Planetary Theory

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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