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73 



OSMANIA UNIVERSITY LIBRARY 

pall No. <T&f- S x 4 <?? /^ Accession No. 
Author 

Title 

This book should be re|jrned on or before the date last marked below. 





PLANETARY THEORY 



LONDON 

Cambridge University Press 

FETTER LANE 

NEW YORK TORONTO 
BOMBAY CALCUTTA MADRAS 

Macmillan 

TOKYO 

Maruzen Company Ltd 
A II rights resewd 



PLANETARY THEORY 



BY 
ERNEST W. 

I'/D/esxor Kniciitns of Mathematics in 
Yale Unit ersitif 

AND 

CLARENCE A. SHOOK 

** ~._ 

Ati^i^tant I'tofensot' of Mathematics in 
Lcli i</h I r itit'<T.'iiltf 



CAMBRIDGE 

AT THE UNIVERSITY PRESS 

1933 



* IN GREAT BR11AIN 



CONTENTS 

Preface ......... page ix 

Chapter I. EQUATIONS OF MOTION . 1 

General introduction 1 

Planetary form of the equations of motion . . 8 

Satellite or stellar form . . . . 11 

Frames of reference . . . . . . 15 

Reference to polar coordinates in the osculating 

plane 22 

True longitude as independent variable . . 28 

Kccentric anomaly as independent variable . . 30 
Reference to the coordinates of the disturbing 

planet 31 

Motion referred to an arbitrary plane of reference 34 

Chapter II. METHODS FOR THE EXPANSION OF A FUNC- 
TION 36 

Expansion by functions of an operator ... 37 
Lagrange's theorem for the expansion of a function 

defined by an implicit equation, and its extension 37 
Transformation of a Fourier expansion from one 

argument to another where the relation between 

the arguments is defined by an implicit equation 40 

Expansion by symbolic operators .... 45 

Product of two Fourier series .... 45 
Fourier expansions of series given in powers of the 

sine and cosine ....... 47 

Function of a Fourier series 49 

Powers of a Fourier series ..... 53 

Cosines and sines of Fourier series ... 54 

Bessel's functions . . . . . . . 55 

Hypergeometric series ...... 56 

Numerical calculation of series .... 59 



vi CONTENTS 

Chapter III. ELLIPTIC MOTION page 62 

Solution of the equations ..... 62 

Relations between the anomalies .... 65 

Kepler's laws 66 

Developments in terms of the eccentric anomaly . 68 

Developments in terms of the true anomaly . . 71 

Developments in terms of the mean anomaly . 72 

Literal developments to the seventh order . . 79 

Development by harmonic analysis ... 80 

Numerical solution of Kepler's equation . . 81 

Chapter IV. DEVELOPMENT OF THE DISTURBING FUNC- 
TION 82 

Development in terms of elliptic elements . . 82 
Expansion by operators in powers of the eccentri- 
cities ........ 87 

Expansion in multiples of the true anomaly . . 89 

Expansion along powers of the inclination . . 91 

Development to the third order .... 93 

Transformation from true to mean anomalies . 94 

Properties of the expansions . . . . . 95 

Calculation of the constant term . . . . 97 

Development in terms of the eccentric anomalies . 99 

Transformation to mean anomalies . . . 100 



The functions of the major axes . . . . 102 

lateral expansion to the second order in terms of 

the mean anomalies . . . . . . 112 

The second term of the disturbing function . . 114 

Chapter V. CANONICAL AND ELLIPTIC VARIABLES . 117 

Contact transformations 118 

Jacobi's partial differential equation and its solu- 
tions. ........ 121 

Jacobian method of solution . . . . . 1 27 

Other canonical and non -canonical sets. . . 131 

The case of attraction proportional to the distance 136 



CONTENTS vii 
Chapter VI. SoLtfTiON OF CANONICAL EQUATIONS . page 138 

General properties of the variables and of the 

disturbing function . . . . . . 138 

Eli mination of a portion of the disturbing function 143 

First approximation . . . , . . 146 

Long period terms . . . . . . 152 

Second approximation . . . . . 156 

Calculation of the second approximations to long 

period and secular terms . . . . . 161 

Summary and special cases . .... 169 



Chapter VII. PLANETARY THEORY IN TERMS OF THE 

ORBITAL TRUE LONGITUDE . . . 174 

Equations of motion and method of solution . 174 
The first approximation . . . . . 17,8 
Solution of the equations . . . . . 185 
Equations for the variations of the elements . 194 
The second approximation . . . . . 198 
Transformation to the time as independent vari- 
able 207 

Approximate formulae for the perturbations of 

the coordinates 210 

Definitions and determination of the constants . 213 



Chapter VIII. RESONANCE 216 

Elementary theory 219 

Solution of a resonance equation . . . 222 

General case of resonance in the perturbation 

problem 226 

A general method for treating resonance cases . 234 

The case e'= T = 238 

The 1 : 2 case 241 

The cases e'^0 246 



viii CONTENTS 

Chapter IX. THE TROJAN GROUP OF ASTEROIDS . page 250 

The triangular solutions ..... 250 

General theory ....... 256 

Elimination of the short period terms . . . 259 

Expansion of the disturbing function . . . 261 

The equation giving the libration . . . 269 

Perturbations of the remaining variables . . 273 

Higher approximations . . . . . 277 

Perturbations by Saturn ..... 280 

Indirect perturbations 283 

Direct action of Saturn ..... 286 

Appendix A. NUMERICAL HARMONIC ANALYSIS . . 289 

Index ......... 801 



PREFACE 

THE chief purpose of this volume is the development of methods 
for the calculation of the general orbit of a planet. If an 
accuracy comparable with that of modern observation is to be 
attained in any particular case, the choice of the method to be 
adopted may be an important factor in the amount of calculation 
to be performed. Not only should the general plan of procedure 
be efficient, but full consideration of tha details of the work 
should be given in advance. We have attempted to anticipate 
the difficulties which arise, not only in the older methods but 
also in those developed here, by setting forth the various devices 
which may be utilised when needed. 

While the developments given below are intended to be 
complete in the sense that they should not require a knowledge 
of the subject drawn from other sources, the volume is not 
supposed to be a substitute for an extended treatise like that of 
Tisserand. It contains, for example, no detailed account of such 
classical theories as those of Leverrier, Hansen and Newcomb. 
It does, however, attempt to indicate that most of the methods 
previously used ultimately reduce to two. One of the methods 
involves a change of the variables to elliptic elements, while the 
other consists of a direct calculation of expressions for the co- 
ordinates. An example of each of these general plans is given 
and worked out in detail. 

Few references to previous work have been made and those 
furnished are merely incidental. It has seemed unnecessary to 
repeat material which the student can find equally well in 
Tisse rand's treatise or in vol. IV of the Encyklopddie der Mathe- 
matischen Wissenschaften. While a critical estimate of the merits 
and demerits of previous works would doubtless be of assistance 
to anyone planning to carry out detailed calculations for the 
theory of a particular planet, in the past the methods which 
have been adopted have been sometimes chosen less on account 
of their efficiency than for other reasons, and the same will 
probably be to some extent true in the future. Nevertheless it 



x PREFACE 

is still advisable to give consideration to each of these plans, 
and we have attempted, by occasional remarks, to aid the 
student in this respect. 

The mathematical processes which are used in developing 
the theories of the planets and satellites from the laws of motion 
are largely formal While mathematical rigour is desirable when 
it can be attained, nearly all progress in the knowledge of the 
effects of these laws would be stopped if complete justification 
of every step in the process were demanded. The use of formal 
processes is justified whenever experience shows that the results, 
riot otherwise obtainable, are useful for the prediction of physical 
phenomena. Thus when calculating with an infinite series 
whose convergence properties are not known, one has to be 
guided by the results obtained; if the series appears to be con- 
verging with sufficient rapidity to yield the needed degree of 
accuracy, there is no choice save that of using the numerical 
values which it gives. We have not attempted to deal with con- 
vergence questions, but have retained throughout the practical 
point of view mentioned in the first sentence of this preface. 

Considerable portions of the volume are new in the sense that 
if they had not been given here, they would have been printed 
in abbreviated form in the current journals. In particular is, 
this true of the last two chapters. The novelty, however, consists 
mainly in the adaptation and further exploitation of previously 
known devices. Some of these extensions owe their effectiveness 
to a recent publication of tables of certain functions*, or to the 
introduction of mechanical computing aids. An example is the 
attention given to development by harmonic analysis. 

The following sentences give a brief summary of the contents 
of the volume. In the first chapter, various forms of the equations 
of motion are derived, other possible forms being suggested. The 
second chapter is a collection of various expansion theorems 
which are or may be needed in the later developments. A short 
account of the essential properties of elliptic motion follows. 
Various methods for developing the disturbing function and 
disturbing forces are set forth in Chapter iv. Chapter v contains 
* See footnote, p. 182. 



PREFACE xi 

the elements of the theory of canonical variables so far as it is 
needed in the later work. This theory is usually difficult for 
a student to grasp, and we have tried to simplify the exposition 
so that he may not only be able to understand it but also to 
make use of it as a tool for investigation. In Chapter VI, it is 
shown how this theory may be efficiently applied to the calcula- 
tion of the orbit of a planet ; the basis of the method is the use 
of the transformation to eliminate the short-period terms as 
a first step, leaving the long-period and secular terms to be dealt 
with separately. 

In Chapter vii, the direct calculation of the coordinates with 
the use of the true orbital longitude as the independent variable 
is developed with sufficient detail for the formation of an approxi- 
mate or of an accurate theory. Chapter VIII contains an attempt 
to place the theory of resonance on a general basis, in a form 
which permits of application to specific problems. The point of 
view taken is mainly that of explaining how this phenomenon 
can be treated mathematically in certain of the cases of its 
presence in the solar system. It appears, however, to give a 
method of approach to the consideration of the question of the 
general stability of the orbits of the planets and thus leads to 
certain aspects of cosmogony. In Chapter IX, the theory is 
applied to the Trojan group of asteroids in a form which it is 
hoped will make the calculations of the orbits of these bodies 
easier than has hitherto been the case. 

The appendix on Harmonic Analysis will be found to contain 
formulae for its application to the development of a given function 
ready for actual use. Most of these formulae have been tried 
out extensively and have been found to render the computations 
easy and accurate, especially when the number of such functions 
to be analysed is great. 

We are indebted to Dr Dirk Brou wer and Mr R. I. Wolff for the 
errata given on p. xii and discovered after the sheets had been 

printed. 

r ERNEST W. BROWN 

CLARENCE A. SHOOK 
1933 



Errata 

p. 26, line 5 from bottom, insert factor , before ^-- . 
* r 2 oT 

p. 28, line 6 from bottom, for u 2 , read uu' 2 _- . 
11 ou du 

p. 28, Equation (4), for $q read . 

2 2tf 2 

p. 28, Equation (5), /or read - .. . 
v ?r ^ 

p. 28, Equation (7), for -T read + l\ 

p. 30, omit lines 12, 13. 

p. 63, Equation (10), /or rtd read andt. 



p. 64, lines 12, 15, for 

p. 66, last line, /or 47r' 2 /x, read /x/47r-. 

p. 68, line 18, insert 2 fc before second formula. 

p. 76, Equation (J), for {-) read ( C -j . 

p. 83, line 18, for a~ read a-. 

p. 87, line 7, for # = exp ^ read x-~exp \f/ *J - 1. 

p. 87, Equation (2), delete the letter^. 



p. 98, line 22, /or ^ 

p. 117, Equation (2), for m v read m. 
p. 118, line 5, for (5) mid (6). 
p. 129, line 6 from bottom, for (3) read (8). 
p. 131, line 14, for ^(-2^)"* m/d ^(-2^)"-. 
p. 131, delete line 15, replacing it by *'Here the choice for S, slightly 
differing from (1), is". 

,._ r o f W , $ S 

p. 145, line 2, /or -_- mid --^- 

p. 157, line 4 from bottom, insert 2 before the last term. 

p. 163, Equation (12), for - 3S read -32/j. 

p. 176, last line, for Dv = l read Dv = l. 

p. 224, line 15, for 8-8(9) read 8'S (10). 

p. 230, line 5, for R read R . 

p. 250, lines 5, 23, for Laplace read Lagrange. 

p. 250, line 19, for configuration read condition. 



p. 253, line 6 from bottom, for read 

r* 

p. 255, line 11, the equations should read 

aB _ db 3 >/3 m' - 8a 3 /t 2 1 (y. m)* _ >/3~ t 
- I 8 ^ "^ 12 + ,I' 



CHAPTER I 

EQUATIONS OF MOTION 

A. GENERAL INTRODUCTION 

I'l. The methods considered in this volume for the investiga- 
tion of the mutual actions of two or more bodies are based wholly 
on Newton's three laws of motion and on his law of gravitation. 
It is assumed that there exist fundamental frames of reference 
with respect to which the laws are exact and that the space in 
which the bodies move is Euclidean. The modern theory of re- 
lativity gives a different approach to the problem, but from 
the point of view taken here, which is chiefly that of deriving 
formulae for the comparison of gravitational theory with obser- 
vation, the numerical difference resulting from the two methods 
of approach is very small, and can be exhibited as a correction 
to the results obtained through the Newtonian approach. These 
corrections, which are near the limit of observational accuracy 
at the present time, will not be considered here. 

A further limitation is the treatment of the motions of the 
bodies as those of particles having masses equal to the actual 
masses. Here again, owing to the theorem that a sphere of 
matter, whose layers of equal density are concentric spheres, 
attracts an outside body as if it were a material particle, and 
also, owing to the fact that most of the bodies with which we 
have to deal are approximately spheres of this character or are 
sufficiently far away from the attracted body that they can be 
so treated, the differences are small. All other possible and actual 
forces, unless they obey the inverse square law and have con- 
stants which can be supposed to be included in the constants 
which we call the masses of the bodies, are neglected. 

1*2. A general knowledge of the masses and relative distances 
of the various bodies from one another has to be assumed 
because the method of treatment to be recommended depends 

B&SPT I 



2 EQUATIONS OF MOTION [OH. i 

on this knowledge. The two principal divisions are the planetary 
problem and the satellite or stellar problem. A third division 
may include the cometary problem and those cases of motions 
which cannot be included in either of the first two divisions. 

In the planetary problem, one mass is very much greater than 
the combined masses of all the other bodies and dominates the 
motion of any one of them to such an extent that during a few 
revolutions the orbit of the latter is not greatly different from 
that which it would describe if the remaining masses did not 
exist. These approximate orbits moreover are ellipses with the 
principal mass in one focus and having minor axes which do not 
differ from their major axes by more than about ten per cent, of 
the latter. Further, the planes of these ellipses tire inclined to 
one another at small angles generally less than 20. The dis- 
tances of the bodies from one another may have any values what- 
ever provided they do not fall below a certain limit. In general, 
the methods of this volume are developed for this case alone. 

In the satellite or stellar problem the distance between two 
of the bodies must be small compared with the distance of 
either from the third; the two nearer bodies circulate round one 
another and their centre of mass circulates round the third body. 
The maintenance of this state of motion requires a limiting 
relation between the masses and distances of the bodies. There 
are also limitations concerning the shapes and positions of the 
orbits similar to those present in the planetary case. In the 
satellite problem, the mass of one of the two nearer bodies is 
small compared with that of the other, and the mass of the latter 
small compared with that of the third body. In the stellar 
problem, the masses are usually of the same order of magnitude. 
The methods adopted to obtain the motions in these two limited 
cases are not applicable to the cometary and other cases. 

The methods developed below give expressions for the co- 
ordinates in terms of the time which serve to give the positions 
of the bodies over long intervals of time: the results are usually 
named the general perturbations. Practically all other cases have 
at present to be treated by the method of special perturbations 



2, 3] INTRODUCTION 3 

which consists in a completely numerical process of calculating 
the orbit over successive small arcs by 'mechanical integration.' 
Practically all the problems which have hitherto been brought 
within the range of observation belong to one of these classes. 
There are numerous problems in the stellar universe in which 
the law of gravitation undoubtedly plays a dominant r61e. At 
the present time the deviations from rectilinear motion have 
not been directly observed, although such deviations have been 
inferred by statistical methods. 

1*3. Since no method for the exact integration of the equations 
of motion of three or more bodies exists, devices for continued 
approximation to the actual motion are used. Sometimes these 
lead rapidly to the desired degree of accuracy; in other problems, 
the process may have to be repeated many times. In most cases, 
the first approximation is taken to be an ellipse and this is 
equivalent to a start with the two-body problem and a con- 
tinuation with the calculation of the disturbing effects produced 
by the attractions of other bodies. 

As far as possible these changes in position or 'perturbations/ 
as they are named, are expressed by sums of periodic terms 
which take the form of sines or cosines of angles directly pro- 
portional to the time, or to some variable which always changes 
in tfye same sense as the time. When, however, the number of 
such terms becomes too great for convenient numerical applica- 
tion, the terms of very long period are replaced by powers of the 
time or other adopted independent variable, and these powers 
are used in combination with the other periodic terms. In any 
case, the expressions which are obtained give reliable results for 
a limited interval of time only; all the periods with which we 
have to deal are determined from observation and therefore 
possess a limited degree of accuracy. 

Expansions of functions in series, especially as sums of sines 
and cosines, thus play a large part in the work. The possibility 
of obtaining these expansions in such forms that numerical 
results may be deduced from them without too much labour, 



4 EQUATIONS OF MOTION [CH. i 

usually depends on the presence of small constants or variables 
'parameters' in the coefficients of the periodic terms, and 
the expansions are partly made along powers of these parameters. 
Their orders of magnitude are important. In general, they 
consist of the eccentricities which rarely rise much above *2, 
of the inclinations of the orbital planes to one another usually 
less than 20, and of the ratios of the distances. 

For the majority of cases, this last ratio lies between *4 and 
*8, and the feet that we are compelled to expand in powers of so 
large a parameter is responsible for many of the difficulties of 
the problems. 

1'4. The most fundamental difficulty, however, is caused by 
approximate or exact 'resonance.' This term refers to those 
cases in which two or more of the periods, which enter into the 
expansions for the coordinates, are nearly or exactly in the ratio 
of two small integers. In the approximate case, large amplitudes 
of certain of the periodic terms, and slow numerical convergence 
to the needed degree of accuracy, are characteristic effects. In 
the cases of exact resonance the form of the solution has to be 
changed. 

In either event, the terms which cause the chief trouble are 
those with periods which are long in comparison with the period 
of revolution of the body round the central mass. Such periodic 
terms may have small coefficients in the equations of motion, 
but the integration of the equations produces small divisors 
which give large coefficients in the coordinates. These small 
divisors demand that the terms affected be carried to a much 
higher degree of accuracy than the remaining terms, and as 
there exists no short method for securing this accuracy, the 
amount of calculation needed in any given planetary problem 
depends mainly on the few, perhaps one or two only, terms of 
long period which are sensible in the observations. The existence 
of such terms in every planetary problem has to be kept in 
mind while divising methods and in carrying them out. The 
method finally chosen should depend mainly, not on the ease 



3-5] ASTRONOMICAL MEASUREMENTS 5 

with which the first approximation may be obtained, but on the 
work required for the final approximation. 

1/5. Astronomical measurements. The only measurements of 
the position of a celestial object which have a precision com- 
parable with gravitational theory are those of angles on the 
celestial sphere. The ultimate planes of reference from which 
these angles are measured along great circles are defined by the 
average positions of the stars (which have observable motions 
relative to one another). For the theory, an origin is needed, 
and this is ultimately the centre of mass of the system. It is 
assumed that these definitions will give a Newtonian frame 
because the stars are so far away that they affect neither the 
motion of the centre of mass nor the relative motions of the 
bodies within the system to an observable extent. 

Time in this frame is measured by the interval between the 
instants when a plane fixed in the earth arid passing through 
its axis of rotation passes through a mark in the sky supposed 
to be fixed relatively to the stars. 

Owing to the rapid motion of the earth about its axis the 
observer finds it convenient to give his measurements with 
respect to the plane of the earth's equator, and to a point on the 
equator defined by its intersection with the ecliptic the plane 
of the earth's orbit round the sun. Both these planes are in 
motion but their motions and positions relative to the ultimate 
stellar frame are supposed to be known. This frame is incon- 
venient for working out gravitational theories on account of its 
large inclination to the planes of motion of most of the bodies 
of the solar system. For this purpose the ecliptic and the point 
on it defined above are used. The motions of these are nearly 
uniform and are easily taken into account. 

In the observer's frame, the angular coordinates are the 
declination measured along a great circle from the object to the 
equator and perpendicular to the latter, and the right ascension, 
the angle between this great circle and that perpendicular to 
the equator and passing through its intersection with the 



6 EQUATIONS OF MOTION [CH. i 

ecliptic. In the computer's frame, the angular coordinates are 
the celestial latitude and longitude similarly measured with 
respect to the ecliptic. As this latter frame is moving the 
ultimate reference is to its position at some give'ri date. Thus 
to transform the results obtained from the gravitational theory 
for the use of the observer, geometrical relations must be com- 
puted, and there is, in addition, a kinematical relation due to 
the motion of the observer's frame. 

It is assumed that a complete gravitational theory of the motion of any 
body in the solar system referred to a frame whose motions are fully known, 
should give the position of the body at any time when the constants of the 
motion have been determined. Differences between the calculated and 
observed positions of the body may be due to defects in the theory, un- 
known motions of the frame of reference, errors in the determination of 
the constants, or errors of observation. The analysis of these differences 
in order to discover their source is a problem involving many difficulties. 
In many cases two or more interpretations are possible and these can only 
be separated by the use of more observations. An outstanding difference, 
for example, between the observed and calculated values of the motion of 
the perihelion of Mercury was variously attributed to the gravitational 
attraction of a ring of matter supposed to surround the sun, to a motion 
of the frame of reference, to defects in the gravitational theory, until the 
theory of relativity furnished an explanation. A marked deviation of the 
moon from its gravitational theory has received an explanation as a varia- 
tion in the rate of rotation of the earth about its axis, through detection 
of similar deviations from the gravitational theories of the observed posi- 
tion of the sun, the satellites of Jupiter, and the planet Mercury. 

1 *6. Observations of bodies in the solar system are usually of two classes. 
Those made with the transit telescope give the instant of passage of the 
body across the meridian of the observer and the angle between its observed 
direction and that of the earth's axis, the time being given by a clock which 
is constantly compared with the transits of stars. Differential observations, 
often made by photography, give the position of the body at any time with 
respect to stars in its neighbourhood, the places of these stars referred to 
the frame being known. Under good conditions, either class of observation 
should give the position with a probable error less than V such a standard 
at least is aimed at in the gravitational theories of the principal bodies in 
the solar system. 

While direct observations of distances cannot in general be made ac- 
curately, their theoretical determination is necessary because we cannot 



6-7] LAW OF GRAVITATION 7 

eliminate them from the equations of motion without introducing compli- 
cations greater than those which the equations already possess. They are 
also needed because the planetary theories use the sun as an origin ; the 
transformation from the observer's position on the earth to an origin in 
the sun requires a knowledge of the variations of these distances. 

Observations of masses or of relative masses are not made : a mass is 
only known to us by its gravitational effects. In the solar system, the 
orders of magnitude are such that the mass of a body has little effect on 
its own motion ; the operation of Kepler's third law, (3 -6), eliminates from 
the angular coordinates the greater part of the mass eftect. When dealing 
with the effect of one planet on another, we need to know the ratio of the 
mass of the disturbing planet to that of the sun. If the planet has a 
satellite whose motion can be calculated, the mass of the planet can be 
found with sufficient accuracy to calculate its effect on any other planet. 
When it has no satellite, its mass can be found only by comparing its cal- 
culated disturbing effects with those furnished by observation. There is of 
course a correlation between the degree of accuracy required to calculate 
the perturbations and that with which the perturbations can be observed. 
Thus the masses of all the major planets except those of Venus and Mercury 
are fairly well determined. The masses of the minor planets can be ob- 
tained only from observation of the light they send with estimates as to 
their albedo and density, since they are too small to exert any observable 
attractive effect on any other body. 

17. The Newtonian law of gravitation. This law states that 
the attractive force between two particles of masses m. m' at a 
distance r apart is along the line joining them and of magnitude 
Cmm'/i*, where C is a constant the 'gravitation constant' 
wfiose value depends only on the units adopted. It is convenient, 
in order to avoid the continual presence of C in the mathematical 
operations, to so choose the units that (7=1. Since m/r 2 has 
then the apparent dimensions of an acceleration, it follows that 
a mass with these units has the apparent dimensions of the cube 
of a length and the inverse square of a time. The adoption of 
this unit called the 'astronomical unit of mass' is closely 
associated with Kepler's third law. This law, with a slight 
modification (3*6) to bring it into accord with the law of gravi- 
tation, states that if ZTT/H be the period of revolution of two 
bodies about one another and if a be their mean distance apart, 
ii* a* = C x sum of masses. 



8 EQUATIONS OF MOTION [CH. i 

In numerical work we need to deal with ratios only. If a 
relation of this kind is used in the equations of motion, the 
latter will be freed from the apparent inconsistency of possessing 
terms having different physical dimensions, and will consist of 
ratios of masses, distances and times only. 

The units of time and distance to be finally chosen, depend 
on the problem under investigation: a choice is not usually 
needed until comparison with observation is to be made. For 
this purpose, the mean solar second, day or year are used as 
units of time, and the mean radius of the earth or the mean 
distance of the earth from the sun as units of distance. 

B, PLANETARY AND SATELLITE OR STELLAR TYPES 
OF THE EQUATIONS OF MOTION 

1*8. The Equations of motion with rectangular coordinates 
A force- function for the motion of any particle, when it exists, 
is defined as a function whose directional derivatives with respect 
to the coordinates of the particle give the components of the 
forces. 

For two particles with masses m, in and at a distance r apart, 
the gravitational force-function is mm' jr. For n + 1 bodies with 
masses mi and mutual distances ?', it is 





* ij 

the summation including each combination of i,j once only. 
This function is independent of the directions in the frame of 
reference which may be used. It may or may not depend on 
the origin chosen. 

In treatises on general mechanics, the potential is denned as a function 
which has the property that its directional derivatives give the reversed 
components of all the forces which act on the system. We shall have oc- 
casion later (e.g. in 1 *9) to construct force-functions which are not potentials 
with reversed signs, since the force-function for the motion of one particle 
is not the same as those for the other particles of the system. 



7-9] PLANETARY FORM 9 

Let , m, f be the coordinates of m if Then, with the defini- 
tion of V given above and under the limitations stated in 1*1, 
the equations of motion are 

d^i W d*m SV d 2 ^ BF 

m i~Ta^> m / J72~ = T~ ' m i~J^ = ^^' -(*) 

at* dgi at* dm at 2 o%i 

Only the relative motions of the bodies are needed and, in 
finding them, the equations are usually given two principal 
forms, depending on the origins chosen. 

1*9. Planetary form of the equations. In this form, one body 
w is chosen as the origin of coordinates, and the motions of the 
remaining bodies relative to m Q are to be determined. 

Let 

ff* = &-?o, l/k^^k-no, s&=6fc-oi / = !, 2, ...w, 

r k 2 = r ok 2 = (KI? + j/fc a + ^A 2 . 
Then r/ = (^ - #,)* + (^ - y^) 2 -h (^ - ^) 2 . 

These definitions, together with the equations of motion, give 



dt 2 



__ _ /p , v- _ '^fc _ v ; y 

""8.'-*U ; >,J >* 3 ~ ; ^ 3 ' 



where J =(= ^- The penultimate term, being equal to the derivative 
of m k jr k with respect to #&, can evidently be combined with that 
of mo/r k . The last term can be written 

9 



in which form it will serve for all three coordinates. 
Hence, if we put 



the equations of motion for m k relative to m Q become 



with similar equations for y k , z k . 



12 EQUATIONS OF MOTION . [OH. I 

so that the new coordinates still depend on the differences of 
the original coordinates. 

Now when 7 is expressed in terms of the coordinates ;, TH, &, 
we have, as a result of changes in the & only, 



But since V contains the ^ only through their differences. 

dV 37 dV 



Whence, combining thia with the previous equation, we have 
d V = j| (rfft - rf>) + ^ (d& - d$>). 

<7i O^2 

But when V is;to be expressed in terms of #, ,7;', we have 



7 , , , 

TT- dx + =-, dx . 

ox ox 

If we substitute the values for dx, dx' in terms of dgi dgQ, 
d^z d^ and equate the coefficients of the latter in the two 
expressions for dV, we obtain 

ar 

3 A'' ' 

...... (4) 

The transformed equations of motion for #, x r become, with 
the help of these results, 



^r, (5) 

dV I ftV , dV\ 

' - - ' - - ~i ^~~ ' 



. 

' y 



12, 13] SATELLITE FORM 13 

For the purposes of calculation, we insert the value of V, 

namely, 

, 7 m^mi 

y . j 

n>l 7*02 

with ?*i 2 2 = .1? + ?/ 2 + 2 = / 2 , 



= (,'i/H 2 ,] +... + ..., 

\ nil + itiz I 



giving 

d 2 x ?) (? + 77i w /M + //? 2 ) ///h w a 

~ I H --- 

\r i r 02 

/ 



For theoretical investigations, the equations are exhibited in 
the Newtonian form by putting 



2 ir / 1/r /ox 

, Kl = ^yU, K, ...... (8) 

tQi~- ^) 
when they become 



with similar equations for ?/, ^, ?/', ^'. 

This form cannot be used, however, if ?% or m% is zero. But 
the equations then revert to the planetary form, with the motion 
of m Q relative to raj elliptic if m 2 = 0, so that the only motion 
which has to be considered is that of ra 2 relative to raj. 

1'13. The equations of motion in the form 1*12 (6), (7) will not be needed 
in the developments of this volume since they are, in general, useful only 
when the distance between mi , m^ is small compared with those between 
wi , mi and wt , ?n 2 , that is when r/r' is small. The initial development of 
V will, however, be shown, in order to exhibit the contrast with the 
developments used in the planetary problem. 

Put xx' +yy' + zz? = rr' cos S, 



14 EQUATIONS OF MOTION [CH. i 

so that S is the angle between ?, r'. Then 

" '2 i 2W *2 ' V*f 2 \ 2 a 

r ()1 - r i -f - rr' cos >S -i- --- --- - ) r 2 , 

?/i 1 -fw 2 Vm^wo/ 

2 '<; 2m 1 , , , / Wi Y 2 ,, 
9 02 2 = r 2 - ?r eos*S-H ---- - ) ? . 

m! + 7?l 2 V> 1 1+ W1 2/ 



Hence, if /^ be the zonal harmonic of degree i with argument ft, we have 

1 -i / l+ ; (_ '*_ TP) 

'01 ' V ,=1 \ T+'S 'V '/' 

JL.UH. 

%2 '* I t 

and thence 

y^'/^wiy Wp(wi-f^a) Wf> * 
r / r t=2 

the terms for /=! having disappeared. 
The force-function 1*12 (6) then becomes 



L. _iilL_ 2 '^vi ( !_\ p I / 

the second term of F being useless. The function 1*12 (7) becomes 



r > 

the first term in this case being useless. 

In each case, if the function be confined to its first term we obtain elliptic 
motion (Chap, in), the remaining portion being that which produces the dis- 
turbing effect. If ?i, a be the mean angular velocity and mean distance in the 
former, and n', a' those in the latter, wo have by Kepler's third law, (3-6), 



By putting z = l, we see that the significant factors of the disturbing 
effects in (1), (2) are 



In the satellite problem, m is much greater than m t and m-! than w 2 , 
and a/ a', n'/w are small, so that both these factors are small. If m Q1 m-j, wi 2 
refer to the sun, earth and moon, respectively, these ratios have the mag- 
nitudes -007, 1-SxlO- 7 . 

In those stellar problems which have up to the present shown observa- 
tional evidence of perturbing effects, ?n , m t , m 2 are of the same order of 
magnitude, but a/a', n'/n are small of the order '2 or less. The disturbing 
effects are chiefly shown in the motions of the apses and nodes. See 
P. Slavenas, "The Stellar Case of the Problem of Three Bodies*." 

* Trans, of Yale Obs. vol. 6, pt. 3. 



13-15] COORDINATE SETS 15 

C. FRAMES OF REFERENCE 

1*14. Choice of variables. In the preceding sections, the 
equations of motion have been referred to certain origins in a 
Newtonian frame with fixed directions for axes. Experience has 
shown, however, that neither rectangular coordinates nor fixed 
axes are convenient for finding the position of the body at any 
time, and that the calculations may be much abbreviated by 
suitable choices of coordinates. Analytically, the deduction of 
these sets of equations may be regarded as nothing more than 
a change from one set of variables to another. But since the 
choice of a set of variables always depends on a knowledge of 
the general characteristics of the motion, it is often useful to 
give a geometrical or dynamical interpretation to the variables 
chosen. 

1'15. For the development of the planetary theory, the 
osculating plane as a principal plane of reference possesses 
certain advantages over all other planes of reference. It is 
defined as a plane passing through the sun and the tangent to 
the orbit of the planet. The plane is in motion but it is found 
in most cases that its motions are either small or slow; that is, its 
deviations from a mean position are either small or require long 
periods of time to become large. This fact can be so used in the 
analytical work as to abbreviate the calculations. 

A second and even more important property of this plane is 
due to the small effect its motion has on the motion of the 
planet within the plane. In many cases this secondary effect 
can be altogether neglected, so that the motion within the plane 
can be treated as though the latter always occupied its mean 
position. These remarks refer equally to both disturbing and 
disturbed planet, that is, to the planet whose motion is supposed 
known and to that which we are finding. The effect of the 
motion of the plane of the former on the latter is usually 
negligible or can be accounted for quite simply. 

All the methods developed in this volume use the osculating 
plane as a plane of reference. 



16 EQUATIONS OP MOTION [CH. i 

116. The choices of coordinates within the plane of reference 
may be placed in two categories. In the first of these, the distance 
of the planet from the sun or some function of this distance is used 
as one coordinate, the second coordinate being the elongation of 
this radius reckoned from some fixed or moving line with the time 
as the third variable in the equations of motion. The roles played 
by the second and third of these variables may be interchanged. 

Two ways of measuring the elongation are used below. One 
is the usual method of using a single symbol to denote the sum 
of two angles measured in different planes : that between the 
radius and the line of intersection of the osculating plane with 
a fixed plane of reference, and that between this latter line and 
a line fixed* in the plane of reference. This symbol is usually 
called the longitude in the orbit or, briefly, the longitude. The 
second is the measurement of the elongation from a line in the 
osculating plane, this line being so defined that its resultant 
velocity is always perpendicular to the osculating plane. This 
second method has the advantage of eliminating the motion of 
the osculating plane from the kinetic reactions within the plane. 
With the use of these methods it is convenient to introduce an 
auxiliary variable which substantially is the angular momentum 
or a function of it. 

1*17. The second category consists of the use of certain 
variables associated with the osculating ellipse. This ellipse is 
defined as the orbit which the body would follow if, at any 
instant, all disturbing forces were annihilated and the body 
continued its motion under the sole attraction of the central 
mass. The definition requires that the velocity in the orbit and 
in the osculating ellipse shall be the same in magnitude and 
direction and therefore that its plane shall be the osculating 
plane at the point. The variables used are those which define 
the size, shape and position of the ellipse, or certain functions of 
them which may or may not contain the time. These functions 
are called the elements of the ellipse. 

The elements which are simplest for descriptive purposes are 
* The word ' fixed ' is used in the Newtonian sense. 



16-18] COORDINATE SETS 17 

the major axis, the eccentricity, the longitude of the axis and 
the time of passage through the nearer apse or position when 
the distance from the focus occupied by the central mass is least ; 
the period of revolution is connected with the major axis by 
Kepler's third law which involves the sum of the masses of the 
two bodies. This sum may be unknown but, as it remains constant, 
the relation between the variations of the major axis and the 
period is always the same. Various combinations of these elements 
and of the two elements which define the position of the osculating 
plane are also used as elements : as we go from point to point of 
the actual orbit these elements will change. According to the 
definition, the changes will depend on the existence of attractions 
other than that of the central mass, and the term ' Variation of 
the Elements' refers to these changes. It will be seen below 
that the methods used to determine them are similar to that 
unfortunately named ' the Variation of Arbitrary Constants ' in 
the theory of differential equations. 

1*18. This geometrical description of the elliptic frame, while 
useful for descriptive purposes, conceals the analytical meaning 
which is essential for a clear understanding of the processes 
involved. Analytically, the elements are nothing else than a new 
set of variables allied to the coordinates by a definite set of 
relations which remain unchanged. Thus the process of forming 
the differential equations satisfied by the elements is precisely 
that of changing from one set of variables to another. 

The description of the process is complicated by the fact 
that the three old variables (the coordinates) are replaced by 
six new variables ; consequently, three relations between the 
latter are at our disposal. If the coordinates be denoted by Xi 
and the new variables by j, and if the relations between them be 

Xi=fi(ai,a 2 , .. .,<,,), t=l, 2, 3, 

then the three additional relations are almost invariably chosen 
to be so defined that they satisfy the equations 



B&SPT 



18 EQUATIONS OF MOTION [CH. i 

As a result of this definition, we have 



It follows that %i, dxi/dt are replaced by f it dfi/dt in the equations 
of motion and that d 2 %i/dt 2 is replaced by 



This last process gives three equations and these with (1) furnish 
the six equations necessary to find the a,-. The forms of the 
functions /i, and consequently those of their partial derivatives, 
remain unchanged and are given by algebraic and trigonometrical 
formulae developed in Chap. III. 

The analytical point of view just given is that which is chiefly 
needed in the development of the equations of motion. This view 
is often obscured by the methods used to obtain the functions fi. 
These methods require the solution of the equations for elliptic 
motion and in this solution the c^ appear as the arbitrary con- 
stants : in the general problem they become the variables. The 
fact is that the solution of the equations for elliptic motion is 
merely a convenient device for finding the functions f t which 
connect the old and new variables. 

The fact that the differential equations satisfied by the new 
variables are all of the first order, together with another property 
to be developed in Chap, v, namely, that the variables can be so 
chosen that the equations have the canonical form, is largely 
responsible for the use that has been made of them in theoretical 
investigations. Their practical value lies in the ease with which 
the equations may be integrated and in the simplicity of the geo- 
metrical interpretations which maybe given to some of the results. 

1*19. Certain methods like those of Hansen and Gylden, not developed 
in this volume, as well as that given in Chap, vn, possess to some extent 
the characteristics of both categories. No sharp division is possible or 
necessary, the sole test being that of convenience for the problem under 
consideration. Whenever a new variable is introduced, it can generally 
be related to some property of the ellipse, but this relation is not usually 
helpful except in so far as it may have led to the choice of the variable. 



18-21] COORDINATE SETS 19 

1*20. The frames of reference should also be regarded as four- 
dimensional in the sense that the time as well as the space co- 
ordinates should enter into consideration in making choices of 
new variables. The observer's demand is for expressions giving 
the space coordinates in terms of the time, but the analyst is 
free to regard any three of them as a function of the fourth and 
to solve the problem according to his choice. If the time be not 
used as the fourth coordinate, that is, as the independent vari- 
able, a final transformation is usually, though not necessarily, 
made to obtain the space coordinates in terms of the time. 
However, practical demands limit the nature of the independent 
variable. The linear coordinates are sums of periodic functions 
of the time, that is of a variable which is unlimited in magnitude 
and whose changes are always in the same sense. Any other 
independent variable which is chosen should have the same pro- 
perty: it may be angular or even areal provided the angles or 
areas are always changing in the same sense. Otherwise trouble- 
some infinities are apt to be introduced. 

1'21. The choice of a method for the solution of any particular problem 
depends on a number of factors which should receive consideration. 

As between the two principal categories described above (1*16 and 1*17), 
the elliptic frame requires the calculation of the expressions for six 
variables as against the three coordinates which are alone needed by the 
observer. On the other hand, the solution of the differential equations is 
much more simple for the elliptic frame than those for the coordinates. 
For this reason, certain sets of equations belonging to the first category 
have been so developed that their solution is as simple as those for the 
elliptic elements. 

The elliptic frame as actually used requires a literal development of the 
disturbing forces in terms of the variables : when high accuracy is needed 
this development may entail very great labour. On the other hand, in 
particular portions of the problem, for example, in the calculation of the 
secular terms, those of very long period and resonance terms, it appears to 
give the needed results more easily than any other method which has had 
extensive trial. 

For theoretical researches, and for the discovery of qualitative properties 
of the motion, the elliptic frame has in general been more fruitful than 
most of the other forms of the equations of motion. This statement refers 
to motions of the general character of those which have been observed 



20 EQUATIONS OF MOTION [OH. i 

rather than to those which are mathematically possible, and to work which 
has been done in the past rather than to what may be accomplished in the 
future. In this connection, it should be remembered that a quantitative 
solution for a particular set of problems is often more easily obtained by 
a procedure different from that which is used to deduce a qualitative result. 

1*22. In making a choice for the solution of a particular problem from 
the various methods which have been proposed or developed, there are 
several factors which should receive consideration. 

1. The question as to whether a literal or numerical development is to 
be made, that is, a development available for several cases or one which is 
applicable to the motion of a single body only. The choice depends, not 
only on the number of cases to which the solution can be applied, but on 
the degree of accuracy with which the initial conditions, that is, the 
arbitrary constants, are known. In cases where the deviations from 
elliptic motion are large, the literal method may involve such extensive 
computations that it becomes practically impossible, even if the infinite 
series used were sufficiently convergent to give the quantitative results 
needed. Sometimes a partly literal and partly numerical method can be 
adopted with but little extra labour. In all numerical methods, provision 
must be made for changes in the arbitrary constants which future obser- 
vations may furnish. 

Some details with reference to the problems of the solar system will 
make these statements more concrete. For the eight major planets the 
elements are known with considerable accuracy so that corrections to them 
need scarcely be considered at the present time as a factor in the choice 
of a method. It is impracticable to use the literal values of the ratios of 
their mean distances from one another owing to the numerical magnitudes 
of these quantities: numerical values must be adopted for these from the 
outset and these involve numerical values for the periods of revolution 
round the sun. Little is gained by the use of literal values ior their 
eccentricities and inclinations, and much labour is saved by using 
numerical values for the constant parts of the angular elements. Thus, 
completely numerical theories are indicated for the major planets. For 
the moon, the ratio of the periods of the moon and sun is the parameter 
along which convergence is least rapid and there is little doubt that its 
numerical value should be used from the outset. Literal values for all the 
other elements can be used with but little additional work. 

For the minor planets, numerical values of the ratios of the mean 
distances are again a necessity, but since there are groups of them in 
which this ratio is nearly the same it is useful to devise methods in which 
this ratio has a given numerical value while the other elements are left 



21,22] COORDINATE SETS 21 

arbitrary. For most of the satellites other than the moon, complete 
numerical theories are indicated. This becomes a practical necessity in 
the cases of the outer satellites of Jupiter where the eccentricities and in- 
clinations have large values, although nothing is gained by using the 
numerical values of the constant parts of the angles. 

In general, planetary problems should be separated from satellite 
problems. In the former convergence is slow along powers of the ratios 
of the mean distances, but rapid along powers of the ratio of the mass of 
the disturbing body to that of the primary; in the latter the case is 
reversed. For the planetary problems the amount of calculation needed 
for the terms dependent on the second and higher powers of the mass 
ratio is nearly always small compared with that needed for the first power, 
except, perhaps, in the case of the mutual perturbations of Jupiter and 
Saturn. The same is true in the asteroid problems, except in the difficult 
resonance cases, on account of the lower degree of accuracy at present 
demanded. 

2. Consideration should be given to the amount of routine computing 
available. In some methods much of the work can be arranged so as to 
be done by routine computers, in others this is not the case. 

3. The liability to errors of computation and the extent to which tests 
may be applied, play some part. It is rare that an extensive theory is 
tested throughout by others than the author, and safeguards against mis- 
takes should be provided as far as possible. 

4. Possibilities for an extension of the work as new needs arise. 

5. An examination of existing developments in order to discover the 
numerical magnitudes which will be involved in the work. 

6. The degree of numerical accuracy aimed at. 

7. The extent to which use can be made of existing numerical or 
literal developments and in particular of those of the disturbing func- 
tion. 

8. The extent to which any peculiarity of the motion may dominate 
the whole work. In most cases this peculiarity is that of approximate 
resonance between two periods, as for example in the great inequality in 
the motions of Saturn and Jupiter, the principal librations in the Trojan 
group of asteroids, and so on. 



22 EQUATIONS OF MOTION [CH. i 

D. VARIOUS FORMS OF THE EQUATIONS OF MOTION 

DEPENDING ON THE USE OF POLAR COORDINATES 

IN THE OSCULATING PLANE 

1*23. Polar coordinates with the time as independent variable. 

The osculating plane is one containing the origin and the 
tangent to the orbit of the disturbed body. It is defined by the 
angle i which it makes with a fixed plane and the angle 6 which 
its line of intersection with that plane the line of nodes 
makes with a fixed line in the same plane. When i is less than 
90, 6 is measured in the same sense as the actual motion; thus 6 
refers to that node at which the body is ascending from below 
to above the fixed plane. In Fig. 1, let 0, S3, P be the points 
where the fixed line, the line of nodes and the radius vector r 
cut the unit sphere with centre at the origin. Let the angle S3 P 
be denoted by v 6. 




Fig. 1. 

The angular velocity of S3 along the fixed plane can be re- 
solved into the components 

d0 . d0 . . 



within and perpendicular to the osculating plane. The latter 
contributes a component (dO/dt) sin i cos (v 6) to the motion 
of P perpendicular to the osculating plane. 

The change of inclination contributes a velocity 

(di/dt) sin (v - 9) 



23] POLAR COORDINATES 23 

to the motion of P perpendicular to the same plane. The 
definition of this plane therefore gives 

-r-sin(v 0) -j- sinicos(u 0) = .......... (1) 

(it dt 

The velocity of P is compounded of its velocity relative to 8 
and the velocity of & . It is therefore 

d, ^d0 . dv r d0 

( v _ m _j_ -7- cos i = -T- 1 -r: , 1 = 1 cos ^. 

dt ^ ' dt dt dt 

Hence the square of the velocity of the planet is 



A geometrical interpretation can be given to a variable v 

defined by 

J dv = dv Ydv. 

This definition makes dv/dt the angular velocity of the radius 
vector in the osculating plane. We can therefore regard v as an 
angle reckoned from a departure point 0' in that plane which 
is such that, as the plane moves, the locus of 0' is perpendicular 
to the trace of the osculating plane on the unit sphere. 

The function T contains four variables. It may be regarded 
as the kinetic energy of a system with four degrees of freedom 
if we suppose the osculating plane to be material and if we add 
a term depending on its mass and motion. Let F be the force- 
function of this dynamical system. We can then apply Lagrange's 
equations to it, with the variables r, v, 9, F, and, after forming 
them, put the mass of the osculating plane equal to zero*. If, 
in the resulting equations, we put 



--- 
dt dt~ dt y 
they can be written 



- r - 

di) ~^' Jt\ ~dt 



dt\di 

* The relation (1) expresses the fact that the material osculating plane is not 
acted on by any forces which do work. 



24 EQUATIONS OF MOTION [OH. i 

In these equations* we substitute for F the force- function 
for the particular problem under consideration. 

If #, y, z be the rectangular coordinates with the fixed plane 
as that of #, y and as the trace of the #-axis, the positive part 
of the y-axis being 90 from reckoned in the same sense as 0, 
and that of the ^-axis being above the plane, we have 

x = r cos (v 0) cos r sin (v 6) sin cos i,\ 
y~r cos (v 0) sin -f r sin (v 0) cos cos i, > ---CO 
2 = ?* sin (t; #) sin i = r sin Z, J 

which show that F is expressible in terms of r, v, } F. The 
definition of L shows that it is the angle between r and its pro- 
jection on the fixed plane or the latitude of the body above this 
plane. 

1*24. Canonical form of the equations. If we put 



so that 

97 ,_/rfY . <? <fr_aT 
~\di) r>' dt~ dr' 

we can write the equations in the form 



.(3) 



or, more compactly, 

dr . Sr - Sr . dr + d(? . Sy - 8 G . dv 



* For other derivations of these equations, see E. W. Brown, "Theory of the 
Trojan Group of Asteroids," Trans, of Yale Obs. vol. 3 (1923), p. 9; C. A. Shook, 
"An Extension of Lagrange's Equations," Bull, of the Amer. Math. Soc. vol. 38 
(1932), p. 135. 



23,24] CANONICAL FORM 25 

If, in the latter, we replace dv by dv Yd0 and QdO . SF by 
d0(SHi rSG), obtained by submitting (2) to a variation S, 
the equations take the canonical form 

dr . Sr - Sr . dr + dO . Sv - S G . dv 



We have here implicitly supposed that F has been replaced by 
Hi/G in F\ T is a function of r, G, r only. 

In general, F will be an explicit function of t as well as of the 
six variables r, r, G, v, HI, 6. By making use of the canonical 
equations (4), we obtain 



a result which is independent of the variables in terms of which 
F, T are expressed, provided that the equations defining any 
change of variables do not contain t explicitly. 

If in (5) we introduce the value of T, and integrate, we obtain 



dt) + ,*-** " 
The equation 1'23 (3) may be written 

<Pr_GP_ SF 

T dt* r 2 ~ r dr 

Eliminating G between this and (6) we have 



A further useful equation may be obtained. When F is 
expressed as a function of 7', v, F, 0, t y we have 

d F dF . dF . dF^ dF * dF 
-rr = ^-r + ^-i;+r-f^-0+-^-. 
dt dr dv ol dv dt 

In this equation replace v by v + F/9 and substitute for 6, F 
their values obtained from 1*23 (5), (6). The result is 

dF 3F . dF . dF /QN 

jT^a-r+a-v+or ................... (8) 

dt dr dv dt ^ ' 



26 EQUATIONS OF MOTION [OH. i 

This equation when R replaces F, where F = p/r + J?, is evidently 
still true. 

1*25. The equations of Encke and Newcomb. Put 

p d'R dR dR dR . dR . 

jP^-fJ?, -jr=-jT--57=-;r r ' + -5- v - 
r at at at or dv 

Equation 1*24 (7) rnay then be written 



When R is neglected, the motion becomes elliptic and r can 
be expressed as a periodic function of t (Chap. in). Let this 
value of r be denoted by r and the complete value by r + Sr. 
On expanding in powers of 8r and neglecting powers of Br beyond 
the first in the left-hand member, we obtain 



(2) 



Elliptic values are substituted for the coordinates in the right- 
hand member which then becomes a function of t. The equation 
can then be integrated and it gives Sr. 
The coordinate v is obtained from 



$-*-/* < 3 > 



where GQ is an arbitrary constant, or, neglecting terms depending 
on the square of the disturbing mass, from 



............. (4) 

dt ?o 7*0 r 2 J dv ^ ' 

Further, 

^ = ^Y r - dR -^Y H?^ 

*""A + d""(ft" l "dvar - d + dwar > 

di dt 

to the same order. 

If then VQ be the value of v in elliptic motion, and VQ + # its 
complete value, we have, to the first order of the disturbing mass, 

d , 20 * 1 [dR j 4 , FdR 

T $ VS --. " S r -- ^ + .......... /5) 

d^ r 8 r 2 J dv V Q 9F v ^ 



24-26] NEWCOMB'S SOLUTION 27 

Since dF/d0, dF/dT contain the disturbing mass as a factor, 
the equations 1*23 (5), (6) are immediately integrable if we put 
r*dv/dt*=* Go in the latter. 

1*26. Newcomb solves the equation for Br in the following manner. He 
notices that when /2=0, the solution of the equation for r contains two 
arbitrary constants, e the eccentricity and w the longitude of perihelion 
(see 3'2 (7)), in addition to the arbitrary constant already present in the 
equation, a constant which is independent of e, w. Two particular 
solutions of 1'25 (2) for 5r with R~Q are obtained by varying e, 07. 
These two solutions may be written 



He then makes use of a well-known method in the solution of a linear 
differential equation of the second order, namely, that if yy\, yy'L are 
two particular solutions of the equation 



a particular solution of the equation 



is 



Cy =y 2 / Qy l dt - ^ I Qy 2 dt, 



where y 2 -~ -y l -J^ 2 (7, a constant. 

In these formulae, I\ ^ are supposed to be known functions of t. 

If X be the eccentric anomaly, we have from 3'2 (16), 3*2 (15), when Xis 
expressed as a function of n, e, e> w, t, 

dX __ sin X __ a sin X 
de ~~ 1 - cos X ~~ r 
Thence, by differentiation of 3*2 (15), we obtain 



Thus cos X e, sin X can be taken as the two particular solutions. They 
give Ctt, the mean motion. 

In the exposition of the application of these formulae to the theories of 
the four inner planets (Amer. Eph. Papers, vol. 3, pt. 5), Newcomb 
apparently puts v = v, for he makes no mention of any difference between 
them. The difference between them, -Jrdtf, which to the first order of 
the disturbing mass may be written 2 sin 2 1 80, is very small because the 



28 EQUATIONS OF MOTION [OH. i 

inclinations of the orbits of these planets to the ecliptic are small. The 
constant and secular parts of this term are absorbed in the constants of 
the mean longitude, so that the only doubt which remains is whether the 
term gives rise to any sensible periodic terms, and if it does, whether these 
have been included in his final results. 

1*27. Equations of motion with the true orbital longitude as 
independent variable. 

These equations are deduced from those of 1*23 by making v 
the independent variable instead of t. The transformation is 
effected by the introduction of new dependent variables u, q, 
defined by the equations 



With these definitions we have 

dr _ dr dv _ ~ du 
~~~ m '~ ' 



d z r d*u du dG _ d*u 1 dO du 

~~~ " 



dG d, *K_^ O d 

dv dt\ dtj dv^ } 

If then we put 

==/*(> + 72), .................. (3) 



^ 4- 2 

so that TS- = MW ur -^- , 

dr du 

and if we change from G to q by means of (2), the equations 
1-23 (3), (4), (5), (6) are transformed to 

d*u dR , dqdu do 2 dR //f . /cx 

-1-5 + ft ? = ? o + 1 ? ;r :j- -T-- -- 2o~'-- (&)> ( 5 ) 
dv 2 * * du * * dv dv dv u 2 dv ^ " \ ' 



TT-T X ^r =1 - r ;r> ( 6 >'( 7 ) 

av 2 \/A' av av \ /> \ / 

) = "^W' dv = 7 2 3T (8)>(9) 



26-28] IN TERMS OF THE LONGITUDE 29 

It is to be noticed that the disturbing function has been denoted 
by fiR instead of by R, so that the mass factor present in the 
new R is the ratio of the disturbing mass to the mass of the sun. 

When we proceed by continued approximation as in Newcomb's method, 
the equation for u is immediately solved when that for q has been inte- 
grated, and it has the advantage of being one with constant coefficients, so 
that the device shown in 1*26, requiring two multiplications of series, is 
not needed. 

When 72 =0 we have const., and the solution of the equation for u is 
(3-2) 

e cos (v or)}, \jq-a (1 -e 2 ). 



Since dq/dv has the disturbing mass as a factor, we can change the 
variable u to MI where n = Uif(q) without losing the easy integrability of 
the equation for HI . The special cases 



have certain advantages which will be pointed out in 7*2. 

It may be noticed also that when the terms in R containing the angle 6 
are neglected, we have Tq ~ ^ = const., so that r can be completely eliminated 
at the outset. 

The variables l/<?, \ju have the dimension of a length. If we introduce 
the constants w , such that n 2 a 3 =fi, and put dt\ for n^dt, Ui for ua Q , 
qi for qa ot RI for a Q R, the constants /LI, , n will disappear from the 
equations and the variables are all ratios. 

1'28. Latitude equation. 
From the equations 1*23 (1), (2), namely, 
di sin (v 6) = d6 sin i cos (v 6), dv = dv (1 cos i) d0, 
we easily deduce 

d {sin i sin (v 6)} = sin i cos (v 0) dv, 

, f . , mi . . x /ix 7 sin i cos i 7/ , 

a sin i cos (v 0)1 = sin i sin (y a) av + -; ; --- -rr d^. 
1 /j 7 sm(v 9) 

Whence, with the help of 1-27 (9), 

d 2 -, \ - - / ^ sin i cos i d0 
T-a + 1 ) sin sin (v - tf ) = -77 - -^ -T~ 
dv 2 / v ' sm (v - ^) dv 

_ sin i cos i q dR _ m 
" = " >( } 



30 EQUATIONS OF MOTION [CH. i 

It will be seen, by differentiation of 4*1 (1) with the help of 
the definitions of A, S, F, that dR/dT contains sin (t; 0) as a 
factor, so that there is no discontinuity in Z when v is a 
multiple of TT. In the section referred to, R is shown to be 
a function of r, r', cos S and 

dR . / /\\ / /\\ vR 
^. _ sin (-(?) an (-<?)g g . 

If L be the latitude of the disturbed planet above the plane 
of reference (1*23), equation (1) may be written 

nZ = , .................. (2) 



so that sin L is obtained from an equation of the same type as 
that for u. 

When Z has been expanded, we obtain v by integrating 

dv -. r, - - / /K 
-7- = 1 Z sm i cos i sm (v u). 
dv / 

1*29. The Equations of motion with the disturbed Eccentric 
Anomaly as independent variable. 

Another variable which gives a linear form to the equation 
for r is X as defined by 

r dX pdt, p = const. 
This variable gives 

dr _ dr d / dr\ _ p a d 2 r 
T dt~ P dX' dt\'dt)'~~rdX*' 
Equation 1'24 (7) with F^p/r + R therefore gives 



- dt .......... (1) 

r dX* r dr J dt ^ ' 



^ = -2^+2^ (2) 

dt\a) dt* dt ................... v ' 



Define a by the equation 

dt\a 
Then equation T24 (6) may be written 

$'+"(*)'-*: ; 

which shows that 2a is the disturbed major axis. 



28-30] RATIAL COORDINATES 31 

From (1), (2) we deduce 



7TT9 a I * I 9 n V^V 

dX 2 p 2 \a / p 2 dr v ' 

If a be an arbitrary constant, the integral of (2) may be written 

11 E> f) r 3 Z? 

1 ^ Jtt < / f>/t 1Tr /^ 

- = -- 2- + Ir^dX, (o) 

and, with the aid of this equation, (4) becomes 
d 2 r LL , . r 2 dR u 



(6) 

71* (t N "' / *"'* '*'*' '*''* l '* ft i ^ / 

The transformation of the remaining equations to the variable 
X as independent variable is effected immediately. 

If ?i , p be defined by the equations /i/? 2 = a , /i=n 2 a 3 , the definition 
of X gives r dXa^n^dt. A reference to Chap, in shows that in undisturbed 
motion, X is the eccentric anomaly. 

If the equation (6) be solved by the method outlined in 1 *26, it will be 
seen that the solution is closely analogous to that of Newcomb, when we 
change the variable from t to X under the integral sign. It has, however, 
the advantage of being exact instead of approximate and is thus adaptable 
to the calculation of the higher approximations. 

These equations, which appear to be new, will not be developed further 
in this volume. The general method of treatment would follow lines similar 
to those adopted when the true longitude is taken as the independent 
variable (Chap. vn). It may, however, be noticed that, since 



in elliptic motion, the equations are integrated without multiplications of 
series when R, rdlt/dr have been expressed in terms of X. The only 
exception is the equation for v an exception common to all methods. 

1*30. Equations of motion referred to the coordinates of the 
disturbing planet. 

T j. i ndv' , t dv , /1X 

Put r = rp, r^--A, P^,=h f , ............ (1) 

so that **S = AVlr = A'Ap ................... (2) 

(At cttV 

A. direct transformation from the variables r, t to p, v' gives 

d?r /dv\ a _ /t' 2 f fflp /dv\ 2 ) 1 dh' dp d?r' 

~ ~ ~ ~ 



d?r /dv\ a _ /t' 2 f fflp /dv\ 2 ) 
dt* ~ r \dt) ~ 7 left/ 2 ~ p \dv') } 



32 EQUATIONS OF MOTION [OH. i 

This equation, with the aid of the definitions 
F, dF 1 W, 



enables us to transform 1*23 (3) to 

c77*~ p w/ "F* J + p h'*~dt* ~/7 ~di d7"" W 

With the definitions (1) of h p ,h', equation 1*23 (4) gives 

A , dh<, h dv = ajp 

rtt p dt dv ' 
Replacing o? by r'*dv'/h' and 7^ by /\/r', we obtain 

p _ * /. / K \ 

dt/ /^ /2 Sw A' dt p ^ ; 

Now let r',v' be the polar coordinates of a disturbing planet 
moving in the plane of reference and satisfying the equations 



The elimination ofd?r'/dt 2 , dh'/dt from (4), (5) by means of (6), 

(7) gives 

dtp (dv\* _ _/ 3i / r^ dF'\ _r^d/^dp 
dv'* P \dv') ~h'* dp P \ k' 2 dr') h' dt dv' 



dp ' 2 J 2 V A' 2 dr' h' dt dv' ' 

...... (8) 



__ 
dv'~h'*dv h' a/ ............................ 

The transformation of the equations for F, 9 is easily made. 

When the disturbing planet moves in an ellipse, we have 
(Chap, in) 

/ 1 aP' af 



so that the equations become 

dtp t dv _ SF 2 

~ ~ dp' 



30, 31] RATIAL COORDINATES 33 

u o /i + JpVcosfv'-w') , , x 

where F 2 = \ % ^ -', (11) 

1 +e cos(i/ w') v ' 

equations which have a form similar to that which obtains when 
t is the independent variable. 

When the motion of the disturbing planet is no longer elliptic, 
we put 



The additional terms in (10) are easily written down : they 
contain the derivatives dR'/dr', dR'/dv'. 

The chief point of interest in this form of the equations of motion arises 
from the fact that F l is independent of /, t explicitly. That this is the 
case is seen from 1*10 when we put 

^ = y, Vj = v', 0, = 0, ^ = r, H? + i, = /i, m k =m'. 

For then F l =/V = ^ + m' \ -------- - - . -- p cos S\ , 

P l(H-// 2 -2pcostf)* J 



where cos 8= (1 - -ir) cos (v- t>') + i r cos (v+v 1 - 20). 

Thus the only way in which the eccentricity of the disturbing planet 
appears in the equations of motion is in the explicit form shown in F% . 
It should be pointed out, however, that if we put m' so that 7^ is re- 
duced to the term ra/p, the solution of the equations for p, v will contain e' ; 
hence, in the second approximation, when we substitute these values in 
the coefficient of m', the development of F l will contain e'. Nevertheless 
the equations appear to have possibilities for usefulness in the discussion 
of certain problems in which it is necessary to take into account the per- 
turbations of the disturbing planet by another planet < the indirect 
effect' of the latter, that is, the effect transmitted through the disturbing 
planet. A case of this kind is the effect of the action of Saturn on Jupiter 
where the latter is disturbing the motion of an asteroid of the Trojan group. 
Another case is the indirect effect of a planet on the motion of the moon. 

1*31. The most important of the various forms of the equations of 
motion, namely, that referred to the * elliptic frame, 5 will be developed in 
Chap. v. In this form the two variables r, v are replaced by four new 
variables which are so chosen that their -first derivatives only appear in the 
equations of motion. The elliptic frame in various forms has been 
extensively used for the calculation of perturbations. In its direct form 
it was used by Leverrier for obtaining the orbits of the major planets and 
in the 'canonical' form by Delaunay for the motion of the moon. In a 
different form it was used by Hansen for both the planets and the moon. 
B&SPT * 



34 EQUATIONS OF MOTION [CH. i 

Another group of methods depends on the use of a uniformly rotating 
frame of reference. This, like most of the methods which have been used 
for actual calculation, was initiated by Euler and adopted by G. W. Hill 
and E. W. Brown for the development of the motion of the moon as dis- 
turbed by the sun. 

For these and other methods not treated here, the reader is referred to 
standard treatises like those of Tisserand and Brown. References up to 
the date of their publication will be found in the articles dealing with 
celestial mechanics in the Ency. Math. Wiss. 

1*32. Motion referred to an arbitrary plane of reference. 

In the preceding developments the plane of reference has 
been that of the motion of the disturbing planet. 

Now let the symbols v, 9, i refer to any fixed arbitrary 
reference plane and let v', & ', i' have the corresponding signi- 
fications for the disturbing planet; the rectangular coordinates 
of the disturbed planet are then given by 1*23(7) and similar 
formulae will hold for the disturbing planet. 

We have rr r cos S xx' 4- yy' + zz' . 

Put i = V 1 so that 

xx 4- yy' is the real part of (x 4 ty) (x iy'). 

Define T, k, I", V by 
F = 1 cos i, 2k = 1 4- cos i, F' = 1 cos i', 2k' = 1 4- cos i'. 

We can then obtain from 1*23(7), 



and similarly 



The product contains v, v' only in the combinations v 4 v'. If 
we form it and, after taking the real part, separate the coefficients 
of the sines and cosines of these two angles we obtain 

cos (v - v') {W 4 i IT' cos (2(9 - 20')} 
4- sin (v - v 1 ) {JIT' sin (2(9 - 2(9')} 
4- cos (v + v') $kT' cos 26' 4- J A?T cos 20} 
4- sin (v 4- v') $ kF sin 20' + J&T sin 20}. 



31, 32] ARBITRARY REFERENCE PLANE 35 

To find cos S we must add zz' jrr to this. The latter can be 
written 

sin i sin i'fcos (v v') cos (6 6') + sin (v fl') sin (0 0') 

- cos (v + v') cos (0 + 6') - sin (v + v') sin (0 + 0')}- 
We can therefore express cos S in the form 

cos 8 = K Q cos (v v' KI) + ^2 cos (v + 1/ jBT 8 )> 

where K = 1, ^ = /i a = ^3 = when i = i' = 0. Since the K's 
are functions of 0, 0', i, i', only, cos S has the same form as in 
the simpler case. Here 

KQ , -Or 1ST K i , K 2 , /V3 

replace 1-^r, W--OT', |T, 20, 

used in the developments of the later chapters. 
In these and many similar cases we replace 

A cos a -h B sin a by (7 cos (a a )> 
where (7, are determined from 

(7cosoo = J., (7 sin <)=-#> 
(7 being in general so taken as to have a positive sign. 



3-2 



CHAPTER II 

METHODS FOR THE EXPANSION 
OF A FUNCTION 

2*1. The greater part of the work of solving any problem in 
celestial mechanics consists of the expansions of various functions 
into sums of periodic terms, mainly because the integrals of 
these functions cannot be obtained conveniently in any other 
way. The majority of these methods, which depend chiefly on 
Taylor's expansion and Fourier's theorem, are well known, but 
there are certain expansions, continually recurring, which require 
much labour. It is the purpose of this chapter to ease the work, 
partly by giving formulae which are ready for immediate appli- 
cation, and partly by so arranging them that the calculations 
may be carried out with the least chance of error. Certain of the 
formulae are intended to be used only when literal expansions are 
required : when the coefficients are numerical the methods of har- 
monic analysis usually give higher accuracy and are less laborious. 

The coefficient of a periodic term in the expansions of most of 
the functions considered here takes the form 

a l '(a + i 2 H-a 2 4 -f ...), (1)' 

where a is a parameter and a , i, a*, ... are integers or fractions. 
It is frequently required to calculate the function as far as some 
definite power of , and to carry one or two coefficients con- 
siderably further. It is this latter need which causes difficulty 
because there is much wasted labour if the whole series be 
carried to this higher power. This fact has to be remembered 
when a choice of any method of expansion is made. 

Expansions in power series are so much easier to perform and 
are so much less subject to error than operations with series of 
periodic functions, that the latter are usually reduced to the 
former by the substitutions 

x = exp. 6 V^T, 2 cos iff = x i -t- or*, 2 V 1 sin iff = x i or*. 

(2) 






1,2] LAGRANGE'S THEOREM 37 

When the coefficient of cos id or sin id has the form (1) the 
work is made easier by the substitutions 

. 6 V 1, p = a 2 , 1 

-\ 2a* v ^Tsin iff = z i -p*z-* t ) 
We then expand in positive powers of p and in positive and 
negative powers of 2. This simple change from the substitution 
(2) not only gives greater freedom in the choice of methods of 
expansion, but aids materially in solving the problem referred 
to on the preceding page. 

Extensive use is made of another device, namely, that of 
expansions of functions of an operator. These usually take the 
form <f>(D).f(x) where D d/dx. It is then always supposed 
that <f> ( D) is developable in the form 



where o, i, a 2 ,... are independent of x, so that 



Operations with functions of D are performed in accordance with 
the rules of ordinary algebra except that functions of D and 
those of x do not follow the commutative law of multiplication. 
The gain is partly in brevity of expression and partly in the 
methods of expansion which are suggested by well-known ex- 
pansions. Thus we can use such forms as exp. Z), log(l + />), 
(1 4- D) n , (1 + a) D , cos J9, etc., each acting 



2'2. Lagrange's theorem for the expansion of a function defined 
by an implicit equation. Let the equation be 

y = a? +a(y) = & + <, .................. (1) 

where a is a parameter and and its derivatives are continuous 
functions of y. The problem \A the expansion of F(y) = jPin powers 
of a with coefficients which are functions of a?. The theorem gives 

2 d 



<> 



38 METHODS FOR EXPANSION OF A FUNCTION [CH. n 

where F x = F(x) 9 (f> x =<f> (x). 

The proof which follows indicates an extension of the theorem 
to several variables. 

We may regard F as a function of a, x. Regarded as a function 
of a, it may be expanded in powers of a by Maclaurin's theorem 
in the form 

F=F Q + a(AF) Q +~ ] (A*F) Q + ... y A=~ t ...... (3) 

where the zero suffix denotes that a is put equal to zero after 
the derivatives have been formed. Evidently F Q = F X . Put 
D = d/dx. The use of the operators A, D implies that the 
functions on which they operate are expressed as functions of #, . 
Operating on (1) with A, D } successively, we have 

Ay = 4> + a^ Aij, Dy=I+a d fDy, 
J ^ dy dy J 

so that Ay = <f) Dy, 

and therefore, for any function y of ?/, 



From this result we can show, by induction, that 

A n F=D n ~ 1 ((f) n DF) ................... (5) 

Assume that (5) is true and operate on it with A. Then, since 
x, a are independent so that D, A are commutative, 

A n+1 F-= D"- 1 {DF. A<f> n + ^A (DF)} 
= D*- 1 {DF. <f>D<f> n 4- <f> n D (AF)}, 

the change in the first term being made by the use of (4) and 
in the second by the commutation of A, D. But, by putting 
g = F in (4) and operating with D we have D(AF) = D (QDF), 
so that the portion under the operator D 11 " 1 is D (<f>DF . <f> n ). 
Hence 



and since the theorem is true for n = 1, it holds universally. 



2, 3] EXTENSION OF LAGRANGE'S THEOREM 39 

Finally, since <, F become <f> x , F x when a = 0, the coefficient 
of a n in (3) becomes the same as that in (2), and the theorem 
is proved. 

Particular case. When F(y) = y, we have 

yy2 ,/ $ ,72 



2*3. Extension of Lagranges theorem. 
If yi = Xi + aai<l>(yi 9 yz t y 3 ) = a;i + aai<l>, i = l, 2, 3, ...(1) 
where the a$ are constants and a is a parameter, and if 



where F and <f> are continuous functions with continuous deriva- 
tives of yi, 2/2> y?.> and further, if 
F x = F(XI, o; 2j ar 8 ), 



then 

2/2, yd^ 



x ) ....... (2) 



The proof follows the same general lines as before. We first 
regard F as a function of a and expand in the form 2*2 (3). 
Next, by differentiation of (1), 



where the first term of the right-hand member is 1 or according 
as i =j or i 4= j. This equation is multiplied by a,- and summed 
for j = 1, 2, 3. The result is 

But from (1) we have 

2JL. 2*,. 

(5) 



40 METHODS FOR EXPANSION OF A FUNCTION [CH. n 

These two sets of equations may be regarded as linear, the 
first set for the determination of J9y t -, and the second for that 
of dyi/da. They are the same except that the absolute terms 
in the latter are </> times those in the former. Hence 



and therefore ^ = <j>DF . 

The remainder of the proof is the same as that in 2*2. 

2*4. The most general case in which 

Hi = -n + Q/i , y* , #0, 1 = 1,2, 3, 

does not seem to be soluble by a simple general formula. It is not difficult, 
however, to obtain the solution as far as a 2 . If we put 

<tx = </>t (#1 , *2 1 # 3)1 D == 2 t <j> lx a ^ , /^ x 
we obtain 



Lagrange's theorem can be used to find X or any function of X in terms 
of g from Kepler's equation X=*g e&\i\X (3'2 (16)), and was probably 
suggested by this problem. The extension may be applied to the Jacobian 
solution of the canonical equations (Chap, v) to find the new variables in 
terms of the old or vice versa, when the disturbing function is confined to 
a single periodic term or to a Fourier set of terms. The more general case 
mentioned above is that of the Jacobian solution where the disturbing 
function contains any periodic terms. 

2'5. Transformation of a Fourier expansion with argument y 
into one with argument #, where y is defined in terms of x by means 
of an implicit equation*. 

Let F(y} be expanded in the form 

F(y) ^(c i Gosiy-{-disiniy) > i = 0,1,2, ...... (1) 

and let y x -f a cf> (y\ ........................ (2) 

where a is a parameter and </> (y) is expressed in the same form 

* E. W. Brown, Proc. Nat. Acad. Sc. Wash. vol. 16 (1930), p. 150. 



3-5] IMPLICIT FUNCTIONS 41 

as F (y). It is required that we obtain the coefficients a if b it 
when F(y) is expressed in the form 

F=^(ai cos ix -h bi sin ix) ................ (3) 

With the help of the notation, 

</> = (x), F=F O), D = d/dx, 
nnd with the use of Lagrange's theorem, (2) gives 

F(y)=F+'S, an ] D n - 1 (^ n DF) t n = l, 2, ....... (4) 

Let -v/r be another function of # expressed as a Fourier series. 
On multiplying both members of (4) by D-v/r, we obtain 



). ...(5) 
The identity, 

2 (<f> n DF), 

shows that, when all three terms are expressed as Fourier series, 
the constant term of the left-hand member is the same as that 
of the last term, the remaining term being the derivative of a 
Fourier series. By repeating this process n 2 times, we deduce 
the fact that the constant terms in the Fourier expansions of 

D^r . D"- 1 (4> tl DF), (- I) n ~ l l) n ^ . <f> n DF 

lire the same. On applying this result to each term of the right- 
hand member of (5) we obtain a series which is the expansion, 
by Taylor's theorem, of 

-DF.^(x-a<f)) ..................... (6) 

in powers ofa<p. 

Hence, the constant term in the Fourier expansion of 



(7) 



when F(y} is expressed in terms of x, is the same as the constant 
term in the Fourier expansion of 

-~F(x). + {x-a$(x)}, ............... (8) 

or in that of F (a). -]-$ [x a $ (so)} ................ (9) 

ax 



42 METHODS FOR EXPANSION OF A FUNCTION [OH. n 

It is to be remembered that ty [x a <(#)} means the result 
obtained by replacing x by x a<j>(x) in ty(x). We obtain (9) 
from (8) by noticing that their difference is the derivative of 
the product of the two functions F(x), ^r {x a<f> (x)}. 

The use of (9) enables us to state the theorem in a slightly 
different form. If we put % (x) for d^r (x)/dx, we have the theorem : 

The constant term in the Fourier expansion of 



where F(y) is expressed in terms of x, is the same as the constant 
term in the Fourier expansion of 

, z=x-a$(x) .......... (11) 



According to this definition, % contains no constant term. But 
if ^ is a constant, (4) shows that the constant term in F(y) is 
the same as that in F + a(f>.DF, which is the same as that in 
F(l aD<f>), so that the theorem still holds when ^ (x) contains 
a constant term. 

Since we are concerned only with the constant terms in (8)> 
(9) or (11), the theorem evidently holds if we replace the letter 
x by the letter y in these three formulae. 

The chief value of this theorem lies in the fact that it removes 
the necessity for solving the implicit equation y = x -f a<f>(y)-m 
order to get y in terms of x. 

The application to the coefficients in (3) is immediate. If we 
put i\fr = sin ix, so that Dty = cos ix, and note that the constant 
term in the product of D-fr by the right-hand member of (3) is 
Ja t -, we find from (8) the result, 

a t - = constant term in r sin {ix za</> (x)} j- F (x). 
i ctx 

...... (12) 

Similarly, by taking Dty = sin ix, we obtain 

bi = constant term in - cos {ix ia < (x)} -y F (x\ . . .(13) 
i dx 

and from (9) 

Oo= constant term in \ 1 a -7- <f> (x) > F (x). . . .(14) 
I ax ) 



5, 6] IMPLICIT FUNCTIONS 43 

When y, x take the values 0, TT together, these last results 
may be deduced from a change of variable in the Fourier in- 
tegral. 

2'6. Extension to two variables. Suppose that we have a second 
pair of variables x', y' , independent of #, y, connected by the 
implicit relation y* = x' + a'<f> (y f ) and that we desire to obtain 
the expansion of F(y, y') in the form 

2,i' {(%' cos (ioc + i'x') + ba> sin (ix + t'V){, ...... (1) 

where i, i' are positive and negative integers. An expansion in 
this form will be called a double Fourier series. 

For the development we adopt a notation similar to that used 
before, namely, 

F= F (x, x'), J} = d/dx, D' = a/a*/, etc. 
A double application of Lagrange's theorem gives 



where the signification when n = or m= is the same as that 
shown by 2'5 (4). This is multiplied by DD'ty(x, x'}, where ty 
is a double Fourier series and the process adopted in 2'5 is then 
fpllowed for each of the variables x, x' . It evidently leads to 
similar results which can be stated in the following theorems. 
The constant term in the doable Fourier expansion of 

F (y ' y 



when F (y, y') is expressed in terms of x, x' y is the same as the 
constant term in the double Fourier expansion of 



-a^( ai ), x' -')}, ...... (2) 

or in that of 

,^{x-a^(x), *'-')) ....... (3) 



And the constant term in the double Fourier expansion of 

F(y,y') X (x,x') ..................... (4) 



4 METHODS FOR EXPANSION OF A FUNCTION [OH. n 
is the same as that in 

F(x, x') % (z, z') -r- . , , , z = x a<f)(x), z' = x' a'<f> (#'). 

...... (5) 

It will be noticed that the double operation removes the negative 
sign present in 2*5 (8). It reappears, however, if we replace 
(5) by 

--F(x,x'}~^{x-a<}>(x\ *' ->' OO}, -(6) 



or by the similar formula in which the derivatives d/dx, d/dx' are 
interchanged. 

In order to apply the theorem to the coefficients in (1), we 
put DD'ty equal to cos (ix -f i'x'), sin (ix + i'x'), successively in 
(2). We obtain 



" ' 



= const, term in . {ix + i'x' ia </> (x) i'a'<$> (x'}\ 
sin 



which hold only when i, i' are both different from zero. 

When i = Q, we use (5) with % (x,x') equal to cos i'x , sin i'x', 
successively, and obtain 

? 0t " = const, term in "*" C S {t V - i'cftf (x')\ 

( r \ /j 



+ sin 



or n 

+ COS 



Jl -a 1 </>(^)| |l -a' ^*'(^)} ^(*X), ...... (8) 

V - i'a'tf (x')} \l-a~<f> (x)\ ? , a , ^ (^, ^). 
r \ /) ^ dx^ ^ ) l tix ^ ' 

...... (9) 

The formulae for a t - , b iQ are similar. 

When i = i' = 0, we use (5) with ^ (x, x 1 ) = 1, and obtain 

aoo = const, term in 41 -a <j> (x)\ \1 -a' , <f>' (x')l F (x, x'). 
^ ox J I c?<^/ J 



6-8] SYMBOLIC OPERATORS 45 

27. Expansion by symbolic operators. 

If p D is expansible in positive integral powers of D and iff (x) 
is expansible in integral powers of x, then 

f(poc) = p D f(x\ where D = x -j- = - r --- . 

j vr / r j \ /> ^ x ^ j g x 

If m be a positive integer, we have 

J)m (0n) _ n m^,n < 

Hence, if p D = a 4- aiD 4- a 2 D 2 4- ... , 

we have p D oc n (px) n . 

If then /(a?) = S n ( b n x n ), n = 0, 1 , . . . , 

we have f(px) = 2 n b n (px) n =p D ^ n b n x n =p D f(x). 

This theorem can evidently be extended to any number of 
variables. Thus 

/(Piffi,j02ff2. ) ss pi Dl .p* D * f(xi,X2, -..) where A = ^3/9^. 
The application of the theorem depends on the possibility of 
expanding p D in powers of D in such a manner that we can, by 
stopping at some definite power, secure a given degree of accuracy. 
This happens when p has the form (1 4- ey) fc , where e is a small 
parameter such that ey is less than unity. We can then use the 
binomial theorem and obtain 



p D = (1 + eyj* - 1 + ey . kD + kD(kD- 1) + .... 

1 . i 

We can also make use of the expansion of Q Z in powers of z if 
we put z = D log p, for then 



and logp has the factor e. 

2*8. Product of two Fourier series. 

(i) Let the series be 

A = a 4- 22a;a* cos {0, .B = 6 + SS^a* cos id, 
Put _ 

z a exp. \l 1, p = a 2 , 2a* cos 16 = z* 4- p^" 1 ', 

5 = &<) 4- 26* 2* ............. CO 



46 METHODS FOR EXPANSION OF A FUNCTION [CH. n 

Then AB = (A + '2a i p i z-*)(Bo + 2,b i p i z- i ) .......... (2) 

Since A, B are even functions of 0, their product will have 

the same property and the coefficient of z* will be the same as 

that of p i z"~ i . It is therefore sufficient to find the coefficient of 

z*, i > 0, and then to replace z i by 2a* cos iff. 

We thus reject all negative powers of z in (2) and it is 

therefore necessary only to find the coefficient of z i in 

A Q B Q + AoZbip'z-* + 



The first term is the product of the power series (1) and from 
this product we select the coefficient of z*. In the second term 
we select the coefficient of piz* t namely, a i+i b i} and sum for 
j = 1, 2, ... ; the third term is treated in a similar manner. 

On performing these operations and replacing p by a 2 , we find 
for the coefficient of z*, that is, of 2a* cos iff 



-_i -f . . . + a^) + 2 

+ 4 (2&<+2 + a+2&2> 4- . . . , ...... (3) 

and for the constant term, 

<*o&o + 2a a ai& 1 +2d*tf a &2 + ................... (4) 

The parameter a may not be present and we then put a = 1. 
In the series with which we have to deal it is usually present 
implicitly, if not explicitly, so that the order of magnitude of 
any coefficient is denoted by its suffix, and in the product by the 
sum of the suffixes of a, b. Thus the arrangement in (3) is made 
as needed, namely, with respect to the orders of the terms, the 
brackets giving successively the terms of orders i y i -f 2, i -f 4, . . . . 

(ii) If 

A' = 22a,a* sin iff, B' = 226<a< sin iff, 

we adopt the same substitution for 0, a and we have 
2 V^l a* sin iff ^z'-p* z~*. 

The product is an even function of ff and is therefore expressed 
in terms of cosines of iff. We have only to find the coefficient 
of z i in 

z-* -f 



8, 9] PRODUCTS OF FOURIER SERIES 47 

in which do = &o 0. Hence we obtain, for the coefficient of 
2a* cos id, 



......... (3) 

and, for the constant term, 



(6) 
(iii) If 

A' = 22a<a* sin {0, # = 6 + 22M* cos i0, 

the product is an odd function of 0. Here we need only the 
coefficient of z l -r- V 1 in 



and we obtain, for the coefficient of 2a* sin {0, since = 0, 
(ai6 t -_i + 2 ^-2 + . . . + a^o) + 2 (- i^+i -H e&f+i&i) 

-f a 4 (-a 2 6^ 2 4-a^ 2 &2), ...(7) 
there being no constant term. 

2'9. Fourier expansion of a series expressed in powers of 
cosines or sines. 

(i) Let 
' G ^Sa.a'cos'fl = 26, (2 cos 6)\ i = 0, 1, 2, ....... (1) 

Since this is an even function of 6 it can be expressed in terms 
of cosines of multiples of 6. With 2=aexp. 0V 1, so that 
2a cos 6 z -f a 2 /z, we have 



on expansion by the binomial theorem. As in 2*8 we need to 
find only the coefficient of z*, i > 0, and to replace z i by 2a* cos iff. 
The selection gives for the coefficient of 2a* cos id in (7, 

2i + (* + 2* ^ 2 " 2 -^ 



1.2.3 



48 METHODS FOR EXPANSION OF A FUNCTION [OH. n 

The constant term in C is obtained by putting i = in this 
formula. 

The result shows that the numerical work will be simplified 
by the use of ^ = a i a < /2 < . The coefficient of 2 cos id is then 



___^ ____ <+ 

(ii) Let 

S = Sa^a* sin 1 ' = 26, (2 sin 0)<, i = 0, 1, 2, . . . . 

The terms in 8 with odd values of i will produce sines of odd 
multiples of and those with even values of i, cosines of even 
multiples of 0. With the same substitution as before, we have 



By a similar procedure, we obtain, for the coefficient of 
2 sin (2i + 1) 0, 

(2^4-5) (2i + 4) 

2i+ 3 H J g 2i+5 

- 6 2t+7 +...!> 



----- r 2 3 

and, for the coefficient of 2 cos 2i0, 



--- T2T3 ---- 

...... (5) 

the constant term being 

4.3, 6.5.4, 



9, 10] FUNCTION OF A FOURIER SERIES 49 

2*10. Expansion of a function of a Fourier series. 
(i) It is desired to obtain the expansion of 

cos 20 +...), ......... (1) 



in the form 

60 + 2&! cos 6 + 26 2 a 2 cos 2(9 4- . . . , 

the assumption being made that /(a -l-#) is expansible in 
powers of x. Evidently f is an even function of 0. 
As before, we put z a exp. V 1, a 2 = p, so that 
2a* cos id = z i -f p*z~*, 

and recall that it is sufficient to find the coefficients of z i and 
then to replace z i by 2a* cos iff for i > 0. 

Put A = mz + a 2 z 2 + ..., -B = ai^ + a 2 ^ + > 

so that (1) becomes /(oo -h ^4. 4- i?). This function will be first 
expanded in powers of p by Taylor's theorem. We obtain 



the suffix denoting that p is put equal to zero. 
Put /i 



and denote derivatives of/i with respect to a Q by accents. Then 
since p is present in f only through B, 



\d P 



Hence, replacing p by a 2 , we obtain 

a 2 "* 

/(a c + ^+-B)=/ 1 + -i/i'4 

+ 5 </' 



It will be noticed that all negative powers of z are shown in 
explicit form in this expression, and that their coefficients are 

B&SPT 4 



50 METHODS FOR EXPANSION OF A FUNCTION [OH. n 

positive powers of z\ thus the lowest power of z required in the 
second, third, ... coefficients are the first, second, .... We plan, 
however, to stop at some definite power of a ; suppose this power 
be the seventh. Then since z i has the factor *, we shall need 
the following expansions: 

coef. of o?jz, from z to 6 ; 
coef. of a 4 /2 2 , from z 2 to z 5 ; 
coef. of a 6 /^ 3 , from z 3 to 4 ; 

and these are all that are needed. The work of expansion is thus 
reduced to operations with positive power series. 

We next expand fj and its derivatives in powers of A, which 
contains a as a factor. As far as a 7 , the result is, with/ set for/(a ), 

' 



, 

! 



a2 (A f" , ^ 2 /'" , 

7\ ^ 2V + '" 



(2a 2 / " + 0!%'") + . + 5 * (2a/o vl 



- 4 (6a s /o" + Oaxa./o'" + ^y o lv ) 4- . . . 



The final step requires the expansions of powers of A in 
powers of z, the results being required to z 1 in the first line, to 
z 6 in the second line, and so on. The highest powers are easily 
formed. We have, in fact, 

A 1 = ajV, A 9 = ai 6 * 6 + 6ai 8 a a * 7 , .... 

The lower powers are conveniently obtained from the binomial 
theorem by treating aiz + ctzz 2 as the first element and the rest 
of the series as the second element. Thus 

A* = (aiz 4- aiz*)' +j (az + a^z 2 y~ 
will serve for j ^ 3. For j = 2, we have 
^2 = aj 2^2 ^ 2a 1 a 2 2 3 + (a 2 2 + 2a!a 

-f- (a 3 2 + 2a!a 5 4- 2a 2 a 4 ) ^ 6 + (2aia 6 -h 



10] FUNCTION OF A FOURIER SERIES 51 

After the insertion of these results, the rejection of negative 
powers of z, and the replacement of z* by 2a i cosiO t we shall 
have the needed expansion. 

An expansion carried out in this manner should, in general, be used 
only when literal series are desired. If the coefficients have numerical 
values, the methods of harmonic analysis lead much more easily and directly 
to the required series. The same remark applies to the function of a sine 
series, expanded in the next paragraph. 

(ii) The expansion of 

f(2ai sin + 2a a sin 2(9 + . . .). 
With the same substitutions as before this becomes 

/(-Vin"4+V^~I/0, 

so that the results of (i) are immediately applicable. In general, 
the resulting series will contain both sines and cosines, but as 
the important applications are confined to those cases in which 
f is either an even function of 0, in which case we shall have cosines 
only, or an odd function of 0, in which we shall have sines only, 
these two applications alone will be considered. 
The expansion (2) can be utilised if in it 

(a) di be replaced by t - V 1 where it occurs explicitly ; 
(/3) A be replaced by - A V- 1 ; 



/odd. 

We thus obtain, after simplifying : 

(a) When f is an even function of 0, the expansion is 
A 2 4 4 A 6 

/ " n ** * 



+ * 2 ^2a 2 / "+^< 



3! 

+ ^(63/." -</-) + ^ 



A \ ^^1 ^"Z./ U 1 ...(**) 

4-2 



52 METHODS FOR EXPANSION OF A FUNCTION [CH. n 

After the expansion in powers of A as in (i) and the rejection of 
negative powers of z y we replace z i by "la 1 cos id. 

(b) When / is an odd function of 6, the expansion is the 
following expression divided by V 1 : 

A3 A5 A7 

4/o'-~/o"' + |\/o*-/v 



- A 6a 1 a 2 / '" + (6 



~ i2o ~ - 3o !o .(5) 

After following the same processes as before we replace z { by 
V^l . 2a J sin id. 

2'11. Fourier expansions o//( + 2ai cos ^), /(a + 2a 5 sin 0). 
When a 2 ,a 8 , ... are zero, a method dependent on the series 

._ . . 1 f, a; 1 



which is closely allied to a Bessel function, may be conveniently 
adopted. 

Taylor's theorem may be written 

/(tt + x ) "= ex P- ^ ./(o), ^ = 3/3a . 
Hence 

/(oo -f ttia? -f a-!^" 1 ) = exp. a^xD . exp. a^x~ l D ./(a ). . . .(2) 
Expand each of the exponentials in powers of x, taking the 
power i -f k in the first and the power k in the second. The 
product will give the terms containing x i if we sum for k. It is 



10-12] FUNCTION OF A FOURIER SERIES 53 

The similar expression for the coefficient of x~* is evidently found 
by interchanging i -f k, i, that is, by interchanging cij , a_j. Hence 

/(o + aix + a__ix~ l ) = 2,- {(&!#)* 4- (a-iflT 1 )*} 

xQdoia-^D'.ffa). ...(3) 

In any application, the operator IPQifaia-iD*) must be used in 
the expanded form (1). 

For the first Fourier expansion put 

tf-i a i > ~ ex p ^ 1 > &'* 4 &'~~ l = 2 cos id. 
The expansion (3) then gives 

/(oo + 2cn cos (9) = 22; !'*& (a! 2 /) 2 ) ./(a ) cos i0, . ..(4) 

where i = 0, 1,2, ..., the factor 2 being omitted when i = 0. 

For the second Fourier expansion, replace ai, c&i by a_i V 1, 
with the same substitute for x. Since 

x* x~ l = 2 V 1 sin id, 
we get cosines for even values of i and sines for odd values, and 



+ 22i (- I)'' 

...... (5) 

with i=0, 1, 2, ..., the factor 2 in the first series being omitted 
when i = 0. 

It may be pointed out that 

0'Qi(-^) = /i(2*), 

where /; is a Bessel function (2*14), so that the operators may 
be expressed by these functions. But since the expression in 
this form involves the presence of imaginaries, it is simpler to 
use the functions Q{. 

212. Expansion of a power of a Fourier series. For 

(oo + 2ai cos d 4- 2&2a 2 cos 20 + . . ,)>' 
we make use of 210 (3) with 



54 METHODS FOR EXPANSION OF A FUNCTION [OH. n 

It is usually found convenient to take out the factor (i Q J, so that, 
in the operations, 6/ = 1 and/o,/o' a **e integers when j is an 
integer. 

For (2a! *an + 2a 2 a 2 sin 20 + . . .)' 

we apply 2*10 (4) or (5) according as j is even or odd. In these 
we put 



when j is a positive integer. 

2'13. Expansion of the cosine and sine of a Fourier sine series. 
(i) The expansion of 



is obtained from 2'10 (4) by putting / = -/</' =yi) lv = ... = 1. 
As far as a 7 this gives 

, A 2 A* A a 2 / . ^l 3 A*\ 



/J4 



After expansion in powers of A as in 210 (i) and the rejection 
of negative powers of z we replace z* by 2 l cos id. 

(ii) The expansion of 

sin (2a! sin 6 -f 2a 2 a 2 sin 204- . . .) 

is obtained from 210 (5) by putting /</ = -/o'" =/o v = . . . = 1. 
This gives for the function to be divided by V 1: 

A* A* A^_ a? (A 2 A* A^\ 

* 3! + 5! "*" 7! ai z V21 + 4! "*" 6!/ 

4 ,M 2 J. 4 \ 



+ (2) 

In the final result, 0* is replaced by V 1 2a l sin i^. 



12-14] BESSEL'S FUNCTIONS 55 

2 14. BesseVs Functions, The Bessel function of the first kind, 
Jj (#), may be defined by the series 



where ^ is a positive integer. For a negative suffix, we define it by 



A comparison of coefficients of x i shows that 

*) = 2 

...... (3) 



2? rf 

y _i (a;) + J, + i (*;) = ~Jj 00, /M 0*) - ^+1 (*) = 2 



and that the differential equation 



is satisfied. 

The properties most useful for our purposes are deduced 
from the fact that Jj(x) is the coefficient of z* in the expansion 

2^(*)^ j-0, 1, .......... (4) 

This result is shown by expanding exp. ##/2, exp. ( xftz) in 
powers of xz, xjz respectively and choosing the coefficient 
of z j in the product of the two series. 

Put 0= exp. g V 1 in (4), so that 21/2 = 2 V 1 sin g. We 
obtain 
exp. (x sin g . V^T) = ^Jj (x) exp. ( j# V^T), j = 0, 1, . . . . 

...... (5) 

In this equation, change the sign of g, put x = ie and multiply 
both members by exp. (ig V 1). We obtain 

exp. {i (g-e sin g) V- 1 j = S, Jy (fc) . exp. {(t - j) gr %/- 1 }. 
The real and imaginary parts of this equation give 

-e sin g)^^ i J i (ie)GOs(i-j)g t ) +1 

^ '~ ' "" 



56 METHODS FOR EXPANSION OF A FUNCTION [CH. n 

The same process may be applied to (5). With the aid of (2) 
the results may also be written 



sin(tf sin<7) = 22,J w+1 (ar) sin (2j + l)g,\ 3 ~ L> *> 

...... (7) 

The Bessel function may also be defined by 

i r* 

j(x)=*- / cos (j< x sin <) d<, ......... (8) 



a result which is deducible by the use of the Fourier theorem 
from (6). 

2*15. The Hyper geometric series. This series, namely, 



...... (1) 

includes certain series which are needed in the development of 
the disturbing function. It satisfies the differential equation, 



F=0, ...(2) 
and admits of many transformations*, two of which give 

F(A, B, C,X)=(I-X)-*F(A, C-B, a~ r ), ...... (3) 

F(A, B, C, x) = (1 -^-*- F(C-A, G - B, C, ). ...(4) 
The differential equation may be used to find the expansion 

of F in powers of y t where x = a -h y. If this substitution be 

made in (2) and if we put 



in the resulting equation, the condition that the coefficient 
of y n shall vanish identically is 

(a 2 - a) (M + 2) a n+2 + {n (2a - 1) + a(4 + fi + 1) - 0} a n+1 



* See A. R. Forsyth, Differential Equations, Chap, vi; Riemann-Weber, Die 
part. Diff.-Gleich. der Math. Phys. vol. 2, pp. 18, 19. 



14-16] HYPERGEOMETRIC SERIES 57 

This recurrence formula can be used to find all the coefficients 
when two of them are known. By direct calculation do, fti can 
be obtained from 



to the required degree of accuracy, and thence a 2 , aa, ... are 
successively obtained. It may be necessary to carry a<>, i to 
two or three additional places of decimals in order to compen- 
sate for the loss of accuracy which the use of the recurrence 
formula may produce. 

The following formulae are immediately proved : 



IT ^ 

~d~c~' ~ C A+1 > n+1 > c+1 ' ............... ^ ' 

where the meaning of the notation is evident. By differentiating 
these equations and substituting for the derivatives in the 
differential equation, we can obtain various equations connecting 
three series for A, A 4-1,^.4-2 with (7, B unchanged, or 
<7, G -f 1, (7+2 with A, B unchanged or with A, B each increased 
by 1, 2 with unchanged, etc. 

2*16. Expansion of (1 ax)~ s (1 /#)"* in positive and nega- 
tive powers of x. 

Let us adopt the notation 

/V _ n(n + 1) ... (n + r--l) /n\'_ m 

\r) ~ rl ' \OJ "" ' ...... ( } 

the accent being used to avoid confusion with the usual notation 



Expand each of the factors by the binomial theorem. The 
product is 



KO'-'H'G'M' '' j -- 1 ' 2 



58 METHODS FOR EXPANSION OF A FUNCTION [CH, n 

The coefficient of x n , n positive, in this product is obtained by 
putting { = n+j in the first factor and summing for all values 
of j. The coefficient is 



n +jj \JJ 

(& -4- n.\ (& 4- n. -L 1 \ /// _i_ 1 \ 

4 - 



' 1.2 

The coefficient of xr n is evidently obtained by interchanging t, s. 

The most important case is that for which s = t: the co- 
efficients of positive and negative powers of x then become equal. 

The part of the coefficient within the parentheses may be 
expressed by the hypergeometric series 

F(t,8 + n, 14- n, a 2 ), 

a form which permits of the immediate application of the trans- 
formations of 2' 15. 

When t s, we obtain in this way the following forms for the 
coefficient of x n or ar n , namely, 



where 



(3) 

(l-)(2-) s(s+l) a* 



_ r s * 
1+n'll-a 2 



+w.)' 1.2 (l 
l-s l+n-s 



n)(2+w)' 1.2 4 +-..|. 

(5) 



When n =0, the coefficient of /(a 2 ) is 1. 



16,17] DEVICES FOR CALCULATION 69 

If x = exp. 6 V 1, the coefficients become those of 2 cos nO 
and of the constant term in the expansion of (1 2a cos 6 + a a )~ 5 . 
(Cf. 4*2.) 

2*17. Devices for the numerical calculation of the values of 
functions defined by power series. The ratio of two consecutive 
coefficients of the hypergeometric series and of many other series 
which occur in celestial mechanics approaches the limit unity, 
and when the variable is near unity, the calculation may be- 
come tedious. The following device is often effective, especially 
when the coefficients are alternately positive and negative. 

The identity 

If x i x \ 2 ) 

a aidJ + Gtefl 2 ... = ., --- ^oH-Aaor- ~ hA. 2 a -r' +-.-K 
1 4- IK ( 1 -f x \\-\-x: ) 

where ...... 

Aa = a -ai, A 2 </o=o -"i H- ^2, A 3 a = "o 3(/i+ 3a 2 s> 

...... (2) 

is easily proved by expansion of the right-hand member in powers 
of x. If OQ, i, . . . are positive, the coefficients Aa , A a ao, . . . may 
form a rapidly decreasing series, and the rate of convergence is 
increased by the fact that #/(! -f x) < x, when x is positive. For 
efficient use the transformation should only be started at a term 
in the given series where the ratio to the succeeding term is less 
than 2. 

A more general form is given by the identity, 



...... (3) 

where Aa , A 2 <7 , ... are defined as before and 

/=l-M? + & a a - ...................... (4) 

For efficient use, the series/ should be a known function such 
that the coefficients of Aao, A 2 ao> form a decreasing series 
This is the case, for example, when 



where X is positive and less than unity. The transformation 
2'15 (3) from 2*15 (1) may be effected by means of this result. 



60 METHODS FOR EXPANSION OF A FUNCTION [CH. n 

The ratio of the (j + 2)th to the (y + l)th term of the series 2'16 (4) may 
be written 



Suppose that we have calculated a few terms of this series, say j 1 of them. 
The remainder of the series may be written 

K( 2 Yfl !l + - ( ~^l -J- ___ ^4-1 (5) 

Ml-"V L I l+ j(j+l)lj + n + ll-S + -J .......... ( > 

The formula (3) is applicable if we put 



In the applications $ has the values , 3, ..., so that if j be suitably 
chosen , A , ... form a rapidly decreasing series. The function / denned 
by (4) with these values of the b % is the hypergeometric series 



which satisfies the differential equation, 

rf*F dF 

(afi + x) ^ +{(2+7) .r+j + n + l} ^ r +^-0 .......... (6) 

A first integral of this equation is 



since F=sl when a 0. The final integral is given by 



Since /*, y are positive integers, the right-hand member can be integrated. 
When the value of F for any particular value of x has been found from 
this equation, the first and second derivatives of F can be obtained 
from (7), (6) and the higher derivatives by successively differentiating (6). 
^l, n = we have xF 



2*18. A device for approximating to the derivative of a series. 

Suppose that the sum of the series 

/S f a +a 1 # + a 2 a? 2 + ........................... (1) 

is known for a particular value x of x, and that we also know the 
coefficients up to a_i. We then know the value of 

2V+ ................ (2) 



17-19] DEVICES FOR CALCULATION 61 

We have 
dS dS 



If we can neglect the last sum in (3) when # = .r , or approximate to it, 
we obtain a corresponding approximation to (dSfdx\. We may get a use- 
ful approximation because it often happens that an approximate law of 
relation between the coefficients a n> a n + i , ... is known and the divisor n will 
assist in diminishing the error of the approximation. 

Another result of these conditions is an approximation to a n from (2) 
better than would be obtained by the approximate law of relation just 
mentioned. 

2*19. Note on the forms of products of Fourier series with 
different arguments. 

If we express two Fourier series in the forms 

So, . 2 cos j0, 2aV . 2 cos j'0', j, / = 0, 1, 2, . . . , 

where it is understood that for j = 0,/ = 0, the factor 2 is omitted, 
then their product may be expressed in the form 

22aX,'. 2 cos (J0 +/0'), 
where it is understood that the portions for j = 0, /= 0, j j' = 

are 

2 a'/ . 2 cos/0', 2oj . 2 cos j#, a o', 

that is, no attention is to be paid to the double sign when 
either j or f is zero, and the factor 2 is omitted only when 

j=j'=o- 

With the same values 

2a,-2 sin j0 . 2^2 cosj'0' = 22aX/ 2 sin (J e j' 6 '\ 
2aj2 sin j0 . 2a y2 sin/0' = T 22aX/2 cos (j0 /0'). 

This method of expression avoids all doubts as to the presence 
of factors of 2, which may occur if we use the form 2o; cos jO 
with a factor J when j = 0. Moreover, it is the natural form 
which arises when we use exponential methods of expansion. 



CHAPTER III 

ELLIPTIC MOTION 

3*1. The relative motion of two bodies under the Newtonian 
law of gravitation is a simple dynamical problem which admits 
a general solution in terms of well-known algebraic and trigo- 
nometric functions. Analytically and geometrically the range of 
the solution is divided into two portions the elliptic and the 
hyperbolic the transition from one to the other giving a special 
case, the parabolic. It is shown in the elementary text-books 
that one method of distinction is given by the relations 

F 2 -2/4/r<0, =0, >0, 

where V is the relative velocity and r the distance apart at any 
time; /JL is the sum of the masses reckoned in astronomical units. 
The first case is that of motion in a closed conic section, an 
ellipse, in which the eccentricity is less than unity ; in the second 
case the conic is a parabola, with the eccentricity equal to unity, 
and in the third case it is a hyperbola with the eccentricity 
greater than unity. When the eccentricity is zero, we have 
circular motion, the limiting case at one end of the range; when 
it is infinite, the motion is rectilinear, the limiting case at the 
other end of the range. 

In this volume we shall be concerned only with the first case, 
and the range will be further limited to values of the eccentricity 
which are small enough for the series, which are developed in 
powers of this quantity, to be used for numerical calculation 
during a certain interval of time without too much labour. Ex- 
pansions in powers of the eccentricity, either implicit or actual, 
are necessary with the methods developed below, and the greater 
part of this chapter consists of the formation of those expansions 
which will be needed in the problem of three bodies. 

3*2. Solution of the Equations for Elliptic Motion. If we put 
#0 in 1*23 (3), (4) and confine the motion to the plane of 



1, 2] SOLUTION OF THE EQUATIONS 63 

reference (see 3*6), these equations take the form 



__ 

dt* \dt ~ r*' dt 
These equations possess the integrals, 

,dv 



where a, h are arbitrary constants. 

The transformation in 1*27 shows that the elimination of t 
between (1), (3), (4) gives, if u = I/?*, 

d?u /j, A fdu\ 2 2 /X/, 1\ 

j~2 + u - T2 = > I T- ) + '* = /2 ( 2w - - ) 
o?w 2 /i 2 Vat;/ /r\ a/ 

...... (5), (6) 

The solution of the linear equation (5) is 

u= T*{ 1 + ecos(t;-cr)}, ............... (7) 

where 0, tn- are arbitrary constants and we suppose that ^ e < 1 
This must satisfy (6). The substitution of (7) in (6) gives 

A 2 = /m(l-e 2 ) ...................... (8) 

1 1 -he cos (v cr) 

Hence - = ----- 7T-- 2 T .................. ( y ) 

? a(l e 2 ) 

so that the maximum and minimum of r are a (I e). Equation 
(9) is that of an ellipse referred to a focus as origin, the nearer 
apse of the ellipse having polar coordinates a (I e), or. 

Let us now introduce the variable X and a constant n defined 



by rdX = ndt, /i = n*a 3 .......... (10), (11) 

The investigation of 1'29 with jR=* shows that the equations 
of motion with X as independent variable are 

d 2 r dv h aVr~e 2 



_ a --, 

...... (12), (13) 

since (8), (11) show that 

A = V/xtt (1 - e 2 } = na* Vl~ e 2 ............. (14) 



64 ELLIPTIC MOTION [OH. nr 

Since the maximum and minimum of r are a (1 e), the solution 

of (12) is 

r = a(l-ecosZ), .................. (15) 

where we define X as having the value when r a (1 e), that 
is, when v= -&. 

Thence, if e be an arbitrary constant, the solution of (10) gives 

y = nt + -<& = X -esinX ............. (16) 

Finally we obtain, from (13), 



The integral can be put into any one of the three forms, 

, . , Vl (Ps'mX , cosX e 

fsm- 1 -, - - v cos" 1 - ------- -., 

J \ e cos A l e cos A 



We therefore obtain, with the help of (15), 

rcos/= a (cos X e), r sin/= a Vl e 2 sin A, 

...... (18), (19) 



With the definitions (16), (17) of </, /, the following results 
are easily deduced: 

df a 2 V L e 2 dr __ ae sin/' dX _ a 
fig** r*~ ' dg ~ v'T^? ' W/~~r' 

...... (21), (22), (23) 

The constants 2a, e are the major axis and eccentricity of the 
ellipse ; w is the longitude of the nearer apse from the initial 
line, and e, 'the epoch,' is the longitude of the body when it is 
passing through the nearer apse, so that it defines the origin of 
the time. The period of revolution is 2?r/n; n is called the mean 
motion. The angles /, X, g are known as the true, eccentric and 
mean anomalies, respectively, and nt + e as the mean longitude, 
usually denoted in this volume by w. 



2-4] THE THREE ANOMALIES 65 

3 3. Frequent use will be made of <, ^, ^, tj defined by 
<jE> = exp./V 1, =* exp. ^ 



e(l+r,*)-2i,, ..................... (4) 

so that 

2 cos t/= < + !/</>*, 2 V-~7 sin t/ ^ - !/<*, etc. 

...... (5) 

By writing (4) in the form 77 = $e + \eif, and applying 
Lagrange's theorem, (2 '2), we can deduce the expansion 



JHJ 

+ .... 

(6) 

The definitions (1), (2), (4) applied to (18), (19) of the previous 
section give 



...... (7), (8) 



and, applied to 3'2 (9), 

(10) 



3'4. An important property of these relations when they are 
expressed as Fourier series is that characteristic of most of the 
expansions in Chap, n, namely, that the coefficient of cosjO or 
sinjO is of the form 



whether 6 be /, X or g. It appeared in the last chapter that if 
the functions with which we started originally possessed this 
property, it was retained under the operations to which they 
were subjected. 

B&SPT 5 



66 ELLIPTIC MOTION [CH. in 

It is evident from equations (15), (16), (20) of 3*2 that r a, 
g - X, f X, when expressed in terms of X, have this property. 
The operations to which they are subjected are those dealt with 
in the previous chapter, partly expansions in powers and partly 
changes of variable, and consequently the property is retained. 
It is apparently not present in such functions as r?cosqf y 
r^sin qf, expressed in terms of g, but it reappears in rPcosq(f-g), 
r p smq(fg), and the latter can always replace the former in 
the applications. 

Series having this property will be named d'Alembert series. 
Operations with such series have been treated in Chap. II. 

The relations 3*3 (9) show that any function of < in terms 
of x> ?; can be at once transformed into the same function of % 
in terms of <, 77. 

3*5. The facts that l/r in terms of / and r in terms of X satisfy linear 
differential equations of the form 



furnish the reason for transforming the equations of motion to the forms 
given in 1*27-1 '29. It will be noticed from 3'3 (7) that r cos/, rsin/' in 
terms of X and rj cos /, r\ sin ^/ in terms of JA', have the same property, 
but no use appears to have been found for this latter pair of expressions. 

3*6. Agreement of results with Kepler's Laws. 

That the motion takes place in a fixed plane is perhaps obvious. It can, 
however, be proved at once from the equations of 1 *23. For if R = 0, F 
depends only on r so that r' 2 dv/dt, r, 6 are constant, and v v. 

Kepler's three laws and Newton's deductions from them can be im- 
mediately illustrated from the equations of 3*2. 

Law II, which states that equal areas are described by the radii of the 
planets about the sun in equal times, i.e., that the rate of description of 
areas is constant, moans that the right-hand member of 3*2 (2) is zero ; it 
follows that the resultant force is along the radius. 

Law I states that the planets move in ellipses with the sun in one focus. 
If we substitute 3-2 (9), (4) in 1-23 (3), the radial force, dF/dr, will be seen 
to vary inversely as the square of the distance. 

Law III, which states that the squares of the periodic times are pro- 
portional to the cubes of the major axes, is not quite exact. The equations 
/x = n 2 a 3 , r = 2T/>i, give 



4-6] KEPLER'S LAWS 67 

where p is the sum of the masses of the sun and planet. The largest 
planet, Jupiter, has a mass less than 1/1000 that of the sun, so that the 
difference between Kepler's third law and the exact statement is small. 
With the observational material used by Kepler, it is not perceptible. 

The laws are, in fact, only approximate descriptions of the actual 
motions : they cease to hold when the mutual attractions of the planets 
are included. The third law, in particular, becomes a mere definition. We 
obtain the mean angular velocity of the planet directly from observation 
and define a certain distance a by means of the equation fj. = n 2 a 3 . The 
object of this definition is the convenience of calculating by means of 
equations in which the terms are obvious ratios of times and lengths. For 
example, the equation, 

acceleration = mass x [length] ~ 2 , 

which is the initial form of a gravitational equation of motion, is trans- 
formed to 

acceleration = [length] 3 [time] ~ J [length] ~ 2 , 

which is obviously correct in its dimensions. 

The constant a so defined is usually called the 'mean distance.' This 
use of the word 'mean' in the sense of 'average' is incorrect, even when 
the motion is elliptic. Equation 3 '2 (15) shows that it is the average distance 
if the eccentric anomaly be taken as the independent variable, but it is 
easily seen from equations 3*2 (9), 3*10 (5) that it is not so when either the 
true longitude or the time is so used; in the latter case, however, 3-11 (2) 
shows that I/a is the mean value of I jr. 

In the actual integration of the equations, it will be seen that the first 
arbitrary constant to appear is that on which the distance depends: the 
mean angular velocity is seen later to be a function of this and of other 
arbitrary constants which have arisen in the integrations. But since, in 
general, we can deduce the mean angular velocity from observation with 
much higher relative accuracy than is possible for the mean distance, it is 
convenient as a final step to adopt it as an arbitrary constant and to 
express the constant of distance in terms of it and of the other arbitrary 
constants. When this is done, a is nothing but an abbreviation for 

</*/')* 

Confusion is often caused by differences in the meanings attached to 

the letter a. Sometimes it means (ji/n 2 )&; at other times it signifies a 
variable or a constant which has this value as a first approximation. The 
confusion exists throughout the literature, and the only remedy is the dis- 
covery of its exact signification wherever it is used. 

Similar confusion is often caused by the use of terms in the problem of 
three bodies, which have a definite meaning in the problem of two bodies. 
Thus the Eccentricity' in the former case may be the coefficient of a 

5-2 



68 ELLIPTIC MOTION [CH. m 

certain periodic term in the expression for one of the coordinates; the 
' mean anomaly' and 'true anomaly 7 are used for certain angles ; and so on. 
A qualifying adjective, like * osculating,' may again alter the meanings. 
There is usually not much trouble when qualitative descriptions are alone 
involved ; in quantitative work, accuracy of definition is essential. 

3*7. Fourier developments in terms of X. 
The logarithm of 3*3 (9) gives, on account of the definitions 
of <t>> X> 

f V- 1 = X V=l + log (1 - rj/ x ) - log (1 - vx ) 

= AV- 1 +17 (*- 1/X) + iV( X a - 1/x 2 ) + .... 
Hence, by the definition of ^, 



j=l,2, .......... (1) 

J 

Again, from 3*3 (7), we have 

(a \ p / 7i\^^ Q 

iT?) x a (i-w)"- a (i-^) - .-(2) 

The binomial theorem gives, for the expansions of the two 
binomial factors, with the usual notation for the binomial 
coefficients, 



The coefficient of ^, with j = 0, 1, 2, ..., in the product is 
obtained by putting j + k for k in the first sum and summing 
the product for all the values of A; for that of ^~~', we proceed 
similarly with the second factor. Thus the product is 



provided the term for j = 0, which is the same in both portions, 
is not repeated. If we write 



with a similar change in the corresponding binomial coefficient 
of the second term, and define S^ Sj by 



6, 7] ECCENTRIC ANOMALY DEVELOPMENTS 



- q. -j - 1 ) ( p + ?) 

" " ~ " "" 



. 
( j Vixj + 27" " ~ ' " ""1.2 



...... (3) 

the required expansion can be put into the form 



where J = 0, 1, 2, ... ; the term for^' = being 



By putting for ^ its value cos qf+ ^l sin g/ in terms of/*, 
and similarly for the powers of ^ in terms of X, and by equating 
the real and imaginary parts, we obtain the expansions of 

r p cos qf, r p sin qf 

as Fourier series with argument X. 

The formulae for Sj, Sj are hypergeometric series and are 
therefore subject to the transformation 2'15 (3). We have, 
in fact, 



Hence, if we put S f = (1 - T/ 2 )?^* T^ so that 

r, = i+ _ . . ^_ i - -j-^2 

4- ( A 



(j + !)(/+ 2) 

(-5) 



70 ELLIPTIC MOTION [CH. ra 

Tf s= value of T t when the sign of q is changed, and remember 
that 




,.. .(6) 

where j = 0, 1, 2, ... , the terms for j = requiring the factor J. 
In the applications, p takes the values 0, 1, 2, ... , and </ 
the values 0, 1, 2, ... . When p^q, the series has a finite 
number of terms. The case p = q is more easily deduced directly 
from 3-3 (7). It gives 



from which ?*^cos pcf), r p smp<f) are at once obtained. 
The particular case p = 0, q = 1, gives 



sin/' = 

For the case q = 0, we have T/ = T i and the development 
contains only cosines of multiples of X. We thus obtain from (6) 

T t .1v*jX (8) 

the factor 2 being omitted when j = 0, and q having the value 
zero in T$. 

The following are important particular cases with <?= 0. When 
p = 1, TJ = 1 and the binomial coefficient is ( iy. Hence 

-= ^JL_ (1 + 27; cos Z + 277* cos 2Z+ ...). ...(9) 
r v 1 - & 

For p = 2, we have 



7,8] TRUE ANOMALY DEVELOPMENTS 71 

Hence 



and = (l-*)-*Srf(l+j</l-*).2coBJX, ...(10) 

the constant term being (1 e 2 ) ~ 3 . 

3'8. Fourier developments in terms off. 
The formula 3*3 (10) gives 



A comparison of the right-hand member with 3' 7 (2) when 
q = 0, shows that the development given for r p in terms of X 
can be used directly by changing the sign of ij, and multiplying 
by (1 -e)-*a- a * We obtain 



where Tj has the value 3*7 (5) with q = 0, and the factor 2 is 
omitted when j = 0. 

For p = 2, we obtain, as in 3*7, 

in which j= 1, 2, 

The series for g in terms of f is found from this last result, 
by combining it with 3*2 (21) which may be written 

ty = J**-^- 
df a **Jl-e 2 ' 

An integration with the arbitrary constant so determined that 
/, g vanish together, gives 



It should be noticed that the coefficient of sin/ in this series is 



72 ELLIPTIC MOTION [CH. in 

The functions of X in terms of/ are obtained from those of/ 
in terms of X by interchanging/, X and changing the sign of ?;, 
according to a remark in 3'4. They are, from 37(1), 

X=f+2zl(-r,y S injf, .7 = 1,2,..., ...... (4) 

and from 37 (4), with p = 0, 

(5) 



where $j, $/ have the values 3*7 (3) with p = 0, and the value for 
j = requires the factor . From these we obtain cos qX, sin qX. 
In particular, from 37 (7), 



FOURIER DEVELOPMENTS IN TERMS OF THE 
MEAN ANOMALY 

3*9, These developments are deduced from those in terms 
of X by means of the implicit relation cj X e sin X. The 
solution of this equation is avoided by making use of the 
theorem of 2*5. Applied to the present case, this theorem states 
that the coefficient of cosjg in the expansion of f(X) as a 
Fourier series with argument y, is the same as the constant 
term in the Fourier expansion of 

__ 9 d 

............ (1) 



and that the constant term in the expansion is the same as that of 
(l-ecosg)f(g) ...................... (2) 

For the coefficient of sin^, replace the first sin in (1) by ' cos/ 
The form of the first factor of (1) shows that developments 
by means of Bessel functions (2*14) will be needed. In these 
developments the parameter ^e is convenient, while in those of 
functions of/ in terms of X, the parameter r) was used. Hence 
in functions of /in terms of # both parameters may appear. 



8-10] MEAN ANOMALY DEVELOPMENTS 73 

310. Expansions for coskX, s'mkX, A", r, rcosf, rsin/, in 
terms of g. 

When/(A) = cosyLY, 3*9(1) can be written 

k k 

7 cos {(j - k)g -je sin g} + . cos {(- j - k)g +je sin g}. 

According to 314 (6), the constant term in the Fourier expansion 
of this expression is 

k k 

.J^. k (-je\ ............... (1) 



forj=l, 2,.... When j = 0, equation 3'9 (2) shows that it is 
zero if k ^ 1 and \ e if k = 1. 

A similar investigation gives the coefficient of sin jg in the 
expansion of sin kX. 

Hence, allowing j to receive both positive and negative values, 
we have, for kl, 

cos , v ^ k , . cos . , i , o /.>\ 

. kX = 2 -J~- k (ie) . 9(7, 9 = -f-l, -f-2, ....... (2) 

sm J sm^ J ~ 

When &=1, it is convenient to make use of the formulae 
214 (3), although those just written are available if we add 
\e to the expansion for coskX. We obtain 

cos X = - \e + 2S r 2 -'- J ? (je) cos jq, 

(* 

sin X 2S - Jj (je) sinjg, 

The expansion for sin A", inserted in the relation 
X = g 4- e sin X, 

gives X = # -f 22 -. J t (je) sinjg, j = l, 2, ....... (4) 

The expansions for r, rcosf, r sin'/ are obtained from (3) by 
the use of the relations 

r a(l ecos X), rcosf =acos X ae, 
r sin/= a (1 - e*fi sin A. 



74 


ELLIPTIC MOTION [OH. m 


They give 




r =a(l H 


}%?) 2a 2 - 2 e-r- Jj (je) cos jg, 
J de 




r cos/*= |a 


1 rf 


j-1,2,.... 

...(5) 


r sin/= 


1 

2a (1 e a )2 ! -- J} (je) sin j^, 





3*11. Expansions for a/r, r 2 /a 2 , a 2 /r 2 ,/, in terms of g. 

For functions which contain a power of r as a factor, it is some- 
times better to replace 3*0 (1) by* 

2cos(jg-je*mg)f(ff)(I-ecosff) .......... (1) 

That the two expressions have the same constant term in the 
Fourier expansions all that we need is evident since we can 
express their difference as the derivative of a Fourier series. 
The expansion of a/r can be obtained from (1) with 



since r = a (1 ecos X), with the aid of 2*14 (6). It can also be 
found from 3'10 (4) with the aid of the relation ajr dX/dg. 
The result is 



(2) 



The fact that the constant term of a/r, expressed in terms of 
is unity is an important property. 
For the expansion of r 2 /a 2 , we have 



2 jzl 6 j._ i(je} +? Jj_ 2 (je)l cos .?>, 
\ / ./ ' 



by 3-10 (2), 310 (3). The use of 214 (3) enables us to write this 
^ = l+f^-2JiJ,-(>)cosj7, j-1,2, ....... (3) 

* The functional f(g) has, of course, no relation to the true anomaly/. 



10, 11] MEAN ANOMALY DEVELOPMENTS 78 

The relation between the expansions (2), (3) is given by 



dt 2 \r aj 

with /L6 = ?^ 2 a 3 . This equation is easily deduced from 3*2 (1) and 
3-2 (3). 

For the expansion of a 2 /r 2 , we make use of 3*11 (1) with 
f(g)**(l ecos<7)~ 2 , so that the coefficient of cosjg is the 
constant term in the Fourier expansion of 

2 cos (jg je sin g) . (I e cos g)~ l (4) 

The expansion of 1/(1 6 cos X) in terms of X, e is given by 
3*7 (9). Hence that of the second factor in terms of g, 77, is 

(1 e 2 )"^ (\ + 22?7 cos ig), i = 1, 2, . . . , 
or (1 e 2 )"^ 2^1*1 cos igr, ^ = 0, 1,2, 

With the use of this result, (4) may be written 
2(1- e 2 )~i 2 17! *' cos {(i +j)g-je sin </}, 

and, by 2*14 (6), the constant term in the Fourier expansion of 
this function is 

2 (1 - e 2 )-* 2 171*1 J i+j (je), i = 0, 1, 2, . . . . 

This is the coefficient of cosjg in the expansion of a a /r 2 . 

The application of 3*9 (2) shows that the constant term in the 
expansion of ti 2 /r 2 is (1 - e 2 )"*, and 2'14 (1) shows that J" t -(0) 
except for i = when it is unity. The change of i into i+j in 
the previous expression for the coefficient of cosjg therefore 
gives 

,,2 



Jr v > A > 

the factor 2 being omitted for the value j = 0. 

The expansion for f is deduced by inserting this result in 
3'2 (21), namely, in 

df_a*(l-e^ 

dg~ r* ' 



76 ELLIPTIC MOTION [OH. in 

arid integrating with the condition that /, g are to vanish 
together. 
,The result is 

sin;//, ............ (6) 



where i = 0, 1, 2, ...; j = l, 2, .... 

While this formula is quite general, it is not very convenient 
for the actual calculation of any coefficient in powers of e or -77, 
partly because either of these parameters must be expressed in 
terms of the other, and partly because there are j + 1 terms of 
the same order in the coefficient of sin jg. The term of lowest 
order in any coefficient is, however, easily found, since for this 
term we can put 



Hence the principal term in the coefficient of sin^'j is 



The portions depending on higher powers of e in this coefficient 
will not be developed in detail. If we adopt the definition 



the coefficient of (e/2) ?+2 will be found to be 

5* 

(9) 

and that of (*/2)J+* 

i{l. 4.^ + 2. 5j 2 + ... + 0'~ 

J _ 



j .7- 



11-13] MEAN ANOMALY DEVELOPMENTS 77 

3*12. Expansion of any power of r by recurrence. 

The expansions of other special functions of r, / may be obtained from 
the equations of motion. Thus, if we put x r cos/, y = r sin/, the equations 
satisfied by x, y are 

d*x \*.x d^y fjti/ 
'd? " "~ 7 3 ~ ' rf? = " yJ ' 

As we have already found the expansions of .r, _?/, these equations give the 
expansions of cos //r 2 , sin//?- 2 . 
Again the equation 



gives the expansion of 1/r* when those of ?-, I//- 2 are known. 
In general, the equation 



which is deducible from (1), (3), (4) of 3 -2, can be used to obtain, by recur- 
rence, the series for r ?) for all values of p, when those for certain values 
have been obtained. 

3'13. Expansions of r p cos qf, r p sin qf. 

These expansions for values of p, q other than those just 
considered can be dealt with by first expanding in multiples of 
X by 3*7 (6), and then using 3*10 (2) to transform to multiples 
of g. The series suffer from the defect of that for f mentioned in 
3 f ll, namely, that there are^' -f 1 terms of the same order in the 
coefficients of cosjg, sin j'gr. 

If, however, we do not need such expansions beyond e 7 , the 
extensive tables given by Cayley * for various values of p, q and 
for other functions are available and will serve for most purposes. 
In cases where this degree of accuracy is not sufficient, numerical 
values are usually used and then the method of numerical 
harmonic analysis (3*17) is available. 

A combination of the results obtained from the literal and 
numerical developments by the method indicated at the end of 
2*18 will give an approximation to the terms of order ^in certain 
of the coefficients. The method developed in that article also 
increases the accuracy of derivatives with respect to e, when they 
are needed. 

* Mem. E.A.S. vol. 29, pp. 191-306; Coll. Paper*, vol. 3. 



78 ELLIPTIC MOTION [OH. m 

3*14. The constant term in the expansion ofr? cos qfin terms ofg. 

By means of the relation 3*2 (9), this function is immediately 
expressed as a function of/. According to the theorem of 2'5, 
the constant term, when it is expressed as a function of g, is the 
same as the constant term of 



expressed as a function of/. Hence, by 3'2 (21), we need the 

constant term of , , , 

r^ +a cos qf -r a 2 (1 - e 2 )*, 

expressed as a function of/. 

The expansion of r^ +a as a function of/is obtained from 3'8 (1) 
by putting -^~2 for^. We thus need the constant term of 

} . 2 V cos j/cos g/ 

where in Tj we put -ju-2 forp, g = 0. There is only one con- 
stant term in this and it is evidently given by j = #, that is, it is 



(1) 
where 

r-i + rJ^ -P^ 

''~ + 1 ' q+l 

(_p_2)(- ? _ 
1.2 



!.(?+!) 



It is evident that 3T ? is a finite series for all negative integral 
values of p ; it becomes unity for p = 1, 2. 

315. The expansion of r? in terms of g. 

The constant term in this expansion is obtained by putting 
2 = in 3-14(1). The coefficient of (e/2)i cos jg is found to be, 
with the help of the notation 3 - ll (8), 



14-16] LITERAL DEVELOPMENTS 79 

and that of (e/2)> +2 cosjg, 



The series in each brace stops at the suffix j. 

3*16. Literal developments to the seventh order. 

The following detailed developments may be found useful for 
reference. The notation 

e = e, 
is used. 

g = y _ 4 e s i n / + (3 C 2 + 2 e * + 3e 6 ) sin 2/~ f c 3 4- 4c 5 4- 8e 7 sin 3/ 



4- (I e 4 4- 6 A sin 4/- ^ 2 c 5 4- 8e 7 ) sin 5/ 

4- 1 e 6 sin 6/- 1 _ 6 c 7 sin 7/ . ..(1) 

o * 7 

+ ^4e - 2e +^ e 5 + ~ c') sin g+ ^5e 2 - ?^ 2 e 4 + - g - c 8 ) sin 2# 

/26 3 43 95 ,\ . ^ /103 4 902 6 \ . 
IT 2" T j sm ff \ 6" T5" j S1 " /; 



/1097 B 5957 _\ . . 1223 6 . 
+ \W C ~ "36" j S1 " ^ 15" S 



S in7, ....... (2) 

- = 1 + 2c 2 - ( 2e - oc 3 + ? c 5 - ^ 9 e 7 ) cos g 
d \ \j t / 

- ~ e 4 + 4e 6 ) cos 2r; - (,3e 3 - ^ e 5 4- '^ 6 J e 7 ) cos 87 



/16 4 128 6 \ . /125 B 4375 7 \ K 
~ \T ~5" j C S 9 "" \ 12 "" "72" j C S ^ 

108 e . 16807 7 . /0 , 

- c 6 cos6^r- --e 7 cos75r ....... (3) 



80 ELLIPTIC MOTION [CH. m 



+ (4e 2 - ~ 6 c 4 + ^ c 6 ) cos ty + (9e 3 - ^ c 5 + ^ e 7 ) cos 
/64 4 1024 6 \ . /625 5 15025 

+ U c TT c cos + " c ~~ 




/n 
(4) 



It is evident that the expressions for/, r are d'Alembert series 
with respect to the association of powers of e with multiples of 
g or tsr (3;4). 

317. Numerical developments by harmonic analysis. 

When the numerical value of e is given, the most rapid and 
accurate method for computing the functions is that of numerical 
harmonic analysis (App. A). This method requires the calculation 
of the functions for a few special values of the independent 
variable. The calculation presents no difficulties when either 
the eccentric or the true anomaly is taken as the independent 
variable; the formulae in 3*2 are available for the purpose. 

When the independent variable is g, the first step is the 
solution of the equation g = X e sin X, for each special value 
of (/. For a low degree of accuracy, tables for the purpose are 
available*: methods for the correction of these values are given. 
For high accuracy, the method given below will be found con- 
venient. 

When the special values of X have been obtained, those of r, 
/ and thence of any functions of r, / are found from 

r = a(l -ecos Z), 
with any one of the formulae 3'2 (18), (19), (20). 

The considerable increase in accuracy obtained with the use of numerical 
harmonic analysis is due to the fact that in most of the series with which 
we have to deal, the rate of convergence along the coefficients Aj, in the 
series SAjCi'cosjg or SJ/o'sin^, is more rapid than that of A 3 expressed 



* See, for example, those of Boquet, Obs. d'Abbadia, Hendaye, and of J. Bau- 
schinger, Tafeln zur I'heor. Astr. Leipzig. 



16, 17] SOLUTION OF KEPLER'S EQUATION 81 

as a series in powers of a 2 , especially for large values of j, unless a is very 
small. A detailed examination of the errors produced in any coefficient by 
the neglect of the higher terms with any given set of special values of g 
will show how this result is obtained*. 

Numerical Solution of Kepler's equation. When high accuracy is required, 
it may be obtained rapidly by a formula obtained as follows. 
Put X~g-\-x, so that Kepler's equation may be written 

x-e sin (g + x) (1) 

Hence 

sin x = x i^+iJotf 5 - -- 

= esin(^+^)~Jtf 3 sm 3 ( t (7 + ^) + T i^e 5 sm 6 (<7-f.r)- (2) 

Calculate (7, # from 

C cos .'0 1 e cos </, sin # = e sin </ (3) 

These give 

C 2 =l 4- e 2 2<? cos #, C sin (. +gr) = sin #, <?sin (# +#) = 8in # . 

(4) 

With the aid of (3), equation (2) may be written 

(7sin(o;~^ )=-ie 3 sm 3 ((7-f.t;) + - I J T ie 5 sm 6 (^ + a;)- (5) 

If e 3 be neglected, we have #=# , the error being of order e 3 /6. If we 
put X XQ in the right-hand member of (5), the maximum error of its first 
term is found to be of order e 6 /46. Hence the formula 

(7 sin (a; # ) = -^sin 3 (# + #o) (6) 

gives x with an error of order e 6 /46 or e 5 /120. 

By the use of (4) alternative forms for calculation are seen to be 

1 e 3 

sin (x - ff ) = - Q(J sin3 #o = - gg* sin3 # 

these giving the same results as (6). Should still higher accuracy be needed, 
it can be obtained by substituting the value of x thus obtained in the right- 
hand member of (5), but this will very rarely be necessary. For e<'14, 
the error of X found from (6) is less than 0"*1. 

* An example will be found in Mon. Not. R.A.S. vol. 88, p. 631. 



B&SPT 



CHAPTER IV 

THE DEVELOPMENT OF THE 
DISTURBING FUNCTION 

4*1. In this chapter are given methods for expressing the 
disturbing function as a sum of periodic terms when for the 
coordinates are substituted their expressions in terms of the 
elliptic elements given in Chap. in. 

The disturbing function for planetary action obtained in 1*10 is 
j _ m' m'r cos $ .-. 

where r, r' are the distances of the two planets from the sun, 
A is the distance between them, S is the angle between the 
radii r, r', and m f is the mass of the disturbing planet. Hence 

We have seen also in 1*10, that if the plane of motion of m' be 
taken as the plane of reference, 

cos S = cos (v 6) cos (v 6) + cos / sin (v 6) sin (v' ( 
cos 2 ^ I cos (v v') + sin 2 J / cos (v + v' 20). 

(3) 

In this formula, / is the angle between the two orbital planes, 
6 is the longitude of the node of the orbital plane of the dis- 
turbed planet from a fixed line in the plane of reference, v' is 
the longitude of the disturbing planet from the same fixed line, 
and v is that of the disturbed planet reckoned to the node and 
then along its orbital plane to the body. 

If cr', ty be the longitudes of the nearer apses reckoned in 
the same manner as v' 9 v, respectively, and if/', /be the true 
anomalies, we have 

The substitution of (4), (3) and (2) in (1) gives R as a function 
of r y r', /', /', w, r', /, 6. The results of Chap. Ill show how 
r, r' 9 f 9 f may be expressed as functions of the true, eccentric 



1,2] GENERAL PLAN 83 

or mean anomalies. There is thus no difficulty in expressing R 
as a function of these angles; the problem is the expansion of 
R into a sum of sines or cosines whose arguments are multiples 
of them. 

The changes necessary when the plane of reference is arbitrary 
are given in 1*32. 

4*2. Suppose that we put the eccentricities e, e' and the in- 
clination 1 equal to zero. Then the true anomalies f, f, the 
eccentric anomalies X, X 1 \ and the mean anomalies g, g' are 
respectively equal and r, r' reduce to a, a'. The disturbing func- 
tion becomes 



with /S Y = v v g 4- TV g' tsr'. 

The first term can be expressed as a cosine Fourier series with 
argument $; the second term is already in the required form, 
Suppose a<a' and put a/a! = a. Then 

R = ^ (1 - 2 cos $ + a 2 )"* - ~ a cos 
ft (/< 

/ 

= {l-Ja+t(2acu8S-a)+...}, 

Co 

on expansion by the binomial theorem. The various powers of 
cos$ can be replaced by cosines of multiples of S which will 
then have coefficients expanded in powers of a 2 ; the general 
form of the expansion is given in 2'16. 

The practical difficulties in connection with this expansion 
are due to the need for using values of a which are frequently 
as large as '7 and to the fact that the coefficients may be needed 
to five or more significant figures. If the literal series were used, 
some dozens of terms in a coefficient would often be needed and 
the work thus become extremely laborious: not infrequently 
also, some eight or ten multiples of S are required. Thus one 
problem is the construction of a set of devices for the rapid 
calculation of these coefficients. 

6-2 



84 DEVELOPMENT OF DISTURBING FUNCTION [CH. iv 

The disappearance of the term a cos S from the expansion has important 
consequences in satellite theory where a is very small. In the planetary 
theory it simply has the effect of diminishing to some extent the terms 
with argument S, so that those with arguments S, 2/S' have coefficients of 
about the same order of magnitude in the coordinates. 

4*3. When the eccentricities and inclination are not zero, the 
only available methods for development depend on expansions, 
implicit or explicit, in powers of these parameters, As far as 
their magnitudes are concerned, the problem is less difficult 
than with a, because they usually have values in the neigh- 
bourhood of *1. In exceptional cases, one or two of them may 
rise to '4 or '5: beyond this limit, the expansions are useless 
for numerical calculation and, in general, the results will have 
doubtful accuracy for values greater than *3. 

A much more far-reaching effect is produced by the introduc- 
tion of multiples of the anomalies, other than those of their 
difference. When the disturbing function is expressed in terms 
of the time, these multiples take the form jg j'g', where j, j' 
are positive integers, and the coefficient of the term which has 
this angle as argument contains the power | j f \ of the eccen- 
tricities or inclination. The coefficient of t in the angle is jn j'ri > 
and when an integration is performed this quantity will appear 
as a divisor. The divisors with the upper sign will tend to 
diminish the coefficient, but those with the lower sign may in- 
crease it. 

Consider the expression 



Since n, n' are observed quantities, we can always find integers 
which will render this expression as small as we wish, so that 
integrals involve discontinuities which may require special treat- 
ment. It has been pointed out, however, that a term with argu- 
ment jg j'g r contains as a factor of its coefficient the power 
\jj'\ of the eccentricities and inclination, so that for large 
values of j, j' t the factor is very small. From the point of view 
of the applications, the cases of interest are those in which j/f 



2-4] VARIOUS METHODS 85 

has such values as , f , f, ..., and in which the expression (1) 
is small. 

Since the coordinates of the planet will contain the integrals 
of such terms, the relative accuracy of the results will be 
diminished unless the corresponding coefficients be taken to 
more places of decimals. It is this requirement which con- 
stitutes the central difficulty in the development of the disturb- 
ing function: a few coefficients of a given order with respect to 
the eccentricities and inclination are needed to a higher degree 
of accuracy than the remainder of the terms of the same order. 
The problem is practical rather than mathematical, namely, the 
avoidance of extensive calculations of numerous terms, only a 
few of which are ultimately retained. 

4*4. In the majority of the older methods, the time is used 
as the independent variable, requiring the disturbing function 
to be expressed in terms of the mean anomalies. There are 
several methods of approach. One is to express it first in terms 
of the true anomalies by means of the equations, 



1 -t- e cos/ ' 1 -h e' cos/' ' 

and, after expansion, to proceed to its expression in terms of the 
mean anomalies by means of the relations developed in 3'10-3'16. 
A second method is the expression of the coordinates in terms 
of the eccentric anomalies, by means of the relations, 

r = a(l ecos X),} 
ro,o$f=a(cosX e), f ............... (2) 

r sin/= a Vl e 2 sin X, ) 

with similar expressions for the disturbing planet, and then to 
express the results in terms of the mean anomalies through the 
use of the implicit relations 

# = Z-esinZ, tf^X'-e'smX' .......... (3) 

Still another method is to proceed straight to the mean 
anomalies from ?, /, /, /' by means of the series developed in 
the later sections of Chap. in. 



86 DEVELOPMENT OF DISTURBING FUNCTION [CH. iv 

If an independent variable other than the time is used, t is 
eliminated through the relations 

vr, g' = n't + e' -<GT', ............ (4) 



so that the coordinates may be expressed in terms of the vari- 
able chosen. 

4*5. A further distinction between the methods arises accord- 
ing as a literal development in powers of e, e', I is made, or as 
numerical values are substituted for the elements of the ellipses 
from the outset. When the time is the independent variable, 
the expansion contains multiples of four angles: g, g' and the 
differences of tzr, or', 0. If the numerical values be used for the 
latter, as well as for e, e\ J, the disturbing function can be ex- 
pressed in the form 

~r Aj,f cos (jff + jy ) + ~ B u sin (jg -f //), 



where j y f receive positive and negative integral values, and the 
A, B are numerical coefficients. The abbreviation of the work 
is evidently very great. On the other hand, in a numerical 
method it is difficult to find a few coefficients of high order to 
more places of decimals without taking the whole, or the greater 
part of the work to the same degree of accuracy. A further loss 
with the numerical method is due to the fact that the derivatives 
of R with respect to the elements, or to some of them, are needed, 
and these require the calculation of at least three functions when 
numerical values are used from the outset. 

A definite set of rules to fit all cases should be avoided if 
much unnecessary calculation is not to be carried out. Each 
case should be examined in some detail, especially the calcula- 
tions needed for the long-period terms, that is, those for which 
jnj'n' is small, and that plan adopted which would seem to give 
the results needed most efficiently for the case in hand. Famili- 
arity with one method is to some extent time-saving but the 
gain does not usually balance the loss when a choice of methods 
is available and advantage is taken of the choice. 



4-6] IN POWERS OF THE ECCENTRICITIES 87 

4'6. General methods for expansion in powers of the eccen- 
tricities. 

The methods adopted here involve the use of the theorem, 

= *, ............ (1) 



proved in 2*7, together with a variation of this theorem found 
by putting __ 

p = exp. E V- 1, x = exp. ty> 

. ^ d Id n d 

so that x = _= , p u = exp. h, , 

dx \f-ld^ ^ r cfyr 

and 

F(exp. E*J- l.exp.^V-l) = exp. #-7-7 -f\exp. ^r V-l). 

...... (2) 

The two formulae take care of all developments along powers 

of the eccentricities, the former, in general, for linear coordinates, 

and the latter for angular coordinates. 
Put 

r = a . function of e, r</ = a' . function of e', r /r ' = a, 

where the functions of e, e' are at our disposal but reduce to 
unity when e = 0, e' = 0, and let 

r = r p, r' = r y, ..................... (3) 

so that p, p' also become unity when e, e' vanish. Then since R 
is a homogeneous function of r, r' of degree 1, and since we 
assume that a < 1, we may write it 



by a double application of (1). The eccentricities, so far as they 
occur through r, r', are contained explicitly in the factors out- 
side the functional sign only; whether they are present in a or 
not is immaterial to the developments of this chapter. 
Again, regarding R as a function ofy, /', and putting 

............... (5) 



88 DEVELOPMENT OF DISTURBING FUNCTION [CH. iv 

where E, E' vanish with e, e r and -^, ^ are independent of e, e', 
and remembering that/*,/' occur only under the signs sine and 
cosine, we have, by (2), 

^l) = F(ex]). E V^l . G xp. ^ V^T) 



Then 

and for two variables, 

(^f). ...(6) 



<ty ' * 

Again the eccentricities are concentrated in the factors outside 
the functional sign. 

Owing to the fact that p is of the form 1 + pi, where | pi | < 1, 
the two forms of expansion pointed out in 2*7 are available. The 
binomial form gives 

1) + ..., ...(7) 



r v- 1 - ri/ * i pi-*-' ! 21 
and the exponential form, 

where 



The latter form is valuable chiefly when the numerical value of 
e is used, since the coefficients in the functions of p are then 
numerical, and numerical harmonic analysis is efficient for the 
expansion of the powers of log p. Similar remarks apply to the 
expansion of p'"^" 1 which should be made in the form 



(9) 



The harmonic analysis is made with the functions 
l/p', logp'/p',... 



6-8] IN TERMS OF THE TRUE ANOMALIES 89 

4'7. There appears to be no escape from the fact that the 
development of the disturbing function requires a five-fold series, 
Developments along powers of e require in reality a double series 
because r,/ require different methods; we may make various 
combinations of them but the duplicity remains. A similar 
statement is true of r',/'. The development along powers of the 
inclination is also double but has been made essentially single 
by the device of including the factor cos 2 J7 in the functions of a 
(cf. 4*13(3); also the last paragraph of 4*31). And finally we 
have the development along powers of a. Out of this six-fold 
development, one-fold of the development can be avoided by the 
proper use of the fact that R is a homogeneous function of r, /. 
With the methods given in this chapter, the development takes 
the form of series along powers of the inclination and multiples 
of the difference of the anomalies, and along powers of the three 
operators D, B y B', the possibility of such expansions being due 
to the fact that any given power of these three operators has 
as a factor of its coefficient the same power of the eccentricities. 

4*8. Expansion of I/ A along powers of e, e' and multiples of 

'. 

This requires the substitution for r, r' of the expressions, 



1+ccos/' l + e'cos/' 

As in 3-3, put = exp./V^l, e(l +<rj 2 ) = 2<rj, and let 

(I-.? 2 ) 2 1 

-- 



-fr 

with similar expressions for r', ?</, //. 
Thence, according to 4*6 (4), 

- - 



where Ai 2 = 1 -I- 2 2a cos >, =-^, u-=a~. ...(4) 

r </a 

The expansions of p D , p f ~ D " 1 into Fourier series with arguments 
f,f and coefficients depending on positive integral powers of JD 



90 DEVELOPMENT OF DISTURBING FUNCTION [CH. iv 

are given by the formulae of 2*1 G. It will be noticed that p D is 
equivalent to the function expanded in 2'16 if we put ?;, D, D, <f> 
for a, s, t, x, respectively. Hence 

...(5) 



with ; = 0, 1, 2, . .., the factor 2 being omitted when^ = 
Similarly, 



In forming the product of these two series, the rules noted in 
2*19 are to be followed. This product, inserted in (3), gives 



where F t F' denote the hypergeometric series in (5), (6) respec- 
tively. 

This is the required expansion. The portions of the coefficient 
which depend on D have to be expanded in positive powers of D 
and these are operators acting on 1/Ai which contains a only in 
the explicit form shown in (4). In these expansions it is important 
to notice that any power of D is always accompanied by at least 
the same power of 77, 77', so that the number of powers of D 
required is the same as the order with respect to the eccentricities 
to which the expansion is to be developed. 

4*9. It is sometimes more convenient to use 2*16 (5) or 2*16 (4) 
for the developments. The necessary changes are easily seen. If 
we use the former, the formula 4'8 (7) will still serve if we put 
a , a' r /1X 

" 



-D-2,j + l, if), ...... (2) 

and multiply the result by (1 ?; 2 )/( 1 7/ 2 ). In adapting tho 
work to this formula, w-3 have put 



-V s ? 



8-13] IN TERMS OF THE TRUE ANOMALIES 91 

If formula 2*16 (4) is to be used, we put 



(4) 

with 

F-F(l-D,D,j + l,-*), 



F' = F2 + D,-D-l,j+l, f_ /2 . . . .(5) 
In adapting to this formula, we have used 



4*10. The operator ^1 = D 4- ^ has one advantage over D which may 
render its use advisable in some problems. This advantage results from 

the fact that when the expansion of the operator ( . J F has been made 

in powers of A, that of ( ., J F' can be immediately written down by 
changing the sign of A and substituting 77' for TJ. 

4*11. The expansion of any function of r, r', A can evidently be made 
by exactly the same methods. We have, for example, 



the expansion of which follows exactly the same plan. 

4'12. The complication of the various values of a over that usually used, 
namely a/a', is more apparent than real, since the numerical value of a is 
always used. Further, as TJ is greater than rf in most asteroid problems, 
and as the convergence is improved by diminishing a, there is an advantage 
with these values over the value a/a'. The slight disadvantage which 
arises when we have to differentiate with respect to e or rj is easily dealt 
with by adding to the derivative with respect to ^, so far as it occurs 
explicitly, the derivative D.da/adri. 

413. Development along powers of the inclination. 

In the previous paragraphs I/A has been developed into cosines 
of multiples of/,/', with coefficients which depend on 77, ??', 1/Ai 
and on the derivatives of 1/Ai with respect to log a. Now 

.................. (1) 



92 DEVELOPMENT OF DISTURBING FUNCTION [CH. nr 

where, by 4*1 (3), 

cos S = cos 2 \I cos (v - v') + sin 2 / cos (v + v'~ 20). . . .(2) 

The general plan requires the expansion of 1/Ai into a double 
Fourier series with arguments v v', v + v' 20, and this might 
be achieved by first expanding into a Fourier series with argu- 
ment S and then expanding cos iS into sums of cosines of 
multiples of these two angles. More rapid convergence with less 
computation can be obtained by making the development depend 
on the Fourier expansion of 1/A , where 

A 2 =l+a 2 - 2a cos 2 17 cos (>-?/), ......... (3) 

rather than on the same function with cos 2 \I replaced by unity. 
With the definition (3) of A , we have 

sin 2 1/ cos (?; + /- 2(9)) ~* 



which is then expanded by the binomial theorem. This expansion 
evidently involves odd negative powers of A accompanied by 
even powers of sin \I. The powers of cos (v + v' 20) are to be 
expressed as cosines of multiples of the angle. Instead of giving 
the general form of this expansion, we set it down as far as the 
eighth power of sin \I which will be sufficient for all practical 
needs. 

Define RI, R 9 , ..., by 



where 2s takes the values 3, 5, 7, .... We then obtain 

-- = RI + RS tan 4 II + J R 9 tan 8 \I 
AI 

tan 2 \I + R? tan 6 \I) cos (v + v f - 20) 



+ (R 5 tan 4 \I + jff 9 tan 8 /) cos 2 (i; + v' - 2(9) 
+ JJ?7 tan 6 |J cos 3 (v + v' - 20) + T \, # 9 tan 8 J/cos 4 (t; + / - 2(9). 

...... (5) 

In this development, it should be noticed that a multiple 
of in the angle is always accompanied by at least the same 
power of tan %I in the coefficient and that the series in any 
coefficient proceeds by powers of tan 4 J/. 



13, 14] IN TERMS OF THE TRUE ANOMALIES 93 

The angles v, v 1 are expressed in terms of the true anomalies 
by means of the relations 

t; = /+tar, v'=f +r' ................... (6) 

The development in powers of the eccentricities contained the 
angles f y f only, while this development contains the angles 
f-.f+n->v', /+/++' -20. 

When the functions R% 8 have been developed as Fourier series, 
and products of cosines replaced by sums of cosines, we shall have 
a development containing multiples of the four angles 

/, /', w -w', v+v'-20, 

and this development will have the property that the difference, 
taken positively, of the multiples of f y f in any angle will be 
accompanied by the same power of e, e', I in the coefficient. 

4'14. Development in multiples off, f to the third order. 
The development of the RZS is given in a later section of 
this chapter, (4*23), in the form 

.fi 28 = W>COsi(/-/ + *r-O, /8 f l -> =&<>, *=0,1,2,.... 

...... (1) 

The coefficients are functions of a, cos* \I only, and the operators 
D> act solely on these coefficients. The value of a given by 4 - 8 (2), 
4'8 (4) is used here. 

By carrying out the various steps outlined above, we obtain 
the following development as far as the third order with respect 
to t), i)', tan \I. 

^ = ( F! 2/3 4 (t) + 2 tan 2 / . F 9 2/3 t M ) cos i (/- 
where 



v'-O 2 (D + 1) + ip/ D(D + I) 2 } cos/ 
+ {27?' (D + 1) + V s D(D + I) 2 + Vl** 8 (D + 1)1 cos/ 
+ (T) 2 + D) {? cos 2/ - 2 W ' cos (/ / ' ) + 7/ a cos 2/' J 
- JT;* (D 3 + 3J5 2 + 2D) cos 3/ + J V 3 (^ 3 - -) cos 3 /' 



94 DEVELOPMENT OF DISTURBING FUNCTION [CH. iv 



F z = cos (j +/' + @) - vjD cos (2/+/' + @) - yD cos (/' 

+ vf(D + l)cos(/+ 2/' + @) + V (D + 1) cos (/+ 8), 
= w + w ' _ 20. 

The double sign means that there are two terms each having 
the coefficient set down. 

The final step, that of expressing the products of the cosines 
in FI, FZ by cosi(ff' + '& 'GT') as cosines of sums and 
differences of the angles, is to be carried out. This is equivalent 
to adding the angle i (ff -f VT r') to each of the angles in 
FI, F B , because /3 a ( " i} = /3 8 (l) . The term in FI independent of 
r, f requires no treatment; i receives all positive and negative 
integral values and zero. 

4*15. Transformation from the true to the mean anomalies. 

The development in terms of the true anomalies consists of 
a sum of terms of the type A cos (jf+ff -f (7), where A depends 
on a, a', e, e', I and G on sr, -57', 6. To transform to a development 
in which the arguments are functions of the mean anomalies, 
we make use of the expansions 

f=g+E = ff+2fi*mig, f = g' -f E' = g' -f 2// sin tf, 
obtained in 311 (6), together with 4'6 (6) which gives 

cos (jf + j'f -f C) = exp. (# 1 + E' |>) cos (jg + $<f + C). 

...... (1) 

The exponential is expanded in powers and products of Ed/dg, 
E'd/dg', and this requires the expression of powers of E, E' as 
Fourier series with arguments g, g f \ the operators d/dg, d'/dg r 
act only on the explicit functions of g, g' and not on E, E' . 
The same result is reached by writing 

cos (jf+ff + C) = cos (jg + j'g' + G) cos (JE +/#') 

- sin (jg +j'g' + C) sin (jE+j'E'\ 

, . COS . r, COS .. ~, 

and expressing 1 . ih. . i & 

* * sin* 7 sm J 

as Fourier series with arguments g, g' respectively, for the 
different values of j, j' needed. The calculations of the functions 



14, 15] CHANGE TO MEAN ANOMALIES 95 

of E, E' needed can be made in series or numerically by har- 
monic analysis. 

Properties of the expansion. Since R is independent of the 
directions of the axes of the frame of reference, it is independent 
of the origin from which the angles used in the expansion are 
measured. Hence the algebraic sum of the multiples of such 
angles present in any term is zero. 

Thus if w y w f be the mean longitudes, and CT, CT', 6 the longi- 
tudes of the perihelia and node, and if any argument in the 

expansion be 

iw + i'w' +JK+J'<GT' + 2hV, ............... (2) 

we have i + i' +j -h ;' + 2h = 0. 

The original form of R was an expression in terms of v, v', r, r' , 
F, 20. It was pointed out in 3*16, that the expansions of v, r in 
terms of g or v -GJ are d'Alembert series as far as the associa- 
tion of powers of e with multiples of cr is concerned, and the 
same is true of v', r' with respect to e 1 ', -CD-'. It follows that R 
has the same properties. Further, the expansion 413 (5) shows 
that R is a d'Alembert series with respect to F, 26. It follows 
that the coefficient of a term with the argument (2) is of order 
j j | -f | j f | -f | 2h | with respect to the eccentricities and inclination. 

But + ' 



Hence, the order of the coefficient of any term in the expansion 
of R is equal to or greater than the algebraic sum of the multiples 
of w, w' present in that term, and the same is true for the 
multiples of g, g' when we put w g + &, w' g' -f r'. 

This property at once gives the lowest order of the coefficient 
of any term in a numerical expansion of R. 

When numerical values of the elements are used, the expression 
for tt'/A in terms of the true anomalies may be put into the form 

*Q tf cos(tf+j'f') + 2S Jt ,*m(jf+j f f), j,/ = 0, 1,..., 
or 2 coBJf(C Jtf cos//' 4- 8 Jtf sin//') 

'jj, cos// v + S' Jtf sin//), j, / = 0, M, . . . . 



96 DEVELOPMENT OF DISTURBING FUNCTION [OH. iv 

The portions in brackets are transformed to multiples of g f by 
the relation /' = g' + E', either by series or by harmonic analysis. 

The series are then re-arranged in the form 
2 cos JY (A jtj . cosjf+ B jtj > sinjyT) 

+ 2 sin JY (A' jtf WBJf+B'jj sin jf), j, j r = 0, 1, 2, . . . , 
and the change to multiples of g is carried out by using the 
relation f=g + E. By following this procedure we can limit the 
additional work required to obtain the coefficients of the long 
period terms to a higher degree of accuracy, owing to certain 
peculiarities in the series for r,/. 

4*16. The value of r in terms off is the series 

r=a aecos/-f 

Actually, this gives series along powers of %e, because the long 
period terms always arise from the expression of the product of 
two cosines as the sum of two cosines; only one of the latter is 
needed more accurately. The same is true of r'. But when we 
substitute for/ in terms of g by means of the series 



we are substantially expanding in powers of e instead of 0; the 
coefficients, which depend on , are in general of the same order 
of magnitude for the series giving r in terms of/, and /in terms 
of g. Further, many of the actual problems are those of asteroids 
disturbed by one of the great planets and the eccentricities of 
the orbits of the latter are small. Thus while the steps up to 
the last have to be carried out to the full degree of accuracy, 
the series converge rapidly. The convergence is slowest in the 
last step, but it is here that we can make selection of the terms 
which have to be accurately computed, the remainder requiring 
a much lower degree of accuracy. 

The calculation of the coefficient of a particular term can also 
be efficiently carried out by the method which follows. 

4'17. Calculation of the coefficient of a particular term. 
For this calculation we can make use of the theorem of 2*6, 
where it is shown that the coefficient of cos (ig + i'g'} in the 



15-18] % THE CONSTANT TERM 97 

expansion of F(f,f) is the same as the constant term, when g, g' 
are expressed in terms of/, /', of one of the expressions 



For the coefficient of sin(ig + i'y'), change cosine to sine in (1) 
and sine to cosine in (2). 

To make use of them we have the relations 



___ ___ 

df~ a ^i- e *~( 

ztani/, ...... (4) 



with similar expressions for accented letters. An alternative to 
(3), (4) is the use of the series 3'8 (3). 

If we make use of the first of the forms, the initial expansion 
of r 2 /r /2 A instead of the expansion of I/ A, would be made: the 
method for doing so is shown in 4*11, and the values p = 2, 
p' ~ 2 in 411 (1) would be used. With this formula the ex- 
pansions 4*9 (2) or 4*9 (5) are recommended. In F we replace D 
by D -f 2 and in F f we replace D by D - 2: the same changes 
must of course be made in the binomial coefficients in 4*8 (7). 

If, however, the plan is used to get a particular coefficient to 
a higher degree of accuracy after a general development of I/ A 
has been made, the second form of (1) is of advantage because 
the development already made will serve; in such cases neither 
i nor i f is zero, so that this form is always available. 

4*18. Calculation of the constant term. 

This is sometimes needed to a high degree of accuracy. Ac- 
cording to the theorem of 2*6 the constant term in the expansion 
of I/A in multiples of g, g' is the same as the constant term in 
the expansion of 

1 dgdg' = rV 2 

A dfdf A . aV a VT^T 2 Vl - e' 2 

B&SPT 7 



98 DEVELOPMENT OF DISTURBING FUNCTION .. [OH. iv 

in multiples of/,/'. The use of the formula 41 1 (1) with p = 2, 
p = 2 is indicated. This expansion does not require the use 
of the relation connecting / with g or /' with g' ; it depends solely 
on expansions along multiples of /, /', and therefore requires 
merely the substitution for r, r' of their expressions in terms of 
/,/'. A literal development to the eighth order with respect to 
the eccentricities and inclination is to be found* in Astr. Jour. 
vol. 40, pp. 35-38. 

419. When harmonic analysis is used to obtain functions of /in terms 
of g and those of/' in terms of #', the computation can be made as follows. 

If multiples of g not higher than the sixth are needed the seven special 
values of g namely, 0, 30, 60, 90, 120, 150, 180, are recommended. If 
two more values be needed, those for 45, 135 can be added, and with two 
fewer, those for 30, 160 can be omitted. It is useful to notice that the 
addition of new values does not require the greater part of the work, which 
is the computation of the special values of the function, to be done again ; 
only certain small portions of the analysis have to be repeated. 

The values of X for the chosen values of g are obtained from Kepler's 
equation 

X*=g + emi\ X, 

by one of the methods given in 3'17. From the relation 

T~~ 
tan J/ 



the special values of /and thence those of any function of /are then ob- 
tained. As / g take the values 0, 180 together, there are only 3, 5 or 7 
special values of / to be computed for each planet. Methods for analysis 
into Fourier series are given at the end of this volume. 

The functions of /needed are cosjf/j shy/ for a number of integral values 
of ,;'. It is more convenient to calculate cos^' (fg\ sin,;' (/ g\ and after- 
wards to deduce the expansions of cos.;/, ainjf by the use of the factors 
coajg, sin^. 

Most of the asteroid problems require the calculation of the perturbations 
by Jupiter and Saturn only. The series for cosji/', sin j/', once computed 
for these two planets will serve for all cases ; small changes in the values 
of the eccentricities are easily made since the power of e which accompanies 
any term is known by the multiple of g in its argument. 

Harmonic analysis is usually so much more accurate with respect to con- 
vergence, and is so much more easily controlled than literal expansions, that 

* E. W. Brown, The Expansion of the Constant term of the Disturbing Function 
to any order. 



18-20] IN TERMS OF ECCENTRIC ANOMALIES 99 

it should be used whenever possible. Where many such analyses are to be 
carried out, a systematic arrangement of the work, by which one operation 
at a time is performed on all the functions to be analysed, permits the 
calculations to be carried out rapidly and accurately. See App. A. 

4*20. Development in terms of the eccentric anomalies. 

The expressions in 4*1 give I/A as a function of r,/, r',/', 
6, /. Also in 3-3 (8) and (9), with the notation < = exp. / V^HT, 
% = exp. X V 1, where X is the eccentric anomaly, it has been 
found that 



with similar expressions for r', </>'. If we put 







we can make use of the theorem of 2*7 with four independent 
variables. 

Now I/A is equal to a function of r/r', <f>, <f>' divided by r'. 
Hence, if A^. ?V be what A becomes when we replace r, r', <, </>' 
by r , r ' } %, %', the theorem gives 



xV-v'x'r*- 1 -*' ....... (i) 

This has to be expanded in powers of the indices. 

The expansion of the first pair of factors is made by the theorem 
of 2-16. It gives 



J 



3 ' ...... (2) 

7-z 



100 DEVELOPMENT OF DISTURBING FUNCTION [OH. iv 

The result is unchanged if we change the sign of V 1, for then 
%, l/x interchange and also B y B\ it is therefore a real Fourier 
series with argument X. 

The product of the second pair of factors is obtained by 
putting - D - 1, *;', ', /, / for D, rj, B, x> j respectively in (2), 
and it has the same properties. 

Since /, 6 are present in A^ in the same way as they were 
present in AI, the expansion of A^ along powers of the inclina- 
tion follows the same plan as that of A x in 4*13. In fact, if we 
put v = X + er, v' = X 1 + *' in 413 (3), 4-13 (5) (taking note of 
the different significations of r , r ', a), the results can be used 
here without further change. 

Newcomb has given (Astr. Eph. Papers, vol. 3) & detailed expansion of 
the disturbing function in terms of the eccentric anomalies, certain portions 
of which are taken to the seventh order with respect to the eccentricities. 
He uses an operator but did not obtain the general formula which permits 
any coefficient to be written down at once. The latter was given by one of 
us (E. W. Brown, Astr. Jour. vol. 40, p. 19, 1930) in terms of the operator 
D and certain integers i, i' and later (Astr. Jour. vol. 40, p. 61, 1930) in the 
improved form shown in the text with the use of the operators Z>, B^ B '. 

4'21. Transformation from eccentric to mean anomalies. 

After the disturbing function has been expanded in cosines 
and sines of multiples of X, X', the transformation to mean 
anomalies can be effected by the formulae of 3'10, which give 

i\ 
cosjX = A H- S - J K _J (tee) cos /eg, 

\ *=1, 2, ...(1) 
sin jX = S< - J-j (tee) sin teg, 

or, if f = exp. g V 1, in the exponential form, 

tf-A + Z.ij^ice).?, (2) 

K 

where -4 = or \e according as j =}= 1 or j = 1 ; j may have any 
positive or negative integral value. A similar set of formulae 
holds for accented letters. 

Since the only terms in the development which give a constant 



20-22] CHANGE TO MEAN ANOMALIES 101 

part are those containing the first multiples of X, X', the 
constant term of the development in terms of g, g' is obtained 
by adding to that in terms of X, X f , the terms 
- \e . coef. of cos X - \e' . coef. of cos X' 

ee f 
+ -T - sum of coef. of cos (X X') ....... (3) 

The method developed in 4*17 for the calculation of the coefficient of a 
particular periodic term and that in 4*18 for the constant term can 
evidently be applied to the transformation from eccentric to mean anomalies. 
For functions of/, /', we substitute functions of X, A'', with 



But the coefficients in the expansions of costjX+j'X'), sin (JX+j'X') in 
terms of g, g' can now be written down in terms of Bessel functions, as 
defined in 2*14. For both periodic and constant terms, this process is 
equivalent to that in the text and is merely a different mode of stating it. 

4*22. A detailed comparison of the relative advantages of a primary 
development in terms of the true or eccentric anomalies appears to favour 
the 'former. In the first place, the expansion in terms of the true anomalies 
requires the use of only one operator Z>, while that in terms of the eccentric 
anomalies requires the use of three operators D y /?, B'. Those three 
operators produce both cosines and sines while, with the operator D alone, 
only cosines are present, and therefore in reducing products of cosines to 
sums of cosines, there will always be pairs of coefficients which are the 
same. 

It might be thought that the change from true to mean anomalies is 
more complicated than that from eccentric to mean, because we cannot use 
general formulae like the Bessel functions to make the change. As a matter 
of fact, the actual labour of making the expansions differs very little in the 
two cases, whether literal or numerical values of the eccentricities be used. 
The developed series in powers of e have to be used in either case, and such 
series are available in the tables of Leverrier and Cayley, if a literal ex- 
pansion is desired. With a numerical expansion by harmonic analysis the 
only additional work is the calculation, for a few special values, of the values 
of/, after those of X have been found, from the equation 3-7 (1). 

A point connected with the rate of convergence along powers of e, and 
rarely mentioned, deserves some stress because the work of calculating the 
coefficient of some particular term to an order of accuracy higher than that 
of the general development can be made lighter by taking it into considera- 
tion. It has been pointed out in 4*16 that the rates of convergence of the 



102 DEVELOPMENT OF DISTURBING FUNCTION [OH. iv 

series for r, / in terms of /, /' are more rapid than those of /, /' in terms 
of g, cf. There is no such difference in the rates of convergence in passing 
directly from r,/, /,/ to X, X' and from X, X' to g, g'. Thus the longer 
development in terms of X, X ' must also have the full accuracy desired 
while the shorter development in terms of/, /' is still more abbreviated by 
the separation of the more rapidly converging series from the more slowly 
converging series. 

The advantage possessed by the expansion in terms of the eccentric 
anomalies in the form given in the text, consists in the fact that it is the 
only method known by which any coefficient in the development of the 
disturbing function can be written down from a general formula ; it con- 
tains the operators /), /?, B', and the Bessel functions. The highest power 
of these operators present in any portion of a coefficient is the same as the 
order of that coefficient with respect to the eccentricitie.s, so that stoppage 
at a given power of the latter involves stoppage at the same power of the 
former. The order of a Bessel function is known from its suffix. But the 
formula suffers from the defect pointed out in the paragraph following 
3*11 (6) for the case of the general expansion of/ in terms of g, namely, 
that ^'-f-1 numerical coefficients have to be added together to obtain any 
part of order ; ; this defect becomes serious when^' is large. 

4*23. The functions of the major axes. 

The development of the previous sections of this chapter 
require the calculation of the coefficients /3 8 {i) defined* by 

/)*-* () 

- (1) 

where t 0, 1, 2, ...; 2*= 1,3, o, ...; &<->-&<>; 
and for 25 = 1, the numerator of the fraction is unity. The 
definitions of T/T, a are immaterial to the work of this section 
provided |a| < 1, We shall need also the derivatives of the 
coefficients with respect to a or to log a. 

It has been pointed out in 4'2 that the magnitude of a in 
general prevents the use of literal series in powers of a as a 
practical method for calculation and we must consequently use 
other devices. In the following paragraphs, transformations of 
the series in powers of a are made for two of the coefficients so 

* This definition of the coefficients without the factor J is so much more con- 
venient than that of Leverrier and others who have followed him that we have 
retained it throughout. See 2*19. 



22-24] FUNCTIONS OF THE MAJOR AXES 103 

that they may be easily and rapidly obtained. It is then shown 
how all the remaining coefficients and their derivatives can be 
deduced from these two by the use of finite formulae. The two 
coefficients to be first found will be those for s = , i= 10, 11, 
for reasons which will appear. The more usual plan has been to 
calculate 

/%" = ^i> & (1) = | (*i - Ei) ............. (2) 

where FI, E\ are the elliptic integrals of the first and the second 
kind, from the tables of Legendre, with cos 2 7=1. The tables of 
Runkle (Smithsonian Contributions, 1855) give certain of these 
coefficients for different values of a; those of Brown and Brouwer 
(Camb. Univ. Press, 1932) have higher accuracy. 

424. The series for &>. 

Define i, * by* 

! aeosH/ 
- 1+ai 2= i + rf . * = cosi/, ............ (1) 

so that 

(1 + a 2 - 2a cos 2 \I cos f )~* = (~-Y( 1 -f i 2 - 2 x cos ^)~ s . . . .(2) 

VCE/C/ 

The last factor may be expanded into a Fourier series by the 
method given in 216. By inserting this expansion in 4'23 (1) 
we obtain 

' ~ i)' ^- 2 cos 



= a t s (a*)-* S illl.Lr_ rtl . /' . 2 cos t>, ...... (3) 

where F is the hypergeometric series given by 



and i takes the values 0, 1, 2, ..., the constant term under the 
sign of summation in (3) that for i = being F(s, s, 1, i 2 ). 

* The value of c^ can be readily found by putting aj = tan A\ , a = tan A, and 
finding A l from sin 2A t K sin 2A . 



104 DEVELOPMENT OF DISTURBING FUNCTION [CH. iv 

By means of the transformation 2*15 (3) we have 

F(8 + i,8,i + l,a l *)~(l-a)-'F(l- 8,8,i + l,- p\ 
where p = i 2 /( 1 - i 2 ), ..................... (5) 

so that 



in which 



s s-1 ,( + !) (s-l)(s-2) 



This last form of the hypergeometric series is evidently useful 
for large values of i, since in this case the earlier coefficients of 
powers of p diminish rapidly. It is true that the series converges 
only when \p\ < 1, that is, when i< 2~*= 707, and that the 
values of the coefficients are sometimes needed for values of i 
larger than this. But we know that the function which the 
series represents has no singularity provided |i| < 1, that is, 
provided p be finite. It is therefore permissible to use the 
method of analytic continuation to obtain expansions in powers 
of p PQ where p Q =(= 0. 

4*25. The calculation o/ai , an. 

For brevity, let us put fi (i) = a t -, so that 

(1 -f a 2 - 2a cos 2 / cos ^r)~* = a + 2Sa t - cos fy t ...(I) 



with 2 2 



The transformation of (3) to series expanded in powers of p p Q 
can be effected by the use of Taylor's series: 



The values of F(p Q ) t F r (p ) are obtained directly from the series, 
and the remaining derivatives from the recurrence formula 



4, 26] 



FUNCTIONS OP THE MAJOR AXES 



105 



2*15(5) deduced from the differential equation satisfied by the 
hypergeometric series*. 

On putting (1 i 2 )^ = #i^~^, we have the following expressions 
for the cases j9 =0, J, 1. The factors 46,189 and 176,358 are 
the respective products 11 . 13 . 17 . 19 and 2 . 13 . 17 . 19 . 21. 

/a~v / \ 9 

aio = 46,189 A/ -- ( * ) multiplied by one of the series, 
V /ca \4/ r J 

+ 1-00000000 

- -02272 727 p 
+ -00213 068 p 2 

- -00034 15 p 3 
+ -00007 47 jo* 

- -000020 p* 
+ -000006 ;> 



a u = 176,358 



tea 

+ 0-98913047 +0'97912 120 

- -02082065 (p-i) - -01926520 (/>-!) 
+ -0017101 (p-V? + '00141681 (V-l) 2 

- -0002304 (p-l)* - -00016 594 (p- 1) 3 
+ -000041 (/>-4) 4 + -00002 515 Qt>- 1) 4 

- -000009 (jo-i) 6 - -00000451 (p'-l) 6 
+ -000002 (/J-i) + -00000091 (^-l) 6 

- -00000020 (p-l) 1 
+ -00000005 (^-l) 8 
00000001 (p-1) 9 

multiplied by one of the series, 



/<*ip(<*i 

V ^ 



10 



+ 1-00000000 

- -02083 333 p 
+ -00180 288 p' 

- -0002683 p 
+ -00005 48 p 

- -00001 4 'p 
00000 1 p ( 



+ 



The series give 
< -82. The table 

a! = -45 
= '25 



+ 0-99000 356 +0*98074 527 

- -01920 796 (p-i) - -01786420 (jo-1) 
+ -00146840 (p-b) 2 + -00123022 (p-1) 2 

- -0001854 (p-l? - -00013597 (p- 1) 3 
+ -0000311 Q0-I) 4 + -00001956 (p-1) 4 

- -000007 (p-) 5 - -00000 335 (p- 1) 6 
+ -000002 (p-$f + "00000 065 (^-1) () 

- -00000014 (jo-1) 7 
+ -00000003 (/>-!)* 

- -00000001 (p-l) 

aio, an to eight significant figures when 



-61 '66 '71 '78 -82 '88 
-50 75 1-00 T50 2'00 3'00 

indicates the series to choose for any given value of i. For 
a x = -88, the error is about one part in 10 5 . 

* E. W. Brown, Mon. Not. R.A.S. vol. 88, pp. 459-465. The numerical series 
given in the text for2? = |, 1 are taken from this paper. Extensions to the cases 
p = 2, 3, 4 are to be found in Mon. Not. vol. 92, pp. 224-7. 



106 DEVELOPMENT OF DISTURBING FUNCTION [OH. iv 



Alternative form for a 10 , a n . The ratios of consecutive coefficients of 
these series all tend to the limit unity, but for large values of i the approach 
is very slow, and after the first two or three terms the ratio changes slowly. 
We can make use of this fact by expressing the series a -\-a l p + a 2 
in the form 



with a suitable choice of A. In this way the following expressions have 
been obtained : 

a 10 = '35239,4104 



where 



a u = '38183,0736 (&) a 
\ K * / 

Sa 10 = (2131 + 245/7 - 19/> 2 )p 2 . 



Q + (1 + 275^), 



These give 10 , r H to six significant figures when 
figures for p<\ are furnished by the expressions, 

a, = l ~p p* - "00034,1 46/> 3 + '00007,469^ 



. Eight significant 



' 829 ^ 3 + '00005,477/J 4 



4'26. Formulae for calculating the /3 (<) t^Ae/i ^M;O consecutive 
coefficients are known. 

The procedure which appears to give numerical results most 
easily requires the use of the following formulae. It is to be 
noticed that as soon as two consecutive coefficients have been 
found, there is no further need of i; the formulae involve only 
a, K in the form 



The proofs are given in the sections which follow. 

For finding the remaining a t - = /3| ( * ) when two of them are 
known, we have the formula, 

a i = ea m - a +1 + - (ca m - 2a <+2 ) ......... (2) 



25, 26] RECURRENCE FORMULAE 107 

for values of i lower than those known, and 



for higher values of i. These formulae are used by putting 
i = 9, 8, ... successively in (2) when a, an have been found; and 
i = 10, 11, ... as for as they are needed in (3). They are deduced 
from the general formula, 



* ~ * &<>, ......... (4) 

1+ S I + S 

by giving to s the value J. 

The values of two consecutive coefficients for other values of 
s are found by putting * = ,,..., successively in the formulae 



- 

- - 4( 6 _2) ~~ "" 
2s) ^ + (2 + 2 - 2s) /9^ +1) " 
' 



The remaining coefficients are found rapidly from the formula 



which may be used backwards or forwards along values of i. 

Sufficient checks on the numerical work are obtained in the 
following manner. When a 9 , a B) ..., a Q have been successively 
computed by means of (2), the value of a can be obtained 
directly from 4'23 (2) with the use of Legendre's tables: this, in 
effect, tests the whole series of cti . 

When /3 f (11) , ..., # f (0) have been found from (5) for the first 
two and from (6) with s = \ for the remaining coefficients, the 
values of y8^ (1) , /3J * can be tested by computing from (5) with 
s = , i = 0. A similar procedure tests the values for s = f , , .... 

In general, there is a loss of less than one significant figure 
in running down from y8 8 (10) to /3 (0) . There is some loss of 
accuracy in the use of the formula (5), but this loss is balanced 
by the fact that the higher values of s are present only with 
the higher powers of the inclination and indirectly of the 
eccentricities. 



108 DEVELOPMENT OF DISTURBING FUNCTION [CH. iv 

4*27. Formulae for the derivatives of the p 8 (i) . 
The first derivative is obtained from 



for all values of i with s ^, and for two consecutive values of i 
with s = f, f, ..., the remaining derivatives being found most 
easily from 

f> = iD/Sf? + D&? ................ (2) 



For the higher derivatives, either of the following formulae in 
which the index (i), being the same throughout, is omitted, may 
be used: 

Z)'+ a & = (D--*" 11 ~ a 



(3) 

= (2*-l)JD'+ l & + (D + lXA+i 
/c 

>,, + 4,(l-- z 



To j are given successively the values 0, 1, 2, ...: the latter 
formula requires the calculation of one fewer set of coefficients 
/3 8 for a given degree of accuracy with respect to the eccen- 
tricities (cf. 4*31). These formulae are used like those which 
precede it. The derivatives for s = J and all needed values of i 
are computed from either (3) or (4). For the remaining values 
of s, they are used for the computation of the derivatives for the 
two highest values of i only, the remaining derivatives being 
found more easily from 

&&? = <&&?+&/*?. (5) 

These alternative methods of computation furnish obvious checks. 

4*28. The proofs of the preceding formulae are obtained by 
treating the fundamental expansion, 

c 8 (1 4- a 2 - 2/ca cos i/r)- = (a*)*-* 2& (i) cos ty, . . .(1) 
where c f i.f ...(*-!), i = 0, 1, 2, ..., 
as an identity. 



27, 28] PROOFS OP RECURRENCE FORMULAE 109 

The derivative of (1) with respect to f gives 
sc 8 (1 4- a 2 2/ca cos ty)-*- 1 . 2/ea sin ty = (*)*"* 2/3 fi (?) i sin if. 

...... (2) 

Replace 5 by s 4- 1 in (1) and insert the result in the left-hand 
member of (2). Since c s +i = sc 8 , we obtain 

2 sin i/r 2/3^ cos ty = 2/3^ i sin ty. 

The left-hand member of this equation may be expressed as a 
sum of sines of multiples of f. Equating the coefficients of 
sin i\|r, we obtain 

' 



which is the formula 4'26 (6). The derivatives of this give 4'27 (2), 
4-27 (5). 

Again, multiply (2) by 1 + a 2 - 2*a cos i/r and insert (1) in the 
left-hand member of the result. After some reduction, we obtain 
Ka S (i + s) $M sin (i + 1) ^ + *a2 (i - s) /3 8 (i) sin (i - 1) ^ 

= (1 +a a )SiA (t) sini>. 

In each of these series i takes all integral values from + oo to 
oo . The selection of the coefficients of sin (i + 1) tjr, from each 
of them, gives 

*a (i + s) &* +ica(i+2- s) ^ +2) = (1 + a 2 ) (i + 1) #' +1) , 
which is the same as 4*26 (4) when we put e = (1 + a 2 )/ tea. 
Finally, the identity 

(1 + a 2 - 2*a cos ^) S ^^ cos ty = 6-a/t 2 ^ cos if, 

obtained from (1), with the same equation when s + 1 is put for 
s, yields by the same procedure as before, 

) ....... (4) 



If we successively eliminate $+i\ff*+^ between this equation 
and (3), we obtain 



Change i into i 1 in the former of these equations and add 
the result to and subtract it from the latter; the two equations 
thus obtained are the same as 4-26 (5). 



110 DEVELOPMENT OF DISTURBING FUNCTION [CH. iv 

4'29. The values of the derivatives are obtained as follows. 
The derivative with respect to a of the logarithm of 4*23 (1) 
gives, after multiplication by a, 

2s (a 2 /ca cos ^) 
1 -f- a 2 - 2*a cos ^ 
<?n a 2 \ 

o ^ JL ^^ 1* / 



2 1 + 2 2/tra cos ty ' 
But, from the definition of Rzs, 



+ a 2 2/ea cos ^ 



Ktt. 



I a 2 
Hence 01^ = ^^+ - ^2*^2 

KCd 

From this equation, by replacing R^ 8 , R^ s ^ by their expansions 
in Fourier series and equating the coefficients of cos ity, we 
obtain 



which is the equation 4'27 (1). 

The application of the operator D j to this with the help of the 
general theorem, 

D'{oflf(a)}=<#(D + qy.f(a), ............ (1) 

furnishes the equation 4*27 (3). 

4'30. The proof of the relation 4'27 (4) is more difficult. If 

we put 

p = 1 + a 2 2a/c cos yfr, .................. (1) 

we have 



(Dp) 2 + \ = (2a 2 - 2a/ccos >|r) 2 4- 4a 2 /c 2 sin 2 ,fr 
= 4V-f 4a a (/c 2 -l), 

s-A = 4a 2 2a/ cos ->|r -f 2a/t cos ty = 4a 2 . 
dy^ 

Also J5V' = 



29-31] PROOFS OF RECURRENCE FORMULAE 111 

Whence, with the aid of the two previous equations, 

= * ( s 

(s -f- 



...... (2) 

Now, the definition (1) and 4'23 (1) give 

P~' -(a*)*- RI. + C. ................... (3) 

and the theorem 4'29 (1), applied to this, gives 



Insert this result in the left-hand member of (2) and eliminate 
/> from the remaining terms of (2) by the use of (3) after re- 
placing s by s + 1, s + 2, therein. With the help of the relations 

o 8+ i = sc 8 , c s+ z = s(s+ 1) c st 
we obtain, after division by suitable factors, 

(D + * -s) 2 

. 

The final step is the insertion of the Fourier series and the 
equating of the coefficients of cos ity. This process gives 



which is easily seen to be the same as equation 4*27 (4) when 
we putj= in the latter. The result may be written 



The application of the operator D i to this equation and the use 
of the theorem 4-29 (1) give 4'27 (4). 

4*31. The statement in 4'27 contrasting the formulae 4*27 (3), 
4*27 (4), may be justified as follows. The continued use of 
4'27 (4) makes D*/3 8 depend on the calculation ot/3 8 +j,{ii+j-i 9 . . .,&. 
If we put j = in this equation, the right-hand member depends 
on Dftg, j3 s +i which require the calculation of &+i, /8 only; the 



112 DEVELOPMENT OF DISTURBING FUNCTION [OH. nr 



factor of /9, f 2 is 4 tan 2 \I (1 + sec 2 /). Since the operator D 2 
is always accompanied by the square of the eccentricities in the 
development of the disturbing function, the effect of this last 
term is of the fourth order. The argument for values of j greater 
than zero in the formula is similar. 

Incidentally, this formula shows why a considerable increase 
in the convergence along powers of the inclination is obtained 
by the insertion of the factor tc. In general, the coefficients /3 8 +} 
tend to increase with j for a given value of i and the factor 4> 
which occurs in this term shows that it will modify the values 
of the derivatives considerably when / is large. The additional 
computation caused by its presence is very small *. 

4'32. The literal expansion of I/A to the second order, in terms 
of the mean anomalies. 

This expansion is obtained from that in terms of the true 
anomalies given in detail in 4*14. The latter contains products 
of cosines which are expressed as sums of cosines as explained in 
that paragraph. In writing the result out to the second order, 
the notation 

tf^a^a-t, 4 = 6, -6-,, /-/' + w -*r' = *-', 

will be used. The result is 

r * 

= sum for all positive and negative values of i, including 

zero, of 
(1 + ^ L 2 + 77 /2 (D + 1) 2 } a< cos i (v - 1/) 

- 27777' (D 2 + D) ai cos (iv - iV + /- /') 

- 277 Da { cos (iv - iv' +/) + 277' (D + 1) a; cos (it; - iv' + /') 
4- T? 2 (D 2 + D) at cos (iv - iv' + 2/) 

+ 77 /2 (D 2 + D) a< cos (iv - tV + 2/ x ) 

- 27777' (D 2 + D) a t cos (iv - iv' +/+/') 

-20) ................ (1) 



* These modifications of the usual formulae in which * = 1, were given by 
E. W. Brown, Mon. Not. R.A.S. vol. 88, pp. 459-465. 



31, 32] TO THE SECOND ORDER 113 

The transformation to mean anomalies is made by means of 
the theorem 415 (1), namely, 

cos (tf+j'f + 0) - exp. (EJ-+E' g2>) cos (jg + j'g' + 0), 

where f-g + E, f'=g' + E'. 

The expression for Z? is given in 3*16. Expressed in terms of 
77 as far as the second order, it is 

E = 4*77 sin g -f 5?? a sin 2jr, 

and we have a similar form for E' . From these we deduce 
-2 = 8?? 2 _ 8?? 2 cos 2( /) EE'=8w' {cos (g - #') - cos (g + #')}. 
With the aid of these formulae and of the expansion 



exp. tf + 

3 

/ 



we obtain the following development: 



= t x sum for i 0, 1, 2, ... of 

* 



{1 -f 7? 2 (D 2 + 4D - 4i a ) + V 2 ( 2 - 2D - 4i a - 3)} a< cos (iw - tw') 

- 2777;' (D 2 -f D - 4i 2 - 2i) a< cos (iw - iw' +g-g') 
H- 2^7 ( D + 2t') a< cos (iw iw 7 4- g) 

+ 27?' (D + 1 - 2i) % cos (iw - I'M;' + g') 

-f ?7 2 (D 2 - 4iD - 3D + 4i 2 + 5i) a< cos (iw - iw' -f 2#) 

+ 7/ /2 (D 2 - 4iD + 5Z> + 4i a - 9i -f 4) a, cos (iw - iw' 4- 20') 

- 27777' (D 2 - 4iD -f D -f 4i 2 - 2i) a< cos (iw - iw' + g + g') 
+ 2tan 2 J/.6 < cos(iw-iw' + w-f w'-20), ............... (2) 

in which the notations 

w, w' = mean longitudes = g + r, ^' -f w' 
are used. 

B&8PT 8 



114 DEVELOPMENT OF DISTURBING FUNCTION [OH. iv 

For convenient reference, we repeat the significations of the 
remaining symbols used in this development: 



{1 H- a 2 2a cos 2 / cos (w tt/)}"~^ = 2a,- cos f (w w') t . . .(3) 
{a cos 2 / (1 + a 2 - 2a cos 2 |/cos (w w')}~ = 26* cos i (w - w') t 

...... (4) 

where i = 0, 1, 2, Also, to the second order, 77 = ^, 77'=^'. 

These definitions show at a glance the differences between 

the development given above and that of Leverrier, given by 

Tisserand* in which r ' = a' r Q = a. D* = .-, ( r- ) , cos 2 A/= 1 in 

j ! \oa/ 

(3), (4), a< is replaced by %a { and b t by |6 rf , and there is no 
factor ^ in the left-hand member of (4). When the necessary 
changes in notation have been made, the two developments will 
be found to agree with one another. 

4*33. The second term of R. 

The first term of the disturbing function, namely, w'/A, is the 
same for the action of either planet on the other, except as to 
the mass factor; the only condition actually used in the develop- 
ment is that accented letters shall refer to the outer planet. 
Omitting the mass factor, the second term is 

r cos 8 r f cos S 

~/r~~ or ^ > 

according as we are dealing with the action of the outer planet 
on the inner or that of the inner on the outer. 

Expressed in rectangular coordinates these expressions are 

xx' + yy r + zz* xx' 4- yy' 4- zz' 
>s > ^3 

The equations for the elliptic motions of the planets are 



* Mte. Ctl. vol. 1, p. 309. 



32, 33] THE SECOND TERM 115 

If we put ///=n' 2 a' 8 , n'dt dg' in the former and /-t = n 2 a 8 , 
dg in the latter, we may write these 



Since a;', y', z' do not contain $r and x, y, z do not contain g' t 
the two cases of the second term of R may be written 



which give a function rr f cos S, symmetrical with respect to the 
two planets, to be developed. 

These forms of the second term show that in the first case, 
there are no terms in the development in terms of the mean 
anomalies which contain the argument g only, and none in the 
second case containing g' only. 

In neither case does the second term produce a constant 
part. 

A quite general development in terms of the mean anomalies 
can be made. With the form 4*1 (3) for cos S, it is evident that 
rr* cos S is a linear function of r cos v, r sin v and therefore of 
r cos^ r sin/, and similarly of r' cos/', r' sin/', and these 
functions can be expressed in terms of the mean anomalies by 
the formulae 310 (5). An easy way to carry out the calculation 
is to use the second form of 4*1 (3) and write 

rr' cos S = A' . r cos/ B' . r sin/ 

A' aiT^'cos. ,. . 9 . T r f cos , , , _ m 
ra , = cosH7 , . (r - v ) + sin 2 A/ , . O + v -20). 
B ^ r sm v ' 2 r sm v ' 

The series for A', B' may be computed by harmonic analysis, or 
in series from 

A' = Cx . r' cos/' + O a . r' sin/', JB' - ~ C 3 . r' sin/' + C 4 . r' cos/', 

where 

Ci,C 3 = COS 2 |J COS (r - r') sin 2 / COS (tsr + r' - 20), 
<7 2 , C f 4 = cos 2 J/sin (r - w') T sin 2 \I sin ( + *? - 20). 

8-2 



116 DEVELOPMENT OF DISTURBING FUNCTION [OH. iv 

4*34. The expansion of r cos S/r' 2 can also be obtained from 
that of I/A in the following manner. If we put f (<) = except 
for the cases i=l, s = ; i = 0, s=|, and for these put the 
value - %a/c, so that 

ai = a_i = b Q = - \aic = - %a cos 2 17, ......... (1) 

we find from 413 (4), 414 (1), that 



With these values 413 (5) becomes 

-r- = a/c cos (v i/) a* tan 2 \I cos (v -f v' 20) = a cos 
^i 

Finally if this be substituted in 4*8 (3) with D = 1 , we have 



Thus the required expansion is obtained from that of I/A by 
putting D = 1 and making the substitution (1). 
To the second order, with 



we obtain from 4*32 (2), the result: 

rcosS acos 2 i/ (/ . rt 2 /ax . . ,. 

-- ~7a- = -- -7a A -{(l~27; 2 -27/ 2 )cos(^-/-ht ! r^- S r / ) 

/ iL 

-f ?; COS (2g ^' -f cr isr') 3?; COS (</' -or -f tir') 

-f 4^ COS (^ - 20' -f tr - -or') + 4^' COS (2^r - 20' 4- or - <cr') 

-f |r; 2 COS (3<7 - g f -f tar - ') + i^ 2 COS (^ 4- flf' - tar + tsr') 

+ |i?' 2 cos (^r + #' + tzr -'cr')+^ 7 -7 ? ' 2 cos(5r~3/-f ^ --cj') 

- 121777' cos (2/ - ar + w') 

+ tan 2 \l cos (gr -h g' -f r + -cr' - 2(9)}. 

The expansion of r' cos S/r 2 is evidently obtained by inter- 
changing the accents. 



CHAPTER V 

CANONICAL AND ELLIPTIC VARIABLES 

A. THE CONTACT TRANSFORMATION 

5*1. Canonical differential equations. 

Let the coordinates of a particle of mass ra at any time t 
be #1, #2 #3- If this particle moves under the force -function 
U (#1, %2, #3, t), the differential equations of its motion are 

d 2 ^_atr -.190 

These are three equations of the second order. We shall now 
express them as six equations of the first order. This is 
accomplished by the introduction of the three new variables 
2/i> ^2> 2/3 > called momenta, defined by 

The kinetic energy is 



Differentiating (2) and substituting in (1) we have 

%-% '-''' 

From (2) and (3) we have 



Since CT and T are independent of y t and < respectively, (4) and 
(5) may be written, with HTU, 



dt 



118 CANONICAL AND ELLIPTIC VARIABLES [OH. v 

A set of differential equations in this form is said to be in the 
canonical or Hamiltonian form, and H is called the Hamiltonian 
function. 

On account of the definitions of x iy y t as independent variables, 
the equations (5) may be expressed in the symbolic form, 

2(<fo,.8y<-dy.&,) = (ft.5ff, (7) 

where the &Cj, Sy t are arbitrary variations of the & iy y iy and SH 
is the consequent variation of H. The definitions of the symbols 
d, S, introduced in this manner, will be made more precise in 
5*3 below. 

5'2. The Contact transformation. 

Let a i9 yt be any 2n variables which satisfy the canonical 
differential equations, 

dxt_m d yi _ dH . m 

dt'dyt' dt~ a*,' *- 1 >4-> w > W 

where H is a function of ic it y,, t only: it does not contain any 
derivatives. A contact transformation is a change from the 2n 
variables x it y i to 2n new variables, p it q it which shall satisfy 
equations of the same form, namely, 

dp, JIT dq i=: _dHr 

dt dq,' dt dp, ' v ~' 

where H' is related to H by an equation to be given below, and 
is expressed as a function of p iy q it t. 

According to a theorem of Jacobi, relations between the old 
and new variables which fulfil the conditions can be expressed 
by the implicit equations* 

dS dS ct r j. j. /o\ 
2/< = g^> Pt^faS S = funct.a? < ,9 < ,t (3) 

The determining function 8 must thus be expressed as a 
function of one set, either the x t or the y t , of the old variables, 
and one set of the new. It must be so chosen that it is possible, 

* No confusion will be caused between this use of the letter S and that in the 
development of the disturbing function. 



1-3] CONTACT TRANSFORMATION 119 

by means of the equations (3), to express the x it y t in terms of 
the pi, q t , or vice versa. This possibility depends on the Jacobian, 



.(4) 



which must not vanish identically. 

The literature connected with canonical equations and contact trans- 
formations is extensive and can be found from the usual sources of 
information. It may be mentioned, however, that while Hamilton appears 
to have first given the canonical forms of the equations of motion (Brit. 
Ass. Report, 1834, p. 513), Lagrange had given the equations for the 
variations of the elliptic elements in 1809 (M4m. de VInst. de Paris, p. 343) 
in this form. The theorem of Jacobi appeared in the Comptes Rendus for 
1837, p. 61, the contact transformation having been introduced by Hamil- 
ton in 1828 (Trans. Roy. Irish A cad. vol. 15, p. 69). 

Of the numerous applications of the theory of contact transformations 
we shall give only those which are necessary for the later developments in 
this volume. In particular, the proof of Jacobi 's theorem, given in 5*3, 
docs not indicate the process of discovery, but it has the advantage of 
showing immediately, not only the relation between //, H f , $, but also the 
method which appears to be most useful in the search for new forms of 
canonical variables. 

5*3. Proof of the Jacobian transformation theorem. 

For the purposes of this proof and of later developments, it is 
desirable to define in more detail the meanings to be attached 
to the symbols d, 8 in equations involving differentials. 

The equations 5*2 (1) imply the existence of solutions of the 
form 

i = x i (*>> %> <%>), Vi = Hi (*> a l> ' ' ' > a 2)> 

and the relations 5*2 (3), the existence of functions, 

which are solutions of 5*2 (3); in these expressions, aj ,..., a an 
are the arbitrary constants of the solution. 

The symbol d attached to any function will always denote 
that when the function has been expressed in terms of t and 
the a ry it is t alone which is varied, while the symbol 8 implies 



120 CANONICAL AND ELLIPTIC VARIABLES [OH. v 

that any or all of the a r are varied but that t is not changed. 
Thus when x it y t are expressed in the forms just set down, 



and thence, when 8 is expressed as a function of x iy q iy t, 



In these expressions for d$, 8$, the meanings to be attached 
to dx iy 8&i are those just given; similar meanings are to be 
attached to dq t , Sq it 

Since the variations denoted by d, S are independent, the 
commutative law, namely, that 8 . d and d . S acting on any 
function produce the same result, is satisfied. 

The proof that the relations 5*2 (3) transform 5*2 (1) into 
5'2 (2) follows. 

Multiply 5*2 (1) by %*, 8^, respectively, and add for all 
values of i. We obtain 



A similar process performed with 5*2 (3) gives 

85, ......... (2) 



*/ j j x * fiS j dS , \ , a/S . 

2 (yidxt+pidqt) = 2 ^ Ac, + g- d^J = dfif - -^ d$, 

......... (3) 

since S may contain t explicitly as well as through a-^q^ 

Operate on (2) with d and on (3) with 8 and subtract. Since 
the operators d, 8 are commutative, we have dBx t = 8d^ t -, 
d8S= 8dS, etc., and all the terms in which both 8, d act on the 

same function disappear. We obtain 

/dS\ 
X (dy ( Se t - dc, 8 y< ) + 2 (dp, S 3< - dg, 8 P< ) - c . 8 ^ J , 



3, 4] JACOBFS THEOREM 121 

which is the same as 



The addition of (1) to this last equation gives 

*(t"--$*)-'(*+D 

Finally, if we define H 1 by means of the equation 



(5) 



and suppose that H' has been expressed in terms of p it q it t, by 
means of the relations 5*2 (3), so that 



the independence of Sp it Sq t furnishes, through the equality of 
the members of (4), (6), 



^ 

dt ~dq t ' dt d Pi ................... 

In the use of this transformation, it is important to remember 
that dS/dt is found from the expression for S in 5*2 (3) and that 
H' is to be expressed in terms of p^ qi, t. 

B. JACOBI'S PARTIAL DIFFERENTIAL EQUATIONS 

5'4. The transformation theorem just proved is a device for 
changing from one set of variables to another, the new variables 
depending on the choice of the determining function 8. One 
such choice is the following. 

Suppose that it is possible to find a form of 8 which will 
make the derivatives dH'/dp iy dH f /dq { zero. (Since H' must 
necessarily be expressed in terms of p^ q if t before these deriva- 
tives are formed, it follows that the derivatives will then be 
identically zero.) The equations 5'2 (2) show that dpjdt, dqjdt 
become zero, and hence that 

Pi = const., qi = const. 



122 CANONICAL AND ELLIPTIC VARIABLES [OH. v 

Jacobi showed that when S is so determined as to satisfy 
these conditions, it is a solution of a certain partial differential 
equation. In the next section, this equation will be found and 
those properties of it, which will be useful later, will be developed. 

Notation. A semi-colon following a collective symbol x i> 
where i 1, 2, ..., n, will denote that any or all of the x t may 
be present, thus 

t) means /Oi, #2, -,#> 0> 
o 



5*5. The equation and its solutions. 

Amongst the values of S which will make p t , ^ constant, 
we seek one which makes H' = 0, that is, one for which 



But H was originally a function of x it y t -, t, and, by 5'2 (3), 
y i = dS/dtti. Thus, we seek a value of S satisfying 



Now the assumption concerning the form of ti was that it 
should be expressed as a function of x it q it t, and the assumption 
is to be retained in (1). But, in the present case, the q t are 
constants by hypothesis. Hence, in order to satisfy (1), we 
need an expression for S which contains x iy t, and n arbitrary 
constants q { . In other words, if we regard (1) as a partial 
differential equation with x iy t as independent variables and 
with S as the dependent variable, we need a solution of the 
equation containing n arbitrary constants. 

When such a solution has been obtained, all that is necessary 
is to interpret the relation 5*2 (3) in the language of the theory 
of differential equations, remembering that these now constitute 
2n relations between the original variables #,-, y if the constants 
Pi,qi, and t. 



4, 5] JACOBIAN SOLUTION 123 

Theorem. A general solution of the equations 



is provided by the equations 

dS 



where p it #, are arbitrary constants, and S is an integral, contain- 
ing n arbitrary constants q* (exclusive of that additive to S) of 
the partial differential equation 



This type of integral is known as a complete integral, for 
the theory of which the reader is referred to treatises on first 
order partial differential equations. For our purposes, it is 
sufficient to state that the Jacobian 5*2 (4) must not vanish 
identically. 

The constant additive to S plays no part because S appears in the 
differential equations and in the solution only through its derivatives, but 
its presence is theoretically necessary since there are n-f 1 independent 
variables in the partial differential equation. 

It may be pointed out that the ordinary method for the solution of 
first order partial differential equations simply leads back to the canonical 
equations, so that nothing is gained by attempting to use it. In the 
applications to celestial mechanics, the form of the function S is usually 
set down from previous knowledge of the form of the solution, and the 
relations 5*2 (3) are then used in various ways ; or, in particular cases, a 
form for S may be suggested by the equation 5'5 (1). 

The set of arbitrary constants which appears in this way is 
known as a canonical set. It is evident that when the function 
S, containing half of them, has been obtained, the remaining 
half are chosen from the relations p t = dS/dqi. In general, there- 
fore, only n of the 2n constants may be chosen arbitrarily. The 
practical demands of the perturbation problem limit the choice 
to very few types. See 5*12, 5*13, 5*14. 



124 CANONICAL AND ELLIPTIC VARIABLES [OH. v 

5 '6. The case where H is independent of t explicitly. 
A start may be made by assuming that 

S^Si+Ct, ........................ (1) 

so that the partial differential equation is reduced to 

<>-<> ................... < 2 > 



which contains n independent variables only. If C be chosen as 
one of the canonical set of constants q iy and t Q be the corresponding 
constant derived from the equation p i = dS/dq i) we have 

<-w +t ......................... < 3 > 

Since Si does not contain t, it follows that the solution will 
contain t, t Q only in the form t t Q . 

The equation (2) shows that H = constant is an integral of the 
equations. This may be proved directly from the canonical 
equations 5'2 (1), by multiplying them by dyi/dt, dxjdt, and 
adding. 

A similar procedure may be adopted when any one of the 
coordinates x t is absent from H. It is evident that each absent 
coordinate permits the writing down of an integral of the 
equations. 

5*7. Application to the perturbation problem. 

The force-functions for the problem of three bodies which 
have been constructed in Chap, i, have usually been divided 
into the sum of two parts, the first of which, taken alone, gives 
elliptic motion. This division would lead to putting 17= UQ + -R, 
where UQ = /A/r. It has been adopted because we can solve the 
equations completely when R = 0, and it has the added advan- 
tage that, since jR usually contains a small factor, it constitutes 
a first approximation to the motion. These considerations, 
however, do not limit the applicability of the following method 
of procedure. 

If U is replaced by UQ + J?, H is replaced by T UQ R, or 



6-8] VARIATION OF ARBITRARIES 126 

by HQ R if HQ = T - UQ, so that the canonical equations will be 
written 

dxi d rr p dy t % (if p\ /i\ 

- = -^(H,-R), -^-(Ht-R) (1) 

Let us transform to a set of new variables p t , q t by means of 
the relations 

ds as ,. 



where S is a solution of the equation 

ds 



The transformation theorem in 5*2 shows that the equations 
satisfied by the new variables are 



dt dqt' dt dp^ 

for in this case we have, by (3), 

, da__x 

/*-/* ^+g- ^ 

The interpretation of this result, usually adopted, is the 
following. If we solve the equations with H = and obtain a 
canonical set of constants p iy q it and consider these constants as 
variables when 2? 4= 0, the equations which they will satisfy are 
those numbered (4); hence the phrase, Variation of Arbitrary 
Constants. This latter point of view is useful in geometrical 
descriptions of the motion, but it sometimes leads to confusion 
and error if it is adopted in the analytical work. As*a matter of 
fact, the equations with R = are solved mainly in order to 
indicate the choice of the new variables, and are not used unless 
we know in advance that a solution can be obtained. 

5 '8. Osculating orbits. 

Another geometrical interpretation of the results of 5*7 is of 
value in the determination of the orbit of a body from observa- 
tions of its position. Let us suppose that the problem of finding 



126 CANONICAL AND ELLIPTIC VARIABLES [CH. v 

the solution of 5*7 (4) has been solved, so that the variables p it 
q t are expressed as functions of t and arbitrary constants. If we 
put t = t Q in this solution, where fo is some particular value of t y 
these variables become constants and thus constitute a solution 
of the equations 5 f 7 (4) when R = 0. Hence at the instant 
t = fo, the variables and the constants have the same value. But 
the coordinates and velocities are expressible in terms of t 
and Pi> q it It follows that the orbits with jffi = 0, R^O, in- 
tersect at t =s t Q and have the same velocities at the point of 
intersection; when this happens the two orbits are said to 
osculate at that point, and the ellipse described with R = is 
called the osculating ellipse*. 

If at the instant t tv, the disturbing forces which arise 
through R were suddenly annihilated, the body would thereafter 
proceed to move in the osculating ellipse. This constitutes 
another definition of this curve. 

In the great majority of cases arising in the solar system, the forces due 
to R are small compared with those present when 72=0, so that the oscu- 
lating ellipse constitutes a good approximation to the orbit at times near 
= <) In the case of a planet, the separation is small during a period of 
one revolution of the planet round the sun. Thus the osculating ellipse 
can be used to predict approximately the place of the body for some time 
before and after the instant t = t (} . 

Ordinarily two coordinates, which give the angular position as seen from 
the earth, are observed ; neither the distance nor the velocities are directly 
observed. From three such observations an osculating ellipse can in 
general be deduced. A position predicted for some other time in order to 
limit the area of search also needs only the two coordinates. There are six 
constants present in the osculating ellipse, and according to the mathe- 
matical theory of the approximate representation of a curve, considerable 
variations may be made in the six constants without altering the two 
needed functions of them very greatly. Thus the elements of an osculating 
orbit may be considerably in error and yet it may furnish a good search 
ephemeris for a considerable interval following its determination. 

* Since the curvatures are not in general the same, the word 'osculating' is 
not used in the same sense as in the theory of curves. 



S, 9] JACOBIAN METHOD OF SOLUTION 127 

C. JACOBIAN METHOD OF SOLUTION 

5*9. Solution of the case of elliptic motion by the Jacobian 
method. 

The force -function in this case is m/i/r, and the canonical 
equations of motion become, after division by m, 

dxi __ dHo dyt _ 9#o 
dt 9y t - ' dt dxt ' 
where 



The division by m is analytically equivalent to putting m = 1 
in the formulae of 5*1. 

According to 5*5, the equation satisfied by S is 



MIS) +l w-., 

^ +*,+*,) ^ 

Transform to the tri-polar coordinates r, TT ~ i, X, so that L 
is the latitude, and X the longitude of the projection of r on the 
plane of reference. The equation becomes 

dS ,. ,~. 



According to the preceding theory, we need an integral of 
(2) containing three arbitrary constants, exclusive of that 
additive to S. Since t> \ enter only through derivatives, it is 
convenient to put 

8=-ait + a*\ + Si, .................. (3) 

where Si is independent of X, t, and i, a a are constants. 
Inserting this value of S in (2) we obtain, after multiplication 
by 2r^ and rearrangement, 

-- 1 ....... <> 



The form of this equation indicates that we can obtain a 
solution by putting Si = S$ + Sz, where S% is a function of r only 



128 CANONICAL AND ELLIPTIC VARIABLES [CH. v 

and $3 that of L ouly, if we equate each member to a constant 
a 2 2 . This procedure gives 

. 2 a 2 2 /9$3\ 2 3 2 
= 2i 4- =- . I =-= i = a* 2 ^-, . 



\dL ) UU&-JL/ 

The integration of these equations will leave two arbitrary 
constants at our disposal since the necessary three arbitrary 
constants i, 2 , a 3 are already present. Let $ 2 vanish for r = ri> 
where TI is the smaller root of the equation 

_ 2jjL a 2 2 ,, /f . 

2ai-f -^ a = 0, (5) 

* /M /V1<fi ' ^ ' 



and let 3 vanish when // = 0. In order that this value of r 
may be positive, it is necessary that both roots be positive; 
hence a must be negative. 

Inserting the values of jS> 2 , 63, $1, thus obtained, in (3) we 
obtain a solution in the required form : 



...... (6) 

The next step is the deduction of the solution of the canonical 
equations by means of the relations & = 9S/3a$, y i 'dSldx iy 
which would seem to demand a return to rectangular coordinates. 
However, we do not need the latter set since the former gives the 
necessary three relations between r, L, X, t, and the relations 
between r, L t \ and #1, # 2 , #3, are independent of a iy 2 , 3 . 

The derivatives of 8 with respect to the a f are obtained 
without carrying out the quadratures in (6), by means of the 
formula 



The derivative with respect to i gives 

ft + 



since the coefficient of dri/da vanishes on account of (5). 



9, 10] ELLIPTIC CANONICAL CONSTANTS 129 

Similarly, 

L 

' 



The relations (7), (8), (9) are those needed, and the a, , /3< con- 
stitute a set of canonical constants. 

5*10. Relations between the set a iy fit and a or n y e, i, e, w, 6. 

Since the integrand of 5*9 (7) must be real, the equation 
5 '9 (5) gives the maximum and minimum values of r. In 3 '2 
these are shown to be a (1 e). Hence from a well-known 
theorem connecting the roots and coefficients of a quadratic 
equation, 2u/2! = 2a, 2 2 /2i = a* 2 (1 e 2 ), giving c&i = 



According to 5*9 (7), /3j is the value of t when r=r 1 =a(l e). 
But the mean anomaly nt + e w is zero for this value of r. 
Hence /3i = (e w)/n. 

Equation 5*9 (9) shows that /3 3 is the value of \ when L = 0, 
that is, when the body is in the plane of reference. Hence 
y3 3 = 0, the longitude of the node. 

Since the integrand of 5*9 (9) must be real, the maximum 
and minimum values of cos L are 3 /a 2 . But the maximum 
and minimum values of L are i, where i is the inclination 
of the plane of the orbit to the plane of reference. Hence 
3 s= 2 cos i. 

Finally, if we put a 3 = a 2 cosi in the last term of 5*9 (3) it 
becomes 

[ L cos LdL __ C L d (sin L) __ . ^ /sin L\ 
Jo (cos 2 Z - cos 2 z')* ~ Jo (sin 2 ! - sin 2 i)* ~ &m \ am?/' 

But if v be the hypotenuse of the right-angled spherical 
triangle in which L is the side opposite to the angle i 9 we have 
sin L = sin i sin v. Thus the above integral is v and the equation 

B&SPT 9 



130 CANONICAL AND ELLIPTIC VARIABLES [CH. v 

5*9 (8) shows that /9 2 is the value of v when r = n. Since v 
is then the angle between the apse and the node, we have 

$2 = & # 

Collecting these results, we obtain the system of canonical 
elements 

i=(e- 




Hence, according to the principles set forth above, the 
equations satisfied by the a,-, & when R 4= 0, are 

2 (da, 8& - d& 8a f ) = eft . 812, (2) 

or, written in extenso, 

^ m 4& p 

<ft d& eft 9a t 

5*11. The fact that 5'9 (7), (8), (9) give elliptic motion rn\y be deduced 
in the following manner. 

If we put r=a aecos A" in (7), and insert the values* of the constants 
found above, we obtain, after integration, 



n 

which, with the aid of the relation a?n?=p, is seen to be the equation con- 
necting the mean and eccentric anomalies. 

Equation 5*9 (8), with the same substitution for r, becomes 

[X dX 

l~e cos X ' 

since v = v 6, f vw, this equation gives the relation between the true 
and eccentric anomalies. 

Finally, the substitution sin Z = sin i sin v in the integral of (9) gives 

sst X - tan*" 1 (cos i tan i/) ; 

since X 6 is the side adjacent to the angle i in the right-angled triangle 
to which reference is made in the text, this equation merely constitutes a 
well-known geometrical relation. 

A logical procedure requires the proof that the motion is elliptic to 
precede the identification of the constants. But as the objective in view is 
the discovery of a set of canonical constants only, we assumed that the 
nature of the motion had been found by the easier method of Chap. in. 
(Cf. 3-2.) 



10-12] DELAUNAY'S ELEMENTS 131 

D. OTHER CANONICAL AND NON-CANONICAL SETS 

5*12. Delaunays canonical elements. 

Changes from one canonical set of elements a i , /^ to another 
such set can sometimes be carried through easily by the 
Jacobian transformation theorem proved in 5*3 if we use as 
the determining function 

S = 2a,A, ........................ (1) 

where the /3 t (or the a t ) are expressed in terms of half the 
variables of the new set. 

Let us take as three of the variables of a new set, I, g, h, 
defined by 



so that i = /*(- 2ax)~^ - t. 

Hence S, expressed in the required form, is 

S = ^(-2a 1 )^l-a l t-^a 2 g + a B h .......... (2) 

If the other three new variables be L y 0, H, we have, by 
5'2 (3), 

r d$ /- ^ \ i / - ^ 9$ rr ^S 

i = _ =/t (_2a 1 ri=V M a ) (?== # = ^ = 3. 

...... (3) 

The remaining three equations are automatically satisfied. Also 

H'-H + -R-*-R + ............. (4) 

Thus this set, which is that used by Delaunay, satisfies the 
equation 



_ ...... (5) 

where L = V 'pa, I = nt + e tsr,! 

...... (6) 



cos 

9-2 



132 CANONICAL AND ELLIPTIC VARIABLES [OH. v 

The variables in each of the two groups are homogeneous; 
L, 0, H are angular momenta or, since the mass acted on has 
been divided out, areal velocities. If we put /JL = n 2 a 8 , the 
common factor "Ja/j, becomes na 2 . 

5*13. The modified Delaunay set. 

If we use the determining function S Ll + Gg + Hh, in the 
form 



the equations of transformation show that L, G L, HG and 
I -f g -f h, g + h, h form a canonical set. We shall denote them 
by c i? w it so that 

d = v^a, ivi = nt -f e = mean longitude,] 

c 2 = Vyu,a (Vl (? 1), w 2 ~ TZ = l n g- of apse, 

c 3 = \/y^a (1 e 2 ) (cos i 1), w^ long, of node. 

...... (1) 

The Hamiltonian function is unaltered and is equal to R -h yu, 2 /2ci 2 . 
When expansion is made in powers of e, i, the element c 2 is 
divisible by e 2 and c$ by i* properties which make this set 
useful in planetary problems. 

5*14. A set given by Poincare'*. 

This set is ci, p%, PB', w, q%, qs> defined by 



p 2 = \/ 2c 2 sin w, q ?i = V 2c 2 cos sr, 
jt) 3 V- 2c 3 sin 6, </ 3 = V 2c 3 cos ^, 

where 02, CB are defined in 5*13 (1). That it is canonical can be 
tested by showing that 

dpiSqz dq^pz = dczBw 2 div z Sc 2t 

with a similar equation for the suffix 3. 

This is a particular case of a general theorem f which states 

* Les Nouv. Mtt. de la Mtc. Cel. vol. 1, p. 30. 
f C. A. Shook, I.e. in 5-5. 



12-15] tfON-CANONICAL SETS 133 

that if two variables p 2 , fa are related to two canonical variables 
C 2 , w 2 in such a manner that the Jacobian, 



p z , qz are also canonical variables. 

The Hamiltonian function is E -f /* 2 /2ci 2 , as before. 

This set is useful because the approximate values of p 2 , q% are 

V/t . e siri -GT, V/xa , e cos t*j, 
when e 8 is neglected, and those of ^> 3 , f/ 3 are 



e 2 ) . i sin 0, V/xa ( 1 e 2 ) . i cos 0, 
when i 3 is neglected, so that the disturbing function is develop- 
able in powers of p 2) p$, 72 , </3> such a development replacing 
powers of e, i, and cosines and sines of multiples of -or, 6. The 
possibility of such a development depends on the association of 
powers of e with multiples of r, and of powers of i with multiples 
of when the angles are expressed in terms of tsr, r', and the 
mean longitudes. See 4*15, 



5*15. The non-canonical set a, e it PI, w, r, 0, ?0A0re M; = H 4- e, 

e x - s/2 (1 - \/l - e*) = ^ + ^e 3 ..., T! = (1 - cos i) Vl - e 2 . 

...... (1) 

The disturbing function is not usually expressed in terms of 
the preceding canonical sets of elements. The angular variables 
are present explicitly, but the remaining elements are mixtures 
of a, e, i which are present explicitly. The same is true of the 
coordinates. For calculation, it is usually easier to adopt elements 
which are more directly related to those which are explicitly 
present in the ordinary developments of R and of the coordinates. 
The equations satisfied by such elements are not canonical. 

The definitions of e\, FI given in (1) show that these variables 
are related to d, c 2 , c 3 by the equations 

c x =V/ua, c 2 =- iei 2 V/xa, c 3 = Ti^pa ....... (2) 

The variables w, or, are the same as those denoted in 5'13 (1) 
by wi, u'2, w^. 



134 



CANONICAL AND ELLIPTIC VARIABLES [CH. v 



The transformation to the new variables is most easily effected 
by forming their variations from (2) and substituting them in 
the left-hand member of the equation, 



The process is quite straightforward. After the variations 
have been formed, it is convenient to use the equation fju = n 2 a 3 
by putting V 'pa = ulna, V/j/a = p/ncP. After rearrangement, we 
obtain for the left-hand member of (3) the expression 



S0 

multiplied by p/no 2 . 

Since /-c 2 /2ci 2 = /x/2a, we have, if R be supposed expressed 
in terms of the new elements, 



-^ -- o 
3o. 2tt 2 



, 

-^-~ SlV + 
mu oiv 



which is substituted in the right-hand member of (3). 

Since the variations of the new elements are, like those of the 
old elements, independent, we can equate their coefficients on 
the two sides of the equation. On solving the six equations 
thus obtained so as to isolate da/dt, dei/dt, ..., we obtain 

1 da _ _ 2 1 dn = 2na dR 

a dt ~ 3 n dt ? ~~ ^6 dw ' 

dw __ 2?ut a 3JB Jiae^dR naTidR 
dt IUL da %IJL dei p, 3Fi 



_ 
3 dt 



dt 



nadR 
/M 9-or 



nae^dR 



90 



naT^R 

p~~ dw ' 



d9 na dR 



The objective in this transformation is the isolation of deriva- 
tives with respect to a, so that the operator D, which plays so 
large a part in the development of the disturbing function, may 



15, 16] NON-CANONICAL SETS 135 

act only on those portions which are specifically set forth in 
Chap. iv. 

Since R has the dimensions, mass divided by distance, and n 
has the dimension of the inverse of a time, the factor p/no? 
reduces all the equations to relations between ratios. 

The right-hand members of all the equations except the second 
contain the small factor present in R. The value of n found from 
the first equation is to be substituted for the term n of the 
second equation before the latter is integrated. 

The relations 

* = *l(l-l*M 2,;^ 1 (l-iV)-*, ......... (5) 

deduced from (1) and from e = 2iy/(l -f rf\ permit R to be easily 
expressed in terms of e\\ ei e, 2rjei are approximately -J-e 3 , 
so that in many problems it would be possible to neglect these 
differences. It is recalled that e y and therefore e\, is present in 
a as defined in the developments of Chap. IV ; it is also present 
in I\. Hence if R be developed as in that chapter, we have to 
substitute for dR/dei in the equations (4), the value derived from 

/dR dRdT lda\ de 

I ^ -f- f=i ^~ 4- DLL . 1 f - , 
\ce 01 oe a ?e/ de\ 

when dT/de is formed from F = I\ (1 e 2 )~^ and de/dei from (5). 
For dR/dTi we have (dR/dT) (1 -e 2 )-*. 

5*16. The non- canonical set a, e y F or i, w or e, w, 6. 

This set is deduced from 5*15 (4) by means of the relations 
5*15 (1) or 5*15 (5), through the usual process of a change of 
variables from e\ to e and TI to F. The result is the following 
set of equations 



__ ^_ 

a dt 3n dt~~ /A dw ' 
dw 2na dR e*Jl e 2 na dR F na dR 

_ ^ r> . _ fl _ 1 ___ _ ___ ______ _ _ _j_ _ - _ __ 

dt IJL da l-fVl e 2 A 6 de Vl - e 2 P ^F' 

de ,- - o na dR e A/1 & na dR 

_ ._. _ V 1 6 _ _ --" __ __ _ 

dt 



136 CANONICAL ANiD ELLIPTIC VARIABLES [CH. v 

dfff __ i~, nadR F na dR 
~dt~ ~ ^e'de vfT:^ 9F ' 



30 



This set is slightly more convenient than the older form in 
which 61, i replace w, F respectively. 

The definition of 61 is usually given by means of the equation 

w nt -f- e = I n dt -f ei , 

so that dei/dt = dw/dt n. And as e occurs in R and in the co- 
ordinates only in the combination nt -f e, we have dR/Sw = dR/de. 
Nothing is gained by the substitution of ei for w, except perhaps 
a separation, in the case of the long-period terms, of those 
portions which have the square of the small divisor as a factor, 
from those which have the first power of this divisor as a factor. 
The substitution of F = 1 cos i for i is advantageous because 
i occurs in the disturbing function only in the form cost, or 
rather cos / when the orbital plane of the disturbing planet is 
taken as the plane of reference. The older form can be at once 
deduced by means of the relations, 

rfF . di -r, dR . ., . dR 



5' 17. The case of attraction proportional to the distance from 
a fixed origin. 

An example of such a gravitational force is that on any one 
of a spherical arrangement of particles with a mass-density 
which is uniform throughout the sphere. Since such an arrange- 
ment cannot be exactly maintained under the Newtonian laws 
of force and gravitation, the force and resulting motion must be 
considered as approximate only. Another similar example is 
that of an arrangement of particles in an ellipsoid of revolution 



16, 17] FORCE VARYING AS THE DISTANCE 137 

with uniform mass-density: the force on a particle in the 
equatorial plane varies directly as the distance from the centre. 

In the spherical case, the force is pmr, where m is the mass 
of the particle, r its distance from the origin and /j, is 4?r/3 times 
the density multiplied by the gravitational constant. In the 
ellipsoidal case, //, depends also on the eccentricity of the 
ellipsoid. 

The force-function for these ideal cases is $fj,mr 2 . Let mR 
denote the force- function for the remaining forces which act on 
m. If #1, # 2 > #s he the rectangular coordinates of m referred to 
fixed axes through the centre of the system, the equations of 
motion will be 



with >* = x + a 

When R = 0, each coordinate describes a harmonic motion 
with period 2?r/n, where n 2 = p, and the orbit is an ellipse whose 
centre is at the origin. 

As before, the elements of this ellipse may be considered as 
new variables for the examination of the case R + 0. A canonical 
set, with Hamiltonian function R, in which /A is replaced by n a , 
is the following: 



1 Q = e -or, y = w 0, h = ff. 

If 1 be replaced by / = nt + e r, the only change needed is 
the replacement of R by R nL as the Hamiltonian function. 

The proof is left to the reader. A modification similar to that 
of 5*13 may also be made. 



CHAPTEK VI 

SOLUTION OF CANONICAL EQUATIONS 

6*1. The main object of this chapter is the development of 
methods of solution for the types of canonical equations which 
have been obtained in the previous chapter. All the methods 
depend fundamentally on the assumption that the variables 
differ from constants by amounts which have as factor the ratio 
of the disturbing mass to that of the primary, and therefore 
that the variables may be developed in powers of this ratio. As 
long as we confine ourselves to the first power of this ratio there 
is little choice between the various methods; substantially, they 
are equivalent. It is when we need to take into account higher 
powers of the ratio that differences appear, mainly in the amount 
of calculation which is necessary to secure the desired accuracy. 

The fundamental idea of the method of Delaunay, namely a 
change of variables such that the new variables are more nearly 
constant than the old ones, is used throughout. But the applica- 
tion of the idea is different from that which Delaunay made to 
the solution of the satellite problem, where the changes of 
variables were very numerous*. Here it is shown that one 
change of variables is, in general, sufficient for the solution of 
the majority of planetary problems provided the new variables 
are suitably chosen. Much of the discussion in the chapter 
hinges on the amount of labour which the development and 
solution of the equations for the new variables requires. 

6*2. Elliptic elements or variables. 

In Chap, v it has been shown that the equations of motion 
can be put into the canonical form, and methods of changing 
from the coordinates and velocities to other sets of variables 
are developed. Particular sets of new variables which are con- 

* See for example, Tisserand, vol. 3, Chaps, xi, xn; E. W. Brown, Lunar 
Theory , Chap. ix. 



1-3] PROPERTIES OF DISTURBING FUNCTION 139 

nected with motion in an ellipse have been chosen and the 
equations satisfied by these new variables have been given in 
canonical form; other sets are given in non-canonical form. 
Various points of view of these new variables, usually called 
the elliptic elements, are given. It was, however, pointed out 
that for the purposes of mathematical development, they should 
be regarded merely as a set of new variables which are connected 
with the coordinates and velocities by a set of equations, the 
latter remaining the same for all values of the variables. 

These new variables have the property of becoming constants 
or linear functions of t when R = 0. In one set, they are all 
constants; in the other sets, one of them is a linear function 
oft. 

The canonical set denoted by a, Wi and developed in 5*13, will 
be used in this chapter. Only slight changes, easily made, are- 
needed if the set GI , p a , p 3 , MI, q z , q*> g iven in 5 'H be used. Since 
R is not in general developed in terms of canonical elements, it 
is shown how the work may be so adapted that the developments 
of R given in Chap. IV may be used. 

63. The disturbing function. 

In Chap. IV, the disturbing function has been expanded into a 
sum of periodic terms on the assumption that the motion of each 
planet is elliptic. This restriction can now be removed, so far as 
the disturbed planet is concerned. Since the development consisted 
in the replacement of the coordinates by their values in terms 
of the time and the elliptic elements, the relations between them 
being those referred to in 6'2, the development is unchanged if 
we consider these elements as variables. 

The relations between the coordinates and velocities and the 
new variables C;, Wi are independent of t explicitly; in elliptic 
motion t is implicitly present in Wi only. Thus the only way in 
which t is explicitly present in R is through the coordinates of 
the disturbing planet, and it appears actually only through 
id' = n't + e' or <f = n't 4- e' - tsr'. If the motion of the disturb- 
ing planet is not elliptic, its deviations from elliptic motion can 



140 SOLUTION OF CANONICAL EQUATIONS [OH. vi 

be expressed as variations of its elements, so that the given 
development of R can still be utilised. The necessary modifica- 
tions of the solution are not difficult; they are exhibited in one 
particular case (6*19 below). 

The possibility of expanding R in power series when the 
canonical elements C;, Wi are used has to be considered. Since 
d = V//,a, it is evident that Ci can replace a without difficulty. 
Also, by 515 (2), we have 



and since, by 515 (1) or 515 (5), we can replace e by e\ and 
F by TI in the developments, it follows that developments in 
powers of e, F are replaceable by developments in powers of 
(-2c 2 /Ci)^, ( c 8 /ci)^. Derivatives with respect to c 2 , C 3 will 
implicitly involve the presence of negative powers of e, F, and 
it will be necessary to show that these negative powers disappear 
from the transformed disturbing function. The difficulty does 
not arise when the variables Ci, p*, p$, w 1} q 2j q^ are used, be- 
cause R and the coordinates are expansible in positive integral 
powers of p 2) p 9) q 2 , 73 (514). 

We have, approximately, p 2 e sin sr, q t = e cos BT; suppose 
that ?o, ^o ftre the undisturbed constant values of e, *&. We then 

get 

e sin w = CQ sin OTO + perturbations, 

e cos r = CQ cos tzr -h perturbations. 

While the perturbations vanish with the disturbing mass, they 
do not in general vanish with e$. Hence, even if the observed 
eccentricity of the disturbed body is so small as to be negligible, 
we cannot assume that e is negligible in finding its perturbations 
by the method of the variation of the elements. Thus, while 
geometrical descriptions of the motion are simple in terms of 
e, -or, the analytical development fundamentally requires the use 
of e sin tzr, e cos w, as convenient variables. The general control 
of the work is, however, much easier with the use of the former 
than with that of the latter variables. In certain cases, particularly 



3, 4] D'ALEMBERT SERIES 141 

those in which CQ is very small, and also in the discussion of the 
secular perturbations, it is advisable to use the variables p%, q% 
rather than c%, iv 2 . Similar remarks apply to js, q$ and to F, 0, 
but the problem for these latter variables is less difficult because 
R is expansible in positive integral powers of F and therefore of 03. 
The solution of the problem of the apparent presence of 
negative powers of e is given in the theorems of 6'4, 6*15 below. 

6*4. D* Alembert series. 

The association of powers of e with multiples of -or, of powers 
of e r with multiples of or', of powers of VF with multiples of 0, 
and of powers of a with multiples of w w', which have been 
pointed out in the development of R (4*15), and in certain of 
the developments of Chap, in (3*3), are so useful that it was 
found convenient to give a designation to such series: we have 
named them d'Alembert series*. 

Certain of the expansions of Chaps. II, ill, have been seen to 
be of this type. Certain other series are easily related to it. 
Thus, sinjf, cosjf are not d'Alembert series with respect to the 
coefficient e and the angle g, but sinj(fff), co8J(f ff) are. 

It is evident that if we have a d'Alembert series with respect 
to a coefficient A and an angle #, the series is formally expansible 
in powers of A sin#, A cos#, and in powers of A exp. #V 1, 
A exp. x V 1. A general property of these series is given in 
the following theorem. 

If f, f are d'Alembert series with respect to the coefficient A 
and the angle x, then the Jacobian, 



d (A 2 , x) 2 A \dA dx 
is also a d'Alembert series with respect to A, x. 
Let _ __ 

y = A exp. x V 1, 2 = A exp. x V 1, yz = A 2 . 

* It appears that d'Alembert was the first to notice this property of the dis- 
turbing function with respect to the eccentricities and longitudes of perihelia, in 
his memoir, "Becherches sur diffe'rens Points importans du Systeme du Monde," 
Mem. Paris Acad. Sc. 1754. 



142 SOLUTION OF CANONICAL EQUATIONS [OH. vi 

The transformation of the Jacobian from the variables A, x to 
the variables y, z gives 

8T_ a/'N 



_ 

2A \ABy Adz) V dy dz ) 

d + z 

d A 



-, 

\dz dy 'by dz / ' 

since yz = A 2 . The definitions of /, /' make them expansible 
in powers of y, z and therefore their derivatives have the same 
property. The Jacobian is therefore a d'Alembert series. 

Thus while the separate terms of the original Jacobian are 
not d'Alembert series, the divisor A disappears from the Jaco- 
bian when it is expressed in terms of A, x. This is the property 
we really require to know. 

6 '5. Other properties of It. 

It is useful to recall certain of these. 

(a) It is homogeneous and of degree - 1 with respect to 
length. The variables GI are homogeneous and of dimensions 
(mass)* (length)* or, with the unit of mass actually used, of 
dimensions (time)" 1 (length) 2 . 

(6) It depends only on relative coordinates and is therefore 
independent of the origin of measurement of the angles. The 
anomalies are by definition independent of such origin. When 
R is expressed in terms of g t g' , -or, txr', 6, the only forms in which 
the latter three angles are present are their differences, usually 
expressed by tzr txr', tar -f & 26. 

(c) It is a function of the coordinates only and is independent 
of the velocities. This property was utilised in forming the 
equations of motion. One result is the equation 

d ^ a V? dR 



4-6] PROPERTIES OF R 143 

which is a consequence of the equation 



a result which is deducible from the canonical equations in 
513, with the help of the relation 7i = 
The relation (1) may also be written 

dR dR dR 



by the use of the canonical equation for ci. 

When the two bodies are supposed to move in fixed ellipses, 
the relation (3) is evidently true, since t occurs only through 
y, g\ the term dR/dt enters through g' only and dR/dwi through 
g = wi TZ only. Its importance is due to the fact that it is still 
true when the elements are variable, but it must be remembered 
that the term dR/dt is still a derivative with respect to t only as 
the latter is present through the coordinates of the disturbing 
planet. 

(d) Since the term ra'/a' in the preliminary expansion of 
R given in 4'2 does riot contain the coordinates of the disturbed 
planet, R has the factor w'a 2 /a' 3 when a' > a. This factor may 
be written (m'/a) (a/a') 3 . Since the undisturbed force-function 
is fji/r, it follows that the disturbing effect of an outer planet 
has as factors the ratio of its mass to that of the sun and the 
cube of the ratio of the distances of the two planets from the 
sun. 

If the outer planet be the disturbed planet, the first term of 
R can be included in the elliptic force-function. We can then 
consider R as having the square of the ratio of the distances as 
a factor instead of the cube. 

6'6. Elimination of a portion R t of the disturbing function. 

In Chap, iv, R has been expanded into a sum of terms having 
the form 

R = 2KcosN, N = ji wi +j 2 iu 2 +j 3 w 3 +ji'wi +jz'^2, (1 ) 
where w\ = n't + e' g' + /, w 2 ' = w', 



144 SOLUTION OF CANONICAL EQUATIONS [CH. vr 

and the j t -, J/ are positive or negative integers or zero, the sign 
of summation referring to their various values. The coefficients 
K are functions of a, e, /, a', e' and therefore of Ci, c 2 , 3, a', e' t 
and they contain the mass of the disturbing planet as a factor. 
Of. 4-15. 



Put R = R t + R c , ........................ (2) 

where R t may contain any or all of the terms for which the 
relations ji j\ = do not hold. Thus for all terms in R t , the 
value of v defined by 

"=jiw+jiV, n = ^/c^ ............... (3) 

is never zero. It is here assumed that n/n r is not in the ratio of 
two integers ; this case, if it does occur, must also be excluded 
from R t it will be considered in Chaps, vui, ix. 

The elimination of R t will be effected by a change of variables 
with the help of the Jacobian transformation theorem (5'3). 
The new variables will be denoted by 

Cto, w#, i=l, 2, 3, 

and the suffix zero will in all cases denote that the old variables 
have been replaced by the new. Thus 

NO = ji MIO 4-^20 +j 3 MM +ji'wi +j*W2 ....... (4) 

Take as the transforming function 

S y = :E;C t ^o-S, S=S~smJV , ......... (5) 

so that S is a function of the three old variables d and the three 
new variables w^; 8 is to contain only those terms present in 
R t with the divisor v appropriate to each term. It will be 
noticed that S can be regarded as the integral of R t taken on 
the supposition that the ellipses are invariable; this fact, how- 
ever, is not at this stage to be regarded as having any physical 
significance since (5) is merely a definition of S. 
The equations 5*2 (3) give 

dS dS _ 



6] ELIMINATION OF S t 145 

as the six relations connecting the old and new variables. Also 



01 01 V \ V 

by the definition (3) of v. But when (5) is substituted in (6), 
the equation for CIQ becomes 

v 
whence by (3) 

dS ^ -rr ~\T M / \ 

~ 7 = - ^ K COS NQ -f ~- 3 (Ci ~ CIQ). 

dc Ci 

The new Hamiltonian function is 



The equations (5), (6) show that Wi WM has m' as a factor. 
Hence cos N cos N has the same factor, and K cosN 
has m' 2 as a factor. Also, since Ci CIQ has m' as a factor, 



) 

...f, 



The last two terms of (9) can therefore be written 
P* 3( Cl -c 10 ) 2 2 , 



(10) 



2cio 2 2c 10 4 ^ T -' 

the first power of Ci CIQ disappearing. Finally, since R c R^ 
has the factor m' 2 , the right-hand member of (9) may be written 

^co-H 9^~~2 + terms w ^ factor m' 2 (11) 

If, therefore, we omit terms having the factor m' 2 , the equations 

satisfied by the new variables are 

/ ^2 v 



These have the same form as the original equations, but the 
terms present in R t have disappeared. 

B&SPT 10 



146 SOLUTION OF CANONICAL EQUATIONS [OH. vi 

At this point it is convenient to indicate the general plan of the remain- 
ing portions of this chapter. Two choices of the terms to be included in R t 
will be made. In one of them, R t contains all the terms for which *>4=0 ; 
R c then contains only those terms which produce the so-called secular 
motions of the elements. In the other, R t contains the short period terms 
only, so that R e contains the long period and secular terms. 

It is for the latter choice that the calculation of the terms in (10) 
dependent on m' 2 will be made. Several methods for carrying out the work 
to this degree of accuracy will be outlined, and certain cases where it is 
possible to obtain numerical results with but little calculation will be 
developed. 

It may be pointed out that the work as far as equation (9) is quite 
general in character, no approximation being involved. It in only after this 
point that we assume the possibility of development in powers of m' and 
proceed to find the first terms of the expansion. 



THE FIRST APPROXIMATION 
6*7. First approocimation by change of variables. 
Let R t contain all the terms for which z>=i=0; then E c is 
independent of Wi, w\ and therefore of t explicitly. 

The first approximation is obtained by neglecting all terms 
which have m' 2 as a factor. The variables c t - , W Z Q therefore 
satisfy 6*6 (12). Since R cQ is obtained from R c merely by 
substituting the new variables for the old, it follows that R cQ 
is independent of WIQ, t. The equation for CJQ therefore gives 

j~ 0, CIQ = const. = ki , 
and the equation for WIQ is 



_ _ 

dt Cio 3 9cio ki* 9cio " 

When m' is neglected, the remaining variables are constant 
and then 

M 2 
^io=r3^-fi, w = aa, W3o = a 3 ; c,- = *i, ...... (1) 

KI 

where a^, ki are arbitrary constants ; these values may be 
substituted in all terms containing the factor m' and therefore in 
R<$ . As R& does not contain WIQ , it follows that all the derivatives 



6,7] FIRST APPROXIMATION 147 

of jRco will become constants. The solution of the equations 
6*6 (12) to the order iri will therefore be 



M -< 2 > 

c<o = ki + ~t, w = < - T^ * * = 2, 8, 
dWi-o dCio 

in which the coefficients of are all constant. 

The values of the new variables in terms of t having been 
obtained, the old variables are to be calculated from the 
equations 6*6 (6) with the value 6*6 (5) for S. But since 
S, Ci c iQ have the factor m' we can replace c t - by c t0 in S 
and in its derivatives. We therefore obtain 



Finally, since the values (1) differ from (2) by terms having 
the factor m', we can replace WM, c^ by their values (1) in S Q 
and in its derivatives. 

These results may be restated in the following manner. If 

It = R -f ^K cos JV, jV =ji Wi +J2 w 2 + JB w 2 -f ji Wi 4- jz w* > 
where R c contains all terms for which ji = ji 0, the values of 
the variables c, w t - to the order in' are given by 



cos N, 

ij? dR c \^ v a (K\ . 

y~ - -5 U - S A si 
^i 3 3ci/ dcAv) 



c ^ v . 

= i -f ( y~ - -5 U - S A sin 

3 ... (4) 

~ t + 2ji cos 

OWi V 



C4 ^ . .. 

w f =a t - - -^ ^ - S - -5- sm Jv, 
dCi // oCi 

where i = 2, 3 ; in all the terms of the right-hand members the 
constants k iy & 2 , &3 are substituted for d, c a , c 3 , the constants 
2, 3 for ^2i ^3> and the value i + /^ 2 /&i 3 for w lf and i/ is 
defined by 

"=ji+ji'' ...................... (5) 



148 SOLUTION OF CANONICAL EQUATIONS [CH. vi 

Equations (4) may be written in the following form. If we put 

K 

^ = *fl c + S - sinjy, .................. (6) 

M 2 

d - ki = Sci, MI - i - -3 1 = Swi, Wi-ai = Swi, . . .(7) 

#1 

d *-, *~g ......................... (8) 

the undisturbed values (1) being substituted in the right-hand 
members. These are called the perturbations of the elements. It 
must be remembered, however, that the coefficient of t in Swi is 
to be included with fjf/ki 3 in comparing with observation. 

6*8. Secular and periodic terms. 

The coefficient of t in w t is deduced from observation and is 
known as the 'observed mean motion.' If we denote it by T?OO> 
we have 



If we define oo by means of the equation oo 3 ^oo 2 /*, we obtain, 

to order m' 

21 9R e \ , 



But since Ci 2 = /x-a, the equation for Ci gives, to the order m', 

Cl 2 A-! 2 /, 2~.K A7 \ 
-* = -M l -f - Sjj cos N , 

/JL /JL \ ki J * V J* 

/, 2 1 aft c 2 . K . 7 \ 

a =oo(l~ -gr + r^i-cos^) . ...(3) 

\ OWoodCi A?i " 1^ /cj-^ 



or 

Thus the mean value of a is not oo but this quantity with a 
small additive portion. 

The coefficients of t in 02, 03, w; 2 , MS are known as the secular 
parts of these variables. An important result is the fact that 
Ci, and therefore a, contains no secular part to the order m'. 

Since the coefficient of t in Wi is the observed mean motion, 
the secular part of Wi is defined as any part which it may have 
depending on t 2 , tf 3 , ... . To the order m' there is no such part. 



7-9] FIRST APPROXIMATION 149 

The coefficients of the periodic terms in w iy w 2 , w$ depend on 
the derivatives with respect to Ci, c?2, c 3 of K/v. Since v is 
independent of c a , c 3 but does depend on GI, we have 
K\ldK 



9 (K\ 

o~ I - I 

oCi \ v / 



* 



since n = jjf/Ci 3 . 

The presence of the square of v as a divisor in Wi but in 
none of the other elements, is of fundamental importance in 
the theory of the long period terms which have small values 
of vjn. The simple manner in which this divisor arises with the 
method of solution adopted here is noticeable. 

6'9. Transformation to the elements a, e\> T\ or #, e, F. 

In the developments of the disturbing function, the angles w^ or quite 
simple linear functions of them, are used, but in place of the c t we find 
the elements a, e 1? r l5 related to them by the formulae 5-15 (2), or the 
elements a, e, r, shown by 5'13 (1). 

If /be any function of the <S' t -, we have, with arbitrary variations of them, 



From 5*15 (2) we deduce 



(2) 

On substituting these in (1) and equating the coefficients of the fic t , we 
obtain 

\ l V /i ^ 

a/_ l_ l^df d__ \_ty' 



The derivatives of A' with respect to the c$ may be found from (3), and 
the differences of ct, e 1 , TX from constants by means of (2), if we put therein 
dc % =*Ci k$. 

Similarly, for the elements a, e, r, we find 

sa-- a Cl , ae= _(lz^ /i 

na ^ \e 




160 SOLUTION OF CANONICAL EQUATIONS [OH. vi 

and 

df _ 1 / df e(l-erf ?/_ r d/1 " 

cci ~~ no? \ da i -j_ (j _ e 2)i 96 (i _ e vfi crj * 

d#2 net? \ e c!e n e 2 )^ c*rj ' 

a/; = i i a/ 

A difficulty occurs in consequence of the presence of the divisor e\ or e 
in the expressions for 9//?c 2 . But this latter derivative is present only in 
the expression 6*7 (4) for w 2 , which in turn only appears in terms having 
the factor e in the expressions for the longitude, latitude, and radius vector 
in terms of the elliptic elements. It is for this reason that it is usual to 
give the perturbation of fisr in the form eStzr. 

This solution of the difficulty is sufficient when we confine ourselves to 
the first power of ra', but further consideration is necessary when we 
proceed to higher powers. The solution is then contained in the theorem 
of 6*4, applied to the development given in 6'15. 

6*10. The perturbations of the coordinates. 

The coordinates are supposed to be expressed in terms of the 
elliptic elements. If, then, we define the perturbations of the 
elements as in 6'7, the perturbations of any coordinate x to the 
first power of m' will be given by 

"~* \dci l d'Wi / ~*\dC{,dtVi dividci/' "' 

The latter form again introduces a function of the type con- 
sidered in 6'4. The periodic part S of i/r is a d'Alembert series 
and the longitude, latitude, and radius vector, when expressed 
in terms of the elliptic elements are d'Alembert series. It follows 
that the periodic parts of the perturbations of these three 
coordinates are also d'Alembert series. It should be remembered, 
however, that Scz/e, eSw% contain perturbations which do not 
vanish with e. 

611. This form of the solution, in which powers as well as periodic 
functions of t are present, is that which is usual in applications to problems 
of the solar system. The results in this form usually have sufficient accuracy 
during the limited intervals over which observations are available. It is 
evident that a continuation of the process to powers of m' beyond the first 



9-11] SECULAR PERTURBATIONS 161 

will lead to terms containing higher powers of t and to the presence of 
periodic functions of t multiplied by powers of t. That it is possible 
formally to express the perturbations wholly by periodic functions of 2, at 
least in a first approximation, may be indicated in the following manner. 

Instead of the variables c 2 , w 2y let us use p 2 > q 2 defined in 5*14, and for 
simplicity let us neglect the inclination. The canonical equations for these 

two variables are 

dp 2 dR dq>2 dR ,-jx 



dt dp% 

The process for eliminating the terms present in R t can still be followed : 
it leads to new variables p 20 , q. 2{) witli the same function 7^, which is 
the portion of R independent of w\, MI'. The development of R as far as 
the second powers of the eccentricities is given by 4-32 (2), and the 'non- 
periodic part,' depending on the eccentricities, is given by putting i = 
in the first line of the expression, and i= 1 in the second line. The 
resulting terms have the form* 

(2) 



ct 

where P, Q are functions of a/a'. Just as before we show that e and there- 
fore a is constant. To the second powers of the eccentricities, we have, by 

5-14, . 

e sin 37 =p 2 /Ci > e cos w = q. 2 /ci . 

Since a is constant we can use units such that /z, a, w, are all unity. If we 
put e' sin v^ = p f , e' cos w' = #', the expression (2) may be written 

{i P (P2 2 + ?2 2 + P' 2 + q' 2 ) + Q (pp' + qq')} m' a. 
With this portion of R the canonical equations (1) give 



leading to 



and furnishing the solution, 

^2= a ~-5p' + C'sm(m'aP^ + />), ^ 2 =- ^ g' + C coa (mf 

where {7, Z) are arbitrary constants. 

With the adopted units, the period of revolution of the planet is 2?r. 
Since aP<l and m'<'001 for the largest planet Jupiter, it follows that the 
period of the periodic parts of p 2 , q 2 is ver j l n g compared with the period 
of revolution of the planet in general, it is greater than 10,000 years. 
Hence, for intervals of a few centuries, we can develop these periodic 

* As a contains e 2 , e' 2 , functions of a must be expanded as far as e 2 , e' 2 in 
powers of these parameters. 



152 SOLUTION OF CANONICAL EQUATIONS [CH. vr 

terms in powers of t and still obtain the needed accuracy without additional 
calculation. 

The practical objection to this method of procedure is the complication 
caused by the introduction of a new argument. Still another argument 
would be introduced by the solution of the equations for p 3 , #3. Thus, four 
arguments would be present. Even with a single disturbing planet and the 
calculation confined to the first power of ra', the work becomes complicated 
as soon as we proceed beyond the second powers of the eccentricities and 
inclination and the labour becomes almost prohibitive in the case, for 
example, of the mutual perturbations of Jupiter and Saturn. 

In satellite theories, the periods are so short that expansions in powers 
of t are impracticable and the four arguments must be retained. On the 
other hand only a few powers of a, and consequently a few values of i, in 
the expansions of Chap, iv, are needed. Part of the compensation thus 
afforded is lost by the fact that many powers of the disturbing mass must 
be retained. 

The general theory of the secular motions of the elements, to which the 
solution just given constitutes an introduction, will not be given in this 
volume. The reader is referred to other treatises, particularly to that of 
Tisserand, vol. 1, Chap, xxvi, and to later work referred to in Ency. Math. 
Wiss. Bd. 5. A warning concerning frequently quoted results, which give 
limits to the eccentricities and inclinations, should be made. These investi- 
gations, in general, take into account the first powers of the disturbing 
masses and the earlier powers of the eccentricities and inclinations only. 
When higher powers are included, approximate resonance conditions have 
to be considered and these may alter the limits extensively over very long 
periods of time. Thus while such results may be used with a fair degree of 
confidence in cosmogonic speculations for a few million years from the 
present time, we have no present knowledge as to the facts over intervals 
of the order of 10 9 years. 

6'12. Long period terms. 

We have seen that in a first approximation we may sub- 
stitute elliptic values for the coordinates in the development of 
the disturbing function. In Chap, iv, this development gives 
any angle N in the form 

i K - wi') jg j'g' k(wi + < - 2(9), 

where i, j,/, k are positive integers or zeros. The coefficient of 
a term with this argument contains the factor (cf. 4*15) 



11, 12] LONG PERIOD TERMS 153 

When elliptic values are used, the coefficient of t in the 
argument is 

v = i (n n') jn j'n r k (n + n'). 

It follows that if we have any argument in which the coefficient 
of t is jin ji'n', the order of its coefficient will be \ji ji , as far 
as powers of the eccentricities and inclination are concerned. 
A long period term is defined as one in which 

vjn (jiii j\n') -r n 

is small compared with unity: the secular terms for which 
ji = j/ = are excluded from this definition. Since it is 
supposed that we are using the methods developed above, n 
should properly be replaced by ??oo, but the suffix may be 
omitted in the discussion. Since n, n' are usually positive we 
have to consider cases where n'/n is nearly equal to the ratio of 
two integers. The word 'small' as used above is indefinite in 
both the theory and its applications; in general, if v/n is less 
than about one-third, so that the period of the term is longer 
than three times that of the revolution of the body round the 
sun, the term would be treated as one of long period. 

Since n, n' are obtained independently from observation it is 
always possible to find terms in R for which v/n is small. The 
critical values of ji, ji are obtained by expanding n'/n as a 
continued fraction. If p/q be any convergent, then for any 
convergent after the firsfc, ji = p, j\ = q will give a long period 
term. But as the order of the coefficient is \p q\ such a term 
may be quite insensible to observation even after receiving 
the factor 1/iA 

For example in the case of Jupiter and Saturn where 
w = 43996" (Saturn), n' - 109256" (Jupiter), these being the 
mean annual motions, we have 

2n ?&'= '483?i, order 1 ; 
5n - 2n' = '0334^, order 3; 
72n - 29n' = - -0162*1, order 43. 
The third of these is obviously insensible. 



154 SOLUTION OF CANONICAL EQUATIONS [CH. vi 

In considering the degree of approximation needed, we can 
make use of the property of a continued fraction which states 
that if p/q, p'/q' are consecutive convergents, p' >p, no fraction 
whose denominator lies between 9, q' gives so close an approxi- 
mation as p/q. Thus if q' q is large, p/q is usually a close 
approximation, v/n is very small and higher convergents are 
unlikely to give sensible coefficients. An apparent exception to 
the argument is the case of multiples of N when the term with 
argument N has a sensible coefficient. This case is dealt with 
in the second approximation (cf. 618). In the case of Jupiter 
and Saturn, the term for which v = Wn 4w/ has a sensible 
coefficient; that for which i> = 15/i 6n' is insensible. 

We have seen that the element Wi is that chiefly affected by a 
long period term since the term in this element has the divisor i> 2 > 
while those in the other elements have the divisor v only. In 
other words, it is the mean longitude which shows the principal 
effect. But there is an associated short period term in the true 
longitude which may have a coefficient comparable in magnitude 
with that of the long period term and which arises in the 
following way. The determination of the perturbations of e, TV, 
or more properly, of ecos w, e sin or substantially requires the 
division of the term in the disturbing function by e (6*9). Thus 
the term in R substantially acquires the divisor ev when it 
is inserted in the true longitude. Now e, ra occur in the 
true longitude principally through the chief elliptic term 
2esin(w 1 cr) and the long period term, therefore, gives two 
terms with motions n v. As v is small, these have nearly the 
period of revolution as periods. One of the two coefficients is 
usually quite small owing to the association of powers of e with 
multiples of tsr: the proof of this statement is simple and is left 
to the reader. 

The fact that terms of very long period are usually of very high order 
would seem to suggest some theoretical limit beyond which such terms 
could always be neglected. But, as far as is known, the observational 
determinations of the constants which give the mean motions are 
substantially independent of those which give the eccentricities, when the 



12, 13] OTHER SOLUTIONS 155 

determination is spread over many revolutions of the planet round the sun. 
Thus the ratio of e j e'*'r k to v or v 2 has no definite limit and a coefficient 
may be very small or very large according to the values chosen for n, ri. 
From the practical point of view the difficulty is surmounted by supposing 
that such terms, having periods very long compared with the interval of 
observation, can be expanded in powers of t. The constant parts of the 
expansions are absorbed in the arbitrary constants of the solution and, in 
the case of the true longitude, the coefficient of t is absorbed in the mean 
motion; the remaining portions are usually quite insensible during the 
interval. 

There is, however, an upper limit because of the fact that when the 
coefficient of the term exceeds a certain magnitude comparable with STT, 
the procedure previously followed becomes invalid : the phenomena of 
resonance then begin to appear. This limitation does not remove the 
difficulty. The form of tho mathematical development has to be changed 
and the argument proceeds on different lines. The complications which 
arise make the problem exceedingly difficult : some indications of them 
will be given in Chap. vin. It may be pointed out that one of the 
difficulties is due to the fact that the period of the term may become 
comparable with the periods of the so-called secular terms (6'8) and that 
it is not then possible to treat them independently even in a first 
approximation. 

6*13. Other forms of solution. 

Instead of the value of S written in 0*6, we might have used 

T* 

t> = ^c tQ w t -f 2 --- sin N. 



Here the rdles of c t -, w t are simply interchanged. A little con- 
sideration will make evident the fact that in a first approximation 
this form of $ gives nothing new: it is only in the second and 
higher approximations that differences appear. 

When the Poincar6 variables p^ q%, /> 3 , q$ are used, the 
expansion of R takes the form 

R = ZK c cosN' + 2 #. sin -ZV', 
where N f = ji^i 4 ji'wi -f /8, 

y8 depending on the constants of the disturbing body only. In 
this form K cy K s are expanded in positive integral powers of 



156 SOLUTION OF CANONICAL EQUATIONS [OH. vi 

p*> </2> pa, #3 and are also functions of ci, a, e*. The transformation 
function to be used is then 

e - 



- sn 



cos #', 



where the suffix zero has the meaning defined in 6*6 except in 
(7if c )o, (K 8 \, where it means that g a , #3 ar e replaced by 20 #3o> 
while p%, p$ are left unchanged. We may also interchange the 
rdles played by p, q. 

THE SECOND APPROXIMATION 

6*14. The calculation of the second approximation to the 
values of the variables may be very laborious if advantage is 
not taken of every feature which may help to shorten it. The 
first step consists of an examination to discover what classes of 
terms will give sensible coefficients or sensible additions to 
coefficients found in the first approximation. Methods for the 
calculation of the sensible portions will then be given and these 
methods will be developed in such a manner that the actual 
computation may be reduced to comparatively few operations. 

There are several devices which can be used to obtain the 
terms dependent on the squares of the disturbing masses. We 
can follow the process of Delaunay which involves continual 
changes of variables until the Hamiltonian function is freed from 
all sensible terms for which the relations ji =*ji = do not hold; 
the equations for the final variables can then be solved by series 
arranged along powers of t. Another plan is the substitution of 
the results of the first approximation in the derivatives of R 
instead of the elliptic values which have been used in finding 
the first approximation ; the equations are then again integrated 
and the new portions of the variables calculated. Another 
device which is sometimes useful is a method of integration by 
parts which makes use of the fact that the derivatives of c\, c%, 
Ca, w% t w& with respect to t have m' as a factor. 

The most useful device is, however, the separation of the 
terms of long period from those of short period and also from 



13-15] SECOND APPROXIMATION 157 

those which are secular. It will be seen that the second 
approximations to the long period and to the secular terms can 
be rendered almost independent of the first approximation to 
the short period terms, so that they can be determined inde- 
pendently of the latter. But the effects of the long period and 
secular terms on the short period terms are usually sensible, and 
it is these effects which become most evident in comparisons 
with observations extending over long intervals of time. 

The methods adopted to prove that these limitations are 
possible are not necessarily the most convenient for the actual 
calculation of the sensible terms, so that more than one of 
the plans for continuing the approximations will be found 
developed in the sections which follow. 

6*15. The Hamiltonian function in terms of CIQ, WIQ. 

The equations 6*6 (9), 6*6 (10) give, for the Hamiltonian 
function of the equations for the new variables, the value 

A - 1 . i (d - cio) 2 + R e + S^Tcos N - Stfcos N , 

^CIQ A CIQ 

...... (1) 

in which powers of c i i - CIQ beyond the second are dropped. This 
last omission is equivalent to stopping at the order ra' 2 . We 
need (1) expressed in terms of c t0 , W& to this order. 

Since 8 has the factor ra', we have c t -=c t o, ^ = ^,-0 when 
ra' = 0. It follows that, as far as the first power of ra', the 
equations 6*6 (6) give 



and that, if /be any function of c if w iy t, 

?\f 7\f 

f(Ci\ Wi\ )=/(c t0 ; w#\ + 2 ^ (c, - Cfl) + ^ (w< - 

to the same order. The substitution of (2) in this last equation 
enables us to write it 



158 SOLUTION OF CANONICAL EQUATIONS [CH. vi 



On applyiug this to %KcosN=R t and to %K cos N 0} we 
obtain 

V V ^ \T 7? i V /^tO 3$0 SRtQ 3$o\ f . 

2<K cos iv = JK tQ + i ( ~ ---- - - ~ , ...... (4) 

\3c i0 9w t0 dwtfdcio/ 

(5) 



CiO O 

Since R already contains the factor ra', these results hold to the 
order m 2 . The same application to the development of R c may 
be made. The second term of (1) evidently has the factor 'in' 2 
and may be replaced by its value in terms of the new variables 
by means of the first of equations (2). 

Thus the Hamiltonian function for the canonical equations 
satisfied by c,o, w t -o to the order m' 2 becomes 

F t + F c ................... (6) 



2c 10 2 
where 



/IT\ 

F 
c 



The expressions (7) contain the factor m' 2 . The application 
of the theorem of 6*4 shows that they are d'Alembert series and 
therefore that they contain no powers of e, F as divisors. 

If R t be defined to contain all the terms in which ji =ji = 
do not simultaneously hold and no others, R c and therefore ^ c0 
will contain only the terms in which these relations do hold. 
Since $o contains the same terms as those present in RM, it 
follows that F c is like R t o in this respect, and therefore that the 
secular portions which depend on the terms in which ^i = jV = 
will arise only from FI, F t , R cQ . 

The Hamiltonian function (6) will, however, be used below 
only to distinguish between the effects of the short period and 
long period terms, and for this purpose R t will be defined to 
contain the short period terms only. The investigation will show 
what portions may be neglected in the actual calculations which 
can then be carried out by a more simple method. 



15, 16] SHORT PERIOD TERMS ' 159 

A direct second approximation to the solution of the equations 
for Ci, w it is easily seen to be given by 



4. - 4. V _ 

Ci " Cio + ^ w*tofl 

9 2 $o 9$o 



These require the formation of the products of the derivatives 
of SQ for each of the six variables. The method given in the 
text confines the formation of such products to those in one 
function, namely (6). 

6*16. Influence of the short period terms in the first approxi- 
mation on the second approximation. 

Let R t and therefore S Q contain only short period terms, so 
that in 6*15 (6) there are no small divisors v or v* tending to 
raise the magnitudes of the terms in FI, F t , F c . Suppose that 
all these functions are expressed as sums of periodic terms. 
These terms will have the same general form with respect to 
the variables c t -o, w lQ that R had with respect to Ci, Wi, that is, 
they have the form K Q cos N Q> where N Q is a linear function of 
the WM, w\, Wz with integral coefficients, and K Q is a function of 
the CM and of the elements of the disturbing planet. 

The terms present in R cQ are all either of long period or 
those for which ji=j = 0, while those arising from F it F t , F c 
are of the same character with additional terms of short period 
but all having the factor m' 2 . 

If the short period terms were again eliminated by a Jacobian 
transformation, the new variables would differ from CM, w^o 
by terms having the factor m' 2 and with no small divisors 
present. As the largest value of m' in the problems of the 
solar system is less than "001, and as an accuracy to *001 of a 
short period coefficient is rarely attainable in comparisons with 
observations, these portions can generally be neglected. 

The long period terms present in J\, F t there are none in 
F e because the product of terms of the form cos (at + a'), 



160 SOLUTION OF CANONICAL EQUATIONS [CH. vi 

cos (-42+ A 1 ), in which a is small and A is of the order of the 
mean motion, gives rise to terms with arguments 
(Aa)t + A'a' 

have the factor m' 2 and are additive to those in R cQ with the 
factor m'. Thus these terms will merely change the coefficients 
of the long period terms by amounts of the order of *001 of their 
values at most, and such changes again are rarely sensible to 
observation. The same result holds for the terms in which 

ji=ji'-0. 

Exception to these statements may arise on account of the 
fact that when we differentiate with respect to c 2 o or 030 a 
divisor of order e 2 or F is, in fact, introduced. In the method 
adopted for calculation it is seen (last paragraph of 6*17) that 
these terms of lower order disappear, so that the general argu- 
ment is not affected by them. 

Thus the short period terms present in the first approximation 
can be altogether neglected in proceeding to a second approxi- 
mation, or at most, only a very few, and those with the largest 
coefficients, need be retained. It follows that the long period 
and secular terms can be obtained to the order m' 2 with sufficient 
accuracy if we neglect at the outset nearly all short period 
terms present in R. 

As the apparent exception mentioned in the text always raises a 
difficulty in the discussion of the canonical equations for the elements, 
further details as to the occurrence of such terms may be of value. 

According to the theorem of 6*4, R c produces d'Alembert series in F e 
and is therefore free from these terms of lower order. Hence they will 
only arise through F t . The divisor e 2 will arise in F t only through the 
product (9/2 w /<to>2o) (9>/ 9c 2o). This may be written P+, where 



' * 2 acgo 8w>2o 2 j 
By the theorem just quoted, P is a d'Alembert series, and is therefore 
free from the exceptional terms. As for , we note that elliptic values are 
to be substituted and that then jR t Q=dS ldt. Hence, since derivatives with 
respect to , w%) , c^ are commutable we deduce 

^"SSiUfcii 



16, 17] LONG PERIOD TERMS 161 

On account of the relation between R, S this suggests that some second 
order terms might have been included in the expression for S which would 
have prevented the occurrence of these terms in the new Hamiltonian 
functions. The fact that another method shows that they ultimately dis- 
appear, indicates that the portions of this character which arise from the 
solution of the equations 6'15 (8) which give c t> w^ in terms of c,- , w^ to 
the second order, will cancel the portions which arise through Q. 

As a matter of fact, even if the method were used for calculation, the 
terms would cause very little trouble. For we are actually interested only 
in the long period terms present in , and the operator djdt introduces 
the small factor v in such terms. The numerical effect of this fact would 
be to cancel to a large extent that produced by the divisor e* or r. 

6*17. Calculation of the second approximation to long period 
and secular terms. 

The fact that the first approximation to the short period terms 
exercises little or no sensible influence on the second approxima- 
tion to the secular and long period terms, enables us to calculate 
the latter as though the former did not exist. Thus the equations 
for CM, w t o become the same as the original equations for c it w t 
would have been if we had omitted all short period terms. For 
the sake of brevity in notation, therefore, we shall omit the suffix 
zero in finding this second approximation, restoring it only at 
the end of the work. 

The equations are 



all short period terms being excluded from R. These equations, 
except that for w\ , may be written 



dci __ dR dw% _ dR dws __ dR 
di^dwi' ~dt~~fo*' dT""""3c3 ....... ^ 

If we differentiate the equation for Wj and make use of the first 

of equations (2) we obtain 



__ d/f^_dR\_ _ V9^_rf/W*\ 

~~ dt W dc J ~ d 4 dw l dt \dcj ....... 



dt* 

The first approximation gives the values of Ci, Wi in the form 
/3 + i* + 2jBcos(^ + "o), ............... (4) 

B&SPT II 



162 SOLUTION OF CANONICAL EQUATIONS [OH. vi 

if we note that the addition of ^TT to i/ will take care of the 
presence of sines. The coefficients &, B all contain the factor 
m' except that of t in Wi which is n w . In the case of Ci, we have 
fti = 0. Finally, B contains the first power of v as a divisor in 
all cases except that of w\ which contains an additional part 
having v 2 as a divisor. 

This first approximation was obtained on the supposition that 
the elliptic elements in R were constant. If we denote the 
difference of the constant and variable values of the elements by 
the symbol 8 (in the case of Wi, the symbol Sw { denotes the 
difference between Wi and its undisturbed value n$ t + const.), 
and if the additional part due to the second approximation be 
denoted by $2 > Taylor's theorem, applied to equations (2), (3), 
gives 

d rs v / d 2 R ^ d 2 R 5, \ . i rt > /e \ 

-7; 820* ^^U -a toj-f ^ a - ocj ), 1= 1, 2, 3, ...(5) 
at \dWidWj dWidCj / \ / 

* 2 o o /PX 

1 = 2,3, ...(6) 



-- 
dCidWj 



- ^ -- r 

Ci 4 J \dwidWj 
d / PR 



...... (7) 

in which j takes the values 1, 2, 3, Since all the terms in the 
right-hand members of these equations have the factor ra' a , 
constant values may be substituted for the elements in the 
derivatives of R. Also since Sc if Sw t are present in a linear form 
only, their various portions may be separately calculated in any 
manner which may be convenient. 

According to 6'7 (8) the periodic parts of 8c,-, 8w$ are found 
from 

* dS X * S /QX 

SC<= , *", g^, ............... (8) 



where tf-SsinJV, R t = 2KcosN, ............ (9) 



17, 18] SECOND ORDER CALCULATION 163 

with N =ji Wi +J2 w 2 +jz w 3 +ji wi +J2 Wt, 

* I I L f\ 2/3 /I A\ 

if = 'I-* ji -(- o* n } v ^ \jy n == u/ I c\ t ........ {L\J ) 

constant values for the elements being used in these expressions. 
Hence 

1 v y v ^C2 ' v uC$ 

(11) 

Swi = -35^- sin^-S-^sin^. (12) 

Although these formulae serve for all periodic terms, we are 
considering only the long period terms in R t . 
The secular parts are given by 

C/jtV/ UxLfl 

OCi == uC% == t , OCa L ~^. 

8 '^ ^ (13) 

Swi = 0, &W2= t -^ - , 8w$ = t ^ - , 

OCz OC$ 

where R c = ^K cos (jiw* +^3^3 +jV^/). 

The substitution of (8) in the right-hand member of (5) gives 



_ ~- 2 y 

* dwi dwj dcj * dwi dc$ dwj " 

According to the theorem of 6*4, this is a d'Alembert series 
since ti, R, dR/dWf are d'Alembert series. The same result is 
evidently true for the series in (7). For i = 2 in (6), we note 
that CzdR/dcz is a d'Alembert series, so that 0282^2 has the same 
character. Similarly c 3 S 2 w 3 is such a series also. Thus the 
presence of the divisor e 2 or F, lowering the orders of certain 
terms in the first approximation, does not affect the equations 
for the second approximation because such terms disappear. 
This is the proof referred to in 6*16. 

6*18. The principal part of the coefficient of a second order 
term. 

Let us consider first the case of a single term of long period. 
In general, the principal part of the coefficient will be that part 



164 SOLUTION OF CANONICAL EQUATIONS [OH. vi 

in which the divisor v occurs to the highest power. This is 
evidently the fourth, obtained by substituting the first term of 
Sw l in 6-17 (12) in the first line of 6'17 (7). It gives 

3K . n 



~ 9jl 3 ~T ~2 S ^ n N COS N' 

Whence, integrating, 

s 9 . 3 K 2 n* . , 7 

^i=^i 3 - j? - a sm2^. 

If, then, we write 

&wi = B sin N, 

for the first term in 6*17 (12), we obtain 

StWi^ffsmZN ................... (1) 

o 

This result is independent of the method by which the 
coefficient B may have been obtained. It gives at once the 
principal part of the second order term with argument 2jV when 
the first order part with argument N is known*. 

It still holds if we include in Sivi all the terms for which ji, 
ji are the same. For these terms may be written in the form 
P cos ( jj wj +ji'wi) + Q sin (j^v t +ji'wi), 

where P, Q are functions of Ci, c 2 , c 3 , w%, iv$, a', e' y w 2 ', and this 
expression may be put into the form 



where /3, jB are independent of Wi, W}'. In this case we put 
j\Wi +jiWi -f /8 for N. It is thus immediately applicable when 
the numerical values of B, $ have been obtained. 

It is not difficult to extend this result to the case in which 
two or more long period terms are present. For two such terms 
in R denoted by K cos N, K cos N, we have 



j 

- 2 = j^K cos N j-K cos N. 



* The result, obtained by a different method, was given by E. W. Brown, 
loc. cit. 7'32. 



18, 19] SECOND ORDER PERIODIC TERMS 165 

If Swi = BcoaN+5 cos N 9 

we have B = - 3Kn/* c l} B = - 3Kn/v 2 d . 

Whence, on integration, 

(2) 



In the general case, we add together all such pairs of terms. 

6*19. Effect of a long period perturbation of the disturbing 
planet. 

The effect of such a perturbation is most marked in wi and 
the principal part of its effect on a term Kcos N in R will be 

= _ j x ' K sin N. a<. 

In the formation of the canonical equations, it is assumed that 
i is independent of the elements of the disturbed planet. Hence 



If we are given 

Swt' = B' sin N' = B' sin (v't + *><>'), 

where v f /n is small, we obtain, by a procedure similar to that 
followed in 6'18, 

(1) 



When we are dealing with the mutual perturbations of two 
planets, there will be terms in Sw/, due to the effect of the 
planet (which we have been calling the disturbed planet) having 
the argument N, and for these terms (1) gives 



This case requires care if it is deduced directly rather than 
by substitution in (1), because Swi contains the elements c^, w it 
and we might be tempted to substitute in R before forming the 

n-3 



166 SOLUTION OF CANONICAL EQUATIONS [OH. vi 

derivatives with respect to c$, Wi. That we cannot do so is seen 
from the statement made above, namely, that the canonical 
equations are formed on the basis that the coordinates of the 
disturbing planet are functions of t only and are independent 
of the elements of the disturbed planet. This basis must be 
retained in the subsequent work. 

A useful exercise for the student is the deduction of these results by 
solving the equations of 6'15. He will find, for example, that the solution 
6*15 (8) of the equations 6'6 (6) to the second order will contribute 
ji B 2 sin 2N to the value of S 2 ^i f r the case considered in 6*18, while 
the solution of the canonical equations of 6'15 contributes I^Z^sin 2^V, 
the sum of these giving the result 6' 18 (1). 

Incidentally, this exercise furnishes a reason for not continuing the 
solution by the Delaunay method. There are two portions to calculate 
instead of one, and each of these portions is large compared with their sum. 

6*20. More accurate determination of second order terms. 

The calculation of the portions of 8 2 fy, S z Wi which have the 
small divisor i> 3 in the terms with arguments 2^ or JV" N, is 
not difficult. In the equations 6*1 7 (5), 6*17 (6) we need to use 
only the portion of Swi which has the divisor zA Thus the 
derivatives of R needed are 



Now the derivatives of R with respect to w i} Ci will have been 
obtained in finding the first approximation, and the derivatives 
(1) can be written down at once, even after numerical values 
have been inserted. 

To calculate S 2 Wi, we nee d the full value of Sivi and the values 
of 8c,-, St^2> Swa f r the terms in the first line of 6*17 (7); in the 
second line of the latter equation, we need the principal part of 
Swi only and we can neglect Sc it Sw 2 > Siv$. 

In 617 (5), (6) we can also neglect Sc t) Sw%, Sw$ and use only 
the principal part of Swi. Thus, in all cases, the terms divided 
by the cube of the small divisor can be obtained with the second 
derivatives (1) only. 



19-21] SECOND ORDER SECULAR TERMS 167 

6*21. Second order secular effects. 

These are produced by inserting the values 6*17 (13) in the 
equations (5), (6), (7) of 6*17. It will be noted that they give 
d'Alembert series in the same sense as the periodic portions 
discussed in the same section. 

There are three classes of terms present in the right-hand 
members of the equations to be considered. 

(a) Terms of the form t multiplied by terms in which 
ji=ji' = 0. 

(6) Terms of the form t multiplied by terms in which v 4= 0. 

(c) Terms in which ji = j/ = which arise when the periodic 
portions of Bc i} Bw i are substituted, these portions having been 
laid aside in the previous sections. 

The integration of the equations with the terms of class (a) 
gives terms factored by t 2 and by m' 2 since constant values may 
be substituted for the elements in the right-hand members. It 
is to be noticed that B z Ci contains no such terms because when 
J! = 0, dR/dwi and its derivatives are zero. The same result is 
true of the terms arising in the first line of 6'17 (7). A constant 
part arises from the terms of the second line which gives a term 
factored by t 2 in S z wi. All these portions are very small since 
no small divisors enter. 

The terms of class (6) give rise to differential equations of the 
form 



-T = tk cos (vt 



an integral of which is 



tk k 

x sin (vt + i/o) + -a cs (vt 4- z> ). 



Terms of this character arise in S%c it S%W2, B^w^. We have seen 
that terms of the second order with the small divisor v 2 can 
usually be neglected, and the terms with the factor t/v will 
rarely be sensible except for large values of t. 



168 SOLUTION OF CANONICAL EQUATIONS [OH. vi 

For 8 2 Wi we have an equation of the type 

d?x ^ , . , . 
-Tig = & cos (vt 4- VQ), 

a particular integral of which is 

tk k 

x -- g cos (yt -f VQ) + 2 -3 sin (i/ -f i/ )- 

The second of these terms is of the same order as those con- 
sidered in 6'20; the first will be sensible for large values of t. 

Class (c) gives rise to terms of the form \t in & 2 c t -, ^2^2, B^WS 
and to terms of the form \t 2 in S 2 ^i- A long period term in 
Swt possesses the small divisor v* so that the resulting term 
in S 2 ^j may become sensible as a result of integration, for large 
values of t. The proof that S 2 Ci contains no such terms is 
furnished as follows. 

Suppose that in the transformation of 6'6, we define R t as 
containing all terms for which v 4= 0, and that instead of solving 
by the method of approximation adopted above, we write down 
the equation for CIQ as given by 6*15 (6). It is 



By definition, ,<$ is independent of w w and FI, F t) F c contain 
no terms factored by t', as the remaining terms have the factor 
?/i /a , we can insert constant values for c t0) lu^, ^30 and the value 
ftoo + const, for WIQ. The right-hand member has then no constant 
term and consequently CIQ has no term factored by t. 

Next, the solution of the equation Ci = Cio + 9$/9wio, to the 
second order, is 



The portion of this under the sign of summation contains the 
factor m' 2 , and since SQ consists wholly of periodic terms, it cannot 
contain any term factored by t. In the second term we can 
substitute the values of c<o, w iQ to the order m'; these have the 
form j3 Q + fat, where # > Pi are constants. Hence, 3$ /9wio will 



21, 22] GENERAL SUMMARY 169 

contain only terms of the form B cos N or Bt cos NQ, where the 
coefficient of t in N is not zero, and B is a constant. Since it 
has been shown that CIQ contains no term of the form t x const., 
it follows that Ci has the same property. It does, however, 
contain terms of the form Bt cos N where the coefficient of t in 
NQ is not zero. 

Finally, the result is true for any function of Ci. For such 
terms can arise only from products of terms of the form t cos N 
with terms of the form cos N^ the former have the factor m' 2 
while the latter have the factor m', so that the product will have 
the factor m' 3 . In particular it is true for a = CI Z //UL and for any 
function of a to the order m' 2 . For remarks on the degree of 
importance to be attached to this well-known result see 7 '29. 

6*22. General Summary. The notation of 6*15 will now be 
resumed. The results in 6*17 to 6*21 give the values of c^o, w t0 
in the form 

dQ const. 4-Scio, Wio = const. -hSw t -o, ......... (1) 

except in the case of WI Q which takes the form 

Wi = T?OO t -f const, -f SWIQ ................... (2) 

The symbols 8c#), Sw; include all long period and secular per- 
turbations as far as the order m' 2 . 

In putting $ = 2 smN , ..................... (3) 

we have included in S only the terms corresponding to the short 
period terms S/f cos Af in R. For such terms we have seen in 
6*7 that a sufficient approximation to the values of c t -, Wi in terms 
of Cio, WM is, in general, given by 



The values of c^o, WM given by (1), (2) are substituted in (4). 
Since $ contains the factor m', it is sufficient to use the values 
of c,-o, WM to the first order in the second terms of (4). 

Thus the short period terms to the second order are found 
with sufficient accuracy by substituting in them the constant 



170 SOLUTION OP CANONICAL EQUATIONS [CH. vi 

values of the elements increased by their secular and long period 
portions. 

A literal development of R is needed to obtain the first 
approximation in order to obtain the first derivatives of jR with 
respect to the elements. The second derivatives of R are needed 
to a lower degree of accuracy, and as far as they are usually 
necessary for the calculation of the second approximation to the 
long period terms, they can be obtained from the first derivatives 
after numerical values have been inscribed therein. 

6*23. Integration by parts. 

A method of integration which can be applied to non-canonical 
as well as to canonical equations for the variations of the elements 
depends on the identity 



smA r ...... (1) 

dt\jf ) dt\N) 

where N is written for dN/dt. 

Suppose that two of the variables chosen be ivi and a (or any 
function of a) and define n by ?i 2 a 3 = />6. Let the remaining 
variables be any functions of a, e, CT, F, which do not contain 
t explicitly. The equation for wi has the form 



(2) 



and the equation for any one of the other variables, including n t 
has the form 

~ = 2PcosAT+Q, ..................... (3) 

where P, Q have the factor ra' in both cases. Hence P, Q may 
be functions of any of the elements except Wi, and MI, t will 
be present only in JV, and in the form jiWi + ji'wi, where 
Wi' = n't + e'. 

It follows that N has the form 



and that N has the factor m' and the form P cos N 4- Q. Since 
cos (N - 90) = sin N, this statement includes terms of the form 
SPsinJV'+Q. 



22-25] SPECIAL CASES 171 

Integrating (3) by the aid of (1) we have 

...(4) 



where # is a constant. Since P has the factor in' and since the 
derivatives of all the elements present in P, N have the same 
factor, the third term has the factor ra' 2 . 

In a first approximation, terms factored by w' 2 are neglected 
and constant values are substituted for the elements in the 
terms factored by ra'. Hence, the first approximation to the 
integral of (3) is 

.r = a- + 2~"sinN, + Qot ................ (5) 

#o 

For a second approximation, the values (5) are substituted in 
the second and fourth terms of (4): in the third term constant 
values of the elements can be used. The integrations may then 
be carried out in the usual manner. 

The first approximation to Wi is obtained from (2) after the 
substitution for n of its first approximation obtained from (5); 
in this approximation a term Q t is not present in (2). The 
second approximation is made in a manner similar to those out- 
lined for the other elements. 

6*24. The case of a single long period term. 

Whenever it is possible to limit the long period terms to a 
single value of jiw +ji'wi and its multiples it is possible by a 
change of variables to eliminate the time from the Hamiltonian 
function, H. This function equated to a constant then constitutes 
an integral of the equations, and by means of this integral Ci 
may be expressed as a function of the other variables and thus 
eliminated from the equations. The manner in which effective 
use can be made of this elimination is shown in Chapter vin 
which treats of resonance but which is equally applicable to 
terms of long period. 

6 '25. The theory outlined in this chapter, in common with all theories 
which depend on the method of the variation of the elements, has a simplicity 
of analytical form which makes it attractive for many theoretical investiga- 



172 SOLUTION OF CANONICAL EQUATIONS [OH. vi 

tions and particularly for those which are concerned with the phenomena 
of resonance. But it is doubtful whether it lends itself most conveniently 
for the calculation of ordinary planetary perturbations. It appears, in 
general, to demand more extensive calculation to secure a given degree of 
accuracy than those methods in which the perturbations of the coordinates 
are obtained directly. 

The chief objection is the necessity for expanding the disturbing function 
literally in order that the derivatives with respect to the elements of the 
disturbed planet may be obtained : there are six of these derivatives to be 
found, as against three functions to be calculated when the forces are 
used. A second objection is the necessity for carrying the expansion in 
powers of the eccentricity of the disturbed planet to one order higher than 
that needed in the coordinates. A third objection is the slow convergence 
along powers of e 2 , e' 2 , r of the series which gives the coefficient of any 
periodic term, especially for those terms which contain high multiples of 
y, (/'. To a large extent this slow convergence disappears where numerical 
values for these elements are used at the outset of the work, particularly 
if the developments are made by harmonic analysis in the manner outlined 
in 3'17. The chief exception to these statements is the theory of the Trojan 
group, but this theory is so different from that of the ordinary planetary 
theory that comparisons are not useful. It is possibly true that all the 
actual cases of resonance or of very near resonance can be treated effectively 
by this method, but some rather extensive comparisons would be needed 
before any reliable statement could be made in this respect. 

The literature on the subject of the application of the method of the 
variation of the elements to the planetary problem is extensive. The 
reader is referred to the standard treatises, particularly to that of Tisserand 
and to the articles in the Ency. Math. Wiss. for the earlier literature. For 
the later work, references and abstracts will be found in the mathematical 
and astronomical publications which summarise the literature annually. 

6*26. Throughout this chapter it has been supposed that the mutual 
perturbations of two planets can be separated, so that in determining the 
motion of one planet that of the other can be supposed to be known. As 
long as we confine our attention to perturbations which depend only on 
the first power of the ratio of the mass of any planet to that of the sun, 
this procedure is justified by the fact that the coordinates of the disturbing 
planet only appear in a function which has the mass of this planet as a 
factor. Hence, any perturbations of these coordinates will produce pertur- 
bations depending on the squares or products of two disturbing masses. 
When we begin to calculate these higher approximations, it is evidently 
necessary to calculate previously the perturbations depending on the first 
powers of the masses for both planets. 



26,26] SPECIAL CASES 173 

But the general problem of three bodies admits of four integrals in 
addition to those arising from the uniform motion of the centre of mass, 
namely, the three integrals of areas or of angular momenta and the energy 
integral. No use has been made of these in the theory developed above 
and the question naturally arises as to whether they can be effectively 
utilised for the abbreviation of the work. In asteroid problems where 
there is a very small mass disturbed by a very large one, the effect of the 
former on the latter is negligible, and the integrals consist chiefly of por- 
tions depending on the large mass, the effect of the portions depending on 
the small mass being relatively small. Thus the integrals are not useful 
in such cases. But when the two planets have masses of the same order 
of magnitude, as, for example, in the case of the mutual perturbations of 
Jupiter and Saturn, the variations of the coordinates in the portions'of the 
integrals due to the two planets have the same order of magnitude, and it 
would seem that this fact should be utilised to abbreviate the calculations. 
It generally appears, however, that the lack of symmetry which their use 
introduces, causes additional difficulties in the calculations. The more 
useful procedure is that of following the usual method for each of the 
planets and later making the integrals serve as tests of the numerical work. 
These tests are particularly valuable for the coefficients of any terms of 
very long period which may be present. 

For theoretical work in the general problem of three bodies, these inte- 
grals have been much discussed. Since there are four of them, the system 
of variables, namely six for each planet, can be reduced from the twelfth 
order to the eighth. 

We shall see in a later chapter that it is not always possible to proceed 
by following the process described at the beginning of this article. It 
breaks down in certain cases of resonance and notably in the case of the 
Trojan group. If, for example, we attempt to determine the action of 
Saturn on a member of this group without taking into account at the 
same time the action of Jupiter, quite erroneous results will be obtained. 
A difficulty of a similar nature occurs in dealing with the motion of a 
satellite disturbed by a planet other than that about which it is circulating : 
it is necessary to take account of the disturbing action of the sun during 
the computation of the disturbance caused by the planet. 



CHAPTER VII 

PLANETARY THEORY IN TERMS OF THE 
ORBITAL TRUE LONGITUDE 

A. EQUATIONS OF MOTION AND METHOD 
FOR SOLUTION 

7*1. The equations of motion have been derived in 1/27. The 
independent variable v is the longitude reckoned from a depar- 
ture point within the osculating plane, while the longitude v is 
reckoned in the usual way from an origin in the plane of reference 
to the node and then along the osculating plane. The radius 
vector is r and i, 6 are the inclination and longitude of the node 
of the osculating plane. The force-function is 

fj./r + /j,R, (1) 

so that jjuR now denotes the disturbing function. The remaining 
definitions and the equations of motion are as follows: 

?/,= -, (c) = r 2 r=l cost, (2) 

r \q/ dt x ' 

I M """" Q == Q "^ "~" 9. ~^\ ~T "" == Q ~*"\ I t-i ~7 " ~7 ) \ <5 ) 



__ _ , 

dv" u 2 dv ' dv~\p u 2 ' dv dv' 

...... W,( 

d 



The latitude L t defined by 

sin L = sin i sin (v 0), .................. (9) 

may be found directly by solving the equation 

(d 2 i\ r sin i cos i a dR .-. x 

^-j-f llsini^-^-T - ^-^^rr ....... (!0) 
dv 2 ) sin (v -0)u 2 dr v ' 



1-3] EQUATIONS OP MOTION 175 

The values of the variables u, v, t, L (or F, 6) are to be deduced 
from these equations in terms of v. 

7'2. It was pointed out in 1'27 that any substitutions of the form 

=/(?!) M-stt^fal) 

leave the essential characteristics of these equations unchanged, namely, 
that they shall be integrable like linear equations with constant coefficients 
when the right-hand members have been expressed in terms of v. A trans- 
formation which renders the equations useful for the treatment of the 
satellite problem is 

q = q t M=z 1 gr 1 4. 

The transformation to the new variables HI, q^ is straightforward. For 
the Ui equation we have, if D be written for djdv, 






so that Du^ disappears. This and the remaining equations become, if 
be put for w 3 3fl/9w, and /? for u*R, 



/>a Ml + Wl - q L = - -- 3 /^ - wi n /> ( =^ 1 j - g 

=-^ ix . Dt = ~,^ -.. Z)y=l-fr/)0, 

</! 2 Wj 4 8y ' ^ ?^ ' ' 

"r "" 3 "^" "" ^* r W "~^7 4 ^F 

When the ratio of the distances is neglected we have R V 'R V and each 
is independent of u\ . This portion constitutes the chief part of the dis- 
turbing function in the satellite problem. 

It will be noticed that Dt is a function of u\ only, and in fact that the 
substitution % = (DQ"~i/i~i eliminates the radius vector from the equations 
of motion. No particular advantage, however, appears to be gained by this 
elimination. 

7*3. The method for solution. 

The equations will be solved by continued approximation. The 
function R contains as a factor m'jp, the ratio of the mass of the 
disturbing planet to the sum of the masses of the sun and the 
disturbed planet. This factor being always small (its maximum 
value is less than '001), the first step is the solution of the equa- 



176 TRUE LONGITUDE AS ARGUMENT [OH. vn 

tions with R = 0. As we have seen in Chap, in, this solution 
gives elliptic motion and the solution will be called the elliptic 
approximation. The results consist of expressions for the variables 
u, t, ..., in terms of v. 

For the first approximation to the disturbed motion, these 
expressions are substituted in the terms which have m'/ft as a 
factor and the equations are solved again. Analytically, the 
process presents no difficulties since, with the exception of the 
equation for t, the right-hand members become functions of v 
only, while the left-hand members are linear with constant 
coefficients. The value of q is first obtained and then those of 
u, t and of the remaining variables. 

The second approximation is similarly obtained by substitut- 
ing the results from the first approximation in the terms which 
have m'/fji as a factor and proceeding as before. It is rarely 
necessary to go beyond this stage in planetary problems and, in 
fact, a second approximation is necessary in general only for 
those terms which, on account of their long periods, have re- 
ceived large factors during the integration of the equations 
giving the first approximation. 

The system of differential equations is one of the seventh order 
while that from which it was derived was of the sixth order 
requiring six arbitrary constants. The additional arbitrary con- 
stant necessary in the new system owing to the differential 
definition of v, will be defined as follows. The final expression 
of v v in terms of v is a sum of periodic terms and powers of 
v : when all these terms are neglected we are to have v = v. In 
general, this is equivalent to putting v = v when m' = 0. But 
there are sometimes periodic terms present whose coefficients 
do not vanish with m' but whose arguments become constant 
when ra' = ; the relation v = v is to hold when these terms are 
suppressed. 

7*4. The elliptic approximation. When R = we have q, F, 
0, i constant and with D = d/dv, 



3, 4] ELLIPTIC APPROXIMATION 177 

In accordance with the definition given in the previous section, 
the last equation gives v = v. The solution of the equation for 
u can be written 

- = u = q + qe cos (v -or), 

where e, ty are arbitrary constants. As we have seen in Chap, ill, 
this is the equation of an ellipse with the origin at one focus. 
If 2a, e, isr be the major axis, eccentricity and longitude of the 
nearer apse of the curve, we have 



Following the notations of Chap, in, namely, n defined by 
n a a 3 = /*, with nt 4- txr as the mean anomaly and X as the 
eccentric anomaly, we obtain t expressed in terms of v by the 
equations [cf. 3'2 (16) and (20)], 

nt 4- e TV = X e sin X, tan J X = I - - ) tan ( v vr), 

\ 1 -f" 6/ 

...... (1) 

or by [cf. 3'8 (3)], 
nt + = v Ef, Ef = 20 sin (v 57) Je 2 sin 2(v r)+ .... 

...... (2) 

The solution of the equation for i gives 

sin Z = sin i sin (v 6). 

The arbitrary constants of the solution are q, e t t*r, i, 0> 6. It 
is, however, more convenient to regard n as one of the funda- 
mental arbitraries since it is determined more directly from 
observation, and to regard q as a function of n, 0, defined by 
means of the equations l/q = a (I e 2 ), n 2 a 8 = /A. 

The adopted definitions of u, q } R give them the dimension 
1 in length. If we put u/a Qy <?/a , ^R/o for these symbols and 
define n by the equation no 2 3 = p> none of the equations except 
that for Dt is altered and the latter becomes n^Dtq^u^. The 
unit of length is at our disposal: it will be found convenient to 
so choose a that T? O is the mean value of the angular velocity 
of the disturbed body which has been adopted. With this 
definition therefore, we put a = a, n^=n in finding the first 
approximation to the perturbations. 



178 TRUE LONGITUDE AS ARGUMENT [OH. vit 

We shall suppose this transformation to have been made so 

that 

u = (1 + e cos/) -r a (1 - e 2 ) 

with a = 1 is the elliptic approximation, to the end of 7'18. 



B. THE FIRST APPROXIMATION TO THE 
PERTURBATIONS 

7' 6. Development of the disturbing function. 

According to the plan outlined in 7 '3, the elliptic approxima- 
tion, that is, the values of the coordinates in terms of v and six 
arbitrary constants, found in 7'4, is to be substituted in the 
derivatives of R which are present in the right-hand members- 
of the equations of motion. 

Since the disturbing function is here denoted by pR we have, 

from 1-10, 

n m' (\ rcosS\ A2 , , ~ 
7? = ( ^ -73- J , A 2 = r 2 + r 2 - 2rr cos S, 

cos S = cos (v 0) cos (v f 6) + cos / sin (v - 0) sin (v' 6) 
= (1 - JF) cos (v - v') -f^F cos (v + v' - 20). 

The disturbing function contains the coordinates r', v' of the 
disturbing body and these must be expressed in terms of v. Since 
the orbit of the disturbing body is used as the plane of reference, 
we have i = /. 

The work is best done in two steps. First, r', v' are expressed 
in terms of t by means of 

1 iWcosfo'-tQ 
r '- a '(i- e '*) 

v' = n 't +e' + 2e' sin (n't +'-*/) + > 

as found in 3*11, and then in terms of v by means of the relations 
similar to 7 '4 (1), (2). 

It is found convenient to introduce the angles 



4, 5] DEVELOPMENT OF R 179 

Evidently f is the true anomaly of the disturbed planet and /i 
is the mean value of the true anomaly of the disturbing planet 
when the latter is expressed in terms of v. The derivatives of 
f t fi are in the ratio n : n', that is, in the ratio of the mean 
motions. 

The disturbing function is ultimately expressed as a sum of 
cosines of multiples of the angles f, /i, vr -or', r -H r' 20, with 
coefficients which depend on a, a', e y e', F. The chief difference 
in the literal form of the expansion from that obtained with t as 
the independent variable is the presence of a/a' in the form of 
powers of n'/n as well as directly. But as these powers of n 9 jn 
occur only in rapidly converging forms they cause little additional 
trouble*. 

The expression for R used above assumes that the plane of motion of the 
disturbing planet is fixed and adopted as the plane of reference. It should 
be pointed out that, as far as perturbations of the first order with respect 
to the masses are concerned, it makes no difference whether this plane is 
fixed or moving. For since its motions are produced solely by other dis- 
turbing bodies, they contain the disturbing masses as factors. But the 
effects of the disturbing body enter the equations of motion only through 
R which has m' as a factor : these motions will therefore produce perturba- 
tions having the product of two disturbing masses as a factor. 

Hence, if we have solved the problem under the assumption that the 
plane of reference is fixed, the solution to the first order still holds when 
we transform to another plane of reference which is actually fixed, the 
motion of the former plane being included in the transformation. In other 
words, we need to take into account only the geometrical effect of the 
motion of the plane of the disturbing body and can neglect its dynamical 
effect on the disturbed body. Actually, these second order dynamical 
effects are, in most cases, so small that they may be neglected in making 
comparisons with observations. 

Another point of a similar character may be mentioned. We are sub- 
stituting constant values for the various elements in the expressions for 
the coordinates in R. To the first order of the disturbing forces it makes 
no theoretical difference what these elements are, whether, for example, they 
are osculating elements at one date or another, or are mean elements derived 

* A method, similar to that just outlined, for developing the disturbing func- 
tion in terms of the true longitude, is given by C. A. Shook, Mon. Not. R.A.S. 
vol. 91 (1981), p. 553. In this paper will be found the literal development to the 
second order with respect to the eccentricities and inclination. 



180 TRUE LONGITUDE AS ARGUMENT [CH. vn 

in some manner, for all these sets differ from one another only by magni- 
tudes of the order of the disturbing forces. But when we compute to the 
second order actual definitions are necessary. In general, we get better 
accuracy when we use mean elements if they are known. Such elements 
are best found after the theory has been completed and their insertion 
usually involves small corrections to those which have been used in forming 
the theory : in most cases such corrections are easily made. 

For these reasons, it is not necessary, in forming the equations for the 
first approximation, to use a separate notation for the elements used in the 
elliptic approximation and for the new values which may be assigned to them 
in the first approximation to the perturbations. 

7*6. Numerical developments of the disturbing forces. The 
following method of calculation is based on the possibility of 
expressing the disturbing function and the disturbing forces in 
the form 

ZiKiCOsi (/-/i + CT ~ r') + ^iKi sin i(/-/i + *r - w'), (1) 
where the coefficients are series of the form 

2j,h ^h cos (3f + A/i) + ^ A'M sin (jf+ j/V/i), . . .(2) 
the latter coefficients containing the numerical values of the 
eccentricities, of the inclination and of 2tsi 20. The coefficients 
in the series (2) are supposed to be calculated by double harmonic 
analysis for each value of i needed. 

The principal reasons for the adoption of this plan are first, the slow 
convergence of the coefficients A*, K{ with increasing values of i, and 
second, the comparatively rapid convergence of the series for AJJ , A ' JtJ , 
so that only a few special values of /, /i are needed for the harmonic 
analysis. No additional calculation is involved by the retention of w w' 
in a literal form as far as the final step. 

Omitting the factor m'/p, we have 
i^ 7 t3 cos S 



_ 
du ~ dr A 3 "/a 

1 r/2 "~ r2 r2 cos ^ 

~~"2A r 2A 8 7' a ' ............ (' 

^-2tf) l ...... (5) 



fr-7] NUMERICAL DEVELOPMENTS 181 

(iR) = - - . r' sin (v - 6) sin ( - 0), . . .(6) 



When 71 (10) is used instead of 7'1 (7), (8), the calculation of 
(5) is needed only for that of (7); (6) is then calculated without 
the factor sin (v - 0). Of. 1'28. 

It will appear below that of these, the development of (7) re- 
quires the highest degree of accuracy. Somewhat lower accuracy 
will serve for (4), and still lower for (5), (6); since (5) contains 
the factor F, these facts require that (3) shall be carried to the 
highest accuracy of all the functions. The most extensive part of 
the work is the development of the functions r 2 / A, r 3 / A 3 , the former 
being needed to a higher degree of accuracy than the latter. 

7*7. Numerical development of the disturbing function. 
Define A, B, by the equations 

.A cos(v - <?-) = cos (v-0), 



A sin (v - - B) = cos / sin (t; - 0).J ......... ^ ' 

The expression for A 2 in 7*5 can then be written 

tf = r 2 + r' 2 -2rr'Acos(v-v'-B) .......... (2) 

From (1) we deduce 

-4 2 ==l-sm 2 /sin 2 (?;-0), ............... (3) 

A sin B = sin 2 J / sin ( 2v - 2(9), 



v - 2(9), ) 
sm 2 0;-<9).j ......... 



Since v=/+'zar, the special values of A y B corresponding to 
the chosen special values of / can be calculated from (3), (4) 
when the numerical values of F, CT are given. 

Next, define ?*i, C by 

(r, 2 + r' 2 ) C 2 = r 2 + r /2 , r^W^rA, ......... (5) 

so that A 2 = C 2 (n 2 + r' 2 - 2r x r' cos (v - v' - B)} ....... (6) 

The special values of r, r' corresponding to special values of 
f, fi having been found, those of ri, C can be obtained con- 
veniently by calculating X, \i from 

T 

tan\= , sin2\i = A sin2X, ............ (7) 

B&SPT IZ 



182 TRUE LONGITUDE AS ARGUMENT [OH. vu 

and then ri, C from 

/ , _ cos \i /ox 

n-rtan^, 6'-, .............. (8) 

equations which will be found to satisfy (5). 
The expression (6) for A 2 gives 



...(10) 



Methods for the expansions of these functions have been 
given in 4'23 and the following sections. The particular form 
which is useful here is that in which we put 

(1 + a 2 - 2a cos ^)- = /8/> + S 2/3/> cos ii/r, . . .(11) 
with a ri/r', tyv v' B, i=l, 2, ..., s = J, f. 

The expressions for the /3 8 (i > are given by 4*24 (6) with K= 1, 
! == a, namely, 

/v* ^v 

/P (i) - (t) ^ (i) _____ ..... _ ,7 (t) 

" " 



where 



t 2.4...2t ->' '-"> 

in which j9 = a 2 /(l a 2 ). 

The methods developed for the calculation of these functions 
depend on the numerical value of a being given. In the present 
case these numerical values are the special values of r*i/r'. The 
efficiency of the method outlined here depends on the existence 
of tables giving the coefficients for different values of a*. 

* The tables of Brown and Brouwer, I.e. p. 103, give Iog2^() for t=0 to 11 
and for s=, f , f to 8 places of decimals and for s = J to 7 places of decimals. 
They are tabulated for values of p at intervals of '01 up to p = 2-5 or a =-83. 
For higher values of p up to p = 6 (a = -93) rapidly converging series are given. 
A separate table for a -90 to -95 is added. 



7, la] NUMERICAL DEVELOPMENTS 183 

In this way, each of the functions 7*6 (3), (4), (5), (6) is expanded 
into a series having the form 

22C (i) cosi(v-v'-B\ ............... (12) 

in which the special values of the coefficients for each required 
value of i have been obtained. 

We next put 

V - v r - B =/> *r /i -or' - BI, 
where Bi=f'-fi + B, 

and calculate the special values of B t . 

The final special values to be computed are those of 



for each value of i. Each of these functions is then analysed into 
a series of the form 

2 t , il L 1>il cos(jf+j 1 f 1 ) + - i , il L' itil sm(jf+j 1 f i ). ...(13) 

The results give series having the form 7*6 (1), (2). After 
the derivative of 7 '6 (3) with respect to -cr' has been formed, 
the numerical value of &'&' is inserted in all of them, the 
various terms having the same multiples of f t f\ in their argu- 
ments are collected and each expression is put into the form 
(13) or, if desired, into the form 

M it h cos (jf + A/I - N it fc ) ............. (14) 

The most important preliminary step is the expression of f f 
in terms of/i,/. 

7'7a. Expression of the true anomaly of the disturbing planet 
in terms of that of the disturbed planet. 

When harmonic analysis is to be used (App. A), the following 
method gives the required transformation rapidly. 

We have, according to previous definitions, 

g=f-E f , flr'=/i-^,=/i-/i, 

where Ef is the equation of the centre with eccentricity e and 
true anomaly f. Also 

/' = g' + E, =/i - 8/i + E (/i - 



184 TRUE LONGITUDE AS ARGUMENT [CH. vn 

where E g > = E (fi S/i) is the equation of the centre of the 
disturbing planet with eccentricity e' and mean anomaly 
g' =/! S/i. If this function be expanded in powers of S/i we 
obtain, if E' (/i) denotes dE(fi)/df 1) 



Since f\ 4- E(fi) =/ is the expression for a true anomaly in 
which e' is the eccentricity and /i takes the place of the mean 
anomaly, we can write the equation in the form 

- df n' \ 1 d*fn' 

- 



The special values of Ef are calculated with special values of 
/ in the usual manner. Those of / with special values of f\ are 
similarly obtained. For the derivatives we have 
d f ^(1 + e'cos/) 2 
dfi~ (I-**)* 



Hence = (1 - e' 2 )~* (1 + e' cos/). 

If we denote the successive derivatives of /with respect to fi 
by the notations /, f, ..., and put e\= e' (1 e /2 )~^, we can 
obtain the following formulae for their successive determination : 



i ei cos/ 

/ 1 
- 



W ~T6 ' 2-J- 

which will be sufficient for all practical needs. 

The calculation of/,/ iv from the series 3'16 (2) will be found 
to be sufficiently accurate in most cases and is rather easier than 
that from the formulae just given. 

The amount of calculation needed in any particular case depends on a 
variety of circumstances. Before undertaking calculation on the general 
plan outlined above, a preliminary survey should be made to find the 



7a,8] SOLUTION OF THE EQUATIONS 185 

order of magnitude of the term with the largest coefficient in the longitude, 
pr, in the present case, in t expressed in terms of the longitude. Usually, 
the term is one having a long period. The order of magnitude with respect 
to the eccentricities and inclination for a term with argument jf+ji f\ is 
\j+ji\* While a rough approximation to the coefficient can be obtained by 
following the method developed in 7*38 below, the degree of accuracy, that 
is, the number of places of decimals needed in the calculation, can be 
found from the number of significant figures needed to obtain this coefficient 
with the required accuracy. 

The accuracy possible with the methods developed above is theoretically 
unlimited, but is practically limited by the accuracy of the tables of the 
coefficients g^\ Those referred to in the footnote on p. 178 are sufficient 
to obtain solutions of practically all the planetary problems in the solar 
system with the accuracy needed at the present time; the determination 
of the great inequality in the motion of Saturn is probably the limit in 
this respect. 

It may be pointed out that since A < 1, C> 1 (p. 178), the inclusion of 
the inclination in r^r' in general tends to diminish this ratio and therefore 
to increase the rate of convergence. Thus, if we can obtain a certain degree 
of accuracy with 7=0, we can obtain at least the same degree of accuracy 
with /=t=0. The method is thus particularly effective for large inclinations. 

The method of procedure outlined above is a general one. The experi- 
enced computer will see various ways in which it may be abbreviated. One 
important choice is the number of special values to be adopted for /, /i . In 
the majority of minor planet problems, the values of /, /j at intervals of 
45 will serve. If the eccentricity of the planet or the inclination is large, 
additional values at intervals of 60 may be used : these additional values 
merely involve corrections of the coefficients in the last part of the process 
that of the harmonic analysis so that all the previous work is fully 
utilised. It is not difficult to settle at the outset the number of places of 
decimals required, but it is not easy to say how many special values should 
be used. The work can be started with the minimum number and others 
can be added afterwards without the loss of the previous calculations. 

7*8. Solution of the equations. 

The methods of the previous sections give the expression 
of R in the form 

JZ-SJCcoB^+jj/i + i), j = 0, 1, 2, ...; ^ = 0, 1, 2, .... 
When a literal development is made the angles k are multiples 
of r r', w + ta' 20, and the coefficients K are functions of 
a/a', e, e', F. In a numerical development the terms having the 



186 TRUE LONGITUDE AS ARGUMENT [OH. vn 

same values of j> ji are gathered together and R and its 
derivatives are expressed in the form 

0o + 2 cos (jf + jj/x) 4- 2/3' sin ( jf + ^/i), 

or in the form _ 

+ 2 cos (j/+ji /i-lto, 

where y9 > A /3', J5, l?i are numerical quantities dependent on all 
the elements ; in these expressions the terms in which j = ji = 
are gathered into the symbol /3 

The constant y8o is independent of v and is implicitly a 
function of the angles w -GT', OT -f TB' 20. The terms present 
in this constant when expressed in a literal form possess the 
property (6'4) associating a power of e in the coefficient with the 
same multiple of -cr in the angle, with similar properties for 
iff', e' and for 20, F. The corresponding properties are obtained 
when j t ji are not both zero by putting cr =1; /, & =vi fi 
and associating powers of e t e' y 20 with the respective multiples of 
/, /i, 20 a statement which is easily seen to be true by referring 
back to the development of R in 4*14. 

Finally, since we have put v = v, we have 

D (Jf + ji/i) = J + h n 'l n = s > 

so that s becomes a divisor of the coefficient when we integrate 
one of these expressions. 

It is evident that Rf(r, /) will possess these same properties. 

7*9. The equation for q. This equation has the form 

J5 s = / 9 -f^cos(j/+j 1 / 1 ) + SyS'sin(j/+j J / ] ), (1) 

and its integral is 
q = ?o + /9ov + 2 f sin ( j/+ jj/i) - S f cos (jf+jtf), . . .(2) 

o o 

where <jo is an arbitrary constant to be defined later. The term 
$ov is the secular part of q. The terms for which s is small com- 
pared with unity are those of long period, the remaining terms 
having 'short 1 periods, that is, periods of the same order of 
magnitude as 2-7T/71, the period of revolution of the planet, and 
shorter. 



8-10] THE EQUATIONS FOR q, u 187 

7*10. The equation for n. The right-hand member has 
already been developed into an expansion of the form 7*9 (1). 
In the left-hand member, the value of q just obtained is sub- 
stituted. The periodic terms in q are added to the terms of the 
right-hand member and the equation takes the form 

jy u + u = q Q -f v + fa cos/+ fa' sin / 

+ /3 MO + 2&* cos (jf + J!/ t ) + 2&/ sin (jf -f ^/i), 

the terms for which ji = 0,^' = 0, 1, simultaneously, being isolated. 
The integral of this equation is 

u = <?o + ov -f /3i v sin / /3 x ' v cos / 4- e c cos / 4- e 8 sin/ 4- t* p , 

...... (1) 



where 



2 -"-8 cos <# + M) + 2 -~r a sin 

JL o 1 o 



as can be seen by submitting each member to the operator 
D 2 -f 1; e c , e a are the arbitrary constants in the solution. 

Now after the substitution u/a for u, q/a for q, the elliptic 
values of u, q are 

__ 1 -f e cos / _ 1 /____ 

with e, -BT as arbitrary constants. As q Q , e c , e s are at our disposal 
we can put 

e =___._ e = o q = 1 



so that the remaining terms would constitute the perturbations. 
These values of e c , e 8 will be adopted, but instead of that for q Q 
we shall put 



where Sq Q is still arbitrary. It will later be defined to be such 
that the mean value otnDt shall be unity. 

We shall next show that u, q may be written in the forms 

...... (2) 



188 TRUE LONGITUDE AS ARGUMENT [OH. vii 
where - 1+ '- **>. -* + ! (3) 



Here is the value of e which has been used in the develop- 
ment of R\ TI, ei are small constants whose squares may be 
neglected. 

The expansion of UQ in powers of e\, -CTI, gives 

__ l+o cos/ 2eo 0_i l-h#o 2 f o 

Comparison of this with (1) shows that if we put 
lO-V) 2 ;,, il-eo'* 



the coefficients of v cos/, vsin/ in (1) will be included in 
The term /3 v will also be included if 



The argument in 7*24 shows that this relation is satisfied, so that 
it constitutes a useful test of the accuracy of a part of the 
numerical calculations. Hence u and q have the form (2). 

The terms #iv, wj.v are called the secular motions of the 
eccentricity and longitude of perihelion. Expressed in time 
they would be e^ni, vrint. 

Since we are neglecting squares of the disturbing force we may insert 
these secular parts in the perturbations w p , q p . If a development in which 
the literal values of e^ w have been retained is made, this can be done by 
replacing / by / s^v and w by or+orjv, and e by e + e v \ in w p , q p a 
procedure which is advantageous as will appear later. Usually, however, 
it is not possible to make these changes because it is customary to use 
numerical developments in order to save labour. 

7*11. The equation for t. Since Sq Q , u p , u q are of the order of 
the disturbing forces, we have, as far as the first order, 



{1 4- e cos(/' 
x 



10-12] THE EQUATION FOR t 189 

In the terms factored by 8q Q , q p , u p we can put <?i = 0, ^ = 0, 
since the products would be of the second order. Hence 



We now determine Sq Q to be such that in the expansion of 
the right-hand member as a sum of periodic terms, the constant 
term shall be unity. 

By 3'8 (3), the constant part of the first term is 1. It will be 
shown in 713 that if we expand (1 4-e cos/)~ 3 into a Fourier 
series the constant term is (1 + ie 2 )(l - eo a )~*- The constant 
term in the coefficient of Sq Q is therefore 

j(l _ ef) - ( 2 + e ?) = - t (1 + e, 2 ). 
The required condition therefore gives 

^of i - 2 x constant term in the expansion of 

1 "T #0 



It is evident that we only need those portions of r p , u p which 
are independent of the argument /i. It is recalled that q p con- 
tains no constant term and that u p contains no term with 
argument/. 

When Sq Q has been found, its value is inserted in (1). The 
second and third terms are then calculated in the form 

(1 + e cos/) & Bq 



the constant term of which should vanish, so that these portions 
give a sum of periodic terms of the form 

3e cos (jf + jifi) + 2&' sin (jf+ j,f t ) . 



712. Integration of the equation for t. If we neglect all per- 
turbations so that e = e Qy ori= 0, we obtain by 3*8 (3) 

nt + e= v-E fy ..................... (1) 



190 TRUE LONGITUDE AS ARGUMENT [OH. vn 

where Ef is the equation of the centre expressed in terms of the 
true anomaly /. To obtain the integral when e\, ts\ are not 
neglected put e Q + e^v, f tsriv for e, f in (1) and differentiate. 
We obtain, if squares of e\, TZ\ are neglected, 



The first two terms of the right-hand member evidently give 

(1 - <? 2 )* {1+ ecos (/- r lV )}- 2 ; 

in the third and fourth terms we can put e e Q , is\ v = 0. Hence 
the integral of the first term of 7*11 (1) gives 

ffiji 1 f r) J? 

nt + = v - E f - eij -~^df+ TI j -j^ d /, 

where is the constant of integration, and the values e^ + eiv 
and / tsrjv are used for the eccentricity and true anomaly in 
the expression for the second term, that is, for the equation of 
the centre. The integral of the third term is given by the 
formula 7*13 (4) below; that of the fourth term is &iEf. 

The remaining terms in the expression 7*11 (1) for nDt, the 
form of which is given at the end of 7 '11, are integrated im- 
mediately. We obtain 
nt + 6 = v - 



+ S sin (jf + j,/,) - 2 i cos 

o o 

where s=j+jin'/n, E f is the equation of the centre with 
eccentricity e Q + eiV and true anomaly/ -B^V and, by 713 (4)> 



= 2 (- 1)* eo 1 " 1 ( 1 + e 2 H r ] cos if, e Q = 1 2 . 

713. The Fourier expansions which are needed above are obtained from 

fl-*) 1 - CD 



[)(l+e COS/) 2 

Differentiation with respect to e gives 



12-14] THE EQUATION FOR r 191 

and differentiation of this result with respect to / provides 



The value of dE/fie can be obtained either from the series for E/ or by 
integrating (2). The latter integration is performed by making use of the 
identities, 

d sin/ __ cos/4-0 8 sin/ _ 2<?4-cos/ 0cos 2 / 

a/" 1 + e cos/ = (1 +e cos/) 2 ' a/* (1+0 cos/) 2 = (1+ecos/) 3 ' 
the sum of the right-hand members being 

1 2-M 2 2 l-i 2 



_ 

(1+ecos/) 3 ~" e (1 + ecos/) 2 "~~e (\+e cos/) 3 ' 
Hence 

^ - sin / sin / 



\l+ecoa/ (1+ecos/) 2 / 



The value ofFfD' 1 (dEf/de) is obtained immediately from this result, and 
it also completes the expansion 7*13 (3). 

The calculation of these expansions by harmonic analysis is at least as 
rapid as by series and is advisable for large values of e. Harmonic analysis 
can also be used to calculate E f when e = e Q + ei\ by calculating the coef- 
ficients with = 00, an d again with e~e Q -\-1cei when k has some convenient 
numerical value (? 100). The difference between the resulting pairs of 
coefficients divided by k gives the factor of v in the coefficient. 

714. The equation for F. The equation 7*1 (7) for F gives, 
on integration, 

F^ + FiV + F^i .................. (1) 

where FI is the constant term in the expansion of ql dR/u 2 dO, 
T p is a sum of periodic terms, and k is an arbitrary constant. 
The value 7*10 (2) for q gives, on expansion to the first order, 



+ q p ) (1 - ,)*, 
whence 

r A (1 - eoT* + 4 % (1 - ^o 2 )* -f (*cio (1 - eoT* 

+ F! (1 - e 2 )-*} v + i* (1 - ^ 2 )* g p + (1 - ^o')"* ^P- 
The first two terms in this expression constitute the constant 
part of F; denote it by F . In terms which have the disturbing 



192 TRUE LONGITUDE AS ARGUMENT [OH. vn 

mass as a factor we can put &= F (l e Q 2 )^. Hence the value 
of F is given by 

r = To + [e l e, Fo (1 - eoV + Fx (1 - * T*} v 



When the disturbing forces are neglected we have r = F , 
which is therefore the value of F used in calculating the per- 
turbations. The secular motion of F is the coefficient of v. All 
the terms contain F as a factor. 

Since F = 1 cos 7, SF = sin 7S7, the equation for F, which 
may be written F = FO 4- SF, gives 7 = 7o + SF/sin 7 , where ST 
contains Fo = sin 2 7 /(l -f cos7 ) as a factor. Hence when 7 is 
small the perturbations of 7 have 7 e as a factor. 

715. The equation for 0. The integral of 7*1 (8) gives 
0=0 + 0iV4-0p, 

where the signification of the symbols is evident. Here is the 
value of used in calculating the perturbations and 0iv is its 
4 secular motion/ 

7*16. The equation for v. Since dO/dv contains the disturbing 
mass as a factor, we can put F = F in 7*1 (6) so the integral is 



no constant being added, in accordance with the definition in 
the last paragraph of 7*3. 

The usual definition of n is to make it the mean value of 
dv/dt rather than of dv/dt as defined in 7'1 (6). This definition 
requires us to replace n by n/(l ~ Fo0i). Since n is not present 
in the coefficients of elliptic motion and since the change may 
be neglected in the perturbations, no further adjustment of the 
value of t in terms of v is necessary. The value of 8q Q (7*11) is 
altered and receives an additional part F 0i/(l +e 2 ); this is 
usually insensible in the value of u and elsewhere. 

The only further change necessary to obtain t in terms of v 
is the replacement of v by v -f F P . 



14-18] THE SMALL DIVISORS 193 

7*17. The equation for sin L. When T0 P can be neglected, we can save 
some labour by integrating the equation for sin L to replace those for r, 6. 
If we treat the right-hand member like that of the equation for u, isolating 
the terms with argument / and also the constant term, the equation takes 
the form 

............... (1) 



where g p denotes the remaining periodic terms with arguments jf+j\f\ . 
The solution, like that for u, may bo written 

sin Z=sin 7 sin (/+ w -^o)+y - 4^i' v cos 



The constants are shown in the above form because we have in ellipti 
motion 

sin L = sin 7 sin (v - #o) 



The terms with factor v may be included in the solution 



if we put 7 1 v = S7, #^ = 0, where 57, $0 are determined from 

5 {sin 7sin (sr - 0)}= -^i'v, 8 (sin 7cos (ET O - ^Hi&v. ...( 4 ) 

718. The small divisors. The divisors s are present in the 
equation for q, and when s is small, that is, when the corre- 
sponding term has a long period compared with 27r/n, the 
coefficient will be increased by the integration. It is again in- 
creased by the same divisor in the integration of the equation 
for t, so that in the expression for t in terms of v or v, the 
divisor s 2 is present. 

The divisor 1 s 1 is small in the expression for u or sin L 
when s is nearly equal to 1, that is, when the period of the 
term is near that of revolution of the disturbed body. It might 
be expected that these terms in combination with the elliptic 
terms would produce terms with the product of small divisors 
5, 1 s 2 in t. It is true that they do so, but such terms in general 
have a factor e 2 as compared with the terms in q from which they 
arise, a result which will be evident when the following method 
of the variation of the elements is used. 



194 TRUE LONGITUDE AS ARGUMENT [OH. vn 

C. EQUATIONS FOR THE VARIATIONS 
OF THE ELEMENTS 

719. Let us return to the original equations of motion in 7*1 
and introduce three new variables a, e, ur to take the place of 
u, q. These new variables have as yet no relation to those denoted 
by the same letters in the previous sections of this chapter. 

We retain the notation D = d/dv and, for brevity of expression, 
introduce new operators defined by 

Dt=Da!l- + De%-+D*r-, D Q =~ ....... (1) 

da de dur 9v ^ ' 

Thus when we are operating on a function of a, e, tsr, v we have 



Since we are replacing two variables by three, one relation 
between the new variables is at our disposal. The three relations 
to be adopted are 

__ 1 -f 0cos (v CT) 1 ~ 1 -f e cos (v -or) __ 

U ~ a (l-e 2 7~~ ' ?== a(l-6)' * ~ a(\-#) ~ ' 

...... (2) 

so that Du = D Q u, 1)^ = 0, .................. (3) 

when u is expressed in terms of the new variables. 
Since Dq = D^q, the third of equations (2) gives 

{1 -f e cos (v cr)} Dq + qD\ {e cos (v )} = 0. 
This, combined with 7*1 (4), gives 

T\ ( / \1 ^ C/./V / > \ 

I>!{C08(V-W)}--^J- ................ (4) 

Next, the equation Du = DQU gives 

Di^ = qe sin (v tsr) ................... (5) 

Hence 

D*u + w g = D {gesin(v &)} -\-u-q 
= DI {qe sin (v -or)} 
= D^ . e sin ( v tsr) ^Z>i {e sin (v -or)}, 
since DO (^ sin (v r)} =u~q. 

But Dq.e sin (v -or) = (Dq/q) . DM. 



19, 20] VARIATIONS OF THE ELEMENTS 195 

Whence, from 71 (3), 

n f / M d R Du'dR ,-. 

DX {* sin (v - r)} = - -= --- =- -5- .......... (6) 

1 n du u* dv 

The equations (4), (6) are those which give the variations of 

y w " 

To obtain the equation satisfied by a, multiply equations 

71 (3), (4) by Du/q, {(Du)* + u 2 }/2q* respectively, and subtract; 
the result may be written 

| W + *_ 2 ) ag + ^ 

{ q ) dv on 

Substituting for u, Da, q from (2), (5), we obtain 

dR dR _ 



which is the equation satisfied by a. 

7*20. The last equation may be transformed into a form which 
is not only more convenient for calculation but which furnishes 
an important theorem concerning the secular terms. 

The disturbing function was originally expressed as a function 
of u, v, P, 6, and of t through its presence in r' y v'. Hence 

T. n dR -p. dR ^ oR nT1 dR ^^ dR *. 

DR = -5- Dv + ^- Dw + ^ DF + ^^ DQ + -^~ Dt. 
dv ou dT cQ dt 

With the use of the expressions 71 (6), (7), (8) for Dv, DT, DO, 
this reduces to 

T-. ,-> dR dR ^ dR r,. 

DR = -~- + *- Du + -zrDt ................ (1) 

dv du dt ^ 

Now t enters into r', v r only in the form n't + e', and this angle 
enters (7'5) into R only through fi in such a manner that 
^n'. Hence 



a result which is true whether R be expressed in terms of the 
old or new variables. Utilising (1), (2), we may write 719 (7) in 
the forms 



, ...... (3) 

\W 

since Dt = qft~ku~ z and since w does not contain /i explicitly. 



196 TRUE LONGITUDE AS ARGUMENT [OH. vn 

7*21. Finally, the expression for Dt, just quoted, gives, in terms, 
of the new variables with n 2 a s = /4 and the expansion 3*8 (3), 



<s7)} 2 n 7i 3v ' 



where Ef is the equation of the centre. Hence 



f^dv- Dodv + const ............. (2) 



The values of n = (/4/a 8 )^, 0, -or, as deduced from the integrals of 
7*19 (4), (6), (7) are to be substituted under the integral signs in 
(2), and the integrations are then to be carried out. 
Equation (2) may be written, since D = D D\, 

f=f!dv-^+ fl) 1 (^dv + const .......... (3) 

Jn n } \nj ' 

7'22. As in the earlier work, it is advisable to expand Rr 2 = R^ 
rather than R. With this change and with the usual abbrevia- 
tion,/= v w, equations 719 (4), (6) can be written 



.sinf=2u 9 ............ (1) 

De . sin/- eDv? . cos/= r 2 - -- Da ( ~ , ...... (2) 

the computation of r 2 9JR/9r being carried out in the manner ex- 
plained in 7*6. From these equations we deduce those for De, 
The equation for a becomes 






The equations for 6, T remain the same, namely 



The latter may be replaced by 



Equation 7 '21 (3) is used to 6nd t. 



21-24] SOLUTION OF VARIATION EQUATIONS 197 

7*23. Solution of the equations. We proceed as before. When 
R is neglected a, e y -or, F, become constants, and the motion is 
elliptic. These constant values are substituted in the expansion 
of jRx, the derivatives of RI and of R being obtained as in 7 '6. 
For the arbitrary constants we use these same values, except 
in the case of I/a, to the elliptic value of which we add a, where 
is so determined that when the equation for Dt has been 
formed, the constant term shall be represented by I/KQ, where 
KO is the observed value of the mean motion. 

The amount of calculation needed with the use of these equations is not 
very much greater than that required in the previous form. The additional 
work is mainly the multiplication of EI by the three terms in u\ namely, 



where q\ e are numerical constants, in order to find I/a, and the multipli- 
cations by cos/, sin/ to find e, w. 

In either case the work is less laborious than when t is used as the 
independent variable, chiefly because, in the latter case, the single terms 
cos/, sin/ have to be replaced by Fourier series containing a number of 
terms corresponding to the highest power of e we need to retain. 

The chief saving of labour in the use of the methods of this chapter over 
those of Chap, iv is due to the avoidance of the formation of derivatives 
with respect to e. There is great advantage in using the numerical value 
of e from the outset and if this be done we cannot find the derivative with 
respect to e. Further, the development, to secure a given degree of accuracy, 
requires the presence of one power of e higher in the latter case than in 
the former. 

7*24. The proofs of certain theorems, quoted earlier, follow 
easily from the equations of variations. 

To show that there are no secular terms of the first order in 
I/a and none of the form tcv 2 in t, we use equations 7*20 (3). To 
the first order, dRi/dfi has the same value whether RI be 
expressed in terms of u, v, t, F, 6, or in terms of f,fi and the 
constant elements. In the latter case, dRi/dfi will have no term 
free from the angle fi and consequently no constant term. Under 
the same conditions, RI has no term proportional to v, so that 
I/a has no term containing the factor v. 

B&SPT 13 



198 TRUE LONGITUDE AS ARGUMENT [OH. vii 

The secular terms in e, & are ei\, e\is. These, substituted in 
Ef, produce terms of the forms v cos if, v sin if, but no term of 
the form tcv to the first order. Since there is no term factored 
by v in 1/n, there will be none of the form /ev 2 in t. 

a . I 

Smce w= 



the substitution e=e + ^i v i Q the former of these terms will produce a 
secular term in u. This differs from the case of u expressed in terms of t, 
where the absolute term is I/a which has no secular part. This fact exhibits 
the artificial nature of the statement that the major axis has no secular 
part. When referred to actual coordinates, the existance of a secular part 
depends on the coordinates used. 

7'25. We next show that the only terms with the divisors s 2 
arise through the variable a. This is proved by 7*22 (6). For the 
integrals giving e, vs contain the first power of s only, so that 
Effn gives rise to this class of terms only. Also Di(Ef/n), which 
depends on Da, De, Dcr, has no divisors and its integral will give 
rise to terms of the same kind. The integral giving a gives rise 
to divisors s and that of I/n, through n 2 a* = p, to divisors s 2 . 

This latter result which was stated in 7*18 is of some assistance in the 
numerical developments of the methods in the previous section. It shows 
that the terms of long period in which s is small are needed in u to a lower 
degree of accuracy than terms with the same argument in q. If we calculate 
the coefficients of the long period terms of dlt/du to a lower degree of 
accuracy than those of dR/dv, but retain the same number of places of 
decimals, the theorem shows that the inaccurate or omitted portions cancel 
one another. The relative degree of accuracy needed must be judged from 
the small values of s present. 

D. THE SECOND APPROXIMATIONS TO THE 

PERTURBATIONS 

7*26. The results of the first approximations are as follows. 
We have obtained u, q, t, v, F, in the forms 



, l-f ecos(f 

where = 



v - E f , /= v - isr , 



24-28] SECOND APPROXIMATION 199 

in which e=*e Q + eiv; E f is the equation of the centre with 
eccentricity e> true anomaly/ OTI v, and o 3 ^o a = /* The constants 
a Q , e 0y <*o> r , #o have been used to compute the terms arising 
from R; e iy TI, 0i, TI are constants whose values have been 
found ; u p , q p , t p , T p , p are sums of periodic terms, u p , q p alone 
containing a constant term, so that Sqo of 7*10 (2) is now included 
in Up, q p . 

If we had solved by the method of the variation of arbitrary 
constants, the forms of t, F, 6, v would have remained the same, 
but for u, q we should have had 

to be substituted in 

u= {1 -f e cos (v tzr)} -f-a(l e 2 ), l/q = a(l e 2 ). 

7*27. To obtain the second approximation, these values of the 
variables, or of the elements, must be substituted in R in the 
place of the constant elements previously used. Whichever plan 
has been adopted in the first approximation, we can and shall 
still use the equations for the variations of the elements in the 
second approximation on account of their greater simplicity for 
both computation and exposition. The exposition will be limited 
by neglecting the variations of F, 6, v v. The effects of these 
variations on the perturbations of the second order are usually 
insensible, but they can be included, if necessary, by the use of 
the methods given for the other variables. 

7'28. Denote any perturbation of the first order by the symbol 
S and one of the second order by & 2 and put p = 1. 
Equation 7*20 (3) gives 



and 



, 1 

37 = 57T~ u 
8/1 8/ x 9M 



~ 

du 8fi 



3 (dRi\ . 


f d 2 Ri 




- /. ( ^ 1 ou 
dfi \ou / 


+ n Ztf 


,...(2) 
,...(3) 






13-2 



200 TBUE LONGITUDE AS ARGUMENT [OH. vn 



Since R^dRi/du, dRi/dfi, R have already been obtained in finding 
the first approximation and are expressed in terms of the angles 
/, fi t their derivatives with respect to /i are immediately ob- 
tained. If the first approximation has been obtained by finding 
the variations of the elements, we replace Sqk, Su by 

M<i-<rt *-+- ....... w 

where U Q = {1 + e Q cos (v - <sr )} 4- a ( 1 - e 2 ). 

It will presently appear that we can usually neglect Sa, Be, Sr 
and therefore &u, Sl} Q u, Sq in finding & 2 e, S 2 ^ When this is the 
case, equations 7'22 (1), (2) give 

rfi Ft 

(5) 
St. 



sin/D (8,) - e cos/D (S 2 *r) = n' f A ( r* d ) - D u 

(oji \ or J 



...... (6) 

The second order derivatives, again being derivatives with 
respect to/i of first order derivatives used in the first approxima- 
tion, are obtained immediately. Further, only a very few terms 
in &u, St have to be considered and the same is true of their 
products by the second derivatives of R or RI. Finally, as these 
variations enter into the right-hand members of the equations in 
a linear form only, we can compute separately the effects due to 
the few terms in Su, St which have to be taken into consideration. 

7*29. Calculation of the effects due to the secular terms in the 
first approximation. To obtain them we put 

eiv, tsr = T!3- 1 v, 

From these we get 

,j 

8u = 5- -i- 
ceo dtjj 

These results are to be substituted in the right-hand members 
of 7-28 (1), (2), (3), (5), (6). They give terms of the form 

fcv -f v SO sin (sv + a). 



28, 29] SECOND APPROXIMATION 201 

First, let us consider the equation 7'28 (1). The only portion 
of this equation which can give a term of the form tev from Su, is 

- D {A (/feu") . S| = - (D + 

since all the other portions are products of series one member 
of which contains /i and the other is independent of fi. For 
terms of this form we therefore have, since R = 



from which the terms of the form /cv are to be chosen. This is 
equivalent to putting e = e Q + e\v y tar = TJT O -I- OTIV in the constant 
term of the expansion of 2R^ if or in 2R. 
The expression 7'28 (2) may be written 

3 /3JZi, , /3JZi\ 3#i 3 , 3#i 3 



The two latter terms are equal to 3(8jR)/3/i. Hence when we 
substitute for frw, &t the portions u p , t p> 7*28 (3) will have no 
constant term and therefore will produce no term of the form 
KV in S 2 a. Evidently, these portions substituted in D (RiiP) 
produce no such term. Hence, the only term of the form /cv 
present in |8 2 (l/a) arises from the constant term in 

l n '_2fi d ^l 

* 90*3/1' 

which in general will not be zero. Thus the theorem, that I/a 
has no secular term of the second order when t is the independent 
variable, is not true when v is the independent variable. It may 
be noted however that it is true for a second order perturbation 
arising from two different disturbing planets. 

The same arguments evidently apply to the right-hand 
members of 7*28 (5), (6) which thus contain secular terms of the 
forms vcos(jf/-hji/i), v sm(jf+jifi), with ji=t=0. Hence these 
portions of 2 e> S 2 tr give no terms of the form #v 2 in the co- 
ordinates. Such terms, however, will arise from Su; when it is 
necessary to calculate them, we shall need those parts of the 



202 TRUE LONGITUDE AS ARGUMENT [OH. vii 

second derivatives of EI with respect to u,v which are independent 
of fi\ their calculation presents little difficulty since the degree 
of accuracy required is quite low. 

The practical importance of the theorem that there are no secular terms 
of the first and second orders with respect to the masses in I/a when t is the 
independent variable, has been much over-estimated. It is a result which 
eliminates certain secular terms from 1/r, but does not do so from r or, in 
general, from other functions of r. There is no particular reason from a 
physical point of view why, in getting a mean value of the deviation of the 
orbit from circularity, we should choose 1/r rather than r as the function 
to be averaged. The fact is that the separation of the deviations of the co- 
ordinates into deviations of a, 0, or, etc. is an artificial one, convenient for 
calculation and description, but one which has no particular physical 
significance. 

7'30. Terms of the form v cos sv, v sin sv are integrated by 
the formulae 

[ , v . 1 f . , v 1 . 

v cos sv dv = - sinsv-i- - cossv, v sin svav = cossv-H - sin sv. 
J s s 2 ) s s 2 

Such terms are present in S 2 a, $ 2 e, S a r and therefore in 8 2 it, 
DS 2 t The succeeding integration necessary to obtain S 2 t will 
introduce the factors v/s 2 , 1/s 3 . 

These terms will usually be insensible except when s is very 
small and even then the only portions which need be retained 
are those in St having the last-named factors. 

7*31. Calculation of the effects due to the periodic terms in 
the first approximation. Only the long period terms need to be 
considered. The right-hand members of the equations in 7*28 
have the form a derivative of R multiplied by Su or by St. In 
Su, the divisor s is present; in &t, the divisors s, s 2 are present. 
The products just referred to will produce products 

COS 5V . COS s' V = ^ COS (S $') V 4- i COS (S -f $') V, 

with similar results for sines of sv, s'v. Long period terms can 
arise in two ways: from short period terms in which ss' is 
small, or from long period terms in which both s, s' are small. 



29-32J LONG PERIOD TERMS 203 

The former will produce small divisors (s s') in Stu and their 
squares in S 2 . These will rarely be sensible. 

For the latter, we have divisors s, s 2 or 5', s' 2 in the right-hand 
members of the equations of 7*28, and therefore the smallest 
divisors present in their integrals will have the forms s 2 (s s'). 
Hence, the small divisors in S 2 , arising from S z a, will have the 
form s 2 (s s'f. Thus, whenever we are able to neglect the 
squares of the small divisors in the second approximation, the 
equations of 7*28 will be sufficient for the calculation of the 
remaining terms. Even then it is usually necessary to consider 
only one or two terms, so that the amount of calculation needed 
is quite limited. 

We shall now show that the chief part of a long period term 
in S 2 that having the divisor s 2 (s s') 2 can be obtained 
immediately from the first approximation. 

7'32. Calculation of the portion of a coefficient in &%t depend- 
ing on the fourth power of the small divisor*. 

We have seen in the previous paragraph that this portion can 
arise only from the term 



substituted in & 2 t= | S 2 ( - ) dv, ..................... (2) 

J \n/ 

where Bt arises from 



<$(-)dv, ..................... (4) 



substituted in 

by the use of the relation n 2 a 8 = /A. 
Suppose that the term in RI is 

a $ cos sv + a /S' sin sv = a Q A cos (sv + ^), sv = 

...... (5) 

* E. W. Brown, Mon. Not. R.A.S. vol. 90, p. 14. 



204 TRUE LONGITUDE AS ARGUMENT [CH. vii 

If this be substituted in (3) and the result integrated, we 
obtain 

-} = 2n' (&] a 4cos(v + ai) .......... (6) 

CL/ \/^/ S 



The equation n 2 a 3 = /x gives 
Whence, from (4), 



g! 3a, 8 l ...................... (7) 

ft 2i? . 






* 

From the substitution of (5) in (1), we have 

Moo cos (sv + !) . S< 
fiv + si) g< 

(sv + s 1 ), ............... (10) 



O 

from (8), (9). The integral gives 

^v-M,) .......... (11) 

\ v \ / 



2 
a 6 

And since {S(l/a)} 2 has the divisor s 2 only, while S 2 (l/a) has 
the divisor s 8 , we have, neglecting the former, 

/1\ 3 do. A 



Finally, 



which, with the use of (11), gives 
*. In'. 



(12) 



32-35] LONG PERIOD TERMS 205 

7'33. If we have two long period terms in RI, namely, 

Hi = a Q A cos (sv 4- si) + a^A' cos (s'v -f $/), 
and if we wish to obtain the terms due to their combinations, it 
is evident from equation 7*32 (10) that we shall have 

= !- B cos (sv + Sl ) . B' sin (' v -f O 

' C o S ( 5 V + 81') . B sin (^v -f si) 
- ' (^ V) s in K*' ) 

O 



the upper sign giving one term and the lower the other. 
The process previously followed gives 

V = \ J Bff -gr^ 1 ' sin [(,' s)v + s l ' s,}. 

In the case s' = s, si'^si, the term arising from the lower 
signs disappears since ji = jV- The remaining term gives 

1 ^' 
8 2 ^ = - _ frBB' sin (2sv + s l + Si). 

O ??Q 

7*34. The same method may be applied to find the effect of 
a perturbation Sit;' of the longitude of the disturbing planet. 
In 7*5 we replace v' by /i + &' 4- equation of the centre, so that 
Siv' may be regarded as an addition to fi. Hence in this case 
the addition to dR/dfi is 



If Sit;' = B\ sin (s'v + ^i') we can therefore utilise the formulae 
7-32 (11), (12) by putting BJ jri for B'/n. The latter formula, in 
particular, gives an additive part to t: 

S 2 t = l^ BBJ sin (25V +s l + $/). 

O UQ 

7 '35. Numerical illustration. The 'great inequality ' of Jupiter and Saturn. 

The periods of revolution of Jupiter and Saturn are. very nearly in the 
ratio of 2 to 5, so that if n, n' be their mean motions, the terms with 
argument (5n' - 2ri) t will have a very long period actually about 70 times 



206 TRUE LONGITUDE AS ARGUMENT [CH. vii 

that of Jupiter. Thus, in the motion of Jupiter disturbed by Saturn, 1/s* 
will have the order 2-5 x 10 7 . 

Suppose that we have calculated the terms with argument (5n f - 2?i) t to 
the first order in the motions of both planets, and that we need the principal 
second order portions, the latter can be obtained immediately from the 
formulae in 7 '32. We shall perform the calculation and compare the 
results with those given by Hill*. Since the latter uses Hansen's method 
it will be necessary to compare it with that of this chapter. 

Denote Hill's notation by (H) and that of this chapter by (v). In elliptic 
motion we have 

(T), v=z + n+E(z)', (v), nt+c = v-JS(f), 

where E(z) is the equation of the centre expressed in terms of the mean 
anomaly z and E(f) is the same expressed in terms of the true anomaly/. 
In disturbed motion we have 

(H\ ws*g+te + vr+ff(2+bi); (v), nt+c=v-E(f) + nto. 
Now we have seen that, for the principal part of a long period perturbation 
of the first order, the portion due to the elliptic periodic terms can be 
neglected since it only produces portions with the divisor s. Hence, very 

nearly, 

n dt = - n fa, ri dt' =-ri &', 

the former for Jupiter and the latter for Saturn. 

There are two resulting second order perturbations in the motion of each 
planet. The first is that which arises as in 7*32 and the second that which 
arises from substituting the disturbed motion of the disturbing planet in 
R. For the latter we have in the motions of Jupiter and Saturn, respectively, 

dv^ridz*, v = noz, 
since there are additions to */, v in the respective disturbing functions. 

Hill gives t 

7i52 = 1196" sin N, n'8z' = - 2908" sin N, 

where N=- bg 1 - 2g + 69, approximately. (There are additional secular parts 
given by Hill arising from the second order terms.) Thus for Jupiter the 
two portions are, in our notation, 

n 8t = - 1 1 96" sin tf, 8v' = - 2908" sin N, 

and in N we can put / for nt + e - w**g and fi for n't + e' -cr' =g\ for the 
reasons stated above. 

If p as 1/206265'', the factor necessary to reduce the coefficient to radians, 
we have, in the notation of 7 '32, B - - 1 1 96/>, B l = - 2908/>, j\ f = 5, n'/n = 2/5 
and the two parts give 

WoM=| . 1196p (f . 1196/3 + 2908p) sin 2N= 12"-3 sin 2N. 

* American Ephemeris Papers, vol. 4 ; Coll. Works, vol. 3. 
t Coll. Works, vol. 3, pp. 560, 568. 



35, 36] TRANSFORMATION FROM v TO t 207 

For the action of Jupiter on Saturn we have similarly, 

M 2 *= -f -2908p(f .2908p + 1196p)sin2iV= - 30" '9 sin 2^. 
Hill's coefficients for this term in n 82, n' 8z' are * 

Jupiter (first + second approx.) - (first approx.)= - 11"*0 1"'4= 12"'4, 
Saturn = 26"'84-3"-4 = 30"'2. 

After the change of sign necessary for accordance with our notation, the 
comparison shows a close agreement between the two sets of results. 

It may be pointed out that the percentage accuracy of the method is 
greater the smaller the value of s. As far as the present demands for 
accuracy in the various problems of the planetary theory are concerned, 
the method will give all second order coefficients with sufficient accuracy. 
It has been shown, however, that the terms depending on the divisor s 3 
can be obtained without great labour. 

E. TRANSFORMATION TO THE TIME AS 

INDEPENDENT VARIABLE 

7'36. Although most of the purposes for which the theory is 
developed can be equally well served whether v or t be used as 
the independent variable, comparisons of different theories will 
be facilitated if we can transform from one to the other without 
too much labour. 

The method gives results in the form 

n t + e = v-E f + P, v = v + Sv, (1), (2) 

where Ef is the equation of the centre expressed in terms of the 
true anomaly /, and P, 8v are the perturbations expressed in 
terms of v by means of 

/-v-*r, / 1=s ( v -) + '-t*' (3), (4) 

The expression of v in terms of t can be deduced from (1) by 
the use of Lagrange's theorem, but the following method of 
carrying out the transformation in two steps demands much 
less calculation. 

Pub n$t -f e tzr = g and write (1) in the form 

g-P=f-E } (5) 

Coll. Works, vol. 3, pp. 195, 107, 561, 569. 



208 TRUE LONGITUDE AS ARGUMENT [OH. vn 

This equation may be regarded as expressing a 'mean anomaly* 
g P in terms of the true anomaly/. The formula 3'11 (6) 
which gives the true anomaly in terms of the mean may there- 
fore be used. It gives 



g - P + 2e sin (g - P) + |e 2 sin 2 (g - P) + . . . 

.., ............ (6) 



by Taylor's theorem. The terms dependent on P 2 will be very 
small and higher powers of P may be neglected. 

The perturbations Pare expressed in terms of/,/i, and their 
expression in terms of t by continued approximation constitutes 
the second step. The first approximation consists in putting in 
them 

f=g+E g , fi = g' + n ^E g , ............... (7) 

where g' = n't -f e' sr'. 

Consider any term of P : 



where B, Si are constants. With the use of (7) this term becomes 
B sin (jg +J 1 g t + BI) cos j(j + ~^J E^ 

+ B cos (jg +j l g' + B l ) sin 



The second factors of these are expressible as Fourier series with 
argument g, either by harmonic analysis or by the respective 
formulae 

l-t(j+Ji^)X' + ..., (j+fi%)E,-..., ...(8) 

for expansions of sines and cosines in terms of the angles. An 
important point to notice is the fact that for long period terms 
j + ji ft'/ 71 i s small, so that the effect of the transformation in 
changing the terms which usually have the largest coefficients 
is small. 

The transformed value of P is substituted in (6), which then 
gives / in terms of t to the first order of the disturbing forces. 



36, 37] TRANSFORMATION FROM v TO t 209 

If this first approximation be denoted by f=*g + Eg + f, a 
second approximation is obtained by replacing E g by E g -f Sf 
in (8), and adding the second order term in (6) with 8/=0. 

For the great majority of the terms in P, powers of the 
eccentricity beyond the first in the transformation may be 
neglected. For all such cases, a perturbation 



in n^t becomes a perturbation 



in v, where s = j ' -f jfi . n'/n. 
For the final step we have 



where the last term contains the perturbations due to the trans- 
formation from an origin in the osculating plane to one in the 
fixed plane. These, having the square of the inclination as 
well as the disturbing mass as factors, are very small, so that 
the values (5), (6) for/,/i with E g = 2esmg will serve. 

The remaining coordinates 1/r, 9, T are transformed in a 
similar manner, that of 1/r being found with the aid of 2*2 (2). 
The value just found for / in terms of t is used in the 
expression for 1/r, while the values (5), (6) will be sufficiently 
accurate for substitution in the expression for 6, T. 

7'37. The method of this chapter is closely allied, as far as its final form 
is concerned, with that of Hansen*. Substantially, his method requires 
the expression of the true longitude in the form 

v=g+m + tff + 2eam(g+tg)+ e' 2 sin2 (# + fy) + ..., 

so that all the perturbations are expressed by adding 8g to the mean 
anomaly #. Equations 7 '36 (1), (2) show that the same thing is done hero, 
with the difference, however, that while Hansen calculated dg in terms of t y 
it is here calculated in terms of v. 

That the theory is more simple than that of Hansen is due to the fact 
that it can be expressed by means of equations which follow forms well 

* The original theory is given in a volume, Fundamenta Nova, etc., Gotha, 
1838. 



210 TRUE LONGITUDE AS ARGUMENT [OH. vn 

known in other dynamical problems. The two principal objections which 
may be urged are, first, the necessity for expressing the true longitude of 
the disturbing planet as a function of that of the disturbed planet, and 
second, the possible need for the final transformation given in 7 '36. As 
we have seen, the latter requires comparatively little additional calculation, 
while it is doubtful whether the former transformation, which is needed 
only in the development of the disturbing forces, requires more labour than 
that in terms of t. As pointed out in Chap, iv, each requires substantially 
four operations, two of which are more simple in the present method than 
in that of Hansen, while one of the remaining operations is more 
complicated. 

There is, however, a special advantage possessed by the present method 
when we are dealing with the perturbations produced by an exterior planet 
on an interior one having a considerable eccentricity. In the transformation 
giving v' as a function of /, the principal elliptic term, 2e sin /, enters only 
with the factor n'/n which in most cases is not much greater than 1/2. 
Thus the powers of this elliptic term have a maximum effect nearer to 
those of e than to those of 20. To some extent this is compensated by the 
factor r 2 which accompanies R in the developments, but the convergence 
is much more easily controlled with the method of this chapter than with 
the usual independent variable. 



F. APPROXIMATE FORMULAE FOR THE 
PERTURBATIONS 

7*38. It is often useful to get an idea of the order of magnitude 
of the perturbations in a given problem. This is particularly the 
case when extensive calculations are to be undertaken to obtain 
the general perturbations accurately; an approximate preliminary 
calculation may save much unnecessary labour. Approximate 
formulae can also be utilised when the interval of time during 
which the results are needed is short or when the constants of the 
orbit are not well known. 

In obtaining such formulae, we shall neglect the inclination, 
so that v = v, and attention can be confined to the equations 
7*1 (3), (4), (5). If in these equations we put 

e 2 ), q = (1 + Sq) q 0j u = (l+e cos/+ Su) q , | 
= f*> n Q t + = (!-#)% (1 -f e cos/)~ 2 + n St, J 

(1) 



17-39] APPROXIMATE FORMULAE 211 

,hey may be written 

.............................. (2) 



(3) 

d *,_ -">*_ f ift o SM \ 

~ 



+ e cos/y/ 
= JSg -28w + (68^-8^)6 cos/ + ....... (4) 

Consider a term with argument <r in the disturbing function 
md suppose that this term gives rise to terms 

A si no- in ^-(r*Rqo), Bcoscr in r z -^- . 

...... (5), (6) 

[n general, A> B will be of the same order of magnitude. 
Substituting (5) in (2) and integrating, we obtain 

* 2^1 da- _ 

gg, -- COS CT, S T- ................ (7) 

* s dv ^ 

With the help of (5), (6), (7), equation (3) becomes 

(& + i) Su = - f + J8) cos a- + \eA cos (<r +/) 
\rtw / \ s / 

\eA cos(ov /), 
}he integral of which furnishes 

2A \coso- eA fcos(<r+/) . cos(cr-/)] 

~"" + "~' 



...... (B) 

jince l-(* l) 2 = + 2s-s 2 . 

The substitution of (7), (8) in (4) and a subsequent integration 
jive n Q St. 



7*39. Let us first neglect the terms in 7*38 (8), (4) which have 
tihe explicit factor e. The remaining terms give 

/2A 
cosr tt--f 

S 



/2A D \ coso- 
-f +JB)= - - a , ..-(1), 

\ o / -I- o 



212 TRUE LONGITUDE AS ARGUMENT [OH. vir 

The transformation to the time as independent variable ia 
immediately made, since with e = 0, we have Sv= n^St, and 
/=#,/! = / in <r. 

These results hold for all terms in the disturbing function 
whether they contain e or not; we have neglected e only where 
it appears explicitly in 7*38 (3), (4) and in the transformation to 
the time as independent variable. Denote the former by the 
suffix zero, and the additional parts factored by the first power of 
e by the suffix unity. To obtain the results to the first power 
of e, it is sufficient to substitute the terms with suffix zero in 
the previously neglected terms. Hence 



n Q j-Sit- 2SiU -h (68 Q u S Q q) e cos/. ............... (5) 



To the same order, the formulae for the transformation to the 
time as independent variable give 

2e sin g -j-nQ$ Q t - n Q Sit, 



...... (6) 

with/=<7,/i g f in all the formulae. 

The terms of chief importance are usually those in which 
s or s 1 is small and the order of magnitude of these is given 
by S Q v = n S Q t. When s 2 is small the additional terms with 
argument a / must be considered. 

7'40. Solution to the first powers of the eccentricities. The 
development of a'/ A a'r cos S/r' z as far as this order is * 
Si {c^ cos i^r e (D 2mi) a f cos (i^r +/) 

+ e' (D + 1 - 2i) di cos (fy +/i)J 
+ a{cos^r + i$(l -f 2rn)cos(^-/) 

+ i* (1 - 2m) cos (t +/) - 2e' cos (^ -/,)}, 
where 

a = a/a', -^ =/ 4- w ~/i *', D = d/a 7 a, m = w'/w, 



* The development to the second order with respect to the eccentricities and 
inclination is given by C. A. Shook, Mon. Not. R.A.S. vol. 91, p. 558. 



39-41] DETERMINATION OF CONSTANTS 213 

and 

, t = 0, 1, 2,.... 



CN- - j /- /. x 

bince r-^- a-^ ) r a (I ecos/), 

or da 

we have 

r 2 ,r = a 2i }Da $ - cos i\Jr e (Z) 4- 1 2rm) /) cos (ti/r 4-/) 

4- e' (D + 1 - 20 Dcti cos ( 

7?2/ 

4- a 2 - -- { cos -v|r + e (1 -f m) cos (^ /) 

4- e (1 - m) cos (^ -f-/) - 2e' cos ( 
- a - Sii {a 4 sin ity e(D + 2 2nn) a< sin (i- 



- a, sn 

{- sin yfr -f ie(3 + 2m) sin (^ -/) 
+ \ e (3 - 2m) sin (^ +/) - 2e' sin 0/r - 

Substitutions from these formulae in 7*39 (1), (2), (3) give 
SO<?K &o u > nS^t, and from the first term in each parenthesis the 
additional terms in 7*39 (4), (5). In most cases values of i 
beyond 4 will not be needed. 



G. FINAL DEFINITIONS AND DETERMINATION 

OF THE CONSTANTS 

7*41. The method of this chapter suggests the following 
definitions of the constants. 

The mean motion n and the epoch e are such that when all 
periodic and secular terms are suppressed, the true longitude 
shall be represented by nt 4- e. If terms dependent on the second 
powers of the masses be neglected there is no difference in the 
values of n, e whether t or v be used as the independent variable, 
and the additions depending on these second powers will usually 
be insensible to observation, 

B&SPT 14 



214 TRUE LONGITUDE AS ARGUMENT [OH. vn 

The constants e$, OTO have been defined to be such that the 
principal elliptic term in OO/T, where n 2 Oo 8 = /i, shall be repre- 
sented by 

2 cos (v cj ). 



Since any definition depends on the specification of some 
particular coefficient in a particular coordinate, a change to 
another definition can always be made. In numerical work it is 
usually sufficient to make any small correction due to an altered 
definition in the elliptic terms only. 

The constants 7 , Q are defined above by making the principal 
term in the latitude equal to sin /o sin (v # ) Here / is the 
inclination of the two orbital planes. The change to i y the 
inclination to any other plane of reference, is made by the 
formulae of 1*32, and the change to i$ is made to correspond. The 
slight difference when t is used as the independent variable will 
be sufficiently accounted for in the determination of the constants 
from observation. 

7*42. A process of approximation is used in the determination of 
the values of the constants from observation. The perturbations 
may be calculated with the osculating elements at some given 
date unless previous work has given elements more nearly 
approximating to the constants of the theory. Thus constructed, 
the theory is compared with the observations, or with a selection 
from them. The differences are assumed to be due to erroneous 
values of the elements and are analysed so as to determine their 
corrections. While the formulae for the perturbations should be 
examined to see whether these corrections make any sensible 
difference in them, it will usually be found sufficiently accurate 
to correct the elliptic terms only. 

7*43. The detailed work connected with the determination of the 
constants, as well as their correct definitions, has to be carried out what- 
ever method be used to calculate the perturbations. The difficulty of 
avoiding error in performing the work can to some extent be lessened by 



41-43] DETERMINATION OF CONSTANTS 215 

carrying it out as far as possible in a systematic way, since few checks of 
its accuracy are available. Most of the work of developing the disturbing 
forces can be done by harmonic analysis in the manner explained in 7 '7, and 
this work has the advantages of being easily systematised and of carrying 
its own checks. The integration of the equations cannot be done in this 
manner, but the steps with the method of this chapter are easy and simple. 
The final step of comparing the calculated results with observation, although 
dismissed here in a few sentences, is or may be as laborious as that of 
calculating the general perturbations, but it is necessary if good values of 
the constants are to be obtained. 



14-2 



CHAPTEE VIII 

RESONANCE 

8*1. Resonance is usually defined as a case of motion in which 
a particle or body, moving or capable of moving with periodic 
motion, is acted on by an external force whose period is the 
same as that of the motion of the body. This definition, while 
it describes the apparent character of the phenomenon, implies 
the existence of certain conditions which are not present in 
actual mechanical systems. 

Let us take the usual illustration, namely the equation 



t ,, 

-r 4- n*x = m sin n t. 
at* 



When n =f n r , we have the solution 





. / , x in IA 

x = c sin (nt -f a) + - -75 &m n t. 

/ a 



But when n n f , the solution is 

x = c sin (nt + a) \ mn't cos n't, 

where, in both cases, c, a are arbitrary constants. 

The illustration is defective because such an equation does 
not arise in any actual mechanical system except as an approxi- 
mation, and because the approximation is valid only when x is 
small. The solution, therefore, breaks down as soon as n n f 
becomes too small. In actual mechanical problems, either the 
left-hand member which, equated to zero, gives the undisturbed 
motion, is not a linear function of x y or else the variable x is 
present in the expression for the disturbing forces, or both of 
these conditions may be present. 

8*2. In the previous chapters we have based our procedure 
on the plan of continued approximation with respect to the 
disturbing mass. In the elliptic approximation this mass was 
neglected. In the first approximation to the disturbance the 



1-3] FAILURE OF PREVIOUS METHODS 217 

elliptic values were substituted in the expressions for the 
disturbing forces, and the equations were again integrated. In 
the second approximation, the new values were substituted for 
the coordinates in the disturbing forces and the equations were 
again integrated. This procedure carried the implication that 
it was possible to develop the perturbations in positive integral 
powers of the disturbing mass, and that the coordinates would 
be expressed as sums of periodic terms. It is true that terms 
with coefficients increasing with the time were admitted, 
but it was seen that this was merely a convenient device 
adopted in order to abbreviate the calculations when the results 
were needed for a limited interval of time only. The terms so 
treated had periods which were long in comparison with the 
interval during which the expressions were to be used for 
comparison with observation. 

In cases of resonance, this procedure fails. The reasons for 
its failure may be exhibited in several ways. That which is most 
fundamental in the mathematical development is due to the 
fact that expansions in powers of the disturbing mass have to 
be replaced by expansions in powers of the square root of of 
some other fractional power of this mass. Further, there is a 
fundamental discontinuity in the passage from non-resonance to 
resonance, which cannot be bridged by any mathematical device, 
since it is a physical characteristic of the motion. 

The principal features of certain of the resonance problems in 
celestial mechanics can be illustrated by the motion of a pen- 
dulum which can make complete revolutions about a horizontal 
axis as well as oscillate about the vertical, and a following 
section (8*5) contains an analysis of these motions made from 
the point of view needed later. 

8*3. We shall be concerned mainly with those cases of resonance which 
occur in the present configuration of the solar system. A certain number 
of such cases are present in the satellite systems of Jupiter and Saturn, 
where the periods of revolution round the planets appear to be very nearly 
in the ratio of two small integers. In the planetary system, we have the 
Trojan group of asteroids whose members circulate round the sun with 



218 RESONANCE [OH. vin 

the same period as Jupiter. The motions of this group are treated in the 
following chapter. The most difficult problem is, however, to find out why, 
amongst the numerous asteroids which circulate between the orbits of Mars 
and Jupiter, none are known having periods exactly twice or three times 
that of Jupiter, or periods in the ratio to that of Jupiter of two small 
integers. A discussion of certain features of this problem will be given 
in this chapter. 

8*4. In the actual cases of observed motions in the solar system, so far 
as they have been developed, we know of no case in which the discontinuity 
referred to in 8*2 is present in an observable form. We have referred to 
resonance as a set of cases in which the periods of revolution are in 
the ratio of two small integers. Since the final expressions for the co- 
ordinates contain all multiples of the frequencies, each pair of these can be 
regarded as a possibility for resonance conditions. But these frequencies 
are observed quantities, namely, those of the mean periods of revolution, 
and since such a pair of observed quantities can always be expressed as 
the ratio of two integers, it would seem that resonance must always be 
present in any three body problem. 

The question goes further than this. It will appear below that the 
phenomena of resonance occur not only when the observed periods are 
exactly in the ratio of two integers but also when these periods are nearly 
in such a ratio. In other words, resonance occurs not only for a pair of 
special values of the periods but also for a range of values and this range 
is finite. One difficulty, namely, the question of the accuracy of our measures 
of the periods, disappears to some extent, but it is replaced by another, 
namely, the consideration of the infinite number of periodic terms which 
must have the resonance property. 

The discontinuity referred to is not a place where either the coordinates 
or the velocities are discontinuous in a physical sense, but is one in which 
an infinitesimal change in one or more of the constants will ultimately 
produce a different type of motion. Thus the computer arrives at a situation 
where he needs a considerable increase in the accuracy with which the 
constants obtained from observation must be known in order to choose 
between two possible routes. And this process appears to continue as the 
approximations follow one another. From his point of view, there can be 
no general solution of the problem of three bodies, that is, there cannot 
exist one. set of formulae giving the coordinates in terms of the time and 
the initial conditions which will serve for more than one set of such initial 
conditions, which will be valid for all time, and which can be used for 
calculation of the position. This conclusion may be a result of the mathe- 
matical devices which he adopts, but is more probably due to an inherent 
difficulty, namely, that of finding expressions which shall be continuous 



3-5] MOTION OF A PENDULUM 219 

functions of the constants which can be determined from observation. Any 
proper solution of the problem requires also the consideration of the limi- 
tations placed on the observer ; it is not solely a mathematical problem. 

8*5. The motion of a pendulum. The fundamental equation 
in resonance problems appears to be 

K*smx = (1) 



This is the same as the equation of motion of a simple pendulum 
of length , if /c 2 = g/l, and if x be the angle which it makes with 
a vertical line drawn downwards at time t. Since the substitu- 
tion, x -f- TT for x, changes the sign attached to # 2 , the equation 
with /c 2 replacing /c 2 gives the same motion as (1). 
The equation has the integral 

Y-C'-f 2* 2 cos#, (2) 

where G is an arbitrary constant : for the motion to be real it is 
necessary that G 4- 2/c 2 5 0. There are three types of motion 
depending on C > 2# 2 , C < 2/c 2 and the intermediate case C 2* 2 . 

(i) (7>2/e 2 . As dxfdt never vanishes in this case, it is 
always either positive or negative, and the pendulum is making 
complete revolutions in one sense or the other. We have 

dx 



f const. 

as the integral. If we put 

dx 



1 _ J^ f 2 " dx^ 

n ~ 2^ Jo (C + 2* 2 c 



n can replace as the arbitrary constant, and the solution can 
be expanded in the form 

K? /e 4 

x nt 4- e H 5 sin (nt -f e) -f 5 ^ sin 2 (nt -f e) + (3) 

n on 

The periodic portion of this series can be regarded as an oscilla- 
tion about the mean state of motion which is revolution with a 
period 27T/W. The half-amplitude of this oscillation is evidently 



220 RESONANCE [OH. vra 

less than TT and it decreases as n increases. It is convenient to 
consider n, e as the arbitrary constants of the motion to be 
determined from the initial conditions. 

(ii) C < 2/e 2 . Here dx/dt = when x = a, where 



The integral can be written in the form 

sin 2 |a -sin 2 |#), 



and x is a periodic function of t oscillating between the values a, 
where a < TT. The solution can be expanded into the series 

C 3 

x = c sm (pt -f /?) + - T^T sin 3 (pt + ft) -f . . . , (4) 



where p K (1 -^c 2 + ...). 

It is convenient here to consider c, /3 as the arbitrary constants, 
since the limit of p as c approaches zero is tc> a quantity inde- 
pendent of the arbitrary constants. 

(iii) (7= 2# 2 . Here dx/dt %K cos -|#, the solution of which 

gives 

x -f TT = 4 tan" 1 cxp. (fct 4- c/o), (5) 

where a<> is one arbitrary constant, the other having a particular 
value. 

When t= <x> , x= TT: at both places dx/dt = 0, d 2 x/dt 2 = 0, 
and it follows by differentiation of (1) that all higher derivatives 
of x vanish. Near this point, while x approaches one of the 
limits TT, t tends to become an indeterminate function of x. It 
should be noted also that x is a discontinuous function of the 
arbitrary constant (7, since the motion is of a quite different type 
according as <7-^2* a from C - 2* 2 > or from C- 2** < 0. This 
result is of course characteristic of unstable equilibrium, but the 
point of view stated here is required in the applications to be 
made below. 

Attention is drawn to the following facts which are obvious 
consequences, but which are needed for the interpretation of 
resonance equations. 



5-7] MOTION OF A PENDULUM 221 

(a) The mean value of dx/dt in (i) is n and in (ii) it is zero. 

(6) As n passes from positive to negative through zero the 
solution given under type (i) is a discontinuous function of n at 
n = 0. With certain initial conditions, there is a range of solutions 
(independently of the time constant) corresponding to the case 
n = 0. This range of solutions constitutes type (ii) and is 
characterised by the constant c or a which is related to it, and 
can have any value between TT. 

(c) In case (i) the series giving the solution proceeds along 
powers of * 2 ; in case (ii) it depends on V/c 2 . There is no analytical 
continuity between the two types of solution, and they cannot 
be represented by one and the same analytic function of t. 

(d) In case (i), the adopted arbitrary constants are the 
period of revolution and the time of passage through the vertical. 
In case (ii) they are the amplitude of the oscillation and time 
at which this oscillation vanishes. 

8'6. A more general type of motion is exhibited by the equation 



with its integral (~\ = C- 



where/(#) is assumed to have an upper limit/(K). We get the same three 
types of motion according as <7>2/(K), C< 2/(fc), C =%/(*). In the first 
case x can take all its possible values ; in the second case it is limited by 
the value given to (7. In the first case also dx\dt never vanishes and it has 
a mean value different from zero ; in the second case x is a periodic function 
of t and the mean value of dx/dt is zero. When #=2/(K), CZf(x) is 
divisible by (x - /O 2 since K is the value of oo which makes f(x) a maximum, 
so that dx/dty d 2 x/dt' z and consequently all higher derivatives of x vanish. 

8'7. The disturbed pendulum. The characteristics of resonance 
phenomena can be well exhibited by considering the equation 

x-72/yi 

-y-2 4- /c 2 sin x = mic 2 sin (x n't e'), ......... (1) 

(it 

which may be regarded as the equation of motion of a pendulum 
disturbed by a periodic force. We shall suppose that m, n', e' are 



222 RESONANCE [OH. vra 

given constants and that m is small compared with unity. We shall 
further suppose that when ra = 0, the pendulum is oscillating 
with a small amplitude, so that only the type (ii) with the 
solution 8*5 (4) is under consideration for the undisturbed motion. 
To solve the equation (1) conveniently, it is advisable to use 
the method of the ' variation of arbitraries ' The method is given 
in the following article for a more general type of equation than 
(1), as it serves to illustrate in detail the plan to be followed in 
cases where the undisturbed motion is periodic, and also the 
nature of the change of variables useful when resonance problems 
in celestial mechanics have to be considered. 

8*8. It is proposed to find the solution of the equation 

when that of d? 4 "^'^" ^ 

is periodic and is known. 

Suppose bhat the solution of (2) has been obtained in the form 

x~x(l,c\ l nt+e, n = func. c, (3) 

where x (I, c) is a Fourier series with argument I and with 
coefficients depending on c, the arbitrary constants being c, e. 
The solution (3) has the following properties. Let 

be regarded as a function of two independent variables I, c and 
let us form ?i 2 3 2 #/9J 2 , substituting the result in 



(4) 



If n be the function of c defined by (3), the variables /, c disappear 
from (4) and the substitution reduces (4) to zero. The disappear- 
ance of I, c is not dependent on their values : they may be any 
functions of t or any variables whatever or constants. 
Let us suppose that they are variable. We have 
dx dx dl ex dc 



7, 8] CHANGE OP VARIABLES 223 

We are about to replace x in (1) by two new variables I, c which 
are related to x by equations (3). Since we are replacing one 
variable by two others, a relation between the new variables is 
at our disposal. Let us so choose it that 

dx dl dx dc _ dx ,~^ 

'~ w ' .................. ( } 



. i dx dx 

then di n m 

Whence, since t is present in (6) only through I, c, n, 
d*x d*x dl 9 / dx\ dc 



Substituting this in (1) and making use of the fact that (4) is 
zero, we have 

dl 9 / dx dc 



x ( \ , / x\ c ,, /(7N 

n W'(dr n ) + dc( n dl)'dt- m( t > .......... (7 > 

The equations (5) and (7) may be regarded as linear equations 
to find dl/dt n, dc/dt. Their solution gives 

dc m dx , dl m dx , . 



, Tj r 9 / dx\ dx (Px dx /r . x 

where K = - (n -^ }. ~r - n ^ . 5- ................ (9) 

dc\ dl/ dl dl 2 dc ^ / 

It is easy to prove that K is a function of c only. For since 
the expression (4) vanishes identically for all values of l y c } its 
partial derivatives with respect to them will do so also. We thus 
obtain two equations between which df'(x)/dx can be eliminated 
and it is found that the result can be expressed in the form 
dK/dl 0, showing that K is independent of I and is therefore a 
function of c only. 

If <' = d<f>/dx, where <f> is a function of x, t, we can express (8) 

in the form 

dc m d6 dl m 9<f> /irkX 

_ __ _ _ _ ij _ _ r / 1 1 1 \ 

dt'Kdr dt~ Kdc' ............ v ' 

where <f> has been expressed in terms of /, c, t by inserting for x 
its value (3) in terms of I, c. 



224 RESONANCE [CH. vra 

Since n t K are functions of c only, we can put (10) into 
canonical form with new variables Ci, JB, defined by 

dci = Kdc, dB = n dc\ = nKdc, 
and the equations then become 



i /r>, \ /D, i\ /-i\ 

~ = -( B + m <t>), ^ -(B + Kf) ....... (11) 

89, Solution of the equations for I, c. When m is small, the 
usual method of approximation is the substitution of constant 
values of j? , Q, Co for n, e, c in the terms factored by m which 
then become functions of t and can be integrated. If we put 

I = / -f /! = n Q t -f f + hj o = Co -f Ci, 
and neglect powers of / l5 Ci beyond the first, we have 

dn\ 



from which Ci and then li are immediately found. 

In cases of resonance this procedure breaks down, and it is 
necessary to proceed as follows. Differentiate the equation 8*8 (9) 
for dl/dt and substitute the expressions for dc/dt, dl/dt in the 
result. We obtain 



^ - A ( - ty\ *? _ -^ ^ - ^. -? 2 ^ 
dt 2 ~dc\ ~K^c) t dt i Kdldc'Tt Kdcdi 

_ m fdn d^ _ _8^> _ 9 2 j> \ 
~'"~ ?l "" 



Since the last line of (2) has the factor m 2 it may, in general, be 
neglected in a first approximation. 

In the applications, I, t are present in </> only as a sum of 
periodic terms with arguments il j (n't -h e'), where ri y e' are 
given constants. When this is the case 

cty^jV cty 
dt i dl ' 



8,9] THE RESONANCE EQUATION 225 

and the first approximation to (2) can be written 

d * 1 , v/- /\a m 9 / * 9< M /Q , 

-T7 2 -f 5 (in - w ) ^ 5- ~ - =-> ai ) - ....... w) 

ar J 7 iKoc \in-jn oil 

The standard type is that in which < has the form 

< = c^ cos ^-f&, li ilj(n't+e), ......... (4) 

where a it b are functions of c only. The equation for l t is then 



If, in a first approximation, we put c = c , n = n Q) K = K Q 
all constants (5) takes the form of the equation for the pen- 
dulum. [If the coefficient of sin l t be negative, we put 
Z. = i7-j(?i' + e') + 7r 

instead of the value (4).] There are therefore the types of 
solution considered in 8*5 . Type (i) is that in which dli/dt is 
never zero so that iri^jri does not vanish. Type (ii) is that in 
which li oscillates about the value [or TT]. 

With type (i), we put li = i(nQt + o)-j(n't + e') = l i0 in the 
second term of (5) and deduce 

9 / cti \ . , /a . 



as a first approximation. 

With type (ii), we choose n , e to be such that 

iiiQ jn' -=* 0, i Q je f = or TT, (7) 

and li is an oscillating function. If the oscillations be small we 
can put sinli = liy n n^> c = Co, K = !Q in order to find a first 
approximation. This gives 

7 / \ <> \ ma i fi n \ /o\ 

I- = X sin (pt 4- Xo), p "If" 13") > () 

X, X<> being arbitrary constants. 

With similar limitations, the equation for c gives 

dc f t 



Whence c = c mt\ - cos (^ + Xo), (9) 



226 RESONANCE [OH. vni 

where c is determined from WQ =jri \ since T? O is a known function 
of CQ. 

The coefficient of the periodic term in (9) is 

X, ..................... (10) 

v ' 



and we thus have the first term of an expansion in powers of m^. 
If the coefficient of ra^ is not large, the assumption that we can 
put c Co in the coefficient of sin I in (5) is justified. 

The difficult cases in celestial mechanics are those which 
depend on the value of c . If (10) becomes infinite as c tends 
to zero, and if the coefficient of the periodic term in (9) is com- 
parable with c , this method of approximation breaks down. The 
analogy of (5) with the pendulum equation no longer exists and 
special devices have to be employed in order to find out whether 
resonance is possible. A case of this kind in which i = l,j = 2 is 
treated below. 

In general, the solution (6) corresponds to the case of an 
ordinary perturbation and (8) to a case of resonance. The 
various features noted in 8*5 as peculiar to the two types of 
solution are present and can be interpreted in the light of our 
knowledge of the motion of the pendulum*. 

8*10. The general case of resonance in the perturbation problem. 
We recall the method of integrating the equations 

S (dc t . Swi - dw t . 8^) = dt.S ( Ai~ 2 + mR), . . .(1) 
where mR now denotes the disturbing function, ra being the 
disturbing mass with that of the sun as unit. 
We had, with a slightly different notation, 

E = RQ + 2Aco*jiN, .................. (2) 

where A was a function of the c t and 

JI'MI +J2W, ...... (3) 

3 ' = n't 4- e', WZ^VT'. 



* A more elementary treatment of resonance with applications to the motions 
of one and two pendulums is given by E. W. Brown, Rice Institute Pamphlets, 
vol. xix, No. 1. Also reprinted separately and issued by the Cambridge University 
Press. 



9, 10] GENERAL CASE 227 

In a first approximation we put dw^dt = WQ and obtained 
integrals for the values of c it f w i which contained the divisors 
j\n*+ji!n'. It was assumed that no one of these divisors 
vanished. 

Let us now suppose that there is one term for which this 
condition does not hold, or rather, in order not to limit the 
argument too much, let us assume that there is one term in 
which neither ji nor j\ is zero but in which j\n + j\n f is so small 
that the approximation is no longer valid, but that we can 
approximate with all the remaining terms. We shall see later 
on that this latter condition cannot hold, but that an approach 
to the solution can be made by supposing that it does hold. 

All these remaining periodic terms can be eliminated by 
changes of variables in the manner explained in 6 '6. We can 
therefore suppose that the equations (1) refer to the new 
variables after such terms have been eliminated and that 

R^Ro + AcosfrN, .................. (4) 

where jiN has the value (3) and R consists of those parts of R 
which are independent of MI, Wjf. 

Let us change the variable Wi to W\ where 

ji Wi =jiwi +ji'wi +J 2 W, ............... (5) 

so that* 

ji8Wi= i /iSwi, jidWi^jidw^+tfridt ....... (6) 

It is easily seen that the left-hand member of (1) merely 
requires the substitution of Wi for Wi if we replace the right- 
hand member by 



c i ji 

Next, replace Ci by a new variable GH defined by 



(8) 



Previously, n$ was defined for the case m = by the relation 

^2 Clo -3 = ft 0) so that the second relation (8) is the same as 

?i o + ji'n' = 0. This definition does not demand that rc shall be 

* The symbols d and 8 have the same signification as in 5 '3. 



228 RESONANCE [CH. vm 

the final mean value of n, since there may be a constant portion 
in en which prevents this. The only condition needed at this 
stage is that CH/CIO shall be small so that the expansion of Ci~ a 
in powers of this ratio shall be possible. We then have 



j^ 



The first two terms of the right-hand member of (9) do not 
contain the variables and may therefore be omitted from (7) ; 
the coefficient of Cn vanishes in virtue of (8). Hence, inserting 

flo = /^io~ 3 =-ji / ?* / /ji> ............... (10) 

we obtain for (7) the expression 

n 



cos 

...... (11) 

The last expression is the characteristic form of the Hamil- 
tonian function for cases of resonance. It is to be remembered 
that no, CIQ are, by definition, functions of n' only and are there- 
fore independent of c,-, Wi, w { . 

8*11. The equations for c n , Wi become 

-^ = --mjiABUij l N', ....................................... (1) 

dQ Cu a c n 2 3^0 d 

-^ - ~ ~ 



-7- - - ~ """ 5~~ - 5 

dt ecu CIQ Ci<f den oc 



n 



...... (2) 

The right-hand members of the equations for c 2 , c 3 , w 2 , w 3 all 
contain the factor m. If then we replace the variables Cu, c 2 , 3 
by Ci, (7 2 , G z , where 

c u = mi(7i, 02=020 + ^^2, c 3 = Cso + m^Cs, ...(3) 
and ^ by m""*! 7 , the equations can be written 



iv, nj GENERAL CASE 

where TF 2 > TF 8 are written for w 2 , w$ to preserve symmetry of 
form. The quantities Cao, CM are now constants which are at our 
disposal. It will be noticed that the factor m has disappeared 
from the coefficient of cosjiN. 

In order to apply these results to actual problems, we need 
to know what the new variables mean in relation to the dis- 
turbed elliptic orbit. We have 

d = (/ia)* c 2 = ci {(1 - e 2 )* - 1}, C B = ci (1 - e a ) } (cos i - 1). 

The replacement of d by Ci -f m^Ci, with the expansion in 
powers of wJCi/Cio, implies that we assume an initial major 
axis 2a and that its variations are small compared with 2oo. 
The factor wi would seem to imply that Ci is not infinite when 
m = 0. Mathematically this is correct, but as the whole problem 
demands that in shall not be zero, we can at present be content 
with the previous statement. 

Next, since in the problems considered e< 1, we have 



The replacement of c% by Cw+ndCt implies that there is a 
value CQ such that (e e } -f- m% is not very great. But care 
is necessary if we contemplate expansions in powers of m^C^Cm. 
For perturbations by Jupiter, m//j, is of the order 10~ 3 so that 
mi is of order '03/xi. Thus expansions in powers of (<? 2 ^20)^ 
will converge too slowly for useful numerical computation if e 
is much less than 1. (See the last paragraph of 8*24.) The same 
difficulty does not occur in the case of c 3 ; for the expansion is 
made in powers of (2 sin Jt) a , so that it involves positive integral 
powers only of Ca CSQ. However, if we contemplate a develop- 
ment of Li in powers of m*, which these changes imply, we may 
be in danger of not obtaining a real approximation if (2 sin Ji) 2 
is comparable with wi. 

These difficulties are actually present in the consideration of 
the motions of the asteroids circulating between the orbits of 
Mars and Jupiter. They play a much smaller part in the 
resonance cases amongst the satellites of Jupiter and Saturn, 
mainly because the disturbing mass-ratios are much smaller. 

B&SPT 15 



230 RESONANCE [OH. vra 



8*12. Let us suppose that expansions in powers of 
m^ OS/CM are possible, and let us further suppose that, in a first 
approximation to the solution of the equations 8*11 (4), we can 
neglect mi 

The coefficient A then becomes a constant, A Q , and R is a 
function of w 2 , w 3 only ; thus C 2 , 0$ disappear from Q and w 2f w 3 
are therefore constant. The remaining equations are 

rt GI /i\ /n\ 

-3?! -\ ............ (1), (2) 



...... (3), (4) 

The first two equations give 



l 710 *r rx 

-,-^T - 3w o ' a smji tf = 0, 

U-JL CIQ 

or, since w a , ^3 are constant, so that dN/dT=dWi/dT, 



This is the pendulum equation previously discussed. In the 
type of solution where jiN makes complete revolutions so that 
dN/dt never vanishes, we have an ordinary perturbation; this 
was expressly excluded from the definition of N. In the second 
type jiN oscillates about the value or TT according as A Q is 
negative or positive. This oscillation is known as a libration. 

In general, therefore, it appears that, under the stated con- 
ditions, such oscillations are possible. If the amplitude of the 
oscillation is small so that we may replace sin^i^V" by j^N or by 
jiN -f TT, we have, after the replacement of dT by its value m^dt, 

tf=Xsin(p* + Xo), if = a>ji*\A* ~m, ...(6) 

CIQ 

where X, Xo are arbitrary constants. 

The frequency p is proportional to the square root of the dis- 
turbing force, while the coefficient and phase are to be determined 
from observation. In all cases except that of the Trojan group 



12] GENERAL CASE 231 

of asteroids in which ji = /, some power of e, e f , F will be 
present in A and it is therefore necessary to consider the 
possibilities of expansions in powers of e^ > e'&, T$ as well as those 
in powers of ra*. 

The value of Ci is given by (2), (6). We find 

Ci = CIQ + m*(7i = Cio - t ^-r p\ cos (pt + X ). . . .(7) 
^oji 

The small factor p in the coefficient of the periodic part of d 
is consistent with the assumption, made earlier, that expansions 
in powers of m^Ci/Cio are possible. It shows further that while 
the libration of N, that is of the angular position of the body, may 
have a finite amplitude, that of GI and therefore of the major axis 
is small. 

Since R Q) w 2 , w 3 are constants, the integrals of (3), (4) are 



2 ~Ci + const., ............ (8) 

C 8 = w**!^+-i 8 ft + const ............. (9) 

dw 3 fr ^ ' 

Now R Q contains w z , w 3 , w 2 ' only in the form of cosines of 
multiples of w a w 2 ', ^2 4- w 2 ' 2w 3 . In order therefore that 
(7 2 , C 9 shall not increase continually with the time, it is necessary 

that 

w 2 = w 2 =^3 ...................... (10) 

Since jiN = j 1 w l + J 2 w 2 + J 3 iv 3 +ji'wi +J 2 W, 

where the sum of all thej iy ji is zero, the condition (10) gives 

jlN = ji(Wi-W2)+ji'(Wi-W2) .......... (11) 

If, however, e f = 0, RQ is a function of the c t only, so that the 
condition (10) is not needed and as w% disappears we have 



where, as usual, 

jl+jl'+J2 + J3 = 0. 

Since the value of Q in 8'10 (11) does not contain the time 
explicitly, the integral Q = const, exists. We have made no 
direct use of this integral in the investigation just given. This 

15-2 



232 RESONANCE [CH. vni 

omission corresponds to that in the case of the pendulum making 
small oscillations where it is more convenient to solve the 
equation directly than through the medium of its first integral. 

8*13. The constants. The general solution of the equations of 
motion requires the presence of six arbitrary constants. When 
the libration ofjiN is zero, and e r = 0, the constants present in 
wi, wi, w 3 are all determinate since w z , w a are given by 812 (10) 
and that in w\ by the condition that j\N must be zero or TT. 
The constant Cu is determined by 8*10 (8). Thus the constants 
CM> Cao on ty are a k our disposal. But as the libration in general 
will exist and as its presence introduces two new arbitrary 
constants, the loss of four arbitrary constants is reduced to a 
loss of two. Since the two conditions 812 (10) disappear when 
e' = 0, there is no loss in this case. Thus when e r = 0, there is a 
finite range of values for each of the arbitrary constants: in 
other words, the resonance cases are not particular solutions, 
but are merely types of solutions in which all the arbitrary 
constants have finite ranges. 

When e 1 4= 0, the question of the ranges of the constants 
cannot be settled by the approximation used above: this 
involved the neglect of terms factored by m but the retention 
of those factored by m%. The conditions 812 (10) may be merely 
limiting values about which oscillations can exist in the same 
manner that N = is a limiting value about which librations 
are possible. The treatment of this case for the Trojan group 
will be found in Chap. ix. 

814. It is evident that the change of variable, Wi to W\, 
eliminates t from all the angles for which the ratio ji/ji is the 
same: all these terms have in fact the resonance property and 
should properly be included with the single term chosen above. 

After the change of variable, the Hamiltonian function does 
not contain the time explicitly and there is an integral of the 
equations, namely, 

M 2 7i V 



2 

1 4/1 



= const. 



12-14] GENERAL CASE 233 

The succeeding change from ci to c n gives, by 8*1 (9), 



. 
c 10 V Cio 2 c 10 3 

This equation may be regarded as determining Cn in terms of 
the remaining variables. A comparison with 8*5 (2) will show 
that Cn plays a role similar to that of dxjdt in the integral for 
the motion of the pendulum, and that the presence of resonance 
depends on the value attributed to the constant. 

It should be pointed out that the investigation given in the 
preceding articles does not prove the existence of resonance; it 
merely shows that so far no condition preventing resonance has 
appeared. 

The illustration afforded by the motion of the pendulum must be 
regarded as showing only the general nature of the problem. Difficulties 
from which it is free appear as soon as we begin to consider even the 
simplest case of actual resonance in the solar system. Some of these arise 
from the fact that the consideration of a single resonance term is not 
sufficient. For example, in the case of the 2 : 1 ratio, the principal terms 
present in the disturbing function are 

A l e cos ( \GI 2w x ' + w 2 ), A x V cos (w\ - 2wj' + 10%). 

In the ordinary planetary theory, the variation of w%' and especially its 
secular part can be neglected in a first approximation and the result may 
be later corrected sufficiently to satisfy the needs of observation. If, how- 
ever, the former angle is oscillating about a mean value, it is necessary to 
consider the nature of the motion of the latter according as it oscillates or 
makes complete revolutions. 

Another difficulty not exhibited by the pendulum is the existence of 
types of motion in which small oscillations do not exist but in which 
oscillations of finite amplitude can exist. In certain simple cases these 
types may be dealt with by the use of the periodic orbit and of variations 
from this orbit. But these methods have heretofore given little or no in- 
formation as to the range of the oscillations and this range may be of 
importance in actual problems. If, for example, the eccentricity of an 
asteroid can become so large under the influence of Jupiter's attraction 
that its orbit can intersect that of Mars, it is only a question of time until 
a close approach to that planet will occur and such a close approach may 
alter the orbit so fundamentally that a completely new investigation of its 
further motion will be needed. A method of approach to the investigation 
of such cases is given below. 



234 RESONANCE [OH. vm 

GENERAL METHOD FOR RESONANCE CASES 
8*15. Certain features of resonance problems have been 
developed in the previous sections of this chapter. In this and 
the following sections a method of procedure applicable to 
certain of the actual cases of resonance in the solar system will 
be given. 

The integers j,j f , for which j^ jiV is very small or zero, 
are usually less than 5. Here n Q , n r are observed mean motions 
whose ratio can be expanded into a continued fraction. If the 
successive convergents be formed, the practical cases of resonance 
are those in which a convergent with small numbers is so close 
to the ratio that the next convergent is a fraction with large 
integers. Since the order of the coefficient with respect to the 
eccentricities and inclination is l^i ^i'l (cf. 4*15), it follows that 
the coefficients corresponding to the higher convergents will be 
very small, and it will be assumed that their effects can be 
neglected in the limited intervals during which it is desired to 
obtain an approximation to the motion. 

The terms for which jin Q j\n' is not very small or zero can be 
eliminated by the method of 6 '6 and the resulting function there- 
fore contains Wi, Wi only in the combinations p(jiWiji'wi), 
where p is a integer. Further, since the new terms produced by 
the elimination of the short period terms have the factor m 2 , 
they may, in general, be neglected. Thus we can take as the 
Hamiltonian function 



mR = ~- + rafi + m2A cos pjiN, ...... (1) 



where R Q contains the terms in the elliptic development of R 
independent of Wi, Wi' y and ji N contains these variables only in 
the combination j\fw\ j\w\) where ji, ji are given integers. 
Any multiples of w^^ WQ, w^ f may also be present in the angles. 
The substitution of W for w\ defined by 



similar to that of 8*10 (5), is made. The equations still remain 
canonical if to (1) we add the term ji'n'ci/ji. 



15, 16] GENERAL METHOD 235 

In view of the relations Ci = (^a)*, n 2 a 3 = p, it is convenient 
to put 

Ci = C ( 1 + z)~l, c = (^a )*, wo 8 (to 3 = /*, NO = ji n'/ji , 

......... (2) 

so that ft = w (l -M), ........................ (3) 

and is a variable which in stable motion must oscillate between 
limits which are small compared with unity. 

With the notation (2), the function (1) with the additional 
termji'n'cj/ji can be written 

rc Co {i (1 + *)* 4- (1 + *)-*} + 01 JR, 
or, on expansion in powers of #, 



and with 2 replacing Ci we have 



As ^ is not present explicitly in R, the expression (4) equated 
to a constant is an integral of the equations. It may be written 
* 2 -* 3 +... +6mjR/w Ctt=(7. ............ (5) 

The symbol z corresponds to m^Ci used in 8*11 and therefore 
has the factor m*: it follows that C can be regarded as having 
the factor m. 

This equation is analogous to the first integral in the motion 
of the pendulum. In a first approximation, it is assumed that 
constant values can be given in R to all the elements except W. 
Retaining only the lowest power of z, we have 

2 r = (C r -6mJfZ/7? Co) i ................... (6) 

This will be expected to furnish at least two types of motion 
depending on the value assigned to C. It will be shown in 8*17 
that we can go a step further and include in the fundamental 
equation, which is that for W, terms depending on m$. 

8'16. Although we are concerned in this chapter with resonance cases, 
it is of some interest to apply 8'15 (6) to cases in which s is small but 
never zero for any value of t. 



236 RESONANCE [OH. vin 

Suppose that mR contains a single periodic term denoted by An c Q cos/! TV 
and that we include the non-periodic portion mR$ in C. As z does not 
vanish, we can expand 8*15 (6) in the form 



If #00 be the observed value of the mean motion, the definition of z gives 
for its mean value 



^oo i ./i^oo-/!^' j/ 1 36mM a v 

A ~ ~~ . \J * 1 A -r-r- ~~~77i> 1~ I .......... I & ) 

n ftitQ \ 16 6 Y2 / 

Since n w is nearly equal to n , they can be interchanged in the coefficients 
of periodic terms. By hypothesis, the value of C given by (2) is small 
compared with unity. Thus the coefficient of cosj\N in (1) receives the 
small divisor C-. Hence the periodic term in z is large compared with the 
term having the same argument in R. 

The canonical equation for w\ , with the definitions of W 9 z, n Q in 8'15, givea 
d W dR 

~dt= n Z - m ^ 

the second term of which can be neglected in comparison with the first. 
Integrating, and making use of (1), we have 



rrr / x , . . , r . 

TF= (9^ - tt ) t + const. - - - sin ^ ^- ^ \~~c~~) J 

Thus the principal perturbation produced in the longitude by a term of 
long period is 



and there is also another long period term with argument 2ji N having a 

coefficient 

- feji (coef. of sinjiN) 2 , 

a result in accordance with that obtained in 6'18 and also in 7-32. 

817. The equation for W. With the definitions in 815, the 
canonical equation for W becomes 

dW dR 



Differentiating with respect to t, we obtain 

dz &R ( 9R\ 

~Tt ~ m ^~^nr \n z-m 5-}-m %-, -jf 

at acidW \ 3cj/ 9ci a at 

_ v (*IL ** 4. & R *?A 

dt)' 



16, 17] FUNDAMENTAL EQUATION 237 

where, in the last term, i has the values 2, 3. Since the derivatives 
of Ci, <?a, c 3 , Wz, w s contain m as a factor, we obtain, on neglecting 
terms in this equation which are factored by m a , 

dz 



But the first of equations 8-10 (1) with 8*10 (7) gives, on 
substituting for Ci its value 815 (2), 



or, neglecting terms of order m 2 , 
dz 3 , t 



Now K is a function of Ci. If we put therein CI = CQ (1 
and expand .R in powers of z, we obtain 



where the notation ( ) implies that c has been substituted 
for c. 

Hence, to the order m', 

PR 



On combining this result with (2), (4), and noticing that 
(d*R/dcidW)o disappears, we obtain 



in which, to the order of the terms retained, we can put 



^^ 

~ n Q dt 

Thus the variable Ci has been eliminated as far as the order 
% and we can write (7) in the form 

3 ( 

+ ~ m ~ ....... 



It is to be remembered that the variables c a , 03, w z , w 3 are still 
present in the last term of (8). 



238 RESONANCE [CH. vm 

818. Another integral can be obtained when e' = 0. Since 
the disturbing function is a function only of the differences of 
the angles Wi,W2,w 9> W> W> w e have 

dR dR , 3-R dR dR _ A m 

T" ^ i ?\ T > T ^ > " ............. (*) 

dwi dw 2 



But by hypothesis the part of R which we are using contains 
wi t MI only in the combinations jiWi ji'wi, so that 



On changing the variable Wi to W y the new disturbing function 
has the same properties. Hence, from (1) and (2), 

I ^ /*_ | j j - t 

\ ji/dW dwz dwz dwj' 

Thence, with the help of the canonical equations 810 (1), we 
obbain by integration 

/, ji\ , f 3 R n / \ 

1 J -r ci -h c 2 -f c 3 = const. m ~ > dt (6) 

\ ji/ J ow 2 

If, in accordance with a previous notation, we put 
and make use of 815 (2), we obtain 



When e' =Q, R is independent of ^2' = ^ ' The last term o 
disappears, and the equation becomes an integral. 

THE CASE <?' = r = 

819. The variables c 3 , w 3 disappear and the canonical system 
reduces to one with four variables. The differences of the angles 
Wi, Wi, w* are present in R and the ratio of the multiples of 
Wi t W is fixed by the resonance condition. Hence a single angle 
N and its multiples are alone present in R. 

The system admits of the two integrals 815 (5), 818 (4). The 
latter enables us to eliminate the variable c. From the former, 



18, 19] THE RESTRICTED CASE 239 

with the equation for dz/dt y we can eliminate N and thus obtain 
an equation giving dz/dt in terms of z. After the integration of 
this equation, giving z in terms of tf, the remaining variables may 
be found without difficulty. The process thus described will be 
followed below but will be simplified by the omission of terms 
known to be small in comparison with those retained. 

We assume that z, e 2 , m are small compared with unity. The 
omission of higher powers gives e\=* e 2 , and from 8*18 (4), 



--z = const. 



or # = E-l\l- J 4Jz % (1) 

where E is a constant. 

Next, if R = RQ -f %A P cospjiN, R 0y A p are functions of c\ or z 
and c 2 only, and by the use of (1) can be expressed as functions 
of z. Since RQ contains only even powers of e, it can be expressed 
as a positive power series in z, and the constant part is all that 
need be retained, although the retention of z, z 2 creates little 
additional labour. The coefficient A p has the form 



where Oo> <#2?-" catl be expressed in series of positive powers of 
z\ the same limitations as those made in the case of RQ permit 
us to retain the constant term only. 

Finally, on the same basis, we put p = 1 and thus reduce R to 
a single periodic term and a constant portion. 

With these limitations, we can put 

R = const. + riQCo Ae J cosjiN. (2) 

On substitution of this in 8*15 (5), we can suppose that the 
constant part is included in C and thus obtain 

2* = G-6mAe J co*jiN. (3) 

With the same limitations, 8*17 (4) becomes 

~~ 7fi = 3? 1 m Ae J sin ji N. (4) 

770 Cut 



240 RESONANCE [CH. vni 

The elimination of jiN between (3), (4) with the help of (1) 
gives 

4 / d? \ 2 



...... (5) 

This equation has the form dzldt={f(z)}^ and gives t as a function 
of z. For values of / less than 5, the integral is of the elliptic 
type and the discussion of (5) or of its integral gives the chief 
characteristics of the motion. 

For the cases of chief interest in the solar system, J is, in fact, 
less than 5, and the equation includes all these cases. The most 
serious limitation is that introduced by the assumption e r = 0. 

8*20. Particular cases. These are classified according to the 
values of /. 

For J = 0, we have jt=ji = 1. This case, that of the Trojan 
group of asteroids, is treated in detail in Chap. ix. It permits of 
numerous simplifications, but the development ofR takes a quite 
different form. 

For /=!, the ratio ji/ji has the values 1/2, 2/8, 3/4,..., in 
the cases of exterior bodies disturbing interior ones, and their 
inverses when interior are disturbing exterior bodies. For these 
ratios, the mean values of a/a' are '64, *76, '82, It is doubtful 
whether the expansions are sufficiently convergent for numerical 
calculations beyond the ratio 4/5. The case 1/2 is discussed in 
detail in the following paragraphs. 

The case J==2*, corresponding to the ratios 1/3, 3/5, 5/7,..., 
is rather more simple than the case J= 1, owing to the fact that 
only even powers of e are present in the formulae. This case 
also arises when we take into account the inclination of the 
orbit. This and the higher values of J are chiefly of interest in 
the applications to asteroids disturbed by Jupiter. 

* See Charlier, Mech. des Himmels, Absch. (1) ; D B. Ames, Mon. Not. R.A.S. 
vol. 92, p. 542. 



19-22] CASE OF THE 2 : 1 RATIO 241 

THE CASE ji = l, ji' = 2 
8*21. Change of scale. If we put 

a)z, - t, ( ~ } e, for z, t, e respectively, 
??o&> \o/ 

where a> = (12A 2 m 2 )%, the equations given in the preceding 
sections become, with the given limitations and with appropriate 
changes in E, C, 

d W dr 

* 



-~> , N= Wl - 2^' + OT = W + <*, 

Cvv Clu 



For an asteroid disturbed by Jupiter with n Q /n f = 2, we have* 
= -000716. For the change of scale we have o> = *0183, 
so that an actual eccentricity '1 has the value *78 in the new 
scale. The values of the variables in the new scale are thus 
comparable with unity. 

8*22. There are two problems. One, that dealing with the 
conditions under which N is an oscillating angle (resonance) or 
a revolving angle (non-resonance). The other, the conditions 
under which z can pass through the value zero. The latter is 
not, in the limited case here treated, strictly a resonance problem, 
but it becomes one when e' ^ and it is applicable to the cases 
of the apparent absence of asteroids for which the osculating 
mean motion is exactly twice that of Jupiter. 

Conditions to be satisfied. 

(i) Since z measures the deviation of the major axis from 
a mean value, it must be an oscillating function and must there- 
fore lie between finite limits which will be denoted by s d: we 
shall choose d to be a positive number so that s + d is the maxi- 
mum value of z and s d the minimum. 

* Mon. Not. R.A.S. vol. 72, p. 619. 



242 RESONANCE [OH. vm 

(ii) The limiting values of z are given by dzjdt = 0, so that 
s d are two of the roots of 

-(z*-C)* = ................... (1) 



(iii) The left-hand member of (1) must be > for all other 
values of z. 

(iv) The convergence of the developments is doubtful if 
the actual eccentricity is greater than about '3; this gives a limit 
2*4 to the variable in the new scale. 

(v) The conditions s = d separate the cases in which z can 
or cannot take the value zero according as s > or s < 0. 

(vi) The equation e 2 = E+2z gives 
de dz d 2 e /de 



_ 
e dt~dt' dt 2 \dt """3?" 

It is necessary to have d 2 z/dt 2 =f= when dzfdt = 0, in order 
that equation 8*21 (1) shall give a determinate value of z for all 
values of t. It follows from the equations just written that 
e y de/dt cannot be zero simultaneously. Since e is not negative, 
it can vanish only if de/dt vanishes simultaneously: hence e is 
never zero. 

Since z = s d are two of the roots of (1), it is easily deduced 
that 

CW + CP--JJ-, E = 4*ffi + -28, ...(2), (3) 

and that 8*21 (1) may be written 

-4(^) 

These results give 



............... (5) 

so that the maximum and minimum of e are given by 

I ...................... (6) 



ZZ-Z4J RlSti'LKLVLXilJ Z:l VADXi '416 

8-23. The identities 



compared with 8*22 (6) and with 

z 2 = s 2 -I- d 2 - n 6 cos N, 

ZiS 

show that if 5 > 0, d > l/4s 2 , the extreme values of N are TT, 0, 
but that if d< 1/4$*, jV" takes the value TT at both extremes, 
Hence, the relation d = l/4s a separates oscillating from revolving 
angles of N, that is, the resonance from the non-resonance case. It 
is easily seen that the same statement is true if s < 0, but that 
in the latter case N takes the value zero at both extremes when 



8*24. The four values of z given by 8*22 (4) when dz/dt are 
z l = s -f d, z% = s d, 



By hypothesis 2 is to lie between z\, z. This condition demands 
that 



or that 3 > 4 > #1 > ^2- 

(a) For 5 > 0, d 2 > 1/5, the descending order of magnitude 
is zi, z%, z$, 24. The condition z 3 < z z gives 

d<2s, d<s-f^-2. 

fr<S 

(6) For 5 > 0, d 2 < 1/5, the roots #3, # 4 are imaginary and dz/dt 
is always real between z = z, z%. Hence for s > the boundary is 
d = s + l/4s 2 , when d < 2s, 
d 2 =l/s, when d> 25. 

If we regard 5, d as the rectangular coordinates of a point on a 
curve, the two conditions are the equations of two bounding 
curves which meet and have the same tangent at d = 2s = 2^. 



244 



RESONANCE 



[OH. vm 



(c) If s < 0, all four roots are real, and the ascending order 
of magnitude of the roots is z Z) zi, #4, z%. This demands that 
2s + d > 0, d<-s- l/4s a . Since d > 0, s < it is easily seen 
that the latter condition includes the former. It also requires 
that s + d < 0, and as we have s d < 0, it follows that z does 
not change sign. Hence, z cannot vanish for s < 0, that is, if z can 
vanish its middle value is positive. 




-1-5 -1-0 -0-5 



1-6 



The conditions that z may vanish are therefore quite compli- 
cated. The boundary of the region consists of the four curves, 



and d, s are both positive. 

On the other hand, the resonance regions for N are simply 
those portions included between the two curves 



for s < and s > 0, where z is real. 

In Fig. 2, z is imaginary in the regions with inclined shading; 
z can be zero in the region with horizontal shading; N oscillates 



24, 25] RESTRICTED 2 : 1 CASE 245 

in the region with vertical shading; the two latter regions 
overlap as shown. 

Passage from s > to s < is effected when the minimum in 
the former case is the same as the maximum in the latter, that 
is, along the boundary d = s 4- l/4s 2 for s > to the boundary 
d = s l/4s a for s < 0. Along these two curves and also along 
d?=l/s, two of the values of z are equal and W jzdt is 
indeterminate near t= oo in the same sense as in the case of 
the pendulum near its highest position. 

The only other case of equal roots for z is that given by d 0. 
When s > and d is small, the second factor of 8'22 (4) is 
approximately 4s 2 -f l/ which is constant. The solution is then 

z = s + d cos (qt -f <?o), <f = s 2 -f l/4s. 

The same solution is available for s < provided s is not too near 
the value given by s 2 -f l/4s = 0. In these cases the mean value 
of e is 1/2 \s\, so that s must not be too small. The case d = 
in which z oscillates about the value s is the resonance case for 
N TT when the libration is zero. 

From 8-22 (2), (3), we deduce 

C 2 - E=(s*-d 



so that C 2 E changes sign at the boundary separating the region in which 
z can be zero from that in which it is never zero. But C 2 also when 
d 2 =8 2 +l/s, a relation which does not enter into the discussion given above. 
It follows that the condition O L =E is not the necessary and sufficient 
condition that z shall take the value zero*. 

In vol. 4, chap. 25 of his Mfaanique Ce'leste, Tisserand treats the resonance 
case by supposing that the eccentricity is equal to e -f Se, where e Q is a 
constant, and he expands in powers of 8e/e Q . With a proper choice of e Q 
this is theoretically possible, since e is essentially positive, but it gives very 
slow convergence in the most important cases those in which e Q is small. 

8*25. Applications. The discussion in the previous sections is 
applicable to the cases of asteroids whose mean motions are 
nearly twice that of Jupiter. The statistical discussions f show 

* For a different and less complete discussion see E. W. Brown, Mon. Not. 
R.A.S. vol. 72, pp. 609-630. 

f These have been numerous. Fairly complete lists are given by S. G. Barton, 
A.J. 702, 838; A. Klose, Mitteil. Univ. Riga, 1928. 

B&SPT 16 



246 RESONANCE [OH. vnr 

that while there are numerous asteroids with mean motions 
somewhat greater and somewhat less than twice that of Jupiter, 
there is none which can be stated with certainty to have the 
relation satisfied within a certain range. This result refers to 
osculating elements. If we omit the short period terms, the 
variable z may be regarded as an element of this nature and the 
vanishing of z corresponds to the exact relation. 

Now we have seen that the limiting case in which z can be 
zero is given by s = d and the maximum value of z is then 2s. 
According to 8*22 (6) the maximum value of e is then 2s -f 1/2$ 
and the least value which this expression can have is given by 
5 = J. The least maximum of e is, therefore, 1'5. On referring 
back to the scale relation in 8'21, we find that this gives a least 
maximum for the eccentricity of *13. 

So far, therefore, nothing has been proved which prevents the 
existence of asteroids which can have an osculating mean motion 
exactly twice that of Jupiter. But it has been shown that if such 
asteroids can exist, the elements, in particular the eccentricity, 
must oscillate through a considerable range of values ; small 
oscillations are impossible. 

The existence of asteroids or satellites in which the angle N 
oscillates is a quite different question. What has been shown is 
that if such orbits exist, the middle value s of z must be different 
from zero. Small oscillations or librations about this value are 
possible. We have, for example, the case of Titan and Hyperion, 
satellites of Saturn, where the ratio is nearly 3 : 4, a case similar 
to that of 1 : 2. 

THE CASES e'4=0 

8*26. These cases are much more difficult, mainly because the 
integral e 2 = E f (1 ji/ji) z no longer exists. But in the cases 
of the ratios j\ j\ + 1, where the principal terms are of the first 
order with respect to the eccentricities, it is possible, in a first 
approximation, to utilise the results obtained above by a change 
of variables. 

For simplicity let us consider the case ji= 1. The additional 
first order term in R has the form n Q CQe'A' cos N' 9 where 



25, 26] EXTENDED 2 : 1 CASE 247 



JV's= W + vr'. Arguments similar to those used above give the 
equations for W, z : 



- *r / At ' *T/\ 

-- , _ __ sm N + e A sm N) 9 

n at n^ at 

z* = C-6mAe cosN- 6m A' e' cos N'. 

Instead of the variables 02, w 2 , let us transform to the variables 
2> (72 defined by 

sin w 2 , #2 = 0i Ci* cos w 2 , 



where, as before, 2 = isi^i 2 - According to 5'14, the equations 
for Ci, W, p 2) <?2 are still canonical. 

Let us change from p 2 , q* to new variables defined by 



sn cr, ^2 = ^2 + ^ cos 
where X is a function of Ci only. We have 



with similar equations for dq^'/dt, Sq%'. 

Now dp2/dt dR/dq z , and JK is a linear function of p 2 , q 2 for the 
only terms we have under consideration. It follows that dp 2 /dt 
does not contain either eccentricity as a factor while dci/dt does 
contain them. The second term in the equation for dp 2 /dt is 
therefore two orders, with respect to the eccentricities, higher 
than the first term and maybe neglected. The canonical equations 
may therefore be written 

dt . $H=dci . 8W-dW. Sci + dp 2 . Sq 2 - dq 2 . Sp 2 

p z ' . Sq 2 ' - dq 2 f . Sp 2 ' 



3 (e f cos OT' dp 2 e' sin cr' dq%) Sci. 

C/Ci 

But the approximation z = dW/dt to the equation dW/dt=dH/dci 
involved the neglect of all parts of It in this equation and this 
is the only way in which the coefficient of 8ci in the canonical 
set just given arises. It follows that the equations for Ci, TF, 
PZ> q* are still canonical. 

16-2 



248 RESONANCE [CH. vm 

These results suggest that we can put 

,A r 

e cos -or = e cos *& + e ? cos -& , 



e sin ts = e sin *r + e -~r sin IB , 

so that __ 

e A sin N -f e'A ' sin N' = 4 sin N, } ^ _ 
e^. cos JV^-f e'^1 ' cos N' = e^. cos N, ] 

and that when we do so, e, N will have the same properties that 
e, N had in the case e' 0. 
In particular, we shall have 



and the limits previously given for e will now apply to e. 

In the cases of the asteroids disturbed by Jupiter we have 
b'/b s=s -36, e' = '048 (loc. cit. 8'24), so that unless e is small the 
additional terms will not give large corrections to the results 
previously obtained as far as the vanishing of z is concerned. 

In the cases of the small oscillations of N or JV, it appears 
that these must take place about the values or TT and that 
& - tar' must oscillate in a similar manner. But the argument, 
based on the assumption that TS' is constant, is not necessarily 
valid if -or' has a mean motion. 

8*27. The methods of this chapter are constructed mainly for the treat- 
ment of those cases of resonance which arise in the solar system. The 
theory of periodic orbits is applicable as a first approximation in certain 
problems : the asteroids which form the Trojan group are examples. In 
general, however, this theory fails, either because the numerical applications 
are too remote or because the restrictions under which the theory is 
developed avoid the very difficulties which the actual problems present. 

The methods given above apply to cases of resonance in which both 
periods of revolution are present. The perihelia and nodes are angles which 
in general revolve and there are possibilities of resonance relations between 
their periods of revolution. In the comparatively short interval of time 
during which observations have been made, such relations are unimportant 
because, with the very long periods involved, expansions in powers of the 



26, 27] APPLICATIONS TO COSMOGONY 249 

time give the required degree of accuracy. Comparable with these are the 
new periods introduced by the librations, and there are, therefore, further 
possibilities for resonance relations. So long as the past history of the 
solar system was supposed to be confined within an interval of 10 8 years, 
deductions as to its initial configuration from its present configuration ap- 
peared to have some degree of value ; the extension of this interval to 10* 
years or longer makes these deductions quite doubtful. The doubt appears 
not so much in the ranges of values possible for the mean distances as in 
the ranges of the eccentricities and inclinations. 

The indications furnished by the theory of resonance as applied to the 
solar system point towards the possibility of occasional large osculating 
eccentricities and inclinations at some time in the future. On the other 
hand, statistical evidence appears to indicate that these elements will tend 
to be confined within narrow limits. A discussion of these and other 
difficulties involved in the attempts to apply the theory to the solar system 
will be found elsewhere*. 

* E. W. Brown, Bull. Amer. Math. Soc. May-June, 1928; PuU. Astro. Soc. 
Pac. Jan. 1932. 



CHAPTEB IX 

THE TROJAN GROUP OF ASTEROIDS 

9*1. The triangular solutions of the problem of three bodies. 

The problem of three bodies does not, in general, admit a 
finite solution in terms of known functions. Laplace, however, 
has shown that there is a solution in which the three bodies 
always occupy the vertices of an equilateral triangle. The plane 
of the triangle is fixed and any two of the bodies describe 
ellipses having the same eccentricity about the third body which 
lies in a focus. Further, if n, a be the mean motion and semi- 
axis major of any one of these ellipses, the relation 

tt 2 a 3 = sum of the masses 

is found to be a necessary consequence of the solution. Other 
sets of finite solutions, in which the bodies are collinear, are 
known but they will not be considered here. 

Small changes from the triangular configuration or in the 
appropriate velocities, or perturbations by other planets, cause 
oscillations about the triangular configuration, provided the 
masses satisfy a certain limiting configuration ; the study of these 
oscillations and the applications of the theory are the objects of 
this chapter. Ten asteroids, each of which, with Jupiter and the 
Sun, apparently satisfies the given conditions, have been dis- 
covered, the first in 1901, more than a century after Laplace 
gave the solution, and the last in the year 1932. They have re- 
ceived names taken from the Iliad of Homer and from this cir- 
cumstance constitute what is usually called the Trojan group. 

We shall first prove the existence of the triangular solutions 
and of small oscillations of a certain kind about this solution ; 
these will indicate some of the characteristic features of the 
motion. A general theory for the motion of an asteroid of the 
Trojan group will then be based on the methods used in 
Chaps. VI and VIII. 



1,2] THE TRIANGULAR SOLUTIONS 251 

The problem differs from that of the ordinary planetary theory 
in several respects. In the first place, the development of the 
disturbing function given in Chap. IV cannot be used because 
the ratio of the mean distances of Jupiter and the asteroid is 
very near unity and that development ultimately depends on 
series in powers of this ratio which do not converge when the 
ratio is unity. In the second place, the motion is a case of 
resonance, since the ratio of the mean motions of the asteroid 
and Jupiter oscillates about the value unity. Thirdly, these 
oscillations, instead of being small, may have very considerable 
amplitudes and require special methods if an accuracy com- 
parable with that of observation is to be secured. Another point, 
brought out in Chap, vm, is the development in powers of the 
square root of the ratio of the mass of Jupiter to that of the Sun, 
instead of in integral powers of this ratio as in the ordinary 
planetary theory ; since the square root of a small fraction is 
much greater than the fraction, the rate of numerical con- 
vergence may be much diminished in consequence. Still another 
peculiar feature is the theory of the long period terms produced 
by other planets, and notably by Saturn. A first approximation 
to their coefficients cannot be obtained by neglecting the action 
of Jupiter, and these coefficients tend to become greatest, not 
when the periods are longest, but when these periods approach 
most nearly to that of the principal libration. 

9*2. Existence of the triangular solutions. 

Since the motion takes place in a fixed plane, the latter may 
be used as the plane of reference. With the use of the equations 
of 1*23, those numbered (5), (6) disappear and v = v. Let us take 
one of the bodies, mass m , as origin and let the coordinates and 
masses of the other two bodies be r, v, mi and r', v' t m'. It is 
then necessary to show that the equations 

*-W d( dv 

Jt) ~dr' 5V 5 



_ 
dt* \dt ~ 9/ ' dt \ dt 



252 THE TROJAN GROUP OF ASTEROIDS [CH. ix 

where, according to 1'9, 1*10, with the inclinations zero, 

^ mo + mi , , fl r cos (v v')) 7 _, 

H ' __ * I AV) ' J .. V / I I ^ I 

J.' ~ ~ in i A ~"~ 7o i j \ *-* / 

r (A r 2 J ^ ' 

, _ 7/? + w' (1 r' cos (v' - v)\ /r . 

jp j_ ^j j v 10) 

r (A r 2 j 

A 2 = r 2 -f r' a - 2rr' cos (v - v'), 

are simultaneously satisfied by r = r' = A, v t/ = 60, with 
elliptic motion for each of the bodies. 

According to 3'2(1), (2), these conditions demand that 



--_ -- 

dr ~ dr' ~ r 2 ' dv "~ fo' ~ 
From (5) we have 

dF _ w Q + m t , (r - r' cos (v-v') cos (v t;' 

-' 



dr 

^ = _ 

dv ~ 

with similar expressions for dF'/dr', dF'/dv'. It is at once 
evident that the equations (7) are satisfied by the given 
relations provided /u = m Q -f mi 4- rti'. 

The elements n> a, are evidently the same for the two 
ellipses with n 2 a 8 = 7W -f m\ 4- w'. For the remaining elements 
we have e 6 / =cr t*r'=4 60. 

9'3. 2%^ equations of variations. 

These equations are defined by giving to the coordinates in 
the general equations of motion small additions to their elliptic 
values, the squares, products and higher powers of these 
additions being neglected. This procedure is not sufficient for 
the calculation of the general perturbations, but it serves to 
indicate their nature to some extent. The actual calculation of 
the perturbations is more easily carried out by quite different 
methods. 

The problem will be limited now and throughout the remainder 
of this chapter by supposing that the mass mi of one body (the 



2, 3] EQUATIONS OF VARIATIONS 253 

asteroid) is so small compared with either of the masses of the 
other two bodies (the Sun and Jupiter), that it can be neglected 
in the equations of motion. We then have F' = (m Q 4- w')/r' and 
the motion of m r relative to mo is elliptic with mo in one focus. 

In the present section, two further limitations will be made. 
The motion of m' relative to mo will be supposed to be circular 
and to receive no disturbance, and the disturbance of mi will be 
supposed to take place within the plane of motion of m', so that 
the problem of the motion of mi is still two-dimensional. 

According to these assumptions, the undisturbed motion of 77*1 
will be circular. Denote this motion by the suffix zero and the 
disturbed values by 

r = r + Sr, v = VQ -f Sv. 

Substitute these values in 9'2(1), (2) and expand in powers ot 
Sr, Sv and of their derivatives, neglecting powers and products 
of these quantities above the first. Since the equations are 
satisfied when 8r, Sv and their derivatives are zero, this pro- 
cedure gives 



, 

-Ji ~J7 ji ^~2 ^-^- Sv, 

dt ) dt dt 2 



d ( 2 d * , O dv <> 5^ \ f & F \ S> , / 92 A 55 

-r. r 2 -j-. Sv + 2r ~-~ Sr = 5-=- Sr + ( 5-0 ) ov. 

dt\ dt dt ) \drdv Jo \9^ 2 / 

These are the 'equations of variations.' 

The second derivatives of .fare formed from 9'2 (5), (6). They 
are 

') m' 3m' , , x)2 



1 8 2 f /I IN , . , /x 

7 5-^-= ~ ( Xa /a ) r sm (v - v ) 

m 9r3v \A 3 r' 3 / v 



The limitations imposed above give 
r = a = r / = A, VQ v' = 60, 



254 THE TROJAN GROUP OF ASTEROIDS [OH. nc 

The substitution of these values in the second derivatives of F 
and in the equations of variations gives for the latter, 



s . 2 m -m ,, 

or zarz ~r A ov n*or = j-= o?* -f -=-- ra ov, 
dt 4a 3 "" 4a 2 

d * .3 A/3 ,, 9m' 

^ ^ r = + "T~2- m ^ r + T 8v. 

dt ~ 4a 2 4a 

These equations being linear with constant coefficients, their 
solution is obtained by assuming 



where A, B, X are constants. The substitution of these values 
gives, after division by e xt , the conditions 



The elimination of the ratio A : between these equations 
gives 



X* + X' 8 n - 2 *- + + T- = 0. 

V a 3 / 4a 8 \ a 3 ; 

The use of the relation ?i 2 a 8 = r??o + ^i / and the introduction 
of m, where 



reduce this to 

X 4 + X 2 /i 2 + ^-w 4 /^ (1 - m) = 0. 

If 27//i(l-wi)<l, 

or m < '04 approximately, the roots are all pure imaginary and 
the motion is oscillatory. Since m < *001 in the case of the Trojan 
group, the condition is easily satisfied. If powers of m beyond 
the first be neglected, the roots are 



so that the periods are 

2-rr -r n V^w, 2w -T- n (1 - 



3] PERIOD OF LIBRATION 255 

With 27T/n = 11*86 years, l/m=1047, the former period is 
148 years and the latter nearly the same as that of revolution of 
the asteroid or of Jupiter round the Sun. 

The oscillation having a long period is a first approximation 
to the effect usually known as the 'libration.' The short period 
oscillation will be seen below to correspond to the principal elliptic 
term in the motion of the asteroid, so that the principal part 
of the motion of the perihelion is %g-mn. 

The ratio B:A for the long period oscillation is given by either 
of the equations (1), (2) with \ = m (27ra/4)i From (2) we find 



aB 3m'-ani-m _ 1 t, 
A ~~ -27a 3 ?i 2 m-9ra' ~~ + T * \/3m' 

with the aid of the relations a?n 2 = m Q + m' = m' jm> Since the 
second of the two terms is large compared with the first, the 
approximate ratio of \aB\ to | A\ is 1 : V3?Ai or 18'7 : 1. As | A\ is 
the amplitude of the oscillation along the radius vector and 
\aB\ that perpendicular to it, it follows that the former is small 
compared with the latter, the ratio being nearly as V8m : 1. 

The features of the motion brought out in this investigation, 
namely, the long period of the libration, the small disturbance 
along the radius vector as compared with that perpendicular to 
it, and the presence of Vm, will be utilised in the general theory 
which follows*. Incidentally, it may be pointed out that they 
are present in all resonance problems occurring in connection 
with planetary motion, as can be shown from the results of 
Chap. vin. 

We now proceed to the general theory of the motion. 

* The small oscillations were first fully treated by E. J. Eouth, Proc. Lond 
Math. Soc. vol. 6 (1875) ; see also his Dynamics of Rigid Bodies, Part n, Chap, in . 
A special case of them is treated by Charlier, Himmels-Mech. vol. 2, Chap. ix. 



256 THE TROJAN GROUP OF ASTEROIDS [OH. DC 

GENERAL THEOKY OF PERTURBATIONS 
DUE TO JUPITER 

9*4. The disturbing function. 

According to 1*10, the force-function for the action of a planet, 
mass m' , on one of mass nil when the sun, mass mo, is taken as 
the origin of coordinates, is 

r 



,- x 
(1) 



where A 2 = r 2 + r' 2 - 2rr' cos 5, ............... (2) 

/S being the angle between the radii r, r'. 

If 8 be eliminated between (1), (2), F can be written in the 
form 

F= mo + mi + m ' + R^ + R, ............ (3) 

r r x 7 

, D ,/l 1 1 A 2 1 v*\ ... 

where Brsm U~r + 27*~ 2?j' ............ (4) 

since the term | m'/r', thus introduced, can play no part in the 
equations of motion which depend only on the derivatives of F 
with respect to the coordinates of mi. 

In the form (4), .72 = when r r' = A. It can also be written 



a form which at once shows that the first derivative of R with 
respect to any coordinate of mi vanishes when A = r = r'. 

The mass mi of the asteroid will be neglected in comparison 
with mo-fw'. The osculating mean motions n t n' and mean 
distances a, a! will then be connected by the relations 



9'5. The equations of motion. 

We start with the variables c^, Wi (defined in 5*13 and used 
in Chap, vi) which satisfy the canonical equations, 



i+R .......... (1) 



4, 6] GENERAL THEORY 257 

Define a new variable r by the relation 

Wi = n' -f e' + r = Wi + T, ............... (2) 

and in (1) replace the variable Wi by r. Since dw-i^n'dt + dr, 
8^ 1= =8r, the equations for the variables c^, T, ^2, ^3 will still 
remain canonical if we add H'CI to R in the right-hand member 
of (1) so that the Hamiltonian function becomes 



(3) 

Since all the d have the dimension \V, arid since a/a' is near 
unity, let us put 

d = Ci (1 4- X) = \Va' (1 + #), C 2 = Ci'^2 > C3 = Ci'# 3 . . . .(4) 

The equations for the variables #, # 2 , # 3 , T, w 2t w 9 still remain 
canonical and can be written 

dx . ST dr . 



Since JK is a homogeneous function of a, a' of degree 1 and 
contains m' as a factor, the Hamiltonian function is now a pure 
ratio like the variables x> # 2 > #3> T, w 2 , w^, and ofR/p has the 
factor m = m'//^. 

It will be seen below that x consists of portions which have 
either the factor m or the factor TI Vm, where TI is the coefficient 
of the principal term in the libration. No case of an asteroid is 
yet known* in which \x\ exceeds *01, and as eccentricities and 
inclinations are in general of the order *1, x will be treated as of 
the same order as the square of the eccentricities. We shall 
carry R as far as the fourth powers of the eccentricities and 
inclination for the long period terms and on this basis it is 
advisable to retain terms of orders x*, mx 2 . In such terms the 
eccentricities and inclination can be neglected, and then the 
retention of terms of order x*, mx 2 presents no serious difficulty, 
but as it will much simplify the exposition to neglect them, they 
will not be retained in the developments which follow. In any 
* Except possibly Anchises discovered in 1931. 



258 THE TEOJAN GROUP OF ASTEROIDS [OH. ix 

case we are going to neglect terms in the Hamiltonian function 
which have the factor w 2 , provided such terms have no small 
divisors, so that the neglect here of the terms of orders # 4 , ma? 
follows a general rule. 

With these limitations, the right-hand member of (5) can be 
expanded in powers of x, and becomes 

n'dt . S (I a* - W + + 2x ~] to order , . . .(6) 

\2 p fJb OCLJ 

where R is the value of R when a = a' or x = and similarly for 
the definition of 3R/9a. The approximation a = a! (1 -f 2#) has 
been used in the expansion of R. 

9*6. Form of the expansion of R. 

As in Chap. IV, it is assumed that R is developable in powers 
of the eccentricities and mutual inclination. The development 
there given is available also here as far as the stage where it is 
reduced to odd powers of 1/A (cf. 4'13), with 



a (JL 

and to derivatives of these powers with respect to a, wi. With 
WI WI=T, A becomes a function of a, F, r, and r is an oscil- 
lating angle. 

The angles present are g, g' , Wi + Wi - 20 or r, w Z) WB, Wi, -cr'. 
Thus R consists of a sum of terms of the form 



where K> K' are functions of x, x%> ^3, r and N of r, w^, w 3> t. The 
particular point to be remembered is that r is present both in 
coefficient and angle because K, K' contain functions of A . 

We shall distinguish between the terms containing w\ after 
the substitutions, g tVi Wz, g' = Wi' w^, WI = T + WI, and 
those independent of Wi. Our preliminary investigation showed 
that dr/dt was small so that the terms in the former class have 
short periods and those in the latter have long periods or are 
constant. The latter class is also distinguished by being inde- 
pendent of t explicitly when the orbit of Jupiter is an ellipse. 



5-7] THE SHORT PERIOD TERMS 259 

These considerations give us the form of the development, but 
the actual development will be carried out by a method quite 
different from that of Chap. IV, chiefly because very considerable 
abbreviations of the work are possible with the use of the special 
properties which R possesses in the case of the Trojan group. 

In general, it appears that we can secure sufficient accuracy 
for observational needs by taking the short period terms to the 
second order with respect to the eccentricities and inclination, 
and the terms independent of iv\ to the fourth order. Classified 
with respect to the arguments g, g' ', w\ 4- iv\ 26 used in Chap. IV, 
the latter are 

Arg. 0*, orders 0, e 2 , e' 2 , e\ e 2 e' 2 , e'\ T 2 ; 

9~9*> orders ee' , e z e f , ee'*\ 
>} 2#-2<7', order eV 2 ; 
2g-(wi + wi'-20), order (?T \ 

9 + o' ~ ( w * + w i ~~ 2#)> or der 00' r 5 

2g' - (w l + wi - 2(9), order e f * T. 



9'7. Elimination of the short period terms. 

The plan adopted is that used in Chap, vi, namely, a change 
of variables which leaves the equations canonical. Owing, how- 
ever, to the fact that the variable r is contained in both coefficients 
and angles, the form of the transformation function S has to be 
modified. Further, use can be made of the fact that with the 
neglect of m 2 , one variable x appears in the differential equations 
only in a linear form as shown in 9*5. 

Let the short period terms be denoted by R t and put 



where, in accordance with 9'5 (6), K, L are independent of x, 
but are functions of # 2 > #3, T, and 

N^j'n't -f multiples of T, w 2> w 9 + const., 
as shown in 9*6. 

* The term of order T is included in A . 



260 THE TROJAN GROUP OF ASTEROIDS [CH. n: 

Let us transform to new variables #o, #20 > w> T o> WM, iu& by 
means of the transformation function /, where 



= sn 



sin jy + cos jy 



jn \jn j 

In this last expression, 

Ki t LI are the values of K, L when TO is put for T ; 

MI, Qi are functions of #, # 2 > #3> T o to be determined; 

NQ is the value of N when T O , 1%) , WSG are put for T, ^2, ^3* 

According to the general theory, the relations connectingTthe 
new and old variables are 

_as = _d_s_ \ 

dx> XQ " X aV ...(2) 

3 Cf Ci Q ^ ' . 

__ == __^_ ' 9 S 

and the equations for the new variables will still be canonical 
provided we add d$/dt to the Hamiltonian function. From (1), 



and 
SS/dt = 2 (^i + #ii) cos A 7 + ^2 (Mi cos A r o - (?i sin N ). . . .(3) 

In performing fche transformation to the new variables, we 
shall neglect terms factored by w 2 . We recall that x has the 
factor m* while 8 has the factor m. Hence, to the order m$, 
from (2), 

) , ^ 3 = *o 3 , ...(4) 



where SQ is the value of 8 when # , ^2o> ^30 are substituted for 
#> ^2> #3 therein. 

Next consider the portion of R independent of t explicitly, 
denoting it by R c . Since the new variables differ from the old 
by terms having the factor m y we can put 



~z cO i "O o 

oa /JL /it da 



7,8] CHANGE OF VARIABLE 261 

where terms factored by m 2 are neglected. (Incidentally, it may 
be noticed that the terms of order m? are all of short period, so 
that even if we were calculating the long period terms with 
terms of order m 2 retained in the Hamil toman function, these 
could still be neglected.) Similarly, we can replace E, t by R^o, 
and dS/dt by dS /dt. 

If then in (4) we put for S Q its value deduced from (1), we 
obtain for the new Hamiltonian function expressed in terms of 
the new variables, 

f * a - 2* 3 + -' R.,0 + 2/r. 8 ^ 
/JL /A da 



- x Q 2 (Mo cos N Q - Q Q sin JV ), . . .(5) 

where, in all cases, the suffix zero denotes that the old variables 
are replaced by the new in the corresponding functions. The 
terms arising from R to have been cancelled by the same terms 
present in dSo/dt, and terms factored by xgm have been omitted. 

Finally, if we determine the coefficients MI, Qi by the relations 

ZKrfN Q 3 a^ 

^ 1 = jV9V ^ = ~JV8^' ............ (6) 

relations which still hold to the required order when the suffix 
zero is inserted, the terms of order m% will disappear from (5) 
and it will be reduced to its first line. 

The remaining portion of this chapter will be devoted to the 
determination of the new variables in terms of the time. After 
this work is done, the old variables will be obtained in terms of 
the time with sufficient accuracy if we substitute #o> #20, #30 for 
x, #2> #3 i n S and its derivatives in equations (2). 

9*8. The expansion of R in powers of e, e', F. 
We have, as in 41 (2) and 110 (2), 



cos 8 = (1 - JP) cos (v - v') + |F cos (v + v'~ 20). 
If we put 

sCt, -<;')> 



B&SPT 17 



262 THE TEOJAN GROUP OF ASTEROIDS [CH. ix 

the expansion of R given by 9'4 (4) in powers of the second 
term of cos S as far as the order F 2 can be put in the form 



' cos 2 ( + '- 20)!, ...(1) 



1 16 r' 3 "A^ 

where RI== ^ + 2 ^*~ 'r ~ 2?* (2) 

The long period terms defined in 9 7 will be calculated as far 
as the fourth order with respect to e, e' y F*, and the short period 
terms to the second order. 

For this purpose, put 

a(l+/3), 



~ 



(4) 



so that RQ, A are the values of Z?i, AI when the eccentricities 
vanish. Taylor's theorem then gives 

a'Ri_a'lt a u'p ?R a'dlto 1 a'p* 2 S 2 .Ro 
r' ~ ~7~ + 7- r " 8 + ? ? IT + 2~7~ a 3 8 + -' 

...... (5) 

which is to be continued to the fourth powers of p, f for the 
long period terms. 

For the calculation of the coefficients of the derivatives of R Q 
in (5), we have, from 316, 



= 1 + (e - |e 3 ) cos# + e 2 cos 2 

.& = 2 (e - ^e 8 ) sin^r + |e 2 sin 2#, 

with similar expressions for a'/r', E'. Since gr = Wi + isr, 
gr' == wi 4- ^r', the long period terms will be those whose argu- 



8, 9] EXPANSION OF B 263 

ments are multiples of g g'. The expressions (6) are sufficient 
to obtain such terms to the fourth order, in spite of the omission 
of terms of the forms e 4 cos 2g, e 3 cos 3g, e* cos 4*g in a/r, and of 
similar forms in E, a'/r', E'. For a term with argument 2g 
must be combined with one with arguments 2g, g + g', or 2g' to 
give terms of the required form and these have coefficients of 
the second order, so that the combination is of the sixth order. 
Similarly for the other terms omitted. Finally, RQ and its 
derivatives have no short period terms, so that it is sufficient to 
omit such terms in the expansions of powers of p, 

The advantage of this mode of development is seen by a 
reference to the results given in 9*9. Four of the coefficients are 
zero, three others are the same except for the numerical factors, 
jwo others have the same property, and two more differ only in 
;he fourth order parts. 

9*9. The coefficients of the derivatives of JK * 

To obtain these expressions put a/r = 1 4- u\ , a'/r' = 1 4- MI, 

so that 

of a of , 

p -j = --- 7 = Ui Ui 
r r ' r T ' 

has no long period term. The functions needed have the form 



where i -f j 4. The calculations appear to be most easily 
carried out by expressing each such product as a sum of terms, 
each of the form PQ', where P is a function of ui, E, and Q' of 
ui, E',r'\ from these products the terms independent of g, g' ', 
and those with arguments g g' , 2g 2g', are easily selected. 
The positive and negative powers of r, r' which are needed and 
the positive powers of E, E 1 can be read off from Cayley's 
tables*. 

Use can be made of the fact that pa'/r', E E' both change 
sign when e, g are interchanged with e' y g', so that terms of 
the forms ee' sin (g g'), e 2 e' 2 sin 2 (g g') cannot be present in 

* Mem. Roy. Astr. Soc. vol. 29, pp. 191-306. 



264 THE TROJAN GROUP OP ASTEROIDS [OH. ix 

products of p/r f , p 3 /r' 3 with f, f 8 (in the fourth order terms the 
divisor r' takes the value a'). The remaining terms have the 
form (eV e' 9 e) sin (g #'), and these disappear on account of 
the relation between the coefficients of cos g, sin g in a/r, E, 
respectively. 

The following results for the terms independent of g, g' and 
for those dependent on g g' and its multiples, to the fourth 
order with respect to e, e', have been obtained : 



- (2ee' + f e a e' - f ee' 8 ) cos (g - #') - f eV 2 cos 2 (g - g'), 



- (Zee' - feV - ee' 8 ) cos (g - g') - ^eV 2 cos Z(g- g'), 
- (3e 3 e' + ee' 3 ) cos (0 - 0') 



+ eV 2 cos 2(g-g'), 
4 ^ a = ( e 4 _ e '4) _ ( 2e 3 e ' _ 2ee '3) cos (^ _ g ') t 



= 3 = (e3e ' + ee/3) sin (sf ~ ^' } ~ e2e ' a sin 



== 
3 r r 6 r 

- (4e 8 e' 4- 4ee' 8 ) cos (g - ^r') + 2eV 2 cos 

These are ready for substitution in 9*8 (5) which is the develop- 
ment of the first term of d'Efn in 9'8 (1). 
To the second order, we have 



. ...(1) 



9, 10] EXPANSION OF R 265 

The calculation of the short period terms to the second order 
presents no difficulty. For the portion m (of IT') jRi, we obtain 

afpjr' = ecosg e' cos g' + e 2 cos 2g e' 2 cos 2#', 
a' f/r' = 2e sin g 2e' sin #' -f f e 2 sin 2$r 

- ee' sin (# + g') - e' 2 sin 2#', 

r' = - a'^/r' = e 2 cos 2g - 2ee' cos (gr -f g') + e' 2 cos 20', 

r' = e 2 sin 2g - 2ee' sin ( + ') + e' 2 sin 2'. 



Up to this point, no use has been made of the fact that a' /a is near unity, 
so that the development just given is quite general, at least as far as the 
second order. In order to deduce the results of 4'32, the expansions of the 
derivatives of R Q in terms of the coefficients A t and cosines of multiples of 
Wi w{ are to be substituted. In making the comparison, the difference 
in the definitions of the symbol a should be remembered. However, the 
expansion to the second order is not difficult whatever the method used ; 
it is in the calculation of the terms of higher orders that the expressions 
become long and complicated, so that for them the method should be 
suited to the problem. 

9'10. Calculation of the derivatives of R Q . 
These derivatives can all be reduced to the calculation of 
derivatives with a = 1. For, according to the definition in 9*5, 



and 

\ oa a==1 

since it has been pointed out (9*7) that the first power of cc is 
sufficient in the expansion of R. The last result still holds if we 
substitute for R Q any one of its derivatives. Since we can neglect 
the fourth order terras in the coefficient of #, this coefficient will 
not need derivatives of R Q beyond the third and the latter are 
already required in the calculation of the term independent of x. 
For the calculation of the derivatives put 



so that by 9'8 (3), (4), 

-o a 1 q 
- 



266 THE TROJAN GROUP OF ASTEROIDS [CH. ix 

Whence, when a = ] , Q 2 = 2 (1 - q), 

T> 1 1 3 dRo 1 1 1 



___ - 

g 8 4 Q ' da 3 ~ 2 g 8 8 Q 

1_1_161 
9a 4 ~Q 6 2Q 3 16Q 



9 1 31 



- 

9T 2 



_ ,3 

~ U ' 



9'11. TAe additional portions of a'Rj/ji depending on F. 

The second term of 9'8 (1) has the factor T and the long 
period terms which it produces are of the fourth order at least, 
since the argument v + v' 20 must be combined with the 
arguments 2g, g + g', or 2g' to produce multiples of Wi - Wi, and 
these terms have the respective factors e 2 , ee', e' 2 . 



10, 11] EXPANSION OF It 267 

In order to expand it, write v + v f 20 in the form 

v - v' 4- 2t/ - 26. 
It is then easy to prove that 

'- 20) 



where, as before, a'R\fr' is expressed as a function of r, #, #'. 
But, with the help of 9'8 (6), 

cos (2v' - 2(9) = cos (2wi' - 2(9) cos 2" - sin (2w/ - 2(9) sin 2" 
= cos (2 Wl ' - 2(9) . (1 - 4e' 2 + 4e' 2 cos 20') 

- sin (2wi' - 2(9) . (4e' sin #' + f e /2 sin 2^'), 
and the only portions of this which will give long period terms 
as far as the fourth order including those with factor F are 

cos(2^i'-2(9)- 2e / cos(2M; l / -0 / - 20) + f e' 2 cos(2w/ - 2#' - 2(9). 

...... (1) 

For sin (2v f 2^), these cosines are changed to sines. 

The first of the three terms of (1) gives long period terms by 
combination with the short period terms of a'Ri/r' having 
arguments 2#, g + g' ', 2g', the second with those having arguments 
g, g', and the last with RQ. The derivatives with respect to F, 
T present no difficulties, since they can be formed directly from 
the results in 9*10. 

For the short period part which is taken to the second order 
only, we put v = Wi,v' w\ , r = r' = a', A x = A = Q. 

The third term of a'Rjp, in 9*8 (1), having the fourth order 
factor F a , gives the single term 3F 2 /16Q 5 . 

The portions due to the factor 1 JF have been retained 
throughout on account of the large numerical multipliers which 
accompany them. As in Chap. IV, their presence causes but little 
additional calculation, and adds considerably to the numerical 
convergence for large values of F. But it is easy if desirable to 



268 THE TROJAN GROUP OF ASTEROIDS [OH. ix 

expand the derivatives of R given in 9*10 in powers of F. For, 
when a = 1, F = 0, 

3Q __ 1 cos r d 2 Q __ I cos 2 r 
~ ~ 



,, , s\ ~ r COS T , -no COS 2 T ,~. 

so that (^--Hr*-..., ......... (2) 

where Q 2 = 2 - 2 cos r. 

9*12. Transformation to the canonical elements. 
The element # 2 is given (cf. 9*5 (4) and 513 (1)) by 

* 2 =(l+tf)f(l-* 2 )*-l}, ............... (1) 

from which e can be expressed in terms of ( x$ and x. But the 
canonical elements # 2 , w 2 will be replaced later by |? 2 , #2> where 

j9 2 = 61 sin w 2 (ci/c/)*, # 2 = ^i cos w; 2 (ci/Ci')i, ...... (2) 



These correspond to the elements in 5'14. 

As w 2 = -cr, and as we neglect powers of x beyond the first, 
these give 



esin tzr = a (l i^ 2 #). ^ costxr = 



which, with the relation g g' = r + tv' tsr, permit us to express 
a'Rijr' in terms of these canonical elements with but little 
additional calculation. 

For the terms containing F, 0, the substitutions 

ps = (2F)i (1 - \# -f J0) sin 0, g 3 = (2F)* (1 - \# -h |a;) cos 0, 

...... (3) 

can be further abbreviated by putting unity for 1 
except in the factor 1 ^F, where we put 



with 6 a = ^2 2 + 9 r 2 2 - 

These changes are not necessary in the solution of the equation 
for T, provided, in forming xdR/da we remember that a is present 



11-13] EQUATION FOR THE LIBRATION 269 

in R through e, F when we transform to the canonical elements. 
The coefficient of x in the expansion given above can be obtained 
with sufficient accuracy from 

/ 3 i 9 ^ 9 \ a'R 
-2a 5 ---!e 5 --r= , 
V da * de oTJ JJL 

and in this expression, R can be limited to the terms of the 
second order. 

913. The equation for T. 

It has been shown in 97 that, by a suitable change of variables, 
the short period terms can be eliminated from the disturbing 
function, and that the equations for the new variables, which 
are distinguished by the suffix zero, have the same form as those 
for the old variables if we simply omit from the latter the short 
period terms. The suffix is unnecessary during the solution of 
the equations for the new variables and will be omitted. 

The Hamiltoniau function, namely, 



Q /->< ^ Tt . C\ ^ V\> C 

An& *Iff& i VI \ 'in* _ 

ft-iX/ t\Aj ~\ JLVg f ^O/ "7T , 

/& IJL oa 

will be denoted by 

f ,-E 2 2#3 -f (7 + 2# F, to order m*. 

According to 97, with its reference to 9'5, <7, Fare independent 
of x, t. 

The equations for #, r are then 

dx dU , 3F , s /1X 

r^- = - 4-2^ -^ , to order wi* (1) 

71 Cl6 OT OT 

~ r =. 3# + 6^ 2F, to order m (2) 

n eft v 7 

Differentiate the last equation with respect to t. Since the 
derivatives of oc, x 2 , X B> w%, w& have the factor m and that of r 
has the factor m^, we obtain 

;7T,- 2 ^ tt> to order mi... (3) 



270 THE TEOJAN GROUP OF ASTEROIDS [OH. ix 

On substituting for dx/dt, dr/dt from (1), (2), we see that the 
portions depending on V disappear and that the equation becomes 



0/ , . . 
ss - 8 < 1 - 4a? ) , to order m* .......... (4) 



The disappearance of V from the equation for r, the solution 
of which is the principal part of the problem, is fortunate because 
it enables us to find T with high accuracy without transforming 
the disturbing function to canonical variables. It is true that 
V is needed in the determination of x, but this is a comparatively 
simple problem when T has been obtained in terms of the time. 

Since x = ^ -, -y- , to order m^ t 

o n at 

the transference of the factor 1 4# to the opposite side of (4), 
where it becomes a factor 1 -f- 4#, gives the somewhat more con- 
venient form of (4): 

1 d 2 r 2 d (I dr\* dU ^ , 3 
"2 Ti2 ~ o -TIT 1 -/ T^ I = - 3 ^~ , to order m*. 
?i 2 dt 2 3 n r/^ \?i di/ dr 

Finally, if we put 

^ = ^ -f- ^ (r -f- const.), 

it is easily seen that the equation reduces to 

-S-2 + 3 ?T =. to order m*. (5) 

/Yl'^x-/-^ A x-1 ' ' ^ ' 



which is the fundamental equation for the determination of the 
libration. 

914. First approximation to r. 

We shall obtain this approximation on the assumption that 
#2> #3> W2> Wa are constants: it will appear later that their 
variable parts are divisible by m%. With this assumption, U 
becomes a function of a single variable r and the equation 9*13 (5) 
admits the integral 

const. = C- 6U, 



13, 14] CALCULATION OF THE LIBRATION 271 

so that n' fc -f const. - J (0 - 6 U)~*dr 

gives the solution. A reversion will give r in terms of fo. 

From a theoretical point of view, this is sufficient to determine 
r, but the process is inconvenient for calculation, and it is better 
to find r as follows. 

Since we know that r oscillates about one of the values 60, 
let us put 

T = 60 + ST, 

and expand the second term of 9*13 (5) in powers of ST. If J7- 
be the ith derivative of U with respect to r with r = 60 
inserted, the equation becomes 



_ 

...... (i) 

When e e' F = 0, the formulae of 9*3 give 



In this case, when powers of Sr beyond the first are neglected, 
(1) reduces to 



the solution of which is 

r = bcos(vn'ti + i>o) = bcos <f>, v* = 2?-m, ...... (2) 

where 6, VQ are arbitrary constants. 

The result agrees with that obtained in 9*10. In the further 
approximations, we are assuming that 6 is a parameter which is 
small enough for expansions in powers of b to be possible. The 
largest known value for b is that in the case of Hector for which 
it is near *3. As z> = *079, we have Ji/& = *008 in this case. 
The statement in 9*5 that x is less than '01 in all known (see 
footnote to 9*5) cases is thus justified. 

In general, when constant values of e, e', F are used, we have, 
for the first approximation, 

= & cos </>, i> 2 =3tf 2 ....... (2') 



272 THE TROJAN GROUP OF ASTEROIDS [OH. ix 

For the next approximation, the value (2') is substituted for 
Sr in the term f ^(Sr) 2 of (1) and the term 3Ci is included. 
The equation becomes 

1 _7X 

T = -3/i-f t/3& 2 -7 3 & 2 cos2<. ...(3) 

The addition to 8r is the particular integral corresponding to 
the right-hand member. It is 

TT 1 7T t TT O JL 

7y A TT A TT- A, _ "1 ^ ' 

For the next approximation, we substitute the sum of (2), (4) 
in the term f f/3(8r) 2 , and (2) in the term Jt/VSr) 3 . The ad- 
ditional terms on the right of (3) are 

- 3C/3 b cos $ ( - -J - ^ yj? b 2 + Y 2 7j 



or - 



The particular integral corresponding to the term with 
argument 30 is 

A * fr 3 ^ 2 

3(/ 2 '9-l 8Vi/2 

In the term with argument </> we can put b cos </> = Sr, since 
it has 6 3 as a factor. On combining it with the first approxima- 
tion we have 



This shows that instead of the value 3C/2 for v 2 , previously used, 
we must put 



in order that ST may still remain periodic. 



14, 15] FURTHER APPROXIMATION 273 

The process is continued as far as may be necessary. Each 
alternate approximation requires an addition to the value of v. 

The process of approximation followed above is that which is usual in 
the case of equations of the type (1). It is to be remembered that the 
value 3C/2 is merely an approximation to v 2 and that the latter must 
contain other terms if Sr is to remain periodic. 

The equation may be solved also by putting 



in (1) and equating to zero the coefficients of cost<. A series of equations 
of condition are obtained which have to be solved by continued approxi- 
mation on the basis that 6, b Q and 6 2 , Z> 3 , ... are quantities of the first, 
second, third, ... orders. The coefficient of cos (f> determines v, and b is an 
arbitrary constant. 

The arbitrary constants 6, V Q replace the usual arbitrary constants %, f 
which become fixed as soon as the triangular solution is adopted. 

9*15. The equations for the remaining variables. 

The development of the disturbing function in 9 f 8 shows that 
the long period terms containing the argument 6 are of the 
fourth order. If we neglect terms of this order, the canonical 
equations give dx^fdt 0, or #3 = const. Further, since 

# 3 = T (1 - e 2 )* d/d' = T (1 - e 2 )* (1 + x\ 

we have, to the same degree of accuracy, F = ,r 3 = const. 

The equation for 6 is found with the aid of 9'12 (2) and is 
integrated after this expression has been developed as a Fourier 
series with argument </> with the solution given in 9*14. 

To the same degree of accuracy, 9*12 (1) shows that # 2 = i^ 2 . 

The variables p 2 > <?a, defined in 912(2), reduce to 

p 2 = esin'cr, g^ ecos tzr ................ (1) 

The limitation enables us to neglect the part of R which 
has a? as a factor. Hence, with the Hamiltonian function in 
9*13, the equations for p t) g 2 become 



- __ 

n'dt~ 9c ' ri dt~ 8s ' 
with the briefer notation s for p% and c for q%. 



274 THE TROJAN GROUP OF ASTEROIDS [CH. ix 

The development of a'Rj^ which contains c, s to the second 
order, is given by 9*9 (1). With g g' = T ta- -f TO-', the substi- 
tution (1), and the notation 



this development may be written 
T {- se' cos (r + r') + ce' sin (T + r')} 

+ JP (s 2 + c 2 -f e' 2 - 2e' s sin (T + -sr') - 2e'c cos (T+ r')}. 
The equations for s, c are therefore 

1 d* 

-, ^ = Te' sin (T + w') - Pe' cos (T + w') + PC, 

n af 

1 r/r- 

-, ^ = 2V cos (T + w') + Pe' sin (T + w') - Ps. 

?l C6t 

The only variable present in T 7 , P is T and we suppose that 
T has been expressed as a function of t by means of the solution 
obtained in 914. Further, as T = 60 + Sr, it is supposed that 
any of the functions of r present can be expanded into Fourier 
series with argument </>. 

9*16. Solution of the equations for s, c. 

As is usual with linear equations, we first find the comple- 
mentary function, which is the solution of 

0, ............ (1) 

^ ' 



-- , ~ 

n dt n dt 

where P = P + PI cos <j> -f P 2 cos 2< + . . 

with </> = vn't -\-VQ\P has the factor m. 
It is at once seen that the solution is 

s = sin (-CT! 4- o)> c = ^ cos (*TI -f 

where d^ildt = P, so that 

p 

'' 



(2) 



where eo> WD are arbitrary constants. We thus have 

(P- \ 

P^n't -f TQ + S ~^ cos *0 ) > 
lv ' 

, p. \ 

C = Q COS ( PfiU't + Wo + S ~ COS !</> J . 



15, 16] EFFECT OF JOVIAN ECCENTRICITY 275 

In the usual language of celestial mechanics P n' is the 
mean motion of the perihelion. It appears then that this part 
of the motion gives a constant ' eccentricity ' and a variable motion 
to the perihelion. Since the coefficients Pi/iv contain the 
factor bm^, the solution to this order can be expressed in the form 

s = Q sin (P^nt -f- WQ) 
p 

+ 2 ~ {sin (P Q n't + ^ + i<f>) - sin (P Q n't + CT O - i<f>)}, 
tv 

with a similar form for c. The mean motion of the perihelion is 
divisible by m, the periodic part being divisible by 6i?A 

The particular integral corresponding to the terms factored 
by e' in 9*15(3) is required. These portions are functions of r 
only and can therefore be expressed by Fourier series with 
argument <. Suppose the solution to be of the form 

s = e r 2 (Si cos {</> + s^ sin if) = e' (s c -f s,), 



,.. 
c = e' S (Ci cos icf> + c^ sin i</>) = e' (c c + c,), 

so that the suffixes c, s denote expansions in cosines and sines 
of multiples of <j> respectively. These are to be substituted in 
9'15(3). The coefficients of cosi</>, sini^> equated to zero will 
give the coefficients s it /, Ci, c/. 

In a first approximation we retain only the terms of lowest 
order with respect to m in r. According to 9'13, fa is then equal 
to t, and r = 60 -f 56* cos i<. The terms factored by e' in 
9*15 (3) are then expressible by cosines of multiples of <. The 
only sines which will be present will be those arising from 
ds c /dt, dcc/dt f Sj, c 8 . We must therefore have 

'), -(5) 



Since dcj>/dt has the factor m*, the required conditions can be 
satisfied only if 

s c , c c = const. + cosines of multiples of <, factor m, 
s, c, = sines of multiples of </>, factor m*. 



276 THE TROJAN GROUP OF ASTEROIDS [CH. ix 

If, then, Po be the constant term in the expansion of P as a 
Fourier series with argument <>, the equations (5) and (6), with 
the notation (4), give 

P Co = const, term in the expansion of 

P cos (T + r') - Tsiu (r + w'), 
PO SQ = const, term in the expansion of 

- P sin (r + vr')-T cos (r + vr') 9 
to the order m. 

Since the coefficients c iy c/, Si, s/ have at least the factor mi, 
the terms PC,, Ps, in (5), (6) can be neglected in finding a 
first approximation to these coefficients. Hence s*, c* are given 

by f 

Sj = {_ p cos (r + or') + Tsin (r 4- tar')} dt, 



and they will have the factor m*. When these have been found, 
the first of (5), (6) give s c , c c , the variable parts of which have 
the factor m. 

The same plan is followed when terms factored by ra$ are 
retained in 9*15(3). The work is simplified by remembering 
that in these expansions the coefficients of cosines of multiples 
of have even powers of m% as factors, while those of sines 
have odd powers as factors. The latter also have & as a factor. 
It follows that the errors of c , S Q as determined in the approxi- 
mation just given, have the factor m, those of c it s t the factor m 2 , 
and those of c/, s t f the factor m*. 

When b = we have T = + 60, T = 0, and P is reduced to a 
constant. The particular integral is then 

s = 6 / sin(60 + w'), c = e'cos(60+'cr / ), ...(7) 

and the complete solution is obtained by adding these to the 
complementary function which is 

(8) 



16, 17] HIGHER APPROXIMATIONS 277 

Since the mean longitudes of the asteroid and Jupiter differ 
only by the constant 60, the terms (7) can be interpreted by 
the statement that Jupiter impresses its elliptic terms on the 
motion of the asteroid. This result might have been anticipated 
from the triangular solution given in 9 '2, for the mass of Jupiter 
is so large compared with that of the asteroid that in this 
solution its elliptic motion will be dominant, 

In the ordinary planetary theory, the terms (8) would be 
expanded in powers of t Thus, neglecting terms factored by m 2 , 
we have 



s = eo sin vr Q -f P^te^ cos 1^0, c = # cos W Q 

The constants (7) might therefore have been supposed to be in- 
cluded in the arbitraries sin TOO, e Q cos tsr , which are to be 
determined from observation. Thus although these constants 
are affected by an error of order m in the first approximation, 
most of the error will be absorbed in the determination of e , 
T O from observation. Owing to the need for further approxima- 
tion, however, they must be kept separated. 

The fact thab s, c differ from constants by terms having the 
factor w* at least, justifies the assumption made in 9*14 that e, CT 
may be treated as constants in finding a first approximation to r. 

9'17. Higher approximations and final results. 

The equations for T, x have been taken to the order m% and 
solved with s, c, p 3 , qz constant. Since the variable parts of the 
latter have the factor 6m*, it follows that the errors of the 
equations for r, x have the factor 6m* and also a factor of the . 
order of the squares of the eccentricities and inclination. The 
error of r has therefore this latter factor and also the factor 6ra*. 

For the next approximation, the variable values of s, c, p Q , q$ 
are inserted in U. Since the resulting addition to T will be 
small, it will in general be sufficient to find it from the following 
equation, deduced from 9*13 (5), 

/I d* .A* .,. , , dU 

~/2 j^s + v *\OT = additional terms m -% - ; 
\n * air ) or 

B&SPT iS 



278 THE TROJAN GROUP OF ASTEROIDS [OH. ix 

the additional part of x can then be obtained. If further 
approximations to the values of the remaining variables are 
needed, they can be obtained in a similar manner. 

The final results give the values of T O , #o> #20, #30 > ^20* ^30> r 
of the variables which replace them, where the suffix zero which 
was dropped according to the statement in the first paragraph 
of 9'13 is now replaced. The values of the original variables are 
then found by substituting these results in 9*7 (2). Now the 
portions dependent on the derivatives of S in these latter 
equations all have the factor m, so that the effect of substituting 
variable for constant values of the variables with suffix zero will 
be very small, with one exception, that of r now called T O , and 
for the latter the portion independent of m will serve. 

9'18. Numerical developments. 

A literal theory in which the expansions are made in powers of b, e , r , 
e' y m* can be formed which will give a close approximation without an 
excessive amount of calculation. Even with so large a value of b as *3, the 
series for r converges rapidly owing to the numerical divisors which the 
integration produces. In this respect the theory of the Trojan group differs 
from the ordinary planetary theory where expansions in powers of a con- 
verge so slowly that they are useless for numerical calculation. 

The work can, however, he greatly simplified when the numerical values 
of the parameters are known. The chief part of r is a Fourier series with 
argument <f> and most of the further calculations consist in the calculation 
of various functions of cos r, sin r. The functions are rapidly calculated if 
harmonic analysis be used ; five, or at most seven, special values of < will 
be sufficient. Analyses of the special values of the functions are needed 
only when they have to be integrated or differentiated. 

Harmonic analysis can also be used conveniently to complete the solution 
of 9'13 (5) when an approximate value of r has been obtained by the 
method of 9*14. Suppose that such an approximate value is 

r = 60 + 2^008 ifa d(t>ldt v, (1) 

and lot the required correction be 

Sr= SbbiCoaiQ-tfoZbii sin fy, 66 1 = 0, (2) 

where we neglect squares of the correction to v. As the arbitrary constant 
can be left unchanged, we can put 56 1 = 66 = 0. With the value (1) of r, the 
function dU/dr is computed by harmonic analysis and compared with 
dt*. Let the sum of these be denoted by 
c 4- *i cos <f> + eg cos 2 < 4- . . . . 



17-19] CONSTANTS OF THE ORBIT 279 

Then a further approximation will be obtained by solving the equation 



in which 3 2 U/dr 2 is computed like 3 U/dr. 

On substitution of the expression (2) for 5r, it will be found that the 
coefficient of t disappears and that the values of 8b t , dv can bo obtained by 
equating to zero the coefficients of cos 1$. The process can be repeated if 
necessary. Since the principal part of 33 2 (7/d r 2 is i> 2 , the principal part of 
Sr is found at once. 

9'19. Determination of the constants from observation. 

The nature of the orbit is usually set forth by giving the values of the 
osculating elements at some given date : these are found from the observa- 
tions by methods which are outside the scope of this volume. A procedure 
for finding the values of the constants used in the theory from these 
elements is contained in the following plan. 

An approximation to the short period terms can be obtained by substitut- 
ing for the elements with suffix zero in the terms arising from S, their 
osculating values. The same procedure is followed with the terms due to 
the action of Saturn determined below. We then obtain an approximation 
to the elements with suffix zero by the use of tho equations which connect 
them with the actual osculating elements. 

By comparison of the elements with suffix zero with the literal series for 
r, x in powers of 6, values of 6, v are obtained. With these the short period 
terms, particularly those dependent on the angle $, can be calculated again 
and the same procedure repeated. At this stage the values of the constants 
attached to the remaining elements can be found with high accuracy. If 
necessary, the whole procedure may be repeated, but it will rarely be 
necessary to do so except perhaps for the constants 6, qf> , which are 
sensitive to small changes in the elements. 

The process does not differ essentially from that which would be followed 
in the ordinary planetary theory if the methods of Chap, vi be used to 
determine the perturbations. In the latter, however, the elements with 
suffix zero contain the long period and secular terms only, and if desired 
we can treat these like the short period terms, using the observed osculating 
elements to find a first approximation to n , c , etc. Thus while the methods 
of Chap, vi have certain disadvantages which have been pointed out in 
6*25, the custom of defining an orbit by giving the values of its osculating 
elements at a given date, makes the determination of the constants of the 
orbit from these values a simple problem. 

In the case of the Trojan group, the following modification gives the 
constants more rapidly. The values of the osculating elements are found 
at several dates by carrying them forward or backward by the method of 

18-2 



280 THE TROJAN GROUP OF ASTEROIDS [OH. ix 

special perturbations. Since all the short period terms have periods ap- 
proximating to that of revolution round the sun or sub-multiples of this 
period, a mean value of an element with suffix zero can be found by analysing 
its values at the various dates into a Fourier series with argument 2rr/w-' 
and choosing the constant term as the first approximation to its value at 
a mean date. 

PERTURBATIONS BY SATURN 

9*20. The calculation of the perturbations of an asteroid of 
the Trojan group by Saturn is difficult because the procedure of 
the ordinary planetary theory cannot be followed. This procedure 
consists in finding the perturbations due to each planet separately, 
then those due to their combined actions, and adding the results. 
Here, in finding the principal perturbations produced by Saturn, 
we cannot; neglect the action of Jupiter, even in a first approxi- 
mation. Thus the problem is one of four bodies rather than of 
three, and in this respect it is similar to that of the action of 
the planets on the moon. 

The disturbing function for the direct attraction of Saturn is 
that given by 1*10 (1). An indirect effect is also produced by 
the action of Saturn in causing Jupiter to deviate from elliptic 
motion, so that, in the disturbing function due to the direct 
action of Jupiter, it is necessary to add to the elliptic elements 
of that planet the perturbations caused by Saturn. An indirect 
effect of the action of Jupiter is also present in the perturbations 
this planet produces on the motion of Saturn. 

It is assumed that the mutual perturbations of Jupiter and 
Saturn on one another are completely known. The largest term 
in the action of Saturn on Jupiter has a coefficient in the 
longitude of Jupiter of nearly 1200" and a period of some 870 
years, this long period being due to the fact that the period of 
revolution of Saturn is nearly 2 times that of Jupiter. Since 
the period of revolution of the asteroid is the same as that of 
Jupiter, we might expect to find a term of similar magnitude in 
the motion of the asteroid. It will be shown, however, that the 
action of Jupiter fundamentally alters the direct effect of the 
action of Saturn, and that the indirect effect produced by Saturn 
is the largest part of the action of that planet. 



19-21] PERTURBATIONS BY SATURN 281 

It will be assumed that these effects are, in general, small 
compared with those which we have been considering in the 
first part of this chapter, and that, in developing the disturbing 
functions due to Saturn, we can put for the coordinates of the 
asteroid their values in terms of the time. These additional 
portions can then be separated into long and short period terms. 
The latter may be eliminated as before by a change of variables 
which will give additional portions to the function 8. We have 
then to consider the effect of small additive terms in the function 
U on the variables with suffix zero; as before this suffix will be 
dropped until the final results for the long period terms have 
been obtained. 

9*21. The equation for Sr and its solution. 
The equation for r given in 9 '18, namely, 

' 



is still true when we add to U the portions due to the actions of 
other planets, provided the conditions laid down remain satisfied. 
These conditions demanded that the principal part of x should 
be given by oo^ ^drfn'dti, and that the terms present in x 
should be large compared with the corresponding terms in U. 

With the conditions laid down in the last paragraph of 9*20, 
the inclusion of the action of Saturn will require the solution of 
equations of the form 



/n . 
(2) 



where t/"has the meaning previously given and A,p,p Q are known 
constants. To simplify the exposition, we shall put ti t', this 
amounts to the neglect of terms of order higher than those 
retained. 

Let T = TO be the solution of (1) and r = T O -f Sr that of (2). If 
squares and higher powers of Sr be neglected, we have 



where in U and its derivatives the value r = T O is inserted. 



282 THE TROJAN GROUP OF ASTEROIDS [CH. ix 

The solution of (1) gave T O as a function of t and of two 
arbitrary constants 6, v$\ if this solution be substituted in (1) 
the constants b y VQ disappear identically. We can therefore 
differentiate (1) with respect to b and VQ and obtain 



. *.. _ 

n'* dt* \9fc / 9T 2 ' db ~ ' n'* dt* \fa) + drf ' fa ~ 

...... (4) 

Hence Sr=|? g T = ^<> .................. (5 ) 

ob di/o 

are particular solutions of (3) when A = 0. 

It follows from a well-known theorem that a particular 
solution of (3), corresponding to A =f 0, is 



(6) 

vy i/J'O J Ul/ 

where 



Equation (6) may be tested by substitution in (3), and equation 
(7) by eliminating d 2 U/dT<? from (4). 

Since T O has been obtained as a Fourier series with argument 
<j> = vn't+ z^o, the derivatives of T O with respect to i/ , b will still 
be Fourier series provided v be independent of b. Actually, v is 
a function of b and dr/db contributes a non-periodic portion 
n't(dv/db)(dT/dvo)', it is easily seen, however, that this non- 
periodic part disappears from (6). 

The principal term in T O T 60 is 6cos(im' + j>o)> where 
z> 2 = 27m/4. With this value of T O we obtain 



a result which might be deduced directly from (3) since in this 
case 33 2 7/3r 2 = v 2 . With the complete value of r, the divisor 
v 2 ~p 2 will be replaced by divisors of the form i*v*-p 2 . 



21, 22] INDIRECT EFFECT OF SATURN 283 

9*22. Indirect perturbations. 

These arise through the substitution for the elements of 
Jupiter their disturbed instead of their elliptic values. We shall 
suppose that the perturbations of the plane of Jupiter's orbit 
can be neglected so that we can still use it as a fixed plane of 
reference. The perturbations to be considered are therefore those 
of a', e', Wi = w', wz = w'. 

These perturbations, chiefly due to Saturn, have the mass of 
Saturn as a factor and are substituted in an expression which 
has the mass of Jupiter as a factor. The resulting short period 
terms will be very small and, in any case, they are supposed to 
have been eliminated by the change of variables. It is found 
that the only terms likely to produce sensible effects are those 
of long period in the motion of Jupiter, producing terms of long 
period in the motion of the asteroid. 

For their calculation, we return to the point in the original 
canonical equations where the transformation r = w w' was 
made (9*5). This transformation still holds if w' is any function 
of t independent of the elements of the asteroid provided that 
the expression Cidw'/dt be added to the Hamiltonian function. 
As before, w' is then no longer present explicitly in the dis- 
turbing function. 

If we denote by 8a f , Se' y SOT' the perturbations of a', e', -or', the 
equations for Ci, T become 

dd dR d*R , . a 2 /? , , d*R . , /1X 

-rr =^- + n ^T/S +;Ta-/Be +3-5- -/ or , .................. (1) 

at dr drda drde 



dr ffjf , d , A dR 

-T7= r 3 -tto ~T* W } "5 -- 
dt \Ci 3 dt J dci 



PR , , PR , , d*R 



a a' > ~^i"a ' -" -> , 
dcida ocide BciPtsr 

...... (2) 

where n$ is the constant term in dw'/dt. 

We have seen in Chap, vi that the variations Sa', 80', STST' 
contain the first powers only of the small divisor which is present 
in the case of a long period term, and these variations, when 
multiplied by the second derivatives of R, will give very small 
terms which can be neglected, at least in a first approximation. 



284 THE TROJAN GROUP OP ASTEROIDS [OH. ix 

If the procedure followed in 9*6 and 911 be then adopted, it is 
easily seen to lead to the equation 



an equation which replaces 9*13 (5). To find the principal term 
we put ti=t. 

Suppose Sw' = B sin (n'pt + p Q ), 

and that we substitute this in (3). According to 9*21 (8) the 
principal part of Sr is given by 

Bp 2 
8r = -%-_ 2 sin (n'pt 4- p Q ). 

But Sw = Sr -f w'. Hence 

J ft 

(4) 



The effect of a perturbation in w' therefore depends on the 
relative magnitudes of v, p. 

If v is large compared with p, (4) gives approximately 

Sw = B sin (n'pt + p Q ) = $w'. 

The result Sw = $w' applies also to a secular term since such a 
term can be expressed as one of very long period. Hence the 
general proposition: 

If the period of a perturbation of Jupiter by Saturn or by any 
other planet is long compared with the period of libration of the 
asteroid (about 150 years), the perturbation of the longitude of 
Jupiter is directly impressed on the longitude of the asteroid. 

The principal perturbation of Jupiter by Saturn has a period 
of some 870 years, so that the indirect perturbation of the 
asteroid differs from the direct perturbation in the motion of 
Jupiter by less than three per cent., although this indirect effect 
is one of the second order with respect to the masses while the 
direct effect on Jupiter is one of the first order. 

Terms in which p is large compared with v have been treated 
as short period effects and therefore do not enter into the dis- 
cussion. 



22, 23] INDIRECT EFFECT OF SATURN 285 

Terms in which p 2 is nearly equal to v* would give rise to much 
larger perturbations in the motion of the asteroid than those 
present in the motion of Jupiter. There are no such terms having 
sensible coefficients in the perturbations produced by Saturn, 
Neptune, which has a period of 164 years, comes nearest to pro- 
ducing such a term. 

9*23. To obtain the perturbations of #, the equation 9*13 (1) 
is used. When pjv is small, Sr is small compared with Sw' and 
therefore (d/dt) ST with (d/dt) Sw'. The latter is, however, large 
compared with dR/dc^ which has the factor ra and has no small 
divisor. Hence, since /^ 2 /Ci 3 = (1 + #)~ 3 n'> 



or, since S% = JSa/a', this gives (d/dt) Sw' = f 8n'. Hence 

8a = S&'. 

The long period terms in the major axis of Jupiter are therefore 
impressed on that of the asteroid. 

For the perturbations of e, r, the equations 9'15 (3), which 
are still true if ef ', BT' are variable, are used. Let e', or' receive 
long period variations 80', SOT'. Then, as before, it may be shown 
that PC, Ps may be neglected. When the libration vanishes, the 
equations for 8s, 8c reduce to 

^8s-P S{e'cos(60 4V)} f ^8c = P S {e'sin ( 60 + w% 

If 27r/n'p be the period of the variations of &e', S-sr', the corre- 
sponding coefficients in Ss, Sc are diminished in the ratio P /p 
which is usually small, so that &, 8c will be negligible. For the 
terms depending on <f> when b =j= 0, the divisor is approximately 
iv if p is small compared with v and the coefficients of such terms 
in 8s, 8c will be still smaller than those just treated. Thus the 
long period variations of e', tzr' are not impressed on the asteroid 
but produce effects which are much smaller than those in the 
motion of Jupiter. 



286 THE TROJAN GROUP OF ASTEROIDS [CH. ix 

9*24. The direct action of Saturn. 

The disturbing function for this action is given by 1'10(1), 
and it can be expanded in terms of the elements of the asteroid 
and of Saturn by one of the methods used in the ordinary 
planetary theory. The expansion, if made in a literal form, is 
available whether the elements be constant or variable. The 
disturbing function, denoted by R' y adds a term a'-R'/A 6 to the 
Hamiltonian function in 9*5 (5). 

The variable x which measures the deviation of Ci/Cj' from unity 
is still small, so that 9*5 (6) still holds when we add R f to R. 

Let the elimination of the short period terms, as made in 9*7, 
include those arising from Saturn. For simplicity, we shall retain 
the lowest power of x only, so that the additional terms in S are 
similar to those found in Chap. vi. The Hamiltonian function 
used in 9*13 therefore becomes 

f a? + -(R e + R,') = f x* + U+ U', 

P 

with the notation adopted there. The equations 9*13 (1), (2) then 
become 

l^'-^J 7 ?E1 I L^ T -__<} _' 

n' dt ~ dr + dr ' n' dt 

and equation 9*13 (3) reduces to 

1 ^-_ql d_d_(dU 
n*d&~~ n'dt dt\dx 

On substituting for dx/dt, this gives 

JL rf2r - ( d JL + dU '\ _ 1 fiV'\ 

n' 2 dt 2 ~ \ dr ~dr) dt \dx ) * 

For a term of long period, the last term of this equation is small 
compared with dU /dr, and may be neglected in a first approxi- 
mation. The equation then reduces to 

J ^j.Q^-.q^ 

'" 8 rfi 8 + *ST~ dr 

The right-hand member of this equation has the mass of 
Saturn as a factor, and we can substitute for the elements of the 



24] DIRECT EFFECT OF SATURN 287 

asteroid the values obtained from the action of Jupiter alone. 
In accordance with the previous work, it will be sufficiently 
accurate to use the series for r and to take all the other elements 
constant. 

Let any term in the right-hand member be denoted by 

A sin(pn't-\-po). 

The method of 9*21 is now available for the solution of the 
equation. The principal part of the addition to r will be given by 

A 



- 2 - ^ 
v p 

The principal long period term due to the action of Saturn will 
have nearly the same period as it produces in the motion of 
Jupiter, namely, about 870 years, and therefore p is small com- 
pared with v. Hence the small divisor due to a long period term 
is much larger than in the ordinary planetary theory and the 
resulting effect much smaller. Thus 

Jupiter not only impresses on the asteroid its own inequalities 
of period long in comparison with that of the libration, but it also 
prevents the asteroid from having any very large terms of this 
nature arising from the action of another planet. 

The substitution of the series for r will also produce terms 
with arguments (p iv)n't + p iv Q . When i = l we shall 
have divisors v 2 (p p) 2 or 2pv approximately. Such terms 
will have the factor 6, and if b is large they may be sensible. 

The theory of the perturbations of Jupiter by Saturn given by Leverrier* 
can be utilised for calculating the perturbations of the asteroid by Saturn, 
since the numerical value of the ratio of the mean distances is the same, 
and since Leverrier gives the contribution of each separate power of the 
eccentricities and inclination, so that the change to those of the asteroid 
can be easily computed. But the convergence along powers of the inclina- 
tion is so slow when the inclination is large, as in the cases of certain 
members of the group, that the value of the coefficient of the principal 
long period term obtained in this way is doubtful. Another difficulty arises 
from the fact that the mean motion of the perihelion of the asteroid is 
comparable with that of the argument of this term so that it cannot be 

* Paris Obt. Mtm. vol. x. 



288 THE TKOJAN GROUP OF ASTEROIDS [OH. nc 

neglected, even in a first approximation : see "Theory of the Trojan Group " 
referred to in 9 '25. 

9 '25. The literature connected with the triangular solutions of the pro- 
blem of three bodies is extensive : much of it is concerned with the possible 
orbits which may be described under different conditions but which have 
no present applications in the solar system. Amongst the earlier develop- 
ments arriving at a more general theory for the asteroids of the Trojan 
group may be mentioned those of L. J. Linders (Stockholm Vet. Ak. Arkiv y 
Bd. 4, No. 20) and W. M. Smart (Mem. Roy. Astron. Soc. vol. 62, pp. 79- 
112, 1918) ; the latter used the method adopted by Delaunay for the 
development of the lunar theory. In a paper by E. W. Brown (Mon. Not. 
Roy. Astron. Soc. vol. 71 (1911), pp. 438-454), the particular periodic 
solution which constitutes the principal part of the libration is shown to be 
a linkage between the orbits of planets outside and inside that of Jupiter 
and those of satellites of Jupiter, the passages between them going through 
the collinear solutions. The linkage bears some resemblance to that which 
joins the two sets of solutions of the equation for the motion of a pendulum. 

A literal development sufficiently complete to give the position of an 
asteroid of the group within a few seconds of arc has been made by 
E. W. Brown ("Theory of the Trojan Group of Asteroids," Trans, of Yale 
Obs. vol. 3, pp. 1-47, 87-133). The method of Chap, vn is used for the 
development of the action of Jupiter : the various problems which arise in 
finding this action are closely analogous to those set forth in this chapter. 
The problems and theorems connected with the action of Saturn are dealt 
with in detail. The theory was applied to the asteroid Achilles. This 
numerical application has been revised by D. Brouwer (Trans, of Yale 
Obs. vol. 6, pt. vn) who has added tables for finding its position at any time. 
W. J. Echert has applied the same theory to Hector (Ib. vol. 6, pt. vi), the 
libration of which runs up to over 20. 



A. APPENDIX ON NUMERICAL HARMONIC 
ANALYSIS 

A'l. Let F c (x) be a periodic function of x, period 2?r, which 
is expansible in the form 

F c (x) = CQ + Cicos x + C2cos 2# + . . . + c n cos nx + ....... (1) 



The problem under consideration is the rapid numerical cal- 
culation of the coefficients c^. It is supposed that F c (x) con- 
tains numerical constants only, and that after some term c n cos nx 
the remainder of the series may be neglected. 

Under the latter condition (1) may be regarded as an 
identity satisfied for all values of x. If then we choose n -f 1 
numerical values of x and calculate the corresponding values 
of F c (x) and of the cosines of the multiples of #, we obtain 
Ti + 1 relations which may be regarded as n + 1 linear equations, 
giving the n + 1 unknowns c , Ci, ... , c n . 

The effectiveness of the method depends on the ease with 
which the special values of F e (as) can be computed and on 
that with which the linear equations may be solved. It is found 
that the questions principally to be considered are the choice 
of the special values of x and the best arrangement of the 
work for finding the c-. 

Similar remarks apply to the development of an odd func- 
tion of x in the form 

F 8 (x)-Sismx + s%sm2x+ ... -f s n smnx+ ..., ...... (2) 

except that n values of x only are needed since there is no con- 
stant term. 

If F(x) contains both even and odd functions of x and if we 
stop at the nth harmonic, 2n + 1 special values of x are needed 

In cases where the calculation of special values of F e (x) and 
of F 9 (x) is needed and where much of the work is the same in 
each case, the same special values of x should be used. 



290 APPENDIX [APP. 

A'2. Choices of the special values of x. If F(x) contains both 
even and odd functions of x, they can be separated by choos- 
ing the values of x in pairs , 2?r . Since 

cos ia = cos i (2?r a), sin ia = sin i (2zr a), 
we have 

F(a) + F(%7r -a) = cosine series, F(a)F(27ra) = sine series. 
In future it will be supposed that this separation has been 
effected so that the forms (1) and (2) of A'l can be considered 
separately. 

Next, since 

F e (d) + F c (7r-a), F 8 (a)-F s (7r-a) 
contain even multiples of x only, while 

F c (a)-F c (7r-a), F 8 (a) + F B (ir-a) 

contain odd multiples only, the choice of pairs of supplementary 
values enables us to separate the equations giving the c t - or the 
Sj into two sets, one containing the coefficients with odd suffixes 
and the other those with even suffixes. 

Finally, choices of a which are multiples of TT/?I, where n is an 
integer, have obvious advantages. The special cases n = 3, 4 or 6 
will suffice for most of the requirements in the plans for develop- 
ing the disturbing function and the disturbing forces outlined 
here. For the rare cases in which eight harmonics are needed, 
it is advisable to add the values # = 45, 135 to those given 
by n = 6, since the work done with six harmonics only can be 
fully utilised and repetition of it is not needed. Schedules for 
this last case are to be found in Trans, of Yale Obs. vol. 6, part 4, 
pp. 61-65. 

A'3. Determination of the nth harmonic in F s . Since F 8 
vanishes for x = 0, TT, these values are useless for the computa- 
tion of the S{. Moreover, since sin nx vanishes when x is any 
multiple of TT/W, it is the coefficient of the rath harmonic which 
is undetermined with this set of values of x. It is sometimes 
possible to estimate its coefficient with sufficient accuracy; where 
this is not the case, one of the following devices may be adopted. 



A-2-A-4] HARMONIC ANALYSIS 291 

For n = 3, the chosen values of a? are 0, 60, 120, 180. The 
value # = 90 may be added to find 53. If this value be used 
also in F C) we can find c 4 and then get higher accuracy for 
c 2 (see A*4). 

For n 4, 6 we may proceed as follows : In most cases the 
calculation of dF s jdx for x = 0, TT will be found to be easy. These 
give the values of % 2s 2 + 3s 3 4s 4 -f . . . which, combined with 
the relation furnished by the other values of x inserted in F 8t will 
give the value of s n . Only one of the two values is necessary, 
but the solution of the equations is simplified by using both : 
they give also the (n + l)th harmonic and higher accuracy to 
the (n l)th harmonic. 

A*4. Errors of the coefficients. When multiples of irfn are used 
for the special values of x, the solution of the linear equations for 
the cosine series actually gives, instead of c t -, i < n, the value of 

d + C 2n -i + Ctn+i + C 4n -i + , 

if we include all the terms of the series. Thus the principal part 
of the error of c t - is the rejected coefficient c 2n -i When i = n, 
we actually determine the value of 



so that the error of c n is c 3/l . Hence the lower the harmonic the 
higher the accuracy with which it is found, with the exception 
of the nth which has an error equal to the coefficient of the 
3nth harmonic, approximately. 

In the case of F s we determine the value of 

Si S^ni 4" fyn+i &4ni "^ 

instead of Si, for i < /?, so that the principal part of the error of s f 
is $2n-i, an d tne same rule with respect to the errors of the 
coefficients of the lower harmonics holds. 

When the harmonic with coefficient s n (which vanishes when 
x is a multiple of TT/U) is found for n odd by using the additional 
value x = 90, it is easily seen that we actually determine 



292 APPENDIX [APP. 

for without its use we find s n _ 2 - s n +2 + , and with its use we 
find s n _ 2 s n + s n + g . . . , so that the principal part of the error of 
$ n is 2s n+2 . If x 90 be also used in finding F c , we determine 
c n _! with an error c n +s and c n +i with an error c n +a, the errors 
of the other coefficients being unaffected. 

When n is even, the use of the additional values dF/dx with 
# = 0, TT gives us 

s n _i + s n+ s + . . . , s n + 2s n + 2 + . . . , s n +i + s n +3 4- . . . , 

so that the principal part of the error of s n _i is s n + 3 and that of 
s n is 2s n+2 , the remaining earlier harmonics being unaffected. 
Thus in the case of F 9 the nth harmonic has the greatest error 
exactly opposite to the case of F c where it had the smallest error. 

A'6. The schedules made out below are given in detail so 
that they may be used without preparation for harmonic analysis 
either for a single function or for many functions. In the latter 
case, the work should be carried out in parallel columns, any 
one step being the performance of the same operation in all the 
columns; since most of the operations are simple additions and 
subtractions, much time can be saved by this reduction to 
routine computation. 

In the cases where the coefficients are known to diminish 
with some degree of regularity along values of i the results 
themselves furnish a test of the accuracy of the work; in any 
case, one or two of the special values may be reproduced from 
the final results with but little labour. 

A'6. Notation for the schedules. The function to be analysed 
is denoted by F and the argument by x. For an even function, 
we put 

F = C + Ci COS X + 02 COS 2x + . . . , 

and for an odd function 

F = SI sin x 4- $2 sin 2x + .... 
For the special values of x y 

0, 180; 90; 45, 135; 30, 150; 60, 120; 



A-4-A-6] HARMONIC ANALYSIS 2 

the corresponding values of F will be denoted respectively by 

ET I7 ' . I7 I7 37 '. XT' JP * . IT* I?'. 

TO, ^0 > ^9; ^4, ^4 3 ^3> ^3 5 ^6, ^6 3 

and, in the case of an odd function, 



The letters A,B,C>... used in the schedules are defined therein 
The first column of each schedule gives the symbols; the 
second column gives their numerical values in the examples 
the third, omitted in the computing forms, is explanatory. The 
error of any coefficient is shown in the final values. Thus c 3 (+ c 9 ] 
indicates that if CQ were known it would be subtracted in order 
to find Cs. 

Since the same function has been used in all the examples 
for the cosine analyses and its derivative in the sine analyses : 
direct comparison of the errors of the various results is possible. 



B&SPT 19 



294 



APPENDIX 



[APP. 



A'7. SCHEDULE FOR 3 HARMONICS. COSINE SERIES. 



Values of x = 0, 180, 60, 120. JP = (l-- 



FQ 


63246 


Addition for 


FQ 


1-26491 


extra value # = 90 


FQ 


83666 




FQ 


1-14018 


\A -94869 


A ~F Q +F ' 


1-89737 2c + 2c 2 


FQ I'OOOOO 


B =FQ+FQ' 


1-97684 2(? -c 2 


J4-/i = 2c 2 - '05131 


n A B 


- -07947 


C 2 - -02566 


D =$C=ct 


- -02649 


D-c 2 =c 4 - -00083 


2cQ=B + J) 


1-95035 








C o( + c 6) '97518 


A f ^FQ-FQ 


-63245 2^ + 2^ 


^( + c 5 ) --31 199 


B 1 =FQ~FQ 


- -30353 C!-2c 3 


c 2 ( + c 6 ) - '02566 


Sc^A' + B' 


- -93598 


6' 3 ( + c 9 ) - '00423 


c\ 


-31199 


c 4 ( - <? 6 ) - -00083 


Zc^c.-B' 


- -00846 




c, ( + * 6 ) 


97518 




Ci(+<? 6 ) 


- -31199 




C 2 (+C 4 ) 


- -02649 




C3 (+c 9 ) 


- -00423 





A'8. SCHEDULE FOR 3 HARMONICS. SINE SERIES. 
Values of x = 60, 120, 90. F = '3 sin x (1 - '6 cos a?)-*. 



F Q ' 



31053 

22787 
30000 

53840 
31085 

01085 

08266 
04772 



31085 
04772 
01085 



A-7-A-10] 



HARMONIC ANALYSIS 



295 



A'9. SCHEDULE FOR 4 HARMONICS. COSINE SERIES. 
Values of # = 0, 180, 90, 45, 135. .F = (l - -6cos#)*. 



^0 


63246 


A' = Ft - F{ - -43466 V2 (^ - c 3 ) 


FQ 


1-26491 


B'^JZA' - -61470 2c!-2c 3 


FI 


75877 


C' = FQ - Fd - -63245 2ci + 2c 3 


Ft 


1-19343 


4e 3 =C'-B' - -01775 


F Q 


1-00000 


4c 1 = C" + 5 / -1-24715 


A ^FQ+FJ 


1-89737 2c +2c 2 +2c 4 


Co(+c 8 ) -97522 


B -J4 


94869 


Cj( + c 7 ) --31 179 


2c 2 =i-F 


-05131 


C 2 ( + C 6 ) - '02566 


C = B+F 9 


1-94869 2c +2c 4 


C 3 ( + C 5 ) - -00444 


D =FI+FI 


1-95220 2c -2o 4 


c 4 (+c 12 ) - -00088 


4c =<7+ D 


3-90089 




4ct = C-V 


- -00351 




A10. 


SCHEDULE FOR 4 OR 5 HARMONICS, SINE SERIES. 


Values of x = 0, 180, 90, 45, 135. F = '3 sin x (1 - '6 cos ar)~*. 


^ 


47434 


A'=F d -F d f -23717 4* 2 +85 4 


^V 


23717 


B' =%A' -11859 2* 2 + 45 4 


^9 


30000 


25 2 =/4-^ 4 ' -10182 


^4 


27957 


454=^'- 25 2 -01677 


Fl 


17775 




* 




*i(-*7) -31169 


A = Ft + FJ 


45732 >J2(8i+8 3 -8 6 ) 


* 2 ( - *e) -05091 


^ =.4W2 


'oJjijoi S\ ~\~ 3g SQ 


s 3 ( + 5 7 ) -01281 


2! =^ + ^9 


62337 


5 4 ( - 25 6 ) -00419 


(7 =5-^, 


02337 25 3 -25 6 


5 5 ( + * 7 ) -00113 


D =F d +F d ' 


71151 2,^ + 653 + 1055 




2^9 


60000 




^ =Z)-2/9 


11151 853 + 855 




# =i^ 


02788 2*3 + 2*5 




4*3 =6^+0 


05125 




4,-?5 -0-G 


00451 





19-2 



APPENDIX 



[APP. 



All. SCHEDULE FOR 6 HARMONICS. COSINE SERIES. 
Values of = 0, 180, 90, 30, 150, 60, 120. jF(l--6cos0)*- 

F, 



A = 
B = 

C = 



=(7+1) 



4c =^+2c 4 



63246 
1-26491 
1-00000 

69311 
1-23272 

83666 
1-14018 

1-89737 
2-00000 
1-92583 
1-97684 
3-89737 
3*90267 4c -2c 4 

- -00530 

- '00177 
3-90090 

- -10263 

- -05101 
-15364 

- -05121 
- -00020 



2 - 4c e 



2c 3 



- -63245 

- -30352 

- -53961 

- -93597 
-93463 



-1-87060 

- -00134 
-31199 ^+5 

- '00847 

97523 
- -31177 
- -02561 

- -00423 
-00089 
- -00022 
-00005 



A-11,A*12] 



HARMONIC ANALYSIS 



A12. SCHEDULE FOR 6 OR 7 HARMONICS. SINE SERIES. 
Values of x = 0, 180, 90, 30, 150, 60, 120. F= '3 sin x (1 - '6 cos )- 



F d 


47434 


A 1 =F 3 -F,' 


09474 


V3(* 2 +* 4 ) 


Fd 


23717 


B 1 =FQ-FQ 


08266 


V3(5 2 -5 4 ) 


F 9 


30000 


C' =24' W3 


10940 


2* a +2*4 


F, 


21642 


D' =25'+V3 


09545 


2* 2 -^*4 


Fj 


12168 


4s 4 =c'-jy 


01395 




FQ 


31053 


4s 2 =C' + D' 


20485 




FQ 


22787 


E 1 =F d -F d ' 


23717 


4*2+8*4+12*0 






0' ='-4*2 


03232 


8*4+12 6 


FZ+FS 


33810 Si + 253 + 55-57 


8*4 


02790 




3*3 ^FS+FS'-I 


^ '03810 


12*6 = 0' -8*4 


00442 




* 3 


01270 






4 , 


A =F^+FQ 


53840 v/3(*!-* 5 + * 7 ) 


*l(-*ll) 


31179 


JV 


B =^ + V 


31085 *!-* 5 + *7 


2 ( - 5 lo) 


05121 




C =* 3 + ^o 


31270 * 1 +* 6 -*7 


a(-*) 


01270 




2! =B+C 


62355 


i(-*) 


00349 




D = C-B 


00185 2$ 5 -2*7 


s 5 ( + * 9 ) 


00103 




E =F d +F d ' 


71151 2! + 6*3 + 10*5 +14 r 


* 6 ( + 2* R ) 


00037 




-E- 2*! 


08796 6*3 + 10*5+14*7 


r( + ^) 


00010 




H --6*3 


-07620 -6*3 








K = -5Z) 


- -00925 -10*5+10*7 








24*7 =G+H+K 


00251 








2*7 


00021 








2* 6 =7) + 2* r 


00206 









298 APPENDIX [APP. 

J\'13. Double Harmonic Analysis. When a function of two 
angles can be expressed in the form 

F (a>, y) = 2 (A jtj > cosjx cos j'y + B jtf sin j# sin j'y 

^, > + G *,f cos j >aj sin ?'y + A*,/ sin ^ cos /2/)> 

where J,j' = 0, 1, 2, ..., the numerical calculation of the co- 
efficients can be reduced to a double application of single 
harmonic analyses. 

First, the choice of pairs of values a, 2?r a separates the 
terms containing cosines of multiples' of x from those contain- 
ipg sines by addition and subtraction. A similar choice for y 
makes a similar separation in each case. Thus the analysis is 
reduced to that of each of the four groups, and values for x, y 
equal to or less than 180 are to be used. 

A*14. Consider the first group given by 



= 42J.4 jj' cos joe cos j'y. 

The special values of A XtV are arranged in a block, each line 
containing those corresponding to a special value of x, and each 
column those corresponding to a special value of y. 

The special values in each column are analysed by one of the 
schedules for cosine analysis and give series 2-4^ cos j#, in each 
of which y has a special value. 

The results are rearranged in a block in which all the numbers 
Aj t y corresponding to a given j are placed in a column, the 
successive columns thus containing the special values of the 
coefficients of cos Ox, cos x, cos 2x, . . . for the special values of y. 

The special values in each column are then analysed by one 
of the schedules for cosine analysis and give the coefficients AJJ>. 

The process for finding the BJJ is the same with the exception 
that the cosine analyses are replaced by sine analyses. 

For Cjj>, the first block of analyses is that for cosines, and the 
second that for sines. 



A-13-A-16] DQUBLE HARMONIC ANALYSIS ^299 

For Djj' t the first block of analyses is that for sines, and the 
second that for cosines. 

The choices of special values of x, y are made on the same 
plans as those developed for single harmonic analysis. It is not 
necessary that^the same choices of values be adopted .for so as 
for y. 

A*15. The derivatives 



are to be used in the sine analyses instead of the zero values of 
the functions F. This is possible because the derivatives of the 
cos jx terms in F disappear from dF/dx when x = 0, TT ; and 
similarly those of the cosj'y terms from dF/dy when 2/ = 0, TT. 

A*16. In the method of development of the disturbing function 
and disturbing forces outlined in 4*15, the separation into the 
four sets of terms is made at the outset. Each function which 
vanishes for g 0, TT or for g' = 0, TT, is replaced by a derivative 
as shown above. 

When the development is made in terms of the angles f>fi 
(4*19), the special values of the first two sets are found together 
and must be separated by addition and subtraction: similarly 
for the third and fourth sets. 

An example in which the calculations are shown in detail 
will be found in the Tables for the Development of The Disturb- 
ing Function, by Brown and Brouwer*. 

* Cambridge University Press, 1933. 



INDEX 



(The numbers refer to pages.) 



Anomaly, true, mean, eccentric, 64 
Approximations 
in terms of time 
first, 146 
second, 156 

in terms of true longitude 
elliptic, 176 
first, 178 
second, 198 

principal part of second, 163, 203 
in Trojan group theory 
first, 270 
second, 277 
Asteroids, Trojan, 250 

resonance effect on, 245 
Astronomical measurements, 5 
Astronomical unit of mass, 7 
Attraction 

Newtonian law of, 7 
proportional to distance, 136 

Bessel's functions, 55 

Canonical differential equations, 117 
Canonical equations of motion, 24 
Canonical set of variables, 123 
Canonical set, Delaunay's, 131 
Canonical set, Poincare's, 132 
Complete integral, 123 
Constants of integration 

in true longitude theory, 213 
in case of resonance, 232 
in Trojan group theory, 279 
Contact transformation, 118 

d'Alembert series, defined, 66 

discussed, 141 
Declination, 5 

Delaunay canonical variables, 131 
Delaunay modified set, 132 
Departure point, 23 
Determining function , 118 
Development, of disturbing function 
(see Contents, Chap. IV) 

properties of development, 95, 139, 
142 

in terms of true longitude, 178 

numerical development, 98, 181, 278 

for Trojan asteroid, 261 
Differential equations 

canonical, 117 

Jacobi's partial, 121 

(see Equations of motion) 



Disturbing function, 10 
for double system, 11 
for satellite problem, 13 
for Trojan asteroid, 256, 258 
(see Development) 

Earth-Moon system, 11 
Eccentric anomaly, 64 

as independent variable, 30 
elliptic expansions in terms of, 68 
disturbing function in terms of, 

99 
Elements of ellipse, 16 (see Elliptic 

variables) 

Elliptic expansions 
in terms of eccentric anomaly, 68 
in terms of true anomaly, 71 
in terms of mean anomaly, 72 
literal, to seventh order, 79 
by harmonic analysis, 80 
Elliptic motion, 62 

fundamental relations, 64, 65 
Jacobi's method, 127 
true longitude theory, 176 
Elliptic variables, 16 
Delaunay's, 131 
Delaunay's modified, 132 
Poincare's, 132 
non-canonical, 133-136 
Encke and Newcomb equations, 26 
Epoch, 64 
Equations of motion 

rectangular coordinates, 8 
planetary form, 9 
satellite form, 11 
polar coordinates, 22 
canonical form, 24 
using as independent variable 
the true longitude, 28, 174 
the eccentric anomaly, 30 
the true longitude of disturbing 

planet, 31 

for Trojan asteroids, 256 
Expansions 

Lagrange's theorem, 37 
by symbolic operators, 45, 87 
of functions of major axes, 102-112 
(see Elliptic expansions and Con- 
tents, Chaps. II and IV) 

Force function, 8 

Fourier expansions (see Contents, 
Chap. II) 



302! 



INDEX 



Gravitation, Newtonian law of, 7 

Harmonic analysis, ^289 * 
for developing disturbing function/ 

98, 180 

for elliptic expansions, 80 
in Trojan group theory, 278 

Hypergeometric series, 56 
* 

Jacobi's method for elliptic motion, 127 

Jacob! 's partial differential equation, 
121 

Jacobi's transformation theorem, 119 

Jupiter's effect 
on Saturn, 205 
on certain asteroids, 245 
on Trojan asteroids, 284, 287 

Kepler's equation 

first used, equation (16), 64 

numerical solution of, 81 
Kepler's laws, 66 

third law discussed, 7 

Lagrange's expansion theorem, 37 

Latitude, 6 

Latitude equation, 29, 193 

Law of gravitation, Newtonian, 7 

Laws, Kepler's, 7, 66 

Libration, 230, 255 

Long period terms 

denned, 153 

second approximation to, 161, 202 

of disturbing planet, 165 

case of a single, 171 
Longitude, 6 

true, as independent variable, 28, 174 

mean ,64 , 

Major axes, functions of, 102-112 

stability of, 198, 202 
Mass, determination of, fy 

astronomical unit of, 7 * 
Mean distance, 66 

Mean motion, anomaly, longitude, 64 
Measurements, astronomical, 5 

Osculating ellipse, 16, 126 
Osculating orbit, 125 
Osculating plane, 15 

Pendulum, motion of, 219 

disturbed motion of, 221 
Perturbations 

general and special, 2 

of coordinates, 150 

mutual, of Jupiter and Saturn, 205 



Fertprbations (wntinued) 
approximate formulae for, 210 

"' effect of resonance on, 226 

of Trojan asteroids by Saturn, 280 

Planetary form of equations, 9 

Planetary problem*' 2 

Poincar6 canonical variables, 132 

Polar coordinate equations of motion, 
22 

Potential function, 8 , / 

Power series, numerical devices, 59, 60 

Reference frames, 15 
Bqsonance, defined, 4, 2J.6 (see Con- 
tents, Chap. VIII) 
Bight ascension, 5 

Satellite form of equations, 11 
Satellite problem, 2 
Saturn's effect 
on Jupiter, 205 
on Trojan asteroids, 284, 287 
Secular terms, 148 
second apprdximation to, 161, 167 
effect of, on second approximation, 

200 

Short period terms 
elimination of, 143, 259 * 
effect of, on second approximation 

159 

Small divisors 

source of, 84 
discussed, 154 

in true longitude theory, 193 
Solution 
of canonical equations, ,*138 (see 

Contents, Chap. VI) 
of true longitude equations, 185 (see 

Contents, Chap. VII) 
of equations of variation, 151, 197, 
253 

Time, method of measuring, 5 

Transformation 
contact, 118 
theorem, Jacobi's, 119 
to elliptic elements, 149, 194 
to time as independent variable, 207 

Triangular solution, 250 

Trojan group, 250 (see Contents, 
Chap. IX) 

True longitude as independent variable, 
174 (see Contents, Chap. VII) 

Variation of arbitrary constants, 17, 

125 
equations of, 149, 194, 252 



CAMBRIDGE : PRINTED BY 

WALTER LEWIS, M.A. 
AT THE UNIVERSITY PRESS