhaps make the compre-
hension of Delaunay 's method easier.
We find first the elliptic values of the coordinates of the Sun and the Moon
and, by means of them, express R in terms of /, g, h, a, e, 7, referring to the
Moon, and of l' t g' + h', a, e', referring to the Sun. We have also definite
relations between L, G y H and a, e, 7.
Choosing out of R the non-periodic part and one of the periodic terms, we
solve the canonical equations for these portions of R and find the values of
I, g, h, a, e, 7 in terms of the time and of V, K, 77, a , e , %', of the latter,
X', K y rj contain three new arbitraries which are the constant parts of /, g, h :
the other three new arbitraries are a , e , y . These values are substituted
in R, v, 1/r, u which then become functions of X', ic, vj, a , e , y . Certain
terms are added to R and, when the change of variables has been made,
it is found that the periodic term considered has disappeared. As I, g, h
occur in the arguments of the periodic terms of the expressions for R and for
the coordinates, and as the periodic terms, introduced by the change of varia-
bles, are small, we can expand each cosine or sine (as in Art. Ill) so as to
free the arguments from periodic terms. The form of the new disturbing
function is similar to that of the old, except that the new g, h now contain the
time explicitly ; this is due to the fact that the action of the Sun causes the
perigee and the node to revolve. We also find the relations between A', G',
H' and a , e , y Q ; the equations for the new variables X', tc, rj, A', G', H'
being canonical, we are ready for the next operation.
We proceed in exactly the same manner. With the relations, just found,
between the new L, G, H and the new a, e, 7 (that is, between the old
A', G' y H' and the old a , e , 70) we take the non-periodic part of the new R and
one of its periodic terms and with these we solve the canonical equations for
the new arbitraries (now considered variable), introducing in the same way six
other arbitraries. By this operation another periodic term of R is eliminated.
Continuing in the same way, we eliminate one of the periodic terms in R at
every operation until R is reduced to a non-periodic part only.
156 THE THEORY OF DELAUNAY. [CHAP. IX
Each change of variables may produce new terms in R and may cause
a reappearance of terms whose arguments are the same combinations of I, g, h,
I', g', h', as those of terms previously eliminated or as that of the term under
consideration; in the first case, they will evidently have the same general
form as before, that is, they will be of the form il + i'g + i"h + i'"l + q ;
in the latter cases, the new coefficients will be of a higher order than the
coefficients of the terms with the same arguments previously eliminated.
The series of operations thus continually raises the order of the coefficients of
the periodic terms in R. We go on with the operations until these co-
efficients become sufficiently small to be neglected. Delaunay has continued
them until he has found all terms in the longitude correctly to the seventh
order inclusive ; in addition, some coefficients are calculated to higher orders
when slow convergence indicated the necessity of carrying the approxima-
tions further.
The number of operations required is very large. Delaunay retains all terms in E up
to the eighth order inclusive. He first carries out 57 operations, by means of which he
eliminates all periodic terms in R which are of an order less than the fourth. The
first operation is that outlined in Art. 195 above ; then follows the elimination of the terms
with arguments , 2/i -f %ff + 2 - 2A' 2g' 21' + 1, etc. those terms whose coefficients are
lowered by the integrations being, in general, considered first. The expression for R (in
which every term produced by the successive changes of the variables is shown separately)
together with the details of these operations occupy the greater part of Vol. i. ; the ex-
pression for R alone occupies pages 119-256.
Vol. ii. opens with the value of R .which remains after the 57 operations have been
carried out : it now contains no periodic term of an order less than the fourth and the
great majority of the terms are of a higher order. He then makes 435 further operations
in order to eliminate these remaining terms. In most of these operations it is not
necessary to change the variables in R : the small changes produced are made in the
coordinates only. There are, however, five periodic terms, arising from changes in
R) to be taken into account and these are eliminated by five further operations. Then
follow the values of the longitude, latitude and parallax with the successive modifications,
written out in full, which they have undergone owing to the 57 + 435 + 5 = 497 operations.
The next chapter is devoted to the further researches into the longitude necessary to carry
some of the coefficients to higher orders ; this demands a recalculation of some of the
operations. In performing them, certain errors are detected and the necessary corrections
are added. Finally he gives the reduced values of the coordinates after the change of
arbitraries (explained in Art. 199 below) has been made, together with the numerical value
of each term in every coefficient, for the case of the Moon.
198. Finally, the disturbing function is reduced to a non-periodic term
B. Since B does not contain I, g, h, the canonical equations give
^_A ^?_n ^-n dl = ^_ dg__3B dh dB
dt~ ' dt = ' dt = dt~dL' dt'dG' dt~dH'
Hence L, G, H and therefore a, e, 7 are unchanged, while we have for I, g, h,
respectively, the values
197-199] FINAL EXPRESSIONS OF COORDINATES. 157
where 1 , g , h Q are the values of dB/dL, dB/dG, dB/dH and (I), (g\ (h) are
arbitraries. Since the previous operation has furnished the connection
between L, G, H and a, e, 7, we can obtain 1 , g Q , h Q as a function of a, e, 7.
The final expressions for v, u, 1/r are therefore obtained as a sum of
periodic terms whose arguments are of the general form
il + %'g + i"h + i"'l' - i" (g f + h'\
and whose coefficients are functions of the constants a, e, 7 introduced by the
last operation ; also, l,g, h are each of the form, t x function of a, e, 7 + const.
Further, v contains the term t x function of a, e, 7 + const, and 1/r contains a
constant term which is a function of a, e, 7. We must now see how the final
I, g, h, a, e, 7 are related to the quantities denoted by those letters in purely
elliptic motion.
The Arbitrary Constants and the Mean Motions of the Perigee
and the Node.
199. The result of any operation was to replace a by a + periodic terms
introduced by the operation : the periodic terms, depending on the action of
the Sun, will be small. Similar remarks apply to e, 7. Hence a, e> 7 at any
stage will differ from their original values (which were arbitraries of the elliptic
solution) by terms depending on the action of the Sun, and their principal
parts will be their elliptic values.
Again, after any operation we find for I, g or h expressions of the form,
Arb. const. + t x function of a, e, 7 + periodic terms.
The new I, g, h are the non-periodic parts of these, so that I is replaced
by 1 4- periodic terms ; similarly for g, h.
At the outset we had I =nt+ e -BT, g = ta- 0, h= (0 being here the
longitude of the node). Hence the relations of the final I, g, h to their
initial values are, when the whole series of operations is completed,
Final l^nt+e-vr + tx function of a, e, 7,
Final grs Q + t x function of a, e, 7,
Final h 9 +tx function of a, e, 7.
The last terms of each of these expressions, depending on the action of
the Sun, are all small.
Since a', e', ri are present and since at any stage n is defined by the
relation p n?a s , the coordinate v can be expressed in the form,
Const. + t x const. -f periodic terms with coefficients depending
on n'/n, 6, e', 7, a/a' and arguments depending on I, I', g,h l'g r h'.
158 THE THEORY OF DELAUNAY. [CHAP. IX
The coordinates u, l/r are expressed by periodic terms of similar form,
the coordinate l/r having further a constant term. We now change the
arbitraries so that they may be denned as in Chap. vm. and consequently be
independent of the method of solution adopted.
200. Since the constant parts of I, g, h are arbitraries, we define 6, 0, ra-
in disturbed (or undisturbed) motion as follows :
e = the constant term in I + g + h, that is, the constant part of the
mean longitude,
w = the constant term in g + h t
6 = the constant term in h.
Again, since n (or a), e, y are arbitraries, we take a new n, e, 7, a defined
as follows :
n = coefficient of t in l + g + h, that is, n is the mean motion in longi-
tude whether we consider disturbed or undisturbed motion ;
e is such that the coefficient of sin I in longitude is the same in
disturbed motion as in undisturbed motion;
7 is such that the coefficient of sin (1 + g) in latitude is the same
as in undisturbed motion ;
a = (//,/n 2 )^, where n has the meaning just defined.
In order to transform to these new arbitraries we equate n to the coeffi-
cient of t in the final non-periodic part of v ; e, y are found by equating the
coefficients of the principal elliptic term in longitude and the principal term
in latitude, found from purely elliptic motion (with e, y as the eccentricity
and the sine of half the inclination, respectively), to the coefficients of the
corresponding terms found by the theory. We have then sufficient equations
to express the old arbitraries in terms of the new and thence all the
coefficients can be expressed in terms of the new arbitraries.
Since l + g+h, I, l+g are respectively the mean longitude, the argument
of the principal elliptic term and the argument of the principal term in
latitude, the mean motions of the perigee and the node are given by the
coefficients of t in the final expressions for l+g+h l = g + h and l + g + h
l-g = fi, respectively; these coefficients of t must also be expressed in
terms of the new a, e, 7, n. They were denoted in de Pontecoulant's theory
by (l-c)n,(l-g)n.
The arguments of all terms are combinations of the four angles I, I', l + g,
l+g + h-l'-g' -h'. Delaunay puts D = l + g + h - l'-g - h' t F= I +g, so
that
199-201] DELAUNAY'S RESULTS. 159
2D = Argument of the Variation,
I = Principal Elliptic term,
I' = Annual Equation,
D Parallactic Inequality,
F = Principal term in Latitude.
These were respectively denoted by 2f, </>, <', rj in de Pontdcoulant's
theory. It is necessary, in Delaunay's final results, to replace a/a' by
(E-M)a/(E + M)a,
in order to take into account the correction obtained in Art. 7.
201. The literal results obtained by Delaunay in using the methods explained above,
far surpass any other complete developments in their general accuracy and the high order
of approximation to which they are carried, although further terms of certain portions, such
as the principal parts of the mean motions of the perigee and the node, have been found to
a greater degree of approximation. The only results which can be at all compared with them
are those of Hansen. The latter, however, confined his attention to numerical develop-
ments by substituting the values of m, <?, y, ef, a/a' at the outset, while Delaunay gives complete
literal results for the three coordinates this being necessary in his method of treatment.
Although the disturbances produced by the Sun are alone treated, the method can be and
indeed has been continued from the point where Delaunay stopped, so as to include the
effects produced by the actions of the planets, the figure of the Earth, etc. (see Chap. xin.).
Had Delaunay lived, it was his intention to complete the lunar theory by a full examination
of all these inequalities and so add a third volume to the two large ones already referred to.
M. Tisserand graphically described Delaunay's work in the following terms* : ' Cette
' theorie est tres interessante au point du vue analytique ; dans la pratique, elle atteint le
* but poursuivi, mais au prix de calculs algebriques effrayants. C'est comme une machine
' aux rouages savamment combines qu'on appliquerait presque indefiniment pour broyer un
4 obstacle, fragments par fragments. On ne saurait trop admirer la patience de 1'auteur,
* qui a consacre" plus de vingt annees de sa vie k 1'execution materielle des calculs algebriques
' qu'il a effectues tout seul.'
* Mec. Gel. vol. in. p. 232.
CHAPTER X.
THE METHOD OF HANSEN.
202. THIS chapter contains an explanation of the methods adopted by
Hansen to solve the lunar problem. In the earlier portion of the chapter
to the end of Art. 223 the various equations to be used are formed in a
perfectly general manner ; the next portion from Arts. 224 to 238 contains
an explanation of the manner in which the approximations, as far as the first
order of the disturbing forces, are carried out. When these have been
grasped, the extensions necessary for the further approximations follow very
easily ; they will be outlined in Arts. 239, 240.
For convenience of reference, the notation is based on that of the
Darlegung*; in the few places where a different notation is adopted in order
to avoid confusion, the differences will be pointed out.
The distinguishing features of Hansen 's method are : (i) the angular
perturbations in the plane of the orbit are added to the mean anomaly of an
auxiliary ellipse placed in the plane of the instantaneous orbit, its major
axis and eccentricity being constant and its perigee moving in a given
manner ; (ii) the radial perturbatioos are determined by finding the ratio of
the actual radius vector to the portion of it cut off by the auxiliary ellipse (;
(iii) the reckoning of longitudes from a departure point (Art. 79) in the
plane of the orbit ; (iv) the discovery and use of one function W to find all
the inequalities in the plane of the orbit ; (v) the perfect generality of the
method which permits, without difficulty, the inclusion of inequalities from
every source ; (vi) the completeness with which the method is worked out
numerically and the close agreement with observation of the tables which
were founded on the theory.
* This title refers to Hansen's paper entitled Darlegung der theoretischen Berechnung der in
den Mondtafeln angewandten Storungen. Abh. der. K. Sachs. Gesell. d. Wissenschaften, Vol. vi.
pp. 91-498, Vol. vii. pp. 1-399. The two parts will be referred to as I., II.
t In the Fundamenta (see footnote, p. 36) Hansen finds the logarithm of this ratio.
202-204] NOTATION. 161
A general explanation of Hansen's method has been given in a note by Delaunay*
and also in two papers by Hansen, Note sur la theorie des perturbations plan&aires and
Bemerkungen iiber die Behandlung der Theorie der Storungen des Mondesi.
203. Hansen's theory is much the most difficult to understand of any of those given up
to the present time, partly on account of the somewhat uncouth form in which it is given
in the Fundamenta and partly on account of the very unusual way in which the perturba-
tions are expressed. It was first published in a series of papers entitled, Disquisitiones circa
theoriam perturbationum quae motum corporum coelestiwn affidunt and Commentatio de cor-
porum coelestium perturbationibus*. The methods, although they are in general there
worked out with special reference to the planetary theory only, are, after a few changes,
equally applicable to the lunar theory : the chief difference being that, in the former,
terms increasing with the time are permitted to be present while, in the latter, they are
eliminated by the introduction of a certain quantity y. In the Fundamenta, which
was published in 1838, the methods, as far as they refer to the Moon's motion, are fully
elaborated and detailed expansions are given in forms ready for calculation. A method
for the solution, on the same lines, of the problem of four bodies is added.
In 1857 Hansen began another series of papers in which the perturbations are
expressed in a similar manner, but the methods of arriving at the equations are much
simpler. The first paper II refers to the planetary theory : the method is the same as in the
Darlegung which followed a few years later. The latter was chiefly published in order
to verify the * Tables de la Lune'l which had been previously formed by an application of
the principles explained in the Fundamenta. As far as p. 212 of the first part, the Darleg-
ung is, however, available for the general development and it will be used for that purpose
here; when it is a question of forming the successive approximations, the Fundamenta
must be referred to. The early parts of the Fundamenta and of the Darlegung, though
expressed in forms very different in appearance, can, with some trouble, be seen to be
equivalent.
204. Change of Notation.
In order that the expressions obtained below may be the same as those
of Hansen, a few changes from the notation of Chaps. I viir. are necessary ;
these changes chiefly affect the results of Art. 82, which will be required
directly.
Replace $, X, 3, R
respectively. The right-hand members of the equations of Art. 82 must
therefore be all multiplied by //,; the results of Art. 75 will remain unaltered.
Also, in Art. 124, we have put R/fi = Rw + R + ...; we shall have now
* Jour, des Savants, 1858, pp. 16, 17.
t A sir. Nach. Vol. xv. Cols. 201-216, Vol. xix. Cols. 33-92.
J These are contained in various numbers of the Astr. Nach. from 1829-1836.
These were published in the early volumes of the Abh. d. K. Sachs. Ges. der Wissensch.
II Auseinandersetzung einer zweckmdssigen Methode zur Berechnung der absoluten Stwungen
der kleinen Planeten. Abh. Vol. v. pp. 1-148.
[ London, 1857. Published by the Government.
B. L. T. 11
162 THE METHOD OF HANSEN. [CHAP. X
where R (l) + R + ... retain the same meanings as before*. The mean
anomalies w, w' will be replaced later by g, g', respectively.
Hansen also uses A in a different sense. He puts
The significations of the other quantities present in the equations of
Art. 82, remain unaltered.
The instantaneous elliptic orbit.
205. The elements of the instantaneous orbit being denoted by a, n, e, e,
CT, 0, i, the disturbing forces by fd$, fj%,, //3, the true anomaly by/, the radius
vector by r, the latus rectum by I, the distance of the Moon from the node
by L, we have from the second, third, fifth and sixth of equations (16), Art. 82,
after replacing therein ^, X, 3 by pty, //,$, ^ and using the expression for h
just given,
=
dt h
-
dt eh
/-, r \ /!
1 + -= sin/L
\ IJ )
. dd 7 - . di T ~
--= h^r sin z, - = hr cos z
(2).
We also have from Art. 77, after putting fiX for X,
/*3> = d (
whence by (1),
The last equation replaces that for da/dt in Art. 82. The equation for dcjdt will not be
required since those functions of the instantaneous elements, which are used in Hansen's
particular method of treatment, do not directly involve e. The method being to find the
perturbations of the mean anomaly, the equation which would be obtained by making e,
vary, is really included in the equation giving the disturbed value of the mean anomaly.
206. In Hansen's method the plane of the Sun's orbit is not necessarily
a fixed one. We take as a fixed plane of reference either the ecliptic at a
given date or the Invariable plane (Art. 28); any fixed plane inclined at a
small angle to the ecliptic will serve at present. As before, we define the
positions of all lines by means of their intersections with the unit sphere.
Let a? be a fixed point on ^Oj the plane of reference.
* Hansen uses instead of R,
204-206] THE INSTANTANEOUS ORBIT. 163
Let X be a departure point (Art. 79) on ZDj the instantaneous orbit of
the Moon. Let n x be the node of the instantaneous orbit with the fixed
. 8.
plane, TT the instantaneous position of its perigee and M the corresponding
position of the Moon's radius vector.
We have xtl^O, 1^ = ^-6, ^M = L.
Let X^ = a, XTT = X, XM = v=
then L=v cr, & 6 = % a.
Also, from Art. 101,
da cos i dO,
and therefore
dy dm dO da- d .. dO
/v = ---- 1 -- __ (1 _
+ ~
_
dt dt dt dt ~ dt dt
Whence, the second of equations (2) gives
.................. (4);
the third and fourth of the same equations become
sin i -j- = h^r sin (v a), -j- = h$r cos(v-a) ............ (5).
.
These three, with the equations for de/dt, dh/dt, given in the previous
article, are all we shall require.
The new element x, like o-, is a pseudo-element and its presence is due to the measuring
of the coordinate v* from a departure point. It is not a complete substitute for w
i since the point X is not completely definite ; in order to make it so, it is necessary to
define the initial position of X. The latter is assumed to be such that when ^=0, X coincides
with x ; hence X is on that orthogonal to the orbit which passes through x (Art. 79).
The equations for de/dt, dh/dt give
d , , , de dh
* See Art. 101, where this coordinate is called v l . In the Fundamenta, p. 37, it is denoted
by v, and in the Darlegung, i. p. 102 by v.
112
164 THE METHOD OF HANSEN. [CHAP. X
But since l/r = l + e cos/, we have
I r O r I r /_ r\ ~
e z 7 = 7 " 1 1. + i } * cos />
r a I r I \ U
and therefore
/7 r / *-\ )
(6).
Let & be any function of t. Multiply (6) by cos (^ -ft) and (4) by
sin ( fii) and subtract : we obtain
{he cos (X-&)) = M $ sin (/+ x -&) + X l + cos
the expression for that function of the instantaneous elements required by
Hansen.
207. The considerations which guided Hansen in his method of treating perturbations
are set forth in a reply* to some ill-founded criticisms by de Ponte'coulant on the Funda-
menta. Hansen remarks that the solution of the equations which give the elements in
terms of the time is very troublesome, requiring that six integrations be performed. But the
quantities really sought are not the variable values of the elements but only three definite
functions of them, namely, the three coordinates. He therefore sought for functions of
these elements, by means of which the coordinates could be found in a more direct manner.
It is true that, in any case, six integrations must be performed and also that some method
of continued approximation must be used, but the ease or difficulty of carrying them out
varies enormously according to the plan of treatment. The most numerous of the in-
equalities in the Moon's motion are those which occur in the plane of the orbit. Hansen
succeeded in obtaining a function W, the equation for which was of the first degree ; when
this function is known in terms of the time, two very simple integrations furnish the in-
equalities in the plane of the orbit.
One point which differentiates Hansen's methods from all others consists in the addition of
the perturbations directly to the mean anomaly of a certain auxiliary ellipse in the plane of
the instantaneous orbit instead of to the true anomaly or to the true longitude on the fixed
plane. This fact is sometimes stated by saying that he uses a variable ti^pe. The
auxiliary ellipse will now be defined : it may be looked upon as the intermediate orbit
adopted by Hanaen.
The Auxiliary Ellipse.
208. Consider an auxiliary ellipse placed in the plane of the Moon's
instantaneous orbit, with one focus at the origin. Let its mean anomaly be
denoted by n Q z y its major axis by 2a , where 7i 2 a 3 = /i (//, = sum of the masses
of the Earth and the Moon), and its eccentricity by e . Throughout the whole
of the theory n Q , a , e are absolutely constant.
' Note sur la th^orie des perturbations planStaires.' Astr. Nach. Vol. xv. Cols. 201-216.
206-209] THE AUXILIARY ELLIPSE. 165
In the auxiliary ellipse, let E be the eccentric anomaly*, / the true
anomaly, r the radius vector. We have then (Art. 32)
r cos/= a (cos E - e ), E e sin E n z, )
_ . - > .............. \y/ 5
r sm/= a v 1 - e<? sin E, n 2 a 3 = p )
these, after the elimination of E t will give r, / as functions of one variable z
and of the constants a ,n ,e .
Also let
= ...... (9);
so that h is the same function of n 0) a , e that h was of n, a, e.
Let the perigee of this ellipse have a forward motion in the plane of the
orbit equal to n y, and let TT O be the longitude of the perigee from the
departure point X at time t = 0. The longitude from X of the point whose
true anomaly is/) will be at time t,
TT O .
This ellipse being used as an intermediate orbit, we shall have initially, z = t + const, or
n z = g. The plane of the orbit is then supposed to be fixed and X will be a fixed point on it.
Also ?i , 2 , e will be the mean motion, major axis and eccentricity, while y is a constant,
as yet indeterminate, depending on the Sun's action in the same manner as did the constant
c introduced in Chap. iv. When the complete action of the Sun is taken into account, the
value of n$ will be g+ periodic terms.
209. So far the only relation between the auxiliary ellipse and the
actual position of the Moon consists in the fact that the former is placed
in the instantaneous plane of the orbit. The connecting link is made by
allowing the point whose true anomaly is / to be on the actual radius vector
of the Moon. This fact, expressed in symbols, is, by Arts. 206, 208,
so that r, f are the radius vector and true anomaly of the point on the
auxiliary ellipse where the actual radius vector of the Moon cuts this ellipse.
When z and the actual position of the Moon are known in terms of the time
and of the constants, the auxiliary ellipse is completely denned.
Let the actual radius vector of the Moon be, as earlier, r and put
When z, v are known in terms of the time and of the constants, the position
of the Moon in its orbit will be known. The problem of motion in the
instantaneous plane therefore consists in the determination of z, v and in
* Hansen in the Darlegung, i. p. 102, where these equations are given, denotes the eccentric
anomaly by e.
166 THE METHOD OF HANSEN. [CHAP. X
the determination of the meanings to be attached to the constants n , a , e Q
and to the arbitraries which arise when the equations for 2, v are integrated.
210. Some idea of Hansen's method can now be given. Suppose that the initial position
of X has been denned and that z, v have been expressed in terms of the time. The equa-
tions (8) will then give r,/; from these, by means of the equations
we can calculate v, r ; y is a certain quantity (which is constant when the solar perturba-
tions only are considered) to be defined during the process of solution so that no terms,
increasing continually in proportion with the time, shall be present in the expressions for
n (fig, v. The first object to be sought is therefore the determination of z t v.
The second object in view is the determination of the motion of the plane of the orbit ;
this is given by the equations (5). But in order to reduce the longitude in the orbit to
that along the plane of reference we must know o-. The latter is found, when i, 6 are
known in terms of the time, from the equation
da- .d6
-j- COS I -=- .
dt dt
Also, when <r, i, 0.are known, the latitude above the plane of reference will be obtainable.
The determination of z, v will be reduced to the consideration of a function W which
will presently be constructed ; the variables or, ?', & will be replaced by three others. The
integration of the equations for W. , 2, v will furnish four arbitrary constants which will be
determined in Art. 231, and those for the variables o-, i, & three further arbitrary constants ;
the latter three will furnish the initial position of X and of the plane of the orbit.
All the equations considered are reduced to the first order. The equations for z, v are
really of the second order, since W is determined by an equation of the first order. The
equations for P, Q, K the variables which ultimately replace o-, i, 6 are each of the first
order, so that the seven arbitrary constants are necessary for the general solution of the
equations. Six constants only are necessary to define the position of the Moon : the
seventh constant is that which defines the initial position of X.
The Equations for z, v.
211. Since v, the longitude in the orbit, is measured from a departure
point, dv/dt has the same form, when expressed in terms of the instantaneous
elements and of the time, in disturbed or in undisturbed motion (Art. 79) ;
hence r z dv/dt = na? Vl e 2 , in disturbed motion. From this and from the
equations of Art. 208 we have, as in Art. 32,
00
= 1 + e cos/, -2^-i = 1 -f e cos/
r
>_ 2/1- _/* -2^~
' dt~ ~h' ^dz~~
dr nae __._ , 7 .^,_ ^ = ^^ sin / = ^ sin/
...(10).
209-212] THE EQUATIONS FOR Z, v. 167
Of these, it is to be remembered that the portions to the left, involving
the letters r, f, a, etc., refer to the instantaneous ellipse ; those to the right,
involving r, f, a Q , etc., refer to the auxiliary ellipse. The connection is fur-
nished by the two values for v and by the fact that the two ellipses lie in the
same plane.
Since f is a function of one variable z which is itself supposed to
be a function of t, we deduce immediately
a dv df dz
= =
Eliminating df/dz from this equation by means of the value given for it in
(10), we obtain
y
Also, from the last of the same equations,
I I * f 1 + ecos /, f " V
l + 2 hJa a ^T^r Hl + J'
since h^l Q = ^ h z l. Substituting in (11) and putting /=/ 4 n^y + TT O %,
we find
dz_
dt~
where Fl-+2 (13).
h h, a,, 1 - e 2
These furnish the required equation for z.
212. To find the equation for v we have, since f is a function of z only,
dr . df dz _ dv
Substituting from (11) for dz/dt and observing that 1 +v = r/r, we obtain
dt dz hr 2 r dt a Vl e<? a o dz
hr dz T dt \/j Q 2 dz \&Q>
Put for-, -j- , -, -=- their values from (10). The first and second terms
r dz r dt
of the latter expression for dvjdt become
1
168 THE METHOD OF HANSEN. [CHAP. X
or, since hfjh? = a(l- e z )fa Q (1 - ef), they are
ix f- e o sin/(l + e cos/) + e sin/(l + e cos/)} ....... (14).
But, differentiating (13) partially with respect to z, we have (since z is
only present explicitly in r, /), after inserting the values of drjdz, dfjdz given
in (10),
dW rt h
= 2
which, since f + n^yt + TT O % =/ n a = h Vl e 2 > becomes
= 2 . j sin /(I + e cos/) ( _ - e sin/) I
= 7i~~a\ ! e s i n /(l + e cos /) - (1 + e o cos /~) e sm /l-
Comparing this with the expression (14) which contains the first two
terms of dvjdt, we obtain
dv_ , dW . , y(l + v) d
di~
the required equation for v.
213. It is easy to see that when the two sets of elements coincide, W,
v, y vanish ; further, if the disturbing forces vanish, dzjdt = 1. The quantities
W, z/, y are therefore at least of the first order of the disturbing forces.
Hence, in the expression (12) for dz/dt, the third term is of the order of the
square of the disturbing forces and it may be neglected in the first ap-
proximation; in the fourth term we can, to the same degree of accuracy,
put n z = n t + const. = g : this amounts to neglecting the disturbing forces
in the coefficient of y. The same remark may be made concerning the
second term in the expression (15) for dv/dt. Hence, the principal parts
of the equations for z, v depend on W and this function must now be
expressed in terms of the disturbing forces.
So far the equations (12), (15) are purely algebraical results obtained by the com-
bination of two sets of elliptic formulae and connected by the single fact that the longitude
in each orbit, reckoned from one origin, is the same. One mean anomaly is therefore
a function of the other, but no supposition, involving any relations between the two sets
of elements, has been made and the results would be equally true for any two sets of
elements in one of which the motion of the perigee is directly proportional to that of the
mean anomaly*.
* See Hansen, Ueber die Antoendung osculirender Elemente als Grundlage der Berechnung der
Storungen eines Planeten, und iiber die unabhttngigen Elemente der "Fundamcnta nova etc." Astr.
Nach. Vol. xvm. Cols. 237-288.
212-215] THE DEFINITION OF W. 169
The equation for W.
214. The expression (13) for W is a function of the variable elements
h, e, x ; it also contains t through the term n yt and through r, /the latter
being functions of z and therefore of t. But since the equations which
express h, e, % in terms of the disturbing forces are given by their differ-
entials, it will be better to form dW/dt and then to find W by a single
integration, instead of performing the three integrations necessary to find
h, e, % directly and then substituting their values in the expression for W.
It will now be shown that, in performing these processes, r, / which
are functions of z and therefore of t, may be considered constant. (See
Art. 104.)
Let T be a constant and let f, p, $ denote the values of z, r, f when
T replaces t : f, p, <j> are then the same functions of the constant T that 2, f, f
are of t Let W denote the value of W when p, $ are put for r,f.
Now the expression (13) for W may be put into the form
W = L! + L 2 r + L 3 r cos/+ Z 4 r sin/
where L lt L 2 , L 3 , L 4 do not contain r, / being functions of h, e } % n yt only.
We may write this
_
since r is a constant. After the integration, T must be put equal to t.
Hence we need only consider the function W, in which p, ^ are constants.
We have, substituting p, ^, f for r,/ z respectively, in equations (10),
= 1 + e cos 0, JP = n,a? V, etc. . . .(16).
. ,
p at;
215. The definition of W gives
W__l A Q t * h P
h h a 1 e*
Therefore, considering p, $ as constant and remembering that a , e , n 0) h ,
TT O are always constant,
dW h, dh 2 5 dh 2 5
) * 00(1 -
where ft = ^ + ?i y + 7T .
e ! -
170 THE METHOD OF HANSEN. [CHAP. X
The equations (3), (7) are immediately applicable. By means of them
we find,
But since p = h?a (1 - e 2 ) = h *a (1 - e 2 ), l = a(l- e 2 ),
- & =f+nyt + TT O - < - w ^ - 7r =/ - <,
we obtain
2 + S1D <- + cos
In order to get the last term of this expression into a suitable form,
differentiate (17) with respect to f. This gives
_/ . . i p ^ p <> e sn < + n y^ H- TT O -
~i=T77 I r r ~T ~r~ ~i~A| TV ^7 7V
h / p d h a d
which, by means of (16), becomes
h \ 1 d _ _ A n a
Substituting for e sin (0 + % 2/ + ^TO ~ %) from this result in the last
expression for d W/dt and rearranging the other terms, we obtain
which is the required equation for W. When W has been found from this
equation and thence, by putting r = t, the value of W, the equations (12),
(15) will give 2, v. In the process of integrating (18), p, <j> are, by Art. 214,
to be considered constant.
Several methods of arriving at this expression for d W/dt have been given. On pp. 41-43
of the Fundamenta, Hansen arrives at it by a direct transformation from the equations of
variations of the elements, but the form obtained is slightly different from (18) above ; the
latter becomes the equation given in the Darlegung, I. p. 107, if dR/dv, dR/dr be substituted
for $>, $. The method given above is based on one by Zech*. In the Darlegung i. pp.
* Neue Ableitung der Hansen' schen Fundamentalformeln fur die Berechnung der Storungen.
Astr. Nach. Vol. XLI. Cols. 129-142, 205-208.
215-216] THE NUMBER AND SIGNIFICATION OF THE CONSTANTS. 171
104-107 * is another investigation obtained directly from the fact that the coordinates and
the velocities have the same form in disturbed and in undisturbed motion. In all these
methods Hansen's theorem, enunciated in Art. 104, is used ; Briinnow t gave develop-
ments of a different form which suggested that this theorem was not necessary (see M. N.
R. A. S. 1895, No. 2). Earlier, Cay ley had also given a method of obtaining the equations
of the Fundamenta which assisted in clearing up several difficult points in that work.
216. Six constants have been introduced with the auxiliary ellipse, namely, , e , n ,y,
TT O and that attached to n z (which in undisturbed motion is of the form n t+c ). These
are not all independent and arbitrary. The two a , w are connected by the equation
ft 2 a 3 =/i, while y will be seen to be a certain constant necessary (like c in Chaps, iv. vn.)
to put the solution into a suitable form. The number of independent constants is therefore
four ; the other three arise from <r, t, 6 (or from P, Q, K\ Hence, as far as the motion in
the plane of the orbit is concerned, we have the necessary number (Art. 210). The four
new constants, which will be introduced when the equations for 0, v are integrated, can be
determined at will, and they will be so determined that the meanings of n$, e , TT O , c in
disturbed motion may be rendered independent of the method of solution adopted. As
numerical values are used by Hansen, it is necessary to know beforehand what signifi-
cations are to be attached to w , e , A . These are, however, better explained when the
equations for z, v have been integrated : the definitions will be found in Art. 231 below.
It is only necessary to state here that h 0) e differ from h, e by quantities of the order of
the disturbing forces.
It will therefore be seen that the elements with suffix zero are not the purely elliptic
values of the instantaneous elements, if we understand by ' purely elliptic values ' those with
which we start. On any development with the latter as a basis, the observed mean
motion, for example, would no longer be denoted by a single letter but would consist of the
purely elliptic value together with a series of constant terms due to the disturbing forces.
This was seen in de Pontecoulant's theory where the new arbitraries arising during the
integrations were used in such a way that the mean motion might be denoted by n. The
same thing occurred in Delaunay's theory, but there it was necessary to make a direct
transformation in the final results. Hansen, like de Pontecoulant, keeps the arbitraries
(denoted by 6, below ) which arise in the integrations, for the purpose of defining w , e .
These remarks are necessary for a clear understanding of the three sets of elements used in
the Fundamenta. There Hansen denotes by (a), (n), etc., the quantities denoted by , n ,
etc., above and by , 7i , etc., the purely elliptic or initial values of a, n, etc. (the latter
being the instantaneous elements), that is, the values of a, n, etc., when the disturbing
forces vanish!!. With the notation used in this chapter, and in the Darlegung^, a , T^,
etc. implicitly contain terms due to the disturbing forces.
* It was also given by Hansen in the Astr. Nach. Vol. LXII. Cols. 273-280, Neue Ableitung
meiner Fundtimentalformeln fiir die. Berechnung der Storungen.
t Saturn etc., mbst einer Ableitung der Hansen'schen Fundamentalformeln. Astr. Nach. Vol.
LXTV. Cols. 259-266.
$ On Hansen's Lunar Theory. Quar. Math. Jour. Vol. I. pp. 112-125 ; A Memoir on the
Problem of Disturbed Elliptic Motion. Mem. B. A. S. Vol. xxvu. pp. 1-29 ; A Supplementary
Memoir on the Problem of Disturbed Elliptic Motion. Mem. B. A. S. Vol. xxvin. pp. 217-234.
These are also found in his collected works Vol. in. pp. 13-24, 270-292, 344-359.
Art. 230.
|| Fundamenta, pp. 62, 64.
U Darlegung, i. p. 102.
172 THE METHOD OF HANSEN. [CHAP. X
It is necessary to point out that n^y is not the mean motion of the Moon's perigee along
the true ecliptic although it accounts for the greater part of this motion. It is the mean
motion of the perigee in the orbit. A small correction, which depends on the mean motion
of the Moon's node along the ecliptic and on the square of the inclination, has to be applied
in order to obtain the mean motion of the perigee along the ecliptic. See Arts. 217, 237.
One great advantage of Hansen's method of computing the longitude in the plane of
the orbit is that the inequalities produced in z by the motion of the plane of the orbit are
necessarily very small. Since the force 3 does not occur in the equations for z, v, the
inequalities produced in these variables by the motion of the plane of the orbit must all
be small quantities of the order of the square of the disturbing forces at least.
The Motion of the Plane of the Moons orbit.
217. Definitions. It is necessary now to define the variables by means
of which the motion of the plane of the Moon's instantaneous orbit is found.
We suppose here that the Sun's orbit is not fixed but that it is moving in a
known manner.
On the unit sphere, let XIM, X'lm f be the orbits of the Moon and the
Sun respectively. Let X' be a departure point on the Sun's orbit, defined in
the same manner as X was. Let Oj, H/ be the ascending nodes of the orbits
on the fixed plane of reference. With the notation used in Art. 206 we
have, if accented letters refer to the Sun's orbit, the following old and new
definitions*:
Fig. 9.
* The angles denoted here by \l/ t \f>' are called by Hansen $, $ respectively. Fundamenta,
p. 84. Darlegung, i. p. 110. The change is made to prevent confusion with the letter used
earlier.
216-217] QUANTITIES DEFINING THE PLANE OF THE ORBIT. 173
Hence (Art. 101)
dor cos i d0, da-' = cos i' d&. \
Also, put p = sin i sin a, p' = sin i' sin a-', > (19).
<? = sin i cos o-, q' sin i' cos <r' )
All the quantities denoted by accented letters, except >//, are supposed
known, since they refer solely to the Sun's orbit.
Let N, K be defined by the equations
..(20),
where w ' denotes the distance of the Sun's apse from X' at time t = : a, rj
will be so defined that N, K contain no terms directly proportional to the
time.
We have
Since X'fl = ^', and since ^T, ^T contain no terms directly proportional to
the time, the quantity n (a 4- ??) represents the mean motion of the argu-
ment T/T', that is, n (a + rj) is the mean motion of the Moons node along the
true ecliptic.
Again, if TT be the perigee of the auxiliary ellipse, the mean motion of
the Moon's perigee, when reckoned along the true ecliptic and then along the
orbit, will be the same as that of TT, when reckoned in the same way. Now
by Art. 208. The mean motion of ty -\Jr is 2n 7;, by the second of equations
(20). Hence, the mean motion of the Moon's perigee along the true ecliptic is
n<> (y - 2??).
In general, y, a, TJ are constant quantities. The actions of the planets, however, produce
small accelerations in the mean motions of the perigee and of the node, that is, they
produce terms dependent on t 2 , t 3 ,... These can be taken into account by putting for n^yt^
n at, nQfjt the integrals n^ydt^ n^adt, n $T}dt, respectively. The differentials of these
quantities with respect to the time will then be still denoted by n y, w a, W O T;, respec-
tively *.
Hansen introduces the quantities $, to denote the angles ^r-o-, ^'-o-'f. They are,
however, merely intermediaries in his development of the equations obtained below : as
they are not necessary in the proof given here, they will not be used in this sense.
He uses the letter , in another place, to denote an entirely different quantity. See
Art. 230 below.
* Fundamenta, pp. 51, 97, 98 ; Darlegung, i. pp. 103, 112.
t Fundamenta, p. 82 ; Darlegung, i. p. 112.
174 THE METHOD OF HANSEN. [CHAP. X
The equations satisfied by P, Q, K.
218. In any spherical triangle ABC whose sides, denoted by a, b, c and
angles, denoted by A, B, G, all vary, we have
dC = dA cos 1) dB cos a + dc sin A sin b ,
db = dc cos A + da cos C + dB sin C sin a,
cfo = dc cos B + db cos (7 + cL4 sin (7 sin b.
To prove these, draw BD, AD' so that the angles CBD, CAD' are each equal to a right
angle. We have, in the triangle ABC,
cos (7= -cos .4 cos ,5 + sin A sin B cose ......... (22),
and therefore, when the sides and angles all vary,
- dC sin C= dA (sin A cos J5 -f cos A sin .B cos c)
-fcZ? (cos .4 sin B + sin ^1 cos B cos c) efc sin A sin J5 sin c. A
The coefficient of dA in this equation is equal to
- cos (90 - A ) cos ( 180 - B) + sin (90 - A ) sin (180-5) cos c
= cos A DB = sin (7 cos b,
by the spherical triangles ABD, ADC. Fig> 10 '
Similarly, by considering the triangles ADB, CDB, we prove that the coefficient of dB
is equal to sin C cos a. Also, since sin B sin c = sin 6 sin C, the coefficient of dc is equal
to sin A sin 6 sin (7. Substituting and dividing by - sin C we obtain the expression given
above for dC.
Again, in the result for dC, put TT- A, n B, IT C for a, b, c and IT -a, n b, n-c for
A, B, C, respectively. We immediately deduce, from the known property of the polar
triangle, the value of dc in terms of da, db dC, and thence, by interchanging the letters,
the values of db, da, given above.
For the triangle ft' All (fig. 9), put A = i', B = 180 - i, C = J, a = ^ - a,
b = r - </, c = - 0'. We obtain
dJ = - di' cos (^' - o-') + di cos (^ - <r) + (d# - d6') sin ^ sin (^ - <r'). . .(23),
cfyr' do-' = (d# dO') cos i v + (cfyr do-) cos J" di sin J" sin (i|r a),
d^r - da- = - (d0 - d<9') cos i + (cty' - da) cos J + cK' sin J sin (^ - ff ').
Substituting for da, da' their values cosidO, cosi' d6', and transposing,
the second and third of these equations become,
dty' d^r cos J=dO (cos i' cos i cos J) cK sin /sin (i|r a)
= c0 sin i sin J" cos (i|r a) cfo' sin / sin (-^ a-),
(fyr - dty' cos J" = dfl 7 (cos i cos i' cos J) -h cfo v sin J"sin (>|/ cr 7 )
= - dO' sin ^ sin Jcos (^ - a') + di' sin /sin (-// - a'):
218] THE MOTION OF THE PLANE OF THE ORBIT. 175
the second line in each case being obtained by the successive application of
the formula (22) to the triangle n^ftjft.
But we have, from (20),
+ d^r') (1 cos J) = - 4 (dN + n Q adt) sin 2 \ J,
4 cos J) = - 4 (dK - n Q rjdt) cos 2 \J.
Therefore, substituting these values in the sum and difference of the two
previous equations, we obtain
dN + n adt = ^ cot \J {di sin (^ cr) dd sin i cos (fy cr)
- di sin (->!r' - cr') + dff sin i' cos (b 1 - er')l,
v (24).
dJ'f - ^cft = i tan i^ { ^ s i n (^ ") + ^ sm l cos Of 1 ~ )
di sin (^' cr') + d6' sin i' cos (i|r'
The equations (23), (24) for dJ, dN, dK are purely geometrical results ;
it is necessary now to introduce the disturbing forces.
We deduce immediately from equations (5),
di dO .
-j- COS (ty CT) + -=- Sin I Sin (l|r cr) = fl$r COS (V ty),
di . de .
-j- sm ( Y cr) -y- sin i cos (>/r cr) = hyr sin ( >lr).
ift <n
Also, differentiating the expressions for p, q' in (19) and remembering
that da = cos i' d0', we obtain
dp' = di' cos i' sin cr' + d#' cos i sin i v cos cr',
c^' = rfi' cos i' cos cr' e0' cos i' sin i' sin cr'.
Whence
c^i' x /x du /// /\ * I &P > do
cos (vr cr ) H -y- sin i sin (^ cr ) = ;, I -4 sin "w 1 H -y- cos
a^ ci^ cos i \ dt dt
di' . dff . 1 / dp' f/ dq" . ,A
r Sin (-Ur CT ) 7- Sin I COS ('ur CT ) = n ~ COS -ur + -y- Sin ^ .
rf^ a^ cos i \ dt dt J
Dividing the equations (23), (24) by dt and using the results just
obtained, we find, since (fig. 9) dO sin i' sin (i|r' </) = dd sin i sin (-\Jr a),
-r. = h%r cos (v ilr) ., f^fr sin -Jr' + -^ cos 1
176 THE METHOD OF HANSEN. [CHAP. X
219. The final transformation is made by changing from the variables
J, N to P, Q, where
P=2siniJsin(^-J\r o ), Q= 2 sin JJcos (N-N.) (26).
In these, N denotes the constant part of N. We deduce
dP = dJ cos J J sin (N - N 9 ) + 2dN sin 1 J cos (A r - N )
= dJ cos J Jsin (JV - #) + Qdtf,
dQ = dJcosJcos(N- N ) - 2dNsm%Jsiu(N- N )
= dJ cos Jsin (JV - JV ) - PdN.
Substituting for dJ, dN their values just obtained, we find
(. cos // + sin
- cos
(27),
where ft' = - ^' - N + N .
The equations (25), (27) for K, P, Q, are those required. The angle ^'
may be eliminated from (25) by means of the equation
2 sin cos * - - f = * '
which follows immediately from the definitions of P, Q, p.
When all the disturbing forces are omitted, we have K, P, Q constant
and therefore N, J constant, for a, 77 are of the order of the disturbing forces.
Now, by definition, N is the constant part of N ; let J" be the constant part
of J and K Q that of K. Hence :
The first approximation to P, Q, K is given by
P = 0, Q = 2 sin J ,/, K = K Q (28).
These values correspond to fixed positions of the orbits of the Sun and
the Moon.
The quantities p, q, defined in Art. 217, have not been used. It is evident from the
definitions that the equations for P, Q, K should be symmetrical (except with regard to
signs) with respect to the quantities referring to the Sun and the Moon ; the parts of these
equations dependent on 3 can, in fact, be exhibited in terms of dp/dt, dq/dt by expressions
similar to those which contain dp'/dt, dq'/dt. The reason for not expressing them in this
form is that the latter are known functions while the former are functions of the quantities
we wish to find. It will be seen from fig. 9 that p, q are the sines of the latitudes of
the points X, Y below the plane of reference and that p', q' are those of X' t Y' below the
same plane.
219-220] DEVELOPMENT OF THE DISTURBING FUNCTION. 177
Since J is small and since NN Q contains only periodic terms dependent on the
disturbing forces, equations (26) show that the principal part played by Q is to bear the
periodic variations in the inclination of the Moon's orbit to the ecliptic. The quantity P
is small and it carries chiefly the perturbations included in N; they are multiplied by the
small quantity sin \J. Also, by equations (21), N - N+K Q -K, N Q -N-K Q +K contain
the periodic parts of the motions of the Moon's node, along the Moon's orbit and along the
ecliptic, respectively ; the difference between these is very small.
The Form of the Development of the Disturbing Function.
220. We have from Art. 107, after replacing R by p>R,
As before, S is the cosine of the angular distance between the Sun and Moon
and therefore, by fig. 9,
$ = cos QM cos lm' + sin IM sin lm' cos J.
In order to obtain a perfectly general development of R, the auxiliary
(not the instantaneous) ellipse, with its variable mean anomaly n z, is used
for the developments in the plane of the orbit, and the instantaneous values
of i, 9, a- (or of the variables replacing these) for those of the plane of the
orbit. For symmetry, we suppose the Sun's motion to be defined also by an
auxiliary ellipse with a mean anomaly n V, the perturbations of its radius
vector being denoted by v and the m^an motion of its auxiliary perigee in
the plane of the orbit by n Q y ' .
To develope R we have
r = r (1 + v), QM = v - >|r =f + n yt + TT O - ^,
r' = r' (1 + v'), Clm' = v f ->^' =/'
by Art. 209. Substituting, we obtain
where
8 = cos (/+ nyt + TT O - -f) cos (
+ sin (/+ n yt + TT O -^) sin (/' + n y't + TT,,' - ^) cos J.
Since the variables /', r', v, referring to the Sun's orbit, are supposed to
be known functions of the time, R is thus expressed as a function of t and of
the unknmvn variables f, r, v, ty, ty', J.
Let <w = n^yt + TT O >|r, to' = n y't -f W - ^ ............... (29),
B. L. T. 12
178 THE METHOD OF HANSEN. [CHAP. X
so that, by fig. 9, CD, co' are the distances of the apses of the auxiliary orbits
from the common node fl. We have then
S = cos (/+ a)) cos (/' + co') + sin (/+ co) sin (/' + co') cos J ... (30).
Now f, r are, by equations (8), the elliptic true anomaly and radius vector
corresponding to a mean anomaly n Q z, with constants a , n Q , e ; in the same
way, /', r' correspond to a mean anomaly n Q 'z' with constants a ', n ', e Q '.
Therefore, putting in'of/pa*'* = m*, we find
where S has the value (30).
Comparing these values of R, S with those given in Art. 124, we see,
by the remarks just made, that the method of development, given in Art. 125,
will be available if we simply replace a, e, a', e', w, w' by a > ^o, </> 0o'> n oZ,
n 'z, respectively, and multiply
It by (1 + z/) 2 /(l + vj, R by (1 + v)*/(l + i/) 4 , etc.
Finally, to take into account the correction noted in Art. 7, we must further
multiply R by (E-M)j(E + M).
If we look at the developments of I//, I/A given in Art. 5, it is not difficult to see that
the general form of the multiplier of RV\ necessary when the force function given in Art. 8
is used, is
ri/ E V* 1 i --tf V+ 1 )
+
This expression was first obtained in an indirect manner by Harzer*.
221. We shall thus have the development of R in a perfectly general
form : it will be expressed as an explicit function of the unknown variables
z t v, co, co', J, and of t through the known variables z' ', v the rest of the
symbols present being absolute constants. It will be shown later how ^}, %
are obtained from this development of R. In order to find R in a form
suitable for the determination of the motion of the plane of the orbit, we
must transform from the variables co, co', J to P, Q, K.
By equation (14) of Art. 124, we see that co and co' will only occur in R
in the form of multiples of co + co', and that a term containing in its argument
ji (co + co'), where j^ is a positive integer, will have its coefficient at least of
the order sin 2 -?'- /. Hence, all terms in R are of the form
m?A sin 2 -?'' j~cos { jn^z +j'n 'z' + ^ (co + co') +// (co co')},
where A contains only integral powers of e Q , ', ajaj, sin 2 JJ, and j, j',^' are
positive or negative integers.
* Ueber die Ruckwerkung der von dem Monde in der Bewegung der Sonne erzeugten Storungen
aufdie Bewegung des Mondes, Astr. Nach. Vol. cxxm. Cols, 193-200.
220-222] EXPRESSION OF R IN TERMS OF z, v, P, Q, K. 179
But, from (29) and (20), we deduce
a, + a,' = n 1 (y + y' + 2a) + 2N, o> - ' = n t (y - y f - 2i?) + 2K ;
and therefore the general term is of the form
mfA sin* JJc
where
=><>* +/wy + ji M (y + y' + 2 ) + %'i-tfo + j/ wo* (y - y' - 2r>) + 2J/JT.
Also, from (26), we have
4 sin 2 J= P 2 + Q 2 , 2 sin 2 iJsin 2 (F- N Q ) = PQ,
4 sin 2 iJcos 2 (# - N 9 ) = Q 2 - P 2 .
The expression of .R as a sum of periodic terms therefore contains the five
unknowns n^z, v, P, Q, K ; of these, the variables n z, K occur in the argu-
ments only and the variables v, P, Q in the coefficients only.
It is to be noticed that, since n Q z, v enter into R only through v, r, we can
express R as a function of the time and of the five variables r, v, P, Q, K.
We then have $ = dR/dr, Xr = dR/dv, and the expressions for the disturbing
forces may therefore be directly inserted into equation (18). But as the
latter has to be solved by continued approximation, this process would
necessitate the expansion of the two expressions dR/dr, dR/dv. In the first
approximation, the latter can be transformed into the differentials of R
with respect to certain quantities present explicitly in the expansion of R.
We shall first find relations between 3 and the partial derivatives of R with
respect to P, Q, K, since the results for these are quite general.
222. The general expression for R is
_ m' ( \
~7U~ r' I" IL
where A 2 = (X - XJ + ( F- YJ + (Z- ZJ = r 2 + r 2 - Zrr'S ;
(X, F, Z), (X' } F', Z f ) being the coordinates referred to any axes.
From the first form of expression we deduce
aR = m (Z'-Z _ Z\ mf_ ( l_ l\ ,
* ~ dZ ~ fju ( A 3 r' s ) n U 3 r' 3 )
when we take the axis of Z perpendicular to the plane of the Moon's
instantaneous orbit. And, from the second form of expression for R,
dR _ m (rr_ r\ _ rnf^ /J_ _1 \ ,
M~7U 3 ~V 2 ;-/, U 3 r'^r-
122
180 THE METHOD OF HANSEN. [CHAP. X
Hence 3 = ' ao >
or, since (fig. 9) Z' = r' sin (v' ^') sin J, the relation becomes
3r = - || sin (?/->//) sin J (31).
223. Again, from Art, 220,
S = cos (v >Jr) cos (v' i//) + sin (v >|r) sin (v r \fr') cos J
and, by equations (20),
Therefore, as N, K enter into S only through -\|r, -\Jr',
r)Sf r)Sf
^ = - 2 cos 2 ^/sin (y v' i/r + <fy'\ = 2 sin 2 J/sin (v + v ty ^r').
Again, since J only occurs in S in its explicit form,
ds
jTr = sm J sm (v >/r) sin (v -\Jr ).
But we easily deduce from (26) by differentiation,
Equating the two values of dS/dJ, we obtain
J'sin(?;-^)sin(^-^ / ) ...... (33).
Also, from the values of dS/dK, dSfiN, we find
^^ cos 2 ^J ^j^ sin 2 J /= - sin 2 J cos (# ^) sin (w' ^'\
C/lV C7-X
Multiply this equation by Q, the first equation for dS/dJ by P sin / and
add. We obtain, after using the values (26) of P and Q in the right hand
members,
= - 2 sin 2 J" sin I J cos (v - i|r - N + N ) sin (v' -\Jr').
But by (32),
222-224] THE FIRST APPROXIMATION TO R. 181
and therefore we have, after dividing by P 2 -f Q 2 = 4 sin 2 ^J,
?W r)Sf
^p cos 2 y - JQ ^ = - sin J cos i/cos (v - ^ - N + # ) s i n (*' - +'). . .(34).
In a similar manner we can deduce
= -sin Jcos i/sinu -
Since P, Q, K enter into .R only through S y we have
dR_dRdS 3R_dRdS dR_dRdS
dP~dSdP' dQ~dSTQ' dK~dSdK'
Whence, multiplying the equations (33), (34), (35) by dR/dS and substituting
in their right hand members the value ofdR/dS given by (31), we find
~ + |=2r3 tan i/ sin (-
= r coe
.(36).
g cos" y + JP = ,-3 cos Km ( - ^ - N + N,)
These results, which are quite general, are put into the form in which
they will be useful in Art. 235.
The First Approximation to R and to the disturbing forces.
224. A limitation of the general value of R will now be made by
supposing the orbit of the Sun in its instantaneous plane to be an ellipse, so
that n^z = g' = n 't + c' (where c ' is a constant) and v =0, r r'. The
perturbations of the solar orbit, thus neglected, only produce small effects on
the motion of the Moon.
The first approximation to R is obtained by substituting for the coordi-
nates their elliptic values, that is, we put
n z=g = n t + c , v = 0, K = K, N=N , J ' /;
whence w + a)' = n t (y + y' + 2a) + 2N ,
co a)' = n t (y y 2-I;) + 2jfif ,
in which y' } being a known constant, is retained : y, a, rj are constants to be
found. We have also, by (26),
P = 0, Q =
The first approximation to R is therefore expressed explicitly as a
function of the time, the arguments being sums of multiples of the four
angles g, g', co, o>', and the coefficients being expanded in powers of e , e ',
182 THE METHOD OF HANSEN. [CHAP. X
sin 2 J Jo, Oo/a '. For the motion of the plane of the orbit when R is
expressed in terms of n z, v, P, Q, K, we also put n z=g, v = Q. The partial
derivatives of R with respect to P, Q must be formed before we give to P
the value and to Q the value 2 sin ^ /. As regards K, we evidently have
In the Fundamenta (p. 81), Hansen used the derivatives of R with regard to p, q as
though R were a function of the four variables r, v, p, q only. The difficulty is merely that
Hansen has attached a meaning to dR/dp, dR/dq which is unusual. The point was cleared
up by Jacobi*.
225. To express the disturbing forces ty , X in terms of the partial
derivatives of R.
Denote by *P > ^o the values of the disturbing forces *p, X when the first
approximation to R is used. In the terms multiplied by quantities of the
order of the disturbing forces, we can put r = r = r ,f=f=f 0> where r ,/ are
the values of r,/when n Q z=g. Since, in the first approximation, e , g enter
into R only through r,f and since g enters here in the same way that nt -f e
entered in the expressions of Art. 75, the second and third of equations (4) of
that article are available. We therefore obtain, with the necessary changes
in notation,
dR dR
By means of these results we can express ^ , X in terms ofdR/de , dR/dg.
Also, since G> enters into R only in the form v + CD, we have
xoov
(38) '
The First Approximation to W.
226. The general process of solution adopted by Hansen is one of
continued approximation. There has been found, in Art. 215, a general
expression for dW/dt which contains $, X. Now W is of the order of the
disturbing forces and all the terms present in the expression for dW/dt are
implicitly functions of z, f, t.
* Auszug zweier Schreiben etc. Crelle, Vol. XLII. pp. 12-31.
224-227] THE FIRST APPROXIMATION TO W. 183
Put
n^z = n t + C + &z = g + n z, n f = n r + c + S? = 7 + w 8f . . .(39),
and, in finding the first approximation to W, neglect 8z, ?. Let p , < >/o>
TT , etc. be the values of p, <,/ W, etc. when for v, n are put #, 7, respec-
tively. Also, as h, r differ from h , r Q by quantities of the order of the
disturbing forces, we can, in the terms multiplied by quantities of that order,
put h = h Q , r = r .
Let
^{cos(/ -<W-i;]+2jY,$, S in(/ -0 ).
The equation for W may be written
7 , "0 -* T /
dt v 1 e
But since y is of the order of the disturbing forces, we can, in the coefficient
of y, neglect W and put h = h . Hence, the first approximation will be
obtainable from
dt
227. We shall now transform T so that the values of ty , $ , given
by (37), (38), may be inserted. The suffix zero, which occurs in every symbol
present in T , will, for the sake of brevity, be omitted until the end of this
article.
With this understanding we have, by the elliptic formulae (16),
na
a(l-e>) a(l-*)
Therefore
r + a$ sin/1
J ]
2p sin <f> ["/a sin/ sin/\ ^ m .~]
^ y [ -- ^ + n ^ Xr-a^cos/
Vl-^LV r l ~ e ' J
3 f/aecos/ l+ecos/\ ^
" )
OS/-MN
1-e 2 /
asin/
184 THE METHOD OF HANSEN. [CHAP. X
We deduce, from (37),
daR _ aXr a?e ~
dg VT^e 2 : Vl-e 2
On the left-hand side of this equation put Xr = dR/dco and substitute in
the second line of the latter expression for T: for the first and third lines,
the equations (37) are available. We obtain, on restoring the suffixes,
^ , - ru cos <f> \/da R 1 da R\ t 2/o sin < da R
~ \/i 2 ~^ J
-O ?: c>0 o
dg e \ a J\ dg
This expression is now very easily calculated from the first approximation
to the value of a^R, for R has been expressed as a sum of periodic terms
whose arguments contain g, co and whose coefficients are functions of e .
The portions dependent on R are thus expressed by means of periodic series
with constant coefficients and with arguments of the form fit + ft'.
228. Let, for a moment,
= _ /pocos^o \ _ e,p^ H
\ a ' J a Vl-e 2
in which the signification of F', G', H' is evident. Since a R is expressible
by means of cosines, and since g, o> occur in the arguments only while e Q
occurs in the coefficients only, F', G' will be developable in sines and H' in
cosines of angles which are all of the form /3 + /3' (/3, /3' constant).
Here, fit + ft is formed of multiples of the angles g, g', co, co', all of which,
owing to the introduction of a, 77, contain t ; also, ft' = when /3 = 0, for
n , n ', etc. are supposed to be incommensurable with one another.
Since p , are the radius vector and true anomaly corresponding to a
mean anomaly 7, we have, by the theorem of Art. 43,
T = 2 2 a,- sin
oo
The first sign of summation refers to the angles fit + /3' and the second to the
integral values of j ; G is the symbol for the general coefficient corresponding
to the angle fit + fi'. The extra labour, caused by the presence of the angle
7 (which does not occur in fit + fi'), is compensated by the ease with which
the other coefficients can be obtained, when the values of C , Ci, CLj, for all
values of fit H- fi', have been calculated. See Art. 43.
When TO has been thus found in terms of the time, we obtain from (40)
227-229] DETERMINATION OF y. 185
But, by equation (19) of Art. 43, we have
Hence
dW
-- 2 2 (7, sin
in ( jy + ft + ff) + 2 r Jfy- jfl; + c/ - c'_;l sin J7,
i Lvl-e 2
where the terms for which /3t + ff = are written separately, their coeffi-
cients being denoted by c/.
229. Integration of the Equation for W and Determination of y.
We have on integration, since 7 is constant during the process,
where the additive arbitrary constant, denoted by ^(7), may be a function
of 7.
Putting r = t and therefore 7 = g, we find
2f
i Lvl e 2
.(41).
Now y was specially introduced in order that expressions of the form
t x periodic term might be eliminated, if possible. Determine y so that the
coefficient of t sin $r vanishes. This gives
<v s + c 1 '-c'_ 1 = (42).
The corollary to the theorem of Art. 43, applied to c/ (/_,-, then shows that
ZyjEiftl-ef + c/ - c'_, = 0.
Hence the coefficient of t siujg vanishes, and therefore all the terms having
the time as a factor disappear from W&
The equation (42) gives a first approximation to y*. That it should be
capable of being determined so that all terms of the forbidden form may be
eliminated, is sufficiently evident from what has been said in previous
chapters. All that now remains is the determination of the form of F(g),
the function which contains the arbitrary constants.
* Fundamenta^ p. 191. The determination of y, given in the Darlegung, i. p. 338, is not
directly applicable here because Hansen is there simply performing a verification of his
tables.
186 THE METHOD OF HANSEN. [CHAP. X
230. Determination of the Form of F(g).
The constants are found by considering the initial form of W. Put
p = a (1 e<?) pe cos 0,
and express W as a linear function of p cos $, p sin ^ ; the coefficients of these
quantities being of the order of the disturbing forces, we can put p , </> for p, 0.
Hence, the expression (17) may be written
- cos
where 5 = - 1 -fr + 2 - 3g , * CB (* -.*-*.)-*.
ft fi n 1 60
T = 2 A e cos (x - n<,yt - ir ) - 6 >p = 2 - ^"(X""^^
..(43).
To)
Since the arbitrary .P (7) is the only constant in the expression for W given
in the previous article, all that is required is to find the form of the constant
part of W . Let the constant part of % (which, by Art. 206, is the distance
XTT) be TT O ; then ^ will contain no constant term.
The constant parts of h, e are as yet undefined : it was merely assumed
(Art. 216) that they differed from h , e by quantities of the order of the
disturbing forces. In Art. 226 these differences were multiplied by
quantities .of the order of the disturbing forces and they were therefore
neglected ; here this does not take place. Let the differences be such that
the constant parts of H, T are denoted by 6, f ; the approximate constant
part of e e is then f and that of h/h Q - 1 is b + \e. (See Art. 234.)
The constant part of TF is therefore
by equation (18), Art. 43. From this we deduce F(g) by putting T = t or 7 = $r.
Since the terms multiplied by t have been made to vanish, the equation
(41) becomes, on the substitution of this value of F(g) y
W = tt 2 S -^ cos (jg + Pt + fi f ) + b-!;2, ~^- j cosjg.
On the subject of the determination of the constants b, and on their general meaning
with respect to k, e and A , e , Hansen's paper, referred to in the foot-note of page 168 above,
will be found of great assistance. See also Fundamenta, pp. 65, 66, 196, Darlegung, i.
pp. 332337, etc.
230-231] DETERMINATION OF Z. 187
The Integration of the Equations for z, v, and the Signification to be
attached to the Constants of the Auxiliary Ellipse.
231. Substituting the first approximation TF for TV in (12) and neglecting
the third term of the equation that term being of the order of the square
of the disturbing forces, we find
s^) ......... (44),
where, in the last term, r has been put for r (since y is of the order of the
disturbing forces) and for r has been inserted its value given by
equation (19) of Art. 43. We can also put for y the value found in
Art. 229.
The non-periodic part of dz/dt will be
1 + b + d,
where 6, d are of the order of the disturbing forces, b being arbitrary and d
containing the rest of the known non-periodic terms in (44). On integration,
it will produce in n Q z the term
There is now an opportunity of defining n . Let it represent the mean
motion of the mean anomaly n^z. With this definition we must determine the
arbitrary 6, so that
b + d = 0.
Again, if d' represent the known coefficients of cos g in (44), the coefficient
of this term will be
where d', f are of the order of the disturbing forces. On integration the
coefficient of sin^r in z will therefore be (d' % 9J2 1 /9e )/w . We define t; so
that the coefficient of sin g in z vanishes. This, as we shall see directly,
amounts to a definition of e .
The value of z is therefore given by
7i * = # + 25 sin
where @t + j3' is an angle of the form
39 +/#' +ji + J>' (J>
the term for which j = I,/ = j x =j 1 ' = having its coefficient zero.
188 THE METHOD OF HANSEN. [CHAP. X
The arbitrary constant present in g is the value of the non-periodic part
of the disturbed mean anomaly at time t 0.
232. The value of n Q , defined above, is the mean motion of the mean anomaly of the
auxiliary ellipse, that is, the mean rate of separation of the Moon from the perigee of the
auxiliary ellipse. To obtain the mean motion of the Moon it is necessary to add to n the
term n Q (y 2?;) which (Art. 217) represents the mean motion of the perigee of the auxiliary
ellipse. Since the mean motion of the Moon is observed directly, in order to obtain n Q for
purposes of computation we must also know the mean motion of the Moon's perigee a
quantity which is found from theory. The latter is, however, capable of being observed
with great accuracy and Hansen, in performing his computations, used a value of n n obtained
from these observed values. The computed value for the motion of the perigee agrees
very nearly with the value obtained directly from observation : the small difference causes no
sensible error in the coefficients of the periodic terms*.
When the value of n^ is inserted in the expansion of / in terms of the mean anomaly,
we obtain
/= n^s -f- ej sin
where e lt e 2J ... are known functions of e Q given by equation (7) of Art. 34. Since
%z = g + n 8z, we find
Although n$z contains no term with the argument g, terms with argument g, other
than e v sin <?, will arise in /, owing to combinations of terms in the powers of n Q 8z with
those of the elliptic development. For instance, the term in n Q 8z with argument 2g will
combine with e 1 cos g to produce a term of argument g in /. These terms, as well as those
of the same argument which arise from the reduction of the longitude in the orbit to that
on the ecliptic, are very small.
Hansen computes n^s with e = '05490079 ; this produces 22637"'15 as the coefficient of
sin g in the expression for the ecliptic longitude. The observed value of this coefficient is,
according to him, 22640"-15 and, in order to produce this coefficient, the value of e Q should
have been '05490807. The difference is very small and it is sufficiently taken into account
by multiplying those terms whose characteristic is e by '05490807/'05490079. Very few
terms need to be thus corrected : the principal one is the evection. (See the papers of
Newcomb referred to in Art. 238 below.)
233. The equation (15) will now serve for the calculation of v. But
since dW/dz = dW/d (where the bar denotes that r is changed into Rafter
the differentiation) and since dW/n d=dW/dy in the first approximation, we
can put d W/dz = n 9 W /dy. Also, to the same degree of accuracy, we can
neglect the product yv and put d/dz = n d/dg. The equation therefore
becomes
dv _ ^ 9TF n y d /r V
dt ' 9y *J\^-e<?dg\Q,J
* Darlegung, I. pp. 173, 348.
231-234] DETERMINATION OF v. 189
The value of 9 W /dy may be obtained from Art. 229 and thence, by putting
T = t, that of the first term of this equation ; therefore, after inserting the
values of y> r 2 /a<? as before, we have
whence, integrating,
v=C-2JjCos(j3t + {3'). .
Here j3t + (3 f is of the same form as before and A is the corresponding
coefficient. The constant C, owing to the relation ?i 2 a s = /*, is not arbitrary :
we proceed to find it.
234. We have, from equations (43), Art. 230,
w + s.f- 1 hQ + 2 h -2X h X h
**T -- ' 6 - b >
where B(h/h ), 8(h /h) denote the differences of hjh Q , h /h from unity. Also,
in the same article, 6, f were defined to be the constant parts of H, T.
Hence
& + f <) = constant part of 28 j- & -^
But, by definition (Art. 211),
W=- 1 -- + 2- - 1 + ecos f
h h a 1 e
1 hi
i/ 2
since ^ 2 Z = A 2 / and r = r (1 + i/). Therefore
in equation which is true generally.
Neglecting, in the first approximation, the terms which are of the order
of the square of the disturbing forces, this equation gives
const, part of 2v = const, part of ( 8 -7 W J
= - i& - i^of - const, part of W .
190 THE METHOD OF HANSEN. [CHAP. X
As b, f and the constant part of W are already known, we can find C from
this result.
Cor. When z, v have been found, the first approximation to hjh can be most easily
calculated by means of equation (11) which, when r has been put equal to r (1 + v), gives
or,
This will be required in the next approximation to z.
It can also be found by integrating the equation dh/dt
The values of P, Q, K as far as the First Order of the
Disturbing Forces.
235. In Arts. 218, 219, the equations satisfied by P, Q, K have been
given in terms of the disturbing force 3 and of certain known quantities p',
q. Further, in Art. 223, expressions for the partial derivatives of R with
respect to P, Q, K have been found in terms of 3- Substitute the results
(36) in the second terms of the right-hand members of equations (25), (27),
and substitute the result (27') in the third term of (25). The equations for
dP/dt, dQ/dt, dKfdt become
dP , /dR ^dR\ coskJ/dv' do' .
dt \dQ dKJ ' cosi' \dt ' dt
dK
(( r\&P ndq'\ , ( r\dq f
il Q i +p l) cos ' i+ ( Q l- p
sin
dt
where p = - ^r N + N 9 = n Q (a. + rj) t + N K &<,', by (21).
These are the general equations for P, Q, K ; they are given on p. 93 of the Fundamenta
and on p. 117 of the first volume of the Darlegungi. The second of equations (21) shows
that the constant part >//</ of ^' is 7r ' -N Q + K Q ; therefore, if we neglect the periodic part
(which is very small) of K, we have // = n Q (a + 17) t - i/r '. Since n (a + r))t is the mean
motion of the node (Art. 217), this result shows that - // is the mean longitude of the ascend -
* Darlegung, i. pp. 164, 165. See also G. W. Hill, Note on Hansen's general Formula for
Perturbations. Amer. Journ. Math. Vol. iv. pp. 256-259.
t Hansen, in the Darlegung, denotes the angle /*' by 6. The change is made to avoid confusion
with the angle ailj.
234-237] DETERMINATION OF P, Q, K. 191
ing node of the Moon's orbit on the ecliptic. It is to be remembered that a, rj are, by
definition, to be so determined that N, K or that P, K contain no terms proportional to
the time.
236. We can immediately show that the secular motion of the ecliptic will only produce
periodic terms in P, Q. Let this motion be given by
p' = b^t cos i' + const., q' = bt cos i' + const. ,
where b lt b^ are constants supposed known. Substituting in (45), we see that the parts
ddPjdt, d8Q/dt, due to these terms, are periodic. The corresponding terms in d8K/dt, being
multiplied by P or Q and therefore by sin \J^ are much smaller and they may be neglected.
If we put K = A" in the expression for j*', the periods of these terms will be seen to be the
same as that of /*', that is, they will be 27r/7i (a + 17) ; this quantity is the period of revolution
of the Moon's node along the ecliptic * (Art. 217).
Another method of finding the effect of vthe motion of the ecliptic will be given in
Chap. xni.
237. It has been seen, in Art. 219, that when the disturbing forces
are neglected,
P = 0, Q = 2siniJo, K = K Q .
Neglect the motion of the ecliptic, that is, consider p', q' as constants.
Put
P=0+8P, Q = 28inf/o + 8Q, K = K + SK.
In the terms containing the disturbing forces, neglect SP, &Q, SK and
put h = h Q .
Let n B = -h<> Ugl cos 2 %J 9 , w D = Jfi ^j sin JJ" ,
cos- -
, r l
sin iJ ,
where the zero suffix indicates that, after R has been differentiated, the
constant values of P, Q, K are to be substituted.
The equations (45) become
SP = - 2n ffl a sin J/o + 5 , SQ 0Q ,
It is not difficult to see, from Art. 221, that B , D Q will be expansible in series
of cosines, and (7 in series of sines of angles depending on the time ; hence
G will contain no constant term.
Let A be the constant term in (dR/dQ\ : the constant term in fl ^o will
be h Q A Q cos*^J and that in n D will be %h A sin JJ . Hence the constant
terms in dSP/dt, dSK/dt are respectively
2r? cc sin \ J Q h Q A Q cos 2 ^J 0) n rj + %h Q A sin %J .
* Fundamenta, p. 94; Darlegung, i. p. 118.
192 THE METHOD OF HANSEN. [CHAP. X
When the equations for SP, SK are integrated, these will be the terms
multiplying the time.
Now a, TI were introduced so that JV, K should contain no terms pro-
portional to the time ; the condition demands that P, K contain no such
terms. We therefore determine a, rj so that the two expressions written
above are zero. Hence
w- = - i^oA cos 2 J J" /sin JJ" , n rj = - ^h A sin 1J" ;
giving 77 = a tan 2 \J^
These equations determine the first approximations to a, rj. The remark-
able relation between them is modified in the higher approximations.
The integration of the equations (45) will furnish for P, Q values
depending on sines, and for K a value depending on cosines of arguments
of the form ftt + &.
238. On integration we can add arbitrary constants to 8P, 8Q, 8K : these
arbitraries, since the necessary number has been already introduced, may be
determined at will. That added to BK merely adds to K and it may
therefore be put zero ; K K Q is thus expressed as a series of sines. The
constant additive to BP will also be put zero, so that P is expressed as a
series of sines. The constant part of Q was 2 sin |J" , where J" was arbitrary ;
to SQ we add a constant K, so that Q is expressed in the form
2 sin ^JQ + K -f series of cosines.
The constant K is used so that, when the latitude has been found, the
coefficient of the principal term (which has as its argument the distance
of the Moon from the node) is sin J Q this coefficient being determined
directly from observation.
A careful investigation of the meanings to be attached to these constants and to those
denned in Art. 232, and a comparison with the constants used by Delaunay, is given by
Newcoinb, Transformation of ffanseris Lunar Theory*. Another paper, Investigation of
Corrections to ffansen's Tables of the Moon ivith Tables for their Application-^ , by the same
author, may also be consulted with advantage.
To find the next Approximation to n^z^ v, P, Q, K.
239. When the disturbing forces were neglected, we had z = g, v = and P, Q, K
constants. Let 8z, v, SP, 8$, 8K be the parts, just found, depending on the first order of the
disturbing forces. In order to find the next approximation to the values of the variables, it
is necessary to substitute in R and in the various functions used, instead of the initial
values of the variables, their initial values increased by the parts just found. This is easily
done by Taylor's theorem in the following manner.
* Astron. Papers for the Amer. Ephemeris, Vol. i. pp. 57-107.
t Papers published by the Commission on the Transit of Venus, Pt. in. pp. 1-51.
237-240] THE THIRD APPROXIMATION. 193
From the value of 8z we deduce, by putting t=T, that of df. Now the expression (18)
for dW/dt is, owing to the general form in which the disturbing function has been
expressed (Art. 220), a function of z, , i/, P, Q, K. Put n (=*y+n 8, and let W be the
value of IF when d=0. We have
and so for any function containing . Expand the expression (18) for d W/dt, in this way. The
factor of y is quite easy to calculate when =y. After putting y=y Q + 8y (where y is the
first approximation to y), the values of JF , A/A , y^ furnished by the previous approximation,
are inserted ; the value of 8y will be afterwards determined so that no terms proportional
to the time shall be present in W. The only difficulty that remains is the calculation of
the terms containing $, , and this arises from the presence of v. It is to be remem-
bered that when =y we have p=po, =<<)
240. Denote the first two terms of d W'/dt by T, so that
T=h,Zr [2 ? cos (/- ft,) - 1 + 2 ^ ao(1 P i g()2) (cos (/- fc) - 1}]+ 2A *> $r sin (/- </> ).
Put for $, 5r their values 9.S/dr, a.S/dw, and let
where ^ denotes the value of R (Art. 220) when v is put zero. Let
so that RW, RP\ ... , as well as R, are independent of v explicitly.
We have evidently, since r=r(l + j/),
2
' etc ' ;
and therefore, if the values of <?, C/, 2 be inserted in T t we obtain
(46),
where T=G+ U+2 ; the numbers in brackets denote that the term Tfl 1 ) of R is alone con-
sidered. A similar expression may be obtained for TM.
Hence f is a function of nz, P, Q, K, and it is independent of i/, h/h ; if T be its value
when 8z, 8P, 5$, 8^ are put zero, we have, by Taylor's theorem,
In this we substitute_the values of 8z, 8P, dQ, 8K furnished by the previous approxi-
mation. Since, in (46), G, U, 2 are all multiplied by small quantities, we can substitute
elliptic values for the quantities present in the expressions which they denote ; v, h/h
receive the values furnished by the previous approximation. As T is the quantity called
T Q in Art. 226, we already have its value. Hence the terms in (18) can all be expressed
in terms of the time when it is desired to obtain the second approximation to W.
B. L. T. 13
194 THE METHOD OF HANSEN. [CHAP. X
In the same way we can isolate v, h in the equations (45) for P, $, K. For example, if
n Q B be that portion of the first of these equations which depends on R, we have
Put
then 3(i)= JW + 5(i) - 1 + (2y + v 2
lAo ^o
which can be treated as before.
All the equations are finally expressed in terms of the time and integrated, the
determination of 8y, da, dy being made as in the first approximation.
Reduction to the Instantaneous Ecliptic.
241. A final step is necessary to obtain the longitude and the latitude
referred to the ecliptic ; the parallax is found from r as in Art. 162.
Draw ME (fig. 9, Art. 217) perpendicular to X'l. Then
X'H= longitude = v, HM = latitude = u.
Also, from the right-angled triangle MHO,,
sin ( v >Jr') cos Z7 = sin(w ^|r)cos J, sin 7= sin J"sin (v -*fr) ...(47).
From the relations (10), (21),
v -^ =/+ n yt
where we now take w to denote only the mean part of its value given by (29).
Also, cos J= 1 - (P 2 + $ 2 )/2, and
y = -nt(a + i,)t-N + K + ir 9 ' t by (21),
= -n (a
where ^ denotes the mean longitude of the node on the ecliptic. Thus
g + a> + yttj denotes the mean longitude of the Moon on the ecliptic. Since
fg, 8N, &K, P, Q contain only periodic terms which are at least of the first
order of small quantities, we can, after substituting their values, obtain r and
u in the usual form.
Hansen finds u from the equation
sin u= sin J sin (/+ o>) + s,
so that s denotes the perturbations of the sine of the latitude ; / may
be expanded in powers of n<$z as in Art. 232.
For the details of the transformations, the reader is referred to the Darlegung I. 9.
They are also to be found in Tisserand, J/&. C41. Vol. IIL, Arts. 153155 of Chap. xvn. :
the chapter referred to consists of an account of Hansen's Darlegung.
CHAPTER XI.
METHOD WITH RECTANGULAR COORDINATES.
242. THE use of rectangular coordinates is an essential feature of the
latest method for the treatment of the solar inequalities in the Moon's
motion. The equations of motion, referred to axes of which two move in
their own plane with uniform angular velocity while the third is fixed, have
already been investigated in Section (iii) of Chapter 11. ; a plan for the
complete solution of these equations by means of series will now be given.
The method of obtaining it is, to a certain extent, one of continued approxi-
mation. The approximations do not, however, proceed according to powers
of the disturbing forces, that is of m, but according to powers of e, e r , y, I /a'
the constants which are naturally present in the coefficients but which, in
the earlier approximations, do not occur explicitly in the arguments. The
advantage of the method used here is due to the possibility of carrying a
coefficient to any degree of accuracy, as far as m is concerned, without
making the large number of calculations which the methods discussed earlier
would entail ; a reference to Art. 154 will show the importance of this.
The theory is adapted to a complete literal development, but the labour
necessary to secure accurate expressions for the coordinates of the Moon will
be best employed by giving to m its numerical value and by leaving the
other constants arbitrary. The fact that the mean motions of the Sun and
Moon are the two constants which have been obtained by observation with the
greatest degree of accuracy, will justify this abbreviation of the work ; any
alteration in their values which future observations might give, must neces-
sarily be very minute, and its effect can therefore be deduced from the literal
developments of Delaunay.
A difficulty which caused some trouble in de Ponte'coulant's theory does
not arise here. In obtaining the approximations to a given order, it was
frequently necessary to consider terms of orders higher than those actually
required, owing to the presence of small divisors. Since the order of a term,
as far as e, e', y, I/a' are concerned, is never lowered by integration, this
132
196 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
proceeding will be unnecessary here, for the method enables us to include all
powers of m when calculating terms of a given order with respect to the four
constants just mentioned: those terms, whose orders with respect to m are
lowered by the integrations, merely present themselves with larger coefficients
than they might otherwise be expected to have.
243. The introduction of this method is due to Dr G. W. Hill who in 1877 published
two important papers entitled ' Researches in the Lunar Theory*' and * On the Part of the
Motion of the Lunar Perigee which is a function of the Mean Motions of the Sun and
Moonf. These papers, besides throwing a new light on the methods of celestial
mechanics, contain entirely new analytical devices for treating the problem of three
bodies ; M. Poincare", in the preface to the first volume of his Mecanique Celeste, remarks,
'Dans cette O3uvre...il est permis d'apercevoir le germe de la plupart des progres que
la Science a faits depuis.'
The first paper is devoted partly to an examination of the equations most useful
for the actual determination of the Moon's motion and of the limits between which the
radius vector must lie, and partly to the determination of the principal parts of those
inequalities in the motion which have been called, in Art. 166, the Variational Class. In
the second paper, the determination of (1 - c) n, the principal part of the motion of the
perigee, is shown to depend on the computation of an infinite determinant and the
numerical value of this quantity is calculated with a high degree of accuracy.
On the publication of these papers, the late Prof. J. C. Adams gave the results of an
investigation which he had made several years before, in order to find the corresponding
part of the motion of the node|. This likewise depends on the calculation of an infinite
determinant of similar form ; owing, however, to the simplicity of the equation from which
it arises, no transformations, like those necessary in the case of the perigee, are required.
The full details of his investigation have not been published.
The further developments which have been made and which directly concern the lunar
theory, will be referred to in the course of the Chapter.
The limitations imposed on the problem are the same as those adopted
in the methods of de Pontecoulant and Delaunay. The disturbing function
used is that of Art. 8, and the orbit of the Sun is an ellipse situated in the
plane of (XY) with the Earth occupying one focus.
The preliminary Limitations imposed on the Equations of Motion.
The Intermediate Orbit.
244. The general equations of motion (17), (18) of Chap. II., referred
to axes moving with uniform angular velocity n' in the plane of reference,
have undergone certain transformations : the forms to be used are there
* Amer. Journ. Math. Vol. i. pp. 5-26, 129-147, 245-260.
t Cambridge U.S.A. 1877 and Acta Math. Vol. vm. pp. 1-36.
These two papers will be respectively referred to below by the titles ' Researches ' and Motion
of the Perigee ' and by the pages of the journals in which they appeared.
$ See footnote, p. 230.
242-245] DEFINITION OF THE INTERMEDIARY. 197
numbered (23), (19), (18). Since it is not possible to solve these equations
directly, it will be necessary to neglect certain terms which are known
to be small. Connected with this limitation is the choice of an inter-
mediate orbit : the intermediary will not be the ellipse or the modified ellipse
chosen by previous lunar theorists but will be defined to be a certain periodic
solution of the differential equations after some, but not all, of those parts
of them due to the Sun's action have been neglected. It is assumed, as
before, that expansions, in powers and products of the small quantities which
will be initially neglected in the differential equations, are possible.
Let the equations (23), (19), (18) of Chap. II. be limited by neglect-
ing the small quantities e', I/a. Then r=a', rS = X and ( Art. 19) 1 = 0.
Further, neglect the latitude of the Moon, so that z0. The equation
(18) disappears and the equations (23), (19) reduce to
Z) 2 (v<r) - DvDff - 2m
(vDa - <rDv) + f m 2 (v + <r) 2 = 0, )
Dv-2mv(r) + %w*(v*-<r 2 )=0 J ......... ' ''
It is advisable to notice that the equations (1) are equivalent to equations (17) of
Art. 19 with Q = and to no others. The second of equations (1) may be written
{D*v + 2mZ)u + f m 2 (v + <r)}/v = (D*<r - 2mZ><r + f m 2 ( v + a-)} /a- = x ,
suppose. The first of them is then
Whence, by differentiation,
v - f m 2 (v + <r) (D v + ZV) = -
Therefore +|: = o or, Const. = x (uo-)=x^,
A.
which proves the equivalence of the two forms.
The constant K, which has disappeared, must be determined in terms
of the arbitraries by a reference to one of the original equations of motion.
See Art. 21.
In order to see the connection between these equations and the ordinary
forms by which the lunar motion is expressed, reference is made to the
expressions collected in Art. 149. When e'=0, l/a' = 0, z = Q (or, in the
notation of Chap. VII., e' = 0, a/a' = 0, 7 = 0), the coefficients of the remaining
periodic terms depend on m, e only, while the arguments depend only on the
angles 2 = 2 (n ri} t + 2 (e e 7 ), </> = cnt -f e -cr. Hence, the equations (1)
will furnish all inequalities which depend only on m and e and will besides
give the value of c so far as this quantity depends on m, e.
245. The Intermediate Orbit is defined to be the path described by the
Moon when we neglect e', a/a', % e', it therefore consists of those terms
in the expression for the Moon's motion which depend only on m. Now e is
198 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
an arbitrary of the solution of the equations which determine the Moon's
motion, and since the equations (1) determine all inequalities which depend
only on ra, e, the intermediate orbit must be a solution of .(1) not containing
the full number of arbitrary constants.
The inequalities which depend only on ra have been considered in case i.
of Chap. vil. In Chap. vm. (Art. 166) they have been more fully examined ;
it was seen that they correspond to a symmetrical closed curve referred
to axes moving with angular velocity n' about the Earth : this closed curve
was called the Variational curve. Since the coordinates in equations (I)
refer to axes moving in the same manner, the Intermediate Orbit is the
Variational curve. If we look at the constants present in the expressions for
the coordinates of a point on the Variational curve, it is seen that only two
of them are arbitrary, namely, n, e, for a was defined by the equation p = n z a 3 .
Hence the Intermediate Orbit is a periodic solution of (1) involving only two
arbitrary constants. The first object in view is the determination of this
orbit in the most effective and accurate manner.
246. The only small quantity not neglected in the equations (1) is now n' or m.
The reason for retaining this small quantity at the outset is derived from a knowledge
of previous results. When the full literal developments in a theory like that of Delaunay
are examined, it is seen directly that if the series which represents any coefficient converges
slowly, the slow convergence takes place chiefly when the series is arranged according to
powers of m and not when it is arranged according to powers of e, e', a/a' or y. It is
therefore of the greatest importance to find an intermediate orbit in which the coefficients
may be obtained with any accuracy desirable as far as m is concerned.
It might have seemed better to consider the complete solution of the equations (1) as the
intermediate orbit ; this evidently involves a determination of c. Later developments
have shown that there is a saving of labour if we find those terms which depend on m, e'
immediately after obtaining the parts which depend on m only, the next step being the
determination of the inequalities which depend on m, e', e. This plan will not be followed
here, because the developments necessary to show the reason for it are too long to be
inserted in this Chapter (see Art. 288) ; the order of procedure will be the same as that of
Chap. vn.
It is evident that if we neglect m in the equations (1), they must reduce to those for
elliptic motion. Since v<r=r 2 , -(n-riYDvD<r=(vQ\.f, the first of them can, in this case,
be deduced by eliminating p./r between equations (2), (3) of Art. 12, after putting R, z
zero. The second equation expresses the fact that equal areas are described in equal times.
(i) The Terms whose Coefficients depend only on m.
The Determination of the Intermediate Orbit.
247. In order to arrive at the forms of the expressions for v, a- when the
Intermediate Orbit is under consideration, recourse may be had to Art. 166.
It was there seen that when the terms dependent on m are alone retained, r,
245-247] (i) INTERMEDIARY. FORM OF SOLUTION. 199
the radius vector, and v, the longitude measured from a fixed line, are given by
a/r = 26-rf cos 2i, v - n't - e = % + 2&' 2i sin 2if , (i = 0, 1, . . . oo ).
Here f = (n n) t 4- e e' and 6^-; 6'^ depend only on ra. But since X, F are
the coordinates referred to axes, of which that of X points towards the mean
place of the Sun, we have
X = rcos(v-n't-), Y=r sm(v- n't-e) ......... (2).
Hence X, Y are of the forms
X= i 4rfcos(2 + l) Y= I
f=0 t=0
Let A ^ = a (o^ + CL^)* 4'* = a (a* - a.^^),
where i receives negative as well as positive values and where we suppose
a = 1 ; a will, however, be retained for symmetry. We have then
Fv /a* . i v JP 1 V* OO ,. -T 00 1 ............ (2 ).
= v
Whence, since v = X +Y V 1 , a- X F V 1 , the forms to be given to
v (T are
v = aSi^ exp. (2i + 1) f V^ , <r = a^iO^ exp. (- 2i - 1) f V-I.
Two arbitrary constants n, t (Arts. 18, 21) have been introduced into
equations (1). Let n have the same meaning as before, namely, the observed
mean motion of the Moon, and let
then (n-ri)(t-to) = (n-
m = n'j(n n') = m/(l m).
As e, n were arbitraries of the solution previously found, n, t are thus
denned to be the arbitraries of that solution of (1) which is considered here.
Finally, since f = exp. (n n) (t ) V 1, (Art. 18), the assumed values
for v, a- may, after putting i 1 for i in the expression for a, be written*
v = a2^{^ 1 , <r = a^a_ 2i _ 2 ^ 1 .................. (3);
or, i/^aSiCfcf* , ^aSiflLrff* .................. (3');
where i = oo . . . + oo .
It is now only a question of so determining the coefficients o^ that these
values for v, a- may satisfy the equations (1).
* In the Researches p. 130, a< takes the place of the coefficient here called aa*. The change
is convenient, firstly, because the chief consideration is the determination of ai/a and secondly,
because it will not now be necessary to introduce another letter for the coefficients of the
parallactic terms (Art. 277).
200 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
248. It must be remembered that the object in view is the determination of the
coefficients a 2i . They have been obtained to the order m* in Chap, vii., but no general
law was forthcoming by which they could be found to any degree of accuracy without
great labour. We have only made use of the results of that Chapter in order to discover
the form of the required solution of equations (1). From the point of view of the lunar
theory, it is very important that the connections between the results of the various theories
should be clearly set forth ; for this reason it has seemed preferable to deduce the forms of
the solutions from our previous knowledge rather than to obtain them by a general consi-
deration of the differential equations.
The latter method has been followed by Poincare in his Mecanique Celeste. In
Chap, in., Vol. i. of his treatise, he has considered the conditions under which the equations
of dynamics admit of periodic solutions. In 39 41, he gives some applications of his
results to the Problem of Three Bodies and, as a special case, he proves that the equations
(1) or rather the equations (25) of Art. 23 above, from which the former were deduced,
do admit of periodic solutions in general. Moreover, in the same Chapter, he shows
that these solutions, for sufficiently small values of the parameter, are in general
developable in series.
The Determination of the Coefficients of the Variational Inequalities.
249. Let j be an integer with the same range of values as i t namely,
from + 00 to oo . Since Df* = if*, we have, from the equations (3),
Dv = a2; (2* + 1) flfcf 1 * 1 , Da = aS; (2* + 1)
vo- = a 2
A.SJ and i extend independently from + 00 to oo, we may put j i I for j.
The expression for vo- then becomes
Similarly,
Dv.Do- = ~ a'SjSi (2
vDo- - a-Dv = - &%?,i (4i - 2j + 2) a
When these results have been substituted in equations (1), the coefficients
of the several powers of f must be equated to zero in order that the values
assumed for v, o- may satisfy the equations. The coefficients of f 2 -> give
[4j 2 + (2t + 1) (2 - 2j + 1) + 4 (2i - j + 1) m + f m
except for the value j = 0, when the second equation is identically satisfied
and the right hand side of the first equation is (7/a 2 .
Multiply the second of these equations by (2m + l)/2j and subtract from
248-251] (i) EQUATIONS OF CONDITION FOR THE COEFFICIENTS. 201
the first ; also, divide the second equation by 4y. Excluding the value j = 0,
the results are
0=2if4#-l-2m
- 1H S.-flu-tf,,.- - 4- -Urn 2 4^2- f2m 4- 1 \l^o^A _. .. .
-(4').
2 + J - 2 (2m + 1)1 2ia 2i a stf _ l _, + jf m 2 - J ^ (2m + l)\2i
= 2* (1 - j + m + 2t)
" 2J 21 2>.
v. ^ ;
m 2
J
For the case j = we have, from the first of equations (4),
an equation which serves to determine C when a, a^ have been found.
250. As =1, these equations show that the coefficients 2i are functions of m only.
Also, it is not difficult to see that # will be of the order m l2il *. For example, put,/ = 1 in
equations (4'), and write down a few terms given by small values of i. We obtain
= (3 - 2m+im 2 ) (a 2 a +a B a_ 2 )+(ll - 2m +m 2 )
...) + ^
- f m 2 (a 2 -|- 2a 2 a_ 2 + 2a 4 a_ 4 + . . . - a_ 2 2 - 2a a_ 4 -
If we suppose a 4 , a_ 4 of higher order than 2 , #_ 2 , these equations show that a 2 , a_ 2
are of the order m 2 at least. We may similarly treat the equations for j=% and deduce
the fact that a 4 , a_ 4 are of the order m 4 , and so on. It will appear presently that, in
finding the 2 i from equations (4'), m cannot be a factor of any of the denominators. We
see further that the equations always occur in pairs and that those of principal im-
portance in finding a^-, a_ 2J -, are obtained by equating the coefficients of f 2 '' to zero.
251. It is now necessary to show how the coefficients may be most
suitably obtained from equations (4 7 ). Owing to the fact that a$ is of the
order | 2j \ at least, they may be found by continued approximation ; but in
order to explain how the approximations are carried out, some further remarks
must be made. Suppose that it be desired to determine a 2 j correctly to the
order | 2j , the values of the coefficients, for values of j numerically smaller
than that considered, being supposed known. The equations (4') show that
we shall have two simultaneous linear equations for o^a 0) ct- 2 jo from which
these quantities may be determined ; the same is true when we desire to
obtain them to any degree of approximation whatever. When we use the
numerical value of m at the outset, these are the simplest equations to deal
with ; but when a literal development in powers of m is required, the labour
will be lessened if, before commencing the calculations, we deduce from the
two equations giving a<>j, a_ 2 j two new equations in which the coefficients of
a_ 2 jfi , a 2 ja are respectively zero.
* If a be any real quantity, | a | denotes that a is to be taken positively whichever sign it may
have.
202 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
The terms containing a 2 ja , a_ 2 ja i n equations (4') are obtained by putting
i = j } i = 0. They are, in the respective equations,
(4j 2 - 1 - 2m + Jm 2
(1 +j + m) a a 2 j + (1 - j + m) a a_ 2j .
If, therefore, we multiply the two equations by
j-l-m, 4j 2 -l -2m + Jm 2 ..................... (6),
respectively, and add the results, the coefficient of a a._ 2 j will be zero and that
of a dtj will become
l-2m + im 2 ) ........................... (6').
Further, if we divide the resulting equation by this expression, the
coefficient of a a,y will be 1.
Multiply then the equations (4') by the two expressions (6), respectively,
add them and divide the resulting equation by (6'). Let it be written
+ [2JJ a 2i a 2j _ 2i _ 2 + (2j,) a 2f a_ 2j _^_ 2 } =0 ...... (7).
The coefficients [2j, 2i], [2j,], (2j,)* will be.
[2j,2i] =
((4j 2 -l-2m+im 2 ) + 4iX^-j)l(j-l-m)+(4/ 2 -l-2m+^m 2 ){2t-( j-l-m)} ^
- j (8j 2 - 2 - 4m + m 2 )
i 8j 2 - 2 - 4m + m 2 + 4 (i-j)(j- 1 -m)
j 8j 2 - 2 - 4m + m 2 '
f . 1 ^ifjm 2 + fm 2 (2m + l)}Q'-l-m)-jm 2 (4; 2 -l
-j 2 (8j 2 -2-4m+m 2 )
s m 3 4j 2 - 8j- 2 - 4(j + 2) m - 9m 2
TBJ^-- 8j 2 -2-4m + m 2
. , j j ;m 2 - f m 2 (2m + 1)] ( j - 1 - m) + m 2 (4j 2 - 1 - 2m +jm 2 )
_j 2 (8j 2 _2-4m + m 2 )
_ 3 m 2 20j 2 -16j + 2 - 4 (5j - 2) m + 9m 2
-
......... (70-
It is evident that [2j, 0] = 0, [2j, 2j] = -l, so that the coefficients of
o2j are respectively 0, 1.
Equation (7) will also serve for the determination of a_ 2 j. For j may
receive both positive and negative values and, if j is put for j, the coeffi-
* In the Researches, p. 134, 135, these are denoted by [j, t], [j], (j) respectively. The change
facilitates the determination of the terms dependent on e, e', etc.
251-253] (i) SOLUTION OF THE EQUATIONS OF CONDITION. 203
cient of a a 2j becomes [ 2j, 0] which is zero, while that of a a_ 2 j is
[2;, 2j] which is equal to 1. The value ,;'=0 is excluded as before.
252. The denominator (6') has its least value when jl and it cannot then vanish
for any real value of m. Also, since 8/ 2 - 2 is never zero, m cannot appear as a divisor : the
statement made in Art. 250, with reference to the order of #, will therefore be true for all
these coefficients.
It will be noticed that we have obtained one equation (7) instead of the two (4') : this
is a necessary consequence of the use of imaginary variables. With such variables it is
possible in general to replace two equations by one, as was seen in Art. 18, where either of
the two equations (14) of that article was a complete substitute for the first two of those
there numbered (12). The equations (4) of Art. 249 do not possess this property because
they arise from the equations (1) which respectively contain real terms and imaginary
terms only (see Art. 21). When, however, the equations (4) are combined in a suitable
way, one equation can be made to replace the two. In either of the equations (4), positive
values of j are sufficient; this can be seen by putting j for j and then changing i into
i-j in the first summations: after the two changes the equations become the same as
before.
253. The determination of the coefficients a. 2 i from the equations (7).
In order to show the method of finding the coefficients a^ from these
equations, the latter will be written out in full with a number of terms
sufficient to find a 2 , a_2, ci 6 , a_ 6 to the ninth order inclusive and a 4> a_ 4 to
the seventh order inclusive. In writing them down, it is only necessary to
recall that [2j,], (2j,) are of the second order and that a zj is of the order 2j \.
In the parts multiplied by [2j,], (2;,), a term a^ a^ occurs twice when i t i' are
different. We find
a a 2 = [2, - 2] a_ 2 a_ 4 + [2, 4] a 4 a 2 + [2J (a 2 + 2a_ 2 a 2 ) + (2,) (a
a a- 2 =[-2, - 4] a_4a_ 2 + [-2, 2]a 2 a 4 +[-2,](a_ 2 2 + 2a _,)+(- 2,) (a 2 + 2a_ 2 a 2 );
a a 4 =[4, 2] a z a. 2 + [4,] (2ooO),
a a_ 4 =[- 4, - 2] a_ 2 a 2 + (- 4,) (2a a 2 );
a a 6 =[6, 2] a z a^ + [6, 4] a 4 a_ 2 + [6,] (a 2 2 + 2a a 4 ),
a a_ 6 =[- 6, - 4] a_ 4 a 2 + [- 6, - 2] a_ 2 a 4 + (- 6,) (a 2 3 + 2a a 4 ).
These are easily solved by continued approximation. Since a = l, we
have, neglecting terms of the sixth and higher orders,
a 2 = [2,], a_ 2 = (-2,), torn 5 .
Whence, neglecting terms of the eighth and higher orders,
a. = [4, 2][2,](- 2,) + 2 [4,][2 J, a_ 4 = [- 4, - 2](- 2,)[2,] + 2 (- 4,)[2J, to m'.
Similarly, using these results, we can obtain a 6 , CL^ correctly to the order m 9 .
204 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
To obtain a 2 , &_ 2 correctly to the order m 9 , it is sufficient to add to the
values of a 2 , a_ 2 just found, the terms previously neglected; these latter are
obtained to the required accuracy if we calculate them with the values of
a 2 , CL_2, a 4 , a_ 4 previously found. For a_ 2 , GL_ 4 , which are of the second and
fourth orders, were respectively obtained correctly to the fifth and seventh
orders : their product is therefore correct to the ninth order ; the same pro-
perty holds for the other terms. Should it be desired to obtain their values
still more approximately, a similar process will serve. We proceed by con-
tinued approximation, using the results of the previous approximation for the
calculation of the terms previously neglected.
If these results be utilised to obtain a numerical development, we can,
at the outset, calculate the coefficients [2;, 2i], [2^'J, (2j,) for all values of j, i
required ; the continued approximations are then very simple. For a literal
development in powers of m, these coefficients may be first expanded in
powers of m to the degree of accuracy ultimately demanded; the approxi-
mations then only entail multiplications of series of such powers.
254. The rapidity with which the values approximate may be grasped from the fact that
each new approximation carries the value of the coefficient under consideration four orders
higher. Also, for each such increase of accuracy, it is only necessary to add four new
terms to the equation for that coefficient, two with factors of the form [2;, 2i], one with
the factor 2 [2/J and one with the factor 2(2;',). The advantage of the method, when
we compare it with a laborious one like that of de Pontecoulant, is very striking. Dr Hill
has calculated the values of 2 i literally* to the order m 9 and numerically f to fifteen places
of decimals. To show the rapidity with which they approximate in the latter case, the value
of a_ 2 the largest coefficient is given below.
He takes '/(w-w') = m= -08084 89338 08312 and finds
1st Approx. = - -00869 58084 99634,
additional terms in the 2nd = -f -00000 00615 51932,
3rd = - -00000 00000 13838,
resulting value of a_ 2 = - -00~869 57469 615407
When literal developments in powers of m are considered, the convergency of the series
obtained depends to a large extent on the system of divisors 2(4/ 2 -l)-4m-{-m 2 . These
divisors increase very rapidly with j. The smallest of them, given by j=l, is
6 4m-t-m 2 ; slow convergence would then chiefly arise from this divisor. Dr Hill inquires
what function of m, of the form m = m/(l-}-ani), will make the expansion, in powers of m,
of the inverse of this divisor converge most quickly. It is easily found that the necessary
value for a is ^; the divisor 6 4m + m 2 gives rise to the divisor 6-t-^in 2 , and new divisors,
which are powers of 1 + Jm, are also introduced. The expansion of the inverse of each of
these converges quickly.
It may be remarked that it is not necessary to repeat the whole series of approximations
in order to have the results expressed in terms of in. In the results expressed in terms of
m, we simply put m = in/(l -m) and then expand in powers of in, stopping at that power
* Researches, pp. 142, 143. t Id. pp. 247, 248.
253-255] (i) DETERMINATION OF THE LINEAR CONSTANT. 205
of m to which the expansions in m were carried. Any of the results, whether they
have been expressed in polar or rectangular coordinates, may be treated thus.
One of the most interesting parts of the Researches (pp. 250 260) is that which
contains the investigation of the forms of the variations! curves for increasing values of m.
When m is much greater than J, it is found to be no longer possible to use the expansion
in powers of m and mechanical quadratures must be employed. See Art. 166 above.
255. Determination of a.
Since K (or p) has disappeared from the equations used above, there must
be relations connecting K, p. with n, a. It will now be shown that these
relations are of the forms, K = a 3 (1 4- powers of in), /* = n 2 a 3 (1 + powers of m).
The first of equations (17) of Art. 19, with H = 0, will be used. This equation
may be written
JftJ
^jy f = (&+ 2mD + f m 2 ) v + f m 2 <r.
Substitute the values (3) of v, <r. Since the result must hold for all
values of f, we can, after the substitution, put f = 1. We have then
v = a- = aS^-, Dv = aSi (2i + 1) a*, D z v = aS* (2i + 1) 2 a* ;
and the result is
*a- 3 (2 t -ari)- 2 = 2i {(2t + 1+ m) 2 + 2m 2 } a*.
But K = p/(n - n') 2 = fji (1 + m) 2 /n 2 , by Art. 19. Hence
a = (l + m)l [2i {(2* + 1 + m) 2 + 2m 2 } a*]-* P^]"* . . . (8).
When the values of the coefficients a^ have been inserted, a will be ex-
pressed in the required form. The quantity p/n* is that usually called
a 3 in the lunar theory. Hence a/a differs from unity by powers of m only,
and it is equal to unity when m = 0.
Since the parallax is observed directly, it will not be generally necessary
to make the transformation from a to a in the coefficients ; if we desire, for
the sake of comparison, to have the results expressed in terms of a, the
transformation can always be delayed till the end of the investigation.
The value of a may be also found by substituting for v, o- as before and equating the
coefficients of x to zero. The results, obtained by this method and by that just given in
detail, will be entirely different in form and their agreement will therefore constitute a
useful verification.
The value of C may be found (if it be required) from (5). Another method for obtaining
it, is given by substituting the values of v, cr in the first of equations (1) and putting f = 1
in the result. For this we have
The two results for (7, being also of different forms, furnish another means of verification.
206 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
256. Transformation to real rectangular and to polar coordinates.
The coefficients aa 2 i having been found, the coordinates may be expressed
in the usual manner by the formulae,
r cos (v nt e) = r cos (v n't f -j;)= -J (v~ l + 0-f),
r sin (v nt e) = r sin (v n't e f ) = J (v~ l <r) V 1.
Hence equations (3') give
r cos (v - nt - e) = a [1 + (a 2 + - 2 ) cos 2f + (a 4 + ^-4) cos 4f +...],
r sin (v ft - e) = a [ (a 2 - a_ 2 ) sin 2f + (a 4 a_ 4 ) sin 4f +...].
If it is desired to express these in polar coordinates, we have
v nt e = tan (v nt e) - Jtan 3 (v nt e) + ....
As tan (v nt e) is a small quantity of the second order at least, the
terms in the right-hand member of this equation can be calculated from
the above values of r cos (v nt - e), r sin (v nt e), by expansion. Also, since
an equation for C has been given, the parallax 1/r may be found from either
of the equations (20), (22) of Art. 20, after we have put z and fl zero therein.
When the numerical value of m has been used, it is simplest to make the
transformations for the true longitude by the method of special values.
(ii) The Terms whose Coefficients depend only on m, e.
The General Solution of Equations (1).
257. We shall deduce the form of the general solution of equations (1)
in the manner of Art. 247. From the results of Chaps. VL, VII., it is evident
that, when all the terms whose coefficients depend only on m, e are considered,
we have
cos t + v-t-6 = + sn
where i ranges from oo to -f oo and p from to oo ; the coefficients depend
only on m, e and b 2 i +pc , b'^ +pc are of the order e p at least.
Hence X, Y will be of the forms
X = ^ZpAx+pc cos {(2i + 1) f +p4>}, )
F=a2 i VlWcSin ((2*'+ l)+jN>) )"
where Ayi +pc , A'^+pc depend only on m, e and are of the order e p at least.
If we allow p to receive negative as well as positive values, the accent of
in Y may be omitted.
256-257] (ii) ELLIPTIC INEQUALITIES. FORM OF SOLUTION. 207
From these values of X, Y, the corresponding expressions for v, <r may
be deduced. After putting - i - 1 for i, and p for p, in the expression for <r,
we obtain
_ ..._ ). no).
V- 1,
Let en = c (n ri\ TZ e = eh (n ri) ;
so that ti replaces the arbitrary OT. We have then
<f) = cut -f e IB = c (n n') (t j), c = c (1 + m).
If we put exp. (n n') (t t^ V^T = ,
we have exp. {(2t -f 1) f + p<f>} V^l = ?* +1 ?/.
Now in order that the values (10) may satisfy the equations (1), it
is necessary to substitute them in the latter and to equate to zero the
coefficients of the various powers of f, . In the process of doing so, it will
be necessary to use the differentials,
Since the value of c will not be substituted in the index of , if we put
= (which corresponds to making t ^ = t ) the equations of condition
will be perfectly general; the only point to remember is that, when we
return to real coordinates, the part of the index of f which contains c
corresponds to the argument c (n n') (t ti). The equations (10) may
therefore be written
or, in the more symmetrical forms,
Substitute the values (11) in (1). For this purpose we have, as in
Art. 249,
Dv = aSi^p (2i + 1 + pc) Ati+pcg** 1 ***, etc. ;
where j t q have the same range of values as i, p, namely, from +00 to oo .
Equating the coefficients of f z J +qc to zero, we obtain
-2/-gc + 2)m + fm 2 ]
_u>4. A "I ft >CI9\
_2-fgc pc "t" a -3l+pc- a - ty 21 2 qc~ PCJ | ^ /'
2_L 9rr\ A A
~P 111 f xlgi+nc^J-zi ZJ-i-pC <JC
=
except for j = = q, when the right hand member of the first equation is (7/a*.
208 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
258. On comparing these equations with (4), we see that the latter would
be the same as the former if the symbols
were respectively replaced by
A, 2
In making the corresponding transformations, we can therefore use the results
previously obtained. Indeed this fact was evident as soon as we had arrived
at the equations (11).
To get the equations corresponding to (4'), multiply the second of equa-
tions (12) by (2m + l)/(2? + qc) and subtract it from the first. The second
equation is to be divided by 2 (2j + qc). It is not necessary to write down
the results, for they are immediately deducible from (4').
A pair of corresponding coefficients will be A^+pc, A^i-pc- H i n order
to isolate A^ +pc , we make the transformations of Art. 251 after putting
2j + qG, Zi +pc for 2^', 2t respectively, we arrive at the equation
=
+ [2j + qc,] Azi +pc A 2j -^+ qc - pc + (2j + qc,) A^^A-^.^^} . . .(13),
the value j = = q being excluded. The coefficients in this equation are
the values of (7') after the changes noted have been made. We evidently
have
[2J + 2C, 2/ + gc]=-l, [2j + 0c, 0] = 0;
so that the coefficient of the term A A^ +pc is 1 and that of A A_^^pQ is 0.
The equations for C and a can be obtained in a similar manner.
The equations of condition (13) are to be solved by continued approxima-
tion, but since there is a double sign of summation involved, they are by no
means so simple as those of Art. 251. We know that the term A^ +pc is
of order e\ p \ at least; the simplest method therefore appears to consist in
finding initially those terms which depend on the first power of e only,
neglecting all higher powers of e, thus restricting q to the values + 1.
If we neglect all powers of e and put A =l, the equations will reduce to (7). We have
then
It must be remembered that when powers of e are not neglected, A# is no longer equal
to a# but differs from it by terms of the order e 2 at least. The A# are the coefficients of
the variational terms when powers of e are not neglected, while the a^ are the coefficients
of the same terms when the parts dependent on e 2 , e 4 ... are neglected.
258-260] (ii) ELLIPTIC INEQUALITIES OF THE FIRST ORDER. 209
The Terms dependent on m and on the First Power of e.
259. As powers of e above the first will be neglected and as the coeffi-
cients of the variational terms contain only even powers of e, we put
so that e t -, e/ are of the form ef(m).
In the equations of condition (13), q takes the values -f 1, 1 only.
When q = 4 1, p has the values 1, in the first two terms, and the values 0,
1 in the third term ; any other values of p will give terms of the orders
e 3 , e 5 ,.... Similarly, when q = 1, we must give to p the values 1, in
the first two terms and the values 0, 1 in the third term. The equations
for obtaining e;, e/ are therefore
= 2< ([2j + c, 2* 4- c] eid^j + [2j 4- c, 2
+ [2j + c,] (eiOj-^ 4 a^j-i^) + (2j + c,) (a tt ^ j ^ r . l + ^CL
= 2; [[2j - c, 2i - c] c'iflto-tf 4- [2j - c, 2
+ [%' - c >] (e> 2 j-2i- 2 4- a^-i-!) 4- (2j - c,) (a^e-j^.^ 4- e
Since ^ receives positive and negative values, the equations of condition
may be put into a more symmetrical form by considering those for j, e'_j
which form a ' pair/ Also, since each term is summed for all values of i, we
can, in any term, put i integer for i. In the second term of the first equa-
tion put i+j for i ; in the second part of the third term, j iI for i ;
and so on. With like changes in the second equation after j has been
put for j, the equations of condition may be written,
(14).
260. There are no other relations satisfied by the unknown coefficients ej,
e_j ( j = 4- x ... - oo ), and therefore the equations (14) only suffice to deter-
mine the ratios of the unknowns to one of them, say to e or e r . Moreover,
since the coefficients, as well as the equations (14), always occur in pairs, some
relation must exist between the equations, and it is necessary to so determine
the unknown constant c that the relation may be satisfied. When c has been
found, the ratios of the unknowns e,-, '_,- may be calculated by the ordinary
methods of approximation. Hence one of them is an arbitrary constant, and
this corresponds to the arbitrary constant which, in other theories, is denoted
by e.
B. L. T. 1*
c, 2i 4- c] CiOrf-tf + [2j 4- c, 2j - 2t]
+ 2 [2jf 4- cj e^-.^ 4- 2 (2j -f c,) e'-^-.y-.,} = 0,
- c, - 2i - c] e'^o^i + [- 2j - c, 2t - 2j] c^o,^
+ 2 [- 2j - cj c'-iOji-^-, + 2 (- 2j - c,) e^a^^} =
210 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
It is not difficult to see that the unknowns *,-, e'_y, like the coefficients a 2y , are, in
general, in descending order of magnitude with ascending values of |/|. Assuming this
fact, suppose that we neglect all powers of m above the second and also the unknowns beyond
those given by the values ./=+!; we shall have six equations to find the ratios of e , e ',
e_ 1} e/, *j, *'_!, namely, those given by j=0, 1, and the relation determining c will be
expressible by a determinant with six rows. If we include the values for which
j= + 2, the determinant will have 10 rows, and so on. Finally, including all the equa-
tions, the determinant will have an infinite number of rows and columns the infinite
number being of the form 4j + 2.
It is possible to approximate to c while the approximations to ,-, '_,- are being carried
out, but this would be very troublesome owing to the way in which this constant is involved
in the coefficients [2/-f c, 2i], etc. Or it might be found by calculating the determinant
after neglecting the smaller terms ; this again would be very laborious owing to the large
number of rows necessary to secure the accuracy required at the present day. In his
paper 'The Motion of the Perigee'*, Dr Hill has shown that c may be made to depend on
a symmetrical infinite determinant in which the number of rows or columns is of the
form 2/ + 1, and further, he has succeeded in representing the value of this determinant
by means of a series whose terms diminish with great rapidity. See Arts. 262 et seq.
The homogeneous form of the equations (14) with respect to the unknowns, shows that
the value of c, when powers of e above the first are neglected, is independent of the
particular definition to be assigned to the eccentricity constant. The remark is made
because, when finding c by de Pontecoulant's method (Art. 139), it appeared to be involved
with the definition there given to e. We also notice that, when powers of e above the
second are neglected, c will be a function of m only a fact which renders the previous
remark obvious from another point of view.
In working out a complete theory by this method, it will be found more convenient to
put A zi + c =f 2 i, A 2i _ e = f 2i , in order that the introduction of a new symbol for the coefficients
of the terms dependent on e and on powers of a/a' may be avoided. These coefficients are
of the form J 2i + 1 + c , A 2i+l _ c .
261. Suppose that c has been obtained in some way, accurately to the
order ultimately required, either as a series in powers of m or numerically.
The coefficients in the equations (14) can then be found and the values of the
unknowns calculated.
When the numerical value of m is used at the outset, the best method of
dealing with these equations is to neglect, in the first instance, the two equa-
tions given by j 0. The equations given by j = 1, + 2, . . . will then furnish
the rest of the unknowns in terms of e , e ', by the usual methods of approxi-
mation. Substituting these in the equations given by the value j = 0, we
shall have two equations to find the ratio 60/60', and if the value of c,
previously obtained, is correct, these should agree. A formula of verification
is thus available. The arbitrary constant may be taken to be e, where
e = e - 6 '.
* See footnote, p. 196.
260-263] (ii) THE MOTION OF THE PERIGEE. 211
It is found that e differs very little from the constant e used by Delaunay
and defined in Arts. 159, 200 above.
The method outlined above will be found more fully developed in Parts I. IV. of a
paper by the author, The Elliptic Inequalities in the Lunar Theory*. The notation used is
slightly different: i and j,p and q are interchanged; for c, m are put c, m; instead of e
the symbol Y is used; the other differences will be obvious. The values of a^, c, as
obtained by Hill, are used, and the results for y/e, e'_//e are computed numerically to 10
places of decimals. The ratio of e to e is found by transforming to polar coordinates and
comparing the resulting coefficient of the principal elliptic inequality in longitude with
that given by purely elliptic motion.
The Determination of the Part of c which depends only on m.
262. The problem to be considered here is the discovery of an equation
which will give c : we are not concerned with the unknown coefficients e t -,
e/. Now a transformation of coordinates will not affect the periods of the
various terms and we are therefore at liberty to choose those coordinates
which will put the problem into the simplest form. It was noticed that, when
the method of Art. 259 was followed, the determinant giving c was of infinite
order and that the number of rows or columns was of the form 4j + 2. It
will be shown here that c may be made to depend on an infinite determinant
with only half that number of rows and columns ; in other words, only half
the number of rows or columns are necessary for a given degree of accuracy.
When, however, the determinant has been found, instead of limiting the
number of rows, we shall show how it may be expanded generally as a series,
and we shall then see to what degree of approximation the series must
be taken in order to secure a given accuracy in the results.
The results of Arts. 244-246 show that the problem may be stated as
follows : Given a periodic solution of a pair of differential equations, to find
the periods of a solution differing but little from the given solution.
263. The equations (1) being equivalent to (25) of Art. 23, let the latter
be written
X _ 2n'F= dF'/dX, Y+ 2n'X = dF'/dT, )
where J? 2 + 7* = F 2 = 2F' + const. = 2/*/r + 9n'*X* + const. J ' '
F' is then independent of t explicitly, and V is the velocity with respect to
ihe moving axes of X, Y.
Let i/r be the angle which the tangent at (X, Y) makes with the X-axis,
and put
* Amer. Journ. Math. Vol. xv. pp. 244-263, 321-338.
142
212 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
then dF'fiT, dF'/dN represent the resolved parts of the forces corresponding
to the force-function F', along the tangent and the normal respectively*.
Since X = Fcos^Jr, F= Fsin>/r, we deduce
f
+sm
__ _ _ a zaF
djr af
dt dT
where the meanings to be attached to d*F'/dN 2 , d*F'/dNdT are evident.
Also, since tan->|r= Y/X and therefore d^rjdt = (YX XY)/V 2 , we obtain
easily from (15), (16),
dF' 1 dF'
dt~ dT V dt '
<">
Let BX, BY be any small variations of X, F, which are such that X + SX,
F+SF, as well as X, F, satisfy the differential equations, and let &T, SN
be the corresponding variations along the tangent and normal to the orbit of
X, F. Neglecting squares of &X, B Y, we have
S^cos-^SF-sin f BX ...(18);
and the equations (16) show that
' ' '""'
All the equations may therefore be submitted to a variation 8. We
have, from (15),
~ BX - 2ri -BY- n'^X = JL*F'
dt 2 dt oX
, 2 ,
- SF+ 2ri ^ BX - ri*BY= ~ BF' -
dt* dt oY
Now the left-hand members of these equations are the accelerations of a
point referred to axes moving with angular velocity ri. Take the sum of
these equations multiplied by cosi/r, sin^, respectively, and also their
sum multiplied by sin >/r, cos ->|r, respectively : the results will be the
accelerations referred to the tangent and normal of the original orbit.
The latter rotate with angular velocity ri + d-^/dt. Since the coordinates,
* It is necessary to define the partial differentials in this way hecause F' is not expressible in
terms of T, N only ; they have the same meaning as if F' were so expressible,
263] (ii) EQUATION FOR THE NORMAL DISPLACEMENT. 213
referred to these axes, are ST, &N, we obtain, by the well-known formula for
the acceleration in the direction of the normal*, and by (16), (18),
or, < -J-; -
I dt^
Now
By means of this result, equation (20) becomes
Also, since SF is the variation of the velocity, relative to the moving
axes, in the direction of the tangent, we have
= di ~~~di
But, by the third of equations (15) and by (17),
fif" df" /d^fr
" dN dT ~ \ dt
Eliminating BV, we find
&T
dt J dt
The equation (20') therefore becomes
where (n - nj 6 = 3 ( ^ + n' } +M /2 -^T 9 (22).
This method of arriviDg at the equation (21) is a modification of one given by Prof.
Adams in his lectures on the Lunar Theory in the academic year 1885-6. A similar
investigation was given independently by Prof. G. H. Darwin in his lectures of 1893-4.
The method used by Dr Hill demands several transformations t ; the variable there called
w is equal to 2&^V 1.
* Tait and Steele, Dynamics of a Particle, Art. 42.
t Motion of the Perigee, pp. 6-9.
214 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
264. Form of the Solution of the Equation for BN.
It will be convenient to express (B) in terms of the rectangular coordinates.
For this purpose we have, as before,
(fy YX-XY X Y
~jt= y* -- > cos^ = T , sm^ = ^.
Whence, substituting for d*F'/dN 2 from (16'), we find
......... (220,
where F 2 = Z 2 + F 2 , F' = p (X 2 + F 8 )-* + \n'*X\
Since the quantities present in % refer to the given periodic solution
(which is the intermediate orbit), it is evident from equations (2') that it is
expressible by cosines of multiples of 2 = 2 (n n) (t 1 ). We can therefore
express in the form
+ 2! cos 2f + 26 2 cos 4f + . . . ,
where , i,... are functions of m only. If -i = i, equation (21) becomes
(23).
This equation is of the form referred to in Art. 146, and it admits of a
solution of a similar nature. In order to retain the connection with the
previous methods, the form of the solution will be deduced from our previous
knowledge of the forms of 8X, SF.
We have, from (18),
- sin ^SX = (XBY- Y8X)/V.
Since X, Y are respectively expressible by means of sines and cosines of odd
multiples of f, and SF, 8X, from (9), by sines and cosines of the angles
(2i + l)f0, the function X8YYSX will be expressible by means of
cosines of the angles 2if + <; also F and 1/F are expressible by means of
cosines of the angles 2if . Hence SN will be expressible by means of cosines
of angles of the form
2i (n - n') (t -t )c(n- n) (t - t,)
= (2t c) (n - n) (t -Qc (n - n'} (t, - t,).
Introducing the operator D, defined as before by means of the relation
D' 2 = d 2 j(n n'}* dt 2 , equation (23) becomes
*)W ........................ (23');
264-265] (ii) FORM OF SOLUTION OF THE EQUATION. 215
where i = + x> ... x, and _i = ;. The general solution, according to
the remarks just made, will be of the form
SN = (2i\>* +c ) exp. c (w - n') (k- ,)
+ (Sib/P*-*) exp. - c (n - n 1 ) (t - O V^l.
On substituting this expression in (23') and equating the coefficients of
the several powers of f to zero, we see that, since i receives negative as well
as positive values, the equations of condition for the coefficients b are the
same as those for the coefficients b'_i. It is therefore only necessary for the
determination of c to consider the integral
(24).
This result, which has been deduced from our previous knowledge of the
form of the solution, is a well known property of equations of the form (23').
All that now remains is the substitution of (24) in (23'), and the deduction
of c from the equations of condition obtained by equating the coefficients of
the various powers of f to zero.
265. In order to bring 8 into a form suitable for calculation, it will be better to
express (22) in terms of the complex variables v, <r. We have
V*=-(n
and
therefore, from (22),
, D<r
where G
the transformation used in Arts. 18, 19.
But, from Art. 18,
9On ^
Z) 2 v + 2m Dv = - a- ,
therefore -75- 1- -^r = ~ TT ^ TT ~^~
JJv L/<T -Ua- vv Ln) Go-
Hence
The last two terms, by the equations of motion, are equal to
2
and, since r 2 = uo-, n-^p = i ^ + |m 2 ;
216 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
. Dv d 2 G , Dv
hence A - ? +--
and therefore
The quantities present in this equation have to be expanded in powers of . Assume
/>V^v = Si^if ai , K/^+m^aSiJ/i^i ........................ (25) ;
then Z) 2 ff/Z>o-=-2 i f7 i t~ 2i , M^^M^
When C/i, J/^ have been found, the calculation of requires only two operations of
any length, namely, the squaring of two series.
We have, after the substitution of the values (3) of v, <r,
2, (2e + 1) 2 a 2i ^ 2i = [2i (2t + 1) a 2i {**] [2 f U 2i ] ;
from which, by equating coefficients, the coefficients U t can be obtained.
The coefficients Mi can be found by treating, in the same way, the equation
mV + m 2 v
When the numerical value of m is used, it is easier to calculate the coefficients U^ M
by the method of special values, after eliminating the second differentials by means of the
equations of motion. It is evident that 6* will be of the order m' 2il at least.
The Equation for c.
266. When the value (24) is substituted in (23'), we have, on
equating the coefficient of f^' +c to zero,
(c -f 2j) 2 b,. - 2*0^ = 0, (i = + oo ... - oo );
or, since _i = ;,
... - ^ - 0^ + {(c
by giving to j the values 0, 1, 2,. . ., we obtain a set of linear homogeneous
equations. Since the unknowns and the equations are infinite in number,
some remarks concerning the treatment of them are necessary.
Suppose that the two series of quantities b , b 1 , b 2 ,..., , lt 2 ,... are
in descending order of magnitude, and let all the coefficients beyond b p , b__p,
P (P a positive integer) be neglected ; we shall have 2p + 1 unknowns and
2p + 1 equations. As the equations are linear and homogeneous with respect
to the unknowns, a relation, which may be expressed by equating to zero a
determinant of 2p + 1 rows, must exist between the equations. We shall
assume that the same results hold when p becomes infinitely great (see the
note at the end of Art. 267).
265-267] (ii) INFINITE DETERMINANT AL EQUATION FOR c. 217
We thus suppose that the ordinary rules for treating a set of linear
homogeneous equations may be used when the unknowns and the equations
are infinite in number. To every positive value of j there corresponds an
equal negative value, and there is one equation for j = : we may therefore
consider that the number of equations is odd and that the coefficient of b , in
the equation given by j = 0, is the central constituent of the determinant
formed by eliminating the unknowns. Let the two equations, obtained by
equating to zero the coefficients of *%+*, be divided by 4J 2 . The
condition that the new series of equations, thus formed, shall be consistent is
expressed by the equation
A (c) = 0,
where A(c) represents the infinite symmetrical determinant,
(c-4) 2 -
!
2
3
4
4 2 -
4 2 - '
(c-2) 2 -
4 2 - '
4 2 - '
2
4 2 - ' '
3
2 2 - '
2
2*-
X
2 2 - '
C 2 -
22 e
2 2 - ''
2
O 2 -,,'
3
2 - '
2 - J
! (
2 - '
c + 2) 2 -
2 - '
2 2 -/
2 2 - J
3
2 s -/
a
2 2 -
2 2 - ""
4 2 - '
4 2 - '
4 2 - '
4 2 - '
4 2 - ""
267. The equation A (c) = may be regarded as an equation for c with
an infinite number of roots which have the following properties :
(i) The roots occur in pairs. For when c is put for c, A (c) remains
unaltered. Hence, if c be a root of the equation, c is another root.
(ii) If CQ be a root, the expressions
c= + c + 2j 0' = - oo... + )
are also roots. Let c + 2 be put for c, and let the divisors 4j 2 - (which
are infinitely great at infinity) be all moved on one place ; the central line
and column are merely altered in position, and the determinant therefore
remains unaltered. The same result holds if c be increased or diminished by
any even integer.
Since these roots are also roots of the equation cos TTC = cos TTC O , we have
A (c) = k (cos TTC cos TTC O ) (26).
218 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
(iii) All the roots of A (c) = are included in the expression c + 2J, that
is, k is independent of c. Since @ , lt 3 ,... do not contain c, the number of
roots cannot be altered by giving any special values to lt <B) 2 , 5 let them
be put zero, and let the resulting value of A (c) be denoted by A (c).
We have evidently
The roots of the equation A (c) = are therefore the same as those of the
equation
COS 7TG COS 7T V = \
whence it follows that the roots of A (c) = are the same as those of the
equation
COS 7TC COS 7TC = 0.
This result, combined with (26), shows that k is independent of c.
(iv) Any root c satisfies the equation
sin 2 i7rc = A(0)sin 2 i7rV^ (28),
where A (0) is the determinant obtained by putting c = in A (c).
Since (26) is true for all values of c, the value of k may be found by
equating the coefficients of the highest power of c in the two members of the
equation. But the highest power of c, contained in A (c), is obtained from
the elements of the leading diagonal only; hence k is independent of lt 2 ,....
Let zeros be put for the latter quantities ; A (c) then reduces to A (c) and c
to V . We therefore have
A (C) = k (COS 7TC - COS 7T V )-
Putting c = in this identity, we obtain
k(l- cos TT Vei) = A (0) = 1, by (27).
Finally, substituting this value of k and putting c = in (26), we deduce
the required equation (28).
Since the substitution of + c 4- 2; for c leaves (28) unaltered, this equation
gives the periods of all the terms in the solution. The period we require is
known to be that of the principal term, and therefore, in obtaining c from the
value of sin 2 -j7TC , we must choose that value nearest to unity.
The only step that now remains, for the determination of c , is the
calculation of A (0). Every element of the leading diagonal of this deter-
minant is unity and the other elements are functions of m which may be
267-268] (ii) CONVERGENCY OF INFINITE DETERMINANT. 219
found according to the methods explained in Art. 266. Putting c = in
A (c), we find that A (0) is equal to
! 2 @ 3
4
* ' 42 ~' V V
3
2 2 - ' ,..'. 2 2 -<H> ' 2 2 - '
IM^ r<- ^ /^ \
2 1 -I f i
22 r'
2 - ' 2 - ' 2 - '
0^- "
2 2 - ' 2 2 - ' 2*- '
K^4 15)3 (5)2 ^1
4 2 - ' 4'- ' 4"-,,' *-<,'
, ...
The complete examioation of the assumptions involved in the preceding results is
outside the limits of this treatise ; they have been considered by Poincar^*. We shall
only give below the conditions which must be fulfilled in order that an infinite determinant
may be expanded according to the ordinary methods.
268. Convergency of an Infinite Determinant.
Let a determinant of 2n + 1 rows or columns be denoted by A n ; the determinant A n is
said to be convergent if it continually approaches a finite and determinate limit as n
approaches to infinity.
Let the element in the ith row above the middle row and in the^'th column to the right
of the middle column be denoted by &,_,-, and the element of the principal diagonal in the
ith row by 1+&, _i, where i t j=+n... -n; in other words, let /3 it y be a non-diagonal
element and 1 +&,_ a diagonal element, the central element being 1 -f-/3 0> . Consider the
continued product f
where, under the sign of summation, the value j-i is excluded. This expression, which
we shall for brevity denote by n n , contains all the terms of the development of A n and
other terms besides. Since, by definition, all the terms of n n have positive signs, A n is less
than n n and therefore, when n is infinite,
Lim. A n < Lim. n n .
But
where the value j= i is not excluded from the second member. Whence, when n becomes
infinite,
Lim. An <Lim. Hi (1 + Si |&,y|).
The second expression is convergent if Si, y (ft, /| con verges J. This condition includes
the convergence of 2 t - [&, _ which is the condition of convergence of the product 1^(1 +&, _i).
* Sur les determinants d'ordre infini. Bull de la Soc. Math, de France, Vol. xiv. pp. 77-90.
t See footnote, p. 201.
See (r. Chrystal, Algebra, Pt. n. p. 137. E. W. Hobson, Trigonometry, Arts. 279-281.
220 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
Hence the limit of A n is finite if the sum of the non-diagonal elements be absolutely convergent
and the product of the elements in the principal diagonal be absolutely convergent.
Again, the expansion of the determinant A H can be deduced from that of A n+p (p a
positive integer), by putting zeros for certain of the elements which are present in the latter
but not present in the former. Hence A n+p A n represents the terms which vanish
when these elements are annulled. But U n + p -U n must contain all these terms (and
possibly others) all affected with a positive sign, and n n+p contains every term present in
n. Hence
But, if 2i,j\fii,j\ be convergent, n n + p -n n and therefore A, l + p -A n can be made less
than any finite quantity by sufficiently increasing n. That is, the conditions given in
italics are sufficient for the convergence of A n when n is infinite*.
The condition sufficient for the convergence of the determinant A(0) can be easily
deduced. The second of the conditions, given above, is evidently satisfied, for all the
elements in the principal diagonal are unity. The first condition is satisfied if
be convergent. The latter series is well known to be convergent. Hence, in order that
A(0) may be convergent, it suffices that 2i|0i| be convergent. We shall assume that
this condition is satisfied.
The Development of an Infinite Determinant.
269. It will be convenient to denote the non-diagonal element $ iti by (i :j], and the
diagonal element 1 +&, _ 4 by (* : - 1) f. The central constituent is then (0 : 0), and the term
in the development arising from the product of the elements of the leading diagonal is
...( : -*)(*-! : -t + l)...(l : -1)(0 : 0)(-1 : 1) (-i : i)....
Any other term in the development may be obtained from this by interchanging any of
the second numbers of the several symbols contained in the above expression, the first
numbers remaining unchanged J.
If one change be made between two of the second numbers, the other elements
remaining the same, the term is said to be produced by one exchange ; for example, if we
interchange - ^ 0, so that the elements (i : - i} (0 : 0) become (i : 0) (0 : - i), the new term
is said to be produced by one exchange.
Two exchanges may be made in two ways. They may either be made amongst three
elements (e.g. we may exchange - 1, and then 0, 1), or they may be made amongst four
elements (e.g. by exchanging i t and also 1, i). Similarly three exchanges may be made
* The proof, when the elements of the principal diagonal are all unity, was given by Poincare
in the paper just referred to. The proof given here is from a paper by H. von Koch, ' Sur les
Determinants Infinis, &c.,' Acta. Math., Vol. xvi. pp. 217-295.
t The explanation will be more easily followed by taking the principal row and principal
column as axes : the positive direction of the 7/-axis being to the right and that of the ar-axis
upwards. The ' coordinates ' of any element are then i, j.
% Of course the development may be also made by interchanging the first numbers, the
second numbers remaining unchanged.
268-270] (ii) DEVELOPMENT OF AN INFINITE DETERMINANT. 221
amongst four, five, or six elements. Any term produced by n exchanges which might, by a
different proceeding, have been produced by n - k exchanges, is excluded from the conside-
ration of the terms produced by n exchanges, since it is considered in the n - k exchanges.
Further, it is evident that the terms produced by n exchanges have a positive or
negative sign according as n is even or odd.
In the following, i is an integer which may have any value, including zero, between
- oc and + oo ; , &', &",... are integers having any values from 1 to + oo .
270. Let all the elements of the leading diagonal be unity. The term in the develop-
ment obtained by using all the elements of the leading diagonal is therefore 1.
Consider any two elements of the leading diagonal
Owing to the fact that k can only be a positive integer greater than zero, these expressions
will always denote different elements, and their product will never denote the same pair of
elements for different pairs of values of i, k.
One exchange between these gives
(i:-i-k\ ({+*:-)
Since all the other elements of the leading diagonal are unity, the expression
-2i2 t (i : -i-k)(i+k : -i), (i= -oo ... + , =l...oo) ...... (29),
gives the terms, in the development of the determinant, obtained by one exchange.
Consider three elements of the leading diagonal
(:-*), (i+k : -i-k\
There are only two possible ways of making two exchanges amongst these three elements,
so that none of them remain in the leading diagonal. They are
(i,-i-k\ (+* : -i-k-k'\
and (i: -i-k-k'), (i+k : -i), (i+k+V : -i-k).
Hence, all the terms arising from two exchanges amongst three elements are
: -i)
: --*)} ...... (29').
Two exchanges amongst the four elements
:-i-k\ (i+k+k' : -i-k-V\ (i+k+k' + k" : --*- V-V\
so that none of them remain in the leading diagonal, are given by exchanging -i, ik
and exchanging -i lc V t i-k-tt-k"; the combinations obtained by exchanging i,
_;__' or -i, -i-k-k'-k" are included. Hence the terms in the development,
obtained by making two exchanges amongst four elements, are
: -i-k-k -V'}(i+k+V + W : - i - Ic -
In a similar manner, we may consider the terms of the development, produced by three
or more exchanges.
222 METHOD WITH KECTANGULAR COORDINATES. [CHAP. XI
Development of A(0).
271. Let us now apply these results to the development of A(0). We have
Let 4i 2 -e =ai, so that Of=a_i.
Now e^=e_jfc is of the order m 2fc at least; it is therefore evident that the order of any
element (i :j) is 2|i+ t /|. Since , k' are always positive and greater than zero, we see
immediately that the terms (29), produced by one exchange, will be of the fourth order at
least ; the terms (29'), (29"), produced by two exchanges, will be of the eighth order at
least; similarly, the terms produced by n exchanges will be of the order 4n at least. We
shall here neglect terms of the 12th and higher orders.
The terms (29) give
torn"
Since Q_i = Q k , etc., the terms obtained from (29') are
since values of k, k' greater than 1 give terms of an order higher than the eleventh.
The terms (29") give
ft,2 fttj ,2 1
to m 11 .
Hence, as far as the eleventh order inclusive, we have
272. The final process consists in replacing the summations, in the expression (30), by
finite terms. For this purpose we have, if k be a finite integer*,
. . . 7 ^ - : j ,
a + l a-l a + l + K a-l k
Here a is never equal to an integer, and, in the method used below, the semi-convergency
of these forms will not affect the values of the functions which they are supposed to
represent.
Let B = 4a 2 ; then a t = 4i 2 - 6 = 4 (i 2 - a 2 ). Hence, decomposing into partial fractions,
B C D
where l/^ = l/.D
Therefore, summing each of the four terms by means of the formulae given above,
1 _ n cot na / 1 1 \ _ n cot no.
i + jfc IQka \2a-k 2a + k) ~ 8a (4a 2 -I 2 )
.(31).
By giving to the values 1, 2, successively, the second and third terms of (30) may be
obtained from this result.
* E, W. Hobson, Trigonometry, Art. 293.
271-273] (ii) ELLIPTIC INEQUALITIES OF HIGHER ORDERS. 223
In the same manner, l/aiai + k ai + k > may be decomposed into partial fractions and
summed for all values of i; then, by giving to , K the values 1, 2, respectively, the
fourth term of (30) may be found.
The last term may be obtained by decomposing lla i a i + l a i+v+l a i + ll f +2 into partial
fractions, as before, and summing for all values of i. The result is
(20a 2 - (V + l) 2 -l},r cot TTO
a (4a 2 - 1) (4a 2 - (V + 1) 2 } {4a 2 -(V + 2)2} (4a 2 - V*) '
Replacing 2a by its value, this may be again decomposed and summed for all values of
k' from 1 to oo . The decomposition is best effected by putting k for k' + 1 ; the expression
can then be exhibited as a function of P. After summation for all values of k from to oo ,
the terms for =0, k=\ must be subtracted.
The results are given by Dr Hill so as to include all terms of an order less than the
sixteenth* As far as the sixth (since 1 -6 is of the first) order, we have found in equa-
tions (30), (31),
Taking that value of c , obtained from (28), which is nearest to unity, this expression for
A (0) gives c = 1-07158 28. The value obtained by Dr Hillf to 15 places of decimals, with
m = -08084 89338 08312, is Co= 1-07158 32774 16012, giving
l-c =l-c /(l + m) = -00857 25730 04864.
He has also obtained c literally, to the order m 11 , from the equations of condition J.
The Terms whose Coefficients depend on m and on the Second and
Higher Powers of e.
273. The Terms of Order e 2 . There are two classes of terms to be considered, namely,
those terms whose characteristics are zero and those whose characteristics are e 2 . The
former are the parts of the Variational Inequalities, that is, of the terms with arguments
2i, whose coefficients are of the order e 2 ; the latter are those Elliptic Inequalities whose
coefficients are of the order e- and whose arguments are of the form 2i + 2$. We shall
only indicate the method of procedure.
(a) The Parts of the Variational Inequalities which are of the Order e 2 .
For these it is necessary to put 2=0 in equation (13), but, instead of neglecting in the
results all terms which depend on e a proceeding which gave the equations (7) we must
give to p such values that terms of order e 2 are included. In the first two terms p therefore
takes the values 0, 1, and in the third term the values 1, 0. The equation obtained
from the coefficient of , 2 ' becomes, on combining those terms multiplied by [2/,] (2j,) and
containing the suffix c, which are equal when j i 1 is put for i, etc.,
except for j=0. It will be noticed that those terms whose factors involve c, are at least
* Motion of the Perigee, p. 32.
t Id. p. 35.
+ G. W. Hill, "Literal Expression for the Motion of the Moon's Perigee," Annals of Math.
(U, S. A.), Vol. ix. pp. 31-41.
224 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
of the second order with respect to e ; hence the known part of c, which depends only on
m and which is denoted above by c , is sufficiently accurate.
Let A 2i =a 2i + 8a 2it
where 2i is the coefficient found in Section (i) and 8a 2i is the new part of the order e 2 .
Also, neglecting powers of e above the second, we have
Substituting these in the equation just given and making use of equations (7), we
obtain, to the order required,
2, {[2;, 2i]
The quantities 2i , f{ , */ having been already found, we have a series of linear equations
for the determination of the coefficients da^ ; but since the value ^'=0 is excluded, these
equations are only sufficient to determine Sa 1 , da 2 , ... in terms of m, e 2 , da Q . It is
necessary to see how 8a may be found.
Owing to the introduction of a in the values of v, <r, one of the members of the product
&AQ is arbitrary. When the inequalities dependent on m only were considered, we put
J =a =l (Art. 258) ; since we are now considering the terms dependent on e 2 as well as
those dependent on rn, the definition of A must be extended so as to cover these terms.
It is found to be most convenient, in making the calculations after the method outlined
here, to define A in the same way as before, so that
The same definition holds when the terms of the orders e 4 , e' 2 , etc. are under consideration ;
the new part of a, of the order e 2 , can then be found by an extension of the method of Art.
255. The equations, which are linear with constant terms, are now sufficient to determine
dai, # 2 v m terms of e 2 , m, and the method of continued approximation may be used
to find the unknowns.
It may be remarked that, when the method, referred to at the end of Art. 287 below, is
used, it is more convenient to define a to be such, that its value obtained in Art. 255 is
unaltered by any of the subsequent approximations ; thus 8a=0, but da is no longer zero.
In this method, the requisite number of equations for finding Sa , da ll ... appear naturally.
(b) The Terms whose Characteristics are e 2 .
These terms have arguments of the form 2^ 20. We therefore put q = -f 2, q = - 2,
successively, in the equation (13), such values being given to p that terms of a higher
order shall not occur. All the terms are directly seen to be at least of the order e 2 . We
then obtain, for the determination of the unknown coefficients, a series of linear equations
which correspond in number to the unknowns, and which contain constant terms ; all the
unknowns can therefore be found by continued approximation. Since all the terms are of
the order e 2 , the value of c to be substituted in the coefficients [2/+2c, 2i], etc. is known
with sufficient accuracy.
274. The Terms of Order e 3 . These again are of two kinds : those with arguments
2^$, 2i30. For the latter we put q= + 3 in the equations (13); all the terms
present are of order e 3 , and the determination of the unknowns proceeds on exactly the
same lines as that of the terms whose characteristics are e 2 , The terms of arguments
273-275] (iii) MEAN PERIOD INEQUALITIES. 225
are troublesome because the value of c, already found, is not sufficiently accurate:
it must be known to the order e 2 .
Put q= + 1, - 1 successively in equations (13) and give to p such values that terms of
the order e 3 may be included. The only coefficients present will be found to be A^, ^21+0
A 2i + c = e i + 8 i , A 2i . = fi ' + 8ej, c = c +8c;
8fi, 8ei are then of order e 3 and da 2i , 8c of order e 2 . The terms which are of the order e
vanish by equations (14); the coefficients ^#+20 ^2i-2c are known by Art. 273 (b). We
have remaining a set of linear equations for the determination of 8c, , 8e/, containing
known terms independent of these quantities.
Since one of the coefficients A^^ was arbitrary, the same must be true of one of the
dfi, 8fi ; this fact is also deducible from the consideration that the number of unknowns
is greater by one than the number of equations. It is most convenient to determine the
arbitrary so that, when all powers of e are included,
so that 8cQ=8(Q. With this assumption the values of S^, 5e/, 8c may be obtained by
continued approximation.
The method of carrying out the approximations outlined in this and in the previous
article may be found in Pts. v. ix. of the paper referred to in Art. 261 above. The
results are obtained numerically as far as m is concerned, the value of e being left arbitrary.
The fact that one of the d<, 8fi is arbitrary implies that some relation free from these
quantities and containing only 8c can be obtained from the equations of condition. The
author has given a definite form to this relation, with several extensions, on pp. 336 338
of a paper entitled Investigations in the Lunar Theory*. The method of obtaining it
will be outlined in connection with the latitude inequalities and the determination of 8g.
See Arts. 284, 285, 288 below.
(iii) The Terms whose Coefficients depend only on m, e'.
275. Since the terms dependent on the solar parallax and on the latitude
are neglected, we put n = fl 2 , z = in the equations (23), (19) of Art. 20.
Hence, it is necessary to add to the right-hand members of the equations (1)
of Art. 245 the terms
respectively. But when z is neglected, f) 2 is of the form (Art. 128),
where A, B, C depend only on the coordinates of the Sun. The terms to be
added to the right-hand members of equations (1) are therefore
- 3 (Au 2 + 2Bu<r + Ccr 2 ) + D- 1 (v 2 DA + 2uo-DB + o- 2 DC), 2O 2 - 2AiA . .(32).
* Amer. Journ. Math. Vol. xvn. pp. 318-358.
B. L. T.
226 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
To find A, B, C we have, by Art. 19,
2 = 3m 2 1^ r 2 S 2 - i (i; + <r) 2 j - m*v
But, by Art. 22,
= -J- (v + o-) cos (v f n't - e) - % V- I (v a) sin (v n't e').
Lett>'-n'$-e'=r'. Then
rS = t/ exp. (- F' V^l) + Jo- exp. (+ F' V^l),
r z S 2 = Ju 2 exp. (- 2F' V^l) + Jo- 2 exp. ( + 2F' V^l) + Jv<r.
Substituting in fl a , we find
whence C + A = f m 2 cos 2F X - l , C - A = f m 2 Vl 8
Since v' = <y' - r' - (w'* + e' - tsr') =/' - w' (Arts. 48, 53), the only functions
which have to be calculated are
a /3 /r /3 , (a /3 // 3 ) cos 2 (/' - u/), (a /3 /r /3 ) sin 2 (/' - /).
These are expanded in powers of e' and in cosines and sines of multiples of w'
after the manner explained in Arts. 39 41. The results for them have been
fully worked out by several investigators*.
a' 3
Assume -^ 1 = 2 p a/ cospw',
^ cos 2 F' - 1 = 2 p /3 p ' cos pw', ^- 3 sin 2 F r = 2 p /3/ sin ^>^,
where p = oo . . . + oo and a'_ p = VL^\ then a p ', {3 p ' are known functions of e'.
Using these expressions, we have
Since the coefficients of * in f m and ^' v=ri are the same, we can put f m for
07I/V-! if we remember that, when returning to real variables, e pw "^~ l is to be
put for ^ m ; the value of m will not be substituted in the index of f. Hence
we can write
* See the references given in Arts. 123, 126,
275-277] (iv) PARALLACTIC INEQUALITIES. 227
These values must be substituted in the terms (32), and the results added
to the right-hand members of equations (1).
276. It is evident that the required solution of the equations is a
particular integral corresponding to the newly added terms. Since e is
neglected, the solution will be of the form
v = a
These values being substituted in the equations, and the coefficients of f#+3 m
equated to zero, we shall have a series of equations of condition to deter-
mine the unknown quantities.
The method is similar to that used in Case (ii). Neglecting, initially,
powers of e higher than the first, the values v = u , a- = <r , given by equations
(3), can be used in the right-hand members, since A, B, C are at least of the
order e. We obtain a series of equations similar to (14); but since their
right-hand members are no longer zero, there is no relation like that which
was necessary to find c : this is otherwise evident, for the index of f in all
these terms is quite known and no new arbitrary constant is required. The
higher approximations proceed in the same way. The inverse operation ZH
will introduce divisors of the form 2j 4- $m ; but as m is assumed to be
incommensurable with any integer, none of these divisors will vanish.
Some results, obtained by the author for ine qualities dependent on ef, will be found in
two Notes in the Hon. Not. R. A. S. Vols. LIV. p. 471, LV. pp. 35.
(iv) The Terms whose Coefficients depend only on m, I/a'.
277. Since e' is neglected we have
therefore the terms to be added to the right-hand members of equations (1)
are respectively
an
O" ~
Also, by Arts. 19, 22, we have rS = X = (v + <r), / = a', and therefore, as
z is neglected,
= \ 4 O 3 + O + 4 wr
a a
The terms to be added to the equations are therefore of the third,
fourth,... degrees in v, cr, corresponding to terms of the first, second,... degrees
with respect to I/a'.
152
228 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
If, in the expressions (33), the values v , <T O , which are odd power
series in f and which correspond to the intermediate orbit, be substituted
for v, a, the terms produced by l p will be odd or even power series in f
according as p is odd or even. The values of v, &, when terms dependent on
the solar parallax are included, will therefore COD tain even as well as odd
powers of ?. Hence we assume
This solution includes the intermediate orbit. When I/a' is neglected, we
have A zi = a 2i and A&_i = 0.
The process is similar to that of Case (iii). First, neglecting I/a' 2 and
higher powers, we find the odd coefficients A*^ ; the parts of A* which are
of the order I/a' 2 are next found ; and so on. Since the equations of condition
at any stage of the process are linear, with known terms in their right-hand
members, no relation, independent of these coefficients, exists and the un-
knowns can be calculated by continued approximation.
For the details of the calculations, two papers by the writer' On the Part of the
Parallactic Inequalities in the Moon's Motion which is a Function of the Mean Motions
of the Sun and Moon*,' and 'On the Determination of a Certain Class of Inequalities in
the Moon's Motion f ' may be consulted. See also G. W. Hill, * The Periodic Solution as
a first approximation in the Lunar Theory J.'
(v) The Terms whose Coefficients depend only on m, 7.
278. When e', I/a' are neglected, fl = 0, and the equations (23), (19), (18)
of Chap. II. become
D 2 (var + z*) - DvDo- - (Dzf- 2m (vDa - rDv ) + f m 2 (v + o-) 2 - 3m 2 * 2 = C, ) .
> '
- 2rm;<7) + f m 2 (u 2 - <r 2 ) =
- z (m 2 + K/r 3 ) = ........................ (35).
The form of the last equation shows that its integral contains a constant
factor which is one arbitrary of the solution. In the case of the Moon this
factor is small : it has been denoted in previous chapters by 7. If, in the first
two equations, terms of the order 7 2 be neglected, they reduce to those of
Art. 244 ; the new parts of v, a are therefore at least of the order 7 2 . We
shall neglect the constant of eccentricity and consider the first approximation
to the solution of equations (34), (35) to be the intermediary.
The procedure is the same as in the previous cases. We suppose the
intermediate orbit known and consider first the terms of the order 7 1 ; these
* Amer. Journ. Math. Vol. xiv. pp. 141-160.
t Mon. Not. R.A.S. Vol. LII. pp. 71-80.
$ Astron, Journ, Vol. xv. pp. 137-143,
277-279] (v) PRINCIPAL INEQUALITIES IN LATITUDE. 229
only occur in z and they are obtained from equations (35) by inserting therein
that value of r which corresponds to the intermediary. The terms whose
coefficients are of order y 2 only occur in u, <r and they are found, when those
of order y l have been obtained, from the equations (34); the terms of the
order y 3 only occur in z and they are obtained from (35) ; and so on.
The solution may be represented by
v =
where v^i, o-^, z^^ represent the terms whose coefficients are of the orders
denoted by the suffixes. Therefore, neglecting powers of 7 above the third
and remembering that r* = vcr + z*, r 2 = u <7 , where r , u , <r refer to the
intermediary, we have
1 = _ 1 _ = 1 fi _ 3
r* '
Substituting, equation (35) becomes
& (z, + z+) - (m 2 + ) (*y + ^) = - f * (u o0 y + o-oiv + zf). . .(36):
\ / '0
the differential equation for the terms in z as far as the order y 3 .
Instead of (35), we may use an equation free from the divisor r 3 . This equation
deduced immediately from equations (17), (18) of Art. 19, after putting Q = is
D (zDv - vDz) + ZmzDv + f m 2 z (v + <r) + m 2 zv =
(or a similar equation in which <r, v, D replace v, o-, .D). This new form will be found useful
for a literal development in powers of m ; the method given in the text is less laborious
when the numerical value of m is used at the outset.
The Terms dependent on m and on the First Power of 7.
Principal Part of the Motion of the Node.
279. When terms above the order 7 1 are neglected, the equation (36)
reduces to
, = (37).
Substituting for m 2 + /c/r Q 3 the series 22$lf t -f ai (Art. 265), in which Jf_i = Jk
i = oo . . . + GO , we obtain
= 0.. ....(37').
Since Mi is of the order m **' at least, this equation is of exactly the same
form as (23') and it may be treated in an exactly similar manner. The two
independent integrals are given by
230 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
and the substitution of either of these will give a series of linear homogeneous
equations, from which the ratios of the unknown coefficients may be found ;
the infinite determinant, formed by eliminating the unknowns, will give g.
280. In order to connect this solution with that obtained in Art. 147,
let the latter be written
z = 2a2 j sin(2jf + 77), (j = - oo ... + oo ).
As with the elliptic inequalities (Art. 257) we put
p = exp. 2f V ^1 = exp. 2 (n - n') (t - t.) V^l,
? 2 g = exp. 77 V"^ = exp. (gnt + e-6) V~l = exp. g (n - ri) (t - t 2 ) V^ 1 ;
so that g = gn/(n n'} = g(\+ m), gt z (n ri) = 6 e.
If it be recollected that t is to be replaced by t a in the part of the index of f
which contains g, the solution may be written in the form
(38),
where K'_j = Kj.
When this is substituted in equation (37') and the coefficient of f 2 -? +g
equated to zero, we have
(2j + g)* K 5 - 22iM^ = 0, (i, j = - oo . . . + oo ). . .(39).
The equations for K'_j, obtained by equating to zero the coefficients of f- 2 .?- g ,
are of the same form. The elimination of the Kj or of the K '_, gives an
infinite determinant V (g). This determinant being of the same form as A (c),
all the results proved in Arts. 266272 will be available if we replace ; by
2Mi and c by g.
The part of g which depends only on m, may therefore be found by taking
the value of g , nearest to unity, given by the equation
sin 2 ^TTgo = V (0) sin 2 \ir \/2M Q ;
where V (0) denotes the determinant A (0) of Art. 267, after (B^ has been
replaced by
The determination of g by this method was first made by Adams*. With the value
m=n'/n = '074801 3 (exactly) or m = -08084 89030 51852, he finds g= 1-08517 13927 46869,
giving g- 1 = -00399 91618 46592. Mr P. H. Cowell has verified this value and he has ob-
tained the literal and numerical values of g , K it as well as those of the terms of orders
* " On the Motion of the Moon's Node in the case when the Orbits of the Sun and Moon are
supposed to have no Eccentricities, and when their Mutual Inclination is supposed to be small,"
Mon. Not. JR. A. S. Vol. xxxvni. pp. 43-49 ; Coll. Works, pp. 181-188.
279-283] (v) TERMS IN LATITUDE OF HIGHER ORDERS. 231
281. When g has been found, the coefficients Kj can be determined in
terms of one of them, say of K , by means of equations (39). We first leave
aside the equation given by j ' = ; when the other coefficients have been
found in terms of K Q , their values should satisfy this equation: it is therefore
useful for purposes of verification.
The coefficient of sin rj, that is, of (^ f~ g )/2V 1 will be 2aJT , since
KQ = KQ. In Art. 147 this was denoted by ay. Hence
ay = 2aJT .
The ratio a : a being previously known, we have the new constant of
latitude in terms of the one used in Chap. vn. The relation will be modified
when terms of the order y 3 are considered.
The Terms of Order y*.
282. It is not necessary to give detailed explanations concerning the
calculation of these terms. They are deduced from equations (34), when z y
has been found, in exactly the same manner as the terms in e 2 were deduced
when those of order e had been found. We give to z the value z y just obtained
and put v = v Q + v y *, <j=fc<r -f0y, rejecting, after the substitution, all terms
of an order higher than y 2 . The solution divides into the parts which depend
on f 2 * and f 2 ** 2 *, respectively, and the coefficients are calculated in the manner
explained in Art. 273. The value g of g is sufficiently approximate.
We suppose then that v, a or iy, oy are known correctly to the order y*.
The Terms of Order y*.
The Part of the Motion of the Node which is of Order y*.
283. Returning to the equation (36), we substitute in its right-hand
member the values of z y , iy, oy, v , <r , r , previously obtained, since this
portion is of the order 7 3 at least. Also, from its symmetry with respect to
v, o-, we see that it will be expressible in terms of (? 2i+g - ~ 2i - g ), (^ +3g - f- 2 ^ 3 *),
with known coefficients. With regard to the left-hand member, m 2 + /e/r 3
is expressible as before in terms of p l + f" 2 *. We cannot, however, put
D' 2 z y - (m 2 + K /r 3 ) z y = 0, for, when the value of z y is substituted, D 2 z y contains
g in its coefficients. Denoting the value of g, to the order 7 2 , by g + Sg, we
see that D*z y will produce terms of the order ySg, that is, of the order y*.
Assume as the solution
z V~i = (z y + ^) V^I = a 2,- {(Kj + SKj) * +g + (K'-j + $%,) *-*}
+ terms dependent on ^' 3g ,
232 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
where K'_j = - Kj and 8IC^ = - SKj. The terms of the second line will be
all of the order 7 3 , since the index of f contains 3g : they can be equated to
the terms of corresponding form on the right-hand side of (36), and the unknown
coefficients found in the usual way. The value g of g is evidently sufficient
for these terms. We shall therefore leave them aside and consider only the
equations of condition obtained by equating the powers of f 9' g to zero.
284. With this understanding we have, since g = g + Sg,
J*z V=l = a 2, (2j + go
= a 2,- (2j + g
+ 2a Sg 2, (2j + go) (Kj p
omitting terms of an order higher than <f.
When the latter expression is substituted in (36), the terms under the
first sign of summation cancel with (m 2 + /t/r 3 ) y , by (37 r ); the remaining
terms are all of order <f. We have therefore
2aSg 2j (2j + g ) (K, &+* + K'-j f^M) + a ^- (2j + g
- a (m 2 + /c/n 3 ) 2^ (SKj ?*+* + SK'
those terms on the right, depending on f^ w , being left aside. Equating to
zero the coefficient of f^' +g and putting m 2 + >e/r 3 = 22iMif 2i , we deduce
2% (2j + g ) *} + (2j + go) 2 8^
= coe/ o/ ^' +g in - 1 ^ Y 7
ar
The number of equations obtained from this by giving to j the values
0, 1, 2,... is one less than the number of unknowns &}, Sg; but since K
was arbitrary, S^T must be also arbitrary and it may be determined at will :
when the arbitrary value, to be given to 8^T > nas been fixed, the number
of unknowns will correspond to the number of equations. As Sg is of the
order 7 2 , while &K is of the order 7 3 , the value of Sg is independent of the
value to be given to &T , and therefore some relation, independent of the
unknowns SKj but involving Sg, must exist between these equations. We
shall now find this relation.
285. The Equation for 8g.
We have
f z v (f 0y + cr u y2 + Y 2 )/ r o 5 = terms of order <f in the expansion
of *v {( v o + *v) (o + v) + Vl" 1 in po wers f 7-
283-286] (v) NEW PART OF THE MOTION OF THE NODE. 233
Let E% 2 = (y + v y2 ) (<J + oy) + 2 y 2 = value of r 2 correct to <f.
Substituting, we see that the right-hand member of (40) is
The part, of order y 3 , of the coef. of f^' +g in the expansion of /cz y V^T/aE% 2 .
Multiply (40) by K, and sum for all values of j. Since M^ = M^, the
terms involving the unknowns SKj on the left-hand side will be
S,2< {(2?
As j, i have the same range of values, we may interchange them in the
second term of this expression which then becomes
2* {(%' + g.) 2 K, - 22, MH K t ] BK,.
This is zero because the coefficient of BKj vanishes for all values of j by
equations (39). Hence all the unknowns BKj disappear.
The result of the summation of the equations (40) is therefore
28g2j (2j + g ) Kf = 2j [Kj x coef., order y 3 , of p+* in icZy
In an exactly similar manner we may find
28g2j (2j + g ) K'*.j = 2^ [K'_j x coef., order y, of *-* in
But since K'-j = Kj, equation (38) may be written
and therefore a}, a^T'_^ are respectively the coefficients of f~ 9 ~ g , f^" +g in
^ v V 1. Whence, adding the two previous equations and putting
?*-} = Kf, we have
4tSg 2; (2j + go) -fiT/ = 2,- [(coe/ off-** in z y ) (coef., order <f, of^ +g in ** Y
or, 48g 2j (2j + go) ^/ = Par#, order T 4 , o/ co/wi. ^erm in KzJja?R* y * . . .(41) ;
where tcz y 2 /a?R 3 y t is supposed to be expanded in powers of y and in cosines of
multiples of the arguments contained therein. Since all the other quantities
present in this equation have been previously found, it constitutes a simple
equation for finding 8g.
286. We have seen that 8K Q may be determined at will. It may be
fixed so that the coefficient of sin rj in z is 2aJT at any stage of the ap-
proximations and therefore in the final results ; hence &fiT = 0. When Sg has
been found, the equations obtained by putting j= + 1, + 2,... in (40) will
enable us to find the coefficients &fifi, BK^,... by continued approximation.
The equation given by j = is then superfluous : it may be used as an equa-
tion of verification for the values of 8g, 8K jf when these have been calculated.
234 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
(vi) Terms dependent on m and on Combinations of e', e, 7, I/a
and their Powers.
287. The developments which have been given above suffice to indicate
the general method by which all the solar inequalities in the motion of the
Moon may be found. The only real difficulties which occur are those arising
when the motions of the Perigee and of the Node are required. The infinite
determinant gives the principal parts of these with an accuracy which leaves
nothing to be desired ; the other smaller portions can all be found by simple
equations in the manner explained in Art. 285.
The coefficients of the terms of any order in e, e', % I/a' will always be
determinable by a set of linear equations when the terms of lower orders in
these constants have been found. It will be noticed from what has gone
before that, at every stage, all powers of m are included, and also that, when a
term of argument + i'% p<f> + qfi jrj is being found, all the terms of argu-
ments (2i i') i~ p(f) qfi JTJ (if = or 1 and i, p, q,j, positive integers or
zeros) are determined at the same time. The process is therefore reduced
to the approximate solution of a certain number of sets of linear equations
the number of such sets depending on the order in e, e', y, I/a' to which it
is desired to take the solution.
After finding the intermediate orbit and the principal parts of the motions of the perigee
and of the node, the use of the equations (19), (23) of Art. 20 for obtaining the other
inequalities is not essential. We may return to one of the original equations (17) of
Art. 19 and develope the functions Kv/r 3 , Kar/r B in powers of dv, da- by putting v = V Q + 8v,
<r = <r +8o-, where dv, 80- denote any of the series of inequalities to be found. This plan of
procedure, which has some advantages, especially in the terms of lower orders in e, e', y, I/a',
is outlined in the memoir referred to in Art. 274. The method, given above, for the deter-
mination of Sg, together with its application to the other parts of the motion of the node
and to the motion of the perigee, will be found in the same place.
Relations between the Higher Parts of the Motions of the Perigee and the Node
and the Non-periodic Part of the Moon's Parallax.
288. The equation (41) is only one particular case of a much more general theorem by
which all the parts of the motions of the perigee and of the node may be found when their
principal parts those depending on m only have been obtained. Modified forms of the
equation will serve for the determination of those parts of g which depend on all powers
and products of e' 2 , I/a' 2 , e 2 , y 2 , so that the motion of the node really depends only on the
solution of an infinite determinant and of a series of simple linear equations with one
unknown. The same is true of the perigee. For example, the part of the motion of the
perigee which is of the order e 2 may be shown to be the value of 8c given by
2ftc2, [(2; + 1 + m + c ) fj 2 + (2; - 1 - m + c ) e' 2 _ y ]
= Const, part, order e*, in the expansion of K (X#x e + Y e iy e )/a?2l 3 e i...(42),
287-289] SOME GENERAL THEOREMS. 235
where ^ , x e , X& and y , ?/ e , y^ denote the parts in X and T whose coefficients are of
orders e, e\ e 2 , respectively, and where
An important extension of the results (41), (42) consists in the fact that they are true
when all powers of e' 2 are included in the various terms. In other words, when we suppose
that all parts of the functions z y , R., which depend on m, e' 2 and on y 1 , y 2 , have been found,
and when g , Kj are replaced by their more accurate values g + ^/(m, e^\ Kj+ePyf (m, eP),
the part of the value of g which depends on y 2 and on all powers and products of m, ef 2 is
given by (41) ; a similar result holds for part of the motion of the perigee given by (42). It
is for this reason that the determination of the parts of the solution which depend only
on m, e' should be the step immediately following the calculation of the intermediate
orbit ; the necessary parts of the motions of the perigee and of the node, which depend on
m, e' 2 , are found by the same method.
289. One general theorem is as follows : Let e, y be the constants of eccentricity and
latitude, and let the coefficients ,-, e'_,-, Kj be expressed in terms of them ; these coefficients
are supposed to be of the form e/(m, e' 2 ), and c , g of the form <f> (m, e' 2 ). The constant
part of l/r will contain the terms
and the motions of the perigee and the node the terms
He 2 +Ky 2 , M^ + Ny 2 ,
respectively, where E, F, G, H, K, M, N are functions of m, eP only. Put
where powers of e' 2 are supposed to be included in c , e,-, etc. It may then be shown that
HT e =6E, KT e =6F = MT y , NT y =6G ........................ (43).
The theorem which lies at the basis of these results is as follows : If X, Y t z (Art. 18)
have been fully calculated to the order ePy^-p, where j9=0, 1, ... 2<? (that is, to the order
2q in e, y), the constant part of l/r can be obtained to the order ePy 8 ***-*, where p = Q,
1, ... 2<? + 2 (that is, to the order 2^+2), without further reference to the equations of
motion, by a purely algebraic formula involving only the values of X, Y, z to the order
calculated. A different and more complete statement of this theorem is, that the terms of
order %q with respect to e, y, in the constant part of the expansion of 3/Rzq in powers of e, y,
are equal to the corresponding terms in the expansion of
where JT M , Y 2q , Z 2q contain the terms as far as the order 2q ; x^ y M > z zv &TG tlie terms in
JT 2(Z , 7 M , z. 2q , whose coefficients are of the order 2q, and 2 2q =X 2 2q + Y^+Z 2 ^^. This
result also holds when, all powers of the solar eccentricity are included ; the only quantity
neglected is the solar parallax.
The proofs of the theorems in this and in the previous Article are too long to be
inserted here; they will be found in Part II. of the author's paper referred to in
Art. 274 above.
236 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI
The first of Adams' two theorems* can be deduced from the result just enunciated, by
putting q = l. It states that, in the constant part of the expression for a/r, there are no
terms of the forms e 2 /( m > e ' 2 )> 7 2 /( m > e ' 2 ) > ner e, a/O*/^ 2 ) 3 may be a function of m, e' 2 , but
it must not contain e, y. His second theorem is immediately obtainable from the equa-
tions (43). It states that
H/K=E/F, M/N=F/G.
These interesting results may be also utilised as equations of verification.
* J. C. Adams, ' Note on a remarkable property of the analytical expression for the constant
term in the reciprocal of the Moon's radius vector," M. N. R. A. S. Vol. xxxvm. pp. 460-472 ;
Coll. Works, pp. 189-204.
CHAPTER XII.
THE PRINCIPAL METHODS.
290. A BRIEF account of the methods which have been used to attack
the lunar problem, other than those considered above, will be given in this
Chapter. In general, only those theories which have been tested by actual
application to the discovery of the perturbations produced by the action
of the Sun will be analysed. The historical order will be adhered to as
far as possible, although the time elapsed between the first announcement
of a plan of treatment and the full publication of the results, frequently
makes it difficult to assign any exact date to a theory.
The history of the lunar theory and of celestial mechanics generally (in
the sense in which these terms are now understood) began in 1687 with the
publication of the Principia\ the portion which especially refers to the
motion of the Moon is contained in Props. 22, 25 35 of Book in. In this
and in the later editions of the same work Newton succeeded in showing that
all the principal periodic inequalities, as well as the mean motions of the
perigee and the node, were due' to the Sun's action, and he added some other
inequalities which had not been previously deduced from the observations.
The result which he obtains for the mean motion of the perigee is only half
its observed value ; it appears, however, from the Manuscripts * of Newton
which have come down to us, that he had later succeeded in obtaining its
value within eight per cent, of the whole. The conciseness of the proofs,
when they are given, makes his work very difficult to follow. It is now
generally recognized that he used his method of fluxions to arrive at many
of the results, afterwards covering up all traces of it by casting them into a
geometrical form ; if this be so, the claim of Clairaut to be the first to apply
analysis to the lunar theory must be somewhat modified. No substantial
advance was made until the publication, more than sixty years later, of
Clairaut' s Theorie de la Lune.
* Catalogue of the Portsmouth Collection of Books and Papers written by or belonging to
Sir Isaac Newton. Cambridge, 1888.
238 THE PRINCIPAL METHODS. [CHAP. XII
For the methods which Newton used, the reader is referred to the Principia and to the
numerous commentaries which have been written on it. An analysis of the Principia
and an account of Newton's life and works published or in manuscript together with
many references, have been given by W. W. Eouse Ball*.
The history of the lunar theory up to the publication of the third volume of the Mfaani-
que Celeste of Laplace and an account of the methods used, have been given by Gautierf.
For further details concerning the theories of Newton, Clairaut, d'Alembert, the method
given by Euler in the Appendix to his first theory, Euler's second theory, and concerning
the theories of Laplace, Damoiseau and Plana, reference may be made to the Me'canique
291. Clairaut' s Theory %.
Clairaut commences by finding the differential equations of motion when
the latitude of the Moon is neglected. He takes the inverse of the radius
vector and the time as dependent variables, the true longitude as the inde-
pendent variable, and he finally arrives at equations which are equivalent
to (8), (9) of Art. 16 ; since the 'latitude is neglected, u^ is, in his theory, the
inverse of the radius vector.
The process of solution is one of continued approximation. He considers
the orbit of the Moon to be primarily an ellipse, but, recognizing that the
apse revolves quickly, he introduces the quantity denoted above by c,
so that the first approximation is the modified ellipse. The forces due to
the Sun's action are expressed in terms of the radii vectores of the Sun and
the Moon and the difference of their longitudes, and thence, by means of the
elliptic formulae, in terms of the true longitude of the Moon somewhat
in the manner explained in Art. 127 above. The calculations are, in certain
cases, carried to the second order of the disturbing forces, and, in particular,
the motion of the perigee is obtained to this degree of accuracy.
The motion of the Moon in latitude is obtained by considering the
equations for the variations of the node and the inclination ; these correspond
to the fifth and sixth of the equations of Art. 82. They are much less
accurately worked out than the motion in the plane of the orbit.
Clairaut was the first to publish a method for the treatment of the lunar theory founded
on the integration of differential equations. The artifice of modifying the first approxima-
tion by the introduction of c, in order to take the motion of the perigee into account from
the outset, is also due to him. The determination of this motion is marked by an event
* An Essay on Newton's Principia. Macmillan, 1893.
t Essai historigue sur le Probleme des Trois Corps. Paris, 1817.
J Vol. m. Chaps, in. vn. , ix.
Announced in 1747. The first edition of the Theorie de la Lune was published in 1752, the
second and most complete edition in 1765 at Paris. The latter is a quarto volume of 162 pages
and it contains Tables for the calculation of the position of the Moon, founded on gravity only.
The first set of tables was published separately in 1754.
290-293] CLAIRAUT. D'ALEMBERT. EULER. 239
which shows what progress astronomers had made in admitting the sufficiency of Newton's
laws to account for all the known phenomena of the celestial motions. It must be remem-
bered that Newton's later determination, as shown in his manuscripts, was then unknown.
It had long been a difficulty that the first approximation to the motion of the perigee
gave only half the observed motion. Clairaut, thinking that the Newtonian law must be
considerably in error, tried the effect of adding a term, of the form /e/r 3 , to the forces
resolved along the radius vector, and he succeeded in showing that it would account for the
observed motion. It then occurred to him to carry his approximation a step further, with
the law of the inverse square only, and to see what effect the evection had when introduced
into the expressions for the disturbing forces. He soon found that the new term was nearly
equal in absolute value to the second term and that the greater part of the motion was thus
accounted for by the law of Newton (see Art. 169).
292. D'Alembert's Theory*.
This is very similar in its general plan to that of Clairaut, but while the
latter worked out his results numerically, d'Alembert considered a literal
development and carried out his computations with more completeness. He
gave the term ^f- m? in the motion of the perigee : Clairaut had only
obtained its numerical value. His method of approaching the modified orbit
is much more logical ; he introduces a part of the Sun's action into the first
approximation by proceeding in a manner analogous to that of Art. 67 above.
D'Alembert made several contributions to the theory. He succeeded in showing that
terms increasing continually with the time can be avoided, and he gave a direct method of
approaching the first approximation. He also recognized the fact that the question of the
convergence of the series obtained ought not to be neglected. He further considered the
effects produced by small divisors and showed that the coordinates might be expressed by
means of only four arguments which were necessarily related to the orders of the coefficients.
Tables are added at the end of his researches.
293. Enters First Theory^.
Euler commences by considering the equations of motion referred to
cylindrical coordinates; these, translated into modern notations, are the
equations given at the beginning of Art. 16. The equation for the latitude
is immediately replaced by two others which are practically those for the
variations of the node and inclination, and they are obtained under the
assumption that the first differential of r tan u, with respect to the time,
has the same form in disturbed and undisturbed motion. The second
equation of motion is integrated and the resulting value of v is substituted
in the first equation which then contains only the differentials of r ly with
respect to the time, and certain integrals depending on the forces.
* Recherches sur differents Points important du Systeme du Monde. Pt. i. (1754) Thtorie de la
Lune, 8vo. LXVIII. + 260 pp. D'Alembert sent in his theory to the Secretary of the Academy
in January, 1751.
t Theoria Motus Lunae, etc. (with an Appendix) 4to. 347 pp. Petrop. 1753.
240 THE PRINCIPAL METHODS. [CHAP. XII
The independent variable is changed from t to the true anomaly/, and for
r is put its elliptic value (in terms of/) multiplied by 1 -f v ; v is then a
small quantity depending on the disturbing forces (cf. Hansen's method,
Art. 209). Euler thus arrives at a differential equation of the second order
in which v is the dependent and / the independent variable.
After expanding the disturbing forces according to powers of e, etc., he
divides them into classes: those independent of e, e', i\ those independent
of e', i; and so on, after the manner explained in Chaps. VIL, xi. In
considering the inequalities of the second class (Art. 132), he finds
v = const. + Cj/+ periodic terms ;
so that the motion of the perigee depends on d. This is not determined
directly. Its observed value is used and Euler then compares the latter with
the value deduced from theory in order to test the Newtonian law. He has
previously assumed that the attraction between the Earth and the Moon is
of the form //,/r 2 - const. ; the constant is shown to be very small and well
within the limits of error caused by the neglect of higher terms in the
approximations.
The numerical values of the constants to be used in the theory which is
numerical as far as ra is concerned and algebraical in respect of the other
constants are determined by the consideration of thirteen eclipses; the
want of definiteness in the meanings to be assigned to the constants, which
affected the results of Clairaut and d'Alembert, is avoided, for Euler uses the
formulaB of his own theory in the calculation of these eclipses.
In the Appendix, an investigation, which practically amounts to the
method of the variation of arbitrary constants, is given and worked out with
some detail. Euler expresses himself as unsatisfied with both the theories he
has explained.
294. Euler s Second Theory*.
The method which Euler has here set forth with much detail is interesting
as the first attempt to employ rectangular coordinates referred to moving
axes in the Lunar Problem. He considers an axis of x revolving with the
mean angular velocity of the Moon in the Ecliptic, that of y being also in
this plane and that of z perpendicular to the plane. Taking the mean distance
of the Moon as a, its coordinates are assumed to be a (1 + x), ay, az, so that
x, y, z are small quantities depending on the solar action and on the lunar
eccentricity and inclination ; a is defined to be such that x contains no constant
term. The equations of motion are found in the usual way, the disturbing
* Theoria Motuum Lunae, nova methodo pertractata una cum Tdbulis astronomicis unde at
quodvis tempus Loca Lunae expedite computari possunt, . . . : J. A. Euler, W. L. Krafft, J. A. Lexell.
Opus dirigente L. Eulero. 4to. 775 pp. Petrop. 1772.
293-294] EULER'S SECOND THEORY. 241
forces being developed in powers of I/r'. It is to be noticed that this
method, like that of Chap. XL, has the advantage of allowing the disturbing
forces to be expressed as homogeneous functions of x, y, z of the first,
second, and higher degrees; but the relation between the degrees of the
homogeneous functions and those of I/a', which was observed in the
equations of Section (iii), Chap, n., does not hold in Euler's method : the
coefficients of these functions, in Euler's theory, are expressible by means
of the two arguments f, iu'.
The independent variable is w' t and Euler puts
n/n = mi + 1 ;
so that 1 + Mi is the ratio of the mean motions of the Moon and the Sun :
observation gives m^ = 12'36.... The forces a (I + a^/r 3 , ay/r 3 , az/r 3 , due only
to the mutual actions of the Earth and the Moon, are expanded in powers
of x, y, z.
The general solution is then supposed to be of the form
with similar expressions for y, z. Here e, i are the two arbitrary constants of
the solution corresponding to the eccentricity and the inclination. Sub-
stituting these values in the differential equations and equating the coefficients
of the various powers and products of e, e , etc. to zero, he obtains a series of
differential equations for the determination of A,B lt .... The parts dependent
on A give the variational inequalities, those dependent on B l) B^... the
elliptic inequalities, and so on. In the determination of B l} the motion of the
perigee arises ; as in his earlier methods, he assumes its value from observa-
tion and verifies his results by means of the calculated value. The motion of
the node is treated in the same manner. The various differential equations
are solved by the method of indeterminate coefficients.
M. Tisserand remarks that Euler's method of dividing up the inequalities into classes
requires some modification when we proceed to terms of higher orders, owing to the fact
that the motions of the perigee and the node contain powers of e 2 , e' 2 ,... ; the arguments
depending on these motions, when expanded so as to put the solution into Euler's form,
would introduce into the coefficients terms depending on the time. A reference to Art. 283
above, will show how this objection to Euler's method may be removed.
Euler's main contributions to the lunar theory are : the application of moving rect-
angular axes ; the method of the variation of arbitrary constants, as given in the appendix
to his first theory ; the use of indeterminate coefficients in the solution of the differential
equations ; a new method for the determination of the constants from observation ; the
formation and solution of equations of condition to determine the constants from observa-
tion when the number of unknowns is less than the number of equations ; the final ex-
pression of the coordinates by means of angles of the form a + fit. He also added to the
subject in many other directions, and much of the progress which has since been made,
may be said to be founded on his results.
B. L. T. 16
242 THE PRINCIPAL METHODS. [CHAP. XII
295. Laplace's Lunar Theory *.
The publication of Laplace's Mecanique Celeste marked a new epoch in the
history of the lunar theory, owing to the general plan of treatment adopted
and to the manner in which it was carried out. Some account of Laplace's
method has already been given in previous chapters. In Section (ii) of
Chap. II. his general equations of motion with the true longitude as inde-
pendent variable and with the time, the inverse of the projected radius vector
and the tangent of the latitude as dependent variables have been obtained.
The first approximation found by neglecting the action of the Sun has
been given in Art. 52, and the manner in which this is modified to prevent
the occurrence of terms proportional to the time, in Art. 70. By means of
the modified ellipse, those parts of the equations of motion which are due to
the action of the Sun are expressed in terms of the true longitude (Art. 127).
The equations can then be integrated. Laplace's method is to assume
the solution to be a sum of periodic terms whose coefficients are unknown, and
to substitute it in the differential equations: in each unknown coefficient
the characteristic is written separately; he thus obtains, on equating the
coefficients of the different periodic terms to zero, a series of equations of
condition by means of which the coefficients can be calculated. The new
values of u^, t, s are then used to find the third approximation. The method
of procedure is similar to that of Chap. VIL, except that, instead of
the equation for z, the second of the equations (11), Art. 16, is used and
solved in the same manner as the first of these equations ; the analysis is,
however, rather more simple owing to the forms of the left-hand members of
the equations for u lt s. Terms up to the second order in e, e, 7, a/a' are
considered and certain terms of higher orders whose coefficients become large,
owing to small divisors, are also included. The approximations are, in general,
taken to the second order of the disturbing forces.
The coefficients are not developed in powers of ra. As soon as the
equations giving the values of c, g and the equations of condition between
the coefficients have been obtained, Laplace substitutes the numerical values
of the constants in all terms ; only the characteristics are left arbitrary, so that
a small change in the numerical values of any of the constants, except m, will
not sensibly affect the coefficients. The theory is therefore a semi-algebraical
one. The value of the coefficient of the principal elliptic term in the expres-
sion of the mean longitude in terms of the true being thus obtained, the
value of e necessary to his theory is deduced from observation ; the constant
7 is found in a similar manner. His constants e, 7 are such that the
* Mecanique Celeste, Pt. n. Book vn. pp. 169303, 4to. Paris, 1802. Several editions have
since appeared ; the latest, now in the course of publication, is in a collection of all Laplace's
works. Laplace's investigations cover a period of thirty years anterior to the publication of
Vol. m. of the Mec. Gel.
295-296] LAPLACE. SECULAR ACCELERATION. 243
coefficient of the principal elliptic term in the expression of u^ in terras of v
and that of the principal term of s in terms of v, are the same as in undis-
turbed motion*.
Finally, the numerical values of the constants are all substituted, and a
reversion of series gives u 1} v, s and thence 1/r, v, u in terms of the time *f*.
296. Besides giving a general treatment of the Lunar Theory, Laplace enriched the
subject with several new discoveries. Of these, the most noted is his explanation of the
cause of the secular acceleration of the Moon's mean motion % a phenomenon which had
been observed many years before and which had been the subject of several prizes offered
by various academies. Laplace, after an attempt to account for it by supposing that a finite
time was necessary for the transmission of the force of gravity, announced in 1787 that it
was due to a slow variation in the eccentricity of the Earth's orbit, and his theoretical deter-
mination agreed almost exactly with the value deduced from the observations. He also
showed that the same cause produced sensible accelerations in the motions of the node and
the perigee ; his results were confirmed by a later examination of the observations.
A curious fact concerning the discovery of the cause of the secular acceleration of the
mean motion, the theoretical value of which remained unquestioned for over sixty years,
was pointed out by J. C. Adams . Laplace and his followers had integrated the equations
of motion as if e' were constant, substituting its variable value in the results, and had then
determined the acceleration to a high degree of approximation. Adams showed that, although
this method of procedure is permissible in a first approximation, it is necessary to introduce
the variability of d into the differential equations themselves when proceeding to higher
orders. He then found that the true theoretical value, which amounted to about 6" per
century in a century, was only a little more than half of the value obtained by Laplace and
Plana and therefore that theory was insufficient to account completely for the observed
value. A controversy, which lasted for several years, arose concerning the validity of
Adams' method ; his value was, however, confirmed at various times by several investi-
gators amongst whom may be mentioned Delaunayll, Plana IT, Lubbock** Cayleyff and
Hansen^. It must be stated, however, that doubts have been raised concerning the
correctness of the value deduced from observation by the researches of Prof. Newcomb
into ancient eclipses. The question turns chiefly on the trustworthiness of the records.
A full discussion of the points at issue is given by Tisserand|| ||.
* See Art. 159 above.
t A portion of Laplace's second approximation and the determinations of c, g to the order wi 3
are given by H. Godfray, Elementary Treatise on the Lunar Theory.
{ See Arts. 319-322 below.
" On the Secular Variation of the Moon's Mean Motion," Phil. Trans. 1853, pp. 397406 ;
M. N. R. A. S. 1853; Coll. Works, pp. 140157.
|i " Sur 1' acceleration seculaire du moyen mouvement de la Lune," Comptes Rendus, Vol. XLVIII.
pp. 137138, 817827.
11 " Memoire sur 1'equation s&mlaire de la Lune," Mem. d. Accad. d. Sc. di Torino, Vol. xvm.
pp. 157.
** " On the Lunar Theory," Mem. E. A. S. Vol. xxx. pp. 4352.
ft " On the Secular Acceleration of the Moon's Mean Motion," M. N. R. A. S. Vol. xxn.
pp. 171231 ; Coll. Works, Vol. m. pp. 522561.
" Sur la controverse relative a 1'equation seculaire de la Lune," par M. Delaunay, Comptes
Rendus, Vol. LXII. pp. 704 707.
" Researches on the Motion of the Moon," Washington Observations, 1875, pp. 1280.
Illl Mecanique Celeste, Vol. in. Chaps, xui, xrx.
162
244 THE PRINCIPAL METHODS. [CHAP. XII
297. The Theory of Damoiseau *.
Damoiseau follows Laplace's method almost exactly. He assumes the
final forms of the expressions for u 1} nt, s in terms of v and substitutes them
directly in the differential equations. A number of equations of condition,
involving the unknown coefficients in a more or less complicated manner, are
thus obtained and these are solved by continued approximation. Numerical
values are used all through and the theory is therefore entirely numerical.
When the coefficients have been obtained, a reversion of series is made in
order to express the coordinates in terms of the time ; this is also done by
the use of indeterminate coefficients a method always available when the
arguments of the required series are known.
The object of the theory appears to be the determination of the coefficients accurately
to one-tenth of a second of arc. For this purpose he carries them to the hundredth of a
second and includes certain sensible terms due to the actions of the planets and to the
figure of the Earth. The results are given very concisely, but the work will be easily
followed after a perusal of Laplace's theory as given in the Me'canique Celeste. The labour
of finding the values of the coefficients may be grasped from the fact that the mere writing
down of the equations of condition occupies half the Memoir. The tables which he
deduced t from the results of this theory were not entirely disused until those of Hansen
appeared.
298. The Theory of Plana +.
This is an extension of a theory worked out by Plana and Carlini and
sent in to compete for a prize offered by the Paris Academy of Sciences
in 1820. A prize was awarded to them and also to Damoiseau for his theory.
The results of Plana and Carlini were not printed, but later Plana issued
the three large volumes referred to in the footnote. The method of Laplace is
used ; Plana, however, instead of substituting numerical values, makes a literal
development in powers of m, e, e', 7, a/ a'. The results are, in general, carried
to the fifth order of small quantities ; certain coefficients, which are expressed
by slowly converging series, are carried to the sixth, seventh and eighth orders.
In point of accuracy, judged by Hansen's theory, it is about equal to that of
de Ponte'coulant and slightly inferior to the numerical theory of Damoiseau ;
the inferiority is partly due to the slow convergence of the series arranged in
powers of m and partly to errors which have crept into the work errors
unavoidable where the developments are of such length and complexity. As
a literal development it has only been completely superseded by Delaunay's
theory.
* " Memoire sur la Theorie de la Lime," Mem. (par divers savants) de VInst. de France,
Vol. i. (1827), pp. 313598.
t Tables de la Lune, formees par la seule theorie de Vattraction et suivant la division de la
circonference en 400 degres, Paris, 1824. Tables... en 360 degres, Paris, 1828.
$ Theorie du Mouvement de la Lune, 8vo. Turin, 1832, Vol. i. 794 pp. ; n. 865 pp. ;
in. 856 pp.
297-300] DAMOISEAU. PLANA. POISSON. LUBBOCK. AIRY. 245
299. The Method of Poisson*.
Poisson proposed to apply the method of the Variation of Arbitrary Con-
stants to the solution of the lunar problem. For this purpose he introduces the
equations of Art. 83 above. The disturbing function is to be expanded by
the purely elliptic values of the coordinates and the result substituted in the
right-hand members of the equations. To obtain the second approximation
to the values of the elements, they are regarded at first as constants in
the right-hand members; the equations may then be solved and the resulting
values of the elements, in terms of the time, are to be used as the basis of a
second approximation by substituting them, instead of their constant values,
in the same parts of the equations. To obtain the solar inequalities in the
Moon's motion, the method in this form is almost useless on account of the
enormous developments which it would entail, and it would not be considered
here were it not for its value in investigating the inequalities arising from
other sources and, in particular, for those inequalities known as ' long-period '
and ' secular.' In fact, Poisson only gives a few calculations as illustrations
of the method. It is chiefly of value in the planetary theory.
300. The Method of Lubbock^.
The publication of Lubbock's researches in the volumes of the Phil.
Trans, between 1830 and 1834, places his method next in historical order ;
they are collected and extended in the pamphlets referred to in the footnote.
The method is the same as that of de Pontecoulant, whose results were not
published until 1846 ; but, from the remarks made by Lubbock and de Ponte-
coulant in their prefaces, it is evident that they had adopted the same plan
independently. Lubbock never carried out his method with any complete-
ness: his published papers contain an explanation of the method, a full
development of the second approximation, and the calculation of the earlier
approximations to the coefficients of certain classes of terms ; his results are
compared with those of Plana.
Next in order come the theories of Hansen and Delaunay which have been already
treated. Finally, mention must be made of Airy's method J, which was rather a verification
of previous results than a complete theory in itself.
Airy proposed to take Delaunay's expressions after numerical values had been sub-
stituted for the constants and, considering each coefficient to need a small unknown
correction, to substitute the results, together with the unknown parts, in the equations of
* ' ' M&noire sur le mouvement de la Lune autour de la Terre," Mem. de VAcad. des Sc. de
VInst. de France, Vol. xm. (1835) pp. 209335. (Bead in 1833.)
t On the Theory of the Moon and on the Perturbations of the Planets, London, 8vo. Pt. i.
(1834), 115 pp., with an Appendix containing Plana's results; Pts. n. (1836), in. (1837), iv. (1840),
417 pp. ; Pt. x. (1861), 94 pp., with tables.
J Numerical Lunar Theoi-y, London, 1886, fol. 178 pp.
246 THE PRINCIPAL METHODS. [CHAP. XII
motion*. He had worked at this for several years but, after the volume containing his
results was published, he discovered a serious omission which altogether invalidated them ;
the large corrections which he had found were necessary to make Delaunay's results satisfy
the equations of motion, were probably due to this unfortunate error. In a letter f to the
Secretary of the Koyal Astronomical Society, he says, " I keep up my attention to the
general subject, but with my advanced age (eighty-eight) and failing strength I can
scarcely hope to bring it to a satisfactory conclusion. I will only further remark that I
believe the plan of action which I had taken up would, if properly used, have led to a
comparatively easy process, arid might in that respect be considered as not destitute of all
value."
301. Tables.
The tables of the Moon's motion which have been formed from the results of theory
alone, in order to calculate the position of the Moon at any time, have already been referred
to, in connection with the theories from which they were deduced. In addition, we may
mention those of Mayer (London, 1770) formed by a combination of theory and observation,
of Mason (London, 1787), which were Mayer's tables improved, of Burg (Paris, 1806), of
Burckhardt (Paris, 1812) and, for the Parallax of the Moon, of Adams (M. J\ T . R. A. S.
Vol. xin. 1853; Nautical Almanac, 1856; Coll Works, pp. 89107). Later efforts in
this direction have been made chiefly for the purpose of correcting Hansen's tables (see
Art. 238).
302. In making a comparison of the various methods of treating the lunar problem,
several considerations enter. There does not appear to be any method which is capable of
furnishing the values of the coordinates with a degree of accuracy comparable with that of
observation, without great labour ; and, in the present state of the lunar theory, looking
only to a practical issue, what is required is rather a verification of the results of previous
methods, say those of Hansen and Delaunay, than new developments. Again, some
methods appear to be most effective for one class of inequalities while other methods give
another class of inequalities most accurately. The question to be discussed is mainly the
relation between the accuracy obtained and the labour expended.
As regards the inequalities produced by the action of the Sun, the methods may be
divided into three classes. The first or algebraical class contains those in which all the
constants are left arbitrary ; the second or numerical, those in which the numerical values
of the constants are substituted at the outset ; the third or semi-algebraical, those in which
the numerical values of some of the constants are substituted at the outset, the others
being left arbitrary : the most useful case of the last class appears to be that in which the
numerical value of the ratio of the mean motions is alone substituted. The advantage of
an algebraical development will be readily recognized. In a numerical development, slow
convergence is to a great extent avoided, but the source of an error is traced with great
difficulty and any change in the values of the arbitraries can not be fully accounted for
without an extended recalculation. The semi-algebraical class, in which the value of m is
alone substituted, appears to possess an accuracy nearly equal to that of a numerical
development, and it has the advantage of leaving those constants arbitrary whose values
are known with least accuracy.
It is difficult to judge of the labour which any particular method will entail, without
performing a considerable part of the calculations by that and by other methods. As far
* An exhaustive analysis and criticism of Airy's method is given by M. Eadau, Bull.
Astronomique, Vol. iv. pp. 274 286.
t " The Numerical Lunar Theory," M. N. E. A. S. Vol. XLIX. p. 2.
300-302] REMARKS ON THE METHODS. 247
as it is possible to estimate, either by general considerations or by the amount of time previous
lunar theorists have spent over their calculations, it may be stated that those methods
which have the true longitude as the independent variable must be altogether excluded if the
solar perturbations are required, owing to the necessary reversion of series. For a complete
algebraical development carried to a greater accuracy than that of Delaunay, none of the
methods given up to the present time seem available without the expenditure of enormous
labour : Delaunay's calculations occupied him for twenty years. If we may judge from
the inequalities computed up to the present time, the methods of Chap. xi. seem
to be best suited to a numerical or semi-algebraic development. It is true that they give
the results expressed in rectangular instead of in polar coordinates, but the labour of
transformation is not excessive in comparison with that expended on the previous
computations, while the accuracy obtained far surpasses that of any other method ; the
transformation of the series, however, would not be necessary for the formation of tables.
The disadvantage of de Pontecoulant's method is the necessity of obtaining the parallax,
with an accuracy much beyond that required for observation, before the longitude can be
found ; this remark applies also to the methods of Chap. XL, but in rather a different way.
Hansen's method labours under the disadvantage of putting the results under a form which
makes comparison with those of other methods difficult. Another consideration which is a
powerful factor, is the question as to how far the ordinary computer, who works by definite
rules only, can be employed in the calculations ; and here the methods of Chap. xi. appear
to have an advantage not possessed by any of the earlier theories.
With reference to the classical treatises on Celestial Mechanics, there is little doubt
that the works of Euler and Laplace will best repay a careful study ; those of Lagrange in
a different direction the general problem of three bodies must also be mentioned. The
ideas upon which aU the later investigations have been built, may be said to have originated
from the works of one or other of these three writers.
CHAPTER XIII.
PLANETARY AND OTHER DISTURBING INFLUENCES.
303. AN explanation of the way in which the principal effects of
planetary action and of the figure of the Earth may be included in the
lunar theory will be given in this chapter. A general plan of integration for
the new terms introduced into the disturbing function will be first explained ;
the discovery and development of the disturbing functions for the direct and
indirect actions of the planets and for the direct effect of the ellipticity of
the Earth then follow, the results being illustrated by applying them to a
few of the principal inequalities. The perturbations produced by the motion
of the ecliptic and by the secular variation of the solar eccentricity are,
owing to their peculiar nature, treated by special methods. In all cases,
the developments will be only given as far as they are necessary for the
purpose of explanation ; references are given to the memoirs in which more
complete investigations may be found. As far as the end of Art. 318, Delau-
nay's notation will be used; the determination of the secular acceleration
being made by the use of de Pontecoulant's equations, we use the notation
of Chap. vn. in Arts. 319322.
The effect of the terrestrial Tides and of the figure of the Moon on the motion of the
latter will not be treated here. The former is considered in the Memoirs of Prof. G. H.
Darwin* in detail ; the chief effect is on the Moon's mean period and mean distance, and
the amount of the correction, within the limits of time during which observations have
been recorded, is very small. As to the latter, it is very doubtful whether it produces
any appreciable effect : Hanseri introduces an empirical periodic term supposed to be due
to the difference between the centre of mass and the centre of figure of the Moon f.
General Method of Integration.
304. The expression of the disturbing causes which affect the motion of
the Moon can, in nearly all cases, be made by inserting additional periodic and
constant terms in the disturbing function. The periods and coefficients of
these terms of the disturbing function the parts which involve the elements
* Phil. Trans. 18791881.
t Darlegung, i. pp. 175, 474479.
303-305] GENERAL METHOD OF INTEGRATION. 249
of the Moon as well as those arising from other sources can generally be
found with an accuracy sufficient for practical purposes ; for this reason, it is
advisable to use a method of integration which shall be adaptable easily to
any periodic term, and such a method, founded on Delaunay's formulae, has
been devised by Dr Hill*. Its value chiefly depends on the fact that the
coefficients of the new terms in the disturbing function are always small and
that, in consequence, it is seldom necessary to consider the changes produced
in the new terms of the disturbing function by those changes of the elements
which occur when any one of the old or new periodic terms is eliminated
by Delaunay's processes. The operations are therefore similar to the majority
of those mentioned in the last paragraph of Art. 197, but it will be seen that
we may use numerical values for the elements of the Moon's orbit and that,
owing to this fact, the operations of Delaunay may be very considerably
abridged. The numerical results given below are those obtained by
Radaui* in a valuable Memoir to which frequent reference will be made. He
introduces a slight modification of Hill's method and his numerical values for
SM differ to a small extent from those of Hill. Periodic terms only will be
discussed ; the changes produced by new non-periodic terms due to the
inequalities considered below, are very small.
It is supposed that the periodic terms, arising solely from the action of
the Sun considered to be moving in an elliptic orbit, have been eliminated,
and that the disturbing function contains only the remaining constant portion
together with the new terms to be considered. It is further supposed that the
operation of Art. 198 above has not been carried out, and that the final change
of constants, which Delaunay makes in order to reduce his expressions to a
suitable form (Art. 200), is as yet not performed. The results required here
are all contained in Delaunay's volumes : the latter will be referred to as in
Chap. IX.
305. Delaunay's canonical equations are (Art. 183)
dL_dR dl _dR
dt~ dl' "" "' dt dL' "" '
Let B be the constant part of R which remains after the periodic terms due
to the Sun have been eliminated ; we then have, by Art. 198,
dB dB dB
* It is contained in pt. in. of his Memoir ' ' On certain Lunar Inequalities due to the action of
Jupiter and discovered by Mr E. Neison," Astron. Papers for Amer. Eph. Vol. in. pp. 373393.
t "Recherches concernant les In6galites planetaires du Mouvement de la Lune," Ann. de
I'Obs. de Paris (Memoires), Vol. xxi. pp. 1114. See also, " Remarques sur certaines inegalites
a longue p&iode du mouvement de la Lune," Bulletin Astrommique, Vol. rx. pp. 137 146,
185212, 245246.
250 PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII
Let BR be the new part of R, and let BL, 8G, BH, Si, Bg, Bh be the new
parts of L, G, H, I, g, h, due to BR; let B1 , Bg , Bh be the new parts of
1 0) g 0) h,) (which are functions of L, G, H), due to BL, BG, BH. The canonical
equations may be written,
*M-J**...,...; * -,-*.... .............. (1).
We choose out one of the periodic terms of BR and put
BR = + A cos (il + i'g + i"h + at + ft) = A cos 6,
where at + is the part of the argument independent of the lunar elements,
and where A is the coefficient, containing three of the six lunar elements,
namely, L, G, H or a, e, 7. We have
= (il, + i'g, + i"h + a.)t + j3' = Mt + ff,
suppose, so that M is the motion of the argument 6.
As A is always very small, we can, with a sufficient approximation,
substitute the value of BR in (1) and integrate on the supposition that
L, G, If, 1 , g 0) h Q are constant when multiplied by the small quantity A.
In this way we find
BL = -Acos0, BG = -Acos0, BH=-Acos0.
M M M
Whence, since a, e, 7, 1 , g Q , h Q are functions of L, G, H,
.da ., da .,da\A ~ .,
The second three of equations (1) give
d ~, dA
Substituting the values of B1 , Bg , Bh and integrating on the supposition that
the lunar elements are constant in the right-hand members, we find
. a dAsmd ^ s , /0 ,
''" " ......
The equations (2), (3) give the new terms to be added to the elements.
306. As regards the calculation of the various quantities present in (2),
(3), the partial differential coefficients of a, e, 7, Z , g Q , h with respect to
L, G, H may be obtained from the expressions given by Delaunay*. Since
* Delaunay, n. pp. 235238.
305-306] EQUATIONS FOR THE VARIATIONS OF THE ELEMENTS. 251
we shall not consider the changes produced in jR by the changes in the
elements, the numerical values of n'/n, e, e, 7, a/a may be substituted in the
results : the numerical values of the lunar constants will not be quite the same
as those used by Delaunay in his final results, because the final transformation
which alters the meaning of n, e, 7, a (Art. 200) has not been made ; the
necessary modifications can be obtained from the formulae given by Delaunay
for the transformations*. The values thus obtained by Radau are
7*771 = 0-0744, e=0-0549, 7 = 00449, a/a' = 0-00257.
To obtain the partial derivatives of A with respect to L, G, H, we have
M_dAda, dAde_d_^d_V_
dL ~ oadL* de dL + dy dL '
in which the partials da/dL, . . . may be numerically calculated in the manner
just explained. These calculations being made once for all, we can obtain
very simple formulas for the determination of the coefficient of any periodic
term.
Instead of g, the mean longitude M Q = I + g + h is introduced, so that
and instead of J/, the ratio p riJM. Put
, p A dA . A dA ., . dA .,, .
A =%W> a S~a =jA ' e Ye =}A > ^=-> A -
After inserting the numerical values of all the known terms, as explained
above, Radau findsf that the equations (2), (3) become
Sa/a = (014901i - 0'000246i" - 0-000006O A cos 0, ^
Se = (1-4215 i - 1'4238 % + 0'00024 i") A' cos 0,
S y = (O-OOOlOi + 0-41203 i' - 0-41370 i") A' cos 0,
8M = {(-3-0576 1 + 0-05601 i' - 0'01124 i")p
- 0-14876J + 0-02551 / + 0-03492 /'} A sin e,
$1 = {(- 3-0826 i + 0-06142 i - 0-03621 i")p
- 0-14901J - 25-891 / - 0'00232 j"\ A' sin 0,
$h = }(_o-03641i + 0-02890 i v - 0-00372 i")p
+ 0-000006J - 0-00435 / + 9'2169 /'} A' sin 6 }
The simplicity of these equations enables us to calculate easily the first
approximation (generally sufficient), according to powers of A or A' y to the
coefficient of any term. Examples will be found below.
* Delaunay, n. p. 800. t Recherches etc., p. 36.
252 PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII
307. To calculate the corresponding terms in the coordinates, it will
often be sufficient to limit their expressions to the principal elliptic and
solar terms, and to find the small changes in the coordinates induced by the
calculated changes in the elements, after inserting the numerical values of all
parts of the coefficients except the characteristics. For example, it is in many
cases sufficient to put
v = M + 2e sin I + (Hie sin (2D - J),
where the numerical part 0'41 arises from the series in powers of ri/n etc.
From this expression, Sv may be obtained by putting D = M n't e',
and causing M , e, I to receive the small increments calculated above.
If this be not sufficiently approximate, we can take the principal terms in
the unreduced values of the coordinates in terms of the elements* and submit
them to a variation S, all the elements being supposed variable.
General method for the Inequalities produced by the
Direct and Indirect Actions of the Planets.
308. The Disturbing Functions.
Let x', y', z' ', r' be the coordinates of the Sun, x, y, z, r those of the Moon,
referred to axes fixed in direction and passing through the Earth, and let S
be the cosine of the angle between r, r'. The disturbing function for the
motion of the Moon, due to the Sun, is, by Art. 107,
In order that the coordinates x', y', z' , r' may be considered to refer to the
motion of the Sun about G (the centre of mass of the Earth and the Moon),
it is necessary to multiply the second term of this expression by
(E-M)I(E + M).
Let f , i), f, D be the coordinates and distance of a planet P, referred to the
same axes. The action of a planet of mass ra", on the motion of the Moon,
will evidently be expressed by a disturbing function of the same form as R,
namely R', where
S' being the cosine of the angle between r, D. In order that f, 77, f, D may
be considered to refer to G as origin, it is necessary to multiply the second
term by (E - M)/(E + M).
* Delaunay, n. Chaps, vii. ix.
307-309] DISTURBING FUNCTIONS FOR PLANETARY ACTION. 253
The ratios r/r', r/D, m"/m are always small and, in most cases, the effect
of R' on the motion of the Moon will be sufficiently accounted for by
considering only the first term of R' ; the other terms will therefore be neg-
lected here. The inequalities produced by R' are said to be due to the direct
action of the planets. To each planet will correspond a function R' ; but
since the terms produced by the combination of two terms, one from each
such function, are generally negligible, it is only necessary to consider one of
these functions, applying it to the case of each planet successively.
The solar inequalities, as far as they arise from the purely elliptic motion
of the Sun, are supposed to have been determined. The actions of the
planets on the motion of the Earth produce small deviations from elliptic
motion in the apparent motion of the Sun : these, being substituted in R,
may be considered as small corrections S#', &/, Sz' to the coordinates #', y', z'.
As these corrections are never large, it will be sufficient, for the inequalities
thus produced in the motion of the Moon, to limit R to its first term. The
lunar inequalities arising in this way are said to be due to the indirect action
of the planets. Since m" is very small compared with m ', it will not be
necessary to consider these variations of x', y', z' in R'. See Art. 310 (d).
309. Separation of the terms in R, R', and their expressions in polar
coordinates.
Confining R, R to their first terms we have, on introducing rectangular
coordinates, \ \, ,
These may be written
l 3/ 2 \ o^-Y^-y*
-71 -- TT + O - - A - /5
* r* J 4 r 5
*) ...... (5),
in which it will be noticed that the coordinates of the Moon are separated
from those of the Sun or of the Planet. It is now necessary to express R, R'
by means of the polar coordinates of the planet and of the Earth (or of Q),
referred to the Sun, and those of the Moon referred to the Earth.
The notation of Chap. IX. will be used whenever it differs from that of
254 PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII
previous chapters. We suppose the ecliptic to* be a fixed plane perpendicular
to the axis of z. As before,
L = Distance of the Moon from its node,
h = Longitude of the node,
7 = Sine of half the inclination of the lunar orbit.
We have then (fig. 5, Art. 73)
x = r cos L cos h r cos i sin L sin h,
y r cos L sin h + r cos i sin L cos h,
z r sin L sin i ;
or x = (1 y 2 ) r cos (L + h) + y 2 r cos (L h),}
y = (l-rf)rsm(L + h)-Y 2 rsm(L-h)\ ................. (6).
z = 2y Vl y 2 r sin i j
Let F' = Longitude of the Earth as seen from the Sun. The longitude of
the Sun, as seen from the Earth, will be V + 180, and therefore
x' = -r cos V, 2/' = -/sin V, z' = Q ............... (7).
The coordinates , 77, f, being those of P relative to E, are those of P
relative to the Sun added to those of the Sun relative to E. The coordi-
nates of P relative to the Sun may be deduced from (6) if we put
y" = Sine of half the inclination of the orbit of P to the ecliptic,
h" = Longitude of its node on the ecliptic,
V" = Longitude of P as seen from the Sun, reckoned along the ecliptic
to its node and then along its orbit,
r" = Solar radius vector of P,
for y, h, L -f A, r, respectively ; the coordinates of the Sun are given by (7).
We therefore obtain
f = - r ' cos V + (1 - y //2 ) r" cos V" + y" 2 / 7 cos ( V" - 2/O,]
77 = - r sin V + (1 - y" 2 ) r" sin V" - y" 2 / 7 sin ( V" - 2A"), I ...... (8).
f = 2y"VT^7'V' sin ( v - h") }
Whence if- = 2 + 7? 2 + ? 2
= D<? + 4y'Vr" sin ( V - h") sin ( V" - h"),
where D<? = r* + r" 2 - 2/r" cos ( V - V") ..................... (9).
It will be unnecessary to consider powers of y" beyond the second ; we shall
therefore have
* = + {cos(F'+F"-2r)-cos(F-F")) ...... (10).
By means of the formulae (6) (10), R, R' can be expressed in terms of
r, r, r", L, V, V" , h, h', h", y, y".
309-310] DEVELOPMENT OF THE DISTURBING FUNCTIONS. 255
310. Development of the Disturbing Functions.
We shall first expand the expressions obtained for the various parts of
the disturbing functions, by substituting elliptic values for the coordinates of
the three bodies, and then show how non-elliptic terms present in these
coordinates may be taken into account. When this has been done, the
disturbing functions are to be expressed as sums of periodic terms.
(a) The portions which depend only on the coordinates of the Moon.
Neglecting powers of 7 beyond the fourth, we obtain, from (6),
i (r - 3<)/r 2 = J (1 - 67* + 67*) + f 7 2 (1 - 7 2 ) cos 2z,
f O 2 - y^/r 2 = f (1 - T 2 ) 2 cos (2z + 2^) + f 7 4 cos (2z - 2h) + f 7* (1 - 7*) cos 2h ;
corresponding expressions may be obtained for f tfy/r 2 , f xz/r 2 ,
As in Chap. IX., let g be the distance of the lunar perigee from the node
and /the true anomaly. Then
and, from the expressions given in Art. 39,
rYa 2 = 1 + 1 e 2 - (2e - Je 3 ) sin I - . . . ,
- 2 cos (2/+ a) = (1 - fe 2 ) cos (2Z + a) + e cos (3^ + a) - 3e cos(J + a) + ...,
d
where a may be any angle.
By giving to a. suitable values, all the five functions J (r 2 3^ 2 ), J (a? 2 i/ 2 ),
etc., present in R, R', can be expressed in series of cosines or sines involving
/, g, h in their arguments and e, y in their coefficients. Moreover, the orders
of the coefficients can be associated with the multiples of /, g, h in the
corresponding arguments, by the rules obtained in Chap. VI. Putting M Q for
I + g + h, it is easily seen that
r 2 - 3* 2 = 2^ e* cos kl + rf ^B^ cos (2JJf c -2h kl),
where k is a positive integer and where A , B Q are coefficients of zero order
containing powers and products of e 2 , T 2 . The other four functions depend-
ing only on the coordinates of the Moon may be similarly treated.
(b) The parts which involve the coordinates of the Sun and
of the Planet in the second degree.
By means of the formulae (8), we find the values of f 2 , ? 2 -if, fr, ff, 17?
expressed as sums of cosines or sines. The arguments of these terms will
contain multiples of V, V", h" , and the coefficients will contain 7" and will
256 PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII
have r' 2 , r" 2 or r'r" as factors. They are expanded in terms of the elliptic
elements of the Sun and of the planet by the formulae
r'/a = 1 + K 2 - e ' cos l> ~ - V ' = M + 2e/ sin l> +
/'/a" = 1 + ie" 2 - e" cos I" - . . ., 7" = Jf " + 2e" sin I" + . . .
where M ', M " denote the mean longitudes of the Earth and the Planet
respectively. The portions of R which depend only on the coordinates of the
Sun, present no difficulty.
In the same manner as with the coordinates of the Moon, the composition
of the argument of any term in the development of these expressions may be
associated with the order of its coefficient, though the connection is by no
means so simple. For instance, it may be shown that
e ' k 'e >>k " cos {2if " -2iM Q " + j (Mj - M,") + 2ih" til' k'T],
where A Q is a coefficient of zero order : k', k" are positive integers or zeros,
and i,j have positive integral or zero values such that i+j < 3.
(c) The parts of R arising from the Divisors D^.
The equation (10) shows that it is only necessary to consider the divisors
/). They are the functions which cause the great difficulty in finding the
planetary inequalities in the Moon's motion; the difficulty is of the same
nature as that encountered in the planetary theory and it arises from the
near equality of /, r" or of ri, n" in the cases of those planets which are not
far from the Earth (see Art. 9).
We can expand 1/A& by means of Legendre's coefficients, in the form
AT* = J V' + V* cos ( v ' ~ V"} + V c <> s 2 (V - V") + ...,
where B q ( b is a homogeneous function of r', r" ; when r' t r" are comparable
with one another in magnitude, these coefficients diminish very slowly and it
becomes frequently necessary to consider terms in which j is a large number*.
In the case of a superior planet, expansion must be made in powers of r'\r"
and, in the case of an inferior planet, in powers of r"/r.
Substituting the values of r', r", V, V", given by (11), it is easily seen
that
' k 'e" k " cos { j (Mj - M Q ") k'l' k'T},
in which A Q is a homogeneous function of a, a" and of zero order with respect
to e', e", and j, k' } k" have positive integral or zero values.
* Radau's method (Recherches, pp. 1731), for abbreviating the calculations in such cases,
should be consulted.
310-311] TERMS DUE TO INDIRECT ACTION. 257
(d) The terms arising in R from non-elliptic terms present in the
coordinates of the Earth, the Planet and the Moon.
Two methods may be used for these terms. We may either consider r, v'
(and also u' y if the terms dependent on the motion of the ecliptic be not
neglected) to receive small increments 8r, &v' and then expand the formula
(3) of Art. 108 in powers of these increments by means of Taylor's theorem,
substituting for r, v their elliptic values and for Sr', Sv the small terms
given by the planetary theory. Or we may suppose the additional terms
to be given as small corrections to the elements of the solar orbit, in which
case the development (5) of Art. 114 will be available after the changes of
notation, necessary to express the result in Delaunay's form (Arts. 123, 180),
have been made. The same methods may also be employed to take into
account any non-elliptic terms present in the coordinates of the planet.
The solar terms present in the coordinates of the Moon cannot, in all
cases, be neglected. In the process of eliminating, by Delaunay's method,
the periodic terms of R which are due to the action of the Sun, the lunar
elements present in R' will be changed at each operation. Instead of
inserting the changes, thus produced, by adding them to the elements, it
will generally be more convenient to suppose that the elliptic values of the
coordinates receive small increments, these increments being the principal
solar terms which occur in the unreduced values of the coordinates, as given
by Delaunay. Numerical values may usually be substituted in all parts of
the coefficients of the new terms, except in the characteristics.
311. After the various processes, outlined above, have been carried out,
it is only necessary to multiply the series obtained for the various parts of
R or R and to express them as sums of cosines of angles. To do this in any
general manner, would involve enormous labour due chiefly to the divisors
z#; and much of the labour would be without result, because the great
majority of the terms have quite insensible coefficients in the coordinates
of the Moon. The plan usually adopted consists in trying to discover the
terms which have long periods and which, in consequence, may have coeffi-
cients large in comparison with their order when the equations of motion
are integrated. Certain short-period terms which are either associated with
these long-period terms, or which have an independent existence in the
disturbing function, must also be' included when there is a possibility of a
large coefficient in one of the coordinates. In every case, the methods by
which R, R have been developed, give the order of the coefficient in the
disturbing function in relation to the eccentricities and inclinations. No
further rules can be given to guide us in the choice of these terms. Many of
them have been indicated by observation : others have been obtained directly
from theory in the course of investigations into the effects of planetary
action.
B. L. T. W
258 PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII
The method here outlined for the treatment of the disturbing function was first given
by Dr Hill* and was afterwards extended and applied to many planetary inequalities
by M. Kadau t. Combined with Hill's method of integration (Arts. 304 307), it forms
the only complete and generally effective method known up to the present time for the
investigation of the planetary inequalities.
There are several ways in which the calculation of the coefficient of a term in R',
with a given argument, may be abridged ; to give an account of them would lead us
outside the limits of this treatise. The reader is referred to Eadau's memoir and also
to Tisserand's Mfaanique C4leste\ which contains an account of this memoir.
We shall illustrate the methods of the previous articles by applying them to the
calculation of two celebrated inequalities one due to the direct action of Venus and the
other due to its indirect action.
312. Example of an inequality due to the Direct action of Venus.
There is a term in R' of period 27T/(Z + I6n' 18n"), where n" is the
mean motion of Venus. Observation furnishes, for the daily motions,
1 = 47033"'97, n 1 = 3548"-19, n" = 5767"'67.
The daily motion of this inequality is therefore 13"'0, giving a period
of 273 years.
It can be shown that the term in R, having this period, is
- 0"-00133w /a a a e cos (I + 16M ' - 18M " + 2h").
We therefore have, on applying the formulae of Art. 306,
p = - 273, A' = - 0"'00133pe = 0"'0199 ;
i = l, i' = = z"; j=2, /=!, j" = 0;
and the equations (4) give
SM Q = 16"-6 sin (I + 16Jlfo' - I&flf " + 2A"),
which also gives the approximate value of the term in longitude, since SI is
nearly equal to BM and since the equations for Sa, Be, By, Bh only give small
coefficients. The more accurate value of the coefficient, when terms of
higher orders are included, is 14"*4||.
This is the largest known periodic inequality in longitude, produced by the action of
the planets. Indeed, according to the table given by Eadau at the end of his memoir, no
other inequality in longitude has a coefficient so great as 1", although there are several
greater than half a second ; the majority of the inequalities are of comparatively short
period either approximating to the lunar month or having a period of a few years.
* "On certain Possible Abbreviations in the Computation of the Long-Period Inequalities of
the Moon's Motion due to the Direct Action of the Planets," Amer. Journ. Math. Vol. vi. pp.
115130.
t Recherches etc.
J Vol. in. Chap. xvni.
Tisserand, Mec. Gel. Vol. in. p. 396.
|| Badau, Recherches etc., p. 64.
311-313] CASE OF INDIRECT ACTION. 259
The inequality just calculated was discovered by Hansen*, who found by theory a
coefficient of 27"'4. He also noticed another inequality with a mean motion 8ri' I3ri
and a coefficient 23"'2. In both cases Hansen was in error ; the former coefficient has
just been seen to be about 14"'4, while Delaunayt and others have shown that the
coefficient of the latter term is less than 0"-004. The values of the coefficients, which
Hansen obtained by a discussion of the observations and which he adopted in his tables,
were 15" -34 and 21"'47, respectively, including the parts due to the indirect action (see
Art. 313).
The Indirect Action of a Planet.
313. A very simple formula can be obtained for this in many cases.
Neglecting the perturbations of the plane of the ecliptic and the ratio of
the parallaxes, we have, by Art. 116,
SR = - (3//)oV - (dR/dv) W.
Let us confine our attention to the term m'r 2 /^ 3 of R (Art. 108), since
most of the larger inequalities of long period will arise from this term. Sub-
stituting in the expression for &R and neglecting the solar eccentricity,
we obtain immediately
Suppose that the solar tables give an inequality 8r' = a' A cos 6, where A,
6 are independent of the lunar elements. We obtain
&R=-| ri' 2 a?A cose.
Using the equations (4), we have ', i', i",j',j" zero and j = 2. Whence
BM 9 = SI = f x 0'149^ sin = %pA sin 6,
approximately.
If we suppose further that the inequality Sr' is of long period and that it
arises principally from a variation &a of a', a direct approximate relation
between &v and SV can be deduced. For (Art. 81)
and therefore, owing to the various conditions assumed above,
But since n' z a' 3 = ra', we have
Za'Sri = - 3n'8a f = - 3n'Sr' = - Ma' A cos 6.
Therefore W = - f sin d = - f pA sin 0.
At
* " Auszug aus einem Briefe," etc. Astr. Nach. Vol. xxv. Cols. 325332 ; "Lettre & M. Arago,"
Comptes Rendus, Vol. xxrv. pp. 795799.
t " Sur TIn6galit6 lunaire 4 longue periode due k 1'action perturbatrice de V6nus et dependant
de 1'argmnent 13Z'-8r," Conn, des Temps, 1863, Additions, pp. 156. The result is given
on p. 46.
172
260 PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII
On combining this result with the value just obtained for BM , we find
Bv = BM = -4rBV' = -}BV,
approximately. In this case we can therefore obtain an approximate idea of
the magnitude of the coefficient in longitude, by dividing the corresponding
inequality in the Earth's longitude by - 7.
For example, the solar tables * give an inequality of period
27r/(13rc / - 8ft") = 239 years,
due to the action of Venus. In longitude, this is
BV = + 1"'92 sin (13Jfo' - 8M " + 132).
Multiplying the coefficient by 4/27, we obtain for the corresponding in-
equality in the Moon's motion, due to the indirect action of Venus,
Bv=- 0"-284 sin (13Jf ' - 8 Jf " + 132) ;
the correct value, as found by Delaunay f, being
Bv = - 0"'272 sin (I3MJ - 8M " + 138).
The inequality having this period, due to the indirect action of Venus,
is therefore much greater than that, with the same period, produced by the
direct action (Art. 312).
For a complete investigation of the inequalities produced by the direct and indirect
actions of the planets, the reader is referred to Kadau's memoir. A large number of
references to the labours of other investigators on the same subject is also given. To these
may be added an important paper by Newcomb}, "Theory of the Inequalities in the
Motion of the Moon produced by the Action of the Planets," in which the whole theory of
the subject is treated in a very general manner.
Inequalities arising from the Figure of the Earth.
314. Let A, B, C be the moments of inertia of the Earth about three
rectangular axes meeting in the centre of mass, and let / be the moment of
inertia about the line connecting this point with the centre of mass of the
Moon. The difference of the attractions on the Moon, of the Earth and of a
spherical body of equal mass, produces a potential
We suppose that one principal axis of the Earth is its polar axis and that
the moments of inertia about the other two axes are equal. Let B = A, and
* Ann. de VObt. de Paris (Mem.), Vol. iv. p. 35. The inequality is given in the form
- l"-283 sin (13M ' - &M ") + l"-425 cos (13M f - SM "). '
t See footnote, p. 259.
J Astron. Papers for Amer. Eph. Vol. v. Pt. in. pp. 97-295.
E. J. Bouth, Rigid Dynamics, Vol. n. Art. 481.
313-315] THE FIGURE OF THE EARTH. 261
let d be the declination of the Moon. We then have
I = A cos 2 d+ <7sin 2 d;
and the new part to be added to the disturbing function is
8R = (2A + C- 3A cos 2 d - 3(7 sin 2 d)
where JJL is the sum of the masses of the Earth and the Moon, and
It is proved in works on the figure of the Earth * that, if the Earth's
surface be supposed to be an equi-potential surface,
where E is the Earth's mass, R its equatoreal radius, a its ellipticity and JB
the ratio of the centrifugal force to gravity at the equator.
The numerical determination of \dd may be made in several ways. It is possible to
find it by the reverse process of comparing the theoretical values of the coefficients of
the principal terms produced by the figure of the Earth on the motion of the Moon, with
those deduced from observation ; owing to the near equality of the periods of these terms
with the periods of certain terms produced by planetary action terms whose coefficients
are not known with a great degree of certainty this method is not capable of very great
accuracy. The value may be deduced from the latter of the formulae given above for /*',
by obtaining a from geodetic measures and /3 from pendulum observations ; this method
involves an assumption concerning the interior constitution of the globe. Thirdly, it may
be obtained from the first formula and Hill so finds itf by a discussion of pendulum
observations, made to find the intensity of gravity at various stations on the surface of the
Earth. A determination has also been made by comparing the observed and the calculated
values of the yearly precession of the Equinoxes.
315. Let x (Fig. 5, Art. 73) be the ascending node of the Ecliptic on the
Equator the place from which longitudes are reckoned and let w^ be the
inclination of these two planes. If / be the pole of the equatoreal plane, we
have zsf = m lt s'Jlf=90 -d, zM=9Q-i7, z / zM=9Q-v.
The triangle z'zM therefore gives
sin d = sin 7 cos o^ + cos u sin o>! sin v.
We also have sin u= sin i sin L,
cos u cos (v h) = cos L, cos u sin (v h) = sin L cos i ;
* e.g. J. H. Pratt, Art. 85.
t In chapter v. of his Memoir " Determination of the Inequalities of the Moon's Motion which
are produced by the Figure of the Earth : a supplement to Delaunay's Lunar Theory," Astron.
Papers for Amer. Eph. Vol. in. pp. 201344.
262 PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII
and therefore
cos u sin v = cos 2 \i sin (L + ti) sin 2 %i sin (L h).
Hence
sin d = cos o^ sin i sin L + cos 2 J i sin o^ sin (L + h) sin 2 ^i sin o^ sin (z h).
Putting sin ^i = y and neglecting powers of 7 beyond the first, we obtain
sin d = sin Wj sin (L + h) + 27 cos o^ sin L ;
= (titf/v*)[% - J sinX 4- i sinX cos 2 (z + &) - 7 sin 2^ (cos & - cos (2z + &)}].
As the method of integration will be that of Arts. 304 307, it is neces-
sary to substitute elliptic values for r and L. We put
r = a (1 e cos I + . . .)> L g + I + 2e sin I +
The terms in SR which will give the greatest coefficients are those of long
period : it is easily seen that, after the substitution of elliptic values, there
is only one such term that with argument h. We therefore take
R = - pk' (y/a 3 ) sin 2^ cos h.
All that now remains is the application of the formulae (4). Since the
diurnal motion of the node is 190"'77, we find
p = n '/h = - (3548"-2 + 190"'77) = - 18-60 ;
= ** = 0, r=l; j = -3, 7 = 0, j" = l;
A'^-pPj^L s i n 2ft,, = 18-60 x ^ - 2 7 sin 2ft, t .
f a? n 2 a? n 2 a 2
Substituting and retaining only three places of decimals in the coefficient of
A' cos h, we obtain
= + 0-6904' cos />, 8Z = + 11184'8infc, Sh = + 9'28GA' sin
Hill finds, by his discussion of pendulum observations *,
(tf/a*) sin 2ft,! = // '072854.
Hence, with the values of 7, n'jn given in Art. 306, 4'= 10 //> 99. The results
(13) then give
S 7 =_4"-55cos/i,
8M Q = + 7">58 sin h, Sl = + l 2"'28 sin h, Sh = + 1 02 /r '0 sin h.
* On p. 340 of the memoir referred to in Art. 314 above. The quantity here called k' sin 2^
is denoted by /3 2 .
315-317] INEQUALITIES DUE TO THE EARTH'S ELLIPTICITY.
316. To find the corresponding inequalities in the coordinates, it is only
necessary to subject Delaunay's results for the elliptic and solar terms to
a variation 8 and to insert the above values. It is sufficient for our purposes
to take, in longitude,
v = M + 2e sin I - y 2 sin 2 (g + I),
8v = 8M + 2eSl cos I - 2y8 7 sin 2(0 + Z) - Z<f (8g + 81) cos 2 (g + I).
When the values of 81, 8y, 8g + 8l = 8M - 8h, are substituted, it will be found
that the first term only gives an inequality so great as V. Hence
In latitude, we have
sin u = sin i sin (g + I).
Putting 7 = sin \ i and neglecting quantities of the order <f, we find
8u = 28y sin (y + l) + 2y (8M - 8h) cos (g + I)
= - 9"10 sin (g + l)cosh- 8"'48 cos (g + I) sin h
= - 8"-79 sin (h + g + l)- 0"'31 sin (g + l- h).
The only inequalities, having coefficients greater than 1" in longitude
and latitude, have therefore arguments equal to the longitude of the node and
to the mean longitude, respectively; the periods are 18'6 years and one mean
sidereal month. The coefficients, as found by Hill* who followed Delaunay's
method exactly, are + 7"*67 and 8"'73, so that the calculations made above
give the values with considerable accuracy. The extensions necessary to find
the coefficients of the other periodic terms can be easily made by the method
used here : there are several of about half a second of arc in magnitude.
Other determinations of the inequalities due to the figure of the Earth are to be found
in the works of Laplace f, de PontecoulantJ, Hansen, Tisserand||, etc.
The Motion of the Ecliptic.
317. Owing to planetary action, the plane of the Earth's orbit, which has
been hitherto considered to be the plane of reference, is not fixed. If the
plane of reference, e.g. the ecliptic at a given date, had been fixed, this
motion of the ecliptic, being very small, would have produced but little effect
on the motion of the Moon when introduced into R. But it is usual to use
the instantaneous ecliptic as the plane for the measurement of longitudes and
* Mem. cit. pp. 341, 342.
t Mec. Gel. Pt. n. Bopk vn. Chap. n. ; Book xvi. Chap. m.
$ Sys. du Monde, Vol. iv. Chap. iv.
Darlegung, i. pp. 459474, n. pp. 273322.
H M6c. Gel. Vol. in. pp. 144149, 155160.
264
PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII
latitudes. Hence the apparent place of the Moon will be affected to an
extent which is comparable with the motion of the ecliptic. Since the
inclination of the lunar orbit is a small quantity, and since the line of in-
tersection of two consecutive positions of the ecliptic has a motion so small
that it may be neglected in comparison with the rotation of the ecliptic, the
principal effect produced by referring the Moon's place to the moving ecliptic
will occur in the latitude of the Moon. The approximate fixture of the node
of the ecliptic reduces the consideration of its motion to that of a small
rotation about a fixed line.
Fig. 11.
318. Let Hf^-M be the position of the lunar orbit, and let X'tliM',
Xi&iMi be the two ecliptics at times t, t + dt. The position of the Moon
will have changed during the interval dt ; but since two small changes may
be calculated by considering their effects separately and adding the results,
we can consider the position of the Moon as unchanged in finding the
apparent change in time dt due to the motion of the ecliptic. Let the
rate of rotation of the ecliptic be denoted by ftri.
Draw
and further
and MM' perpendicular to X' ' M f . As before, we put
L, X'M'=v t M'M = u-
since X', X are now departure points and li/ is fixed with reference
to them.
We have
dh=SlQ = QjQ cot i = ftn'dt cot i sin (h - ti).
Also, by considering a point on the ecliptic 90 in advance of 1, we obtain
- di = ftn'St sin (90 + h-h') = ftndt cos (h - h f ).
The equations for h, i are therefore
^ = ftri cot i sin (h - h'\ ~ = -ftri cos (h - h').
ctt dt
When the motion of the ecliptic is neglected, h = h t + const. Since ft is
a very small coefficient, we may integrate the equations on the supposition
317-319] EFFECT OF THE MOTION OF THE ECLIPTIC. 265
that h has this value and that i is constant in their right-hand members.
The new parts of h, i are therefore given by
h = - Oe/i'/Ao) cot i cos (A - h'), Si = - O&i'/Ao) sin (h - h').
Further, &(g + l)= - flQ 1 = - COS * "
cos i sin i
The latitude is given by the equation, sin u= sin i sin (g + ). Hence
S*7 cos u = Si cos i sin (# + Z) + 8 (g + ) sin i cos (# + I)
= (fln'/ho) {- cos i sin (# + Z) sin (h - h') + cos (g + Q cos (h - h')},
which, by considering the triangles M^lSl^, MlfM-[ t becomes
The period of the inequality is therefore the same as that of the Moon.
The annual motion of the ecliptic is 0"*48, and the node of the Moon's orbit
makes a revolution in 18'6 years. Hence
therefore ffn'/ht = (XH8 x 18'6 -h 618 = l"-42,
giving 8u= l //> 42 cos (v h').
The corresponding inequality in longitude is much smaller. Its period is
that of the mean motion of the node and its coefficient is less than one-third
of a second. The calculation of it presents some difficulties and requires a
more extended investigation.
The above method of investigation was given by Adams in a " Note on the Inequality
in the Moon's Latitude which is due to the secular change of the Plane of the Ecliptic *."
A more complete investigation by Hill will be found in a paper " On the Lunar Inequalities
produced by the Motion of the Ecliptic f. :> Reference may also be made to Hansen|,
Radau, Tisserandll.
The Secular Acceleration of the Moon's Mean Motion.
319. The action of the planets produces a slow variation in the eccen-
tricity of the Earth's orbit which is usually expressed in the form,
e' = e ' - at + at 2 -f ....
The coefficients a, a', ... are quite insensible in the motion of the Earth,
* M. N. R. A. S. Vol XLI. pp. 385403. Coll. Works, pp. 231252. Godfrey's Lunar
Theory, Art. 113.
t Annals of Math. (U. S. A.), Vol. i. pp. 510, 2531, 5258.
J Darlegung, i. pp. 118120, 490491.
"Influence du Deplacement seculaire de 1'Ecliptique," Bull. Astron. Vol. ix. pp. 363373.
|| Mec. Gel. Vol. m. pp. 136140, 160164.
266 PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII
but the first of them produces in the longitude of the Moon an effect which
is easily noticeable when observations, extending over a hundred years or
more, are discussed. When this expression, instead of e', is introduced into
the disturbing function, it is evident that terms of the forms V cos (at + b)
will be introduced into R and therefore into the coordinates of the Moon.
In order to make the investigation as brief as possible, we shall make two
assumptions which have been verified by actual calculation. The first is that
the only sensible coefficient in the expression for e' is a and that the squares
and higher powers of a may be neglected ; the second is that terms of the
form t sin (at + b), which will arise in the final expressions for the coordinates,
have coefficients so small that they may be neglected. It is required, therefore,
to find what non-periodic terms are produced in the coordinates by the term
at. The method of Chapter VII. will be used for the investigation.
320. Let us consider how non-periodic terms were produced in the
right-hand member of the equation (1), Art. 130. It was seen in Chap.
VII. that the first approximation to the coefficient of any term in &u was
obtained by simply considering the corresponding term in BR : terms of a
lower characteristic could be neglected. As only non-periodic terms are re-
quired here, and as it was shown in Art. 116 that d'R/dt contained no such
terms, we have, from equation (2), Art. 130, neglecting a /a',
P = r dR/dr + const. = 2R + const.
There is only one non -periodic term in R which need be considered,
namely, that containing e' 2 . We have therefore, from Art. 114,
P = f raV 2 /a + const.
Equation (1), Art. 130, then becomes, since e, y are neglected,
d? m z
a 3 ^j-8u Su = % e' z + const.
at* a
Putting e' = e Q f at and neglecting squares of a, this equation furnishes
aS^ = -fmV 2 + const., or, aSu = fra 2 (e^ - e" 2 ) + const.,
giving the inequality produced in the parallax.
321. Next, consider the equation (5), Art. 131, for the longitude.
Neglecting e, 7, a/a', we obtain
iL g = l^dt s
dt a? a 2 J d a
319-322] THE SECULAR ACCELERATION. 267
The integral in the right-hand member gives no term free from sines or
cosines. We therefore get, on substituting for &u, and putting h no?,
J 7
8v = + const. + f m*n (e ' z - e'*).
Now n was defined so that all constant terms in this equation should
vanish : this definition will be retained. As e ' 2 e' z contains t as a factor,
the term involving this quantity cannot be made to vanish. Hence, by a
suitable determination of Sh, we find, on integration,
The additive constant is put zero, according to the remarks of Art. 158.
The general expression for the longitude therefore becomes
v = nt + e + f ra a ji I (e ' a e' 2 ) dt + periodic terms
J
= nt -f e -f 1 mn'edcd? + periodic terms.
Let the unit of time be one Julian year ; n' will then be the angle de-
scribed by the Sun in one year. The planetary theory gives, for the epoch
1850-0*,
e ' = 0-016771, a = 0-0000004245, n = 1295977", m = 0'07480.
The term in v, involving &, is therefore f
+ 10"'35(*/100) 2 .
The presence of this term in the longitude is usually expressed by saying
that the mean angular velocity of the Moon is not quite constant but has
a secular acceleration of 10"'35 per century ; the more correct statement
being that the mean motion is increasing at the rate of 2 x 10"*35 per century
in a century.
322. This is approximately the value found by Laplace and it requires
considerable modification when we proceed to higher powers of m and to the
terms dependent on e 2 , T 2 , etc. It is in the second and higher approximations
that the difficulty of the subject arises. To obtain the next approximation,
it is necessary to consider not only the non -periodic part of R but also those
periodic terms which, in combination with periodic terms of equal arguments,
may produce non-periodic terms in the longitude equation : and it is to
be remembered, when integrating the equations of motion, that e' is variable.
For instance, to get the next approximation in powers of m by this method,
* Ann. de VObs. de Paris (Mem.), VoL iv. p. 102.
t The result obtained for this term by Adams and Delaunay (see Art. 296) is + 10"-66, owing
to the use of a slightly different value for a.
PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII
it is necessary to retain (i) the parts of R of characteristic zero, (ii) the
non-periodic term of the order e' z , (iii) the periodic terms of characteristic
e', (iv) the values of r', v' as far as the order e' 2 in the non-periodic terms and
to the order e in the periodic terms. The details of the next approximation
are too extended to be given here ; they may be found, calculated to the
order m 4 after this method, in a memoir by Cayley (see Art. 296 above).
The value to the order m s is
_ 3771 m 4 _ 340_4! m 5) n / (g o '2 _
The coefficients corresponding to the second and third terms are 2"'27,
1"'54, causing the acceleration to be diminished to 6"*5. Delaunay * finds
6" '11 to be the complete theoretical value.
It is necessary to make one further remark. The value of e' does not
always continue to diminish ; after a period of about 24000 years it will have
reached its minimum value and begin to increase again, attaining a maximum
after the lapse of another period of similar magnitude. Were it not for this
fact, the period of revolution of the Moon would go on increasing and its
distance diminishing until it was brought within the limits of the terrestrial
atmosphere. The periodic nature of the variation of e' prevents this small
inequality from producing any great change in the relations between the
Moon and the Earth.
The observable effect on the mean distance is quite insensible. This will be readily
understood when it is mentioned that the annual mean approach of the Moon to the
Earth, due to this cause, is less than one inch ; in 200 years, the mean distance will be
only fourteen feet smaller than at the present time, corresponding to a change in the
parallax of less than one twenty-thousandth of a second of arc.
Since the expressions of c, g contain terms dependent on e' 2 , direct observations of the
motions of the Perigee and the Node will show similar secular changes. The values of
c, g contain the terms f wV 2 , + f mV 2 . The first approximations to the secular
accelerations of these motions will therefore be
respectively. The first of these is very much altered by the further approximations.
Delaunay f finds -40"'0 and +6" '8 for their complete values.
For further references on the subject of the secular acceleration, see Art. 296.
* "Calcul de I'acc61eration s6culaire du moyen mouvement de la lune," Comptes Rendus,
Vol. XLVIII. pp. 817827.
t " Calcul des variations seculaires des moyens mouvements du perigee et du nceud de 1'orbite
de la lune," Comptes Rendus, Vol. XLIX. pp. 309314.
TABLES OF NOTATION
AND
INDICES
I. REFERENCE TABLE OF NOTATION.
The numbers following the symbols refer to the articles in which the symbols are first
used. Brackets denote that the symbol is defined and used with that definition only, in the
articles which accompany it. Symbols which are defined and used in one article only, or in
two consecutive articles only, are not generally included. The new symbols occurring only
in articles 308-313 are also omitted. The letters used in the figures are not included.
General Notation i.-vm., XL, xn.
(1-178, 242-302, 319-322).
Delaunay ix., xm.
(179-201,303-318).
Hansen x.
(202-241).
a 12, a' 19, a* 247, A' 22, A { Al 247,
(4 64-66), (A A' 111, 115), a 247,
A 128.
6rf 134, 6 7 rf 166, & 22, (B 135, 158),
b; b/ 264, B 128.
c 68, a 139, G 20, G' 22, c 257, c
267, C 128.
d' 12, D 18, A 20.
e 32, e' 53, (* ' 319-322), E 3, E 32,
e 261, Eq. 39.
/ 32, /' 110, F 3, (F 3-8), (F, F,'
5-9), F 18.
g 68, 9i 147, g 279, go 280.
h 12, Ao 77, (# 94-98).
i 44.
Ji ( ) 37.
JT 22, Ki K{ 279, K 128.
I 32, z Zj 44.
m 114, wj 124, m' 3, M 3, ^ 265,
m 18.
n 18 and 48, n' 19, iV 263.
(Pi 94-104), P 130, $! 16, $ 75.
(# 94-104).
a 192, 4 183,
A' 306.
5 #! 183.
c (7 183.
D200.
e 192.
P200.
183, (0) 184,
186, 0' 190.
183, (A) 184,
hi h s Hi H c
186, ff 190.
i ' i" i'" 180.
jj'j" 306.
#184.
I L I 179, fc l s
Z t -Z c 186, (0
195, z 309.
Oo 208, a 7 220.
6230.
Co Co ^^4, Cy Cj Cj
0233.
e 208, ej 220, ^
_208.
/ 208, /' 220, /
225.
g 208, / 224.
A 204, A 208.
i 7 217.
J217, J 219.
K 217, # 219.
Z 208.
m 1 220.
n 208, < 220, ^
217, N 219.
p p 7 217, P 219, $
204, $ 225.
5 2 7 217, Q 219.
r 208, r 7 220, r
225, jR 204.
5241.
REFERENCE TABLE OF NOTATION.
271
r r 3, (n 5-7), r, 12, r 130 ; R 8, ft
43, -K (< > 124.
5 12, S 3, (ft 5-7), (8 94-98).
* the time, t 18, ^ 257, 2 280, (T
94-98), (T 165-172), ^ 16, X 75.
u 130, u, 16, /i 265, u 15.
^12, ?/22, v 131, F263.
w 32, w' 110.
x y z x y' z 3, z^ +l 278, Z Y X' Y
is, (3eg)3*'g)'3' 3-5), 3iie, 3
75.
(a 134, 158), ! 02 a 3 84, (a 319-
322).
A A & 84.
7 45, 7l 122.
A 3, A (c) 266, v (g) 28 -
6 48, e 22, ej 73, e* e/ 259.
ris.
T; O 45, 77 68.
6 44, (ft ft ft 73-80), 263, B
264.
K 19.
(X 33, 36).
111.
-GT 45, -a/ 53.
o- 18, (7 276, ^ 278.
v 18, v 276, u^i 278.
<f> 68, </>' 110.
^ 263.
w a)' 124, H 19, % 19.
q 180.
T 226, X 204, $
R 179, ft 183,
225.
R' 189, R"
v 206, v' 217, (v
190, (#308-
241).
312).
F211, TT214, W
7 179, 70 192.
226, F 229.
77 189, (17 308-
y 208, y' 220.
310).
z 208, y 220, 3
6 183, ft
204.
ft (*)$ c 186.
a 217.
*189.
(ft 206, 215), ft ft'
X A 189, X' A'
228.
190.
7 226.
< 188, <' 191.
?214.
77 217.
#, M, /*, m',
(9'217.
r, r', v, u, s,
/ 219, (/*! 241).
ft n', a', e', a?,
z/ 209, v 220.
/> J > ^f ^
f H 230.
y retain the
7r 208, TiV 217.
same mean-
J5 214, po 226.
ings as in the
o- 206, a-' 217.
first column.
r214.
n, a, e, (and 7)
T230.
refer first to
$ 214, < 226.
the instanta-
v 206.
neous orbit.
i|r ^ 217, 230.
In Art. 200,
w' 220.
they denote
the arbitrary
#, Jf, IL, m', r, r',
constants.
A, u, ft <*>, ft
For change of
retain the same
notation, see
meanings as in the
Art. 179.
first column.
f, E, L refer here to
the instantaneous
orbit.
For change of nota-
tion, see Art. 204.
II. GENERAL SCHEME OF NOTATION.
The symbols connected with the lunar orbit refer to undisturbed elliptic motion in
Chap. HI. and to the Instantaneous Ellipse in Chap, v.; their meanings in Chaps, VIL, vm.,
together with those of other symbols, will be found in Table III. Accented letters, in general,
refer to the solar orbit.
Symbols.
Significations.
E, M, m f
P
r, r, A
S
x, y, z
X, Y, z
v, <r
u
V, V
u
s
F
R
ft $, 3
ft, Si, 3!
n, ri
a, a
e, e f
tzr, TV'
e
, w
E
L
I
h
r
D
Masses of Earth, Moon and Sun.
E + M. (fi = 1 in de Pontecoulant's theory.)
Distances of Moon and Sun from Earth, and of Moon from Sun.
Cosine of angle between r, r'.
Coordinates of Moon referred to Fixed axes through Earth.
Coordinates of Moon referred to Moving axes through Earth.
l/r.
Longitudes of Moon and Sun reckoned on Ecliptic.
Latitude of Moon above Ecliptic.
Tan u.
Force Function for Motion of Moon.
Disturbing Function arising from solar action.
Solar Forces on Moon, along r, perpendicular to r in plane of
orbit, and perpendicular to plane of orbit.
Solar Forces on Moon, along the projection of r, perpendicular
to projection of r in Ecliptic, and perpendicular to Ecliptic.
Mean Motions of Moon and Sun.
Denned by equations //, = n?a s , m' + /z = n' 2 a' s .
Eccentricities of lunar and solar orbits.
Longitudes of Epochs of Mean Motions.
Longitudes of lunar and solar Perigees.
Longitude of lunar Node.
Inclination of lunar orbit to Ecliptic.
Denned by equations 7 = tan i, ^ = sin \i.
Lunar and solar True Anomalies.
Mean
Eccentric Anomaly.
Angular distance of Moon from its Node.
Latus Rectum of Moon's orbit.
Rate of description of areas by Moon in Ecliptic.
Exp. {(rc-n'^ + e-e'J V^l.
Ill COMPARATIVE TABLE OF NOTATION.
In the first column is the notation used by de Pontecoulant, the second and third columns
contain the corresponding symbols used by Delaunay and Hansen. The fourth column
contains the final definitions; those in square brackets refer to the methods of Chap. xi.
De Pont.
Del
Hansen*.
Definitions.
n
n
^(1+y-ft,)
Observed Mean Motion.
en
/
n o
Mean Motion of Mean Anomaly.
(l-c)n
9o + h
n (y - 2rj)
Perigee. [c = c/(l-m)J
(l-g)n
^0
- n (a + vj)
Node. [g = g/(l-m).]
ri
ri
w ' + njy
Observed Mean Motion of Sun.
y
Mean Motion of Solar Mean Anomaly.
*
I
9
Arg. of Principal Ell. Term in Long.
<t>'
r
9'
Annual Equation.
t
D
g+w-g'-w'
Half Arg. of Variation.
*}
F
g+a)
Arg. of Principal Term in Lat.
m
m
m = n'/n, m? = n' 2 K 2 (* + /V m ')-
[m=n7(n-O-]
a
a
a o
a^ 2 = fj, = a 3 n 2 .
[For a, see Arts. 255, 273.]
a'
a!
a.'
a' 3 ri 2 = m', do 3 n ' 2 m' + ^b.
*t
e
Ecc., defined by Principal Ell. Term in
Long. [For e, see Arts. 261, 274.]
e o
Ecc., defined by Aux. Ellipse.
e
e'
Solar Eccentricity.
e d
Solar Ecc., defined by Solar Aux. Ellipse.
rf
Tan i, defined by Principal Term in z/a.
7,t
7
Sinii Lat.
J.
Inclin., sin 7.
[For K , see Arts. 281, 286.]
f.n.'
True and Mean Anomalies of Aux. Ell.
9> n $ z
Mean and Periodic parts of n z.
s
Sin U sin J Q sin (/+ ay).
r
Radius vector of Aux. Ellipse.
1 + v
r/f.
* Hansen leaves out the zero suffix when the earlier developments have been completed,
t These are not the definitions actually assigned by de Pontecoulant. See Art. 159.
B. L. T. 18
INDEX OF AUTHORS QUOTED.
(The numbers refer to the pages.)
Adams 10, 24, 196, 213, 230, 236, 243, 246, Hansen 36, 74, 75, 76, 77, 91, 123, 131,
265, 267. 160-194, 243, 246, 248, 259, 263, 265.
Airy 131, 245, 246. Harzer 178.
Andoyer 47. Hayward 67.
Arago 259. Hill 10, 45, 47, 114, 125, 190, 196-213, 223,
228, 249, 258, 261, 262, 263, 265.
Ball (W. W. E.) 238.
Binet 75.
Breen 131.
Briinnow 171.
Bruns 27.
Burckhardt 246.
Burg 246.
Lagrange 73, 75, 77, 247.
Laplace 16, 19, 28, 43, 87, 92, 238, 242, 243,
244, 247, 263, 267.
Leverrier 131, 260, 267.
Lexell 240.
Lubbock 243, 245.
Hobson 32, 219, 222.
Jacobi 25, 66, 74, 75, 77, 135, 182.
Koch, von 220.
Krafft 240.
Carlini 244.
Cayley 33, 36, 43, 60, 76, 77, 92, 171, 243,
268.
Cheyne 64, 73.
Chrystal 219.
Clairaut 237, 238.
Cowell 230.
Mason 246.
Mayer 246.
Damoiseau 238, 244.
Darwin 213, 248.
D'Alembert 238, 239.
Delaunay 87, 89, 123, 125, 126, 128, 130, 133, Newton 127, 237, 238, 239.
137, 138, 155, 156, 161, 243, 249-252, 259,
260, 267, 268. Plana 238, 243, 244.
De PontScoulant 16, 28, 86, 87, 112, 113, 114, Poincare 10, 27, 46, 53, 66, 77, 125, 135, 196,
Neison 249.
Newcomb 123, 131, 188, 192, 243, 246, 260.
127, 245, 263.
Donkin 76.
Dziobek 28, 66, 73, 77, 135.
Euler 238, 239, 240, 247.
Forsyth 47, 50.
Frost 75.
Gautier 238.
Godfray 243, 265.
Gogou 87.
Greatheed 32.
Gylden 10, 45, 47.
Hamilton 77,
200, 219, 220.
Poisson 77, 245.
Pratt 261.
Eadau 134, 246, 249, 251, 258, 260, 265.
Routh 1, 28, 53, 55, 59, 68, 69, 73, 260.
Tait and Steele 17, 32, 213.
Thomson and Tait 77.
Tisserand 28, 36, 43, 73, 77, 108, 134, 155,
159, 194, 238, 241, 243, 258, 263, 265.
Todhunter 33, 79.
Williamson 32.
Zech 170 f
GENERAL INDEX.
(The numbers refer to the pages.)
The following abbreviations are frequently used: P. for de Pont6coulant's method, D. for
Delaunay's method, H. for Hansen's method, R. and rect. coor. method for the
method of Chapter xi. These abbreviations refer only to the methods as set forth in
the text.
Acceleration, dimensions of, with astronomical
units, 1 ;
secular, 243, 265 (see secular).
Accelerative effect, term 'force' used for, 1.
Action of the Sun, causes the perigee and node
to revolve, 53 ;
D'Alembert's method of introducing the,
in the first approximation, 239.
Action of the planets (see planetary).
Adams, equations used in proving the theorems
of, connecting the parallax with the motions
of the perigee and the node, 24 ;
statements of the theorems, 236 ;
method of, for the motion of the node,
230;
for the secular acceleration, 243 ;
for the motion of the ecliptic, 263 et seq.
Airy, numerical lunar theory of, 245.
Algebraic uniform integrals, number of, in the
problem of three bodies, 27.
Algebraic development, de Pontecoulant's, 113 ;
d'Alembert's, 239 ;
Delaunay's, 159 ;
Lubbock's, 245 ;
Plana's, 244;
contrasted with semi-algebraic and nu-
merical, 246.
Analysis, claim of Clairaut in first using, 237.
Angular coordinate, periodicity of, defined, 49.
Angular coordinates, final forms of the three
lunar, 121.
Annual equation, defined, 129 ;
order of coefficient of, lowered by inte-
gration, 104 ;
coefficients of, 129, 130;
notation for the argument of (D.), 159.
Anomaly, eccentric, defined, 30 ;
expansion of functions of, in terms of
the mean anomaly, 34 ;
of Hansen's auxiliary ellipse, 165.
Anomaly, mean, defined, 30 ;
in terms of the true, 31 ;
in elliptic motion, 40 ;
convergence of series in terms of, 43 ;
development of disturbing function in
terms of, 89 et seq. ;
used as a variable (D.), 136;
of the auxiliary ellipse (H.), 165 ;
perturbations added to, 160, 164 ;
equation for, 167 ;
integration of, 187 ;
considered constant in the inte-
grations, 169 ;
arbitrary constant present in, 171 ;
used in the development of the
disturbing function, 177 et seq. ;
elliptic value of, 181 ;
form of expression for the disturbed,
187;
definition of mean motion of, in
disturbed motion, 187 ;
remarks on, 188 ;
third approximation to, 192;
of the Sun, notation for, 177 ;
182
276
GENERAL INDEX.
of the Sun, used by Euler as independent
variable, 241.
Anomaly, true, defined, 30 ;
in terms of the mean, 32 ;
symbolic formula for, 33 ;
expansion of functions of, in terms of
the mean, 34 et seq. ;
Hansen's method for, 36 ;
used by Euler as independent variable,
240.
Approximation, solution by continued, 47 (see
continued) ;
second, to the disturbing function, 87 ;
second, and third (P.), 95;
second (H.), 181 et seq.;
third (H.), 192;
method of (K.), 195;
rapidity of (R.), 204.
Apse (see perigee).
Arbitrary constants in elliptic motion, 40, 41 ;
connection between those present in
de Pontecoulant's equations and,
41;
any function of, called Elements, 48 ;
meanings to be attached to the, in
Jacobi's method, 71 ;
in disturbed motion, difficulties in the
interpretation of, 16 ;
interpretation of, 115 et seq.;
numerical values of the solar, 123 ;
of the lunar, 124 et seq. ;
references to memoirs contain-
ing the, 131;
in de Pontecoulant's method, number of,
introduced and necessary, 16;
equation to determine the extra
constant, 16;
definitions of, used by de Ponte"-
coulant, 113, 119;
in Laplace's equations, number of, intro-
duced and necessary, 19 ;
method of defining the, 119, 243 ;
in Delaunay's method, used as new
variables, 139, 142;
transformation of the final, 158 ;
final definitions of, 158 ;
a change of, . explains an apparent
error in the results, 87 ;
in Hansen's method, seven necessary,
166;
number introduced into the equa-
tions, 171;
their significations, 171 ;
for the motion of the orbital plane,
176;
determination of, in W., 186;
definitions of, 187, 188, 192 ;
remarks on the, 188 ;
in rect. coor. method, number introduced
and necessary, 22 ;
definitions of the, introduced, 199 ;
final definitions of the new, intro-
duced, 210, 225, 231, 233 ;
numerical values of the, obtained by
Euler, 240;
his method for the determinations
of the, 241 ;
(see also constants, elements) ;
variation of the (see variation).
Areas, integrals of, in the problem of three
bodies, 27 ;
rate of description of, in elliptic motion,
40, 41, 67, 136.
Argument, mean, of latitude defined, 41 ;
in the disturbing function, whose motion
is equal to the mean motion, 85.
Arguments, connections between coefficients
and, in elliptic expansions, 36;
when the ellipse is inclined to the
plane of reference, 40;
in the disturbing function, 82 ;
for planetary actions, 255, 256;
the expressions for the coordinates con-
tain multiples of only four, 84 ;
discovered by d'Alembert, 239 ;
form of, in de Pontecoulant's method,
in the disturbing function, 81 ;
in the final expressions for the co-
ordinates, 110;
connection of, with Laplace's method,
131;
for terms with coefficients not con-
taining m as a factor, 87 ;
in Delaunay's method, 137;
in the disturbing function 89, 138 ;
after any operation, unaltered, 156 ;
final, 157 ;
notation for, in the results, 159 ;
in Hansen's method, in the disturbing
function, 91, 178;
in the disturbed mean anomaly, 187 ;
in the disturbed radius vector, 189 ;
in the motion of the orbital plane,
192;
in rect. coor. method, 199, 206, 227,
228, 230, 231;
used by Euler, 241;
in Laplace's theory, functions of the
true longitude, 132, 242 ;
in the disturbing function, 92 ;
GENERAL INDEX.
277
coefficients of the time in, incommen-
surable, and will not vanish unless
the arguments vanish, 49, 81, 184.
Ascending node, defined, 41 (see node) ;
of the ecliptic on the equator, the origin
for reckoning longitudes, 261.
Astronomical unit of mass, defined, 1.
Attraction, Newton's law of, 1;
Gaussian constant of, 1 ;
law of, for spherical bodies, 2.
Auxiliary ellipse in Hansen's method, defined,
164;
used as an intermediary, 164 ;
relation of, to the actual position of the
Moon, 165 ;
formulae referring to, 166 ;
coordinates of, considered constant, 169 ;
constants of, 171;
signification of, 187 ;
used for development of disturbing func-
tion, 177 et seq. ;
the solar, 177.
Axes, rectangular (see rectangular);
Euler's formulae, for rotations of, 55 ;
of the variational curve, 125, 127.
Bessel's functions, defined, 33;
used for elliptic expansions, 33 et seq.
Bodies, the celestial, considered as particles, 2 ;
problem of three, of p (see problem).
Canonical constants, defined, 66 ;
Delaunay's, 64;
dynamical and geometrical meanings
of, 67;
produced by Jacobi's dynamical method,
73;
initial coordinates and velocities form a
system of, 76.
Canonical system of equations, Delaunay's,
deduced, 65 ;
obtained by Jacobi's method, 72;
transformation from a,to a, tangential,66;
Lagrange's, 76;
Hansen's extension of, 76 ;
in Delaunay's method, defect of the
first, 134;
second system of, transformation to,
to avoid the presence of the time
as a factor, 136 ;
integration of the, 139 et seq. ;
nature of the solution, 144 ;
the arbitrary constants of the solu-
tion give a new, 143 ;
second system, to avoid the
presence of the time as a
factor, 147;
third system, to correspond
with the previous second
system, 149;
Hill's method of using Delaunay's, for
small disturbances, 249 et seq.
Centre, equation of, defined, 35 ;
expansions of functions of, 36.
Centre of mass of the Earth and Moon, the
Sun's force-function relative to, 5 ;
motion of, considered an ellipse, 6;
correction to the disturbing function
when the motion is referred to the
Earth instead of to the, 8 (see cor-
rection).
Change of position due to changes in the ele-
ments, general formulae for, 56 ;
zero in the motion, 59.
Characteristic, defined, 82;
connection with the argument, 82 ;
unaltered by the integration of the
radius- and longitude-equations, 86 ;
diminished one order, by substitution
in the latitude-equation, 86;
left arbitrary in Laplace's method, 242;
and in finding the variations of the co-
ordinates in Hill's method for small
disturbances, 252.
Clairaut, lunar theory of, 238 ;
showed that the observed and theoretical
motions of the perigee agree, 239.
Classes, division of the inequalities into, P., 95;
not made by de Pontecoulant, 112 ;
in rect. coor. method, 198 ;
by Euler, 240, 241.
Coefficients, orders of, defined, 80;
denoted by the index, 80;
form of the, in the disturbing function,
81,82;
connection between arguments and, 82 ;
discovered by d'Alembert, 239 ;
in the planetary disturbing func-
tions, 255, 256 ;
characteristic parts of, defined, 82 (see
characteristic) ;
of the time in the arguments, will not
vanish unless the argument vanishes
and assumed incommensurable, 49,
81, 184;
effect produced on the orders of, in the
coordinates by the integrations, 84
et seq. ;
certain, to be left indeterminate until
the third approximation, 86, 110;
278
GENERAL INDEX.
of the same order in the second and
third approximation, 86 ;
some properties of, 86 ;
orders of, in the successive approxima-
tions, 95 ;
division of the inequalities, into classes
according to the orders of, 95 (see
slow convergence of the series repre-
senting the, 113;
the particular, used to determine the
arbitraries, 119 et seq.;
conversion of the, into seconds of arc, 121 ;
numerical values of certain lunar, 124
et seq.;
magnitudes of the, 131 ;
hi Delaunay's theory, form of the, in the
disturbing function, 138 ;
in the solution of the canonical
equations, 144;
in the calculation of any operation,
150 et seq. ;
relations between the new and old,
155;
of the time, in the arguments, 157,
158;
in Hansen's theory, in the disturbing
function, 181 et seq. ;
of inequalities due, to Venus, 258, 260 ;
to the figure of the Earth, 263 ;
to the motion of the ecliptic, 265.
Comparison, of theoretical and observed values,
a test, 123;
of the motion of the perigee, 188,
239;
of the secular acceleration, 243 ;
of certain inequalities, to determine
the figure of the Earth, 261;
of the systems of notation used, 136,
161, 273 ;
of Delaunay's results with Hansen's,
159;
of the values of the various methods, 246.
Complex variables, used in the lunar theory,
20 (see rectangular coordinates).
Condition, equations of, for the variational
coefficients, 200 et seq. ;
method of solution, 203 ;
rapidity of approximations in,
204;
for the elliptic inequalities, 207, 208 ;
of the first order, 209;
method of solution, 210;
of the second order, 223, 224;
for finding the motion of the perigee, 216 ;
for inequalities in latitude, of the first
order, 230;
of the third order, 232 ;
when the number of unknowns is greater
than the number of equations, 122 ;
first used by Euler, 241.
Condition, that an infinite system of linear
homogeneous equations be consistent, 217 ;
of convergency, of elliptic series, 43 ;
of an infinite determinant, 219.
Connection between, arguments and coefficients,
in elliptic expansions, 36, 40 ;
in the disturbing function, 82 ;
for planetary action, 255, 256;
the developments of the disturbing func-
tions of P., D., H., 89,92;
the auxiliary and instantaneous ellipses
(H.), 165;
(see relations).
Constant, Gaussian, of attraction, 1 ;
of mean motion, in elliptic motion, 40;
in disturbed motion, P., 97, 118;
D., 158; H., 188; B., 199;
numerical value, 124 ;
when the secular acceleration is
considered, 267;
of epoch, in elliptic motion, 40;
in disturbed motion, P., 97, 118;
D., 158; B., 199;
when the secular acceleration is
considered, 267;
the linear, in elliptic motion, 40 ;
in disturbed motion, P., 95, 98,
119; D., 158; H., 171, 189; B.,
205, 224;
numerical value of, 124;
remarks on, 120 ;
of eccentricity, in elliptic motion, 40 ;
in disturbed motion, P., 102, 119;
D., 158; H., 187, 188; B., 210,
225;
used by Laplace and de Pont<-
coulant, 113, 119, 243 ;
numerical value, 128; H., 188;
of latitude, in elliptic motion, 41 ;
in disturbed motion, P., 108, 120,
130; D., 158; H., 192, 194; B.,
231, 233;
in Laplace's method, 243;
numerical value, 130 ;
of epoch of mean longitude of perigee,
in elliptic motion, 40;
in disturbed motion, P., 120; D.,
158; B., 207;
of epoch of mean anomaly (H.), 181 ;
GENERAL INDEX.
279
of epoch of mean longitude of node, in
elliptic motion, 41 ;
in disturbed motion, P., 121; D.,
158; H., 173, 192; E., 230;
of energy (E.), determination, 205;
used for verification, 205.
Constant parts of the functions used, form of
the, 86.
Constants, introduced and necessary in P.'s
equations, 16;
in Laplace's equations, 19 ;
in Hansen's method, 166;
in rect. coor. method, 22 ;
definitions of the, introduced,
199;
interpretation of, 16, 115 et seq.;
determination by observation of, 121 '
et seq. ;
Euler's method, 240;
the solar, 42 ;
numerical values of, 123 ;
numerical values of the lunar, 124 et seq. ;
references to memoirs containing
the determination of the, 131 ;
(see also constant, arbitrary, elements);
in Hansen's method, of the auxiliary
ellipse, defined, 164;
their significations, 187, 188;
determination of the arbitrary, in
W., 186;
their significations, 187 ;
for the motion of the orbital plane,
176;
their significations, 192 ;
in the problem of three bodies, the ten,
26;
unreduced numerical values of De-
launay's, 251;
for the figure of the Earth, methods for
the numerical determination of, 261 ;
variation of arbitrary (see variation).
Controversy concerning the secular acceleration,
243.
Continued approximation, solution by, method
of, 47;
applied to de Pontecoulant's equa-
tions, 49;
to Hansen's method, 181, 182;
to rect. coor. method, 195.
Convergence, of Bessel's functions assumed, 33 ;
conditions of, for elliptic series, 43;
for an infinite determinant, 219 ;
slow, of series for the coordinates, 113;
a particular case of, 114;
indicates the rect. coor. method, 198;
change of parameter to improve,
114, 204;
avoided by using the numerical
value of w, 246 ;
question of, recognised by d'Alembert,
239.
Coordinates, referred to fixed axes, 13 ;
to moving rectangular axes, 19;
used by Euler, 240;
elliptic expressions for, 41 ;
forms of, used by Delaunay, 138 ;
forms necessary for the, 45 ;
modified elliptic expressions for, 52;
and velocities have the same forms in
disturbed and undisturbed motion, in
the method of the variation of arbi-
trary constants, 48, 58, 72;
in Jacobi's method, 68;
ideal, defined, 74;
general conditions for the existence
of, 75;
Euler's formulae for transformation of, 75 ;
initial velocities and, form a canonical
system, 76;
expression of disturbing function in
terms of fixed rectangular, 8, 79, 136;
in the case of planetary actions,
253;
transformation to polar, 254 ;
of moving rectangular, 20 et seq.,
179;
expressions for, in de Pontecoulant's
method, 110;
forms of the three angular, of the Moon,
121;
forms of expressions for, after Delaunay's
operations, 157 et seq.;
of the auxiliary ellipse, can be con-
sidered constant (H.), 169;
cylindrical, used by Euler, 239.
Corrections, to be made to the force-functions
for the solar and lunar motions, 5-8 ;
to take into account the Moon's mass,
to the disturbing function, 8, 178, 252 ;
general form of, 178;
to the parallactic inequalities, 126, 159 ;
to Kepler's laws, to account for the
Earth's mass, 42, 90;
to be applied to the tables, 123;
to Hansen's eccentricity, 188;
to Newton's law found unnecessary, by
Clairaut, 239;
by Euler, 240 ;
to Laplace's value for the secular accele-
ration, 243.
280
GENERAL INDEX.
Curve, the elliptic, 29 et seq. (see elliptic).
Cylindrical coordinates, used by Euler, 239.
D'Alembert, method and discoveries of, 239.
Damoiseau, method of, 244.
Darlegung, full title of the, 160;
object of publication, 161;
determination of y in, not available, 185.
Definitions of the constants (see constant).
Delaunay, theory of, canonical system of ele-
ments and equations, 64, 134 ;
signification of, 67 ;
deduced by Jacobi's method, 72 ;
problem solved in, 133;
development of the disturbing function,
88;
comparison with other methods, 89,
92;
form of, in Delaunay's notation, 137 ;
canonical equations used in, 136;
notation used, 136;
meanings of variables, 137 ;
elliptic expressions for the coordi-
nates in, 138;
method of integration, 139 et seq. ;
general procedure, 155 ;
analysis of the theory, 156 ;
changes of the arbitraries and final form
of the results, 158;
correction to, to account for the Moon's
mass, 159;
numerical values of solar and lunar
constants used in, 123 et seq. ;
Airy's method of verification of the re-
sults of, 245 ;
compared with other theories, 247 ;
Hill's method of continuing the, for small
disturbances, 248 et seq.
Departure point, defined, 60 ;
curve described by a, cuts the orbital
plane orthogonally, 60 ;
longitudes reckoned from a, have the
same form in disturbed and undis-
turbed motion, 60 ;
use of a, introduces a pseudo-element,
75;
and an arbitrary constant, 163 ;
when ecliptic is in motion, 264 ;
used in Hansen's method, 160, 163.
Dependent variables used in the various
methods, 12 (see variables).
De Pontecoulant's method, variables used in, 12;
equations of motion, 13 et seq. ;
arbitrary constants introduced into,
and necessary for, 16 ;
solution of, when the solar action is
neglected, 41 ;
solved by continued approximation,
49;
modified intermediary for, 52 ;
development of the disturbing function,
82;
deducible from Delaunay's, 89;
and from Hansen's, 92 ;
unit of mass used in, 82 ;
the effects produced by the integrations,
84 et seq. ;
method for the higher approximations,
87;
preparation of the equations for the
second and higher approximations,
93 et seq. ;
details of the second, and of parts
of the third, approximation, 96 et
seq.;
summary of the results, 110;
analysis of, as contained in the Systeme
du Monde, 112 et seq. ;
slow convergence of the series for the
coefficients, 113;
meanings to be attached to the arbitraries
in, 117 et seq. ;
definition of the eccentricity used
by de Pontecoulant, 119 ;
comparison of the arguments in Laplace's
method, with those in, 131 ;
form of solutions for rect. coor. method
deduced from, 199, 207, 230 ;
similar to Lubbock's, 245 ;
compared with other methods, 247 ;
first approximation to the secular accele-
ration by, 266 et seq.
Derivatives of the disturbing function (see
disturbing).
Determinant, of a system of linear homogeneous
equations, 210 et seq. (see infinite).
Development, of the disturbing function (see
disturbing) ;
of an infinite determinant, 221 et seq.
Differences between theory and observation
(see comparison).
Direct action of a planet, defined, 253 (see
planetary).
Discoveries of, Newton, 237 ;
Clairaut, 238 ;
d'Alembert, 239 ;
Euler, 241 ;
Laplace, 243 ;
Adams, with reference to the secular
acceleration, 243.
GENERAL INDEX.
281
Distance, mean, defined, 120;
method of finding the, from observation,
123;
numerical value of the, 124 ;
effects of the secular acceleration on the,
268.
Distance, constant of mean, in disturbed
motion, P., 119; D., 158; H., 171, 189; E.,
205, 224 ;
Euler's, 240.
Distances, ratio of, the disturbing function first
developed in powers of, 5 et seq. ;
large in the planetary theory, 9;
of the second order, 80 ;
(see parallactic).
Distribution of mass of the Moon, Hansen's
empirical term to account for a supposed
non-uniform, 248.
Disturbances, Hill's method of integrating for
small, 248 et seq.
Disturbed body, defined, 9 ;
mass of a, relative to the primary, 9.
Disturbed elliptic orbit, coordinates and veloci-
ties have the same form in the undisturbed
and, 58 ;
also, longitudes reckoned from a
departure point, 60 ;
(see variation of arbitrary constants).
Disturbing body, defined, 9;
mass of, relative to the primary, 9.
Disturbing forces, derivatives of the disturbing
function in terms of, 57 ;
rate of rotation of the orbital plane due
to, 60;
variations of the elements in terms of, 63 ;
in Hansen's theory, 163 ;
deduced from the development of the
disturbing function (P.), 83;
higher approximations to, 88 ;
hi Hansen's theory, 179 et seq., 182;
higher approximations to, 193 ;
deduced direct from the force function in
Laplace's theory, 92 ;
in Hansen's theory, notation for, 161 ;
equation for W in terms of the, 170 ;
for the motion of the orbital plane,
in terms of the, 176.
Disturbing function, due to the Sun's action,
defined, 8 ;
for planetary actions, 252 ;
separation of the terms, 253 ;
expression hi polar coordinates, 254 ;
for the figure of the Earth, 261.
Disturbing function, derivatives of the, with
respect to the polar coordinates, 14, 15, 58 ;
to the elements in terms of the forces,
57;
in Hansen's method, 180 et seq.,
182;
to the major axis, remarks on, 66 ;
equations for the elements in terms of, 64 ;
coefficients of, independent of the
time, 73 ;
properties of, 83 ;
higher approximations to, 88.
Disturbing function, development of, in powers
of the ratio of the distances, 5, 20, 79 ;
in the planetary and lunar theories,
9;
for planetary action, 252 ;
in terms of the elliptic elements, 80 ;
properties of, 81, 86 ;
higher approximations to, 87 ;
in de Pontdcoulant's theory, 80 ;
result, 82;
parts of, for certain inequalities,
96, 100, 103, 104, 107 ;
deduced directly, 111 ;
in Delaunay's theory, method for, 88 ;
form used, 137 ;
one term of, used for integration, 139 ;
form of, with the new variables, 145 ;
effect of an operation on, 150 ;
relations between the new and old,
152;
reduced to a non-periodic term, 156 ;
in Hansen's theory, method for, 89 ;
form of, 177 ;
first approximation to, 181 ;
higher approximations to, 192 ;
in rect. coor. method, 20, 92 ;
terms in, for certain inequalities,
225, 227 ;
in Laplace's theory, 92 ;
for planetary action, 255 et seq. ;
term in, due to Venus, 258 ;
due to the variation of the solar
eccentricity, 266 ;
for the figure of the Earth, 262.
Divergent series may represent functions, 53.
Division of inequalities into classes (see classes).
Divisors, effect of, on the orders of coefficients,
84 et seq., 95, 96 et seq.;
on the variational inequalities, 204 ;
on the mean-period inequalities, 227 ;
on planetary inequalities, 257.
Dynamical methods of Hamilton, Jacobi and
Lagrange, 67 et seq.
Eccentric anomaly (see anomaly).
282
GENERAL INDEX.
Eccentricities, considered of the first order, 80 ;
used as parameters in expansions, 30, 80 ;
connection between the arguments and
powers of, in the disturbing functions,
36, 82, 255, 256 ;
forms in which the, occur, 86.
Eccentricity, lunar, equation for the variation
of, 61 ; H., 162 ;
constant of, in disturbed motion, P.,
102, 119; D., 158; H., 187, 188; B.,
119, 210, 225 ;
determined from the principal ellip-
tic term, 123 ;
relation of, to that used in
rect. coor. method, 211 ;
numerical value, 128; H., 188;
used by de Pont6coulant and Laplace,
113, 119, 243 ;
relation of, to Delaunay's variables, 138;
presence of, as a denominator avoided
(D.). 154;
inequalities dependent on (see elliptic
inequalities).
Eccentricity, solar, numerical value of, 123 ;
how the, is included in the motions of
the perigee and node, 234 ;
variation of, the cause of the secular
acceleration, 243, 265 ;
periodic nature of, 268.
Eclipses, used by Euler to determine the arbi-
traries, 240 ;
ancient, and the secular acceleration,
243.
Ecliptic, considered fixed, 13 ;
in Hansen's method, movable, 162 ;
quantities defining the motion of,
172;
reduction of expressions to, 194 ;
effect of secular motion of (H. ), 191 ;
Adams' method, 263 ;
principal inequality due to, 265 ;
ascending node of, on equator, the origin
for reckoning longitudes, 261.
Elements, definitions of, 41 ;
extended, 48, 74 ;
of the lunar orbit, 41 ;
of the solar orbit, 42 ;
of the instantaneous orbit, 48 ;
coordinates of the Sun and Moon in
terms' of the, and of the time, 41, 42,
138;
of the true longitude, 42 ;
development of the disturbing functions
in terms of the (see disturbing func-
tion) ;
not used in the rect. coor. method,
92;
derivatives of the disturbing function
with respect to, in terms of the dis-
turbing forces, 57 ;
change of position due to variability of,
56;
is zero in the actual motion, 59 ;
equations for the variations of, in terms
of the disturbing forces, 63 ;
required in Hansen's method,
162 et seq. ;
in terms of the derivatives of the
disturbing function, 64 ;
Lagrange's, 73 ;
canonical systems of, 64 et seq. (see
canonical) ;
pseudo-, definition and properties of, 74,
75;
meanings of Delaunay's, 67, 136 ;
purely elliptic values of, defined (H. ) , 171 ;
used in the Fundamenta, 171 ;
relations of the final constants to the
elliptic (D.), 158 ;
Eadau's numerical equations for the,
for small disturbances, 251 ;
variations of, due to motion of ecliptic,
264.
Ellipse, motion of the Sun considered an, 6 ;
formulae and expansions connected with
the, 29 et seq. (see elliptic) ;
used as an intermediary, 46 ;
also, when modified, 52;
used by Clairaut, 238 ;
instantaneous, 48; H., 162;
auxiliary (H.), 164.
Elliptic elements (see elements).
Elliptic expansions, in terms of the true ano-
maly, 31 ;
in terms of the mean anomaly, 32 ;
by Bessel's functions, 33 et seq. ;
Hansen's theorem concerning, 36 ;
when the plane is inclined, 38 et seq. ;
for the coordinates, 41, 138;
convergence of, 43.
Elliptic inequalities, defined, 128 ;
determination of, de Pontecoulant's me-
thod, 100 et seq. ;
rect. coor. method, 206 ;
of the first order, 209 ;
of higher orders, 224 ;
terms in the disturbing function
for, 112.
Elliptic motion, formulas and expansions con-
nected with, 40 et seq. ;
GENERAL INDEX.
283
method of including the effects of the
solar and planetary deviations from,
253, 257.
Elliptic term, principal, in longitude, used to
define the eccentricity, 102, 119, 123, 128,158;
observed value of the coefficient of,
127;
period of, 128 ;
combination of, with the evection,
128.
Ellipticity of the Earth, 260 (see figure of the
Earth).
Empirical term, Hansen's, supposed to be due
to the non-uniform distribution of the Moon's
mass, 248.
Energy, integral of, in the problem of p bodies,
26;
in the lunar theory, 14, 18, 22 ;
apparent inconsistency of, 26 ;
used as a means of verification, 205.
Epoch of the mean longitude, defined, 120;
equations for the variation of, 62, 64 ;
not used by Hansen, 162 ;
constant of, in disturbed motion, 118
etc. (see constant).
Epochs of mean longitudes of perigee and
node, defined, 120, 121;
constants of, 120 etc. (see constant).
Equation of the centre, defined, 35;
expansions containing, 36.
Equation, determinantal, for the motion of the
perigee, 217 ;
of the node, 230.
Equations of condition (see condition).
Equations of motion, de Pont6coulant's, 13
et seq.;
solution of, when the solar action is
neglected, 41;
effects produced by the integration
of, 84 et seq. ;
preparation of, for the second and
higher approximations, 93 et seq. ;
Laplace's, 17 et seq. ;
solution, neglecting the Sun's action,
42;
Hansen's, for radius and longitude, 167,
168, 170 ;
for the plane of the orbit, 190 ;
in rect. coor. method, 19 et seq. ;
simplified forms of, for the inter-
mediary and the elliptic inequali-
ties, 24, 197, 211;
for mean-period inequalities, 225 ;
for parallactic inequalities, 227 ;
for inequalities in latitude, 228 ;
for Adams' researches, 24 ;
simplified to obtain a first approximation,
44;
referred to polar coordinates, 59 ;
Hamilton's, 68;
Jacobi's solution of elliptic, 69 et seq. ;
Clairaut's, 238 ;
Euler's, 239, 240 ;
for the problem of three bodies, 25 et seq. ;
the ten first integrals of, 26;
cases when the, are integrable, 28.
Equations for the variations of the elements,
59 et seq. (see elements, variation).
Equations, linear (see linear).
Equator, ascending node of ecliptic on, origin
for reckoning longitudes, 261.
Equinoxes, precession of, used to determine the
figure of the Earth, 261.
Equivalence of the two forms of the equations
for the intermediary (R.), 197.
Error in Airy's theory, 246.
Euler, methods of analysis of the, 239, 240 ;
contributions of, 241 ;
formulae of, for rotations, 55 ;
for transformation of coordinates, 75.
Evection, defined, 128 ;
order lowered by integration, 101 ;
coefficient and period of, 128 ;
combination of, with the principal ellip-
tic term, 129 ;
effect of, on the motion of the perigee,
discovered by Clairaut, 239.
Existence of integrals in the problem of three
bodies, 27.
Expansions (see elliptic, disturbing function,
etc.).
Expressions for the coordinates, in undisturbed
motion, 41 ;
form to be given to, 45;
effects of small divisors on, 84 et seq. ;
facts concerning, 86, 87 ;
obtained by de Ponte'coulant's method,
110;
slow convergence of, 113 ;
form of, D., 158; H., 166, 194;
Euler's, 240, 241 ;
Laplace's, 243 ;
(see coordinates).
Figure of the Earth, disturbing function for,
261;
numerical determination of, 261 ;
principal inequalities due to, 263.
Figure of the Moon, Hansen's empirical term,
supposed to be due to, 248.
284
GENERAL INDEX.
Force, used instead of accelerative effect of, 1.
Forces, relative to the Earth, 3 ;
on the Moon relative to theJEarth and
on the Sun relative to the centre of
mass of the Earth and Moon, 5 ;
disturbing (see disturbing).
Force-function used by Laplace, 92.
Force-functions for the lunar and solar motions,
3;
second form of, 5 ;
corrections to, 6-8.
Form (see disturbing, expressions, etc.).
Function, disturbing (see disturbing).
Functions, Bessel's, defined, 33 ;
used in elliptic expansions, 34 et seq.
Fundamenta, fuU title of the, 36 ;
contents of, 161;
elements used in, 171.
Gaussian constant of attraction, 1.
Geodetic measures and pendulum observations
used to determine the figure of the Earth,
261.
Hamilton's dynamical method, 67 et seq.
Hansen, methods of, for elliptic expansions,
36;
theorem of, concerning elliptic expan-
sions, 36 et seq. ;
extension of, of method for the variation
of arbitrary constants, 76 ;
theorem concerning, 77;
method of, for the development of the
disturbing function, 89 et seq. ;
two inequalities of, due to Venus, 259.
Hansen's theory, features of, 160, 164, 166 ;
history of, 161 ;
notation for, 161 ;
instantaneous ellipse, 162 ;
auxiliary ellipse, 164 et seq. ;
relation of instantaneous to, 165 ;
disturbing function, form of, 177 et seq.;
first approximation to, 181 ;
derivatives of, in terms of the forces,
179 et seq., 182;
motion in the orbital plane, equations
for, 167 et seq. ;
introduction of T, 169 ;
the function W, 169 ;
first approximation to, 182 et
seq.;
integration of the equations, 185 et
seq.;
the arbitrary constants, 171, 187,
188;
motion of the orbital plane, definitions
for, 172 et seq. ;
equations for, 174 et seq., 190;
integration of the equations, 191 ;
the arbitrary constants, 192;
third and higher approximations, 192 et
seq.;
reduction to true ecliptic, 194.
Hill, equations used in the researches of, 24,
197;
particular solution of, 24 ;
method of, for the variational inequali-
ties, 196 et seq. ;
for the motion of the perigee, 211
et seq. ;
for adapting Delaunay's theory to
small disturbances, 248 et seq. ;
for separating the terms in the
planetary disturbing functions,
253.
History of the lunar theory since Newton, 237
etseq.;
of Hansen's theory, 161.
Ideal coordinates, defined, 74;
general conditions for, 75.
Inclination, equation for the variation of, 61;
first used by Euler, 239 ;
when the ecliptic is in motion, 264.
Inclination, sine of half or tangent of, a
parameter in elliptic motion, 39 ;
considered of the first order, 80 ;
a parameter in the development of
the disturbing function, 80 ;
connection between arguments and
powers of, 82;
properties of, in the coordinates,
86;
(see constant of latitude, latitude).
Incommensurable, coefficients of the time in
the arguments assumed, 49, 81, 184.
Independent variable (see variable).
Indeterminate coefficients in the second ap-
proximation (P.), 86, 110;
method of solution by, first used by
Euler, 241.
Index of a coefficient denotes the order, 80.
Indirect action of a planet, 253 (see planetary).
Inequalities, division into classes, 95, 198,
241;
variational, 96, 125, 198 (see varia-
tional) ;
elliptic, 100, 128, 209, 224 (see elliptic);
mean-period, 103, 129, 225 (see mean
period) ;
GENERAL INDEX.
285
parallactic, 104, 127, 227 (see paral-
lactic) ;
principal, in latitude, 106, 130, 228 (see
latitude) ;
of higher orders, P., 109; B., 234;
special, deduced directly from the dis-
turbing function, 111;
long- and short- period, defined, 85 ;
method of calculating small, 248 et seq. ;
planetary, 252 et seq. ;
due to Venus, 258, 260;
to the motion of the ecliptic, 265 ;
to the variation of the solar eccen-
tricity, 266, 267;
due to the figure of the Earth, 263 ;
principal, obtained by Newton, 237.
Infinite determinant, to find the motion of the
perigee, 217 ;
properties of, 217 et seq. ;
to find the nodal motion, 230 ;
convergency of, 219;
development of, 220 ;
application to the perigee, 222.
Instantaneous axis, the radius vector, rate of
rotation of the orbit about, 60.
Instantaneous ellipse, defined, 48 ;
the intermediary when the method of
the variation of arbitraries is used, 48 ;
relations between, and the auxiliary
(H.), 165;
(see variation, Hansen).
Integrable, case when Hill's equations are, 24 ;
cases when the equations for the pro-
blem of three bodies are, 28.
Integrals, the ten first, in the problem of three
or p bodies, 26 et seq. ;
Jacobian (see velocity, Jacobian).
Integration by continued approximation (see
continued).
Integration, small divisors introduced by, 84
et seq. ;
effects of, on the orders of coefficients,
86;
of the prepared equations (P.), 96 et seq.;
of canonical equations with one periodic
term of the disturbing function (D.),
139 et seq. ;
in particular cases, 153, 156 ;
mean anomaly constant in (H.), 169 ;
of equations, for mean anomaly and
radius vector (H.), 185 et seq. ;
for motion of orbital plane, 191 ; '
of equations of motion, by Clairaut,
238;
method of Laplace, 242 ;
with a variable solar eccentricity,
243, 267;
Hill*s general method of, for small dis
turbances, 248 et seq.
Intermediary, defined, 45;
in the various methods, 46 et seq. ; P.,
52; D., 134; H., 165; B., 197; La-
place, 53 ;
modification of, 51, 238, 239.
Intermediate orbit (see intermediary).
Interpretation of arbitraries (see arbitrary).
Invariable plane, defined, 27 ;
as a fixed plane of reference, 27, 162.
Jacobi, dynamical method of, 68 ;
elliptic motion by, 69 ;
produces a system of canonical con-
stants, 73.
Jacobian integral, when the solar eccentricity
is neglected, 25;
in the problem of three bodies, 26 ;
(see velocity).
Jupiter, large inequality in motion of, 10.
Kepler's laws, approximate representation of
motions of planets and satellites by, 9 ;
correction to, due to Earth's mass, 42, 90.
Lagrange, equations for the variation of arbi-
traries of, 73 ;
canonical system of, 76.
Laplace, condition of convergence of, for el-
liptic series, 43;
discoveries of, 243 ;
value of the secular acceleration of, 267.
Laplace's method, equations of motion for, 17
et seq. ;
solution of, when the solar action is
neglected, 42;
intermediary for, 53 ;
development of force-function for, 92 ;
form of solution compared with P., 131 ;
definition of eccentricity, 119, 243 ;
analysis, of, 242.
Latitude, argument of, defined, 41;
in disturbed motion, P., 52; D.,
159; H.,194;
constant of, in disturbed motion, P., 108,
120, 130; D., 158; H., 192, 194; E.,
231, 233; Laplace, 243;
used by de Pont6coulant, 113 ;
numerical value of, 130 ;
of the first order, 80 ;
development of the disturbing func-
tion in powers of, 82 ;
286
GENERAL INDEX.
connections between arguments and
powers of, 81;
form of expression for, 121 ; H., 194;
long-period inequalities in, 85 ;
magnitudes of coefficients in, 131 ;
perturbations of, Hansen's form for, 194;
principal inequalities in, determination
of, P., 106 et seq.; E., 228 et seq.;
due to the figure of the Earth, 263;
to the motion of the ecliptic, 265 ;
principal term in, used for the determi-
nation of the constant, 108, 120, 123,
130, 192, 231, 233, 243;
coefficients and period of, 130 ;
tangent of, expression for, in elliptic
motion, in terms of the time, 41 ;
of the true longitude, 42 ;
in disturbed motion (P.), Ill;
terms not containing ra in, 87.
Latitude-equation, defined (P.), 16;
effect of integration of, on the orders of
coefficients, 84 et seq. ;
not used in the calculations, 94 ;
Euler replaces, by two equations, 239.
Legendre's coefficients, expansions by, of the
disturbing functions, 79, 256.
Limitations of the lunar theory, 2; D., 133;
E., 196.
Line, fixed in the orbital plane, defined, 60.
Linear constant (see distance, constant).
Linear equations to find the motions of the
perigee and node, 108, 213, 229.
Linear equations arising in the second approxi-
mation, 50, 53.
Linear homogeneous equations, determinant of
an infinite number of, 210, 216, 230.
Longitude, derivatives of the disturbing func-
tion with respect to, 14 et seq. , 58, 83, 179.
Longitude of epoch, perigee, node (see epoch,
perigee, node).
Longitude, mean, in elliptic motion, 40; D.
137;
in disturbed motion, P., 97, 118; D.,
158; H., 194; (see mean motion).
Longitude, true, expression for, in elliptic
motion, 41; D., 138;
in disturbed motion, 110;
form of expression for, P., 121 ; D., 157;
H., 194; Euler, 240;
independent variable the, theories using,
12, 242, 244 ;
remarks on, 247;
equations with, 17 et seq. ;
elliptic motion with, 42 ;
intermediary, 53 ;
motions of perigee and node with,
131;
magnitudes of coefficients in, 131 ;
terms in, long- and short-period, 85 ;
not containing m as factor, 87 ;
due to figure of the Earth, 263 ;
to motion of ecliptic, 265 ;
to secular acceleration, 267;
transformation to find (E.), 206.
Longitudes, origin for reckoning, 261 ;
reckoned from a departure point, pro-
perty of, 60;
introduce pseudo-elements, 75 ;
used by Hansen, 160.
Longitude-equation, defined, 16;
effect of integration of, on the orders of
coefficients, 84 et seq. ;
prepared form of (P.), 94.
Long-period inequalities, defined, 85 ;
found best by the method of the varia-
tion of arbitraries, 66, 245 ;
due to planetary action, 257.
Lubbock, method of, 245.
Lunar theory, a particular case of the problem
of three bodies, 2 ;
limitations initially assigned, 2;
distinction between the, and the planet-
ary, 8-10, 66;
variables used in the various methods,
12;
analysis of the methods given by
Airy, 245;
Clairaut, 238 ;
Damoiseau, 244;
d'Alembert, 239 ;
de Pontecoulant, 112 et seq.;
Delaunay, 156;
Euler, 239, 240;
Hansen, 166;
Laplace, 242 ;
Lubbock, 245;
Newton, 237 ;
Plana, 244;
Poisson, 245;
Eectangular coordinates, 195 et seq.;
another method, 234 ;
comparison of the methods, 246;
tables deduced from (see tables).
Magnitudes of the coefficients, 131.
Major axis, equation for the variation of, 61 ;
derivative of the disturbing function
with respect to, 66;
relation of, to Delaunay's elements, 138 ;
(see distance).
GENERAL INDEX.
287
Mass, astronomical unit of, denned, 1 ;
unit of, used (P.), 82;
of the Moon, of the Earth, correction
necessary to include the (see correc-
tion);
methods for determination of, 127.
Mean anomaly (see anomaly).
Mean argument of latitute, denned, 41 (see
latitude).
Mean distance (see distance).
Mean motion, in an ellipse, 40 ;
in Delaunay's notation, 138;
in the disturbed orbit, denned, 63 ;
in disturbed motion, P., 97, 118; D.,
158; H., 188; B., 199;
obtained from observation, 123;
numerical value of the, 124;
of the solar, 123;
term in the disturbing function having
a period equal to that of the, 85, 86 ;
secular acceleration of, 243, 265 et seq.
Mean motions, ratio of the, assumed incom-
mensurable, 49;
considered of the first order, 80;
square of, a factor of the disturbing
function, 82;
a factor of the terms in the coordi-
nates due to the Sun, 86;
cases of exception, 87 ;
numerical value of, 124;
inequalities dependent only on (see
variational) ;
of the perigee, node, mean anomaly (see
perigee, node, anomaly);
of two planets, nearly commensurable, 9.
Mean period, obtained by observation, 123 ;
numerical value of the, 124 ;
of the solar, 123.
Mean period inequalities, defined, 130;
determination of, P., 103; B., 225;
terms in the disturbing function for,
112.
Modification of intermediary, 51 et seq. (see
intermediary).
Motion, oscillation about a steady, 47, 52, 211 ;
of the Moon, effect of, on the motion of
the centre of mass of the Earth and
Moon, 6 ;
of the Sun, referred to the centre of
mass of the Earth and Moon, as-
sumed to be known, 4, 6 ;
Kepler's laws an approximate repre-
sentation of, 9 ;
(see ecliptic, elliptic, equations, perigee,
node, etc.).
Newton, law of attraction of, 1 ;
sufficiency of, to account for the
motion of the perigee, 239 ;
tested by Euler, 240 ;
results and discoveries of, 127, 237.
Node, ascending, defined, 41 ;
of the ecliptic on the equator, the
origin for reckoning longitudes,
261;
generally assumed to be in motion, in
the intermediary, 46 ;
made to revolve by the Sun's action, 53;
period of revolution of, 130 ;
distance of, from the perigee (H.), 177.
Node, longitude of the, in elliptic motion, 41 ;
equation for the variation of, 61, 64 ;
used by Euler, 239 ;
due to the motion of the ecliptic,
264;
notation for (D.), 137 ;
mean, on the ecliptic (H.), 194 ;
epoch of the, defined, 121.
Node, mean motion of, determination of, P.,
109; D., 158; H., 192; B., 230, 233; by
Newton, 237;
notation for (H.), 173 ;
numerical value of, 130 ;
Adams', of the principal part of,
230;
a test of the theory, 123 ;
higher parts of, equations for, 233, 234 ;
in Euler's method, 241 ;
connections of, with the constant
part of the parallax, 235 ;
in Laplace's method, 131;
secular acceleration of, 243, 268.
Notation, in Delaunay's theory, 136, 248 ;
in the operations, 152 ;
for the arguments, in the final
results, 159;
in Hansen's theory, 161, 172 ;
in the Fundamenta, 171 ;
tables of, 270-273.
Numerical orders, defined, 80.
Numerical theories, Clairaut's, 238 ;
Damoiseau's, 244 ;
Hansen's. 171 ;
Airy's, 245 ;
contrasted with other methods, 246.
Numerical values, of the lunar constants, 124
et seq. ;
difficulties in the determination of,
115 et seq. ;
determination of, by observation,
121 et seq. ;
288
GENERAL INDEX.
references to memoirs with, 131 ;
of Hansen, 188 ;
of Delaunay, unreduced, 251 ;
of the principal coefficients and periods,
124 et seq.;
magnitudes of, 131 ;
rapid approximation to (E.), 204 ;
of the motion of the perigee, 128, 223 ;
of the node, 130, 230 ;
of the coefficients in Eadau's equations,
for small disturbances, 251 ;
of inequalities due, to Venus, 258-260 ;
to the figure of the Earth, 262, 263 ;
to the motion of the ecliptic, 265 ;
of the secular acceleration, 267, 268 ;
of the solar constants, 123.
Nutation of the Earth's axis, used to determine
the ratio of the masses of the Earth and
Moon, 127.
Observation, determination of the constants
by, 121 et seq. ;
references to memoirs contain-
ing, 131 ;
method of Euler for, 240, 241 ;
theoretical motion of the perigee recon-
ciled with, 239 ;
tables deduced from, 246 ;
coefficients of Hansen's inequalities as
obtained from, 259 ;
determination of the ellipticity constant
from, 261;
the secular change in parallax insensible
to, 268.
Observed mean motion (see mean motion,
perigee).
Operation (D.), method for the calculation of
an, 150 et seq. ;
particular cases of, 153 et seq. ;
the final, 156 ;
effect of an, on the variables, 157.
Orders, of parameters, defined, 80 ;
of coefficients in the disturbing function,
connection between arguments and,
82;
for planetary action, 255 et seq. ;
effect on, produced by the integra-
tions, 84 et seq. ;
highest, given by Delaunay, 89 ;
least, of the solar terms in the coordi-
nates, 86 ;
of certain terms in the higher approxi-
mations, 86 ;
of the coefficients in the successive
approximations (P.), 95 et seq.;
to which the theories are carried, P., 113 ;
D., 133; E., 204, 211, 223, 230;
other methods, 237 et seq.
Origin, of coordinates considered to be the
Earth, 8 ;
for reckoning longitudes, 261.
Oscillations about steady motion, examples of,
47, 52, 211.
Parallactic inequalities, defined, 127;
determination of, P., 104 ; E., 227 ;
terms for, deduced directly from the
disturbing function, 111 ;
effect of, on the variational curve, 127 ;
correction to, for the Moon's mass, 126,
159.
Parallactic inequality, defined, 125 ;
order of, lowered by integration, 104 ;
period and coefficient of, 126 ;
used to determine the Sun's parallax,
127;
notation for argument of (D.), 159.
Parallax of the Moon, determination of, from
the inverse of the radius vector, 121;
mean value of the, 124 ;
magnitudes of coefficients in, 131 ;
secular inequality in, 266 ;
effect on the mean, 268 ;
(see radius vector).
Parallax of the Sun, determined by the paral-
lactic inequality, 127 ;
mean value of, 123.
Parameter, change of, to improve convergence,
114, 204.
Parameters, orders of, used, 80 (see orders).
Particles, Earth, Moon and Sun considered
as, 2.
Particular integrals, forms of, in the second
approximation, 50, 227.
Pendulum observations used to determine the
figure of the Earth, 261.
Perigee, generally assumed to be in motion in
the intermediary, 46 ;
made to revolve by the Sun's action, 53 ;
period of revolution of, 128;
distance of, from the node (H.), 177.
Perigee, longitude of, in elliptic motion, 41;
equations for the variation of, 62, 64;
notation for (D.), 137 ;
epoch of the mean, defined, 120.
Perigee, mean motion of the, determined, P.,
101, 103; D., 158; H., 185, 192; E., 218,
223, 234 ;
by Laplace's method, 131 ;
by Newton, 237 ;
GENERAL INDEX.
289
to the second order, numerically by
Clairaut, 238 ;
incident connected with, 239 ;
algebraically by d'Alembert,
239;
by Euler, 240 ;
of the auxiliary ellipse (H.), 165 ;
determined, 185 ;
in rect. coor. method,' determinantal
equation for principal part of, 217;
simple equation for, 218;
the higher parts of, 225, 234 ;
connections of, with the con-
stant part of the parallax,
235;
higher parts of, in Euler's method, 241 ;
numerical value of, 128;
Hill's, for the principal part, 223 ;
observed, used by Hansen, 188 ;
by Euler, 240 ;
a test of the theory, 123;
secular acceleration of, 243, 268.
Period, the mean, obtained from observation,
123;
numerical value of, 124 ;
of the Sun, 123 ;
of the mean motion of the perigee, 128 ;
of the node, 130 ;
of the variational curve, 125 ;
mean-, inequalities (see mean period).
Periods, of oscillations about a steady motion,
47, 52, 211 ;
case of an infinite number of, 217 ;
of the principal inequalities, 124 et seq. ;
due to planetary action, 257, 258 ;
to Venus, 258, 260;
to the figure of the Earth, 263 ;
to the motion of the ecliptic, 265.
Periodic functions, defined, 45 ;
used to represent the coordinates, 45 ;
time as a factor of, to be avoided, 45
(see time).
Periodic solution, defined, 46 ;
used as an intermediary, in general, 46;
Hill's (E.), 198 et seq.
Periodicity, of an angular coordinate, defined,
49;
of the variation of the solar eccentricity,
268.
Perturbations, of the solar orbit included (H. ),
172;
in the disturbing function, 177 ;
neglected, in the first approximation,
181;
of the ecliptic (H.), 191 ;
B. L. T.
Adams' method for, 264 ;
of latitude, Hansen's method of ex-
pressing, 194.
Plana and Carlini, theory of, 244.
Plane of orbit, line fixed in, defined, 60 ;
properties of, 60;
rate of rotation of, due to the disturbing
forces, 60 ;
quantities defining (H.), 172;
equations of motion for, in terms of the
disturbing forces (H.), 176 ;
in terms of the derivatives of the
disturbing function, 190.
Plane of reference, the plane of the Sun's orbit
supposed fixed, 13 ;
the invariable plane as a, 27 ;
Hansen's, 162.
Planetary action, effect of, on the apparent
solar motion, 6 ;
direct and indirect, disturbing functions
for, 252 ;
separation of the terms in, 253 ;
expressions by polar coordinates,
254;
developments in terms of the elliptic
elements, 255, 256 ;
nature of the terms in, 257 ;
direct, inequality due to Venus, 258 ;
indirect, methods of including, 253, 257 ;
case of, 259 ;
inequality due to Venus, 260 ;
motion of the ecliptic, 263 et seq. ;
secular acceleration, 265 et seq.
Planetary theory, distinction between the lunar
and the, 8-10;
variation of arbitrary constants used in
the, 66, 245;
Hansen's, 161 ;
theorem at the basis of, 77.
Poisson, method proposed by, 245.
Polar coordinates, transformation from rect-
angular to (E.), 206 (see coordinates).
Potential due to the figure of the Earth, 260 ;
(see force-function).
Precession of the Equinoxes, used to determine
the figure of the Earth, 261.
Primary, defined, 9 ;
mass of, compared to that of the dis-
turbed body, 9.
Principal elliptic inequality (see elliptic}.
Principal function, defined, 68 ;
satisfies a partial differential equation, 69 ;
used for equations of elliptic motion,
69 et seq.;
and in the variation of arbitraries, 71.
19
290
GENERAL INDEX.
Principal inequality in latitude (see latitude).
Problem of p bodies, defined, 2 ;
force-function for, 11.
Problem of three bodies, defined, 2 ;
lunar theory, a case of, 2 ;
limitations of, 2 ;
equations of motion for, 25;
the ten first integrals, 26;
the number of variables reduced
by, 28.
Pseudo-element, defined, 74;
derivative with respect to a, 74 ;
occurs when longitudes are reckoned
from a departure point, 75;
introduces another arbitrary, 75 ;
used by Hansen, 163.
Radau, numerical equations of, for small dis-
turbances, 251;
application to various inequalities,
258, 259, 262.
Radius-equation, defined, 16;
effect produced on the orders of coeffi-
cients by the integration of, 84 et seq. ;
prepared form of (P.), 93;
constant parts of, omitted, 100.
Radius vector, elliptic expression for, 41 ;
as an instantaneous axis, rate of rota-
tion of the orbit about, 60 ;
derivative of disturbing function with
respect to the, 14 et seq., 83, 179;
terms not containing m in, 87 ;
constant part of, 120 (see distance) ;
Hansen's method of computing, 160 ;
relation of, to the, of the auxiliary
ellipse (H.), 165;
equation for, 168;
solution, 189.
Radius vector, inverse of the elliptic value of,
35; D., 138;
in disturbed motion (P.), 110;
form of expression for (D.), 158;
constant part of, 120;
connections of, with the motions of
perigee and node, 234 et seq. ;
general theorem concerning, 235 ;
determination of the parallax from the,
121;
transformation to find the (R.), 206;
projection of the, theories using as a
dependent variable, 17, 238 et seq. ;
(see parallax).
Radius vector of the Sun, perturbations of, H.,
177, 259.
Ratio (see mean motions, distances, mass).
Rectangular axes, moving with the mean solar
angular velocity, 19 ;
moving with the mean lunar angular
velocity, used by Euler, 240.
Rectangular coordinates, method with, equa-
tions of motion, 19 et seq.;
particular cases of, 24;
development of the disturbing function,
20, 92;
elliptic series not used in, 92 ;
origin of, and limitations imposed on,
196;
intermediate orbit, 46, 197 ;
determination of, 198 et seq. ;
transformation to polars, 206 ;
elliptic inequalities, 206 et seq. ;
of the first order, 209 ;
of higher orders, 223;
motion of the perigee, method for, 211
et seq. ;
determinantal equation for, 217 ;
simple equation for, 218;
value obtained from, 223 ;
parts of, of higher orders, 225, 234 ;
mean period inequalities, 225;
parallactic inequalities, 227;
inequalities in latitude, 228 et seq. ;
of the first order, 229 ;
of higher orders, 231 ;
motion of the node, 230 ;
equation for part of, of the second
order, 233;
of higher orders, 235 ;
another mode of development, 234 ;
theorems in connection with, 235, 236 ;
compared with other methods, 247.
Reduction, the, defined, 39;
terms constituting the, in elliptic mo-
tion, 39.
Reference, plane of (see plane).
Relations between the, developments of the
disturbing functions, 89, 92 ;
constants, in the various methods, 116 ;
inR. and P., 205,211, 231;
solutions, in R. and P., 199, 207, 230;
elliptic elements and Delaunay's vari-
ables, 138;
old and new variables, after any opera-
tion (D.), 149;
disturbing functions, after any operation
(D.), 152;
motions of perigee and node and the
constant part of the parallax, 235;
coefficients of a solar and the resulting
lunar inequality, 259.
GENERAL INDEX.
291
Relative forces (see force, force-function).
Results, summary of (P.), HO;
Delaunay's, form of, after the operations,
157;
correction to, to account for the
Moon's mass, 159;
comparison of, with Hansen's, 159 ;
Hansen's, form of, 188 ;
reduction of, to the ecliptic, 194.
Reversion of series, necessary in Laplace's
method, 243;
theories requiring, useless when great
accuracy is needed, 247.
Roots of an infinite determinantal equation,
properties of, 217, 218.
Rotation of orbits due to the disturbing forces,
rate of, 60.
Saturn, large inequality in the motion of, 10.
Seconds of arc, reduction of coefficients to, 121.
Secular acceleration, defined, 267 ;
cause of, 265 ;
first approximation to, determined, 266,
267;
controversy concerning, 243 ;
of the perigee, of the node, 243, 268.
Secular terms, avoided by a modification of the
intermediary, 53;
treated by the variation of arbitraries,
66, 245.
Semi-algebraical theories, rect. coor., 195;
of Euler, Laplace, 240-242;
value of, 246.
Separation of terms in the planetary disturb-
ing functions, 253.
Series (see coordinates, convergence).
Short-period terms due to planetary action, 257.
Signification of the constants, elements (see
constants, elements).
Small divisors introduced by integration, 85 et
seq.
Solution, when the solar action is neglected,
of de Pontecoulant's equations, 41 ;
of Laplace's equations, 42 ;
form to be given to the general, 45 ;
a periodic, defined, 46 ;
used as an intermediary, 46;
by continued approximation (see con-
tinued approximation);
of a certain linear equation, 108 ;
nature of, of canonical equations (D.),
144;
methods of deducing the (R.), 200;
deduction of the (R.), from de Ponte-
coulant's results, 199, 207, 230;
of differential equations by indetermi-
nate coefficients, first used by Euler,
241.
Summary of results (P.), 110.
Symbolic formula for the true anomaly in
terms of the mean, 33.
Tables, of the Moon's motion, references to,
123, 161, 192, 238 et seq., 246;
of the notation used, 270-273.
Tangential transformation from one canonical
system to another, 66.
Theorem, of Hansen, relative to elliptic ex-
pansions, 36 ;
application, 184, 185;
relative to any function of the
elements and the time, 77;
concerning the Moon's parallax, 235;
of Adams, 236.
Theory, lunar, planetary (see lunar, planetary).
Three bodies, problem of (see problem).
Tides, used to determine the ratio of the
masses of the Earth and Moon, 127;
effect of, on the lunar motion, 248.
Time, expression for, hi terms of the true
longitude, in undisturbed motion, 42;
not present explicitly, in the equations
for the variable elements, 73 ;
or in Delaunay's formulae, 138 ;
coefficient of, in the true longitude, 97;
coefficients of, in the arguments, assumed
incommensurable and will not vanish
unless the arguments vanish, 49, 81,
184;
introduction of a constant (H.), 169;
terms increasing with, produced by the
secular acceleration, 266.
Time as a factor of periodic terms, to be
avoided if possible, 10, 45;
how it may occur, 50;
modification of intermediary to avoid, 52 ;
explanation of apparent occurrence of, in
the second approximation, 85 et seq. ;
removed from the equation for the
epoch, 63;
in Delaunay's method, 134;
transformation to avoid, 135 ;
appears again, 145 ;
transformation to avoid, 147 ;
in Hansen's method, how avoided, 166,
173;
neglected in the secular acceleration,
266.
Transcendental uniform integrals, in the prob-
lem of three bodies, limited number of, 27.
292
GENERAL INDEX.
Transformation, tangential, denned, 66 ;
method of, used in Delaunay's theory,
134;
conditions of the possibility of, 135;
applications of, 136, 142, 147, 149 ;
to new constants (D.), 158;
from rectangular to polar coordinates
(E.) 5 206.
Triangle, variations of the sides and angles of
a spherical, 174.
True anomaly (see anomaly).
True longitude (see longitude).
Undisturbed elliptic motion, 40 et seq.
Unit of mass, astronomical, defined, 1 ;
used in de Ponte"coulant's method, 82.
Variables, used in the various methods, 12,
238 et seq. ;
number of, in the problem of three
bodies, 28;
change of, in Delaunay's method, 135,
136, 147, 148;
used in Hansen's method, 166, 172, 179.
Variation of arbitrary constants, method of,
47;
coordinates and velocities have the same
form, for disturbed and undisturbed
motion in, 48, 58, 72;
application of, elementary, 54 et seq. ;
Jacobi's method, 71 ;
Lagrange's method, 73;
given by Euler, 241 ;
suggested by Poisson, 245 ;
remarks on use of, in the lunar and
planetary theories, 66, 245;
(see variations).
Variation of the solar eccentricity, the cause of
the secular acceleration, 243, 265.
Variation, the, defined, 124 ;
period and coefficient of, 124, 125 ;
Newton's value, 127 ;
notation for the argument of (D.), 159.
Variations of the elements, change of position
produced by, 56;
zero in the motion, 59;
equations for, found, 59 et seq. ;
in terms of the forces, 63 ;
in terms of the derivatives of the
disturbing function, 64;
Delaunay's canonical equations for, 64;
in the form used, 136 ;
remarks on, 66;
deduced by Jacobi's method, 72 ;
use of pseudo-elements in, 74 ;
Lagrange's canonical system, 76 ;
Hansen's extension, 76 ;
equations for, used by Hanson, 162 ;
for small disturbances, Eadau's numerical
equations, 251 ;
due to the motion of the ecliptic, 264.
Variations of the sides and angles of a spherical
triangle, 174.
Variational curve, defined, 125 ;
for different values of m, 125 ;
effect of the parallactic inequalities on,
127;
Newton's ratio of the axes of, 127 ;
used as an intermediary (R.), 198;
a periodic solution of Hill's equa-
tions, 198.
Variational inequalities, defined, 125 ;
determination of (P.), 96 et seq. ;
terms for, deduced directly from the
disturbing function, 111 ;
in rect. coor. method, equations for,
196;
form of solution, 199 ;
determination of coefficients, 200 et
seq. ;
rapidity of the approximations, 204 ;
transformation to polar coordinates,
205;
parts of, of higher orders, 223, 231.
Velocity^ expression for the square of, with
de Pontecoulant's equations, 14 ;
with Laplace's equations, 18 ;
with rect. coor. method, 22, 25 ;
in elliptic motion, 40.
Velocities and coordinates, having the same
form in disturbed and undisturbed motion,
48, 58, 72;
initial, a canonical system, 76.
Venus, ratio of distance of, from the Sun to
that of the Earth, 9;
inequality due to the direct action of,
258;
to the indirect action, 260 ;
Hansen's two inequalities due to, 259.
Verification, equations for (R.), 205, 236;
of Delaunay's results, Airy's method for,
245.
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