TEXT FLY WITHIN THE BOOK ONLY DO 164413 73 OSMANIA UNIVERSITY LIBRARY pall No. <T&f- S x 4 <?? /^ Accession No. Author Title This book should be re|jrned on or before the date last marked below. PLANETARY THEORY LONDON Cambridge University Press FETTER LANE NEW YORK TORONTO BOMBAY CALCUTTA MADRAS Macmillan TOKYO Maruzen Company Ltd A II rights resewd PLANETARY THEORY BY ERNEST W. I'/D/esxor Kniciitns of Mathematics in Yale Unit ersitif AND CLARENCE A. SHOOK ** ~._ Ati^i^tant I'tofensot' of Mathematics in Lcli i</h I r itit'<T.'iiltf CAMBRIDGE AT THE UNIVERSITY PRESS 1933 * IN GREAT BR11AIN CONTENTS Preface ......... page ix Chapter I. EQUATIONS OF MOTION . 1 General introduction 1 Planetary form of the equations of motion . . 8 Satellite or stellar form . . . . 11 Frames of reference . . . . . . 15 Reference to polar coordinates in the osculating plane 22 True longitude as independent variable . . 28 Kccentric anomaly as independent variable . . 30 Reference to the coordinates of the disturbing planet 31 Motion referred to an arbitrary plane of reference 34 Chapter II. METHODS FOR THE EXPANSION OF A FUNC- TION 36 Expansion by functions of an operator ... 37 Lagrange's theorem for the expansion of a function defined by an implicit equation, and its extension 37 Transformation of a Fourier expansion from one argument to another where the relation between the arguments is defined by an implicit equation 40 Expansion by symbolic operators .... 45 Product of two Fourier series .... 45 Fourier expansions of series given in powers of the sine and cosine ....... 47 Function of a Fourier series 49 Powers of a Fourier series ..... 53 Cosines and sines of Fourier series ... 54 Bessel's functions . . . . . . . 55 Hypergeometric series ...... 56 Numerical calculation of series .... 59 vi CONTENTS Chapter III. ELLIPTIC MOTION page 62 Solution of the equations ..... 62 Relations between the anomalies .... 65 Kepler's laws 66 Developments in terms of the eccentric anomaly . 68 Developments in terms of the true anomaly . . 71 Developments in terms of the mean anomaly . 72 Literal developments to the seventh order . . 79 Development by harmonic analysis ... 80 Numerical solution of Kepler's equation . . 81 Chapter IV. DEVELOPMENT OF THE DISTURBING FUNC- TION 82 Development in terms of elliptic elements . . 82 Expansion by operators in powers of the eccentri- cities ........ 87 Expansion in multiples of the true anomaly . . 89 Expansion along powers of the inclination . . 91 Development to the third order .... 93 Transformation from true to mean anomalies . 94 Properties of the expansions . . . . . 95 Calculation of the constant term . . . . 97 Development in terms of the eccentric anomalies . 99 Transformation to mean anomalies . . . 100 The functions of the major axes . . . . 102 lateral expansion to the second order in terms of the mean anomalies . . . . . . 112 The second term of the disturbing function . . 114 Chapter V. CANONICAL AND ELLIPTIC VARIABLES . 117 Contact transformations 118 Jacobi's partial differential equation and its solu- tions. ........ 121 Jacobian method of solution . . . . . 1 27 Other canonical and non -canonical sets. . . 131 The case of attraction proportional to the distance 136 CONTENTS vii Chapter VI. SoLtfTiON OF CANONICAL EQUATIONS . page 138 General properties of the variables and of the disturbing function . . . . . . 138 Eli mination of a portion of the disturbing function 143 First approximation . . . , . . 146 Long period terms . . . . . . 152 Second approximation . . . . . 156 Calculation of the second approximations to long period and secular terms . . . . . 161 Summary and special cases . .... 169 Chapter VII. PLANETARY THEORY IN TERMS OF THE ORBITAL TRUE LONGITUDE . . . 174 Equations of motion and method of solution . 174 The first approximation . . . . . 17,8 Solution of the equations . . . . . 185 Equations for the variations of the elements . 194 The second approximation . . . . . 198 Transformation to the time as independent vari- able 207 Approximate formulae for the perturbations of the coordinates 210 Definitions and determination of the constants . 213 Chapter VIII. RESONANCE 216 Elementary theory 219 Solution of a resonance equation . . . 222 General case of resonance in the perturbation problem 226 A general method for treating resonance cases . 234 The case e'= T = 238 The 1 : 2 case 241 The cases e'^0 246 viii CONTENTS Chapter IX. THE TROJAN GROUP OF ASTEROIDS . page 250 The triangular solutions ..... 250 General theory ....... 256 Elimination of the short period terms . . . 259 Expansion of the disturbing function . . . 261 The equation giving the libration . . . 269 Perturbations of the remaining variables . . 273 Higher approximations . . . . . 277 Perturbations by Saturn ..... 280 Indirect perturbations 283 Direct action of Saturn ..... 286 Appendix A. NUMERICAL HARMONIC ANALYSIS . . 289 Index ......... 801 PREFACE THE chief purpose of this volume is the development of methods for the calculation of the general orbit of a planet. If an accuracy comparable with that of modern observation is to be attained in any particular case, the choice of the method to be adopted may be an important factor in the amount of calculation to be performed. Not only should the general plan of procedure be efficient, but full consideration of tha details of the work should be given in advance. We have attempted to anticipate the difficulties which arise, not only in the older methods but also in those developed here, by setting forth the various devices which may be utilised when needed. While the developments given below are intended to be complete in the sense that they should not require a knowledge of the subject drawn from other sources, the volume is not supposed to be a substitute for an extended treatise like that of Tisserand. It contains, for example, no detailed account of such classical theories as those of Leverrier, Hansen and Newcomb. It does, however, attempt to indicate that most of the methods previously used ultimately reduce to two. One of the methods involves a change of the variables to elliptic elements, while the other consists of a direct calculation of expressions for the co- ordinates. An example of each of these general plans is given and worked out in detail. Few references to previous work have been made and those furnished are merely incidental. It has seemed unnecessary to repeat material which the student can find equally well in Tisse rand's treatise or in vol. IV of the Encyklopddie der Mathe- matischen Wissenschaften. While a critical estimate of the merits and demerits of previous works would doubtless be of assistance to anyone planning to carry out detailed calculations for the theory of a particular planet, in the past the methods which have been adopted have been sometimes chosen less on account of their efficiency than for other reasons, and the same will probably be to some extent true in the future. Nevertheless it x PREFACE is still advisable to give consideration to each of these plans, and we have attempted, by occasional remarks, to aid the student in this respect. The mathematical processes which are used in developing the theories of the planets and satellites from the laws of motion are largely formal While mathematical rigour is desirable when it can be attained, nearly all progress in the knowledge of the effects of these laws would be stopped if complete justification of every step in the process were demanded. The use of formal processes is justified whenever experience shows that the results, riot otherwise obtainable, are useful for the prediction of physical phenomena. Thus when calculating with an infinite series whose convergence properties are not known, one has to be guided by the results obtained; if the series appears to be con- verging with sufficient rapidity to yield the needed degree of accuracy, there is no choice save that of using the numerical values which it gives. We have not attempted to deal with con- vergence questions, but have retained throughout the practical point of view mentioned in the first sentence of this preface. Considerable portions of the volume are new in the sense that if they had not been given here, they would have been printed in abbreviated form in the current journals. In particular is, this true of the last two chapters. The novelty, however, consists mainly in the adaptation and further exploitation of previously known devices. Some of these extensions owe their effectiveness to a recent publication of tables of certain functions*, or to the introduction of mechanical computing aids. An example is the attention given to development by harmonic analysis. The following sentences give a brief summary of the contents of the volume. In the first chapter, various forms of the equations of motion are derived, other possible forms being suggested. The second chapter is a collection of various expansion theorems which are or may be needed in the later developments. A short account of the essential properties of elliptic motion follows. Various methods for developing the disturbing function and disturbing forces are set forth in Chapter iv. Chapter v contains * See footnote, p. 182. PREFACE xi the elements of the theory of canonical variables so far as it is needed in the later work. This theory is usually difficult for a student to grasp, and we have tried to simplify the exposition so that he may not only be able to understand it but also to make use of it as a tool for investigation. In Chapter VI, it is shown how this theory may be efficiently applied to the calcula- tion of the orbit of a planet ; the basis of the method is the use of the transformation to eliminate the short-period terms as a first step, leaving the long-period and secular terms to be dealt with separately. In Chapter vii, the direct calculation of the coordinates with the use of the true orbital longitude as the independent variable is developed with sufficient detail for the formation of an approxi- mate or of an accurate theory. Chapter VIII contains an attempt to place the theory of resonance on a general basis, in a form which permits of application to specific problems. The point of view taken is mainly that of explaining how this phenomenon can be treated mathematically in certain of the cases of its presence in the solar system. It appears, however, to give a method of approach to the consideration of the question of the general stability of the orbits of the planets and thus leads to certain aspects of cosmogony. In Chapter IX, the theory is applied to the Trojan group of asteroids in a form which it is hoped will make the calculations of the orbits of these bodies easier than has hitherto been the case. The appendix on Harmonic Analysis will be found to contain formulae for its application to the development of a given function ready for actual use. Most of these formulae have been tried out extensively and have been found to render the computations easy and accurate, especially when the number of such functions to be analysed is great. We are indebted to Dr Dirk Brou wer and Mr R. I. Wolff for the errata given on p. xii and discovered after the sheets had been printed. r ERNEST W. BROWN CLARENCE A. SHOOK 1933 Errata p. 26, line 5 from bottom, insert factor , before ^-- . * r 2 oT p. 28, line 6 from bottom, for u 2 , read uu' 2 _- . 11 ou du p. 28, Equation (4), for $q read . 2 2tf 2 p. 28, Equation (5), /or read - .. . v ?r ^ p. 28, Equation (7), for -T read + l\ p. 30, omit lines 12, 13. p. 63, Equation (10), /or rtd read andt. p. 64, lines 12, 15, for p. 66, last line, /or 47r' 2 /x, read /x/47r-. p. 68, line 18, insert 2 fc before second formula. p. 76, Equation (J), for {-) read ( C -j . p. 83, line 18, for a~ read a-. p. 87, line 7, for # = exp ^ read x-~exp \f/ *J - 1. p. 87, Equation (2), delete the letter^. p. 98, line 22, /or ^ p. 117, Equation (2), for m v read m. p. 118, line 5, for (5) mid (6). p. 129, line 6 from bottom, for (3) read (8). p. 131, line 14, for ^(-2^)"* m/d ^(-2^)"-. p. 131, delete line 15, replacing it by *'Here the choice for S, slightly differing from (1), is". ,._ r o f W , $ S p. 145, line 2, /or -_- mid --^- p. 157, line 4 from bottom, insert 2 before the last term. p. 163, Equation (12), for - 3S read -32/j. p. 176, last line, for Dv = l read Dv = l. p. 224, line 15, for 8-8(9) read 8'S (10). p. 230, line 5, for R read R . p. 250, lines 5, 23, for Laplace read Lagrange. p. 250, line 19, for configuration read condition. p. 253, line 6 from bottom, for read r* p. 255, line 11, the equations should read aB _ db 3 >/3 m' - 8a 3 /t 2 1 (y. m)* _ >/3~ t - I 8 ^ "^ 12 + ,I' CHAPTER I EQUATIONS OF MOTION A. GENERAL INTRODUCTION I'l. The methods considered in this volume for the investiga- tion of the mutual actions of two or more bodies are based wholly on Newton's three laws of motion and on his law of gravitation. It is assumed that there exist fundamental frames of reference with respect to which the laws are exact and that the space in which the bodies move is Euclidean. The modern theory of re- lativity gives a different approach to the problem, but from the point of view taken here, which is chiefly that of deriving formulae for the comparison of gravitational theory with obser- vation, the numerical difference resulting from the two methods of approach is very small, and can be exhibited as a correction to the results obtained through the Newtonian approach. These corrections, which are near the limit of observational accuracy at the present time, will not be considered here. A further limitation is the treatment of the motions of the bodies as those of particles having masses equal to the actual masses. Here again, owing to the theorem that a sphere of matter, whose layers of equal density are concentric spheres, attracts an outside body as if it were a material particle, and also, owing to the fact that most of the bodies with which we have to deal are approximately spheres of this character or are sufficiently far away from the attracted body that they can be so treated, the differences are small. All other possible and actual forces, unless they obey the inverse square law and have con- stants which can be supposed to be included in the constants which we call the masses of the bodies, are neglected. 1*2. A general knowledge of the masses and relative distances of the various bodies from one another has to be assumed because the method of treatment to be recommended depends B&SPT I 2 EQUATIONS OF MOTION [OH. i on this knowledge. The two principal divisions are the planetary problem and the satellite or stellar problem. A third division may include the cometary problem and those cases of motions which cannot be included in either of the first two divisions. In the planetary problem, one mass is very much greater than the combined masses of all the other bodies and dominates the motion of any one of them to such an extent that during a few revolutions the orbit of the latter is not greatly different from that which it would describe if the remaining masses did not exist. These approximate orbits moreover are ellipses with the principal mass in one focus and having minor axes which do not differ from their major axes by more than about ten per cent, of the latter. Further, the planes of these ellipses tire inclined to one another at small angles generally less than 20. The dis- tances of the bodies from one another may have any values what- ever provided they do not fall below a certain limit. In general, the methods of this volume are developed for this case alone. In the satellite or stellar problem the distance between two of the bodies must be small compared with the distance of either from the third; the two nearer bodies circulate round one another and their centre of mass circulates round the third body. The maintenance of this state of motion requires a limiting relation between the masses and distances of the bodies. There are also limitations concerning the shapes and positions of the orbits similar to those present in the planetary case. In the satellite problem, the mass of one of the two nearer bodies is small compared with that of the other, and the mass of the latter small compared with that of the third body. In the stellar problem, the masses are usually of the same order of magnitude. The methods adopted to obtain the motions in these two limited cases are not applicable to the cometary and other cases. The methods developed below give expressions for the co- ordinates in terms of the time which serve to give the positions of the bodies over long intervals of time: the results are usually named the general perturbations. Practically all other cases have at present to be treated by the method of special perturbations 2, 3] INTRODUCTION 3 which consists in a completely numerical process of calculating the orbit over successive small arcs by 'mechanical integration.' Practically all the problems which have hitherto been brought within the range of observation belong to one of these classes. There are numerous problems in the stellar universe in which the law of gravitation undoubtedly plays a dominant r61e. At the present time the deviations from rectilinear motion have not been directly observed, although such deviations have been inferred by statistical methods. 1*3. Since no method for the exact integration of the equations of motion of three or more bodies exists, devices for continued approximation to the actual motion are used. Sometimes these lead rapidly to the desired degree of accuracy; in other problems, the process may have to be repeated many times. In most cases, the first approximation is taken to be an ellipse and this is equivalent to a start with the two-body problem and a con- tinuation with the calculation of the disturbing effects produced by the attractions of other bodies. As far as possible these changes in position or 'perturbations/ as they are named, are expressed by sums of periodic terms which take the form of sines or cosines of angles directly pro- portional to the time, or to some variable which always changes in tfye same sense as the time. When, however, the number of such terms becomes too great for convenient numerical applica- tion, the terms of very long period are replaced by powers of the time or other adopted independent variable, and these powers are used in combination with the other periodic terms. In any case, the expressions which are obtained give reliable results for a limited interval of time only; all the periods with which we have to deal are determined from observation and therefore possess a limited degree of accuracy. Expansions of functions in series, especially as sums of sines and cosines, thus play a large part in the work. The possibility of obtaining these expansions in such forms that numerical results may be deduced from them without too much labour, 4 EQUATIONS OF MOTION [CH. i usually depends on the presence of small constants or variables 'parameters' in the coefficients of the periodic terms, and the expansions are partly made along powers of these parameters. Their orders of magnitude are important. In general, they consist of the eccentricities which rarely rise much above *2, of the inclinations of the orbital planes to one another usually less than 20, and of the ratios of the distances. For the majority of cases, this last ratio lies between *4 and *8, and the feet that we are compelled to expand in powers of so large a parameter is responsible for many of the difficulties of the problems. 1'4. The most fundamental difficulty, however, is caused by approximate or exact 'resonance.' This term refers to those cases in which two or more of the periods, which enter into the expansions for the coordinates, are nearly or exactly in the ratio of two small integers. In the approximate case, large amplitudes of certain of the periodic terms, and slow numerical convergence to the needed degree of accuracy, are characteristic effects. In the cases of exact resonance the form of the solution has to be changed. In either event, the terms which cause the chief trouble are those with periods which are long in comparison with the period of revolution of the body round the central mass. Such periodic terms may have small coefficients in the equations of motion, but the integration of the equations produces small divisors which give large coefficients in the coordinates. These small divisors demand that the terms affected be carried to a much higher degree of accuracy than the remaining terms, and as there exists no short method for securing this accuracy, the amount of calculation needed in any given planetary problem depends mainly on the few, perhaps one or two only, terms of long period which are sensible in the observations. The existence of such terms in every planetary problem has to be kept in mind while divising methods and in carrying them out. The method finally chosen should depend mainly, not on the ease 3-5] ASTRONOMICAL MEASUREMENTS 5 with which the first approximation may be obtained, but on the work required for the final approximation. 1/5. Astronomical measurements. The only measurements of the position of a celestial object which have a precision com- parable with gravitational theory are those of angles on the celestial sphere. The ultimate planes of reference from which these angles are measured along great circles are defined by the average positions of the stars (which have observable motions relative to one another). For the theory, an origin is needed, and this is ultimately the centre of mass of the system. It is assumed that these definitions will give a Newtonian frame because the stars are so far away that they affect neither the motion of the centre of mass nor the relative motions of the bodies within the system to an observable extent. Time in this frame is measured by the interval between the instants when a plane fixed in the earth arid passing through its axis of rotation passes through a mark in the sky supposed to be fixed relatively to the stars. Owing to the rapid motion of the earth about its axis the observer finds it convenient to give his measurements with respect to the plane of the earth's equator, and to a point on the equator defined by its intersection with the ecliptic the plane of the earth's orbit round the sun. Both these planes are in motion but their motions and positions relative to the ultimate stellar frame are supposed to be known. This frame is incon- venient for working out gravitational theories on account of its large inclination to the planes of motion of most of the bodies of the solar system. For this purpose the ecliptic and the point on it defined above are used. The motions of these are nearly uniform and are easily taken into account. In the observer's frame, the angular coordinates are the declination measured along a great circle from the object to the equator and perpendicular to the latter, and the right ascension, the angle between this great circle and that perpendicular to the equator and passing through its intersection with the 6 EQUATIONS OF MOTION [CH. i ecliptic. In the computer's frame, the angular coordinates are the celestial latitude and longitude similarly measured with respect to the ecliptic. As this latter frame is moving the ultimate reference is to its position at some give'ri date. Thus to transform the results obtained from the gravitational theory for the use of the observer, geometrical relations must be com- puted, and there is, in addition, a kinematical relation due to the motion of the observer's frame. It is assumed that a complete gravitational theory of the motion of any body in the solar system referred to a frame whose motions are fully known, should give the position of the body at any time when the constants of the motion have been determined. Differences between the calculated and observed positions of the body may be due to defects in the theory, un- known motions of the frame of reference, errors in the determination of the constants, or errors of observation. The analysis of these differences in order to discover their source is a problem involving many difficulties. In many cases two or more interpretations are possible and these can only be separated by the use of more observations. An outstanding difference, for example, between the observed and calculated values of the motion of the perihelion of Mercury was variously attributed to the gravitational attraction of a ring of matter supposed to surround the sun, to a motion of the frame of reference, to defects in the gravitational theory, until the theory of relativity furnished an explanation. A marked deviation of the moon from its gravitational theory has received an explanation as a varia- tion in the rate of rotation of the earth about its axis, through detection of similar deviations from the gravitational theories of the observed posi- tion of the sun, the satellites of Jupiter, and the planet Mercury. 1 *6. Observations of bodies in the solar system are usually of two classes. Those made with the transit telescope give the instant of passage of the body across the meridian of the observer and the angle between its observed direction and that of the earth's axis, the time being given by a clock which is constantly compared with the transits of stars. Differential observations, often made by photography, give the position of the body at any time with respect to stars in its neighbourhood, the places of these stars referred to the frame being known. Under good conditions, either class of observation should give the position with a probable error less than V such a standard at least is aimed at in the gravitational theories of the principal bodies in the solar system. While direct observations of distances cannot in general be made ac- curately, their theoretical determination is necessary because we cannot 6-7] LAW OF GRAVITATION 7 eliminate them from the equations of motion without introducing compli- cations greater than those which the equations already possess. They are also needed because the planetary theories use the sun as an origin ; the transformation from the observer's position on the earth to an origin in the sun requires a knowledge of the variations of these distances. Observations of masses or of relative masses are not made : a mass is only known to us by its gravitational effects. In the solar system, the orders of magnitude are such that the mass of a body has little effect on its own motion ; the operation of Kepler's third law, (3 -6), eliminates from the angular coordinates the greater part of the mass eftect. When dealing with the effect of one planet on another, we need to know the ratio of the mass of the disturbing planet to that of the sun. If the planet has a satellite whose motion can be calculated, the mass of the planet can be found with sufficient accuracy to calculate its effect on any other planet. When it has no satellite, its mass can be found only by comparing its cal- culated disturbing effects with those furnished by observation. There is of course a correlation between the degree of accuracy required to calculate the perturbations and that with which the perturbations can be observed. Thus the masses of all the major planets except those of Venus and Mercury are fairly well determined. The masses of the minor planets can be ob- tained only from observation of the light they send with estimates as to their albedo and density, since they are too small to exert any observable attractive effect on any other body. 17. The Newtonian law of gravitation. This law states that the attractive force between two particles of masses m. m' at a distance r apart is along the line joining them and of magnitude Cmm'/i*, where C is a constant the 'gravitation constant' wfiose value depends only on the units adopted. It is convenient, in order to avoid the continual presence of C in the mathematical operations, to so choose the units that (7=1. Since m/r 2 has then the apparent dimensions of an acceleration, it follows that a mass with these units has the apparent dimensions of the cube of a length and the inverse square of a time. The adoption of this unit called the 'astronomical unit of mass' is closely associated with Kepler's third law. This law, with a slight modification (3*6) to bring it into accord with the law of gravi- tation, states that if ZTT/H be the period of revolution of two bodies about one another and if a be their mean distance apart, ii* a* = C x sum of masses. 8 EQUATIONS OF MOTION [CH. i In numerical work we need to deal with ratios only. If a relation of this kind is used in the equations of motion, the latter will be freed from the apparent inconsistency of possessing terms having different physical dimensions, and will consist of ratios of masses, distances and times only. The units of time and distance to be finally chosen, depend on the problem under investigation: a choice is not usually needed until comparison with observation is to be made. For this purpose, the mean solar second, day or year are used as units of time, and the mean radius of the earth or the mean distance of the earth from the sun as units of distance. B, PLANETARY AND SATELLITE OR STELLAR TYPES OF THE EQUATIONS OF MOTION 1*8. The Equations of motion with rectangular coordinates A force- function for the motion of any particle, when it exists, is defined as a function whose directional derivatives with respect to the coordinates of the particle give the components of the forces. For two particles with masses m, in and at a distance r apart, the gravitational force-function is mm' jr. For n + 1 bodies with masses mi and mutual distances ?', it is * ij the summation including each combination of i,j once only. This function is independent of the directions in the frame of reference which may be used. It may or may not depend on the origin chosen. In treatises on general mechanics, the potential is denned as a function which has the property that its directional derivatives give the reversed components of all the forces which act on the system. We shall have oc- casion later (e.g. in 1 *9) to construct force-functions which are not potentials with reversed signs, since the force-function for the motion of one particle is not the same as those for the other particles of the system. 7-9] PLANETARY FORM 9 Let , m, f be the coordinates of m if Then, with the defini- tion of V given above and under the limitations stated in 1*1, the equations of motion are d^i W d*m SV d 2 ^ BF m i~Ta^> m / J72~ = T~ ' m i~J^ = ^^' -(*) at* dgi at* dm at 2 o%i Only the relative motions of the bodies are needed and, in finding them, the equations are usually given two principal forms, depending on the origins chosen. 1*9. Planetary form of the equations. In this form, one body w is chosen as the origin of coordinates, and the motions of the remaining bodies relative to m Q are to be determined. Let ff* = &-?o, l/k^^k-no, s&=6fc-oi / = !, 2, ...w, r k 2 = r ok 2 = (KI? + j/fc a + ^A 2 . Then r/ = (^ - #,)* + (^ - y^) 2 -h (^ - ^) 2 . These definitions, together with the equations of motion, give dt 2 __ _ /p , v- _ '^fc _ v ; y ""8.'-*U ; >,J >* 3 ~ ; ^ 3 ' where J =(= ^- The penultimate term, being equal to the derivative of m k jr k with respect to #&, can evidently be combined with that of mo/r k . The last term can be written 9 in which form it will serve for all three coordinates. Hence, if we put the equations of motion for m k relative to m Q become with similar equations for y k , z k . 12 EQUATIONS OF MOTION . [OH. I so that the new coordinates still depend on the differences of the original coordinates. Now when 7 is expressed in terms of the coordinates ;, TH, &, we have, as a result of changes in the & only, But since V contains the ^ only through their differences. dV 37 dV Whence, combining thia with the previous equation, we have d V = j| (rfft - rf>) + ^ (d& - d$>). <7i O^2 But when V is;to be expressed in terms of #, ,7;', we have 7 , , , TT- dx + =-, dx . ox ox If we substitute the values for dx, dx' in terms of dgi dgQ, d^z d^ and equate the coefficients of the latter in the two expressions for dV, we obtain ar 3 A'' ' ...... (4) The transformed equations of motion for #, x r become, with the help of these results, ^r, (5) dV I ftV , dV\ ' - - ' - - ~i ^~~ ' . ' y 12, 13] SATELLITE FORM 13 For the purposes of calculation, we insert the value of V, namely, , 7 m^mi y . j n>l 7*02 with ?*i 2 2 = .1? + ?/ 2 + 2 = / 2 , = (,'i/H 2 ,] +... + ..., \ nil + itiz I giving d 2 x ?) (? + 77i w /M + //? 2 ) ///h w a ~ I H --- \r i r 02 / For theoretical investigations, the equations are exhibited in the Newtonian form by putting 2 ir / 1/r /ox , Kl = ^yU, K, ...... (8) tQi~- ^) when they become with similar equations for ?/, ^, ?/', ^'. This form cannot be used, however, if ?% or m% is zero. But the equations then revert to the planetary form, with the motion of m Q relative to raj elliptic if m 2 = 0, so that the only motion which has to be considered is that of ra 2 relative to raj. 1'13. The equations of motion in the form 1*12 (6), (7) will not be needed in the developments of this volume since they are, in general, useful only when the distance between mi , m^ is small compared with those between wi , mi and wt , ?n 2 , that is when r/r' is small. The initial development of V will, however, be shown, in order to exhibit the contrast with the developments used in the planetary problem. Put xx' +yy' + zz? = rr' cos S, 14 EQUATIONS OF MOTION [CH. i so that S is the angle between ?, r'. Then " '2 i 2W *2 ' V*f 2 \ 2 a r ()1 - r i -f - rr' cos >S -i- --- --- - ) r 2 , ?/i 1 -fw 2 Vm^wo/ 2 '<; 2m 1 , , , / Wi Y 2 ,, 9 02 2 = r 2 - ?r eos*S-H ---- - ) ? . m! + 7?l 2 V> 1 1+ W1 2/ Hence, if /^ be the zonal harmonic of degree i with argument ft, we have 1 -i / l+ ; (_ '*_ TP) '01 ' V ,=1 \ T+'S 'V '/' JL.UH. %2 '* I t and thence y^'/^wiy Wp(wi-f^a) Wf> * r / r t=2 the terms for /=! having disappeared. The force-function 1*12 (6) then becomes L. _iilL_ 2 '^vi ( !_\ p I / the second term of F being useless. The function 1*12 (7) becomes r > the first term in this case being useless. In each case, if the function be confined to its first term we obtain elliptic motion (Chap, in), the remaining portion being that which produces the dis- turbing effect. If ?i, a be the mean angular velocity and mean distance in the former, and n', a' those in the latter, wo have by Kepler's third law, (3-6), By putting z = l, we see that the significant factors of the disturbing effects in (1), (2) are In the satellite problem, m is much greater than m t and m-! than w 2 , and a/ a', n'/w are small, so that both these factors are small. If m Q1 m-j, wi 2 refer to the sun, earth and moon, respectively, these ratios have the mag- nitudes -007, 1-SxlO- 7 . In those stellar problems which have up to the present shown observa- tional evidence of perturbing effects, ?n , m t , m 2 are of the same order of magnitude, but a/a', n'/n are small of the order '2 or less. The disturbing effects are chiefly shown in the motions of the apses and nodes. See P. Slavenas, "The Stellar Case of the Problem of Three Bodies*." * Trans, of Yale Obs. vol. 6, pt. 3. 13-15] COORDINATE SETS 15 C. FRAMES OF REFERENCE 1*14. Choice of variables. In the preceding sections, the equations of motion have been referred to certain origins in a Newtonian frame with fixed directions for axes. Experience has shown, however, that neither rectangular coordinates nor fixed axes are convenient for finding the position of the body at any time, and that the calculations may be much abbreviated by suitable choices of coordinates. Analytically, the deduction of these sets of equations may be regarded as nothing more than a change from one set of variables to another. But since the choice of a set of variables always depends on a knowledge of the general characteristics of the motion, it is often useful to give a geometrical or dynamical interpretation to the variables chosen. 1'15. For the development of the planetary theory, the osculating plane as a principal plane of reference possesses certain advantages over all other planes of reference. It is defined as a plane passing through the sun and the tangent to the orbit of the planet. The plane is in motion but it is found in most cases that its motions are either small or slow; that is, its deviations from a mean position are either small or require long periods of time to become large. This fact can be so used in the analytical work as to abbreviate the calculations. A second and even more important property of this plane is due to the small effect its motion has on the motion of the planet within the plane. In many cases this secondary effect can be altogether neglected, so that the motion within the plane can be treated as though the latter always occupied its mean position. These remarks refer equally to both disturbing and disturbed planet, that is, to the planet whose motion is supposed known and to that which we are finding. The effect of the motion of the plane of the former on the latter is usually negligible or can be accounted for quite simply. All the methods developed in this volume use the osculating plane as a plane of reference. 16 EQUATIONS OP MOTION [CH. i 116. The choices of coordinates within the plane of reference may be placed in two categories. In the first of these, the distance of the planet from the sun or some function of this distance is used as one coordinate, the second coordinate being the elongation of this radius reckoned from some fixed or moving line with the time as the third variable in the equations of motion. The roles played by the second and third of these variables may be interchanged. Two ways of measuring the elongation are used below. One is the usual method of using a single symbol to denote the sum of two angles measured in different planes : that between the radius and the line of intersection of the osculating plane with a fixed plane of reference, and that between this latter line and a line fixed* in the plane of reference. This symbol is usually called the longitude in the orbit or, briefly, the longitude. The second is the measurement of the elongation from a line in the osculating plane, this line being so defined that its resultant velocity is always perpendicular to the osculating plane. This second method has the advantage of eliminating the motion of the osculating plane from the kinetic reactions within the plane. With the use of these methods it is convenient to introduce an auxiliary variable which substantially is the angular momentum or a function of it. 1*17. The second category consists of the use of certain variables associated with the osculating ellipse. This ellipse is defined as the orbit which the body would follow if, at any instant, all disturbing forces were annihilated and the body continued its motion under the sole attraction of the central mass. The definition requires that the velocity in the orbit and in the osculating ellipse shall be the same in magnitude and direction and therefore that its plane shall be the osculating plane at the point. The variables used are those which define the size, shape and position of the ellipse, or certain functions of them which may or may not contain the time. These functions are called the elements of the ellipse. The elements which are simplest for descriptive purposes are * The word ' fixed ' is used in the Newtonian sense. 16-18] COORDINATE SETS 17 the major axis, the eccentricity, the longitude of the axis and the time of passage through the nearer apse or position when the distance from the focus occupied by the central mass is least ; the period of revolution is connected with the major axis by Kepler's third law which involves the sum of the masses of the two bodies. This sum may be unknown but, as it remains constant, the relation between the variations of the major axis and the period is always the same. Various combinations of these elements and of the two elements which define the position of the osculating plane are also used as elements : as we go from point to point of the actual orbit these elements will change. According to the definition, the changes will depend on the existence of attractions other than that of the central mass, and the term ' Variation of the Elements' refers to these changes. It will be seen below that the methods used to determine them are similar to that unfortunately named ' the Variation of Arbitrary Constants ' in the theory of differential equations. 1*18. This geometrical description of the elliptic frame, while useful for descriptive purposes, conceals the analytical meaning which is essential for a clear understanding of the processes involved. Analytically, the elements are nothing else than a new set of variables allied to the coordinates by a definite set of relations which remain unchanged. Thus the process of forming the differential equations satisfied by the elements is precisely that of changing from one set of variables to another. The description of the process is complicated by the fact that the three old variables (the coordinates) are replaced by six new variables ; consequently, three relations between the latter are at our disposal. If the coordinates be denoted by Xi and the new variables by j, and if the relations between them be Xi=fi(ai,a 2 , .. .,<,,), t=l, 2, 3, then the three additional relations are almost invariably chosen to be so defined that they satisfy the equations B&SPT 18 EQUATIONS OF MOTION [CH. i As a result of this definition, we have It follows that %i, dxi/dt are replaced by f it dfi/dt in the equations of motion and that d 2 %i/dt 2 is replaced by This last process gives three equations and these with (1) furnish the six equations necessary to find the a,-. The forms of the functions /i, and consequently those of their partial derivatives, remain unchanged and are given by algebraic and trigonometrical formulae developed in Chap. III. The analytical point of view just given is that which is chiefly needed in the development of the equations of motion. This view is often obscured by the methods used to obtain the functions fi. These methods require the solution of the equations for elliptic motion and in this solution the c^ appear as the arbitrary con- stants : in the general problem they become the variables. The fact is that the solution of the equations for elliptic motion is merely a convenient device for finding the functions f t which connect the old and new variables. The fact that the differential equations satisfied by the new variables are all of the first order, together with another property to be developed in Chap, v, namely, that the variables can be so chosen that the equations have the canonical form, is largely responsible for the use that has been made of them in theoretical investigations. Their practical value lies in the ease with which the equations may be integrated and in the simplicity of the geo- metrical interpretations which maybe given to some of the results. 1*19. Certain methods like those of Hansen and Gylden, not developed in this volume, as well as that given in Chap, vn, possess to some extent the characteristics of both categories. No sharp division is possible or necessary, the sole test being that of convenience for the problem under consideration. Whenever a new variable is introduced, it can generally be related to some property of the ellipse, but this relation is not usually helpful except in so far as it may have led to the choice of the variable. 18-21] COORDINATE SETS 19 1*20. The frames of reference should also be regarded as four- dimensional in the sense that the time as well as the space co- ordinates should enter into consideration in making choices of new variables. The observer's demand is for expressions giving the space coordinates in terms of the time, but the analyst is free to regard any three of them as a function of the fourth and to solve the problem according to his choice. If the time be not used as the fourth coordinate, that is, as the independent vari- able, a final transformation is usually, though not necessarily, made to obtain the space coordinates in terms of the time. However, practical demands limit the nature of the independent variable. The linear coordinates are sums of periodic functions of the time, that is of a variable which is unlimited in magnitude and whose changes are always in the same sense. Any other independent variable which is chosen should have the same pro- perty: it may be angular or even areal provided the angles or areas are always changing in the same sense. Otherwise trouble- some infinities are apt to be introduced. 1'21. The choice of a method for the solution of any particular problem depends on a number of factors which should receive consideration. As between the two principal categories described above (1*16 and 1*17), the elliptic frame requires the calculation of the expressions for six variables as against the three coordinates which are alone needed by the observer. On the other hand, the solution of the differential equations is much more simple for the elliptic frame than those for the coordinates. For this reason, certain sets of equations belonging to the first category have been so developed that their solution is as simple as those for the elliptic elements. The elliptic frame as actually used requires a literal development of the disturbing forces in terms of the variables : when high accuracy is needed this development may entail very great labour. On the other hand, in particular portions of the problem, for example, in the calculation of the secular terms, those of very long period and resonance terms, it appears to give the needed results more easily than any other method which has had extensive trial. For theoretical researches, and for the discovery of qualitative properties of the motion, the elliptic frame has in general been more fruitful than most of the other forms of the equations of motion. This statement refers to motions of the general character of those which have been observed 20 EQUATIONS OF MOTION [OH. i rather than to those which are mathematically possible, and to work which has been done in the past rather than to what may be accomplished in the future. In this connection, it should be remembered that a quantitative solution for a particular set of problems is often more easily obtained by a procedure different from that which is used to deduce a qualitative result. 1*22. In making a choice for the solution of a particular problem from the various methods which have been proposed or developed, there are several factors which should receive consideration. 1. The question as to whether a literal or numerical development is to be made, that is, a development available for several cases or one which is applicable to the motion of a single body only. The choice depends, not only on the number of cases to which the solution can be applied, but on the degree of accuracy with which the initial conditions, that is, the arbitrary constants, are known. In cases where the deviations from elliptic motion are large, the literal method may involve such extensive computations that it becomes practically impossible, even if the infinite series used were sufficiently convergent to give the quantitative results needed. Sometimes a partly literal and partly numerical method can be adopted with but little extra labour. In all numerical methods, provision must be made for changes in the arbitrary constants which future obser- vations may furnish. Some details with reference to the problems of the solar system will make these statements more concrete. For the eight major planets the elements are known with considerable accuracy so that corrections to them need scarcely be considered at the present time as a factor in the choice of a method. It is impracticable to use the literal values of the ratios of their mean distances from one another owing to the numerical magnitudes of these quantities: numerical values must be adopted for these from the outset and these involve numerical values for the periods of revolution round the sun. Little is gained by the use of literal values ior their eccentricities and inclinations, and much labour is saved by using numerical values for the constant parts of the angular elements. Thus, completely numerical theories are indicated for the major planets. For the moon, the ratio of the periods of the moon and sun is the parameter along which convergence is least rapid and there is little doubt that its numerical value should be used from the outset. Literal values for all the other elements can be used with but little additional work. For the minor planets, numerical values of the ratios of the mean distances are again a necessity, but since there are groups of them in which this ratio is nearly the same it is useful to devise methods in which this ratio has a given numerical value while the other elements are left 21,22] COORDINATE SETS 21 arbitrary. For most of the satellites other than the moon, complete numerical theories are indicated. This becomes a practical necessity in the cases of the outer satellites of Jupiter where the eccentricities and in- clinations have large values, although nothing is gained by using the numerical values of the constant parts of the angles. In general, planetary problems should be separated from satellite problems. In the former convergence is slow along powers of the ratios of the mean distances, but rapid along powers of the ratio of the mass of the disturbing body to that of the primary; in the latter the case is reversed. For the planetary problems the amount of calculation needed for the terms dependent on the second and higher powers of the mass ratio is nearly always small compared with that needed for the first power, except, perhaps, in the case of the mutual perturbations of Jupiter and Saturn. The same is true in the asteroid problems, except in the difficult resonance cases, on account of the lower degree of accuracy at present demanded. 2. Consideration should be given to the amount of routine computing available. In some methods much of the work can be arranged so as to be done by routine computers, in others this is not the case. 3. The liability to errors of computation and the extent to which tests may be applied, play some part. It is rare that an extensive theory is tested throughout by others than the author, and safeguards against mis- takes should be provided as far as possible. 4. Possibilities for an extension of the work as new needs arise. 5. An examination of existing developments in order to discover the numerical magnitudes which will be involved in the work. 6. The degree of numerical accuracy aimed at. 7. The extent to which use can be made of existing numerical or literal developments and in particular of those of the disturbing func- tion. 8. The extent to which any peculiarity of the motion may dominate the whole work. In most cases this peculiarity is that of approximate resonance between two periods, as for example in the great inequality in the motions of Saturn and Jupiter, the principal librations in the Trojan group of asteroids, and so on. 22 EQUATIONS OF MOTION [CH. i D. VARIOUS FORMS OF THE EQUATIONS OF MOTION DEPENDING ON THE USE OF POLAR COORDINATES IN THE OSCULATING PLANE 1*23. Polar coordinates with the time as independent variable. The osculating plane is one containing the origin and the tangent to the orbit of the disturbed body. It is defined by the angle i which it makes with a fixed plane and the angle 6 which its line of intersection with that plane the line of nodes makes with a fixed line in the same plane. When i is less than 90, 6 is measured in the same sense as the actual motion; thus 6 refers to that node at which the body is ascending from below to above the fixed plane. In Fig. 1, let 0, S3, P be the points where the fixed line, the line of nodes and the radius vector r cut the unit sphere with centre at the origin. Let the angle S3 P be denoted by v 6. Fig. 1. The angular velocity of S3 along the fixed plane can be re- solved into the components d0 . d0 . . within and perpendicular to the osculating plane. The latter contributes a component (dO/dt) sin i cos (v 6) to the motion of P perpendicular to the osculating plane. The change of inclination contributes a velocity (di/dt) sin (v - 9) 23] POLAR COORDINATES 23 to the motion of P perpendicular to the same plane. The definition of this plane therefore gives -r-sin(v 0) -j- sinicos(u 0) = .......... (1) (it dt The velocity of P is compounded of its velocity relative to 8 and the velocity of & . It is therefore d, ^d0 . dv r d0 ( v _ m _j_ -7- cos i = -T- 1 -r: , 1 = 1 cos ^. dt ^ ' dt dt dt Hence the square of the velocity of the planet is A geometrical interpretation can be given to a variable v defined by J dv = dv Ydv. This definition makes dv/dt the angular velocity of the radius vector in the osculating plane. We can therefore regard v as an angle reckoned from a departure point 0' in that plane which is such that, as the plane moves, the locus of 0' is perpendicular to the trace of the osculating plane on the unit sphere. The function T contains four variables. It may be regarded as the kinetic energy of a system with four degrees of freedom if we suppose the osculating plane to be material and if we add a term depending on its mass and motion. Let F be the force- function of this dynamical system. We can then apply Lagrange's equations to it, with the variables r, v, 9, F, and, after forming them, put the mass of the osculating plane equal to zero*. If, in the resulting equations, we put --- dt dt~ dt y they can be written - r - di) ~^' Jt\ ~dt dt\di * The relation (1) expresses the fact that the material osculating plane is not acted on by any forces which do work. 24 EQUATIONS OF MOTION [OH. i In these equations* we substitute for F the force- function for the particular problem under consideration. If #, y, z be the rectangular coordinates with the fixed plane as that of #, y and as the trace of the #-axis, the positive part of the y-axis being 90 from reckoned in the same sense as 0, and that of the ^-axis being above the plane, we have x = r cos (v 0) cos r sin (v 6) sin cos i,\ y~r cos (v 0) sin -f r sin (v 0) cos cos i, > ---CO 2 = ?* sin (t; #) sin i = r sin Z, J which show that F is expressible in terms of r, v, } F. The definition of L shows that it is the angle between r and its pro- jection on the fixed plane or the latitude of the body above this plane. 1*24. Canonical form of the equations. If we put so that 97 ,_/rfY . <? <fr_aT ~\di) r>' dt~ dr' we can write the equations in the form .(3) or, more compactly, dr . Sr - Sr . dr + d(? . Sy - 8 G . dv * For other derivations of these equations, see E. W. Brown, "Theory of the Trojan Group of Asteroids," Trans, of Yale Obs. vol. 3 (1923), p. 9; C. A. Shook, "An Extension of Lagrange's Equations," Bull, of the Amer. Math. Soc. vol. 38 (1932), p. 135. 23,24] CANONICAL FORM 25 If, in the latter, we replace dv by dv Yd0 and QdO . SF by d0(SHi rSG), obtained by submitting (2) to a variation S, the equations take the canonical form dr . Sr - Sr . dr + dO . Sv - S G . dv We have here implicitly supposed that F has been replaced by Hi/G in F\ T is a function of r, G, r only. In general, F will be an explicit function of t as well as of the six variables r, r, G, v, HI, 6. By making use of the canonical equations (4), we obtain a result which is independent of the variables in terms of which F, T are expressed, provided that the equations defining any change of variables do not contain t explicitly. If in (5) we introduce the value of T, and integrate, we obtain dt) + ,*-** " The equation 1'23 (3) may be written <Pr_GP_ SF T dt* r 2 ~ r dr Eliminating G between this and (6) we have A further useful equation may be obtained. When F is expressed as a function of 7', v, F, 0, t y we have d F dF . dF . dF^ dF * dF -rr = ^-r + ^-i;+r-f^-0+-^-. dt dr dv ol dv dt In this equation replace v by v + F/9 and substitute for 6, F their values obtained from 1*23 (5), (6). The result is dF 3F . dF . dF /QN jT^a-r+a-v+or ................... (8) dt dr dv dt ^ ' 26 EQUATIONS OF MOTION [OH. i This equation when R replaces F, where F = p/r + J?, is evidently still true. 1*25. The equations of Encke and Newcomb. Put p d'R dR dR dR . dR . jP^-fJ?, -jr=-jT--57=-;r r ' + -5- v - r at at at or dv Equation 1*24 (7) rnay then be written When R is neglected, the motion becomes elliptic and r can be expressed as a periodic function of t (Chap. in). Let this value of r be denoted by r and the complete value by r + Sr. On expanding in powers of 8r and neglecting powers of Br beyond the first in the left-hand member, we obtain (2) Elliptic values are substituted for the coordinates in the right- hand member which then becomes a function of t. The equation can then be integrated and it gives Sr. The coordinate v is obtained from $-*-/* < 3 > where GQ is an arbitrary constant, or, neglecting terms depending on the square of the disturbing mass, from ............. (4) dt ?o 7*0 r 2 J dv ^ ' Further, ^ = ^Y r - dR -^Y H?^ *""A + d""(ft" l "dvar - d + dwar > di dt to the same order. If then VQ be the value of v in elliptic motion, and VQ + # its complete value, we have, to the first order of the disturbing mass, d , 20 * 1 [dR j 4 , FdR T $ VS --. " S r -- ^ + .......... /5) d^ r 8 r 2 J dv V Q 9F v ^ 24-26] NEWCOMB'S SOLUTION 27 Since dF/d0, dF/dT contain the disturbing mass as a factor, the equations 1*23 (5), (6) are immediately integrable if we put r*dv/dt*=* Go in the latter. 1*26. Newcomb solves the equation for Br in the following manner. He notices that when /2=0, the solution of the equation for r contains two arbitrary constants, e the eccentricity and w the longitude of perihelion (see 3'2 (7)), in addition to the arbitrary constant already present in the equation, a constant which is independent of e, w. Two particular solutions of 1'25 (2) for 5r with R~Q are obtained by varying e, 07. These two solutions may be written He then makes use of a well-known method in the solution of a linear differential equation of the second order, namely, that if yy\, yy'L are two particular solutions of the equation a particular solution of the equation is Cy =y 2 / Qy l dt - ^ I Qy 2 dt, where y 2 -~ -y l -J^ 2 (7, a constant. In these formulae, I\ ^ are supposed to be known functions of t. If X be the eccentric anomaly, we have from 3'2 (16), 3*2 (15), when Xis expressed as a function of n, e, e> w, t, dX __ sin X __ a sin X de ~~ 1 - cos X ~~ r Thence, by differentiation of 3*2 (15), we obtain Thus cos X e, sin X can be taken as the two particular solutions. They give Ctt, the mean motion. In the exposition of the application of these formulae to the theories of the four inner planets (Amer. Eph. Papers, vol. 3, pt. 5), Newcomb apparently puts v = v, for he makes no mention of any difference between them. The difference between them, -Jrdtf, which to the first order of the disturbing mass may be written 2 sin 2 1 80, is very small because the 28 EQUATIONS OF MOTION [OH. i inclinations of the orbits of these planets to the ecliptic are small. The constant and secular parts of this term are absorbed in the constants of the mean longitude, so that the only doubt which remains is whether the term gives rise to any sensible periodic terms, and if it does, whether these have been included in his final results. 1*27. Equations of motion with the true orbital longitude as independent variable. These equations are deduced from those of 1*23 by making v the independent variable instead of t. The transformation is effected by the introduction of new dependent variables u, q, defined by the equations With these definitions we have dr _ dr dv _ ~ du ~~~ m '~ ' d z r d*u du dG _ d*u 1 dO du ~~~ " dG d, *K_^ O d dv dt\ dtj dv^ } If then we put ==/*(> + 72), .................. (3) ^ 4- 2 so that TS- = MW ur -^- , dr du and if we change from G to q by means of (2), the equations 1-23 (3), (4), (5), (6) are transformed to d*u dR , dqdu do 2 dR //f . /cx -1-5 + ft ? = ? o + 1 ? ;r :j- -T-- -- 2o~'-- (&)> ( 5 ) dv 2 * * du * * dv dv dv u 2 dv ^ " \ ' TT-T X ^r =1 - r ;r> ( 6 >'( 7 ) av 2 \/A' av av \ /> \ / ) = "^W' dv = 7 2 3T (8)>(9) 26-28] IN TERMS OF THE LONGITUDE 29 It is to be noticed that the disturbing function has been denoted by fiR instead of by R, so that the mass factor present in the new R is the ratio of the disturbing mass to the mass of the sun. When we proceed by continued approximation as in Newcomb's method, the equation for u is immediately solved when that for q has been inte- grated, and it has the advantage of being one with constant coefficients, so that the device shown in 1*26, requiring two multiplications of series, is not needed. When 72 =0 we have const., and the solution of the equation for u is (3-2) e cos (v or)}, \jq-a (1 -e 2 ). Since dq/dv has the disturbing mass as a factor, we can change the variable u to MI where n = Uif(q) without losing the easy integrability of the equation for HI . The special cases have certain advantages which will be pointed out in 7*2. It may be noticed also that when the terms in R containing the angle 6 are neglected, we have Tq ~ ^ = const., so that r can be completely eliminated at the outset. The variables l/<?, \ju have the dimension of a length. If we introduce the constants w , such that n 2 a 3 =fi, and put dt\ for n^dt, Ui for ua Q , qi for qa ot RI for a Q R, the constants /LI, , n will disappear from the equations and the variables are all ratios. 1'28. Latitude equation. From the equations 1*23 (1), (2), namely, di sin (v 6) = d6 sin i cos (v 6), dv = dv (1 cos i) d0, we easily deduce d {sin i sin (v 6)} = sin i cos (v 0) dv, , f . , mi . . x /ix 7 sin i cos i 7/ , a sin i cos (v 0)1 = sin i sin (y a) av + -; ; --- -rr d^. 1 /j 7 sm(v 9) Whence, with the help of 1-27 (9), d 2 -, \ - - / ^ sin i cos i d0 T-a + 1 ) sin sin (v - tf ) = -77 - -^ -T~ dv 2 / v ' sm (v - ^) dv _ sin i cos i q dR _ m " = " >( } 30 EQUATIONS OF MOTION [CH. i It will be seen, by differentiation of 4*1 (1) with the help of the definitions of A, S, F, that dR/dT contains sin (t; 0) as a factor, so that there is no discontinuity in Z when v is a multiple of TT. In the section referred to, R is shown to be a function of r, r', cos S and dR . / /\\ / /\\ vR ^. _ sin (-(?) an (-<?)g g . If L be the latitude of the disturbed planet above the plane of reference (1*23), equation (1) may be written nZ = , .................. (2) so that sin L is obtained from an equation of the same type as that for u. When Z has been expanded, we obtain v by integrating dv -. r, - - / /K -7- = 1 Z sm i cos i sm (v u). dv / 1*29. The Equations of motion with the disturbed Eccentric Anomaly as independent variable. Another variable which gives a linear form to the equation for r is X as defined by r dX pdt, p = const. This variable gives dr _ dr d / dr\ _ p a d 2 r T dt~ P dX' dt\'dt)'~~rdX*' Equation 1'24 (7) with F^p/r + R therefore gives - dt .......... (1) r dX* r dr J dt ^ ' ^ = -2^+2^ (2) dt\a) dt* dt ................... v ' Define a by the equation dt\a Then equation T24 (6) may be written $'+"(*)'-*: ; which shows that 2a is the disturbed major axis. 28-30] RATIAL COORDINATES 31 From (1), (2) we deduce 7TT9 a I * I 9 n V^V dX 2 p 2 \a / p 2 dr v ' If a be an arbitrary constant, the integral of (2) may be written 11 E> f) r 3 Z? 1 ^ Jtt < / f>/t 1Tr /^ - = -- 2- + Ir^dX, (o) and, with the aid of this equation, (4) becomes d 2 r LL , . r 2 dR u (6) 71* (t N "' / *"'* '*'*' '*''* l '* ft i ^ / The transformation of the remaining equations to the variable X as independent variable is effected immediately. If ?i , p be defined by the equations /i/? 2 = a , /i=n 2 a 3 , the definition of X gives r dXa^n^dt. A reference to Chap, in shows that in undisturbed motion, X is the eccentric anomaly. If the equation (6) be solved by the method outlined in 1 *26, it will be seen that the solution is closely analogous to that of Newcomb, when we change the variable from t to X under the integral sign. It has, however, the advantage of being exact instead of approximate and is thus adaptable to the calculation of the higher approximations. These equations, which appear to be new, will not be developed further in this volume. The general method of treatment would follow lines similar to those adopted when the true longitude is taken as the independent variable (Chap. vn). It may, however, be noticed that, since in elliptic motion, the equations are integrated without multiplications of series when R, rdlt/dr have been expressed in terms of X. The only exception is the equation for v an exception common to all methods. 1*30. Equations of motion referred to the coordinates of the disturbing planet. T j. i ndv' , t dv , /1X Put r = rp, r^--A, P^,=h f , ............ (1) so that **S = AVlr = A'Ap ................... (2) (At cttV A. direct transformation from the variables r, t to p, v' gives d?r /dv\ a _ /t' 2 f fflp /dv\ 2 ) 1 dh' dp d?r' ~ ~ ~ ~ d?r /dv\ a _ /t' 2 f fflp /dv\ 2 ) dt* ~ r \dt) ~ 7 left/ 2 ~ p \dv') } 32 EQUATIONS OF MOTION [OH. i This equation, with the aid of the definitions F, dF 1 W, enables us to transform 1*23 (3) to c77*~ p w/ "F* J + p h'*~dt* ~/7 ~di d7"" W With the definitions (1) of h p ,h', equation 1*23 (4) gives A , dh<, h dv = ajp rtt p dt dv ' Replacing o? by r'*dv'/h' and 7^ by /\/r', we obtain p _ * /. / K \ dt/ /^ /2 Sw A' dt p ^ ; Now let r',v' be the polar coordinates of a disturbing planet moving in the plane of reference and satisfying the equations The elimination ofd?r'/dt 2 , dh'/dt from (4), (5) by means of (6), (7) gives dtp (dv\* _ _/ 3i / r^ dF'\ _r^d/^dp dv'* P \dv') ~h'* dp P \ k' 2 dr') h' dt dv' dp ' 2 J 2 V A' 2 dr' h' dt dv' ' ...... (8) __ dv'~h'*dv h' a/ ............................ The transformation of the equations for F, 9 is easily made. When the disturbing planet moves in an ellipse, we have (Chap, in) / 1 aP' af so that the equations become dtp t dv _ SF 2 ~ ~ dp' 30, 31] RATIAL COORDINATES 33 u o /i + JpVcosfv'-w') , , x where F 2 = \ % ^ -', (11) 1 +e cos(i/ w') v ' equations which have a form similar to that which obtains when t is the independent variable. When the motion of the disturbing planet is no longer elliptic, we put The additional terms in (10) are easily written down : they contain the derivatives dR'/dr', dR'/dv'. The chief point of interest in this form of the equations of motion arises from the fact that F l is independent of /, t explicitly. That this is the case is seen from 1*10 when we put ^ = y, Vj = v', 0, = 0, ^ = r, H? + i, = /i, m k =m'. For then F l =/V = ^ + m' \ -------- - - . -- p cos S\ , P l(H-// 2 -2pcostf)* J where cos 8= (1 - -ir) cos (v- t>') + i r cos (v+v 1 - 20). Thus the only way in which the eccentricity of the disturbing planet appears in the equations of motion is in the explicit form shown in F% . It should be pointed out, however, that if we put m' so that 7^ is re- duced to the term ra/p, the solution of the equations for p, v will contain e' ; hence, in the second approximation, when we substitute these values in the coefficient of m', the development of F l will contain e'. Nevertheless the equations appear to have possibilities for usefulness in the discussion of certain problems in which it is necessary to take into account the per- turbations of the disturbing planet by another planet < the indirect effect' of the latter, that is, the effect transmitted through the disturbing planet. A case of this kind is the effect of the action of Saturn on Jupiter where the latter is disturbing the motion of an asteroid of the Trojan group. Another case is the indirect effect of a planet on the motion of the moon. 1*31. The most important of the various forms of the equations of motion, namely, that referred to the * elliptic frame, 5 will be developed in Chap. v. In this form the two variables r, v are replaced by four new variables which are so chosen that their -first derivatives only appear in the equations of motion. The elliptic frame in various forms has been extensively used for the calculation of perturbations. In its direct form it was used by Leverrier for obtaining the orbits of the major planets and in the 'canonical' form by Delaunay for the motion of the moon. In a different form it was used by Hansen for both the planets and the moon. B&SPT * 34 EQUATIONS OF MOTION [CH. i Another group of methods depends on the use of a uniformly rotating frame of reference. This, like most of the methods which have been used for actual calculation, was initiated by Euler and adopted by G. W. Hill and E. W. Brown for the development of the motion of the moon as dis- turbed by the sun. For these and other methods not treated here, the reader is referred to standard treatises like those of Tisserand and Brown. References up to the date of their publication will be found in the articles dealing with celestial mechanics in the Ency. Math. Wiss. 1*32. Motion referred to an arbitrary plane of reference. In the preceding developments the plane of reference has been that of the motion of the disturbing planet. Now let the symbols v, 9, i refer to any fixed arbitrary reference plane and let v', & ', i' have the corresponding signi- fications for the disturbing planet; the rectangular coordinates of the disturbed planet are then given by 1*23(7) and similar formulae will hold for the disturbing planet. We have rr r cos S xx' 4- yy' + zz' . Put i = V 1 so that xx 4- yy' is the real part of (x 4 ty) (x iy'). Define T, k, I", V by F = 1 cos i, 2k = 1 4- cos i, F' = 1 cos i', 2k' = 1 4- cos i'. We can then obtain from 1*23(7), and similarly The product contains v, v' only in the combinations v 4 v'. If we form it and, after taking the real part, separate the coefficients of the sines and cosines of these two angles we obtain cos (v - v') {W 4 i IT' cos (2(9 - 20')} 4- sin (v - v 1 ) {JIT' sin (2(9 - 2(9')} 4- cos (v + v') $kT' cos 26' 4- J A?T cos 20} 4- sin (v 4- v') $ kF sin 20' + J&T sin 20}. 31, 32] ARBITRARY REFERENCE PLANE 35 To find cos S we must add zz' jrr to this. The latter can be written sin i sin i'fcos (v v') cos (6 6') + sin (v fl') sin (0 0') - cos (v + v') cos (0 + 6') - sin (v + v') sin (0 + 0')}- We can therefore express cos S in the form cos 8 = K Q cos (v v' KI) + ^2 cos (v + 1/ jBT 8 )> where K = 1, ^ = /i a = ^3 = when i = i' = 0. Since the K's are functions of 0, 0', i, i', only, cos S has the same form as in the simpler case. Here KQ , -Or 1ST K i , K 2 , /V3 replace 1-^r, W--OT', |T, 20, used in the developments of the later chapters. In these and many similar cases we replace A cos a -h B sin a by (7 cos (a a )> where (7, are determined from (7cosoo = J., (7 sin <)=-#> (7 being in general so taken as to have a positive sign. 3-2 CHAPTER II METHODS FOR THE EXPANSION OF A FUNCTION 2*1. The greater part of the work of solving any problem in celestial mechanics consists of the expansions of various functions into sums of periodic terms, mainly because the integrals of these functions cannot be obtained conveniently in any other way. The majority of these methods, which depend chiefly on Taylor's expansion and Fourier's theorem, are well known, but there are certain expansions, continually recurring, which require much labour. It is the purpose of this chapter to ease the work, partly by giving formulae which are ready for immediate appli- cation, and partly by so arranging them that the calculations may be carried out with the least chance of error. Certain of the formulae are intended to be used only when literal expansions are required : when the coefficients are numerical the methods of har- monic analysis usually give higher accuracy and are less laborious. The coefficient of a periodic term in the expansions of most of the functions considered here takes the form a l '(a + i 2 H-a 2 4 -f ...), (1)' where a is a parameter and a , i, a*, ... are integers or fractions. It is frequently required to calculate the function as far as some definite power of , and to carry one or two coefficients con- siderably further. It is this latter need which causes difficulty because there is much wasted labour if the whole series be carried to this higher power. This fact has to be remembered when a choice of any method of expansion is made. Expansions in power series are so much easier to perform and are so much less subject to error than operations with series of periodic functions, that the latter are usually reduced to the former by the substitutions x = exp. 6 V^T, 2 cos iff = x i -t- or*, 2 V 1 sin iff = x i or*. (2) 1,2] LAGRANGE'S THEOREM 37 When the coefficient of cos id or sin id has the form (1) the work is made easier by the substitutions . 6 V 1, p = a 2 , 1 -\ 2a* v ^Tsin iff = z i -p*z-* t ) We then expand in positive powers of p and in positive and negative powers of 2. This simple change from the substitution (2) not only gives greater freedom in the choice of methods of expansion, but aids materially in solving the problem referred to on the preceding page. Extensive use is made of another device, namely, that of expansions of functions of an operator. These usually take the form <f>(D).f(x) where D d/dx. It is then always supposed that <f> ( D) is developable in the form where o, i, a 2 ,... are independent of x, so that Operations with functions of D are performed in accordance with the rules of ordinary algebra except that functions of D and those of x do not follow the commutative law of multiplication. The gain is partly in brevity of expression and partly in the methods of expansion which are suggested by well-known ex- pansions. Thus we can use such forms as exp. Z), log(l + />), (1 4- D) n , (1 + a) D , cos J9, etc., each acting 2'2. Lagrange's theorem for the expansion of a function defined by an implicit equation. Let the equation be y = a? +a(y) = & + <, .................. (1) where a is a parameter and and its derivatives are continuous functions of y. The problem \A the expansion of F(y) = jPin powers of a with coefficients which are functions of a?. The theorem gives 2 d <> 38 METHODS FOR EXPANSION OF A FUNCTION [CH. n where F x = F(x) 9 (f> x =<f> (x). The proof which follows indicates an extension of the theorem to several variables. We may regard F as a function of a, x. Regarded as a function of a, it may be expanded in powers of a by Maclaurin's theorem in the form F=F Q + a(AF) Q +~ ] (A*F) Q + ... y A=~ t ...... (3) where the zero suffix denotes that a is put equal to zero after the derivatives have been formed. Evidently F Q = F X . Put D = d/dx. The use of the operators A, D implies that the functions on which they operate are expressed as functions of #, . Operating on (1) with A, D } successively, we have Ay = 4> + a^ Aij, Dy=I+a d fDy, J ^ dy dy J so that Ay = <f) Dy, and therefore, for any function y of ?/, From this result we can show, by induction, that A n F=D n ~ 1 ((f) n DF) ................... (5) Assume that (5) is true and operate on it with A. Then, since x, a are independent so that D, A are commutative, A n+1 F-= D"- 1 {DF. A<f> n + ^A (DF)} = D*- 1 {DF. <f>D<f> n 4- <f> n D (AF)}, the change in the first term being made by the use of (4) and in the second by the commutation of A, D. But, by putting g = F in (4) and operating with D we have D(AF) = D (QDF), so that the portion under the operator D 11 " 1 is D (<f>DF . <f> n ). Hence and since the theorem is true for n = 1, it holds universally. 2, 3] EXTENSION OF LAGRANGE'S THEOREM 39 Finally, since <, F become <f> x , F x when a = 0, the coefficient of a n in (3) becomes the same as that in (2), and the theorem is proved. Particular case. When F(y) = y, we have yy2 ,/ $ ,72 2*3. Extension of Lagranges theorem. If yi = Xi + aai<l>(yi 9 yz t y 3 ) = a;i + aai<l>, i = l, 2, 3, ...(1) where the a$ are constants and a is a parameter, and if where F and <f> are continuous functions with continuous deriva- tives of yi, 2/2> y?.> and further, if F x = F(XI, o; 2j ar 8 ), then 2/2, yd^ x ) ....... (2) The proof follows the same general lines as before. We first regard F as a function of a and expand in the form 2*2 (3). Next, by differentiation of (1), where the first term of the right-hand member is 1 or according as i =j or i 4= j. This equation is multiplied by a,- and summed for j = 1, 2, 3. The result is But from (1) we have 2JL. 2*,. (5) 40 METHODS FOR EXPANSION OF A FUNCTION [CH. n These two sets of equations may be regarded as linear, the first set for the determination of J9y t -, and the second for that of dyi/da. They are the same except that the absolute terms in the latter are </> times those in the former. Hence and therefore ^ = <j>DF . The remainder of the proof is the same as that in 2*2. 2*4. The most general case in which Hi = -n + Q/i , y* , #0, 1 = 1,2, 3, does not seem to be soluble by a simple general formula. It is not difficult, however, to obtain the solution as far as a 2 . If we put <tx = </>t (#1 , *2 1 # 3)1 D == 2 t <j> lx a ^ , /^ x we obtain Lagrange's theorem can be used to find X or any function of X in terms of g from Kepler's equation X=*g e&\i\X (3'2 (16)), and was probably suggested by this problem. The extension may be applied to the Jacobian solution of the canonical equations (Chap, v) to find the new variables in terms of the old or vice versa, when the disturbing function is confined to a single periodic term or to a Fourier set of terms. The more general case mentioned above is that of the Jacobian solution where the disturbing function contains any periodic terms. 2'5. Transformation of a Fourier expansion with argument y into one with argument #, where y is defined in terms of x by means of an implicit equation*. Let F(y} be expanded in the form F(y) ^(c i Gosiy-{-disiniy) > i = 0,1,2, ...... (1) and let y x -f a cf> (y\ ........................ (2) where a is a parameter and </> (y) is expressed in the same form * E. W. Brown, Proc. Nat. Acad. Sc. Wash. vol. 16 (1930), p. 150. 3-5] IMPLICIT FUNCTIONS 41 as F (y). It is required that we obtain the coefficients a if b it when F(y) is expressed in the form F=^(ai cos ix -h bi sin ix) ................ (3) With the help of the notation, </> = (x), F=F O), D = d/dx, nnd with the use of Lagrange's theorem, (2) gives F(y)=F+'S, an ] D n - 1 (^ n DF) t n = l, 2, ....... (4) Let -v/r be another function of # expressed as a Fourier series. On multiplying both members of (4) by D-v/r, we obtain ). ...(5) The identity, 2 (<f> n DF), shows that, when all three terms are expressed as Fourier series, the constant term of the left-hand member is the same as that of the last term, the remaining term being the derivative of a Fourier series. By repeating this process n 2 times, we deduce the fact that the constant terms in the Fourier expansions of D^r . D"- 1 (4> tl DF), (- I) n ~ l l) n ^ . <f> n DF lire the same. On applying this result to each term of the right- hand member of (5) we obtain a series which is the expansion, by Taylor's theorem, of -DF.^(x-a<f)) ..................... (6) in powers ofa<p. Hence, the constant term in the Fourier expansion of (7) when F(y} is expressed in terms of x, is the same as the constant term in the Fourier expansion of -~F(x). + {x-a$(x)}, ............... (8) or in that of F (a). -]-$ [x a $ (so)} ................ (9) ax 42 METHODS FOR EXPANSION OF A FUNCTION [OH. n It is to be remembered that ty [x a <(#)} means the result obtained by replacing x by x a<j>(x) in ty(x). We obtain (9) from (8) by noticing that their difference is the derivative of the product of the two functions F(x), ^r {x a<f> (x)}. The use of (9) enables us to state the theorem in a slightly different form. If we put % (x) for d^r (x)/dx, we have the theorem : The constant term in the Fourier expansion of where F(y) is expressed in terms of x, is the same as the constant term in the Fourier expansion of , z=x-a$(x) .......... (11) According to this definition, % contains no constant term. But if ^ is a constant, (4) shows that the constant term in F(y) is the same as that in F + a(f>.DF, which is the same as that in F(l aD<f>), so that the theorem still holds when ^ (x) contains a constant term. Since we are concerned only with the constant terms in (8)> (9) or (11), the theorem evidently holds if we replace the letter x by the letter y in these three formulae. The chief value of this theorem lies in the fact that it removes the necessity for solving the implicit equation y = x -f a<f>(y)-m order to get y in terms of x. The application to the coefficients in (3) is immediate. If we put i\fr = sin ix, so that Dty = cos ix, and note that the constant term in the product of D-fr by the right-hand member of (3) is Ja t -, we find from (8) the result, a t - = constant term in r sin {ix za</> (x)} j- F (x). i ctx ...... (12) Similarly, by taking Dty = sin ix, we obtain bi = constant term in - cos {ix ia < (x)} -y F (x\ . . .(13) i dx and from (9) Oo= constant term in \ 1 a -7- <f> (x) > F (x). . . .(14) I ax ) 5, 6] IMPLICIT FUNCTIONS 43 When y, x take the values 0, TT together, these last results may be deduced from a change of variable in the Fourier in- tegral. 2'6. Extension to two variables. Suppose that we have a second pair of variables x', y' , independent of #, y, connected by the implicit relation y* = x' + a'<f> (y f ) and that we desire to obtain the expansion of F(y, y') in the form 2,i' {(%' cos (ioc + i'x') + ba> sin (ix + t'V){, ...... (1) where i, i' are positive and negative integers. An expansion in this form will be called a double Fourier series. For the development we adopt a notation similar to that used before, namely, F= F (x, x'), J} = d/dx, D' = a/a*/, etc. A double application of Lagrange's theorem gives where the signification when n = or m= is the same as that shown by 2'5 (4). This is multiplied by DD'ty(x, x'}, where ty is a double Fourier series and the process adopted in 2'5 is then fpllowed for each of the variables x, x' . It evidently leads to similar results which can be stated in the following theorems. The constant term in the doable Fourier expansion of F (y ' y when F (y, y') is expressed in terms of x, x' y is the same as the constant term in the double Fourier expansion of -a^( ai ), x' -')}, ...... (2) or in that of ,^{x-a^(x), *'-')) ....... (3) And the constant term in the double Fourier expansion of F(y,y') X (x,x') ..................... (4) 4 METHODS FOR EXPANSION OF A FUNCTION [OH. n is the same as that in F(x, x') % (z, z') -r- . , , , z = x a<f)(x), z' = x' a'<f> (#'). ...... (5) It will be noticed that the double operation removes the negative sign present in 2*5 (8). It reappears, however, if we replace (5) by --F(x,x'}~^{x-a<}>(x\ *' ->' OO}, -(6) or by the similar formula in which the derivatives d/dx, d/dx' are interchanged. In order to apply the theorem to the coefficients in (1), we put DD'ty equal to cos (ix -f i'x'), sin (ix + i'x'), successively in (2). We obtain " ' = const, term in . {ix + i'x' ia </> (x) i'a'<$> (x'}\ sin which hold only when i, i' are both different from zero. When i = Q, we use (5) with % (x,x') equal to cos i'x , sin i'x', successively, and obtain ? 0t " = const, term in "*" C S {t V - i'cftf (x')\ ( r \ /j + sin or n + COS Jl -a 1 </>(^)| |l -a' ^*'(^)} ^(*X), ...... (8) V - i'a'tf (x')} \l-a~<f> (x)\ ? , a , ^ (^, ^). r \ /) ^ dx^ ^ ) l tix ^ ' ...... (9) The formulae for a t - , b iQ are similar. When i = i' = 0, we use (5) with ^ (x, x 1 ) = 1, and obtain aoo = const, term in 41 -a <j> (x)\ \1 -a' , <f>' (x')l F (x, x'). ^ ox J I c?<^/ J 6-8] SYMBOLIC OPERATORS 45 27. Expansion by symbolic operators. If p D is expansible in positive integral powers of D and iff (x) is expansible in integral powers of x, then f(poc) = p D f(x\ where D = x -j- = - r --- . j vr / r j \ /> ^ x ^ j g x If m be a positive integer, we have J)m (0n) _ n m^,n < Hence, if p D = a 4- aiD 4- a 2 D 2 4- ... , we have p D oc n (px) n . If then /(a?) = S n ( b n x n ), n = 0, 1 , . . . , we have f(px) = 2 n b n (px) n =p D ^ n b n x n =p D f(x). This theorem can evidently be extended to any number of variables. Thus /(Piffi,j02ff2. ) ss pi Dl .p* D * f(xi,X2, -..) where A = ^3/9^. The application of the theorem depends on the possibility of expanding p D in powers of D in such a manner that we can, by stopping at some definite power, secure a given degree of accuracy. This happens when p has the form (1 4- ey) fc , where e is a small parameter such that ey is less than unity. We can then use the binomial theorem and obtain p D = (1 + eyj* - 1 + ey . kD + kD(kD- 1) + .... 1 . i We can also make use of the expansion of Q Z in powers of z if we put z = D log p, for then and logp has the factor e. 2*8. Product of two Fourier series. (i) Let the series be A = a 4- 22a;a* cos {0, .B = 6 + SS^a* cos id, Put _ z a exp. \l 1, p = a 2 , 2a* cos 16 = z* 4- p^" 1 ', 5 = &<) 4- 26* 2* ............. CO 46 METHODS FOR EXPANSION OF A FUNCTION [CH. n Then AB = (A + '2a i p i z-*)(Bo + 2,b i p i z- i ) .......... (2) Since A, B are even functions of 0, their product will have the same property and the coefficient of z* will be the same as that of p i z"~ i . It is therefore sufficient to find the coefficient of z*, i > 0, and then to replace z i by 2a* cos iff. We thus reject all negative powers of z in (2) and it is therefore necessary only to find the coefficient of z i in A Q B Q + AoZbip'z-* + The first term is the product of the power series (1) and from this product we select the coefficient of z*. In the second term we select the coefficient of piz* t namely, a i+i b i} and sum for j = 1, 2, ... ; the third term is treated in a similar manner. On performing these operations and replacing p by a 2 , we find for the coefficient of z*, that is, of 2a* cos iff -_i -f . . . + a^) + 2 + 4 (2&<+2 + a+2&2> 4- . . . , ...... (3) and for the constant term, <*o&o + 2a a ai& 1 +2d*tf a &2 + ................... (4) The parameter a may not be present and we then put a = 1. In the series with which we have to deal it is usually present implicitly, if not explicitly, so that the order of magnitude of any coefficient is denoted by its suffix, and in the product by the sum of the suffixes of a, b. Thus the arrangement in (3) is made as needed, namely, with respect to the orders of the terms, the brackets giving successively the terms of orders i y i -f 2, i -f 4, . . . . (ii) If A' = 22a,a* sin iff, B' = 226<a< sin iff, we adopt the same substitution for 0, a and we have 2 V^l a* sin iff ^z'-p* z~*. The product is an even function of ff and is therefore expressed in terms of cosines of iff. We have only to find the coefficient of z i in z-* -f 8, 9] PRODUCTS OF FOURIER SERIES 47 in which do = &o 0. Hence we obtain, for the coefficient of 2a* cos id, ......... (3) and, for the constant term, (6) (iii) If A' = 22a<a* sin {0, # = 6 + 22M* cos i0, the product is an odd function of 0. Here we need only the coefficient of z l -r- V 1 in and we obtain, for the coefficient of 2a* sin {0, since = 0, (ai6 t -_i + 2 ^-2 + . . . + a^o) + 2 (- i^+i -H e&f+i&i) -f a 4 (-a 2 6^ 2 4-a^ 2 &2), ...(7) there being no constant term. 2'9. Fourier expansion of a series expressed in powers of cosines or sines. (i) Let ' G ^Sa.a'cos'fl = 26, (2 cos 6)\ i = 0, 1, 2, ....... (1) Since this is an even function of 6 it can be expressed in terms of cosines of multiples of 6. With 2=aexp. 0V 1, so that 2a cos 6 z -f a 2 /z, we have on expansion by the binomial theorem. As in 2*8 we need to find only the coefficient of z*, i > 0, and to replace z i by 2a* cos iff. The selection gives for the coefficient of 2a* cos id in (7, 2i + (* + 2* ^ 2 " 2 -^ 1.2.3 48 METHODS FOR EXPANSION OF A FUNCTION [OH. n The constant term in C is obtained by putting i = in this formula. The result shows that the numerical work will be simplified by the use of ^ = a i a < /2 < . The coefficient of 2 cos id is then ___^ ____ <+ (ii) Let S = Sa^a* sin 1 ' = 26, (2 sin 0)<, i = 0, 1, 2, . . . . The terms in 8 with odd values of i will produce sines of odd multiples of and those with even values of i, cosines of even multiples of 0. With the same substitution as before, we have By a similar procedure, we obtain, for the coefficient of 2 sin (2i + 1) 0, (2^4-5) (2i + 4) 2i+ 3 H J g 2i+5 - 6 2t+7 +...!> ----- r 2 3 and, for the coefficient of 2 cos 2i0, --- T2T3 ---- ...... (5) the constant term being 4.3, 6.5.4, 9, 10] FUNCTION OF A FOURIER SERIES 49 2*10. Expansion of a function of a Fourier series. (i) It is desired to obtain the expansion of cos 20 +...), ......... (1) in the form 60 + 2&! cos 6 + 26 2 a 2 cos 2(9 4- . . . , the assumption being made that /(a -l-#) is expansible in powers of x. Evidently f is an even function of 0. As before, we put z a exp. V 1, a 2 = p, so that 2a* cos id = z i -f p*z~*, and recall that it is sufficient to find the coefficients of z i and then to replace z i by 2a* cos iff for i > 0. Put A = mz + a 2 z 2 + ..., -B = ai^ + a 2 ^ + > so that (1) becomes /(oo -h ^4. 4- i?). This function will be first expanded in powers of p by Taylor's theorem. We obtain the suffix denoting that p is put equal to zero. Put /i and denote derivatives of/i with respect to a Q by accents. Then since p is present in f only through B, \d P Hence, replacing p by a 2 , we obtain a 2 "* /(a c + ^+-B)=/ 1 + -i/i'4 + 5 </' It will be noticed that all negative powers of z are shown in explicit form in this expression, and that their coefficients are B&SPT 4 50 METHODS FOR EXPANSION OF A FUNCTION [OH. n positive powers of z\ thus the lowest power of z required in the second, third, ... coefficients are the first, second, .... We plan, however, to stop at some definite power of a ; suppose this power be the seventh. Then since z i has the factor *, we shall need the following expansions: coef. of o?jz, from z to 6 ; coef. of a 4 /2 2 , from z 2 to z 5 ; coef. of a 6 /^ 3 , from z 3 to 4 ; and these are all that are needed. The work of expansion is thus reduced to operations with positive power series. We next expand fj and its derivatives in powers of A, which contains a as a factor. As far as a 7 , the result is, with/ set for/(a ), ' , ! a2 (A f" , ^ 2 /'" , 7\ ^ 2V + '" (2a 2 / " + 0!%'") + . + 5 * (2a/o vl - 4 (6a s /o" + Oaxa./o'" + ^y o lv ) 4- . . . The final step requires the expansions of powers of A in powers of z, the results being required to z 1 in the first line, to z 6 in the second line, and so on. The highest powers are easily formed. We have, in fact, A 1 = ajV, A 9 = ai 6 * 6 + 6ai 8 a a * 7 , .... The lower powers are conveniently obtained from the binomial theorem by treating aiz + ctzz 2 as the first element and the rest of the series as the second element. Thus A* = (aiz 4- aiz*)' +j (az + a^z 2 y~ will serve for j ^ 3. For j = 2, we have ^2 = aj 2^2 ^ 2a 1 a 2 2 3 + (a 2 2 + 2a!a -f- (a 3 2 + 2a!a 5 4- 2a 2 a 4 ) ^ 6 + (2aia 6 -h 10] FUNCTION OF A FOURIER SERIES 51 After the insertion of these results, the rejection of negative powers of z, and the replacement of z* by 2a i cosiO t we shall have the needed expansion. An expansion carried out in this manner should, in general, be used only when literal series are desired. If the coefficients have numerical values, the methods of harmonic analysis lead much more easily and directly to the required series. The same remark applies to the function of a sine series, expanded in the next paragraph. (ii) The expansion of f(2ai sin + 2a a sin 2(9 + . . .). With the same substitutions as before this becomes /(-Vin"4+V^~I/0, so that the results of (i) are immediately applicable. In general, the resulting series will contain both sines and cosines, but as the important applications are confined to those cases in which f is either an even function of 0, in which case we shall have cosines only, or an odd function of 0, in which we shall have sines only, these two applications alone will be considered. The expansion (2) can be utilised if in it (a) di be replaced by t - V 1 where it occurs explicitly ; (/3) A be replaced by - A V- 1 ; /odd. We thus obtain, after simplifying : (a) When f is an even function of 0, the expansion is A 2 4 4 A 6 / " n ** * + * 2 ^2a 2 / "+^< 3! + ^(63/." -</-) + ^ A \ ^^1 ^"Z./ U 1 ...(**) 4-2 52 METHODS FOR EXPANSION OF A FUNCTION [CH. n After the expansion in powers of A as in (i) and the rejection of negative powers of z y we replace z i by "la 1 cos id. (b) When / is an odd function of 6, the expansion is the following expression divided by V 1 : A3 A5 A7 4/o'-~/o"' + |\/o*-/v - A 6a 1 a 2 / '" + (6 ~ i2o ~ - 3o !o .(5) After following the same processes as before we replace z { by V^l . 2a J sin id. 2'11. Fourier expansions o//( + 2ai cos ^), /(a + 2a 5 sin 0). When a 2 ,a 8 , ... are zero, a method dependent on the series ._ . . 1 f, a; 1 which is closely allied to a Bessel function, may be conveniently adopted. Taylor's theorem may be written /(tt + x ) "= ex P- ^ ./(o), ^ = 3/3a . Hence /(oo -f ttia? -f a-!^" 1 ) = exp. a^xD . exp. a^x~ l D ./(a ). . . .(2) Expand each of the exponentials in powers of x, taking the power i -f k in the first and the power k in the second. The product will give the terms containing x i if we sum for k. It is 10-12] FUNCTION OF A FOURIER SERIES 53 The similar expression for the coefficient of x~* is evidently found by interchanging i -f k, i, that is, by interchanging cij , a_j. Hence /(o + aix + a__ix~ l ) = 2,- {(&!#)* 4- (a-iflT 1 )*} xQdoia-^D'.ffa). ...(3) In any application, the operator IPQifaia-iD*) must be used in the expanded form (1). For the first Fourier expansion put tf-i a i > ~ ex p ^ 1 > &'* 4 &'~~ l = 2 cos id. The expansion (3) then gives /(oo + 2cn cos (9) = 22; !'*& (a! 2 /) 2 ) ./(a ) cos i0, . ..(4) where i = 0, 1,2, ..., the factor 2 being omitted when i = 0. For the second Fourier expansion, replace ai, c&i by a_i V 1, with the same substitute for x. Since x* x~ l = 2 V 1 sin id, we get cosines for even values of i and sines for odd values, and + 22i (- I)'' ...... (5) with i=0, 1, 2, ..., the factor 2 in the first series being omitted when i = 0. It may be pointed out that 0'Qi(-^) = /i(2*), where /; is a Bessel function (2*14), so that the operators may be expressed by these functions. But since the expression in this form involves the presence of imaginaries, it is simpler to use the functions Q{. 212. Expansion of a power of a Fourier series. For (oo + 2ai cos d 4- 2&2a 2 cos 20 + . . ,)>' we make use of 210 (3) with 54 METHODS FOR EXPANSION OF A FUNCTION [OH. n It is usually found convenient to take out the factor (i Q J, so that, in the operations, 6/ = 1 and/o,/o' a **e integers when j is an integer. For (2a! *an + 2a 2 a 2 sin 20 + . . .)' we apply 2*10 (4) or (5) according as j is even or odd. In these we put when j is a positive integer. 2'13. Expansion of the cosine and sine of a Fourier sine series. (i) The expansion of is obtained from 2'10 (4) by putting / = -/</' =yi) lv = ... = 1. As far as a 7 this gives , A 2 A* A a 2 / . ^l 3 A*\ /J4 After expansion in powers of A as in 210 (i) and the rejection of negative powers of z we replace z* by 2 l cos id. (ii) The expansion of sin (2a! sin 6 -f 2a 2 a 2 sin 204- . . .) is obtained from 210 (5) by putting /</ = -/o'" =/o v = . . . = 1. This gives for the function to be divided by V 1: A* A* A^_ a? (A 2 A* A^\ * 3! + 5! "*" 7! ai z V21 + 4! "*" 6!/ 4 ,M 2 J. 4 \ + (2) In the final result, 0* is replaced by V 1 2a l sin i^. 12-14] BESSEL'S FUNCTIONS 55 2 14. BesseVs Functions, The Bessel function of the first kind, Jj (#), may be defined by the series where ^ is a positive integer. For a negative suffix, we define it by A comparison of coefficients of x i shows that *) = 2 ...... (3) 2? rf y _i (a;) + J, + i (*;) = ~Jj 00, /M 0*) - ^+1 (*) = 2 and that the differential equation is satisfied. The properties most useful for our purposes are deduced from the fact that Jj(x) is the coefficient of z* in the expansion 2^(*)^ j-0, 1, .......... (4) This result is shown by expanding exp. ##/2, exp. ( xftz) in powers of xz, xjz respectively and choosing the coefficient of z j in the product of the two series. Put 0= exp. g V 1 in (4), so that 21/2 = 2 V 1 sin g. We obtain exp. (x sin g . V^T) = ^Jj (x) exp. ( j# V^T), j = 0, 1, . . . . ...... (5) In this equation, change the sign of g, put x = ie and multiply both members by exp. (ig V 1). We obtain exp. {i (g-e sin g) V- 1 j = S, Jy (fc) . exp. {(t - j) gr %/- 1 }. The real and imaginary parts of this equation give -e sin g)^^ i J i (ie)GOs(i-j)g t ) +1 ^ '~ ' "" 56 METHODS FOR EXPANSION OF A FUNCTION [CH. n The same process may be applied to (5). With the aid of (2) the results may also be written sin(tf sin<7) = 22,J w+1 (ar) sin (2j + l)g,\ 3 ~ L> *> ...... (7) The Bessel function may also be defined by i r* j(x)=*- / cos (j< x sin <) d<, ......... (8) a result which is deducible by the use of the Fourier theorem from (6). 2*15. The Hyper geometric series. This series, namely, ...... (1) includes certain series which are needed in the development of the disturbing function. It satisfies the differential equation, F=0, ...(2) and admits of many transformations*, two of which give F(A, B, C,X)=(I-X)-*F(A, C-B, a~ r ), ...... (3) F(A, B, C, x) = (1 -^-*- F(C-A, G - B, C, ). ...(4) The differential equation may be used to find the expansion of F in powers of y t where x = a -h y. If this substitution be made in (2) and if we put in the resulting equation, the condition that the coefficient of y n shall vanish identically is (a 2 - a) (M + 2) a n+2 + {n (2a - 1) + a(4 + fi + 1) - 0} a n+1 * See A. R. Forsyth, Differential Equations, Chap, vi; Riemann-Weber, Die part. Diff.-Gleich. der Math. Phys. vol. 2, pp. 18, 19. 14-16] HYPERGEOMETRIC SERIES 57 This recurrence formula can be used to find all the coefficients when two of them are known. By direct calculation do, fti can be obtained from to the required degree of accuracy, and thence a 2 , aa, ... are successively obtained. It may be necessary to carry a<>, i to two or three additional places of decimals in order to compen- sate for the loss of accuracy which the use of the recurrence formula may produce. The following formulae are immediately proved : IT ^ ~d~c~' ~ C A+1 > n+1 > c+1 ' ............... ^ ' where the meaning of the notation is evident. By differentiating these equations and substituting for the derivatives in the differential equation, we can obtain various equations connecting three series for A, A 4-1,^.4-2 with (7, B unchanged, or <7, G -f 1, (7+2 with A, B unchanged or with A, B each increased by 1, 2 with unchanged, etc. 2*16. Expansion of (1 ax)~ s (1 /#)"* in positive and nega- tive powers of x. Let us adopt the notation /V _ n(n + 1) ... (n + r--l) /n\'_ m \r) ~ rl ' \OJ "" ' ...... ( } the accent being used to avoid confusion with the usual notation Expand each of the factors by the binomial theorem. The product is KO'-'H'G'M' '' j -- 1 ' 2 58 METHODS FOR EXPANSION OF A FUNCTION [CH, n The coefficient of x n , n positive, in this product is obtained by putting { = n+j in the first factor and summing for all values of j. The coefficient is n +jj \JJ (& -4- n.\ (& 4- n. -L 1 \ /// _i_ 1 \ 4 - ' 1.2 The coefficient of xr n is evidently obtained by interchanging t, s. The most important case is that for which s = t: the co- efficients of positive and negative powers of x then become equal. The part of the coefficient within the parentheses may be expressed by the hypergeometric series F(t,8 + n, 14- n, a 2 ), a form which permits of the immediate application of the trans- formations of 2' 15. When t s, we obtain in this way the following forms for the coefficient of x n or ar n , namely, where (3) (l-)(2-) s(s+l) a* _ r s * 1+n'll-a 2 +w.)' 1.2 (l l-s l+n-s n)(2+w)' 1.2 4 +-..|. (5) When n =0, the coefficient of /(a 2 ) is 1. 16,17] DEVICES FOR CALCULATION 69 If x = exp. 6 V 1, the coefficients become those of 2 cos nO and of the constant term in the expansion of (1 2a cos 6 + a a )~ 5 . (Cf. 4*2.) 2*17. Devices for the numerical calculation of the values of functions defined by power series. The ratio of two consecutive coefficients of the hypergeometric series and of many other series which occur in celestial mechanics approaches the limit unity, and when the variable is near unity, the calculation may be- come tedious. The following device is often effective, especially when the coefficients are alternately positive and negative. The identity If x i x \ 2 ) a aidJ + Gtefl 2 ... = ., --- ^oH-Aaor- ~ hA. 2 a -r' +-.-K 1 4- IK ( 1 -f x \\-\-x: ) where ...... Aa = a -ai, A 2 </o=o -"i H- ^2, A 3 a = "o 3(/i+ 3a 2 s> ...... (2) is easily proved by expansion of the right-hand member in powers of x. If OQ, i, . . . are positive, the coefficients Aa , A a ao, . . . may form a rapidly decreasing series, and the rate of convergence is increased by the fact that #/(! -f x) < x, when x is positive. For efficient use the transformation should only be started at a term in the given series where the ratio to the succeeding term is less than 2. A more general form is given by the identity, ...... (3) where Aa , A 2 <7 , ... are defined as before and /=l-M? + & a a - ...................... (4) For efficient use, the series/ should be a known function such that the coefficients of Aao, A 2 ao> form a decreasing series This is the case, for example, when where X is positive and less than unity. The transformation 2'15 (3) from 2*15 (1) may be effected by means of this result. 60 METHODS FOR EXPANSION OF A FUNCTION [CH. n The ratio of the (j + 2)th to the (y + l)th term of the series 2'16 (4) may be written Suppose that we have calculated a few terms of this series, say j 1 of them. The remainder of the series may be written K( 2 Yfl !l + - ( ~^l -J- ___ ^4-1 (5) Ml-"V L I l+ j(j+l)lj + n + ll-S + -J .......... ( > The formula (3) is applicable if we put In the applications $ has the values , 3, ..., so that if j be suitably chosen , A , ... form a rapidly decreasing series. The function / denned by (4) with these values of the b % is the hypergeometric series which satisfies the differential equation, rf*F dF (afi + x) ^ +{(2+7) .r+j + n + l} ^ r +^-0 .......... (6) A first integral of this equation is since F=sl when a 0. The final integral is given by Since /*, y are positive integers, the right-hand member can be integrated. When the value of F for any particular value of x has been found from this equation, the first and second derivatives of F can be obtained from (7), (6) and the higher derivatives by successively differentiating (6). ^l, n = we have xF 2*18. A device for approximating to the derivative of a series. Suppose that the sum of the series /S f a +a 1 # + a 2 a? 2 + ........................... (1) is known for a particular value x of x, and that we also know the coefficients up to a_i. We then know the value of 2V+ ................ (2) 17-19] DEVICES FOR CALCULATION 61 We have dS dS If we can neglect the last sum in (3) when # = .r , or approximate to it, we obtain a corresponding approximation to (dSfdx\. We may get a use- ful approximation because it often happens that an approximate law of relation between the coefficients a n> a n + i , ... is known and the divisor n will assist in diminishing the error of the approximation. Another result of these conditions is an approximation to a n from (2) better than would be obtained by the approximate law of relation just mentioned. 2*19. Note on the forms of products of Fourier series with different arguments. If we express two Fourier series in the forms So, . 2 cos j0, 2aV . 2 cos j'0', j, / = 0, 1, 2, . . . , where it is understood that for j = 0,/ = 0, the factor 2 is omitted, then their product may be expressed in the form 22aX,'. 2 cos (J0 +/0'), where it is understood that the portions for j = 0, /= 0, j j' = are 2 a'/ . 2 cos/0', 2oj . 2 cos j#, a o', that is, no attention is to be paid to the double sign when either j or f is zero, and the factor 2 is omitted only when j=j'=o- With the same values 2a,-2 sin j0 . 2^2 cosj'0' = 22aX/ 2 sin (J e j' 6 '\ 2aj2 sin j0 . 2a y2 sin/0' = T 22aX/2 cos (j0 /0'). This method of expression avoids all doubts as to the presence of factors of 2, which may occur if we use the form 2o; cos jO with a factor J when j = 0. Moreover, it is the natural form which arises when we use exponential methods of expansion. CHAPTER III ELLIPTIC MOTION 3*1. The relative motion of two bodies under the Newtonian law of gravitation is a simple dynamical problem which admits a general solution in terms of well-known algebraic and trigo- nometric functions. Analytically and geometrically the range of the solution is divided into two portions the elliptic and the hyperbolic the transition from one to the other giving a special case, the parabolic. It is shown in the elementary text-books that one method of distinction is given by the relations F 2 -2/4/r<0, =0, >0, where V is the relative velocity and r the distance apart at any time; /JL is the sum of the masses reckoned in astronomical units. The first case is that of motion in a closed conic section, an ellipse, in which the eccentricity is less than unity ; in the second case the conic is a parabola, with the eccentricity equal to unity, and in the third case it is a hyperbola with the eccentricity greater than unity. When the eccentricity is zero, we have circular motion, the limiting case at one end of the range; when it is infinite, the motion is rectilinear, the limiting case at the other end of the range. In this volume we shall be concerned only with the first case, and the range will be further limited to values of the eccentricity which are small enough for the series, which are developed in powers of this quantity, to be used for numerical calculation during a certain interval of time without too much labour. Ex- pansions in powers of the eccentricity, either implicit or actual, are necessary with the methods developed below, and the greater part of this chapter consists of the formation of those expansions which will be needed in the problem of three bodies. 3*2. Solution of the Equations for Elliptic Motion. If we put #0 in 1*23 (3), (4) and confine the motion to the plane of 1, 2] SOLUTION OF THE EQUATIONS 63 reference (see 3*6), these equations take the form __ dt* \dt ~ r*' dt These equations possess the integrals, ,dv where a, h are arbitrary constants. The transformation in 1*27 shows that the elimination of t between (1), (3), (4) gives, if u = I/?*, d?u /j, A fdu\ 2 2 /X/, 1\ j~2 + u - T2 = > I T- ) + '* = /2 ( 2w - - ) o?w 2 /i 2 Vat;/ /r\ a/ ...... (5), (6) The solution of the linear equation (5) is u= T*{ 1 + ecos(t;-cr)}, ............... (7) where 0, tn- are arbitrary constants and we suppose that ^ e < 1 This must satisfy (6). The substitution of (7) in (6) gives A 2 = /m(l-e 2 ) ...................... (8) 1 1 -he cos (v cr) Hence - = ----- 7T-- 2 T .................. ( y ) ? a(l e 2 ) so that the maximum and minimum of r are a (I e). Equation (9) is that of an ellipse referred to a focus as origin, the nearer apse of the ellipse having polar coordinates a (I e), or. Let us now introduce the variable X and a constant n defined by rdX = ndt, /i = n*a 3 .......... (10), (11) The investigation of 1'29 with jR=* shows that the equations of motion with X as independent variable are d 2 r dv h aVr~e 2 _ a --, ...... (12), (13) since (8), (11) show that A = V/xtt (1 - e 2 } = na* Vl~ e 2 ............. (14) 64 ELLIPTIC MOTION [OH. nr Since the maximum and minimum of r are a (1 e), the solution of (12) is r = a(l-ecosZ), .................. (15) where we define X as having the value when r a (1 e), that is, when v= -&. Thence, if e be an arbitrary constant, the solution of (10) gives y = nt + -<& = X -esinX ............. (16) Finally we obtain, from (13), The integral can be put into any one of the three forms, , . , Vl (Ps'mX , cosX e fsm- 1 -, - - v cos" 1 - ------- -., J \ e cos A l e cos A We therefore obtain, with the help of (15), rcos/= a (cos X e), r sin/= a Vl e 2 sin A, ...... (18), (19) With the definitions (16), (17) of </, /, the following results are easily deduced: df a 2 V L e 2 dr __ ae sin/' dX _ a fig** r*~ ' dg ~ v'T^? ' W/~~r' ...... (21), (22), (23) The constants 2a, e are the major axis and eccentricity of the ellipse ; w is the longitude of the nearer apse from the initial line, and e, 'the epoch,' is the longitude of the body when it is passing through the nearer apse, so that it defines the origin of the time. The period of revolution is 2?r/n; n is called the mean motion. The angles /, X, g are known as the true, eccentric and mean anomalies, respectively, and nt + e as the mean longitude, usually denoted in this volume by w. 2-4] THE THREE ANOMALIES 65 3 3. Frequent use will be made of <, ^, ^, tj defined by <jE> = exp./V 1, =* exp. ^ e(l+r,*)-2i,, ..................... (4) so that 2 cos t/= < + !/</>*, 2 V-~7 sin t/ ^ - !/<*, etc. ...... (5) By writing (4) in the form 77 = $e + \eif, and applying Lagrange's theorem, (2 '2), we can deduce the expansion JHJ + .... (6) The definitions (1), (2), (4) applied to (18), (19) of the previous section give ...... (7), (8) and, applied to 3'2 (9), (10) 3'4. An important property of these relations when they are expressed as Fourier series is that characteristic of most of the expansions in Chap, n, namely, that the coefficient of cosjO or sinjO is of the form whether 6 be /, X or g. It appeared in the last chapter that if the functions with which we started originally possessed this property, it was retained under the operations to which they were subjected. B&SPT 5 66 ELLIPTIC MOTION [CH. in It is evident from equations (15), (16), (20) of 3*2 that r a, g - X, f X, when expressed in terms of X, have this property. The operations to which they are subjected are those dealt with in the previous chapter, partly expansions in powers and partly changes of variable, and consequently the property is retained. It is apparently not present in such functions as r?cosqf y r^sin qf, expressed in terms of g, but it reappears in rPcosq(f-g), r p smq(fg), and the latter can always replace the former in the applications. Series having this property will be named d'Alembert series. Operations with such series have been treated in Chap. II. The relations 3*3 (9) show that any function of < in terms of x> ?; can be at once transformed into the same function of % in terms of <, 77. 3*5. The facts that l/r in terms of / and r in terms of X satisfy linear differential equations of the form furnish the reason for transforming the equations of motion to the forms given in 1*27-1 '29. It will be noticed from 3'3 (7) that r cos/, rsin/' in terms of X and rj cos /, r\ sin ^/ in terms of JA', have the same property, but no use appears to have been found for this latter pair of expressions. 3*6. Agreement of results with Kepler's Laws. That the motion takes place in a fixed plane is perhaps obvious. It can, however, be proved at once from the equations of 1 *23. For if R = 0, F depends only on r so that r' 2 dv/dt, r, 6 are constant, and v v. Kepler's three laws and Newton's deductions from them can be im- mediately illustrated from the equations of 3*2. Law II, which states that equal areas are described by the radii of the planets about the sun in equal times, i.e., that the rate of description of areas is constant, moans that the right-hand member of 3*2 (2) is zero ; it follows that the resultant force is along the radius. Law I states that the planets move in ellipses with the sun in one focus. If we substitute 3-2 (9), (4) in 1-23 (3), the radial force, dF/dr, will be seen to vary inversely as the square of the distance. Law III, which states that the squares of the periodic times are pro- portional to the cubes of the major axes, is not quite exact. The equations /x = n 2 a 3 , r = 2T/>i, give 4-6] KEPLER'S LAWS 67 where p is the sum of the masses of the sun and planet. The largest planet, Jupiter, has a mass less than 1/1000 that of the sun, so that the difference between Kepler's third law and the exact statement is small. With the observational material used by Kepler, it is not perceptible. The laws are, in fact, only approximate descriptions of the actual motions : they cease to hold when the mutual attractions of the planets are included. The third law, in particular, becomes a mere definition. We obtain the mean angular velocity of the planet directly from observation and define a certain distance a by means of the equation fj. = n 2 a 3 . The object of this definition is the convenience of calculating by means of equations in which the terms are obvious ratios of times and lengths. For example, the equation, acceleration = mass x [length] ~ 2 , which is the initial form of a gravitational equation of motion, is trans- formed to acceleration = [length] 3 [time] ~ J [length] ~ 2 , which is obviously correct in its dimensions. The constant a so defined is usually called the 'mean distance.' This use of the word 'mean' in the sense of 'average' is incorrect, even when the motion is elliptic. Equation 3 '2 (15) shows that it is the average distance if the eccentric anomaly be taken as the independent variable, but it is easily seen from equations 3*2 (9), 3*10 (5) that it is not so when either the true longitude or the time is so used; in the latter case, however, 3-11 (2) shows that I/a is the mean value of I jr. In the actual integration of the equations, it will be seen that the first arbitrary constant to appear is that on which the distance depends: the mean angular velocity is seen later to be a function of this and of other arbitrary constants which have arisen in the integrations. But since, in general, we can deduce the mean angular velocity from observation with much higher relative accuracy than is possible for the mean distance, it is convenient as a final step to adopt it as an arbitrary constant and to express the constant of distance in terms of it and of the other arbitrary constants. When this is done, a is nothing but an abbreviation for </*/')* Confusion is often caused by differences in the meanings attached to the letter a. Sometimes it means (ji/n 2 )&; at other times it signifies a variable or a constant which has this value as a first approximation. The confusion exists throughout the literature, and the only remedy is the dis- covery of its exact signification wherever it is used. Similar confusion is often caused by the use of terms in the problem of three bodies, which have a definite meaning in the problem of two bodies. Thus the Eccentricity' in the former case may be the coefficient of a 5-2 68 ELLIPTIC MOTION [CH. m certain periodic term in the expression for one of the coordinates; the ' mean anomaly' and 'true anomaly 7 are used for certain angles ; and so on. A qualifying adjective, like * osculating,' may again alter the meanings. There is usually not much trouble when qualitative descriptions are alone involved ; in quantitative work, accuracy of definition is essential. 3*7. Fourier developments in terms of X. The logarithm of 3*3 (9) gives, on account of the definitions of <t>> X> f V- 1 = X V=l + log (1 - rj/ x ) - log (1 - vx ) = AV- 1 +17 (*- 1/X) + iV( X a - 1/x 2 ) + .... Hence, by the definition of ^, j=l,2, .......... (1) J Again, from 3*3 (7), we have (a \ p / 7i\^^ Q iT?) x a (i-w)"- a (i-^) - .-(2) The binomial theorem gives, for the expansions of the two binomial factors, with the usual notation for the binomial coefficients, The coefficient of ^, with j = 0, 1, 2, ..., in the product is obtained by putting j + k for k in the first sum and summing the product for all the values of A; for that of ^~~', we proceed similarly with the second factor. Thus the product is provided the term for j = 0, which is the same in both portions, is not repeated. If we write with a similar change in the corresponding binomial coefficient of the second term, and define S^ Sj by 6, 7] ECCENTRIC ANOMALY DEVELOPMENTS - q. -j - 1 ) ( p + ?) " " ~ " "" . ( j Vixj + 27" " ~ ' " ""1.2 ...... (3) the required expansion can be put into the form where J = 0, 1, 2, ... ; the term for^' = being By putting for ^ its value cos qf+ ^l sin g/ in terms of/*, and similarly for the powers of ^ in terms of X, and by equating the real and imaginary parts, we obtain the expansions of r p cos qf, r p sin qf as Fourier series with argument X. The formulae for Sj, Sj are hypergeometric series and are therefore subject to the transformation 2'15 (3). We have, in fact, Hence, if we put S f = (1 - T/ 2 )?^* T^ so that r, = i+ _ . . ^_ i - -j-^2 4- ( A (j + !)(/+ 2) (-5) 70 ELLIPTIC MOTION [CH. ra Tf s= value of T t when the sign of q is changed, and remember that ,.. .(6) where j = 0, 1, 2, ... , the terms for j = requiring the factor J. In the applications, p takes the values 0, 1, 2, ... , and </ the values 0, 1, 2, ... . When p^q, the series has a finite number of terms. The case p = q is more easily deduced directly from 3-3 (7). It gives from which ?*^cos pcf), r p smp<f) are at once obtained. The particular case p = 0, q = 1, gives sin/' = For the case q = 0, we have T/ = T i and the development contains only cosines of multiples of X. We thus obtain from (6) T t .1v*jX (8) the factor 2 being omitted when j = 0, and q having the value zero in T$. The following are important particular cases with <?= 0. When p = 1, TJ = 1 and the binomial coefficient is ( iy. Hence -= ^JL_ (1 + 27; cos Z + 277* cos 2Z+ ...). ...(9) r v 1 - & For p = 2, we have 7,8] TRUE ANOMALY DEVELOPMENTS 71 Hence and = (l-*)-*Srf(l+j</l-*).2coBJX, ...(10) the constant term being (1 e 2 ) ~ 3 . 3'8. Fourier developments in terms off. The formula 3*3 (10) gives A comparison of the right-hand member with 3' 7 (2) when q = 0, shows that the development given for r p in terms of X can be used directly by changing the sign of ij, and multiplying by (1 -e)-*a- a * We obtain where Tj has the value 3*7 (5) with q = 0, and the factor 2 is omitted when j = 0. For p = 2, we obtain, as in 3*7, in which j= 1, 2, The series for g in terms of f is found from this last result, by combining it with 3*2 (21) which may be written ty = J**-^- df a **Jl-e 2 ' An integration with the arbitrary constant so determined that /, g vanish together, gives It should be noticed that the coefficient of sin/ in this series is 72 ELLIPTIC MOTION [CH. in The functions of X in terms of/ are obtained from those of/ in terms of X by interchanging/, X and changing the sign of ?;, according to a remark in 3'4. They are, from 37(1), X=f+2zl(-r,y S injf, .7 = 1,2,..., ...... (4) and from 37 (4), with p = 0, (5) where $j, $/ have the values 3*7 (3) with p = 0, and the value for j = requires the factor . From these we obtain cos qX, sin qX. In particular, from 37 (7), FOURIER DEVELOPMENTS IN TERMS OF THE MEAN ANOMALY 3*9, These developments are deduced from those in terms of X by means of the implicit relation cj X e sin X. The solution of this equation is avoided by making use of the theorem of 2*5. Applied to the present case, this theorem states that the coefficient of cosjg in the expansion of f(X) as a Fourier series with argument y, is the same as the constant term in the Fourier expansion of __ 9 d ............ (1) and that the constant term in the expansion is the same as that of (l-ecosg)f(g) ...................... (2) For the coefficient of sin^, replace the first sin in (1) by ' cos/ The form of the first factor of (1) shows that developments by means of Bessel functions (2*14) will be needed. In these developments the parameter ^e is convenient, while in those of functions of/ in terms of X, the parameter r) was used. Hence in functions of /in terms of # both parameters may appear. 8-10] MEAN ANOMALY DEVELOPMENTS 73 310. Expansions for coskX, s'mkX, A", r, rcosf, rsin/, in terms of g. When/(A) = cosyLY, 3*9(1) can be written k k 7 cos {(j - k)g -je sin g} + . cos {(- j - k)g +je sin g}. According to 314 (6), the constant term in the Fourier expansion of this expression is k k .J^. k (-je\ ............... (1) forj=l, 2,.... When j = 0, equation 3'9 (2) shows that it is zero if k ^ 1 and \ e if k = 1. A similar investigation gives the coefficient of sin jg in the expansion of sin kX. Hence, allowing j to receive both positive and negative values, we have, for kl, cos , v ^ k , . cos . , i , o /.>\ . kX = 2 -J~- k (ie) . 9(7, 9 = -f-l, -f-2, ....... (2) sm J sm^ J ~ When &=1, it is convenient to make use of the formulae 214 (3), although those just written are available if we add \e to the expansion for coskX. We obtain cos X = - \e + 2S r 2 -'- J ? (je) cos jq, (* sin X 2S - Jj (je) sinjg, The expansion for sin A", inserted in the relation X = g 4- e sin X, gives X = # -f 22 -. J t (je) sinjg, j = l, 2, ....... (4) The expansions for r, rcosf, r sin'/ are obtained from (3) by the use of the relations r a(l ecos X), rcosf =acos X ae, r sin/= a (1 - e*fi sin A. 74 ELLIPTIC MOTION [OH. m They give r =a(l H }%?) 2a 2 - 2 e-r- Jj (je) cos jg, J de r cos/*= |a 1 rf j-1,2,.... ...(5) r sin/= 1 2a (1 e a )2 ! -- J} (je) sin j^, 3*11. Expansions for a/r, r 2 /a 2 , a 2 /r 2 ,/, in terms of g. For functions which contain a power of r as a factor, it is some- times better to replace 3*0 (1) by* 2cos(jg-je*mg)f(ff)(I-ecosff) .......... (1) That the two expressions have the same constant term in the Fourier expansions all that we need is evident since we can express their difference as the derivative of a Fourier series. The expansion of a/r can be obtained from (1) with since r = a (1 ecos X), with the aid of 2*14 (6). It can also be found from 3'10 (4) with the aid of the relation ajr dX/dg. The result is (2) The fact that the constant term of a/r, expressed in terms of is unity is an important property. For the expansion of r 2 /a 2 , we have 2 jzl 6 j._ i(je} +? Jj_ 2 (je)l cos .?>, \ / ./ ' by 3-10 (2), 310 (3). The use of 214 (3) enables us to write this ^ = l+f^-2JiJ,-(>)cosj7, j-1,2, ....... (3) * The functional f(g) has, of course, no relation to the true anomaly/. 10, 11] MEAN ANOMALY DEVELOPMENTS 78 The relation between the expansions (2), (3) is given by dt 2 \r aj with /L6 = ?^ 2 a 3 . This equation is easily deduced from 3*2 (1) and 3-2 (3). For the expansion of a 2 /r 2 , we make use of 3*11 (1) with f(g)**(l ecos<7)~ 2 , so that the coefficient of cosjg is the constant term in the Fourier expansion of 2 cos (jg je sin g) . (I e cos g)~ l (4) The expansion of 1/(1 6 cos X) in terms of X, e is given by 3*7 (9). Hence that of the second factor in terms of g, 77, is (1 e 2 )"^ (\ + 22?7 cos ig), i = 1, 2, . . . , or (1 e 2 )"^ 2^1*1 cos igr, ^ = 0, 1,2, With the use of this result, (4) may be written 2(1- e 2 )~i 2 17! *' cos {(i +j)g-je sin </}, and, by 2*14 (6), the constant term in the Fourier expansion of this function is 2 (1 - e 2 )-* 2 171*1 J i+j (je), i = 0, 1, 2, . . . . This is the coefficient of cosjg in the expansion of a a /r 2 . The application of 3*9 (2) shows that the constant term in the expansion of ti 2 /r 2 is (1 - e 2 )"*, and 2'14 (1) shows that J" t -(0) except for i = when it is unity. The change of i into i+j in the previous expression for the coefficient of cosjg therefore gives ,,2 Jr v > A > the factor 2 being omitted for the value j = 0. The expansion for f is deduced by inserting this result in 3'2 (21), namely, in df_a*(l-e^ dg~ r* ' 76 ELLIPTIC MOTION [OH. in arid integrating with the condition that /, g are to vanish together. ,The result is sin;//, ............ (6) where i = 0, 1, 2, ...; j = l, 2, .... While this formula is quite general, it is not very convenient for the actual calculation of any coefficient in powers of e or -77, partly because either of these parameters must be expressed in terms of the other, and partly because there are j + 1 terms of the same order in the coefficient of sin jg. The term of lowest order in any coefficient is, however, easily found, since for this term we can put Hence the principal term in the coefficient of sin^'j is The portions depending on higher powers of e in this coefficient will not be developed in detail. If we adopt the definition the coefficient of (e/2) ?+2 will be found to be 5* (9) and that of (*/2)J+* i{l. 4.^ + 2. 5j 2 + ... + 0'~ J _ j .7- 11-13] MEAN ANOMALY DEVELOPMENTS 77 3*12. Expansion of any power of r by recurrence. The expansions of other special functions of r, / may be obtained from the equations of motion. Thus, if we put x r cos/, y = r sin/, the equations satisfied by x, y are d*x \*.x d^y fjti/ 'd? " "~ 7 3 ~ ' rf? = " yJ ' As we have already found the expansions of .r, _?/, these equations give the expansions of cos //r 2 , sin//?- 2 . Again the equation gives the expansion of 1/r* when those of ?-, I//- 2 are known. In general, the equation which is deducible from (1), (3), (4) of 3 -2, can be used to obtain, by recur- rence, the series for r ?) for all values of p, when those for certain values have been obtained. 3'13. Expansions of r p cos qf, r p sin qf. These expansions for values of p, q other than those just considered can be dealt with by first expanding in multiples of X by 3*7 (6), and then using 3*10 (2) to transform to multiples of g. The series suffer from the defect of that for f mentioned in 3 f ll, namely, that there are^' -f 1 terms of the same order in the coefficients of cosjg, sin j'gr. If, however, we do not need such expansions beyond e 7 , the extensive tables given by Cayley * for various values of p, q and for other functions are available and will serve for most purposes. In cases where this degree of accuracy is not sufficient, numerical values are usually used and then the method of numerical harmonic analysis (3*17) is available. A combination of the results obtained from the literal and numerical developments by the method indicated at the end of 2*18 will give an approximation to the terms of order ^in certain of the coefficients. The method developed in that article also increases the accuracy of derivatives with respect to e, when they are needed. * Mem. E.A.S. vol. 29, pp. 191-306; Coll. Paper*, vol. 3. 78 ELLIPTIC MOTION [OH. m 3*14. The constant term in the expansion ofr? cos qfin terms ofg. By means of the relation 3*2 (9), this function is immediately expressed as a function of/. According to the theorem of 2'5, the constant term, when it is expressed as a function of g, is the same as the constant term of expressed as a function of/. Hence, by 3'2 (21), we need the constant term of , , , r^ +a cos qf -r a 2 (1 - e 2 )*, expressed as a function of/. The expansion of r^ +a as a function of/is obtained from 3'8 (1) by putting -^~2 for^. We thus need the constant term of } . 2 V cos j/cos g/ where in Tj we put -ju-2 forp, g = 0. There is only one con- stant term in this and it is evidently given by j = #, that is, it is (1) where r-i + rJ^ -P^ ''~ + 1 ' q+l (_p_2)(- ? _ 1.2 !.(?+!) It is evident that 3T ? is a finite series for all negative integral values of p ; it becomes unity for p = 1, 2. 315. The expansion of r? in terms of g. The constant term in this expansion is obtained by putting 2 = in 3-14(1). The coefficient of (e/2)i cos jg is found to be, with the help of the notation 3 - ll (8), 14-16] LITERAL DEVELOPMENTS 79 and that of (e/2)> +2 cosjg, The series in each brace stops at the suffix j. 3*16. Literal developments to the seventh order. The following detailed developments may be found useful for reference. The notation e = e, is used. g = y _ 4 e s i n / + (3 C 2 + 2 e * + 3e 6 ) sin 2/~ f c 3 4- 4c 5 4- 8e 7 sin 3/ 4- (I e 4 4- 6 A sin 4/- ^ 2 c 5 4- 8e 7 ) sin 5/ 4- 1 e 6 sin 6/- 1 _ 6 c 7 sin 7/ . ..(1) o * 7 + ^4e - 2e +^ e 5 + ~ c') sin g+ ^5e 2 - ?^ 2 e 4 + - g - c 8 ) sin 2# /26 3 43 95 ,\ . ^ /103 4 902 6 \ . IT 2" T j sm ff \ 6" T5" j S1 " /; /1097 B 5957 _\ . . 1223 6 . + \W C ~ "36" j S1 " ^ 15" S S in7, ....... (2) - = 1 + 2c 2 - ( 2e - oc 3 + ? c 5 - ^ 9 e 7 ) cos g d \ \j t / - ~ e 4 + 4e 6 ) cos 2r; - (,3e 3 - ^ e 5 4- '^ 6 J e 7 ) cos 87 /16 4 128 6 \ . /125 B 4375 7 \ K ~ \T ~5" j C S 9 "" \ 12 "" "72" j C S ^ 108 e . 16807 7 . /0 , - c 6 cos6^r- --e 7 cos75r ....... (3) 80 ELLIPTIC MOTION [CH. m + (4e 2 - ~ 6 c 4 + ^ c 6 ) cos ty + (9e 3 - ^ c 5 + ^ e 7 ) cos /64 4 1024 6 \ . /625 5 15025 + U c TT c cos + " c ~~ /n (4) It is evident that the expressions for/, r are d'Alembert series with respect to the association of powers of e with multiples of g or tsr (3;4). 317. Numerical developments by harmonic analysis. When the numerical value of e is given, the most rapid and accurate method for computing the functions is that of numerical harmonic analysis (App. A). This method requires the calculation of the functions for a few special values of the independent variable. The calculation presents no difficulties when either the eccentric or the true anomaly is taken as the independent variable; the formulae in 3*2 are available for the purpose. When the independent variable is g, the first step is the solution of the equation g = X e sin X, for each special value of (/. For a low degree of accuracy, tables for the purpose are available*: methods for the correction of these values are given. For high accuracy, the method given below will be found con- venient. When the special values of X have been obtained, those of r, / and thence of any functions of r, / are found from r = a(l -ecos Z), with any one of the formulae 3'2 (18), (19), (20). The considerable increase in accuracy obtained with the use of numerical harmonic analysis is due to the fact that in most of the series with which we have to deal, the rate of convergence along the coefficients Aj, in the series SAjCi'cosjg or SJ/o'sin^, is more rapid than that of A 3 expressed * See, for example, those of Boquet, Obs. d'Abbadia, Hendaye, and of J. Bau- schinger, Tafeln zur I'heor. Astr. Leipzig. 16, 17] SOLUTION OF KEPLER'S EQUATION 81 as a series in powers of a 2 , especially for large values of j, unless a is very small. A detailed examination of the errors produced in any coefficient by the neglect of the higher terms with any given set of special values of g will show how this result is obtained*. Numerical Solution of Kepler's equation. When high accuracy is required, it may be obtained rapidly by a formula obtained as follows. Put X~g-\-x, so that Kepler's equation may be written x-e sin (g + x) (1) Hence sin x = x i^+iJotf 5 - -- = esin(^+^)~Jtf 3 sm 3 ( t (7 + ^) + T i^e 5 sm 6 (<7-f.r)- (2) Calculate (7, # from C cos .'0 1 e cos </, sin # = e sin </ (3) These give C 2 =l 4- e 2 2<? cos #, C sin (. +gr) = sin #, <?sin (# +#) = 8in # . (4) With the aid of (3), equation (2) may be written (7sin(o;~^ )=-ie 3 sm 3 ((7-f.t;) + - I J T ie 5 sm 6 (^ + a;)- (5) If e 3 be neglected, we have #=# , the error being of order e 3 /6. If we put X XQ in the right-hand member of (5), the maximum error of its first term is found to be of order e 6 /46. Hence the formula (7 sin (a; # ) = -^sin 3 (# + #o) (6) gives x with an error of order e 6 /46 or e 5 /120. By the use of (4) alternative forms for calculation are seen to be 1 e 3 sin (x - ff ) = - Q(J sin3 #o = - gg* sin3 # these giving the same results as (6). Should still higher accuracy be needed, it can be obtained by substituting the value of x thus obtained in the right- hand member of (5), but this will very rarely be necessary. For e<'14, the error of X found from (6) is less than 0"*1. * An example will be found in Mon. Not. R.A.S. vol. 88, p. 631. B&SPT CHAPTER IV THE DEVELOPMENT OF THE DISTURBING FUNCTION 4*1. In this chapter are given methods for expressing the disturbing function as a sum of periodic terms when for the coordinates are substituted their expressions in terms of the elliptic elements given in Chap. in. The disturbing function for planetary action obtained in 1*10 is j _ m' m'r cos $ .-. where r, r' are the distances of the two planets from the sun, A is the distance between them, S is the angle between the radii r, r', and m f is the mass of the disturbing planet. Hence We have seen also in 1*10, that if the plane of motion of m' be taken as the plane of reference, cos S = cos (v 6) cos (v 6) + cos / sin (v 6) sin (v' ( cos 2 ^ I cos (v v') + sin 2 J / cos (v + v' 20). (3) In this formula, / is the angle between the two orbital planes, 6 is the longitude of the node of the orbital plane of the dis- turbed planet from a fixed line in the plane of reference, v' is the longitude of the disturbing planet from the same fixed line, and v is that of the disturbed planet reckoned to the node and then along its orbital plane to the body. If cr', ty be the longitudes of the nearer apses reckoned in the same manner as v' 9 v, respectively, and if/', /be the true anomalies, we have The substitution of (4), (3) and (2) in (1) gives R as a function of r y r', /', /', w, r', /, 6. The results of Chap. Ill show how r, r' 9 f 9 f may be expressed as functions of the true, eccentric 1,2] GENERAL PLAN 83 or mean anomalies. There is thus no difficulty in expressing R as a function of these angles; the problem is the expansion of R into a sum of sines or cosines whose arguments are multiples of them. The changes necessary when the plane of reference is arbitrary are given in 1*32. 4*2. Suppose that we put the eccentricities e, e' and the in- clination 1 equal to zero. Then the true anomalies f, f, the eccentric anomalies X, X 1 \ and the mean anomalies g, g' are respectively equal and r, r' reduce to a, a'. The disturbing func- tion becomes with /S Y = v v g 4- TV g' tsr'. The first term can be expressed as a cosine Fourier series with argument $; the second term is already in the required form, Suppose a<a' and put a/a! = a. Then R = ^ (1 - 2 cos $ + a 2 )"* - ~ a cos ft (/< / = {l-Ja+t(2acu8S-a)+...}, Co on expansion by the binomial theorem. The various powers of cos$ can be replaced by cosines of multiples of S which will then have coefficients expanded in powers of a 2 ; the general form of the expansion is given in 2'16. The practical difficulties in connection with this expansion are due to the need for using values of a which are frequently as large as '7 and to the fact that the coefficients may be needed to five or more significant figures. If the literal series were used, some dozens of terms in a coefficient would often be needed and the work thus become extremely laborious: not infrequently also, some eight or ten multiples of S are required. Thus one problem is the construction of a set of devices for the rapid calculation of these coefficients. 6-2 84 DEVELOPMENT OF DISTURBING FUNCTION [CH. iv The disappearance of the term a cos S from the expansion has important consequences in satellite theory where a is very small. In the planetary theory it simply has the effect of diminishing to some extent the terms with argument S, so that those with arguments S, 2/S' have coefficients of about the same order of magnitude in the coordinates. 4*3. When the eccentricities and inclination are not zero, the only available methods for development depend on expansions, implicit or explicit, in powers of these parameters, As far as their magnitudes are concerned, the problem is less difficult than with a, because they usually have values in the neigh- bourhood of *1. In exceptional cases, one or two of them may rise to '4 or '5: beyond this limit, the expansions are useless for numerical calculation and, in general, the results will have doubtful accuracy for values greater than *3. A much more far-reaching effect is produced by the introduc- tion of multiples of the anomalies, other than those of their difference. When the disturbing function is expressed in terms of the time, these multiples take the form jg j'g', where j, j' are positive integers, and the coefficient of the term which has this angle as argument contains the power | j f \ of the eccen- tricities or inclination. The coefficient of t in the angle is jn j'ri > and when an integration is performed this quantity will appear as a divisor. The divisors with the upper sign will tend to diminish the coefficient, but those with the lower sign may in- crease it. Consider the expression Since n, n' are observed quantities, we can always find integers which will render this expression as small as we wish, so that integrals involve discontinuities which may require special treat- ment. It has been pointed out, however, that a term with argu- ment jg j'g r contains as a factor of its coefficient the power \jj'\ of the eccentricities and inclination, so that for large values of j, j' t the factor is very small. From the point of view of the applications, the cases of interest are those in which j/f 2-4] VARIOUS METHODS 85 has such values as , f , f, ..., and in which the expression (1) is small. Since the coordinates of the planet will contain the integrals of such terms, the relative accuracy of the results will be diminished unless the corresponding coefficients be taken to more places of decimals. It is this requirement which con- stitutes the central difficulty in the development of the disturb- ing function: a few coefficients of a given order with respect to the eccentricities and inclination are needed to a higher degree of accuracy than the remainder of the terms of the same order. The problem is practical rather than mathematical, namely, the avoidance of extensive calculations of numerous terms, only a few of which are ultimately retained. 4*4. In the majority of the older methods, the time is used as the independent variable, requiring the disturbing function to be expressed in terms of the mean anomalies. There are several methods of approach. One is to express it first in terms of the true anomalies by means of the equations, 1 -t- e cos/ ' 1 -h e' cos/' ' and, after expansion, to proceed to its expression in terms of the mean anomalies by means of the relations developed in 3'10-3'16. A second method is the expression of the coordinates in terms of the eccentric anomalies, by means of the relations, r = a(l ecos X),} ro,o$f=a(cosX e), f ............... (2) r sin/= a Vl e 2 sin X, ) with similar expressions for the disturbing planet, and then to express the results in terms of the mean anomalies through the use of the implicit relations # = Z-esinZ, tf^X'-e'smX' .......... (3) Still another method is to proceed straight to the mean anomalies from ?, /, /, /' by means of the series developed in the later sections of Chap. in. 86 DEVELOPMENT OF DISTURBING FUNCTION [CH. iv If an independent variable other than the time is used, t is eliminated through the relations vr, g' = n't + e' -<GT', ............ (4) so that the coordinates may be expressed in terms of the vari- able chosen. 4*5. A further distinction between the methods arises accord- ing as a literal development in powers of e, e', I is made, or as numerical values are substituted for the elements of the ellipses from the outset. When the time is the independent variable, the expansion contains multiples of four angles: g, g' and the differences of tzr, or', 0. If the numerical values be used for the latter, as well as for e, e\ J, the disturbing function can be ex- pressed in the form ~r Aj,f cos (jff + jy ) + ~ B u sin (jg -f //), where j y f receive positive and negative integral values, and the A, B are numerical coefficients. The abbreviation of the work is evidently very great. On the other hand, in a numerical method it is difficult to find a few coefficients of high order to more places of decimals without taking the whole, or the greater part of the work to the same degree of accuracy. A further loss with the numerical method is due to the fact that the derivatives of R with respect to the elements, or to some of them, are needed, and these require the calculation of at least three functions when numerical values are used from the outset. A definite set of rules to fit all cases should be avoided if much unnecessary calculation is not to be carried out. Each case should be examined in some detail, especially the calcula- tions needed for the long-period terms, that is, those for which jnj'n' is small, and that plan adopted which would seem to give the results needed most efficiently for the case in hand. Famili- arity with one method is to some extent time-saving but the gain does not usually balance the loss when a choice of methods is available and advantage is taken of the choice. 4-6] IN POWERS OF THE ECCENTRICITIES 87 4'6. General methods for expansion in powers of the eccen- tricities. The methods adopted here involve the use of the theorem, = *, ............ (1) proved in 2*7, together with a variation of this theorem found by putting __ p = exp. E V- 1, x = exp. ty> . ^ d Id n d so that x = _= , p u = exp. h, , dx \f-ld^ ^ r cfyr and F(exp. E*J- l.exp.^V-l) = exp. #-7-7 -f\exp. ^r V-l). ...... (2) The two formulae take care of all developments along powers of the eccentricities, the former, in general, for linear coordinates, and the latter for angular coordinates. Put r = a . function of e, r</ = a' . function of e', r /r ' = a, where the functions of e, e' are at our disposal but reduce to unity when e = 0, e' = 0, and let r = r p, r' = r y, ..................... (3) so that p, p' also become unity when e, e' vanish. Then since R is a homogeneous function of r, r' of degree 1, and since we assume that a < 1, we may write it by a double application of (1). The eccentricities, so far as they occur through r, r', are contained explicitly in the factors out- side the functional sign only; whether they are present in a or not is immaterial to the developments of this chapter. Again, regarding R as a function ofy, /', and putting ............... (5) 88 DEVELOPMENT OF DISTURBING FUNCTION [CH. iv where E, E' vanish with e, e r and -^, ^ are independent of e, e', and remembering that/*,/' occur only under the signs sine and cosine, we have, by (2), ^l) = F(ex]). E V^l . G xp. ^ V^T) Then and for two variables, (^f). ...(6) <ty ' * Again the eccentricities are concentrated in the factors outside the functional sign. Owing to the fact that p is of the form 1 + pi, where | pi | < 1, the two forms of expansion pointed out in 2*7 are available. The binomial form gives 1) + ..., ...(7) r v- 1 - ri/ * i pi-*-' ! 21 and the exponential form, where The latter form is valuable chiefly when the numerical value of e is used, since the coefficients in the functions of p are then numerical, and numerical harmonic analysis is efficient for the expansion of the powers of log p. Similar remarks apply to the expansion of p'"^" 1 which should be made in the form (9) The harmonic analysis is made with the functions l/p', logp'/p',... 6-8] IN TERMS OF THE TRUE ANOMALIES 89 4'7. There appears to be no escape from the fact that the development of the disturbing function requires a five-fold series, Developments along powers of e require in reality a double series because r,/ require different methods; we may make various combinations of them but the duplicity remains. A similar statement is true of r',/'. The development along powers of the inclination is also double but has been made essentially single by the device of including the factor cos 2 J7 in the functions of a (cf. 4*13(3); also the last paragraph of 4*31). And finally we have the development along powers of a. Out of this six-fold development, one-fold of the development can be avoided by the proper use of the fact that R is a homogeneous function of r, /. With the methods given in this chapter, the development takes the form of series along powers of the inclination and multiples of the difference of the anomalies, and along powers of the three operators D, B y B', the possibility of such expansions being due to the fact that any given power of these three operators has as a factor of its coefficient the same power of the eccentricities. 4*8. Expansion of I/ A along powers of e, e' and multiples of '. This requires the substitution for r, r' of the expressions, 1+ccos/' l + e'cos/' As in 3-3, put = exp./V^l, e(l +<rj 2 ) = 2<rj, and let (I-.? 2 ) 2 1 -- -fr with similar expressions for r', ?</, //. Thence, according to 4*6 (4), - - where Ai 2 = 1 -I- 2 2a cos >, =-^, u-=a~. ...(4) r </a The expansions of p D , p f ~ D " 1 into Fourier series with arguments f,f and coefficients depending on positive integral powers of JD 90 DEVELOPMENT OF DISTURBING FUNCTION [CH. iv are given by the formulae of 2*1 G. It will be noticed that p D is equivalent to the function expanded in 2'16 if we put ?;, D, D, <f> for a, s, t, x, respectively. Hence ...(5) with ; = 0, 1, 2, . .., the factor 2 being omitted when^ = Similarly, In forming the product of these two series, the rules noted in 2*19 are to be followed. This product, inserted in (3), gives where F t F' denote the hypergeometric series in (5), (6) respec- tively. This is the required expansion. The portions of the coefficient which depend on D have to be expanded in positive powers of D and these are operators acting on 1/Ai which contains a only in the explicit form shown in (4). In these expansions it is important to notice that any power of D is always accompanied by at least the same power of 77, 77', so that the number of powers of D required is the same as the order with respect to the eccentricities to which the expansion is to be developed. 4*9. It is sometimes more convenient to use 2*16 (5) or 2*16 (4) for the developments. The necessary changes are easily seen. If we use the former, the formula 4'8 (7) will still serve if we put a , a' r /1X " -D-2,j + l, if), ...... (2) and multiply the result by (1 ?; 2 )/( 1 7/ 2 ). In adapting tho work to this formula, w-3 have put -V s ? 8-13] IN TERMS OF THE TRUE ANOMALIES 91 If formula 2*16 (4) is to be used, we put (4) with F-F(l-D,D,j + l,-*), F' = F2 + D,-D-l,j+l, f_ /2 . . . .(5) In adapting to this formula, we have used 4*10. The operator ^1 = D 4- ^ has one advantage over D which may render its use advisable in some problems. This advantage results from the fact that when the expansion of the operator ( . J F has been made in powers of A, that of ( ., J F' can be immediately written down by changing the sign of A and substituting 77' for TJ. 4*11. The expansion of any function of r, r', A can evidently be made by exactly the same methods. We have, for example, the expansion of which follows exactly the same plan. 4'12. The complication of the various values of a over that usually used, namely a/a', is more apparent than real, since the numerical value of a is always used. Further, as TJ is greater than rf in most asteroid problems, and as the convergence is improved by diminishing a, there is an advantage with these values over the value a/a'. The slight disadvantage which arises when we have to differentiate with respect to e or rj is easily dealt with by adding to the derivative with respect to ^, so far as it occurs explicitly, the derivative D.da/adri. 413. Development along powers of the inclination. In the previous paragraphs I/A has been developed into cosines of multiples of/,/', with coefficients which depend on 77, ??', 1/Ai and on the derivatives of 1/Ai with respect to log a. Now .................. (1) 92 DEVELOPMENT OF DISTURBING FUNCTION [CH. nr where, by 4*1 (3), cos S = cos 2 \I cos (v - v') + sin 2 / cos (v + v'~ 20). . . .(2) The general plan requires the expansion of 1/Ai into a double Fourier series with arguments v v', v + v' 20, and this might be achieved by first expanding into a Fourier series with argu- ment S and then expanding cos iS into sums of cosines of multiples of these two angles. More rapid convergence with less computation can be obtained by making the development depend on the Fourier expansion of 1/A , where A 2 =l+a 2 - 2a cos 2 17 cos (>-?/), ......... (3) rather than on the same function with cos 2 \I replaced by unity. With the definition (3) of A , we have sin 2 1/ cos (?; + /- 2(9)) ~* which is then expanded by the binomial theorem. This expansion evidently involves odd negative powers of A accompanied by even powers of sin \I. The powers of cos (v + v' 20) are to be expressed as cosines of multiples of the angle. Instead of giving the general form of this expansion, we set it down as far as the eighth power of sin \I which will be sufficient for all practical needs. Define RI, R 9 , ..., by where 2s takes the values 3, 5, 7, .... We then obtain -- = RI + RS tan 4 II + J R 9 tan 8 \I AI tan 2 \I + R? tan 6 \I) cos (v + v f - 20) + (R 5 tan 4 \I + jff 9 tan 8 /) cos 2 (i; + v' - 2(9) + JJ?7 tan 6 |J cos 3 (v + v' - 20) + T \, # 9 tan 8 J/cos 4 (t; + / - 2(9). ...... (5) In this development, it should be noticed that a multiple of in the angle is always accompanied by at least the same power of tan %I in the coefficient and that the series in any coefficient proceeds by powers of tan 4 J/. 13, 14] IN TERMS OF THE TRUE ANOMALIES 93 The angles v, v 1 are expressed in terms of the true anomalies by means of the relations t; = /+tar, v'=f +r' ................... (6) The development in powers of the eccentricities contained the angles f y f only, while this development contains the angles f-.f+n->v', /+/++' -20. When the functions R% 8 have been developed as Fourier series, and products of cosines replaced by sums of cosines, we shall have a development containing multiples of the four angles /, /', w -w', v+v'-20, and this development will have the property that the difference, taken positively, of the multiples of f y f in any angle will be accompanied by the same power of e, e', I in the coefficient. 4'14. Development in multiples off, f to the third order. The development of the RZS is given in a later section of this chapter, (4*23), in the form .fi 28 = W>COsi(/-/ + *r-O, /8 f l -> =&<>, *=0,1,2,.... ...... (1) The coefficients are functions of a, cos* \I only, and the operators D> act solely on these coefficients. The value of a given by 4 - 8 (2), 4'8 (4) is used here. By carrying out the various steps outlined above, we obtain the following development as far as the third order with respect to t), i)', tan \I. ^ = ( F! 2/3 4 (t) + 2 tan 2 / . F 9 2/3 t M ) cos i (/- where v'-O 2 (D + 1) + ip/ D(D + I) 2 } cos/ + {27?' (D + 1) + V s D(D + I) 2 + Vl** 8 (D + 1)1 cos/ + (T) 2 + D) {? cos 2/ - 2 W ' cos (/ / ' ) + 7/ a cos 2/' J - JT;* (D 3 + 3J5 2 + 2D) cos 3/ + J V 3 (^ 3 - -) cos 3 /' 94 DEVELOPMENT OF DISTURBING FUNCTION [CH. iv F z = cos (j +/' + @) - vjD cos (2/+/' + @) - yD cos (/' + vf(D + l)cos(/+ 2/' + @) + V (D + 1) cos (/+ 8), = w + w ' _ 20. The double sign means that there are two terms each having the coefficient set down. The final step, that of expressing the products of the cosines in FI, FZ by cosi(ff' + '& 'GT') as cosines of sums and differences of the angles, is to be carried out. This is equivalent to adding the angle i (ff -f VT r') to each of the angles in FI, F B , because /3 a ( " i} = /3 8 (l) . The term in FI independent of r, f requires no treatment; i receives all positive and negative integral values and zero. 4*15. Transformation from the true to the mean anomalies. The development in terms of the true anomalies consists of a sum of terms of the type A cos (jf+ff -f (7), where A depends on a, a', e, e', I and G on sr, -57', 6. To transform to a development in which the arguments are functions of the mean anomalies, we make use of the expansions f=g+E = ff+2fi*mig, f = g' -f E' = g' -f 2// sin tf, obtained in 311 (6), together with 4'6 (6) which gives cos (jf + j'f -f C) = exp. (# 1 + E' |>) cos (jg + $<f + C). ...... (1) The exponential is expanded in powers and products of Ed/dg, E'd/dg', and this requires the expression of powers of E, E' as Fourier series with arguments g, g f \ the operators d/dg, d'/dg r act only on the explicit functions of g, g' and not on E, E' . The same result is reached by writing cos (jf+ff + C) = cos (jg + j'g' + G) cos (JE +/#') - sin (jg +j'g' + C) sin (jE+j'E'\ , . COS . r, COS .. ~, and expressing 1 . ih. . i & * * sin* 7 sm J as Fourier series with arguments g, g' respectively, for the different values of j, j' needed. The calculations of the functions 14, 15] CHANGE TO MEAN ANOMALIES 95 of E, E' needed can be made in series or numerically by har- monic analysis. Properties of the expansion. Since R is independent of the directions of the axes of the frame of reference, it is independent of the origin from which the angles used in the expansion are measured. Hence the algebraic sum of the multiples of such angles present in any term is zero. Thus if w y w f be the mean longitudes, and CT, CT', 6 the longi- tudes of the perihelia and node, and if any argument in the expansion be iw + i'w' +JK+J'<GT' + 2hV, ............... (2) we have i + i' +j -h ;' + 2h = 0. The original form of R was an expression in terms of v, v', r, r' , F, 20. It was pointed out in 3*16, that the expansions of v, r in terms of g or v -GJ are d'Alembert series as far as the associa- tion of powers of e with multiples of cr is concerned, and the same is true of v', r' with respect to e 1 ', -CD-'. It follows that R has the same properties. Further, the expansion 413 (5) shows that R is a d'Alembert series with respect to F, 26. It follows that the coefficient of a term with the argument (2) is of order j j | -f | j f | -f | 2h | with respect to the eccentricities and inclination. But + ' Hence, the order of the coefficient of any term in the expansion of R is equal to or greater than the algebraic sum of the multiples of w, w' present in that term, and the same is true for the multiples of g, g' when we put w g + &, w' g' -f r'. This property at once gives the lowest order of the coefficient of any term in a numerical expansion of R. When numerical values of the elements are used, the expression for tt'/A in terms of the true anomalies may be put into the form *Q tf cos(tf+j'f') + 2S Jt ,*m(jf+j f f), j,/ = 0, 1,..., or 2 coBJf(C Jtf cos//' 4- 8 Jtf sin//') 'jj, cos// v + S' Jtf sin//), j, / = 0, M, . . . . 96 DEVELOPMENT OF DISTURBING FUNCTION [OH. iv The portions in brackets are transformed to multiples of g f by the relation /' = g' + E', either by series or by harmonic analysis. The series are then re-arranged in the form 2 cos JY (A jtj . cosjf+ B jtj > sinjyT) + 2 sin JY (A' jtf WBJf+B'jj sin jf), j, j r = 0, 1, 2, . . . , and the change to multiples of g is carried out by using the relation f=g + E. By following this procedure we can limit the additional work required to obtain the coefficients of the long period terms to a higher degree of accuracy, owing to certain peculiarities in the series for r,/. 4*16. The value of r in terms off is the series r=a aecos/-f Actually, this gives series along powers of %e, because the long period terms always arise from the expression of the product of two cosines as the sum of two cosines; only one of the latter is needed more accurately. The same is true of r'. But when we substitute for/ in terms of g by means of the series we are substantially expanding in powers of e instead of 0; the coefficients, which depend on , are in general of the same order of magnitude for the series giving r in terms of/, and /in terms of g. Further, many of the actual problems are those of asteroids disturbed by one of the great planets and the eccentricities of the orbits of the latter are small. Thus while the steps up to the last have to be carried out to the full degree of accuracy, the series converge rapidly. The convergence is slowest in the last step, but it is here that we can make selection of the terms which have to be accurately computed, the remainder requiring a much lower degree of accuracy. The calculation of the coefficient of a particular term can also be efficiently carried out by the method which follows. 4'17. Calculation of the coefficient of a particular term. For this calculation we can make use of the theorem of 2*6, where it is shown that the coefficient of cos (ig + i'g'} in the 15-18] % THE CONSTANT TERM 97 expansion of F(f,f) is the same as the constant term, when g, g' are expressed in terms of/, /', of one of the expressions For the coefficient of sin(ig + i'y'), change cosine to sine in (1) and sine to cosine in (2). To make use of them we have the relations ___ ___ df~ a ^i- e *~( ztani/, ...... (4) with similar expressions for accented letters. An alternative to (3), (4) is the use of the series 3'8 (3). If we make use of the first of the forms, the initial expansion of r 2 /r /2 A instead of the expansion of I/ A, would be made: the method for doing so is shown in 4*11, and the values p = 2, p' ~ 2 in 411 (1) would be used. With this formula the ex- pansions 4*9 (2) or 4*9 (5) are recommended. In F we replace D by D -f 2 and in F f we replace D by D - 2: the same changes must of course be made in the binomial coefficients in 4*8 (7). If, however, the plan is used to get a particular coefficient to a higher degree of accuracy after a general development of I/ A has been made, the second form of (1) is of advantage because the development already made will serve; in such cases neither i nor i f is zero, so that this form is always available. 4*18. Calculation of the constant term. This is sometimes needed to a high degree of accuracy. Ac- cording to the theorem of 2*6 the constant term in the expansion of I/A in multiples of g, g' is the same as the constant term in the expansion of 1 dgdg' = rV 2 A dfdf A . aV a VT^T 2 Vl - e' 2 B&SPT 7 98 DEVELOPMENT OF DISTURBING FUNCTION .. [OH. iv in multiples of/,/'. The use of the formula 41 1 (1) with p = 2, p = 2 is indicated. This expansion does not require the use of the relation connecting / with g or /' with g' ; it depends solely on expansions along multiples of /, /', and therefore requires merely the substitution for r, r' of their expressions in terms of /,/'. A literal development to the eighth order with respect to the eccentricities and inclination is to be found* in Astr. Jour. vol. 40, pp. 35-38. 419. When harmonic analysis is used to obtain functions of /in terms of g and those of/' in terms of #', the computation can be made as follows. If multiples of g not higher than the sixth are needed the seven special values of g namely, 0, 30, 60, 90, 120, 150, 180, are recommended. If two more values be needed, those for 45, 135 can be added, and with two fewer, those for 30, 160 can be omitted. It is useful to notice that the addition of new values does not require the greater part of the work, which is the computation of the special values of the function, to be done again ; only certain small portions of the analysis have to be repeated. The values of X for the chosen values of g are obtained from Kepler's equation X*=g + emi\ X, by one of the methods given in 3'17. From the relation T~~ tan J/ the special values of /and thence those of any function of /are then ob- tained. As / g take the values 0, 180 together, there are only 3, 5 or 7 special values of / to be computed for each planet. Methods for analysis into Fourier series are given at the end of this volume. The functions of /needed are cosjf/j shy/ for a number of integral values of ,;'. It is more convenient to calculate cos^' (fg\ sin,;' (/ g\ and after- wards to deduce the expansions of cos.;/, ainjf by the use of the factors coajg, sin^. Most of the asteroid problems require the calculation of the perturbations by Jupiter and Saturn only. The series for cosji/', sin j/', once computed for these two planets will serve for all cases ; small changes in the values of the eccentricities are easily made since the power of e which accompanies any term is known by the multiple of g in its argument. Harmonic analysis is usually so much more accurate with respect to con- vergence, and is so much more easily controlled than literal expansions, that * E. W. Brown, The Expansion of the Constant term of the Disturbing Function to any order. 18-20] IN TERMS OF ECCENTRIC ANOMALIES 99 it should be used whenever possible. Where many such analyses are to be carried out, a systematic arrangement of the work, by which one operation at a time is performed on all the functions to be analysed, permits the calculations to be carried out rapidly and accurately. See App. A. 4*20. Development in terms of the eccentric anomalies. The expressions in 4*1 give I/A as a function of r,/, r',/', 6, /. Also in 3-3 (8) and (9), with the notation < = exp. / V^HT, % = exp. X V 1, where X is the eccentric anomaly, it has been found that with similar expressions for r', </>'. If we put we can make use of the theorem of 2*7 with four independent variables. Now I/A is equal to a function of r/r', <f>, <f>' divided by r'. Hence, if A^. ?V be what A becomes when we replace r, r', <, </>' by r , r ' } %, %', the theorem gives xV-v'x'r*- 1 -*' ....... (i) This has to be expanded in powers of the indices. The expansion of the first pair of factors is made by the theorem of 2-16. It gives J 3 ' ...... (2) 7-z 100 DEVELOPMENT OF DISTURBING FUNCTION [OH. iv The result is unchanged if we change the sign of V 1, for then %, l/x interchange and also B y B\ it is therefore a real Fourier series with argument X. The product of the second pair of factors is obtained by putting - D - 1, *;', ', /, / for D, rj, B, x> j respectively in (2), and it has the same properties. Since /, 6 are present in A^ in the same way as they were present in AI, the expansion of A^ along powers of the inclina- tion follows the same plan as that of A x in 4*13. In fact, if we put v = X + er, v' = X 1 + *' in 413 (3), 4-13 (5) (taking note of the different significations of r , r ', a), the results can be used here without further change. Newcomb has given (Astr. Eph. Papers, vol. 3) & detailed expansion of the disturbing function in terms of the eccentric anomalies, certain portions of which are taken to the seventh order with respect to the eccentricities. He uses an operator but did not obtain the general formula which permits any coefficient to be written down at once. The latter was given by one of us (E. W. Brown, Astr. Jour. vol. 40, p. 19, 1930) in terms of the operator D and certain integers i, i' and later (Astr. Jour. vol. 40, p. 61, 1930) in the improved form shown in the text with the use of the operators Z>, B^ B '. 4'21. Transformation from eccentric to mean anomalies. After the disturbing function has been expanded in cosines and sines of multiples of X, X', the transformation to mean anomalies can be effected by the formulae of 3'10, which give i\ cosjX = A H- S - J K _J (tee) cos /eg, \ *=1, 2, ...(1) sin jX = S< - J-j (tee) sin teg, or, if f = exp. g V 1, in the exponential form, tf-A + Z.ij^ice).?, (2) K where -4 = or \e according as j =}= 1 or j = 1 ; j may have any positive or negative integral value. A similar set of formulae holds for accented letters. Since the only terms in the development which give a constant 20-22] CHANGE TO MEAN ANOMALIES 101 part are those containing the first multiples of X, X', the constant term of the development in terms of g, g' is obtained by adding to that in terms of X, X f , the terms - \e . coef. of cos X - \e' . coef. of cos X' ee f + -T - sum of coef. of cos (X X') ....... (3) The method developed in 4*17 for the calculation of the coefficient of a particular periodic term and that in 4*18 for the constant term can evidently be applied to the transformation from eccentric to mean anomalies. For functions of/, /', we substitute functions of X, A'', with But the coefficients in the expansions of costjX+j'X'), sin (JX+j'X') in terms of g, g' can now be written down in terms of Bessel functions, as defined in 2*14. For both periodic and constant terms, this process is equivalent to that in the text and is merely a different mode of stating it. 4*22. A detailed comparison of the relative advantages of a primary development in terms of the true or eccentric anomalies appears to favour the 'former. In the first place, the expansion in terms of the true anomalies requires the use of only one operator Z>, while that in terms of the eccentric anomalies requires the use of three operators D y /?, B'. Those three operators produce both cosines and sines while, with the operator D alone, only cosines are present, and therefore in reducing products of cosines to sums of cosines, there will always be pairs of coefficients which are the same. It might be thought that the change from true to mean anomalies is more complicated than that from eccentric to mean, because we cannot use general formulae like the Bessel functions to make the change. As a matter of fact, the actual labour of making the expansions differs very little in the two cases, whether literal or numerical values of the eccentricities be used. The developed series in powers of e have to be used in either case, and such series are available in the tables of Leverrier and Cayley, if a literal ex- pansion is desired. With a numerical expansion by harmonic analysis the only additional work is the calculation, for a few special values, of the values of/, after those of X have been found, from the equation 3-7 (1). A point connected with the rate of convergence along powers of e, and rarely mentioned, deserves some stress because the work of calculating the coefficient of some particular term to an order of accuracy higher than that of the general development can be made lighter by taking it into considera- tion. It has been pointed out in 4*16 that the rates of convergence of the 102 DEVELOPMENT OF DISTURBING FUNCTION [OH. iv series for r, / in terms of /, /' are more rapid than those of /, /' in terms of g, cf. There is no such difference in the rates of convergence in passing directly from r,/, /,/ to X, X' and from X, X' to g, g'. Thus the longer development in terms of X, X ' must also have the full accuracy desired while the shorter development in terms of/, /' is still more abbreviated by the separation of the more rapidly converging series from the more slowly converging series. The advantage possessed by the expansion in terms of the eccentric anomalies in the form given in the text, consists in the fact that it is the only method known by which any coefficient in the development of the disturbing function can be written down from a general formula ; it con- tains the operators /), /?, B', and the Bessel functions. The highest power of these operators present in any portion of a coefficient is the same as the order of that coefficient with respect to the eccentricitie.s, so that stoppage at a given power of the latter involves stoppage at the same power of the former. The order of a Bessel function is known from its suffix. But the formula suffers from the defect pointed out in the paragraph following 3*11 (6) for the case of the general expansion of/ in terms of g, namely, that ^'-f-1 numerical coefficients have to be added together to obtain any part of order ; ; this defect becomes serious when^' is large. 4*23. The functions of the major axes. The development of the previous sections of this chapter require the calculation of the coefficients /3 8 {i) defined* by /)*-* () - (1) where t 0, 1, 2, ...; 2*= 1,3, o, ...; &<->-&<>; and for 25 = 1, the numerator of the fraction is unity. The definitions of T/T, a are immaterial to the work of this section provided |a| < 1, We shall need also the derivatives of the coefficients with respect to a or to log a. It has been pointed out in 4'2 that the magnitude of a in general prevents the use of literal series in powers of a as a practical method for calculation and we must consequently use other devices. In the following paragraphs, transformations of the series in powers of a are made for two of the coefficients so * This definition of the coefficients without the factor J is so much more con- venient than that of Leverrier and others who have followed him that we have retained it throughout. See 2*19. 22-24] FUNCTIONS OF THE MAJOR AXES 103 that they may be easily and rapidly obtained. It is then shown how all the remaining coefficients and their derivatives can be deduced from these two by the use of finite formulae. The two coefficients to be first found will be those for s = , i= 10, 11, for reasons which will appear. The more usual plan has been to calculate /%" = ^i> & (1) = | (*i - Ei) ............. (2) where FI, E\ are the elliptic integrals of the first and the second kind, from the tables of Legendre, with cos 2 7=1. The tables of Runkle (Smithsonian Contributions, 1855) give certain of these coefficients for different values of a; those of Brown and Brouwer (Camb. Univ. Press, 1932) have higher accuracy. 424. The series for &>. Define i, * by* ! aeosH/ - 1+ai 2= i + rf . * = cosi/, ............ (1) so that (1 + a 2 - 2a cos 2 \I cos f )~* = (~-Y( 1 -f i 2 - 2 x cos ^)~ s . . . .(2) VCE/C/ The last factor may be expanded into a Fourier series by the method given in 216. By inserting this expansion in 4'23 (1) we obtain ' ~ i)' ^- 2 cos = a t s (a*)-* S illl.Lr_ rtl . /' . 2 cos t>, ...... (3) where F is the hypergeometric series given by and i takes the values 0, 1, 2, ..., the constant term under the sign of summation in (3) that for i = being F(s, s, 1, i 2 ). * The value of c^ can be readily found by putting aj = tan A\ , a = tan A, and finding A l from sin 2A t K sin 2A . 104 DEVELOPMENT OF DISTURBING FUNCTION [CH. iv By means of the transformation 2*15 (3) we have F(8 + i,8,i + l,a l *)~(l-a)-'F(l- 8,8,i + l,- p\ where p = i 2 /( 1 - i 2 ), ..................... (5) so that in which s s-1 ,( + !) (s-l)(s-2) This last form of the hypergeometric series is evidently useful for large values of i, since in this case the earlier coefficients of powers of p diminish rapidly. It is true that the series converges only when \p\ < 1, that is, when i< 2~*= 707, and that the values of the coefficients are sometimes needed for values of i larger than this. But we know that the function which the series represents has no singularity provided |i| < 1, that is, provided p be finite. It is therefore permissible to use the method of analytic continuation to obtain expansions in powers of p PQ where p Q =(= 0. 4*25. The calculation o/ai , an. For brevity, let us put fi (i) = a t -, so that (1 -f a 2 - 2a cos 2 / cos ^r)~* = a + 2Sa t - cos fy t ...(I) with 2 2 The transformation of (3) to series expanded in powers of p p Q can be effected by the use of Taylor's series: The values of F(p Q ) t F r (p ) are obtained directly from the series, and the remaining derivatives from the recurrence formula 4, 26] FUNCTIONS OP THE MAJOR AXES 105 2*15(5) deduced from the differential equation satisfied by the hypergeometric series*. On putting (1 i 2 )^ = #i^~^, we have the following expressions for the cases j9 =0, J, 1. The factors 46,189 and 176,358 are the respective products 11 . 13 . 17 . 19 and 2 . 13 . 17 . 19 . 21. /a~v / \ 9 aio = 46,189 A/ -- ( * ) multiplied by one of the series, V /ca \4/ r J + 1-00000000 - -02272 727 p + -00213 068 p 2 - -00034 15 p 3 + -00007 47 jo* - -000020 p* + -000006 ;> a u = 176,358 tea + 0-98913047 +0'97912 120 - -02082065 (p-i) - -01926520 (/>-!) + -0017101 (p-V? + '00141681 (V-l) 2 - -0002304 (p-l)* - -00016 594 (p- 1) 3 + -000041 (/>-4) 4 + -00002 515 Qt>- 1) 4 - -000009 (jo-i) 6 - -00000451 (p'-l) 6 + -000002 (/J-i) + -00000091 (^-l) 6 - -00000020 (p-l) 1 + -00000005 (^-l) 8 00000001 (p-1) 9 multiplied by one of the series, /<*ip(<*i V ^ 10 + 1-00000000 - -02083 333 p + -00180 288 p' - -0002683 p + -00005 48 p - -00001 4 'p 00000 1 p ( + The series give < -82. The table a! = -45 = '25 + 0-99000 356 +0*98074 527 - -01920 796 (p-i) - -01786420 (jo-1) + -00146840 (p-b) 2 + -00123022 (p-1) 2 - -0001854 (p-l? - -00013597 (p- 1) 3 + -0000311 Q0-I) 4 + -00001956 (p-1) 4 - -000007 (p-) 5 - -00000 335 (p- 1) 6 + -000002 (p-$f + "00000 065 (^-1) () - -00000014 (jo-1) 7 + -00000003 (/>-!)* - -00000001 (p-l) aio, an to eight significant figures when -61 '66 '71 '78 -82 '88 -50 75 1-00 T50 2'00 3'00 indicates the series to choose for any given value of i. For a x = -88, the error is about one part in 10 5 . * E. W. Brown, Mon. Not. R.A.S. vol. 88, pp. 459-465. The numerical series given in the text for2? = |, 1 are taken from this paper. Extensions to the cases p = 2, 3, 4 are to be found in Mon. Not. vol. 92, pp. 224-7. 106 DEVELOPMENT OF DISTURBING FUNCTION [OH. iv Alternative form for a 10 , a n . The ratios of consecutive coefficients of these series all tend to the limit unity, but for large values of i the approach is very slow, and after the first two or three terms the ratio changes slowly. We can make use of this fact by expressing the series a -\-a l p + a 2 in the form with a suitable choice of A. In this way the following expressions have been obtained : a 10 = '35239,4104 where a u = '38183,0736 (&) a \ K * / Sa 10 = (2131 + 245/7 - 19/> 2 )p 2 . Q + (1 + 275^), These give 10 , r H to six significant figures when figures for p<\ are furnished by the expressions, a, = l ~p p* - "00034,1 46/> 3 + '00007,469^ . Eight significant ' 829 ^ 3 + '00005,477/J 4 4'26. Formulae for calculating the /3 (<) t^Ae/i ^M;O consecutive coefficients are known. The procedure which appears to give numerical results most easily requires the use of the following formulae. It is to be noticed that as soon as two consecutive coefficients have been found, there is no further need of i; the formulae involve only a, K in the form The proofs are given in the sections which follow. For finding the remaining a t - = /3| ( * ) when two of them are known, we have the formula, a i = ea m - a +1 + - (ca m - 2a <+2 ) ......... (2) 25, 26] RECURRENCE FORMULAE 107 for values of i lower than those known, and for higher values of i. These formulae are used by putting i = 9, 8, ... successively in (2) when a, an have been found; and i = 10, 11, ... as for as they are needed in (3). They are deduced from the general formula, * ~ * &<>, ......... (4) 1+ S I + S by giving to s the value J. The values of two consecutive coefficients for other values of s are found by putting * = ,,..., successively in the formulae - - - 4( 6 _2) ~~ "" 2s) ^ + (2 + 2 - 2s) /9^ +1) " ' The remaining coefficients are found rapidly from the formula which may be used backwards or forwards along values of i. Sufficient checks on the numerical work are obtained in the following manner. When a 9 , a B) ..., a Q have been successively computed by means of (2), the value of a can be obtained directly from 4'23 (2) with the use of Legendre's tables: this, in effect, tests the whole series of cti . When /3 f (11) , ..., # f (0) have been found from (5) for the first two and from (6) with s = \ for the remaining coefficients, the values of y8^ (1) , /3J * can be tested by computing from (5) with s = , i = 0. A similar procedure tests the values for s = f , , .... In general, there is a loss of less than one significant figure in running down from y8 8 (10) to /3 (0) . There is some loss of accuracy in the use of the formula (5), but this loss is balanced by the fact that the higher values of s are present only with the higher powers of the inclination and indirectly of the eccentricities. 108 DEVELOPMENT OF DISTURBING FUNCTION [CH. iv 4*27. Formulae for the derivatives of the p 8 (i) . The first derivative is obtained from for all values of i with s ^, and for two consecutive values of i with s = f, f, ..., the remaining derivatives being found most easily from f> = iD/Sf? + D&? ................ (2) For the higher derivatives, either of the following formulae in which the index (i), being the same throughout, is omitted, may be used: Z)'+ a & = (D--*" 11 ~ a (3) = (2*-l)JD'+ l & + (D + lXA+i /c >,, + 4,(l-- z To j are given successively the values 0, 1, 2, ...: the latter formula requires the calculation of one fewer set of coefficients /3 8 for a given degree of accuracy with respect to the eccen- tricities (cf. 4*31). These formulae are used like those which precede it. The derivatives for s = J and all needed values of i are computed from either (3) or (4). For the remaining values of s, they are used for the computation of the derivatives for the two highest values of i only, the remaining derivatives being found more easily from &&? = <&&?+&/*?. (5) These alternative methods of computation furnish obvious checks. 4*28. The proofs of the preceding formulae are obtained by treating the fundamental expansion, c 8 (1 4- a 2 - 2/ca cos i/r)- = (a*)*-* 2& (i) cos ty, . . .(1) where c f i.f ...(*-!), i = 0, 1, 2, ..., as an identity. 27, 28] PROOFS OP RECURRENCE FORMULAE 109 The derivative of (1) with respect to f gives sc 8 (1 4- a 2 2/ca cos ty)-*- 1 . 2/ea sin ty = (*)*"* 2/3 fi (?) i sin if. ...... (2) Replace 5 by s 4- 1 in (1) and insert the result in the left-hand member of (2). Since c s +i = sc 8 , we obtain 2 sin i/r 2/3^ cos ty = 2/3^ i sin ty. The left-hand member of this equation may be expressed as a sum of sines of multiples of f. Equating the coefficients of sin i\|r, we obtain ' which is the formula 4'26 (6). The derivatives of this give 4'27 (2), 4-27 (5). Again, multiply (2) by 1 + a 2 - 2*a cos i/r and insert (1) in the left-hand member of the result. After some reduction, we obtain Ka S (i + s) $M sin (i + 1) ^ + *a2 (i - s) /3 8 (i) sin (i - 1) ^ = (1 +a a )SiA (t) sini>. In each of these series i takes all integral values from + oo to oo . The selection of the coefficients of sin (i + 1) tjr, from each of them, gives *a (i + s) &* +ica(i+2- s) ^ +2) = (1 + a 2 ) (i + 1) #' +1) , which is the same as 4*26 (4) when we put e = (1 + a 2 )/ tea. Finally, the identity (1 + a 2 - 2*a cos ^) S ^^ cos ty = 6-a/t 2 ^ cos if, obtained from (1), with the same equation when s + 1 is put for s, yields by the same procedure as before, ) ....... (4) If we successively eliminate $+i\ff*+^ between this equation and (3), we obtain Change i into i 1 in the former of these equations and add the result to and subtract it from the latter; the two equations thus obtained are the same as 4-26 (5). 110 DEVELOPMENT OF DISTURBING FUNCTION [CH. iv 4'29. The values of the derivatives are obtained as follows. The derivative with respect to a of the logarithm of 4*23 (1) gives, after multiplication by a, 2s (a 2 /ca cos ^) 1 -f- a 2 - 2*a cos ^ <?n a 2 \ o ^ JL ^^ 1* / 2 1 + 2 2/tra cos ty ' But, from the definition of Rzs, + a 2 2/ea cos ^ Ktt. I a 2 Hence 01^ = ^^+ - ^2*^2 KCd From this equation, by replacing R^ 8 , R^ s ^ by their expansions in Fourier series and equating the coefficients of cos ity, we obtain which is the equation 4'27 (1). The application of the operator D j to this with the help of the general theorem, D'{oflf(a)}=<#(D + qy.f(a), ............ (1) furnishes the equation 4*27 (3). 4'30. The proof of the relation 4'27 (4) is more difficult. If we put p = 1 + a 2 2a/c cos yfr, .................. (1) we have (Dp) 2 + \ = (2a 2 - 2a/ccos >|r) 2 4- 4a 2 /c 2 sin 2 ,fr = 4V-f 4a a (/c 2 -l), s-A = 4a 2 2a/ cos ->|r -f 2a/t cos ty = 4a 2 . dy^ Also J5V' = 29-31] PROOFS OF RECURRENCE FORMULAE 111 Whence, with the aid of the two previous equations, = * ( s (s -f- ...... (2) Now, the definition (1) and 4'23 (1) give P~' -(a*)*- RI. + C. ................... (3) and the theorem 4'29 (1), applied to this, gives Insert this result in the left-hand member of (2) and eliminate /> from the remaining terms of (2) by the use of (3) after re- placing s by s + 1, s + 2, therein. With the help of the relations o 8+ i = sc 8 , c s+ z = s(s+ 1) c st we obtain, after division by suitable factors, (D + * -s) 2 . The final step is the insertion of the Fourier series and the equating of the coefficients of cos ity. This process gives which is easily seen to be the same as equation 4*27 (4) when we putj= in the latter. The result may be written The application of the operator D i to this equation and the use of the theorem 4-29 (1) give 4'27 (4). 4*31. The statement in 4'27 contrasting the formulae 4*27 (3), 4*27 (4), may be justified as follows. The continued use of 4'27 (4) makes D*/3 8 depend on the calculation ot/3 8 +j,{ii+j-i 9 . . .,&. If we put j = in this equation, the right-hand member depends on Dftg, j3 s +i which require the calculation of &+i, /8 only; the 112 DEVELOPMENT OF DISTURBING FUNCTION [OH. nr factor of /9, f 2 is 4 tan 2 \I (1 + sec 2 /). Since the operator D 2 is always accompanied by the square of the eccentricities in the development of the disturbing function, the effect of this last term is of the fourth order. The argument for values of j greater than zero in the formula is similar. Incidentally, this formula shows why a considerable increase in the convergence along powers of the inclination is obtained by the insertion of the factor tc. In general, the coefficients /3 8 +} tend to increase with j for a given value of i and the factor 4> which occurs in this term shows that it will modify the values of the derivatives considerably when / is large. The additional computation caused by its presence is very small *. 4'32. The literal expansion of I/A to the second order, in terms of the mean anomalies. This expansion is obtained from that in terms of the true anomalies given in detail in 4*14. The latter contains products of cosines which are expressed as sums of cosines as explained in that paragraph. In writing the result out to the second order, the notation tf^a^a-t, 4 = 6, -6-,, /-/' + w -*r' = *-', will be used. The result is r * = sum for all positive and negative values of i, including zero, of (1 + ^ L 2 + 77 /2 (D + 1) 2 } a< cos i (v - 1/) - 27777' (D 2 + D) ai cos (iv - iV + /- /') - 277 Da { cos (iv - iv' +/) + 277' (D + 1) a; cos (it; - iv' + /') 4- T? 2 (D 2 + D) at cos (iv - iv' + 2/) + 77 /2 (D 2 + D) a< cos (iv - tV + 2/ x ) - 27777' (D 2 + D) a t cos (iv - iv' +/+/') -20) ................ (1) * These modifications of the usual formulae in which * = 1, were given by E. W. Brown, Mon. Not. R.A.S. vol. 88, pp. 459-465. 31, 32] TO THE SECOND ORDER 113 The transformation to mean anomalies is made by means of the theorem 415 (1), namely, cos (tf+j'f + 0) - exp. (EJ-+E' g2>) cos (jg + j'g' + 0), where f-g + E, f'=g' + E'. The expression for Z? is given in 3*16. Expressed in terms of 77 as far as the second order, it is E = 4*77 sin g -f 5?? a sin 2jr, and we have a similar form for E' . From these we deduce -2 = 8?? 2 _ 8?? 2 cos 2( /) EE'=8w' {cos (g - #') - cos (g + #')}. With the aid of these formulae and of the expansion exp. tf + 3 / we obtain the following development: = t x sum for i 0, 1, 2, ... of * {1 -f 7? 2 (D 2 + 4D - 4i a ) + V 2 ( 2 - 2D - 4i a - 3)} a< cos (iw - tw') - 2777;' (D 2 -f D - 4i 2 - 2i) a< cos (iw - iw' +g-g') H- 2^7 ( D + 2t') a< cos (iw iw 7 4- g) + 27?' (D + 1 - 2i) % cos (iw - I'M;' + g') -f ?7 2 (D 2 - 4iD - 3D + 4i 2 + 5i) a< cos (iw - iw' -f 2#) + 7/ /2 (D 2 - 4iD + 5Z> + 4i a - 9i -f 4) a, cos (iw - iw' 4- 20') - 27777' (D 2 - 4iD -f D -f 4i 2 - 2i) a< cos (iw - iw' + g + g') + 2tan 2 J/.6 < cos(iw-iw' + w-f w'-20), ............... (2) in which the notations w, w' = mean longitudes = g + r, ^' -f w' are used. B&8PT 8 114 DEVELOPMENT OF DISTURBING FUNCTION [OH. iv For convenient reference, we repeat the significations of the remaining symbols used in this development: {1 H- a 2 2a cos 2 / cos (w tt/)}"~^ = 2a,- cos f (w w') t . . .(3) {a cos 2 / (1 + a 2 - 2a cos 2 |/cos (w w')}~ = 26* cos i (w - w') t ...... (4) where i = 0, 1, 2, Also, to the second order, 77 = ^, 77'=^'. These definitions show at a glance the differences between the development given above and that of Leverrier, given by Tisserand* in which r ' = a' r Q = a. D* = .-, ( r- ) , cos 2 A/= 1 in j ! \oa/ (3), (4), a< is replaced by %a { and b t by |6 rf , and there is no factor ^ in the left-hand member of (4). When the necessary changes in notation have been made, the two developments will be found to agree with one another. 4*33. The second term of R. The first term of the disturbing function, namely, w'/A, is the same for the action of either planet on the other, except as to the mass factor; the only condition actually used in the develop- ment is that accented letters shall refer to the outer planet. Omitting the mass factor, the second term is r cos 8 r f cos S ~/r~~ or ^ > according as we are dealing with the action of the outer planet on the inner or that of the inner on the outer. Expressed in rectangular coordinates these expressions are xx' + yy r + zz* xx' 4- yy' 4- zz' >s > ^3 The equations for the elliptic motions of the planets are * Mte. Ctl. vol. 1, p. 309. 32, 33] THE SECOND TERM 115 If we put ///=n' 2 a' 8 , n'dt dg' in the former and /-t = n 2 a 8 , dg in the latter, we may write these Since a;', y', z' do not contain $r and x, y, z do not contain g' t the two cases of the second term of R may be written which give a function rr f cos S, symmetrical with respect to the two planets, to be developed. These forms of the second term show that in the first case, there are no terms in the development in terms of the mean anomalies which contain the argument g only, and none in the second case containing g' only. In neither case does the second term produce a constant part. A quite general development in terms of the mean anomalies can be made. With the form 4*1 (3) for cos S, it is evident that rr* cos S is a linear function of r cos v, r sin v and therefore of r cos^ r sin/, and similarly of r' cos/', r' sin/', and these functions can be expressed in terms of the mean anomalies by the formulae 310 (5). An easy way to carry out the calculation is to use the second form of 4*1 (3) and write rr' cos S = A' . r cos/ B' . r sin/ A' aiT^'cos. ,. . 9 . T r f cos , , , _ m ra , = cosH7 , . (r - v ) + sin 2 A/ , . O + v -20). B ^ r sm v ' 2 r sm v ' The series for A', B' may be computed by harmonic analysis, or in series from A' = Cx . r' cos/' + O a . r' sin/', JB' - ~ C 3 . r' sin/' + C 4 . r' cos/', where Ci,C 3 = COS 2 |J COS (r - r') sin 2 / COS (tsr + r' - 20), <7 2 , C f 4 = cos 2 J/sin (r - w') T sin 2 \I sin ( + *? - 20). 8-2 116 DEVELOPMENT OF DISTURBING FUNCTION [OH. iv 4*34. The expansion of r cos S/r' 2 can also be obtained from that of I/A in the following manner. If we put f (<) = except for the cases i=l, s = ; i = 0, s=|, and for these put the value - %a/c, so that ai = a_i = b Q = - \aic = - %a cos 2 17, ......... (1) we find from 413 (4), 414 (1), that With these values 413 (5) becomes -r- = a/c cos (v i/) a* tan 2 \I cos (v -f v' 20) = a cos ^i Finally if this be substituted in 4*8 (3) with D = 1 , we have Thus the required expansion is obtained from that of I/A by putting D = 1 and making the substitution (1). To the second order, with we obtain from 4*32 (2), the result: rcosS acos 2 i/ (/ . rt 2 /ax . . ,. -- ~7a- = -- -7a A -{(l~27; 2 -27/ 2 )cos(^-/-ht ! r^- S r / ) / iL -f ?; COS (2g ^' -f cr isr') 3?; COS (</' -or -f tir') -f 4^ COS (^ - 20' -f tr - -or') + 4^' COS (2^r - 20' 4- or - <cr') -f |r; 2 COS (3<7 - g f -f tar - ') + i^ 2 COS (^ 4- flf' - tar + tsr') + |i?' 2 cos (^r + #' + tzr -'cr')+^ 7 -7 ? ' 2 cos(5r~3/-f ^ --cj') - 121777' cos (2/ - ar + w') + tan 2 \l cos (gr -h g' -f r + -cr' - 2(9)}. The expansion of r' cos S/r 2 is evidently obtained by inter- changing the accents. CHAPTER V CANONICAL AND ELLIPTIC VARIABLES A. THE CONTACT TRANSFORMATION 5*1. Canonical differential equations. Let the coordinates of a particle of mass ra at any time t be #1, #2 #3- If this particle moves under the force -function U (#1, %2, #3, t), the differential equations of its motion are d 2 ^_atr -.190 These are three equations of the second order. We shall now express them as six equations of the first order. This is accomplished by the introduction of the three new variables 2/i> ^2> 2/3 > called momenta, defined by The kinetic energy is Differentiating (2) and substituting in (1) we have %-% '-''' From (2) and (3) we have Since CT and T are independent of y t and < respectively, (4) and (5) may be written, with HTU, dt 118 CANONICAL AND ELLIPTIC VARIABLES [OH. v A set of differential equations in this form is said to be in the canonical or Hamiltonian form, and H is called the Hamiltonian function. On account of the definitions of x iy y t as independent variables, the equations (5) may be expressed in the symbolic form, 2(<fo,.8y<-dy.&,) = (ft.5ff, (7) where the &Cj, Sy t are arbitrary variations of the & iy y iy and SH is the consequent variation of H. The definitions of the symbols d, S, introduced in this manner, will be made more precise in 5*3 below. 5'2. The Contact transformation. Let a i9 yt be any 2n variables which satisfy the canonical differential equations, dxt_m d yi _ dH . m dt'dyt' dt~ a*,' *- 1 >4-> w > W where H is a function of ic it y,, t only: it does not contain any derivatives. A contact transformation is a change from the 2n variables x it y i to 2n new variables, p it q it which shall satisfy equations of the same form, namely, dp, JIT dq i=: _dHr dt dq,' dt dp, ' v ~' where H' is related to H by an equation to be given below, and is expressed as a function of p iy q it t. According to a theorem of Jacobi, relations between the old and new variables which fulfil the conditions can be expressed by the implicit equations* dS dS ct r j. j. /o\ 2/< = g^> Pt^faS S = funct.a? < ,9 < ,t (3) The determining function 8 must thus be expressed as a function of one set, either the x t or the y t , of the old variables, and one set of the new. It must be so chosen that it is possible, * No confusion will be caused between this use of the letter S and that in the development of the disturbing function. 1-3] CONTACT TRANSFORMATION 119 by means of the equations (3), to express the x it y t in terms of the pi, q t , or vice versa. This possibility depends on the Jacobian, .(4) which must not vanish identically. The literature connected with canonical equations and contact trans- formations is extensive and can be found from the usual sources of information. It may be mentioned, however, that while Hamilton appears to have first given the canonical forms of the equations of motion (Brit. Ass. Report, 1834, p. 513), Lagrange had given the equations for the variations of the elliptic elements in 1809 (M4m. de VInst. de Paris, p. 343) in this form. The theorem of Jacobi appeared in the Comptes Rendus for 1837, p. 61, the contact transformation having been introduced by Hamil- ton in 1828 (Trans. Roy. Irish A cad. vol. 15, p. 69). Of the numerous applications of the theory of contact transformations we shall give only those which are necessary for the later developments in this volume. In particular, the proof of Jacobi 's theorem, given in 5*3, docs not indicate the process of discovery, but it has the advantage of showing immediately, not only the relation between //, H f , $, but also the method which appears to be most useful in the search for new forms of canonical variables. 5*3. Proof of the Jacobian transformation theorem. For the purposes of this proof and of later developments, it is desirable to define in more detail the meanings to be attached to the symbols d, 8 in equations involving differentials. The equations 5*2 (1) imply the existence of solutions of the form i = x i (*>> %> <%>), Vi = Hi (*> a l> ' ' ' > a 2)> and the relations 5*2 (3), the existence of functions, which are solutions of 5*2 (3); in these expressions, aj ,..., a an are the arbitrary constants of the solution. The symbol d attached to any function will always denote that when the function has been expressed in terms of t and the a ry it is t alone which is varied, while the symbol 8 implies 120 CANONICAL AND ELLIPTIC VARIABLES [OH. v that any or all of the a r are varied but that t is not changed. Thus when x it y t are expressed in the forms just set down, and thence, when 8 is expressed as a function of x iy q iy t, In these expressions for d$, 8$, the meanings to be attached to dx iy 8&i are those just given; similar meanings are to be attached to dq t , Sq it Since the variations denoted by d, S are independent, the commutative law, namely, that 8 . d and d . S acting on any function produce the same result, is satisfied. The proof that the relations 5*2 (3) transform 5*2 (1) into 5'2 (2) follows. Multiply 5*2 (1) by %*, 8^, respectively, and add for all values of i. We obtain A similar process performed with 5*2 (3) gives 85, ......... (2) */ j j x * fiS j dS , \ , a/S . 2 (yidxt+pidqt) = 2 ^ Ac, + g- d^J = dfif - -^ d$, ......... (3) since S may contain t explicitly as well as through a-^q^ Operate on (2) with d and on (3) with 8 and subtract. Since the operators d, 8 are commutative, we have dBx t = 8d^ t -, d8S= 8dS, etc., and all the terms in which both 8, d act on the same function disappear. We obtain /dS\ X (dy ( Se t - dc, 8 y< ) + 2 (dp, S 3< - dg, 8 P< ) - c . 8 ^ J , 3, 4] JACOBFS THEOREM 121 which is the same as The addition of (1) to this last equation gives *(t"--$*)-'(*+D Finally, if we define H 1 by means of the equation (5) and suppose that H' has been expressed in terms of p it q it t, by means of the relations 5*2 (3), so that the independence of Sp it Sq t furnishes, through the equality of the members of (4), (6), ^ dt ~dq t ' dt d Pi ................... In the use of this transformation, it is important to remember that dS/dt is found from the expression for S in 5*2 (3) and that H' is to be expressed in terms of p^ qi, t. B. JACOBI'S PARTIAL DIFFERENTIAL EQUATIONS 5'4. The transformation theorem just proved is a device for changing from one set of variables to another, the new variables depending on the choice of the determining function 8. One such choice is the following. Suppose that it is possible to find a form of 8 which will make the derivatives dH'/dp iy dH f /dq { zero. (Since H' must necessarily be expressed in terms of p^ q if t before these deriva- tives are formed, it follows that the derivatives will then be identically zero.) The equations 5'2 (2) show that dpjdt, dqjdt become zero, and hence that Pi = const., qi = const. 122 CANONICAL AND ELLIPTIC VARIABLES [OH. v Jacobi showed that when S is so determined as to satisfy these conditions, it is a solution of a certain partial differential equation. In the next section, this equation will be found and those properties of it, which will be useful later, will be developed. Notation. A semi-colon following a collective symbol x i> where i 1, 2, ..., n, will denote that any or all of the x t may be present, thus t) means /Oi, #2, -,#> 0> o 5*5. The equation and its solutions. Amongst the values of S which will make p t , ^ constant, we seek one which makes H' = 0, that is, one for which But H was originally a function of x it y t -, t, and, by 5'2 (3), y i = dS/dtti. Thus, we seek a value of S satisfying Now the assumption concerning the form of ti was that it should be expressed as a function of x it q it t, and the assumption is to be retained in (1). But, in the present case, the q t are constants by hypothesis. Hence, in order to satisfy (1), we need an expression for S which contains x iy t, and n arbitrary constants q { . In other words, if we regard (1) as a partial differential equation with x iy t as independent variables and with S as the dependent variable, we need a solution of the equation containing n arbitrary constants. When such a solution has been obtained, all that is necessary is to interpret the relation 5*2 (3) in the language of the theory of differential equations, remembering that these now constitute 2n relations between the original variables #,-, y if the constants Pi,qi, and t. 4, 5] JACOBIAN SOLUTION 123 Theorem. A general solution of the equations is provided by the equations dS where p it #, are arbitrary constants, and S is an integral, contain- ing n arbitrary constants q* (exclusive of that additive to S) of the partial differential equation This type of integral is known as a complete integral, for the theory of which the reader is referred to treatises on first order partial differential equations. For our purposes, it is sufficient to state that the Jacobian 5*2 (4) must not vanish identically. The constant additive to S plays no part because S appears in the differential equations and in the solution only through its derivatives, but its presence is theoretically necessary since there are n-f 1 independent variables in the partial differential equation. It may be pointed out that the ordinary method for the solution of first order partial differential equations simply leads back to the canonical equations, so that nothing is gained by attempting to use it. In the applications to celestial mechanics, the form of the function S is usually set down from previous knowledge of the form of the solution, and the relations 5*2 (3) are then used in various ways ; or, in particular cases, a form for S may be suggested by the equation 5'5 (1). The set of arbitrary constants which appears in this way is known as a canonical set. It is evident that when the function S, containing half of them, has been obtained, the remaining half are chosen from the relations p t = dS/dqi. In general, there- fore, only n of the 2n constants may be chosen arbitrarily. The practical demands of the perturbation problem limit the choice to very few types. See 5*12, 5*13, 5*14. 124 CANONICAL AND ELLIPTIC VARIABLES [OH. v 5 '6. The case where H is independent of t explicitly. A start may be made by assuming that S^Si+Ct, ........................ (1) so that the partial differential equation is reduced to <>-<> ................... < 2 > which contains n independent variables only. If C be chosen as one of the canonical set of constants q iy and t Q be the corresponding constant derived from the equation p i = dS/dq i) we have <-w +t ......................... < 3 > Since Si does not contain t, it follows that the solution will contain t, t Q only in the form t t Q . The equation (2) shows that H = constant is an integral of the equations. This may be proved directly from the canonical equations 5'2 (1), by multiplying them by dyi/dt, dxjdt, and adding. A similar procedure may be adopted when any one of the coordinates x t is absent from H. It is evident that each absent coordinate permits the writing down of an integral of the equations. 5*7. Application to the perturbation problem. The force-functions for the problem of three bodies which have been constructed in Chap, i, have usually been divided into the sum of two parts, the first of which, taken alone, gives elliptic motion. This division would lead to putting 17= UQ + -R, where UQ = /A/r. It has been adopted because we can solve the equations completely when R = 0, and it has the added advan- tage that, since jR usually contains a small factor, it constitutes a first approximation to the motion. These considerations, however, do not limit the applicability of the following method of procedure. If U is replaced by UQ + J?, H is replaced by T UQ R, or 6-8] VARIATION OF ARBITRARIES 126 by HQ R if HQ = T - UQ, so that the canonical equations will be written dxi d rr p dy t % (if p\ /i\ - = -^(H,-R), -^-(Ht-R) (1) Let us transform to a set of new variables p t , q t by means of the relations ds as ,. where S is a solution of the equation ds The transformation theorem in 5*2 shows that the equations satisfied by the new variables are dt dqt' dt dp^ for in this case we have, by (3), , da__x /*-/* ^+g- ^ The interpretation of this result, usually adopted, is the following. If we solve the equations with H = and obtain a canonical set of constants p iy q it and consider these constants as variables when 2? 4= 0, the equations which they will satisfy are those numbered (4); hence the phrase, Variation of Arbitrary Constants. This latter point of view is useful in geometrical descriptions of the motion, but it sometimes leads to confusion and error if it is adopted in the analytical work. As*a matter of fact, the equations with R = are solved mainly in order to indicate the choice of the new variables, and are not used unless we know in advance that a solution can be obtained. 5 '8. Osculating orbits. Another geometrical interpretation of the results of 5*7 is of value in the determination of the orbit of a body from observa- tions of its position. Let us suppose that the problem of finding 126 CANONICAL AND ELLIPTIC VARIABLES [CH. v the solution of 5*7 (4) has been solved, so that the variables p it q t are expressed as functions of t and arbitrary constants. If we put t = t Q in this solution, where fo is some particular value of t y these variables become constants and thus constitute a solution of the equations 5 f 7 (4) when R = 0. Hence at the instant t = fo, the variables and the constants have the same value. But the coordinates and velocities are expressible in terms of t and Pi> q it It follows that the orbits with jffi = 0, R^O, in- tersect at t =s t Q and have the same velocities at the point of intersection; when this happens the two orbits are said to osculate at that point, and the ellipse described with R = is called the osculating ellipse*. If at the instant t tv, the disturbing forces which arise through R were suddenly annihilated, the body would thereafter proceed to move in the osculating ellipse. This constitutes another definition of this curve. In the great majority of cases arising in the solar system, the forces due to R are small compared with those present when 72=0, so that the oscu- lating ellipse constitutes a good approximation to the orbit at times near = <) In the case of a planet, the separation is small during a period of one revolution of the planet round the sun. Thus the osculating ellipse can be used to predict approximately the place of the body for some time before and after the instant t = t (} . Ordinarily two coordinates, which give the angular position as seen from the earth, are observed ; neither the distance nor the velocities are directly observed. From three such observations an osculating ellipse can in general be deduced. A position predicted for some other time in order to limit the area of search also needs only the two coordinates. There are six constants present in the osculating ellipse, and according to the mathe- matical theory of the approximate representation of a curve, considerable variations may be made in the six constants without altering the two needed functions of them very greatly. Thus the elements of an osculating orbit may be considerably in error and yet it may furnish a good search ephemeris for a considerable interval following its determination. * Since the curvatures are not in general the same, the word 'osculating' is not used in the same sense as in the theory of curves. S, 9] JACOBIAN METHOD OF SOLUTION 127 C. JACOBIAN METHOD OF SOLUTION 5*9. Solution of the case of elliptic motion by the Jacobian method. The force -function in this case is m/i/r, and the canonical equations of motion become, after division by m, dxi __ dHo dyt _ 9#o dt 9y t - ' dt dxt ' where The division by m is analytically equivalent to putting m = 1 in the formulae of 5*1. According to 5*5, the equation satisfied by S is MIS) +l w-., ^ +*,+*,) ^ Transform to the tri-polar coordinates r, TT ~ i, X, so that L is the latitude, and X the longitude of the projection of r on the plane of reference. The equation becomes dS ,. ,~. According to the preceding theory, we need an integral of (2) containing three arbitrary constants, exclusive of that additive to S. Since t> \ enter only through derivatives, it is convenient to put 8=-ait + a*\ + Si, .................. (3) where Si is independent of X, t, and i, a a are constants. Inserting this value of S in (2) we obtain, after multiplication by 2r^ and rearrangement, -- 1 ....... <> The form of this equation indicates that we can obtain a solution by putting Si = S$ + Sz, where S% is a function of r only 128 CANONICAL AND ELLIPTIC VARIABLES [CH. v and $3 that of L ouly, if we equate each member to a constant a 2 2 . This procedure gives . 2 a 2 2 /9$3\ 2 3 2 = 2i 4- =- . I =-= i = a* 2 ^-, . \dL ) UU&-JL/ The integration of these equations will leave two arbitrary constants at our disposal since the necessary three arbitrary constants i, 2 , a 3 are already present. Let $ 2 vanish for r = ri> where TI is the smaller root of the equation _ 2jjL a 2 2 ,, /f . 2ai-f -^ a = 0, (5) * /M /V1<fi ' ^ ' and let 3 vanish when // = 0. In order that this value of r may be positive, it is necessary that both roots be positive; hence a must be negative. Inserting the values of jS> 2 , 63, $1, thus obtained, in (3) we obtain a solution in the required form : ...... (6) The next step is the deduction of the solution of the canonical equations by means of the relations & = 9S/3a$, y i 'dSldx iy which would seem to demand a return to rectangular coordinates. However, we do not need the latter set since the former gives the necessary three relations between r, L, X, t, and the relations between r, L t \ and #1, # 2 , #3, are independent of a iy 2 , 3 . The derivatives of 8 with respect to the a f are obtained without carrying out the quadratures in (6), by means of the formula The derivative with respect to i gives ft + since the coefficient of dri/da vanishes on account of (5). 9, 10] ELLIPTIC CANONICAL CONSTANTS 129 Similarly, L ' The relations (7), (8), (9) are those needed, and the a, , /3< con- stitute a set of canonical constants. 5*10. Relations between the set a iy fit and a or n y e, i, e, w, 6. Since the integrand of 5*9 (7) must be real, the equation 5 '9 (5) gives the maximum and minimum values of r. In 3 '2 these are shown to be a (1 e). Hence from a well-known theorem connecting the roots and coefficients of a quadratic equation, 2u/2! = 2a, 2 2 /2i = a* 2 (1 e 2 ), giving c&i = According to 5*9 (7), /3j is the value of t when r=r 1 =a(l e). But the mean anomaly nt + e w is zero for this value of r. Hence /3i = (e w)/n. Equation 5*9 (9) shows that /3 3 is the value of \ when L = 0, that is, when the body is in the plane of reference. Hence y3 3 = 0, the longitude of the node. Since the integrand of 5*9 (9) must be real, the maximum and minimum values of cos L are 3 /a 2 . But the maximum and minimum values of L are i, where i is the inclination of the plane of the orbit to the plane of reference. Hence 3 s= 2 cos i. Finally, if we put a 3 = a 2 cosi in the last term of 5*9 (3) it becomes [ L cos LdL __ C L d (sin L) __ . ^ /sin L\ Jo (cos 2 Z - cos 2 z')* ~ Jo (sin 2 ! - sin 2 i)* ~ &m \ am?/' But if v be the hypotenuse of the right-angled spherical triangle in which L is the side opposite to the angle i 9 we have sin L = sin i sin v. Thus the above integral is v and the equation B&SPT 9 130 CANONICAL AND ELLIPTIC VARIABLES [CH. v 5*9 (8) shows that /9 2 is the value of v when r = n. Since v is then the angle between the apse and the node, we have $2 = & # Collecting these results, we obtain the system of canonical elements i=(e- Hence, according to the principles set forth above, the equations satisfied by the a,-, & when R 4= 0, are 2 (da, 8& - d& 8a f ) = eft . 812, (2) or, written in extenso, ^ m 4& p <ft d& eft 9a t 5*11. The fact that 5'9 (7), (8), (9) give elliptic motion rn\y be deduced in the following manner. If we put r=a aecos A" in (7), and insert the values* of the constants found above, we obtain, after integration, n which, with the aid of the relation a?n?=p, is seen to be the equation con- necting the mean and eccentric anomalies. Equation 5*9 (8), with the same substitution for r, becomes [X dX l~e cos X ' since v = v 6, f vw, this equation gives the relation between the true and eccentric anomalies. Finally, the substitution sin Z = sin i sin v in the integral of (9) gives sst X - tan*" 1 (cos i tan i/) ; since X 6 is the side adjacent to the angle i in the right-angled triangle to which reference is made in the text, this equation merely constitutes a well-known geometrical relation. A logical procedure requires the proof that the motion is elliptic to precede the identification of the constants. But as the objective in view is the discovery of a set of canonical constants only, we assumed that the nature of the motion had been found by the easier method of Chap. in. (Cf. 3-2.) 10-12] DELAUNAY'S ELEMENTS 131 D. OTHER CANONICAL AND NON-CANONICAL SETS 5*12. Delaunays canonical elements. Changes from one canonical set of elements a i , /^ to another such set can sometimes be carried through easily by the Jacobian transformation theorem proved in 5*3 if we use as the determining function S = 2a,A, ........................ (1) where the /3 t (or the a t ) are expressed in terms of half the variables of the new set. Let us take as three of the variables of a new set, I, g, h, defined by so that i = /*(- 2ax)~^ - t. Hence S, expressed in the required form, is S = ^(-2a 1 )^l-a l t-^a 2 g + a B h .......... (2) If the other three new variables be L y 0, H, we have, by 5'2 (3), r d$ /- ^ \ i / - ^ 9$ rr ^S i = _ =/t (_2a 1 ri=V M a ) (?== # = ^ = 3. ...... (3) The remaining three equations are automatically satisfied. Also H'-H + -R-*-R + ............. (4) Thus this set, which is that used by Delaunay, satisfies the equation _ ...... (5) where L = V 'pa, I = nt + e tsr,! ...... (6) cos 9-2 132 CANONICAL AND ELLIPTIC VARIABLES [OH. v The variables in each of the two groups are homogeneous; L, 0, H are angular momenta or, since the mass acted on has been divided out, areal velocities. If we put /JL = n 2 a 8 , the common factor "Ja/j, becomes na 2 . 5*13. The modified Delaunay set. If we use the determining function S Ll + Gg + Hh, in the form the equations of transformation show that L, G L, HG and I -f g -f h, g + h, h form a canonical set. We shall denote them by c i? w it so that d = v^a, ivi = nt -f e = mean longitude,] c 2 = Vyu,a (Vl (? 1), w 2 ~ TZ = l n g- of apse, c 3 = \/y^a (1 e 2 ) (cos i 1), w^ long, of node. ...... (1) The Hamiltonian function is unaltered and is equal to R -h yu, 2 /2ci 2 . When expansion is made in powers of e, i, the element c 2 is divisible by e 2 and c$ by i* properties which make this set useful in planetary problems. 5*14. A set given by Poincare'*. This set is ci, p%, PB', w, q%, qs> defined by p 2 = \/ 2c 2 sin w, q ?i = V 2c 2 cos sr, jt) 3 V- 2c 3 sin 6, </ 3 = V 2c 3 cos ^, where 02, CB are defined in 5*13 (1). That it is canonical can be tested by showing that dpiSqz dq^pz = dczBw 2 div z Sc 2t with a similar equation for the suffix 3. This is a particular case of a general theorem f which states * Les Nouv. Mtt. de la Mtc. Cel. vol. 1, p. 30. f C. A. Shook, I.e. in 5-5. 12-15] tfON-CANONICAL SETS 133 that if two variables p 2 , fa are related to two canonical variables C 2 , w 2 in such a manner that the Jacobian, p z , qz are also canonical variables. The Hamiltonian function is E -f /* 2 /2ci 2 , as before. This set is useful because the approximate values of p 2 , q% are V/t . e siri -GT, V/xa , e cos t*j, when e 8 is neglected, and those of ^> 3 , f/ 3 are e 2 ) . i sin 0, V/xa ( 1 e 2 ) . i cos 0, when i 3 is neglected, so that the disturbing function is develop- able in powers of p 2) p$, 72 , </3> such a development replacing powers of e, i, and cosines and sines of multiples of -or, 6. The possibility of such a development depends on the association of powers of e with multiples of r, and of powers of i with multiples of when the angles are expressed in terms of tsr, r', and the mean longitudes. See 4*15, 5*15. The non-canonical set a, e it PI, w, r, 0, ?0A0re M; = H 4- e, e x - s/2 (1 - \/l - e*) = ^ + ^e 3 ..., T! = (1 - cos i) Vl - e 2 . ...... (1) The disturbing function is not usually expressed in terms of the preceding canonical sets of elements. The angular variables are present explicitly, but the remaining elements are mixtures of a, e, i which are present explicitly. The same is true of the coordinates. For calculation, it is usually easier to adopt elements which are more directly related to those which are explicitly present in the ordinary developments of R and of the coordinates. The equations satisfied by such elements are not canonical. The definitions of e\, FI given in (1) show that these variables are related to d, c 2 , c 3 by the equations c x =V/ua, c 2 =- iei 2 V/xa, c 3 = Ti^pa ....... (2) The variables w, or, are the same as those denoted in 5'13 (1) by wi, u'2, w^. 134 CANONICAL AND ELLIPTIC VARIABLES [CH. v The transformation to the new variables is most easily effected by forming their variations from (2) and substituting them in the left-hand member of the equation, The process is quite straightforward. After the variations have been formed, it is convenient to use the equation fju = n 2 a 3 by putting V 'pa = ulna, V/j/a = p/ncP. After rearrangement, we obtain for the left-hand member of (3) the expression S0 multiplied by p/no 2 . Since /-c 2 /2ci 2 = /x/2a, we have, if R be supposed expressed in terms of the new elements, -^ -- o 3o. 2tt 2 , -^-~ SlV + mu oiv which is substituted in the right-hand member of (3). Since the variations of the new elements are, like those of the old elements, independent, we can equate their coefficients on the two sides of the equation. On solving the six equations thus obtained so as to isolate da/dt, dei/dt, ..., we obtain 1 da _ _ 2 1 dn = 2na dR a dt ~ 3 n dt ? ~~ ^6 dw ' dw __ 2?ut a 3JB Jiae^dR naTidR dt IUL da %IJL dei p, 3Fi _ 3 dt dt nadR /M 9-or nae^dR 90 naT^R p~~ dw ' d9 na dR The objective in this transformation is the isolation of deriva- tives with respect to a, so that the operator D, which plays so large a part in the development of the disturbing function, may 15, 16] NON-CANONICAL SETS 135 act only on those portions which are specifically set forth in Chap. iv. Since R has the dimensions, mass divided by distance, and n has the dimension of the inverse of a time, the factor p/no? reduces all the equations to relations between ratios. The right-hand members of all the equations except the second contain the small factor present in R. The value of n found from the first equation is to be substituted for the term n of the second equation before the latter is integrated. The relations * = *l(l-l*M 2,;^ 1 (l-iV)-*, ......... (5) deduced from (1) and from e = 2iy/(l -f rf\ permit R to be easily expressed in terms of e\\ ei e, 2rjei are approximately -J-e 3 , so that in many problems it would be possible to neglect these differences. It is recalled that e y and therefore e\, is present in a as defined in the developments of Chap. IV ; it is also present in I\. Hence if R be developed as in that chapter, we have to substitute for dR/dei in the equations (4), the value derived from /dR dRdT lda\ de I ^ -f- f=i ^~ 4- DLL . 1 f - , \ce 01 oe a ?e/ de\ when dT/de is formed from F = I\ (1 e 2 )~^ and de/dei from (5). For dR/dTi we have (dR/dT) (1 -e 2 )-*. 5*16. The non- canonical set a, e y F or i, w or e, w, 6. This set is deduced from 5*15 (4) by means of the relations 5*15 (1) or 5*15 (5), through the usual process of a change of variables from e\ to e and TI to F. The result is the following set of equations __ ^_ a dt 3n dt~~ /A dw ' dw 2na dR e*Jl e 2 na dR F na dR _ ^ r> . _ fl _ 1 ___ _ ___ ______ _ _ _j_ _ - _ __ dt IJL da l-fVl e 2 A 6 de Vl - e 2 P ^F' de ,- - o na dR e A/1 & na dR _ ._. _ V 1 6 _ _ --" __ __ _ dt 136 CANONICAL ANiD ELLIPTIC VARIABLES [CH. v dfff __ i~, nadR F na dR ~dt~ ~ ^e'de vfT:^ 9F ' 30 This set is slightly more convenient than the older form in which 61, i replace w, F respectively. The definition of 61 is usually given by means of the equation w nt -f- e = I n dt -f ei , so that dei/dt = dw/dt n. And as e occurs in R and in the co- ordinates only in the combination nt -f e, we have dR/Sw = dR/de. Nothing is gained by the substitution of ei for w, except perhaps a separation, in the case of the long-period terms, of those portions which have the square of the small divisor as a factor, from those which have the first power of this divisor as a factor. The substitution of F = 1 cos i for i is advantageous because i occurs in the disturbing function only in the form cost, or rather cos / when the orbital plane of the disturbing planet is taken as the plane of reference. The older form can be at once deduced by means of the relations, rfF . di -r, dR . ., . dR 5' 17. The case of attraction proportional to the distance from a fixed origin. An example of such a gravitational force is that on any one of a spherical arrangement of particles with a mass-density which is uniform throughout the sphere. Since such an arrange- ment cannot be exactly maintained under the Newtonian laws of force and gravitation, the force and resulting motion must be considered as approximate only. Another similar example is that of an arrangement of particles in an ellipsoid of revolution 16, 17] FORCE VARYING AS THE DISTANCE 137 with uniform mass-density: the force on a particle in the equatorial plane varies directly as the distance from the centre. In the spherical case, the force is pmr, where m is the mass of the particle, r its distance from the origin and /j, is 4?r/3 times the density multiplied by the gravitational constant. In the ellipsoidal case, //, depends also on the eccentricity of the ellipsoid. The force-function for these ideal cases is $fj,mr 2 . Let mR denote the force- function for the remaining forces which act on m. If #1, # 2 > #s he the rectangular coordinates of m referred to fixed axes through the centre of the system, the equations of motion will be with >* = x + a When R = 0, each coordinate describes a harmonic motion with period 2?r/n, where n 2 = p, and the orbit is an ellipse whose centre is at the origin. As before, the elements of this ellipse may be considered as new variables for the examination of the case R + 0. A canonical set, with Hamiltonian function R, in which /A is replaced by n a , is the following: 1 Q = e -or, y = w 0, h = ff. If 1 be replaced by / = nt + e r, the only change needed is the replacement of R by R nL as the Hamiltonian function. The proof is left to the reader. A modification similar to that of 5*13 may also be made. CHAPTEK VI SOLUTION OF CANONICAL EQUATIONS 6*1. The main object of this chapter is the development of methods of solution for the types of canonical equations which have been obtained in the previous chapter. All the methods depend fundamentally on the assumption that the variables differ from constants by amounts which have as factor the ratio of the disturbing mass to that of the primary, and therefore that the variables may be developed in powers of this ratio. As long as we confine ourselves to the first power of this ratio there is little choice between the various methods; substantially, they are equivalent. It is when we need to take into account higher powers of the ratio that differences appear, mainly in the amount of calculation which is necessary to secure the desired accuracy. The fundamental idea of the method of Delaunay, namely a change of variables such that the new variables are more nearly constant than the old ones, is used throughout. But the applica- tion of the idea is different from that which Delaunay made to the solution of the satellite problem, where the changes of variables were very numerous*. Here it is shown that one change of variables is, in general, sufficient for the solution of the majority of planetary problems provided the new variables are suitably chosen. Much of the discussion in the chapter hinges on the amount of labour which the development and solution of the equations for the new variables requires. 6*2. Elliptic elements or variables. In Chap, v it has been shown that the equations of motion can be put into the canonical form, and methods of changing from the coordinates and velocities to other sets of variables are developed. Particular sets of new variables which are con- * See for example, Tisserand, vol. 3, Chaps, xi, xn; E. W. Brown, Lunar Theory , Chap. ix. 1-3] PROPERTIES OF DISTURBING FUNCTION 139 nected with motion in an ellipse have been chosen and the equations satisfied by these new variables have been given in canonical form; other sets are given in non-canonical form. Various points of view of these new variables, usually called the elliptic elements, are given. It was, however, pointed out that for the purposes of mathematical development, they should be regarded merely as a set of new variables which are connected with the coordinates and velocities by a set of equations, the latter remaining the same for all values of the variables. These new variables have the property of becoming constants or linear functions of t when R = 0. In one set, they are all constants; in the other sets, one of them is a linear function oft. The canonical set denoted by a, Wi and developed in 5*13, will be used in this chapter. Only slight changes, easily made, are- needed if the set GI , p a , p 3 , MI, q z , q*> g iven in 5 'H be used. Since R is not in general developed in terms of canonical elements, it is shown how the work may be so adapted that the developments of R given in Chap. IV may be used. 63. The disturbing function. In Chap. IV, the disturbing function has been expanded into a sum of periodic terms on the assumption that the motion of each planet is elliptic. This restriction can now be removed, so far as the disturbed planet is concerned. Since the development consisted in the replacement of the coordinates by their values in terms of the time and the elliptic elements, the relations between them being those referred to in 6'2, the development is unchanged if we consider these elements as variables. The relations between the coordinates and velocities and the new variables C;, Wi are independent of t explicitly; in elliptic motion t is implicitly present in Wi only. Thus the only way in which t is explicitly present in R is through the coordinates of the disturbing planet, and it appears actually only through id' = n't + e' or <f = n't 4- e' - tsr'. If the motion of the disturb- ing planet is not elliptic, its deviations from elliptic motion can 140 SOLUTION OF CANONICAL EQUATIONS [OH. vi be expressed as variations of its elements, so that the given development of R can still be utilised. The necessary modifica- tions of the solution are not difficult; they are exhibited in one particular case (6*19 below). The possibility of expanding R in power series when the canonical elements C;, Wi are used has to be considered. Since d = V//,a, it is evident that Ci can replace a without difficulty. Also, by 515 (2), we have and since, by 515 (1) or 515 (5), we can replace e by e\ and F by TI in the developments, it follows that developments in powers of e, F are replaceable by developments in powers of (-2c 2 /Ci)^, ( c 8 /ci)^. Derivatives with respect to c 2 , C 3 will implicitly involve the presence of negative powers of e, F, and it will be necessary to show that these negative powers disappear from the transformed disturbing function. The difficulty does not arise when the variables Ci, p*, p$, w 1} q 2j q^ are used, be- cause R and the coordinates are expansible in positive integral powers of p 2) p 9) q 2 , 73 (514). We have, approximately, p 2 e sin sr, q t = e cos BT; suppose that ?o, ^o ftre the undisturbed constant values of e, *&. We then get e sin w = CQ sin OTO + perturbations, e cos r = CQ cos tzr -h perturbations. While the perturbations vanish with the disturbing mass, they do not in general vanish with e$. Hence, even if the observed eccentricity of the disturbed body is so small as to be negligible, we cannot assume that e is negligible in finding its perturbations by the method of the variation of the elements. Thus, while geometrical descriptions of the motion are simple in terms of e, -or, the analytical development fundamentally requires the use of e sin tzr, e cos w, as convenient variables. The general control of the work is, however, much easier with the use of the former than with that of the latter variables. In certain cases, particularly 3, 4] D'ALEMBERT SERIES 141 those in which CQ is very small, and also in the discussion of the secular perturbations, it is advisable to use the variables p%, q% rather than c%, iv 2 . Similar remarks apply to js, q$ and to F, 0, but the problem for these latter variables is less difficult because R is expansible in positive integral powers of F and therefore of 03. The solution of the problem of the apparent presence of negative powers of e is given in the theorems of 6'4, 6*15 below. 6*4. D* Alembert series. The association of powers of e with multiples of -or, of powers of e r with multiples of or', of powers of VF with multiples of 0, and of powers of a with multiples of w w', which have been pointed out in the development of R (4*15), and in certain of the developments of Chap, in (3*3), are so useful that it was found convenient to give a designation to such series: we have named them d'Alembert series*. Certain of the expansions of Chaps. II, ill, have been seen to be of this type. Certain other series are easily related to it. Thus, sinjf, cosjf are not d'Alembert series with respect to the coefficient e and the angle g, but sinj(fff), co8J(f ff) are. It is evident that if we have a d'Alembert series with respect to a coefficient A and an angle #, the series is formally expansible in powers of A sin#, A cos#, and in powers of A exp. #V 1, A exp. x V 1. A general property of these series is given in the following theorem. If f, f are d'Alembert series with respect to the coefficient A and the angle x, then the Jacobian, d (A 2 , x) 2 A \dA dx is also a d'Alembert series with respect to A, x. Let _ __ y = A exp. x V 1, 2 = A exp. x V 1, yz = A 2 . * It appears that d'Alembert was the first to notice this property of the dis- turbing function with respect to the eccentricities and longitudes of perihelia, in his memoir, "Becherches sur diffe'rens Points importans du Systeme du Monde," Mem. Paris Acad. Sc. 1754. 142 SOLUTION OF CANONICAL EQUATIONS [OH. vi The transformation of the Jacobian from the variables A, x to the variables y, z gives 8T_ a/'N _ 2A \ABy Adz) V dy dz ) d + z d A -, \dz dy 'by dz / ' since yz = A 2 . The definitions of /, /' make them expansible in powers of y, z and therefore their derivatives have the same property. The Jacobian is therefore a d'Alembert series. Thus while the separate terms of the original Jacobian are not d'Alembert series, the divisor A disappears from the Jaco- bian when it is expressed in terms of A, x. This is the property we really require to know. 6 '5. Other properties of It. It is useful to recall certain of these. (a) It is homogeneous and of degree - 1 with respect to length. The variables GI are homogeneous and of dimensions (mass)* (length)* or, with the unit of mass actually used, of dimensions (time)" 1 (length) 2 . (6) It depends only on relative coordinates and is therefore independent of the origin of measurement of the angles. The anomalies are by definition independent of such origin. When R is expressed in terms of g t g' , -or, txr', 6, the only forms in which the latter three angles are present are their differences, usually expressed by tzr txr', tar -f & 26. (c) It is a function of the coordinates only and is independent of the velocities. This property was utilised in forming the equations of motion. One result is the equation d ^ a V? dR 4-6] PROPERTIES OF R 143 which is a consequence of the equation a result which is deducible from the canonical equations in 513, with the help of the relation 7i = The relation (1) may also be written dR dR dR by the use of the canonical equation for ci. When the two bodies are supposed to move in fixed ellipses, the relation (3) is evidently true, since t occurs only through y, g\ the term dR/dt enters through g' only and dR/dwi through g = wi TZ only. Its importance is due to the fact that it is still true when the elements are variable, but it must be remembered that the term dR/dt is still a derivative with respect to t only as the latter is present through the coordinates of the disturbing planet. (d) Since the term ra'/a' in the preliminary expansion of R given in 4'2 does riot contain the coordinates of the disturbed planet, R has the factor w'a 2 /a' 3 when a' > a. This factor may be written (m'/a) (a/a') 3 . Since the undisturbed force-function is fji/r, it follows that the disturbing effect of an outer planet has as factors the ratio of its mass to that of the sun and the cube of the ratio of the distances of the two planets from the sun. If the outer planet be the disturbed planet, the first term of R can be included in the elliptic force-function. We can then consider R as having the square of the ratio of the distances as a factor instead of the cube. 6'6. Elimination of a portion R t of the disturbing function. In Chap, iv, R has been expanded into a sum of terms having the form R = 2KcosN, N = ji wi +j 2 iu 2 +j 3 w 3 +ji'wi +jz'^2, (1 ) where w\ = n't + e' g' + /, w 2 ' = w', 144 SOLUTION OF CANONICAL EQUATIONS [CH. vr and the j t -, J/ are positive or negative integers or zero, the sign of summation referring to their various values. The coefficients K are functions of a, e, /, a', e' and therefore of Ci, c 2 , 3, a', e' t and they contain the mass of the disturbing planet as a factor. Of. 4-15. Put R = R t + R c , ........................ (2) where R t may contain any or all of the terms for which the relations ji j\ = do not hold. Thus for all terms in R t , the value of v defined by "=jiw+jiV, n = ^/c^ ............... (3) is never zero. It is here assumed that n/n r is not in the ratio of two integers ; this case, if it does occur, must also be excluded from R t it will be considered in Chaps, vui, ix. The elimination of R t will be effected by a change of variables with the help of the Jacobian transformation theorem (5'3). The new variables will be denoted by Cto, w#, i=l, 2, 3, and the suffix zero will in all cases denote that the old variables have been replaced by the new. Thus NO = ji MIO 4-^20 +j 3 MM +ji'wi +j*W2 ....... (4) Take as the transforming function S y = :E;C t ^o-S, S=S~smJV , ......... (5) so that S is a function of the three old variables d and the three new variables w^; 8 is to contain only those terms present in R t with the divisor v appropriate to each term. It will be noticed that S can be regarded as the integral of R t taken on the supposition that the ellipses are invariable; this fact, how- ever, is not at this stage to be regarded as having any physical significance since (5) is merely a definition of S. The equations 5*2 (3) give dS dS _ 6] ELIMINATION OF S t 145 as the six relations connecting the old and new variables. Also 01 01 V \ V by the definition (3) of v. But when (5) is substituted in (6), the equation for CIQ becomes v whence by (3) dS ^ -rr ~\T M / \ ~ 7 = - ^ K COS NQ -f ~- 3 (Ci ~ CIQ). dc Ci The new Hamiltonian function is The equations (5), (6) show that Wi WM has m' as a factor. Hence cos N cos N has the same factor, and K cosN has m' 2 as a factor. Also, since Ci CIQ has m' as a factor, ) ...f, The last two terms of (9) can therefore be written P* 3( Cl -c 10 ) 2 2 , (10) 2cio 2 2c 10 4 ^ T -' the first power of Ci CIQ disappearing. Finally, since R c R^ has the factor m' 2 , the right-hand member of (9) may be written ^co-H 9^~~2 + terms w ^ factor m' 2 (11) If, therefore, we omit terms having the factor m' 2 , the equations satisfied by the new variables are / ^2 v These have the same form as the original equations, but the terms present in R t have disappeared. B&SPT 10 146 SOLUTION OF CANONICAL EQUATIONS [OH. vi At this point it is convenient to indicate the general plan of the remain- ing portions of this chapter. Two choices of the terms to be included in R t will be made. In one of them, R t contains all the terms for which *>4=0 ; R c then contains only those terms which produce the so-called secular motions of the elements. In the other, R t contains the short period terms only, so that R e contains the long period and secular terms. It is for the latter choice that the calculation of the terms in (10) dependent on m' 2 will be made. Several methods for carrying out the work to this degree of accuracy will be outlined, and certain cases where it is possible to obtain numerical results with but little calculation will be developed. It may be pointed out that the work as far as equation (9) is quite general in character, no approximation being involved. It in only after this point that we assume the possibility of development in powers of m' and proceed to find the first terms of the expansion. THE FIRST APPROXIMATION 6*7. First approocimation by change of variables. Let R t contain all the terms for which z>=i=0; then E c is independent of Wi, w\ and therefore of t explicitly. The first approximation is obtained by neglecting all terms which have m' 2 as a factor. The variables c t - , W Z Q therefore satisfy 6*6 (12). Since R cQ is obtained from R c merely by substituting the new variables for the old, it follows that R cQ is independent of WIQ, t. The equation for CJQ therefore gives j~ 0, CIQ = const. = ki , and the equation for WIQ is _ _ dt Cio 3 9cio ki* 9cio " When m' is neglected, the remaining variables are constant and then M 2 ^io=r3^-fi, w = aa, W3o = a 3 ; c,- = *i, ...... (1) KI where a^, ki are arbitrary constants ; these values may be substituted in all terms containing the factor m' and therefore in R<$ . As R& does not contain WIQ , it follows that all the derivatives 6,7] FIRST APPROXIMATION 147 of jRco will become constants. The solution of the equations 6*6 (12) to the order iri will therefore be M -< 2 > c<o = ki + ~t, w = < - T^ * * = 2, 8, dWi-o dCio in which the coefficients of are all constant. The values of the new variables in terms of t having been obtained, the old variables are to be calculated from the equations 6*6 (6) with the value 6*6 (5) for S. But since S, Ci c iQ have the factor m' we can replace c t - by c t0 in S and in its derivatives. We therefore obtain Finally, since the values (1) differ from (2) by terms having the factor m', we can replace WM, c^ by their values (1) in S Q and in its derivatives. These results may be restated in the following manner. If It = R -f ^K cos JV, jV =ji Wi +J2 w 2 + JB w 2 -f ji Wi 4- jz w* > where R c contains all terms for which ji = ji 0, the values of the variables c, w t - to the order in' are given by cos N, ij? dR c \^ v a (K\ . y~ - -5 U - S A si ^i 3 3ci/ dcAv) c ^ v . = i -f ( y~ - -5 U - S A sin 3 ... (4) ~ t + 2ji cos OWi V C4 ^ . .. w f =a t - - -^ ^ - S - -5- sm Jv, dCi // oCi where i = 2, 3 ; in all the terms of the right-hand members the constants k iy & 2 , &3 are substituted for d, c a , c 3 , the constants 2, 3 for ^2i ^3> and the value i + /^ 2 /&i 3 for w lf and i/ is defined by "=ji+ji'' ...................... (5) 148 SOLUTION OF CANONICAL EQUATIONS [CH. vi Equations (4) may be written in the following form. If we put K ^ = *fl c + S - sinjy, .................. (6) M 2 d - ki = Sci, MI - i - -3 1 = Swi, Wi-ai = Swi, . . .(7) #1 d *-, *~g ......................... (8) the undisturbed values (1) being substituted in the right-hand members. These are called the perturbations of the elements. It must be remembered, however, that the coefficient of t in Swi is to be included with fjf/ki 3 in comparing with observation. 6*8. Secular and periodic terms. The coefficient of t in w t is deduced from observation and is known as the 'observed mean motion.' If we denote it by T?OO> we have If we define oo by means of the equation oo 3 ^oo 2 /*, we obtain, to order m' 21 9R e \ , But since Ci 2 = /x-a, the equation for Ci gives, to the order m', Cl 2 A-! 2 /, 2~.K A7 \ -* = -M l -f - Sjj cos N , /JL /JL \ ki J * V J* /, 2 1 aft c 2 . K . 7 \ a =oo(l~ -gr + r^i-cos^) . ...(3) \ OWoodCi A?i " 1^ /cj-^ or Thus the mean value of a is not oo but this quantity with a small additive portion. The coefficients of t in 02, 03, w; 2 , MS are known as the secular parts of these variables. An important result is the fact that Ci, and therefore a, contains no secular part to the order m'. Since the coefficient of t in Wi is the observed mean motion, the secular part of Wi is defined as any part which it may have depending on t 2 , tf 3 , ... . To the order m' there is no such part. 7-9] FIRST APPROXIMATION 149 The coefficients of the periodic terms in w iy w 2 , w$ depend on the derivatives with respect to Ci, c?2, c 3 of K/v. Since v is independent of c a , c 3 but does depend on GI, we have K\ldK 9 (K\ o~ I - I oCi \ v / * since n = jjf/Ci 3 . The presence of the square of v as a divisor in Wi but in none of the other elements, is of fundamental importance in the theory of the long period terms which have small values of vjn. The simple manner in which this divisor arises with the method of solution adopted here is noticeable. 6'9. Transformation to the elements a, e\> T\ or #, e, F. In the developments of the disturbing function, the angles w^ or quite simple linear functions of them, are used, but in place of the c t we find the elements a, e 1? r l5 related to them by the formulae 5-15 (2), or the elements a, e, r, shown by 5'13 (1). If /be any function of the <S' t -, we have, with arbitrary variations of them, From 5*15 (2) we deduce (2) On substituting these in (1) and equating the coefficients of the fic t , we obtain \ l V /i ^ a/_ l_ l^df d__ \_ty' The derivatives of A' with respect to the c$ may be found from (3), and the differences of ct, e 1 , TX from constants by means of (2), if we put therein dc % =*Ci k$. Similarly, for the elements a, e, r, we find sa-- a Cl , ae= _(lz^ /i na ^ \e 160 SOLUTION OF CANONICAL EQUATIONS [OH. vi and df _ 1 / df e(l-erf ?/_ r d/1 " cci ~~ no? \ da i -j_ (j _ e 2)i 96 (i _ e vfi crj * d#2 net? \ e c!e n e 2 )^ c*rj ' a/; = i i a/ A difficulty occurs in consequence of the presence of the divisor e\ or e in the expressions for 9//?c 2 . But this latter derivative is present only in the expression 6*7 (4) for w 2 , which in turn only appears in terms having the factor e in the expressions for the longitude, latitude, and radius vector in terms of the elliptic elements. It is for this reason that it is usual to give the perturbation of fisr in the form eStzr. This solution of the difficulty is sufficient when we confine ourselves to the first power of ra', but further consideration is necessary when we proceed to higher powers. The solution is then contained in the theorem of 6*4, applied to the development given in 6'15. 6*10. The perturbations of the coordinates. The coordinates are supposed to be expressed in terms of the elliptic elements. If, then, we define the perturbations of the elements as in 6'7, the perturbations of any coordinate x to the first power of m' will be given by "~* \dci l d'Wi / ~*\dC{,dtVi dividci/' "' The latter form again introduces a function of the type con- sidered in 6'4. The periodic part S of i/r is a d'Alembert series and the longitude, latitude, and radius vector, when expressed in terms of the elliptic elements are d'Alembert series. It follows that the periodic parts of the perturbations of these three coordinates are also d'Alembert series. It should be remembered, however, that Scz/e, eSw% contain perturbations which do not vanish with e. 611. This form of the solution, in which powers as well as periodic functions of t are present, is that which is usual in applications to problems of the solar system. The results in this form usually have sufficient accuracy during the limited intervals over which observations are available. It is evident that a continuation of the process to powers of m' beyond the first 9-11] SECULAR PERTURBATIONS 161 will lead to terms containing higher powers of t and to the presence of periodic functions of t multiplied by powers of t. That it is possible formally to express the perturbations wholly by periodic functions of 2, at least in a first approximation, may be indicated in the following manner. Instead of the variables c 2 , w 2y let us use p 2 > q 2 defined in 5*14, and for simplicity let us neglect the inclination. The canonical equations for these two variables are dp 2 dR dq>2 dR ,-jx dt dp% The process for eliminating the terms present in R t can still be followed : it leads to new variables p 20 , q. 2{) witli the same function 7^, which is the portion of R independent of w\, MI'. The development of R as far as the second powers of the eccentricities is given by 4-32 (2), and the 'non- periodic part,' depending on the eccentricities, is given by putting i = in the first line of the expression, and i= 1 in the second line. The resulting terms have the form* (2) ct where P, Q are functions of a/a'. Just as before we show that e and there- fore a is constant. To the second powers of the eccentricities, we have, by 5-14, . e sin 37 =p 2 /Ci > e cos w = q. 2 /ci . Since a is constant we can use units such that /z, a, w, are all unity. If we put e' sin v^ = p f , e' cos w' = #', the expression (2) may be written {i P (P2 2 + ?2 2 + P' 2 + q' 2 ) + Q (pp' + qq')} m' a. With this portion of R the canonical equations (1) give leading to and furnishing the solution, ^2= a ~-5p' + C'sm(m'aP^ + />), ^ 2 =- ^ g' + C coa (mf where {7, Z) are arbitrary constants. With the adopted units, the period of revolution of the planet is 2?r. Since aP<l and m'<'001 for the largest planet Jupiter, it follows that the period of the periodic parts of p 2 , q 2 is ver j l n g compared with the period of revolution of the planet in general, it is greater than 10,000 years. Hence, for intervals of a few centuries, we can develop these periodic * As a contains e 2 , e' 2 , functions of a must be expanded as far as e 2 , e' 2 in powers of these parameters. 152 SOLUTION OF CANONICAL EQUATIONS [CH. vr terms in powers of t and still obtain the needed accuracy without additional calculation. The practical objection to this method of procedure is the complication caused by the introduction of a new argument. Still another argument would be introduced by the solution of the equations for p 3 , #3. Thus, four arguments would be present. Even with a single disturbing planet and the calculation confined to the first power of ra', the work becomes complicated as soon as we proceed beyond the second powers of the eccentricities and inclination and the labour becomes almost prohibitive in the case, for example, of the mutual perturbations of Jupiter and Saturn. In satellite theories, the periods are so short that expansions in powers of t are impracticable and the four arguments must be retained. On the other hand only a few powers of a, and consequently a few values of i, in the expansions of Chap, iv, are needed. Part of the compensation thus afforded is lost by the fact that many powers of the disturbing mass must be retained. The general theory of the secular motions of the elements, to which the solution just given constitutes an introduction, will not be given in this volume. The reader is referred to other treatises, particularly to that of Tisserand, vol. 1, Chap, xxvi, and to later work referred to in Ency. Math. Wiss. Bd. 5. A warning concerning frequently quoted results, which give limits to the eccentricities and inclinations, should be made. These investi- gations, in general, take into account the first powers of the disturbing masses and the earlier powers of the eccentricities and inclinations only. When higher powers are included, approximate resonance conditions have to be considered and these may alter the limits extensively over very long periods of time. Thus while such results may be used with a fair degree of confidence in cosmogonic speculations for a few million years from the present time, we have no present knowledge as to the facts over intervals of the order of 10 9 years. 6'12. Long period terms. We have seen that in a first approximation we may sub- stitute elliptic values for the coordinates in the development of the disturbing function. In Chap, iv, this development gives any angle N in the form i K - wi') jg j'g' k(wi + < - 2(9), where i, j,/, k are positive integers or zeros. The coefficient of a term with this argument contains the factor (cf. 4*15) 11, 12] LONG PERIOD TERMS 153 When elliptic values are used, the coefficient of t in the argument is v = i (n n') jn j'n r k (n + n'). It follows that if we have any argument in which the coefficient of t is jin ji'n', the order of its coefficient will be \ji ji , as far as powers of the eccentricities and inclination are concerned. A long period term is defined as one in which vjn (jiii j\n') -r n is small compared with unity: the secular terms for which ji = j/ = are excluded from this definition. Since it is supposed that we are using the methods developed above, n should properly be replaced by ??oo, but the suffix may be omitted in the discussion. Since n, n' are usually positive we have to consider cases where n'/n is nearly equal to the ratio of two integers. The word 'small' as used above is indefinite in both the theory and its applications; in general, if v/n is less than about one-third, so that the period of the term is longer than three times that of the revolution of the body round the sun, the term would be treated as one of long period. Since n, n' are obtained independently from observation it is always possible to find terms in R for which v/n is small. The critical values of ji, ji are obtained by expanding n'/n as a continued fraction. If p/q be any convergent, then for any convergent after the firsfc, ji = p, j\ = q will give a long period term. But as the order of the coefficient is \p q\ such a term may be quite insensible to observation even after receiving the factor 1/iA For example in the case of Jupiter and Saturn where w = 43996" (Saturn), n' - 109256" (Jupiter), these being the mean annual motions, we have 2n ?&'= '483?i, order 1 ; 5n - 2n' = '0334^, order 3; 72n - 29n' = - -0162*1, order 43. The third of these is obviously insensible. 154 SOLUTION OF CANONICAL EQUATIONS [CH. vi In considering the degree of approximation needed, we can make use of the property of a continued fraction which states that if p/q, p'/q' are consecutive convergents, p' >p, no fraction whose denominator lies between 9, q' gives so close an approxi- mation as p/q. Thus if q' q is large, p/q is usually a close approximation, v/n is very small and higher convergents are unlikely to give sensible coefficients. An apparent exception to the argument is the case of multiples of N when the term with argument N has a sensible coefficient. This case is dealt with in the second approximation (cf. 618). In the case of Jupiter and Saturn, the term for which v = Wn 4w/ has a sensible coefficient; that for which i> = 15/i 6n' is insensible. We have seen that the element Wi is that chiefly affected by a long period term since the term in this element has the divisor i> 2 > while those in the other elements have the divisor v only. In other words, it is the mean longitude which shows the principal effect. But there is an associated short period term in the true longitude which may have a coefficient comparable in magnitude with that of the long period term and which arises in the following way. The determination of the perturbations of e, TV, or more properly, of ecos w, e sin or substantially requires the division of the term in the disturbing function by e (6*9). Thus the term in R substantially acquires the divisor ev when it is inserted in the true longitude. Now e, ra occur in the true longitude principally through the chief elliptic term 2esin(w 1 cr) and the long period term, therefore, gives two terms with motions n v. As v is small, these have nearly the period of revolution as periods. One of the two coefficients is usually quite small owing to the association of powers of e with multiples of tsr: the proof of this statement is simple and is left to the reader. The fact that terms of very long period are usually of very high order would seem to suggest some theoretical limit beyond which such terms could always be neglected. But, as far as is known, the observational determinations of the constants which give the mean motions are substantially independent of those which give the eccentricities, when the 12, 13] OTHER SOLUTIONS 155 determination is spread over many revolutions of the planet round the sun. Thus the ratio of e j e'*'r k to v or v 2 has no definite limit and a coefficient may be very small or very large according to the values chosen for n, ri. From the practical point of view the difficulty is surmounted by supposing that such terms, having periods very long compared with the interval of observation, can be expanded in powers of t. The constant parts of the expansions are absorbed in the arbitrary constants of the solution and, in the case of the true longitude, the coefficient of t is absorbed in the mean motion; the remaining portions are usually quite insensible during the interval. There is, however, an upper limit because of the fact that when the coefficient of the term exceeds a certain magnitude comparable with STT, the procedure previously followed becomes invalid : the phenomena of resonance then begin to appear. This limitation does not remove the difficulty. The form of tho mathematical development has to be changed and the argument proceeds on different lines. The complications which arise make the problem exceedingly difficult : some indications of them will be given in Chap. vin. It may be pointed out that one of the difficulties is due to the fact that the period of the term may become comparable with the periods of the so-called secular terms (6'8) and that it is not then possible to treat them independently even in a first approximation. 6*13. Other forms of solution. Instead of the value of S written in 0*6, we might have used T* t> = ^c tQ w t -f 2 --- sin N. Here the rdles of c t -, w t are simply interchanged. A little con- sideration will make evident the fact that in a first approximation this form of $ gives nothing new: it is only in the second and higher approximations that differences appear. When the Poincar6 variables p^ q%, /> 3 , q$ are used, the expansion of R takes the form R = ZK c cosN' + 2 #. sin -ZV', where N f = ji^i 4 ji'wi -f /8, y8 depending on the constants of the disturbing body only. In this form K cy K s are expanded in positive integral powers of 156 SOLUTION OF CANONICAL EQUATIONS [OH. vi p*> </2> pa, #3 and are also functions of ci, a, e*. The transformation function to be used is then e - - sn cos #', where the suffix zero has the meaning defined in 6*6 except in (7if c )o, (K 8 \, where it means that g a , #3 ar e replaced by 20 #3o> while p%, p$ are left unchanged. We may also interchange the rdles played by p, q. THE SECOND APPROXIMATION 6*14. The calculation of the second approximation to the values of the variables may be very laborious if advantage is not taken of every feature which may help to shorten it. The first step consists of an examination to discover what classes of terms will give sensible coefficients or sensible additions to coefficients found in the first approximation. Methods for the calculation of the sensible portions will then be given and these methods will be developed in such a manner that the actual computation may be reduced to comparatively few operations. There are several devices which can be used to obtain the terms dependent on the squares of the disturbing masses. We can follow the process of Delaunay which involves continual changes of variables until the Hamiltonian function is freed from all sensible terms for which the relations ji =*ji = do not hold; the equations for the final variables can then be solved by series arranged along powers of t. Another plan is the substitution of the results of the first approximation in the derivatives of R instead of the elliptic values which have been used in finding the first approximation ; the equations are then again integrated and the new portions of the variables calculated. Another device which is sometimes useful is a method of integration by parts which makes use of the fact that the derivatives of c\, c%, Ca, w% t w& with respect to t have m' as a factor. The most useful device is, however, the separation of the terms of long period from those of short period and also from 13-15] SECOND APPROXIMATION 157 those which are secular. It will be seen that the second approximations to the long period and to the secular terms can be rendered almost independent of the first approximation to the short period terms, so that they can be determined inde- pendently of the latter. But the effects of the long period and secular terms on the short period terms are usually sensible, and it is these effects which become most evident in comparisons with observations extending over long intervals of time. The methods adopted to prove that these limitations are possible are not necessarily the most convenient for the actual calculation of the sensible terms, so that more than one of the plans for continuing the approximations will be found developed in the sections which follow. 6*15. The Hamiltonian function in terms of CIQ, WIQ. The equations 6*6 (9), 6*6 (10) give, for the Hamiltonian function of the equations for the new variables, the value A - 1 . i (d - cio) 2 + R e + S^Tcos N - Stfcos N , ^CIQ A CIQ ...... (1) in which powers of c i i - CIQ beyond the second are dropped. This last omission is equivalent to stopping at the order ra' 2 . We need (1) expressed in terms of c t0 , W& to this order. Since 8 has the factor ra', we have c t -=c t o, ^ = ^,-0 when ra' = 0. It follows that, as far as the first power of ra', the equations 6*6 (6) give and that, if /be any function of c if w iy t, ?\f 7\f f(Ci\ Wi\ )=/(c t0 ; w#\ + 2 ^ (c, - Cfl) + ^ (w< - to the same order. The substitution of (2) in this last equation enables us to write it 158 SOLUTION OF CANONICAL EQUATIONS [CH. vi On applyiug this to %KcosN=R t and to %K cos N 0} we obtain V V ^ \T 7? i V /^tO 3$0 SRtQ 3$o\ f . 2<K cos iv = JK tQ + i ( ~ ---- - - ~ , ...... (4) \3c i0 9w t0 dwtfdcio/ (5) CiO O Since R already contains the factor ra', these results hold to the order m 2 . The same application to the development of R c may be made. The second term of (1) evidently has the factor 'in' 2 and may be replaced by its value in terms of the new variables by means of the first of equations (2). Thus the Hamiltonian function for the canonical equations satisfied by c,o, w t -o to the order m' 2 becomes F t + F c ................... (6) 2c 10 2 where /IT\ F c The expressions (7) contain the factor m' 2 . The application of the theorem of 6*4 shows that they are d'Alembert series and therefore that they contain no powers of e, F as divisors. If R t be defined to contain all the terms in which ji =ji = do not simultaneously hold and no others, R c and therefore ^ c0 will contain only the terms in which these relations do hold. Since $o contains the same terms as those present in RM, it follows that F c is like R t o in this respect, and therefore that the secular portions which depend on the terms in which ^i = jV = will arise only from FI, F t , R cQ . The Hamiltonian function (6) will, however, be used below only to distinguish between the effects of the short period and long period terms, and for this purpose R t will be defined to contain the short period terms only. The investigation will show what portions may be neglected in the actual calculations which can then be carried out by a more simple method. 15, 16] SHORT PERIOD TERMS ' 159 A direct second approximation to the solution of the equations for Ci, w it is easily seen to be given by 4. - 4. V _ Ci " Cio + ^ w*tofl 9 2 $o 9$o These require the formation of the products of the derivatives of SQ for each of the six variables. The method given in the text confines the formation of such products to those in one function, namely (6). 6*16. Influence of the short period terms in the first approxi- mation on the second approximation. Let R t and therefore S Q contain only short period terms, so that in 6*15 (6) there are no small divisors v or v* tending to raise the magnitudes of the terms in FI, F t , F c . Suppose that all these functions are expressed as sums of periodic terms. These terms will have the same general form with respect to the variables c t -o, w lQ that R had with respect to Ci, Wi, that is, they have the form K Q cos N Q> where N Q is a linear function of the WM, w\, Wz with integral coefficients, and K Q is a function of the CM and of the elements of the disturbing planet. The terms present in R cQ are all either of long period or those for which ji=j = 0, while those arising from F it F t , F c are of the same character with additional terms of short period but all having the factor m' 2 . If the short period terms were again eliminated by a Jacobian transformation, the new variables would differ from CM, w^o by terms having the factor m' 2 and with no small divisors present. As the largest value of m' in the problems of the solar system is less than "001, and as an accuracy to *001 of a short period coefficient is rarely attainable in comparisons with observations, these portions can generally be neglected. The long period terms present in J\, F t there are none in F e because the product of terms of the form cos (at + a'), 160 SOLUTION OF CANONICAL EQUATIONS [CH. vi cos (-42+ A 1 ), in which a is small and A is of the order of the mean motion, gives rise to terms with arguments (Aa)t + A'a' have the factor m' 2 and are additive to those in R cQ with the factor m'. Thus these terms will merely change the coefficients of the long period terms by amounts of the order of *001 of their values at most, and such changes again are rarely sensible to observation. The same result holds for the terms in which ji=ji'-0. Exception to these statements may arise on account of the fact that when we differentiate with respect to c 2 o or 030 a divisor of order e 2 or F is, in fact, introduced. In the method adopted for calculation it is seen (last paragraph of 6*17) that these terms of lower order disappear, so that the general argu- ment is not affected by them. Thus the short period terms present in the first approximation can be altogether neglected in proceeding to a second approxi- mation, or at most, only a very few, and those with the largest coefficients, need be retained. It follows that the long period and secular terms can be obtained to the order m' 2 with sufficient accuracy if we neglect at the outset nearly all short period terms present in R. As the apparent exception mentioned in the text always raises a difficulty in the discussion of the canonical equations for the elements, further details as to the occurrence of such terms may be of value. According to the theorem of 6*4, R c produces d'Alembert series in F e and is therefore free from these terms of lower order. Hence they will only arise through F t . The divisor e 2 will arise in F t only through the product (9/2 w /<to>2o) (9>/ 9c 2o). This may be written P+, where ' * 2 acgo 8w>2o 2 j By the theorem just quoted, P is a d'Alembert series, and is therefore free from the exceptional terms. As for , we note that elliptic values are to be substituted and that then jR t Q=dS ldt. Hence, since derivatives with respect to , w%) , c^ are commutable we deduce ^"SSiUfcii 16, 17] LONG PERIOD TERMS 161 On account of the relation between R, S this suggests that some second order terms might have been included in the expression for S which would have prevented the occurrence of these terms in the new Hamiltonian functions. The fact that another method shows that they ultimately dis- appear, indicates that the portions of this character which arise from the solution of the equations 6'15 (8) which give c t> w^ in terms of c,- , w^ to the second order, will cancel the portions which arise through Q. As a matter of fact, even if the method were used for calculation, the terms would cause very little trouble. For we are actually interested only in the long period terms present in , and the operator djdt introduces the small factor v in such terms. The numerical effect of this fact would be to cancel to a large extent that produced by the divisor e* or r. 6*17. Calculation of the second approximation to long period and secular terms. The fact that the first approximation to the short period terms exercises little or no sensible influence on the second approxima- tion to the secular and long period terms, enables us to calculate the latter as though the former did not exist. Thus the equations for CM, w t o become the same as the original equations for c it w t would have been if we had omitted all short period terms. For the sake of brevity in notation, therefore, we shall omit the suffix zero in finding this second approximation, restoring it only at the end of the work. The equations are all short period terms being excluded from R. These equations, except that for w\ , may be written dci __ dR dw% _ dR dws __ dR di^dwi' ~dt~~fo*' dT""""3c3 ....... ^ If we differentiate the equation for Wj and make use of the first of equations (2) we obtain __ d/f^_dR\_ _ V9^_rf/W*\ ~~ dt W dc J ~ d 4 dw l dt \dcj ....... dt* The first approximation gives the values of Ci, Wi in the form /3 + i* + 2jBcos(^ + "o), ............... (4) B&SPT II 162 SOLUTION OF CANONICAL EQUATIONS [OH. vi if we note that the addition of ^TT to i/ will take care of the presence of sines. The coefficients &, B all contain the factor m' except that of t in Wi which is n w . In the case of Ci, we have fti = 0. Finally, B contains the first power of v as a divisor in all cases except that of w\ which contains an additional part having v 2 as a divisor. This first approximation was obtained on the supposition that the elliptic elements in R were constant. If we denote the difference of the constant and variable values of the elements by the symbol 8 (in the case of Wi, the symbol Sw { denotes the difference between Wi and its undisturbed value n$ t + const.), and if the additional part due to the second approximation be denoted by $2 > Taylor's theorem, applied to equations (2), (3), gives d rs v / d 2 R ^ d 2 R 5, \ . i rt > /e \ -7; 820* ^^U -a toj-f ^ a - ocj ), 1= 1, 2, 3, ...(5) at \dWidWj dWidCj / \ / * 2 o o /PX 1 = 2,3, ...(6) -- dCidWj - ^ -- r Ci 4 J \dwidWj d / PR ...... (7) in which j takes the values 1, 2, 3, Since all the terms in the right-hand members of these equations have the factor ra' a , constant values may be substituted for the elements in the derivatives of R. Also since Sc if Sw t are present in a linear form only, their various portions may be separately calculated in any manner which may be convenient. According to 6'7 (8) the periodic parts of 8c,-, 8w$ are found from * dS X * S /QX SC<= , *", g^, ............... (8) where tf-SsinJV, R t = 2KcosN, ............ (9) 17, 18] SECOND ORDER CALCULATION 163 with N =ji Wi +J2 w 2 +jz w 3 +ji wi +J2 Wt, * I I L f\ 2/3 /I A\ if = 'I-* ji -(- o* n } v ^ \jy n == u/ I c\ t ........ {L\J ) constant values for the elements being used in these expressions. Hence 1 v y v ^C2 ' v uC$ (11) Swi = -35^- sin^-S-^sin^. (12) Although these formulae serve for all periodic terms, we are considering only the long period terms in R t . The secular parts are given by C/jtV/ UxLfl OCi == uC% == t , OCa L ~^. 8 '^ ^ (13) Swi = 0, &W2= t -^ - , 8w$ = t ^ - , OCz OC$ where R c = ^K cos (jiw* +^3^3 +jV^/). The substitution of (8) in the right-hand member of (5) gives _ ~- 2 y * dwi dwj dcj * dwi dc$ dwj " According to the theorem of 6*4, this is a d'Alembert series since ti, R, dR/dWf are d'Alembert series. The same result is evidently true for the series in (7). For i = 2 in (6), we note that CzdR/dcz is a d'Alembert series, so that 0282^2 has the same character. Similarly c 3 S 2 w 3 is such a series also. Thus the presence of the divisor e 2 or F, lowering the orders of certain terms in the first approximation, does not affect the equations for the second approximation because such terms disappear. This is the proof referred to in 6*16. 6*18. The principal part of the coefficient of a second order term. Let us consider first the case of a single term of long period. In general, the principal part of the coefficient will be that part 164 SOLUTION OF CANONICAL EQUATIONS [OH. vi in which the divisor v occurs to the highest power. This is evidently the fourth, obtained by substituting the first term of Sw l in 6-17 (12) in the first line of 6'17 (7). It gives 3K . n ~ 9jl 3 ~T ~2 S ^ n N COS N' Whence, integrating, s 9 . 3 K 2 n* . , 7 ^i=^i 3 - j? - a sm2^. If, then, we write &wi = B sin N, for the first term in 6*17 (12), we obtain StWi^ffsmZN ................... (1) o This result is independent of the method by which the coefficient B may have been obtained. It gives at once the principal part of the second order term with argument 2jV when the first order part with argument N is known*. It still holds if we include in Sivi all the terms for which ji, ji are the same. For these terms may be written in the form P cos ( jj wj +ji'wi) + Q sin (j^v t +ji'wi), where P, Q are functions of Ci, c 2 , c 3 , w%, iv$, a', e' y w 2 ', and this expression may be put into the form where /3, jB are independent of Wi, W}'. In this case we put j\Wi +jiWi -f /8 for N. It is thus immediately applicable when the numerical values of B, $ have been obtained. It is not difficult to extend this result to the case in which two or more long period terms are present. For two such terms in R denoted by K cos N, K cos N, we have j - 2 = j^K cos N j-K cos N. * The result, obtained by a different method, was given by E. W. Brown, loc. cit. 7'32. 18, 19] SECOND ORDER PERIODIC TERMS 165 If Swi = BcoaN+5 cos N 9 we have B = - 3Kn/* c l} B = - 3Kn/v 2 d . Whence, on integration, (2) In the general case, we add together all such pairs of terms. 6*19. Effect of a long period perturbation of the disturbing planet. The effect of such a perturbation is most marked in wi and the principal part of its effect on a term Kcos N in R will be = _ j x ' K sin N. a<. In the formation of the canonical equations, it is assumed that i is independent of the elements of the disturbed planet. Hence If we are given Swt' = B' sin N' = B' sin (v't + *><>'), where v f /n is small, we obtain, by a procedure similar to that followed in 6'18, (1) When we are dealing with the mutual perturbations of two planets, there will be terms in Sw/, due to the effect of the planet (which we have been calling the disturbed planet) having the argument N, and for these terms (1) gives This case requires care if it is deduced directly rather than by substitution in (1), because Swi contains the elements c^, w it and we might be tempted to substitute in R before forming the n-3 166 SOLUTION OF CANONICAL EQUATIONS [OH. vi derivatives with respect to c$, Wi. That we cannot do so is seen from the statement made above, namely, that the canonical equations are formed on the basis that the coordinates of the disturbing planet are functions of t only and are independent of the elements of the disturbed planet. This basis must be retained in the subsequent work. A useful exercise for the student is the deduction of these results by solving the equations of 6'15. He will find, for example, that the solution 6*15 (8) of the equations 6'6 (6) to the second order will contribute ji B 2 sin 2N to the value of S 2 ^i f r the case considered in 6*18, while the solution of the canonical equations of 6'15 contributes I^Z^sin 2^V, the sum of these giving the result 6' 18 (1). Incidentally, this exercise furnishes a reason for not continuing the solution by the Delaunay method. There are two portions to calculate instead of one, and each of these portions is large compared with their sum. 6*20. More accurate determination of second order terms. The calculation of the portions of 8 2 fy, S z Wi which have the small divisor i> 3 in the terms with arguments 2^ or JV" N, is not difficult. In the equations 6*1 7 (5), 6*17 (6) we need to use only the portion of Swi which has the divisor zA Thus the derivatives of R needed are Now the derivatives of R with respect to w i} Ci will have been obtained in finding the first approximation, and the derivatives (1) can be written down at once, even after numerical values have been inserted. To calculate S 2 Wi, we nee d the full value of Sivi and the values of 8c,-, St^2> Swa f r the terms in the first line of 6*17 (7); in the second line of the latter equation, we need the principal part of Swi only and we can neglect Sc it Sw 2 > Siv$. In 617 (5), (6) we can also neglect Sc t) Sw%, Sw$ and use only the principal part of Swi. Thus, in all cases, the terms divided by the cube of the small divisor can be obtained with the second derivatives (1) only. 19-21] SECOND ORDER SECULAR TERMS 167 6*21. Second order secular effects. These are produced by inserting the values 6*17 (13) in the equations (5), (6), (7) of 6*17. It will be noted that they give d'Alembert series in the same sense as the periodic portions discussed in the same section. There are three classes of terms present in the right-hand members of the equations to be considered. (a) Terms of the form t multiplied by terms in which ji=ji' = 0. (6) Terms of the form t multiplied by terms in which v 4= 0. (c) Terms in which ji = j/ = which arise when the periodic portions of Bc i} Bw i are substituted, these portions having been laid aside in the previous sections. The integration of the equations with the terms of class (a) gives terms factored by t 2 and by m' 2 since constant values may be substituted for the elements in the right-hand members. It is to be noticed that B z Ci contains no such terms because when J! = 0, dR/dwi and its derivatives are zero. The same result is true of the terms arising in the first line of 6'17 (7). A constant part arises from the terms of the second line which gives a term factored by t 2 in S z wi. All these portions are very small since no small divisors enter. The terms of class (6) give rise to differential equations of the form -T = tk cos (vt an integral of which is tk k x sin (vt + i/o) + -a cs (vt 4- z> ). Terms of this character arise in S%c it S%W2, B^w^. We have seen that terms of the second order with the small divisor v 2 can usually be neglected, and the terms with the factor t/v will rarely be sensible except for large values of t. 168 SOLUTION OF CANONICAL EQUATIONS [OH. vi For 8 2 Wi we have an equation of the type d?x ^ , . , . -Tig = & cos (vt 4- VQ), a particular integral of which is tk k x -- g cos (yt -f VQ) + 2 -3 sin (i/ -f i/ )- The second of these terms is of the same order as those con- sidered in 6'20; the first will be sensible for large values of t. Class (c) gives rise to terms of the form \t in & 2 c t -, ^2^2, B^WS and to terms of the form \t 2 in S 2 ^i- A long period term in Swt possesses the small divisor v* so that the resulting term in S 2 ^j may become sensible as a result of integration, for large values of t. The proof that S 2 Ci contains no such terms is furnished as follows. Suppose that in the transformation of 6'6, we define R t as containing all terms for which v 4= 0, and that instead of solving by the method of approximation adopted above, we write down the equation for CIQ as given by 6*15 (6). It is By definition, ,<$ is independent of w w and FI, F t) F c contain no terms factored by t', as the remaining terms have the factor ?/i /a , we can insert constant values for c t0) lu^, ^30 and the value ftoo + const, for WIQ. The right-hand member has then no constant term and consequently CIQ has no term factored by t. Next, the solution of the equation Ci = Cio + 9$/9wio, to the second order, is The portion of this under the sign of summation contains the factor m' 2 , and since SQ consists wholly of periodic terms, it cannot contain any term factored by t. In the second term we can substitute the values of c<o, w iQ to the order m'; these have the form j3 Q + fat, where # > Pi are constants. Hence, 3$ /9wio will 21, 22] GENERAL SUMMARY 169 contain only terms of the form B cos N or Bt cos NQ, where the coefficient of t in N is not zero, and B is a constant. Since it has been shown that CIQ contains no term of the form t x const., it follows that Ci has the same property. It does, however, contain terms of the form Bt cos N where the coefficient of t in NQ is not zero. Finally, the result is true for any function of Ci. For such terms can arise only from products of terms of the form t cos N with terms of the form cos N^ the former have the factor m' 2 while the latter have the factor m', so that the product will have the factor m' 3 . In particular it is true for a = CI Z //UL and for any function of a to the order m' 2 . For remarks on the degree of importance to be attached to this well-known result see 7 '29. 6*22. General Summary. The notation of 6*15 will now be resumed. The results in 6*17 to 6*21 give the values of c^o, w t0 in the form dQ const. 4-Scio, Wio = const. -hSw t -o, ......... (1) except in the case of WI Q which takes the form Wi = T?OO t -f const, -f SWIQ ................... (2) The symbols 8c#), Sw; include all long period and secular per- turbations as far as the order m' 2 . In putting $ = 2 smN , ..................... (3) we have included in S only the terms corresponding to the short period terms S/f cos Af in R. For such terms we have seen in 6*7 that a sufficient approximation to the values of c t -, Wi in terms of Cio, WM is, in general, given by The values of c^o, WM given by (1), (2) are substituted in (4). Since $ contains the factor m', it is sufficient to use the values of c,-o, WM to the first order in the second terms of (4). Thus the short period terms to the second order are found with sufficient accuracy by substituting in them the constant 170 SOLUTION OP CANONICAL EQUATIONS [CH. vi values of the elements increased by their secular and long period portions. A literal development of R is needed to obtain the first approximation in order to obtain the first derivatives of jR with respect to the elements. The second derivatives of R are needed to a lower degree of accuracy, and as far as they are usually necessary for the calculation of the second approximation to the long period terms, they can be obtained from the first derivatives after numerical values have been inscribed therein. 6*23. Integration by parts. A method of integration which can be applied to non-canonical as well as to canonical equations for the variations of the elements depends on the identity smA r ...... (1) dt\jf ) dt\N) where N is written for dN/dt. Suppose that two of the variables chosen be ivi and a (or any function of a) and define n by ?i 2 a 3 = />6. Let the remaining variables be any functions of a, e, CT, F, which do not contain t explicitly. The equation for wi has the form (2) and the equation for any one of the other variables, including n t has the form ~ = 2PcosAT+Q, ..................... (3) where P, Q have the factor ra' in both cases. Hence P, Q may be functions of any of the elements except Wi, and MI, t will be present only in JV, and in the form jiWi + ji'wi, where Wi' = n't + e'. It follows that N has the form and that N has the factor m' and the form P cos N 4- Q. Since cos (N - 90) = sin N, this statement includes terms of the form SPsinJV'+Q. 22-25] SPECIAL CASES 171 Integrating (3) by the aid of (1) we have ...(4) where # is a constant. Since P has the factor in' and since the derivatives of all the elements present in P, N have the same factor, the third term has the factor ra' 2 . In a first approximation, terms factored by w' 2 are neglected and constant values are substituted for the elements in the terms factored by ra'. Hence, the first approximation to the integral of (3) is .r = a- + 2~"sinN, + Qot ................ (5) #o For a second approximation, the values (5) are substituted in the second and fourth terms of (4): in the third term constant values of the elements can be used. The integrations may then be carried out in the usual manner. The first approximation to Wi is obtained from (2) after the substitution for n of its first approximation obtained from (5); in this approximation a term Q t is not present in (2). The second approximation is made in a manner similar to those out- lined for the other elements. 6*24. The case of a single long period term. Whenever it is possible to limit the long period terms to a single value of jiw +ji'wi and its multiples it is possible by a change of variables to eliminate the time from the Hamiltonian function, H. This function equated to a constant then constitutes an integral of the equations, and by means of this integral Ci may be expressed as a function of the other variables and thus eliminated from the equations. The manner in which effective use can be made of this elimination is shown in Chapter vin which treats of resonance but which is equally applicable to terms of long period. 6 '25. The theory outlined in this chapter, in common with all theories which depend on the method of the variation of the elements, has a simplicity of analytical form which makes it attractive for many theoretical investiga- 172 SOLUTION OF CANONICAL EQUATIONS [OH. vi tions and particularly for those which are concerned with the phenomena of resonance. But it is doubtful whether it lends itself most conveniently for the calculation of ordinary planetary perturbations. It appears, in general, to demand more extensive calculation to secure a given degree of accuracy than those methods in which the perturbations of the coordinates are obtained directly. The chief objection is the necessity for expanding the disturbing function literally in order that the derivatives with respect to the elements of the disturbed planet may be obtained : there are six of these derivatives to be found, as against three functions to be calculated when the forces are used. A second objection is the necessity for carrying the expansion in powers of the eccentricity of the disturbed planet to one order higher than that needed in the coordinates. A third objection is the slow convergence along powers of e 2 , e' 2 , r of the series which gives the coefficient of any periodic term, especially for those terms which contain high multiples of y, (/'. To a large extent this slow convergence disappears where numerical values for these elements are used at the outset of the work, particularly if the developments are made by harmonic analysis in the manner outlined in 3'17. The chief exception to these statements is the theory of the Trojan group, but this theory is so different from that of the ordinary planetary theory that comparisons are not useful. It is possibly true that all the actual cases of resonance or of very near resonance can be treated effectively by this method, but some rather extensive comparisons would be needed before any reliable statement could be made in this respect. The literature on the subject of the application of the method of the variation of the elements to the planetary problem is extensive. The reader is referred to the standard treatises, particularly to that of Tisserand and to the articles in the Ency. Math. Wiss. for the earlier literature. For the later work, references and abstracts will be found in the mathematical and astronomical publications which summarise the literature annually. 6*26. Throughout this chapter it has been supposed that the mutual perturbations of two planets can be separated, so that in determining the motion of one planet that of the other can be supposed to be known. As long as we confine our attention to perturbations which depend only on the first power of the ratio of the mass of any planet to that of the sun, this procedure is justified by the fact that the coordinates of the disturbing planet only appear in a function which has the mass of this planet as a factor. Hence, any perturbations of these coordinates will produce pertur- bations depending on the squares or products of two disturbing masses. When we begin to calculate these higher approximations, it is evidently necessary to calculate previously the perturbations depending on the first powers of the masses for both planets. 26,26] SPECIAL CASES 173 But the general problem of three bodies admits of four integrals in addition to those arising from the uniform motion of the centre of mass, namely, the three integrals of areas or of angular momenta and the energy integral. No use has been made of these in the theory developed above and the question naturally arises as to whether they can be effectively utilised for the abbreviation of the work. In asteroid problems where there is a very small mass disturbed by a very large one, the effect of the former on the latter is negligible, and the integrals consist chiefly of por- tions depending on the large mass, the effect of the portions depending on the small mass being relatively small. Thus the integrals are not useful in such cases. But when the two planets have masses of the same order of magnitude, as, for example, in the case of the mutual perturbations of Jupiter and Saturn, the variations of the coordinates in the portions'of the integrals due to the two planets have the same order of magnitude, and it would seem that this fact should be utilised to abbreviate the calculations. It generally appears, however, that the lack of symmetry which their use introduces, causes additional difficulties in the calculations. The more useful procedure is that of following the usual method for each of the planets and later making the integrals serve as tests of the numerical work. These tests are particularly valuable for the coefficients of any terms of very long period which may be present. For theoretical work in the general problem of three bodies, these inte- grals have been much discussed. Since there are four of them, the system of variables, namely six for each planet, can be reduced from the twelfth order to the eighth. We shall see in a later chapter that it is not always possible to proceed by following the process described at the beginning of this article. It breaks down in certain cases of resonance and notably in the case of the Trojan group. If, for example, we attempt to determine the action of Saturn on a member of this group without taking into account at the same time the action of Jupiter, quite erroneous results will be obtained. A difficulty of a similar nature occurs in dealing with the motion of a satellite disturbed by a planet other than that about which it is circulating : it is necessary to take account of the disturbing action of the sun during the computation of the disturbance caused by the planet. CHAPTER VII PLANETARY THEORY IN TERMS OF THE ORBITAL TRUE LONGITUDE A. EQUATIONS OF MOTION AND METHOD FOR SOLUTION 7*1. The equations of motion have been derived in 1/27. The independent variable v is the longitude reckoned from a depar- ture point within the osculating plane, while the longitude v is reckoned in the usual way from an origin in the plane of reference to the node and then along the osculating plane. The radius vector is r and i, 6 are the inclination and longitude of the node of the osculating plane. The force-function is fj./r + /j,R, (1) so that jjuR now denotes the disturbing function. The remaining definitions and the equations of motion are as follows: ?/,= -, (c) = r 2 r=l cost, (2) r \q/ dt x ' I M """" Q == Q "^ "~" 9. ~^\ ~T "" == Q ~*"\ I t-i ~7 " ~7 ) \ <5 ) __ _ , dv" u 2 dv ' dv~\p u 2 ' dv dv' ...... W,( d The latitude L t defined by sin L = sin i sin (v 0), .................. (9) may be found directly by solving the equation (d 2 i\ r sin i cos i a dR .-. x ^-j-f llsini^-^-T - ^-^^rr ....... (!0) dv 2 ) sin (v -0)u 2 dr v ' 1-3] EQUATIONS OP MOTION 175 The values of the variables u, v, t, L (or F, 6) are to be deduced from these equations in terms of v. 7'2. It was pointed out in 1'27 that any substitutions of the form =/(?!) M-stt^fal) leave the essential characteristics of these equations unchanged, namely, that they shall be integrable like linear equations with constant coefficients when the right-hand members have been expressed in terms of v. A trans- formation which renders the equations useful for the treatment of the satellite problem is q = q t M=z 1 gr 1 4. The transformation to the new variables HI, q^ is straightforward. For the Ui equation we have, if D be written for djdv, so that Du^ disappears. This and the remaining equations become, if be put for w 3 3fl/9w, and /? for u*R, />a Ml + Wl - q L = - -- 3 /^ - wi n /> ( =^ 1 j - g =-^ ix . Dt = ~,^ -.. Z)y=l-fr/)0, </! 2 Wj 4 8y ' ^ ?^ ' ' "r "" 3 "^" "" ^* r W "~^7 4 ^F When the ratio of the distances is neglected we have R V 'R V and each is independent of u\ . This portion constitutes the chief part of the dis- turbing function in the satellite problem. It will be noticed that Dt is a function of u\ only, and in fact that the substitution % = (DQ"~i/i~i eliminates the radius vector from the equations of motion. No particular advantage, however, appears to be gained by this elimination. 7*3. The method for solution. The equations will be solved by continued approximation. The function R contains as a factor m'jp, the ratio of the mass of the disturbing planet to the sum of the masses of the sun and the disturbed planet. This factor being always small (its maximum value is less than '001), the first step is the solution of the equa- 176 TRUE LONGITUDE AS ARGUMENT [OH. vn tions with R = 0. As we have seen in Chap, in, this solution gives elliptic motion and the solution will be called the elliptic approximation. The results consist of expressions for the variables u, t, ..., in terms of v. For the first approximation to the disturbed motion, these expressions are substituted in the terms which have m'/ft as a factor and the equations are solved again. Analytically, the process presents no difficulties since, with the exception of the equation for t, the right-hand members become functions of v only, while the left-hand members are linear with constant coefficients. The value of q is first obtained and then those of u, t and of the remaining variables. The second approximation is similarly obtained by substitut- ing the results from the first approximation in the terms which have m'/fji as a factor and proceeding as before. It is rarely necessary to go beyond this stage in planetary problems and, in fact, a second approximation is necessary in general only for those terms which, on account of their long periods, have re- ceived large factors during the integration of the equations giving the first approximation. The system of differential equations is one of the seventh order while that from which it was derived was of the sixth order requiring six arbitrary constants. The additional arbitrary con- stant necessary in the new system owing to the differential definition of v, will be defined as follows. The final expression of v v in terms of v is a sum of periodic terms and powers of v : when all these terms are neglected we are to have v = v. In general, this is equivalent to putting v = v when m' = 0. But there are sometimes periodic terms present whose coefficients do not vanish with m' but whose arguments become constant when ra' = ; the relation v = v is to hold when these terms are suppressed. 7*4. The elliptic approximation. When R = we have q, F, 0, i constant and with D = d/dv, 3, 4] ELLIPTIC APPROXIMATION 177 In accordance with the definition given in the previous section, the last equation gives v = v. The solution of the equation for u can be written - = u = q + qe cos (v -or), where e, ty are arbitrary constants. As we have seen in Chap, ill, this is the equation of an ellipse with the origin at one focus. If 2a, e, isr be the major axis, eccentricity and longitude of the nearer apse of the curve, we have Following the notations of Chap, in, namely, n defined by n a a 3 = /*, with nt 4- txr as the mean anomaly and X as the eccentric anomaly, we obtain t expressed in terms of v by the equations [cf. 3'2 (16) and (20)], nt 4- e TV = X e sin X, tan J X = I - - ) tan ( v vr), \ 1 -f" 6/ ...... (1) or by [cf. 3'8 (3)], nt + = v Ef, Ef = 20 sin (v 57) Je 2 sin 2(v r)+ .... ...... (2) The solution of the equation for i gives sin Z = sin i sin (v 6). The arbitrary constants of the solution are q, e t t*r, i, 0> 6. It is, however, more convenient to regard n as one of the funda- mental arbitraries since it is determined more directly from observation, and to regard q as a function of n, 0, defined by means of the equations l/q = a (I e 2 ), n 2 a 8 = /A. The adopted definitions of u, q } R give them the dimension 1 in length. If we put u/a Qy <?/a , ^R/o for these symbols and define n by the equation no 2 3 = p> none of the equations except that for Dt is altered and the latter becomes n^Dtq^u^. The unit of length is at our disposal: it will be found convenient to so choose a that T? O is the mean value of the angular velocity of the disturbed body which has been adopted. With this definition therefore, we put a = a, n^=n in finding the first approximation to the perturbations. 178 TRUE LONGITUDE AS ARGUMENT [OH. vit We shall suppose this transformation to have been made so that u = (1 + e cos/) -r a (1 - e 2 ) with a = 1 is the elliptic approximation, to the end of 7'18. B. THE FIRST APPROXIMATION TO THE PERTURBATIONS 7' 6. Development of the disturbing function. According to the plan outlined in 7 '3, the elliptic approxima- tion, that is, the values of the coordinates in terms of v and six arbitrary constants, found in 7'4, is to be substituted in the derivatives of R which are present in the right-hand members- of the equations of motion. Since the disturbing function is here denoted by pR we have, from 1-10, n m' (\ rcosS\ A2 , , ~ 7? = ( ^ -73- J , A 2 = r 2 + r 2 - 2rr cos S, cos S = cos (v 0) cos (v f 6) + cos / sin (v - 0) sin (v' 6) = (1 - JF) cos (v - v') -f^F cos (v + v' - 20). The disturbing function contains the coordinates r', v' of the disturbing body and these must be expressed in terms of v. Since the orbit of the disturbing body is used as the plane of reference, we have i = /. The work is best done in two steps. First, r', v' are expressed in terms of t by means of 1 iWcosfo'-tQ r '- a '(i- e '*) v' = n 't +e' + 2e' sin (n't +'-*/) + > as found in 3*11, and then in terms of v by means of the relations similar to 7 '4 (1), (2). It is found convenient to introduce the angles 4, 5] DEVELOPMENT OF R 179 Evidently f is the true anomaly of the disturbed planet and /i is the mean value of the true anomaly of the disturbing planet when the latter is expressed in terms of v. The derivatives of f t fi are in the ratio n : n', that is, in the ratio of the mean motions. The disturbing function is ultimately expressed as a sum of cosines of multiples of the angles f, /i, vr -or', r -H r' 20, with coefficients which depend on a, a', e y e', F. The chief difference in the literal form of the expansion from that obtained with t as the independent variable is the presence of a/a' in the form of powers of n'/n as well as directly. But as these powers of n 9 jn occur only in rapidly converging forms they cause little additional trouble*. The expression for R used above assumes that the plane of motion of the disturbing planet is fixed and adopted as the plane of reference. It should be pointed out that, as far as perturbations of the first order with respect to the masses are concerned, it makes no difference whether this plane is fixed or moving. For since its motions are produced solely by other dis- turbing bodies, they contain the disturbing masses as factors. But the effects of the disturbing body enter the equations of motion only through R which has m' as a factor : these motions will therefore produce perturba- tions having the product of two disturbing masses as a factor. Hence, if we have solved the problem under the assumption that the plane of reference is fixed, the solution to the first order still holds when we transform to another plane of reference which is actually fixed, the motion of the former plane being included in the transformation. In other words, we need to take into account only the geometrical effect of the motion of the plane of the disturbing body and can neglect its dynamical effect on the disturbed body. Actually, these second order dynamical effects are, in most cases, so small that they may be neglected in making comparisons with observations. Another point of a similar character may be mentioned. We are sub- stituting constant values for the various elements in the expressions for the coordinates in R. To the first order of the disturbing forces it makes no theoretical difference what these elements are, whether, for example, they are osculating elements at one date or another, or are mean elements derived * A method, similar to that just outlined, for developing the disturbing func- tion in terms of the true longitude, is given by C. A. Shook, Mon. Not. R.A.S. vol. 91 (1981), p. 553. In this paper will be found the literal development to the second order with respect to the eccentricities and inclination. 180 TRUE LONGITUDE AS ARGUMENT [CH. vn in some manner, for all these sets differ from one another only by magni- tudes of the order of the disturbing forces. But when we compute to the second order actual definitions are necessary. In general, we get better accuracy when we use mean elements if they are known. Such elements are best found after the theory has been completed and their insertion usually involves small corrections to those which have been used in forming the theory : in most cases such corrections are easily made. For these reasons, it is not necessary, in forming the equations for the first approximation, to use a separate notation for the elements used in the elliptic approximation and for the new values which may be assigned to them in the first approximation to the perturbations. 7*6. Numerical developments of the disturbing forces. The following method of calculation is based on the possibility of expressing the disturbing function and the disturbing forces in the form ZiKiCOsi (/-/i + CT ~ r') + ^iKi sin i(/-/i + *r - w'), (1) where the coefficients are series of the form 2j,h ^h cos (3f + A/i) + ^ A'M sin (jf+ j/V/i), . . .(2) the latter coefficients containing the numerical values of the eccentricities, of the inclination and of 2tsi 20. The coefficients in the series (2) are supposed to be calculated by double harmonic analysis for each value of i needed. The principal reasons for the adoption of this plan are first, the slow convergence of the coefficients A*, K{ with increasing values of i, and second, the comparatively rapid convergence of the series for AJJ , A ' JtJ , so that only a few special values of /, /i are needed for the harmonic analysis. No additional calculation is involved by the retention of w w' in a literal form as far as the final step. Omitting the factor m'/p, we have i^ 7 t3 cos S _ du ~ dr A 3 "/a 1 r/2 "~ r2 r2 cos ^ ~~"2A r 2A 8 7' a ' ............ (' ^-2tf) l ...... (5) fr-7] NUMERICAL DEVELOPMENTS 181 (iR) = - - . r' sin (v - 6) sin ( - 0), . . .(6) When 71 (10) is used instead of 7'1 (7), (8), the calculation of (5) is needed only for that of (7); (6) is then calculated without the factor sin (v - 0). Of. 1'28. It will appear below that of these, the development of (7) re- quires the highest degree of accuracy. Somewhat lower accuracy will serve for (4), and still lower for (5), (6); since (5) contains the factor F, these facts require that (3) shall be carried to the highest accuracy of all the functions. The most extensive part of the work is the development of the functions r 2 / A, r 3 / A 3 , the former being needed to a higher degree of accuracy than the latter. 7*7. Numerical development of the disturbing function. Define A, B, by the equations .A cos(v - <?-) = cos (v-0), A sin (v - - B) = cos / sin (t; - 0).J ......... ^ ' The expression for A 2 in 7*5 can then be written tf = r 2 + r' 2 -2rr'Acos(v-v'-B) .......... (2) From (1) we deduce -4 2 ==l-sm 2 /sin 2 (?;-0), ............... (3) A sin B = sin 2 J / sin ( 2v - 2(9), v - 2(9), ) sm 2 0;-<9).j ......... Since v=/+'zar, the special values of A y B corresponding to the chosen special values of / can be calculated from (3), (4) when the numerical values of F, CT are given. Next, define ?*i, C by (r, 2 + r' 2 ) C 2 = r 2 + r /2 , r^W^rA, ......... (5) so that A 2 = C 2 (n 2 + r' 2 - 2r x r' cos (v - v' - B)} ....... (6) The special values of r, r' corresponding to special values of f, fi having been found, those of ri, C can be obtained con- veniently by calculating X, \i from T tan\= , sin2\i = A sin2X, ............ (7) B&SPT IZ 182 TRUE LONGITUDE AS ARGUMENT [OH. vu and then ri, C from / , _ cos \i /ox n-rtan^, 6'-, .............. (8) equations which will be found to satisfy (5). The expression (6) for A 2 gives ...(10) Methods for the expansions of these functions have been given in 4'23 and the following sections. The particular form which is useful here is that in which we put (1 + a 2 - 2a cos ^)- = /8/> + S 2/3/> cos ii/r, . . .(11) with a ri/r', tyv v' B, i=l, 2, ..., s = J, f. The expressions for the /3 8 (i > are given by 4*24 (6) with K= 1, ! == a, namely, /v* ^v /P (i) - (t) ^ (i) _____ ..... _ ,7 (t) " " where t 2.4...2t ->' '-"> in which j9 = a 2 /(l a 2 ). The methods developed for the calculation of these functions depend on the numerical value of a being given. In the present case these numerical values are the special values of r*i/r'. The efficiency of the method outlined here depends on the existence of tables giving the coefficients for different values of a*. * The tables of Brown and Brouwer, I.e. p. 103, give Iog2^() for t=0 to 11 and for s=, f , f to 8 places of decimals and for s = J to 7 places of decimals. They are tabulated for values of p at intervals of '01 up to p = 2-5 or a =-83. For higher values of p up to p = 6 (a = -93) rapidly converging series are given. A separate table for a -90 to -95 is added. 7, la] NUMERICAL DEVELOPMENTS 183 In this way, each of the functions 7*6 (3), (4), (5), (6) is expanded into a series having the form 22C (i) cosi(v-v'-B\ ............... (12) in which the special values of the coefficients for each required value of i have been obtained. We next put V - v r - B =/> *r /i -or' - BI, where Bi=f'-fi + B, and calculate the special values of B t . The final special values to be computed are those of for each value of i. Each of these functions is then analysed into a series of the form 2 t , il L 1>il cos(jf+j 1 f 1 ) + - i , il L' itil sm(jf+j 1 f i ). ...(13) The results give series having the form 7*6 (1), (2). After the derivative of 7 '6 (3) with respect to -cr' has been formed, the numerical value of &'&' is inserted in all of them, the various terms having the same multiples of f t f\ in their argu- ments are collected and each expression is put into the form (13) or, if desired, into the form M it h cos (jf + A/I - N it fc ) ............. (14) The most important preliminary step is the expression of f f in terms of/i,/. 7'7a. Expression of the true anomaly of the disturbing planet in terms of that of the disturbed planet. When harmonic analysis is to be used (App. A), the following method gives the required transformation rapidly. We have, according to previous definitions, g=f-E f , flr'=/i-^,=/i-/i, where Ef is the equation of the centre with eccentricity e and true anomaly f. Also /' = g' + E, =/i - 8/i + E (/i - 184 TRUE LONGITUDE AS ARGUMENT [CH. vn where E g > = E (fi S/i) is the equation of the centre of the disturbing planet with eccentricity e' and mean anomaly g' =/! S/i. If this function be expanded in powers of S/i we obtain, if E' (/i) denotes dE(fi)/df 1) Since f\ 4- E(fi) =/ is the expression for a true anomaly in which e' is the eccentricity and /i takes the place of the mean anomaly, we can write the equation in the form - df n' \ 1 d*fn' - The special values of Ef are calculated with special values of / in the usual manner. Those of / with special values of f\ are similarly obtained. For the derivatives we have d f ^(1 + e'cos/) 2 dfi~ (I-**)* Hence = (1 - e' 2 )~* (1 + e' cos/). If we denote the successive derivatives of /with respect to fi by the notations /, f, ..., and put e\= e' (1 e /2 )~^, we can obtain the following formulae for their successive determination : i ei cos/ / 1 - W ~T6 ' 2-J- which will be sufficient for all practical needs. The calculation of/,/ iv from the series 3'16 (2) will be found to be sufficiently accurate in most cases and is rather easier than that from the formulae just given. The amount of calculation needed in any particular case depends on a variety of circumstances. Before undertaking calculation on the general plan outlined above, a preliminary survey should be made to find the 7a,8] SOLUTION OF THE EQUATIONS 185 order of magnitude of the term with the largest coefficient in the longitude, pr, in the present case, in t expressed in terms of the longitude. Usually, the term is one having a long period. The order of magnitude with respect to the eccentricities and inclination for a term with argument jf+ji f\ is \j+ji\* While a rough approximation to the coefficient can be obtained by following the method developed in 7*38 below, the degree of accuracy, that is, the number of places of decimals needed in the calculation, can be found from the number of significant figures needed to obtain this coefficient with the required accuracy. The accuracy possible with the methods developed above is theoretically unlimited, but is practically limited by the accuracy of the tables of the coefficients g^\ Those referred to in the footnote on p. 178 are sufficient to obtain solutions of practically all the planetary problems in the solar system with the accuracy needed at the present time; the determination of the great inequality in the motion of Saturn is probably the limit in this respect. It may be pointed out that since A < 1, C> 1 (p. 178), the inclusion of the inclination in r^r' in general tends to diminish this ratio and therefore to increase the rate of convergence. Thus, if we can obtain a certain degree of accuracy with 7=0, we can obtain at least the same degree of accuracy with /=t=0. The method is thus particularly effective for large inclinations. The method of procedure outlined above is a general one. The experi- enced computer will see various ways in which it may be abbreviated. One important choice is the number of special values to be adopted for /, /i . In the majority of minor planet problems, the values of /, /j at intervals of 45 will serve. If the eccentricity of the planet or the inclination is large, additional values at intervals of 60 may be used : these additional values merely involve corrections of the coefficients in the last part of the process that of the harmonic analysis so that all the previous work is fully utilised. It is not difficult to settle at the outset the number of places of decimals required, but it is not easy to say how many special values should be used. The work can be started with the minimum number and others can be added afterwards without the loss of the previous calculations. 7*8. Solution of the equations. The methods of the previous sections give the expression of R in the form JZ-SJCcoB^+jj/i + i), j = 0, 1, 2, ...; ^ = 0, 1, 2, .... When a literal development is made the angles k are multiples of r r', w + ta' 20, and the coefficients K are functions of a/a', e, e', F. In a numerical development the terms having the 186 TRUE LONGITUDE AS ARGUMENT [OH. vn same values of j> ji are gathered together and R and its derivatives are expressed in the form 0o + 2 cos (jf + jj/x) 4- 2/3' sin ( jf + ^/i), or in the form _ + 2 cos (j/+ji /i-lto, where y9 > A /3', J5, l?i are numerical quantities dependent on all the elements ; in these expressions the terms in which j = ji = are gathered into the symbol /3 The constant y8o is independent of v and is implicitly a function of the angles w -GT', OT -f TB' 20. The terms present in this constant when expressed in a literal form possess the property (6'4) associating a power of e in the coefficient with the same multiple of -cr in the angle, with similar properties for iff', e' and for 20, F. The corresponding properties are obtained when j t ji are not both zero by putting cr =1; /, & =vi fi and associating powers of e t e' y 20 with the respective multiples of /, /i, 20 a statement which is easily seen to be true by referring back to the development of R in 4*14. Finally, since we have put v = v, we have D (Jf + ji/i) = J + h n 'l n = s > so that s becomes a divisor of the coefficient when we integrate one of these expressions. It is evident that Rf(r, /) will possess these same properties. 7*9. The equation for q. This equation has the form J5 s = / 9 -f^cos(j/+j 1 / 1 ) + SyS'sin(j/+j J / ] ), (1) and its integral is q = ?o + /9ov + 2 f sin ( j/+ jj/i) - S f cos (jf+jtf), . . .(2) o o where <jo is an arbitrary constant to be defined later. The term $ov is the secular part of q. The terms for which s is small com- pared with unity are those of long period, the remaining terms having 'short 1 periods, that is, periods of the same order of magnitude as 2-7T/71, the period of revolution of the planet, and shorter. 8-10] THE EQUATIONS FOR q, u 187 7*10. The equation for n. The right-hand member has already been developed into an expansion of the form 7*9 (1). In the left-hand member, the value of q just obtained is sub- stituted. The periodic terms in q are added to the terms of the right-hand member and the equation takes the form jy u + u = q Q -f v + fa cos/+ fa' sin / + /3 MO + 2&* cos (jf + J!/ t ) + 2&/ sin (jf -f ^/i), the terms for which ji = 0,^' = 0, 1, simultaneously, being isolated. The integral of this equation is u = <?o + ov -f /3i v sin / /3 x ' v cos / 4- e c cos / 4- e 8 sin/ 4- t* p , ...... (1) where 2 -"-8 cos <# + M) + 2 -~r a sin JL o 1 o as can be seen by submitting each member to the operator D 2 -f 1; e c , e a are the arbitrary constants in the solution. Now after the substitution u/a for u, q/a for q, the elliptic values of u, q are __ 1 -f e cos / _ 1 /____ with e, -BT as arbitrary constants. As q Q , e c , e s are at our disposal we can put e =___._ e = o q = 1 so that the remaining terms would constitute the perturbations. These values of e c , e 8 will be adopted, but instead of that for q Q we shall put where Sq Q is still arbitrary. It will later be defined to be such that the mean value otnDt shall be unity. We shall next show that u, q may be written in the forms ...... (2) 188 TRUE LONGITUDE AS ARGUMENT [OH. vii where - 1+ '- **>. -* + ! (3) Here is the value of e which has been used in the develop- ment of R\ TI, ei are small constants whose squares may be neglected. The expansion of UQ in powers of e\, -CTI, gives __ l+o cos/ 2eo 0_i l-h#o 2 f o Comparison of this with (1) shows that if we put lO-V) 2 ;,, il-eo'* the coefficients of v cos/, vsin/ in (1) will be included in The term /3 v will also be included if The argument in 7*24 shows that this relation is satisfied, so that it constitutes a useful test of the accuracy of a part of the numerical calculations. Hence u and q have the form (2). The terms #iv, wj.v are called the secular motions of the eccentricity and longitude of perihelion. Expressed in time they would be e^ni, vrint. Since we are neglecting squares of the disturbing force we may insert these secular parts in the perturbations w p , q p . If a development in which the literal values of e^ w have been retained is made, this can be done by replacing / by / s^v and w by or+orjv, and e by e + e v \ in w p , q p a procedure which is advantageous as will appear later. Usually, however, it is not possible to make these changes because it is customary to use numerical developments in order to save labour. 7*11. The equation for t. Since Sq Q , u p , u q are of the order of the disturbing forces, we have, as far as the first order, {1 4- e cos(/' x 10-12] THE EQUATION FOR t 189 In the terms factored by 8q Q , q p , u p we can put <?i = 0, ^ = 0, since the products would be of the second order. Hence We now determine Sq Q to be such that in the expansion of the right-hand member as a sum of periodic terms, the constant term shall be unity. By 3'8 (3), the constant part of the first term is 1. It will be shown in 713 that if we expand (1 4-e cos/)~ 3 into a Fourier series the constant term is (1 + ie 2 )(l - eo a )~*- The constant term in the coefficient of Sq Q is therefore j(l _ ef) - ( 2 + e ?) = - t (1 + e, 2 ). The required condition therefore gives ^of i - 2 x constant term in the expansion of 1 "T #0 It is evident that we only need those portions of r p , u p which are independent of the argument /i. It is recalled that q p con- tains no constant term and that u p contains no term with argument/. When Sq Q has been found, its value is inserted in (1). The second and third terms are then calculated in the form (1 + e cos/) & Bq the constant term of which should vanish, so that these portions give a sum of periodic terms of the form 3e cos (jf + jifi) + 2&' sin (jf+ j,f t ) . 712. Integration of the equation for t. If we neglect all per- turbations so that e = e Qy ori= 0, we obtain by 3*8 (3) nt + e= v-E fy ..................... (1) 190 TRUE LONGITUDE AS ARGUMENT [OH. vn where Ef is the equation of the centre expressed in terms of the true anomaly /. To obtain the integral when e\, ts\ are not neglected put e Q + e^v, f tsriv for e, f in (1) and differentiate. We obtain, if squares of e\, TZ\ are neglected, The first two terms of the right-hand member evidently give (1 - <? 2 )* {1+ ecos (/- r lV )}- 2 ; in the third and fourth terms we can put e e Q , is\ v = 0. Hence the integral of the first term of 7*11 (1) gives ffiji 1 f r) J? nt + = v - E f - eij -~^df+ TI j -j^ d /, where is the constant of integration, and the values e^ + eiv and / tsrjv are used for the eccentricity and true anomaly in the expression for the second term, that is, for the equation of the centre. The integral of the third term is given by the formula 7*13 (4) below; that of the fourth term is &iEf. The remaining terms in the expression 7*11 (1) for nDt, the form of which is given at the end of 7 '11, are integrated im- mediately. We obtain nt + 6 = v - + S sin (jf + j,/,) - 2 i cos o o where s=j+jin'/n, E f is the equation of the centre with eccentricity e Q + eiV and true anomaly/ -B^V and, by 713 (4)> = 2 (- 1)* eo 1 " 1 ( 1 + e 2 H r ] cos if, e Q = 1 2 . 713. The Fourier expansions which are needed above are obtained from fl-*) 1 - CD [)(l+e COS/) 2 Differentiation with respect to e gives 12-14] THE EQUATION FOR r 191 and differentiation of this result with respect to / provides The value of dE/fie can be obtained either from the series for E/ or by integrating (2). The latter integration is performed by making use of the identities, d sin/ __ cos/4-0 8 sin/ _ 2<?4-cos/ 0cos 2 / a/" 1 + e cos/ = (1 +e cos/) 2 ' a/* (1+0 cos/) 2 = (1+ecos/) 3 ' the sum of the right-hand members being 1 2-M 2 2 l-i 2 _ (1+ecos/) 3 ~" e (1 + ecos/) 2 "~~e (\+e cos/) 3 ' Hence ^ - sin / sin / \l+ecoa/ (1+ecos/) 2 / The value ofFfD' 1 (dEf/de) is obtained immediately from this result, and it also completes the expansion 7*13 (3). The calculation of these expansions by harmonic analysis is at least as rapid as by series and is advisable for large values of e. Harmonic analysis can also be used to calculate E f when e = e Q + ei\ by calculating the coef- ficients with = 00, an d again with e~e Q -\-1cei when k has some convenient numerical value (? 100). The difference between the resulting pairs of coefficients divided by k gives the factor of v in the coefficient. 714. The equation for F. The equation 7*1 (7) for F gives, on integration, F^ + FiV + F^i .................. (1) where FI is the constant term in the expansion of ql dR/u 2 dO, T p is a sum of periodic terms, and k is an arbitrary constant. The value 7*10 (2) for q gives, on expansion to the first order, + q p ) (1 - ,)*, whence r A (1 - eoT* + 4 % (1 - ^o 2 )* -f (*cio (1 - eoT* + F! (1 - e 2 )-*} v + i* (1 - ^ 2 )* g p + (1 - ^o')"* ^P- The first two terms in this expression constitute the constant part of F; denote it by F . In terms which have the disturbing 192 TRUE LONGITUDE AS ARGUMENT [OH. vn mass as a factor we can put &= F (l e Q 2 )^. Hence the value of F is given by r = To + [e l e, Fo (1 - eoV + Fx (1 - * T*} v When the disturbing forces are neglected we have r = F , which is therefore the value of F used in calculating the per- turbations. The secular motion of F is the coefficient of v. All the terms contain F as a factor. Since F = 1 cos 7, SF = sin 7S7, the equation for F, which may be written F = FO 4- SF, gives 7 = 7o + SF/sin 7 , where ST contains Fo = sin 2 7 /(l -f cos7 ) as a factor. Hence when 7 is small the perturbations of 7 have 7 e as a factor. 715. The equation for 0. The integral of 7*1 (8) gives 0=0 + 0iV4-0p, where the signification of the symbols is evident. Here is the value of used in calculating the perturbations and 0iv is its 4 secular motion/ 7*16. The equation for v. Since dO/dv contains the disturbing mass as a factor, we can put F = F in 7*1 (6) so the integral is no constant being added, in accordance with the definition in the last paragraph of 7*3. The usual definition of n is to make it the mean value of dv/dt rather than of dv/dt as defined in 7'1 (6). This definition requires us to replace n by n/(l ~ Fo0i). Since n is not present in the coefficients of elliptic motion and since the change may be neglected in the perturbations, no further adjustment of the value of t in terms of v is necessary. The value of 8q Q (7*11) is altered and receives an additional part F 0i/(l +e 2 ); this is usually insensible in the value of u and elsewhere. The only further change necessary to obtain t in terms of v is the replacement of v by v -f F P . 14-18] THE SMALL DIVISORS 193 7*17. The equation for sin L. When T0 P can be neglected, we can save some labour by integrating the equation for sin L to replace those for r, 6. If we treat the right-hand member like that of the equation for u, isolating the terms with argument / and also the constant term, the equation takes the form ............... (1) where g p denotes the remaining periodic terms with arguments jf+j\f\ . The solution, like that for u, may bo written sin Z=sin 7 sin (/+ w -^o)+y - 4^i' v cos The constants are shown in the above form because we have in ellipti motion sin L = sin 7 sin (v - #o) The terms with factor v may be included in the solution if we put 7 1 v = S7, #^ = 0, where 57, $0 are determined from 5 {sin 7sin (sr - 0)}= -^i'v, 8 (sin 7cos (ET O - ^Hi&v. ...( 4 ) 718. The small divisors. The divisors s are present in the equation for q, and when s is small, that is, when the corre- sponding term has a long period compared with 27r/n, the coefficient will be increased by the integration. It is again in- creased by the same divisor in the integration of the equation for t, so that in the expression for t in terms of v or v, the divisor s 2 is present. The divisor 1 s 1 is small in the expression for u or sin L when s is nearly equal to 1, that is, when the period of the term is near that of revolution of the disturbed body. It might be expected that these terms in combination with the elliptic terms would produce terms with the product of small divisors 5, 1 s 2 in t. It is true that they do so, but such terms in general have a factor e 2 as compared with the terms in q from which they arise, a result which will be evident when the following method of the variation of the elements is used. 194 TRUE LONGITUDE AS ARGUMENT [OH. vn C. EQUATIONS FOR THE VARIATIONS OF THE ELEMENTS 719. Let us return to the original equations of motion in 7*1 and introduce three new variables a, e, ur to take the place of u, q. These new variables have as yet no relation to those denoted by the same letters in the previous sections of this chapter. We retain the notation D = d/dv and, for brevity of expression, introduce new operators defined by Dt=Da!l- + De%-+D*r-, D Q =~ ....... (1) da de dur 9v ^ ' Thus when we are operating on a function of a, e, tsr, v we have Since we are replacing two variables by three, one relation between the new variables is at our disposal. The three relations to be adopted are __ 1 -f 0cos (v CT) 1 ~ 1 -f e cos (v -or) __ U ~ a (l-e 2 7~~ ' ?== a(l-6)' * ~ a(\-#) ~ ' ...... (2) so that Du = D Q u, 1)^ = 0, .................. (3) when u is expressed in terms of the new variables. Since Dq = D^q, the third of equations (2) gives {1 -f e cos (v cr)} Dq + qD\ {e cos (v )} = 0. This, combined with 7*1 (4), gives T\ ( / \1 ^ C/./V / > \ I>!{C08(V-W)}--^J- ................ (4) Next, the equation Du = DQU gives Di^ = qe sin (v tsr) ................... (5) Hence D*u + w g = D {gesin(v &)} -\-u-q = DI {qe sin (v -or)} = D^ . e sin ( v tsr) ^Z>i {e sin (v -or)}, since DO (^ sin (v r)} =u~q. But Dq.e sin (v -or) = (Dq/q) . DM. 19, 20] VARIATIONS OF THE ELEMENTS 195 Whence, from 71 (3), n f / M d R Du'dR ,-. DX {* sin (v - r)} = - -= --- =- -5- .......... (6) 1 n du u* dv The equations (4), (6) are those which give the variations of y w " To obtain the equation satisfied by a, multiply equations 71 (3), (4) by Du/q, {(Du)* + u 2 }/2q* respectively, and subtract; the result may be written | W + *_ 2 ) ag + ^ { q ) dv on Substituting for u, Da, q from (2), (5), we obtain dR dR _ which is the equation satisfied by a. 7*20. The last equation may be transformed into a form which is not only more convenient for calculation but which furnishes an important theorem concerning the secular terms. The disturbing function was originally expressed as a function of u, v, P, 6, and of t through its presence in r' y v'. Hence T. n dR -p. dR ^ oR nT1 dR ^^ dR *. DR = -5- Dv + ^- Dw + ^ DF + ^^ DQ + -^~ Dt. dv ou dT cQ dt With the use of the expressions 71 (6), (7), (8) for Dv, DT, DO, this reduces to T-. ,-> dR dR ^ dR r,. DR = -~- + *- Du + -zrDt ................ (1) dv du dt ^ Now t enters into r', v r only in the form n't + e', and this angle enters (7'5) into R only through fi in such a manner that ^n'. Hence a result which is true whether R be expressed in terms of the old or new variables. Utilising (1), (2), we may write 719 (7) in the forms , ...... (3) \W since Dt = qft~ku~ z and since w does not contain /i explicitly. 196 TRUE LONGITUDE AS ARGUMENT [OH. vn 7*21. Finally, the expression for Dt, just quoted, gives, in terms, of the new variables with n 2 a s = /4 and the expansion 3*8 (3), <s7)} 2 n 7i 3v ' where Ef is the equation of the centre. Hence f^dv- Dodv + const ............. (2) The values of n = (/4/a 8 )^, 0, -or, as deduced from the integrals of 7*19 (4), (6), (7) are to be substituted under the integral signs in (2), and the integrations are then to be carried out. Equation (2) may be written, since D = D D\, f=f!dv-^+ fl) 1 (^dv + const .......... (3) Jn n } \nj ' 7'22. As in the earlier work, it is advisable to expand Rr 2 = R^ rather than R. With this change and with the usual abbrevia- tion,/= v w, equations 719 (4), (6) can be written .sinf=2u 9 ............ (1) De . sin/- eDv? . cos/= r 2 - -- Da ( ~ , ...... (2) the computation of r 2 9JR/9r being carried out in the manner ex- plained in 7*6. From these equations we deduce those for De, The equation for a becomes The equations for 6, T remain the same, namely The latter may be replaced by Equation 7 '21 (3) is used to 6nd t. 21-24] SOLUTION OF VARIATION EQUATIONS 197 7*23. Solution of the equations. We proceed as before. When R is neglected a, e y -or, F, become constants, and the motion is elliptic. These constant values are substituted in the expansion of jRx, the derivatives of RI and of R being obtained as in 7 '6. For the arbitrary constants we use these same values, except in the case of I/a, to the elliptic value of which we add a, where is so determined that when the equation for Dt has been formed, the constant term shall be represented by I/KQ, where KO is the observed value of the mean motion. The amount of calculation needed with the use of these equations is not very much greater than that required in the previous form. The additional work is mainly the multiplication of EI by the three terms in u\ namely, where q\ e are numerical constants, in order to find I/a, and the multipli- cations by cos/, sin/ to find e, w. In either case the work is less laborious than when t is used as the independent variable, chiefly because, in the latter case, the single terms cos/, sin/ have to be replaced by Fourier series containing a number of terms corresponding to the highest power of e we need to retain. The chief saving of labour in the use of the methods of this chapter over those of Chap, iv is due to the avoidance of the formation of derivatives with respect to e. There is great advantage in using the numerical value of e from the outset and if this be done we cannot find the derivative with respect to e. Further, the development, to secure a given degree of accuracy, requires the presence of one power of e higher in the latter case than in the former. 7*24. The proofs of certain theorems, quoted earlier, follow easily from the equations of variations. To show that there are no secular terms of the first order in I/a and none of the form tcv 2 in t, we use equations 7*20 (3). To the first order, dRi/dfi has the same value whether RI be expressed in terms of u, v, t, F, 6, or in terms of f,fi and the constant elements. In the latter case, dRi/dfi will have no term free from the angle fi and consequently no constant term. Under the same conditions, RI has no term proportional to v, so that I/a has no term containing the factor v. B&SPT 13 198 TRUE LONGITUDE AS ARGUMENT [OH. vii The secular terms in e, & are ei\, e\is. These, substituted in Ef, produce terms of the forms v cos if, v sin if, but no term of the form tcv to the first order. Since there is no term factored by v in 1/n, there will be none of the form /ev 2 in t. a . I Smce w= the substitution e=e + ^i v i Q the former of these terms will produce a secular term in u. This differs from the case of u expressed in terms of t, where the absolute term is I/a which has no secular part. This fact exhibits the artificial nature of the statement that the major axis has no secular part. When referred to actual coordinates, the existance of a secular part depends on the coordinates used. 7'25. We next show that the only terms with the divisors s 2 arise through the variable a. This is proved by 7*22 (6). For the integrals giving e, vs contain the first power of s only, so that Effn gives rise to this class of terms only. Also Di(Ef/n), which depends on Da, De, Dcr, has no divisors and its integral will give rise to terms of the same kind. The integral giving a gives rise to divisors s and that of I/n, through n 2 a* = p, to divisors s 2 . This latter result which was stated in 7*18 is of some assistance in the numerical developments of the methods in the previous section. It shows that the terms of long period in which s is small are needed in u to a lower degree of accuracy than terms with the same argument in q. If we calculate the coefficients of the long period terms of dlt/du to a lower degree of accuracy than those of dR/dv, but retain the same number of places of decimals, the theorem shows that the inaccurate or omitted portions cancel one another. The relative degree of accuracy needed must be judged from the small values of s present. D. THE SECOND APPROXIMATIONS TO THE PERTURBATIONS 7*26. The results of the first approximations are as follows. We have obtained u, q, t, v, F, in the forms , l-f ecos(f where = v - E f , /= v - isr , 24-28] SECOND APPROXIMATION 199 in which e=*e Q + eiv; E f is the equation of the centre with eccentricity e> true anomaly/ OTI v, and o 3 ^o a = /* The constants a Q , e 0y <*o> r , #o have been used to compute the terms arising from R; e iy TI, 0i, TI are constants whose values have been found ; u p , q p , t p , T p , p are sums of periodic terms, u p , q p alone containing a constant term, so that Sqo of 7*10 (2) is now included in Up, q p . If we had solved by the method of the variation of arbitrary constants, the forms of t, F, 6, v would have remained the same, but for u, q we should have had to be substituted in u= {1 -f e cos (v tzr)} -f-a(l e 2 ), l/q = a(l e 2 ). 7*27. To obtain the second approximation, these values of the variables, or of the elements, must be substituted in R in the place of the constant elements previously used. Whichever plan has been adopted in the first approximation, we can and shall still use the equations for the variations of the elements in the second approximation on account of their greater simplicity for both computation and exposition. The exposition will be limited by neglecting the variations of F, 6, v v. The effects of these variations on the perturbations of the second order are usually insensible, but they can be included, if necessary, by the use of the methods given for the other variables. 7'28. Denote any perturbation of the first order by the symbol S and one of the second order by & 2 and put p = 1. Equation 7*20 (3) gives and , 1 37 = 57T~ u 8/1 8/ x 9M ~ du 8fi 3 (dRi\ . f d 2 Ri - /. ( ^ 1 ou dfi \ou / + n Ztf ,...(2) ,...(3) 13-2 200 TBUE LONGITUDE AS ARGUMENT [OH. vn Since R^dRi/du, dRi/dfi, R have already been obtained in finding the first approximation and are expressed in terms of the angles /, fi t their derivatives with respect to /i are immediately ob- tained. If the first approximation has been obtained by finding the variations of the elements, we replace Sqk, Su by M<i-<rt *-+- ....... w where U Q = {1 + e Q cos (v - <sr )} 4- a ( 1 - e 2 ). It will presently appear that we can usually neglect Sa, Be, Sr and therefore &u, Sl} Q u, Sq in finding & 2 e, S 2 ^ When this is the case, equations 7'22 (1), (2) give rfi Ft (5) St. sin/D (8,) - e cos/D (S 2 *r) = n' f A ( r* d ) - D u (oji \ or J ...... (6) The second order derivatives, again being derivatives with respect to/i of first order derivatives used in the first approxima- tion, are obtained immediately. Further, only a very few terms in &u, St have to be considered and the same is true of their products by the second derivatives of R or RI. Finally, as these variations enter into the right-hand members of the equations in a linear form only, we can compute separately the effects due to the few terms in Su, St which have to be taken into consideration. 7*29. Calculation of the effects due to the secular terms in the first approximation. To obtain them we put eiv, tsr = T!3- 1 v, From these we get ,j 8u = 5- -i- ceo dtjj These results are to be substituted in the right-hand members of 7-28 (1), (2), (3), (5), (6). They give terms of the form fcv -f v SO sin (sv + a). 28, 29] SECOND APPROXIMATION 201 First, let us consider the equation 7'28 (1). The only portion of this equation which can give a term of the form tev from Su, is - D {A (/feu") . S| = - (D + since all the other portions are products of series one member of which contains /i and the other is independent of fi. For terms of this form we therefore have, since R = from which the terms of the form /cv are to be chosen. This is equivalent to putting e = e Q + e\v y tar = TJT O -I- OTIV in the constant term of the expansion of 2R^ if or in 2R. The expression 7'28 (2) may be written 3 /3JZi, , /3JZi\ 3#i 3 , 3#i 3 The two latter terms are equal to 3(8jR)/3/i. Hence when we substitute for frw, &t the portions u p , t p> 7*28 (3) will have no constant term and therefore will produce no term of the form KV in S 2 a. Evidently, these portions substituted in D (RiiP) produce no such term. Hence, the only term of the form /cv present in |8 2 (l/a) arises from the constant term in l n '_2fi d ^l * 90*3/1' which in general will not be zero. Thus the theorem, that I/a has no secular term of the second order when t is the independent variable, is not true when v is the independent variable. It may be noted however that it is true for a second order perturbation arising from two different disturbing planets. The same arguments evidently apply to the right-hand members of 7*28 (5), (6) which thus contain secular terms of the forms vcos(jf/-hji/i), v sm(jf+jifi), with ji=t=0. Hence these portions of 2 e> S 2 tr give no terms of the form #v 2 in the co- ordinates. Such terms, however, will arise from Su; when it is necessary to calculate them, we shall need those parts of the 202 TRUE LONGITUDE AS ARGUMENT [OH. vii second derivatives of EI with respect to u,v which are independent of fi\ their calculation presents little difficulty since the degree of accuracy required is quite low. The practical importance of the theorem that there are no secular terms of the first and second orders with respect to the masses in I/a when t is the independent variable, has been much over-estimated. It is a result which eliminates certain secular terms from 1/r, but does not do so from r or, in general, from other functions of r. There is no particular reason from a physical point of view why, in getting a mean value of the deviation of the orbit from circularity, we should choose 1/r rather than r as the function to be averaged. The fact is that the separation of the deviations of the co- ordinates into deviations of a, 0, or, etc. is an artificial one, convenient for calculation and description, but one which has no particular physical significance. 7'30. Terms of the form v cos sv, v sin sv are integrated by the formulae [ , v . 1 f . , v 1 . v cos sv dv = - sinsv-i- - cossv, v sin svav = cossv-H - sin sv. J s s 2 ) s s 2 Such terms are present in S 2 a, $ 2 e, S a r and therefore in 8 2 it, DS 2 t The succeeding integration necessary to obtain S 2 t will introduce the factors v/s 2 , 1/s 3 . These terms will usually be insensible except when s is very small and even then the only portions which need be retained are those in St having the last-named factors. 7*31. Calculation of the effects due to the periodic terms in the first approximation. Only the long period terms need to be considered. The right-hand members of the equations in 7*28 have the form a derivative of R multiplied by Su or by St. In Su, the divisor s is present; in &t, the divisors s, s 2 are present. The products just referred to will produce products COS 5V . COS s' V = ^ COS (S $') V 4- i COS (S -f $') V, with similar results for sines of sv, s'v. Long period terms can arise in two ways: from short period terms in which ss' is small, or from long period terms in which both s, s' are small. 29-32J LONG PERIOD TERMS 203 The former will produce small divisors (s s') in Stu and their squares in S 2 . These will rarely be sensible. For the latter, we have divisors s, s 2 or 5', s' 2 in the right-hand members of the equations of 7*28, and therefore the smallest divisors present in their integrals will have the forms s 2 (s s'). Hence, the small divisors in S 2 , arising from S z a, will have the form s 2 (s s'f. Thus, whenever we are able to neglect the squares of the small divisors in the second approximation, the equations of 7*28 will be sufficient for the calculation of the remaining terms. Even then it is usually necessary to consider only one or two terms, so that the amount of calculation needed is quite limited. We shall now show that the chief part of a long period term in S 2 that having the divisor s 2 (s s') 2 can be obtained immediately from the first approximation. 7'32. Calculation of the portion of a coefficient in &%t depend- ing on the fourth power of the small divisor*. We have seen in the previous paragraph that this portion can arise only from the term substituted in & 2 t= | S 2 ( - ) dv, ..................... (2) J \n/ where Bt arises from <$(-)dv, ..................... (4) substituted in by the use of the relation n 2 a 8 = /A. Suppose that the term in RI is a $ cos sv + a /S' sin sv = a Q A cos (sv + ^), sv = ...... (5) * E. W. Brown, Mon. Not. R.A.S. vol. 90, p. 14. 204 TRUE LONGITUDE AS ARGUMENT [CH. vii If this be substituted in (3) and the result integrated, we obtain -} = 2n' (&] a 4cos(v + ai) .......... (6) CL/ \/^/ S The equation n 2 a 3 = /x gives Whence, from (4), g! 3a, 8 l ...................... (7) ft 2i? . * From the substitution of (5) in (1), we have Moo cos (sv + !) . S< fiv + si) g< (sv + s 1 ), ............... (10) O from (8), (9). The integral gives ^v-M,) .......... (11) \ v \ / 2 a 6 And since {S(l/a)} 2 has the divisor s 2 only, while S 2 (l/a) has the divisor s 8 , we have, neglecting the former, /1\ 3 do. A Finally, which, with the use of (11), gives *. In'. (12) 32-35] LONG PERIOD TERMS 205 7'33. If we have two long period terms in RI, namely, Hi = a Q A cos (sv 4- si) + a^A' cos (s'v -f $/), and if we wish to obtain the terms due to their combinations, it is evident from equation 7*32 (10) that we shall have = !- B cos (sv + Sl ) . B' sin (' v -f O ' C o S ( 5 V + 81') . B sin (^v -f si) - ' (^ V) s in K*' ) O the upper sign giving one term and the lower the other. The process previously followed gives V = \ J Bff -gr^ 1 ' sin [(,' s)v + s l ' s,}. In the case s' = s, si'^si, the term arising from the lower signs disappears since ji = jV- The remaining term gives 1 ^' 8 2 ^ = - _ frBB' sin (2sv + s l + Si). O ??Q 7*34. The same method may be applied to find the effect of a perturbation Sit;' of the longitude of the disturbing planet. In 7*5 we replace v' by /i + &' 4- equation of the centre, so that Siv' may be regarded as an addition to fi. Hence in this case the addition to dR/dfi is If Sit;' = B\ sin (s'v + ^i') we can therefore utilise the formulae 7-32 (11), (12) by putting BJ jri for B'/n. The latter formula, in particular, gives an additive part to t: S 2 t = l^ BBJ sin (25V +s l + $/). O UQ 7 '35. Numerical illustration. The 'great inequality ' of Jupiter and Saturn. The periods of revolution of Jupiter and Saturn are. very nearly in the ratio of 2 to 5, so that if n, n' be their mean motions, the terms with argument (5n' - 2ri) t will have a very long period actually about 70 times 206 TRUE LONGITUDE AS ARGUMENT [CH. vii that of Jupiter. Thus, in the motion of Jupiter disturbed by Saturn, 1/s* will have the order 2-5 x 10 7 . Suppose that we have calculated the terms with argument (5n f - 2?i) t to the first order in the motions of both planets, and that we need the principal second order portions, the latter can be obtained immediately from the formulae in 7 '32. We shall perform the calculation and compare the results with those given by Hill*. Since the latter uses Hansen's method it will be necessary to compare it with that of this chapter. Denote Hill's notation by (H) and that of this chapter by (v). In elliptic motion we have (T), v=z + n+E(z)', (v), nt+c = v-JS(f), where E(z) is the equation of the centre expressed in terms of the mean anomaly z and E(f) is the same expressed in terms of the true anomaly/. In disturbed motion we have (H\ ws*g+te + vr+ff(2+bi); (v), nt+c=v-E(f) + nto. Now we have seen that, for the principal part of a long period perturbation of the first order, the portion due to the elliptic periodic terms can be neglected since it only produces portions with the divisor s. Hence, very nearly, n dt = - n fa, ri dt' =-ri &', the former for Jupiter and the latter for Saturn. There are two resulting second order perturbations in the motion of each planet. The first is that which arises as in 7*32 and the second that which arises from substituting the disturbed motion of the disturbing planet in R. For the latter we have in the motions of Jupiter and Saturn, respectively, dv^ridz*, v = noz, since there are additions to */, v in the respective disturbing functions. Hill gives t 7i52 = 1196" sin N, n'8z' = - 2908" sin N, where N=- bg 1 - 2g + 69, approximately. (There are additional secular parts given by Hill arising from the second order terms.) Thus for Jupiter the two portions are, in our notation, n 8t = - 1 1 96" sin tf, 8v' = - 2908" sin N, and in N we can put / for nt + e - w**g and fi for n't + e' -cr' =g\ for the reasons stated above. If p as 1/206265'', the factor necessary to reduce the coefficient to radians, we have, in the notation of 7 '32, B - - 1 1 96/>, B l = - 2908/>, j\ f = 5, n'/n = 2/5 and the two parts give WoM=| . 1196p (f . 1196/3 + 2908p) sin 2N= 12"-3 sin 2N. * American Ephemeris Papers, vol. 4 ; Coll. Works, vol. 3. t Coll. Works, vol. 3, pp. 560, 568. 35, 36] TRANSFORMATION FROM v TO t 207 For the action of Jupiter on Saturn we have similarly, M 2 *= -f -2908p(f .2908p + 1196p)sin2iV= - 30" '9 sin 2^. Hill's coefficients for this term in n 82, n' 8z' are * Jupiter (first + second approx.) - (first approx.)= - 11"*0 1"'4= 12"'4, Saturn = 26"'84-3"-4 = 30"'2. After the change of sign necessary for accordance with our notation, the comparison shows a close agreement between the two sets of results. It may be pointed out that the percentage accuracy of the method is greater the smaller the value of s. As far as the present demands for accuracy in the various problems of the planetary theory are concerned, the method will give all second order coefficients with sufficient accuracy. It has been shown, however, that the terms depending on the divisor s 3 can be obtained without great labour. E. TRANSFORMATION TO THE TIME AS INDEPENDENT VARIABLE 7'36. Although most of the purposes for which the theory is developed can be equally well served whether v or t be used as the independent variable, comparisons of different theories will be facilitated if we can transform from one to the other without too much labour. The method gives results in the form n t + e = v-E f + P, v = v + Sv, (1), (2) where Ef is the equation of the centre expressed in terms of the true anomaly /, and P, 8v are the perturbations expressed in terms of v by means of /-v-*r, / 1=s ( v -) + '-t*' (3), (4) The expression of v in terms of t can be deduced from (1) by the use of Lagrange's theorem, but the following method of carrying out the transformation in two steps demands much less calculation. Pub n$t -f e tzr = g and write (1) in the form g-P=f-E } (5) Coll. Works, vol. 3, pp. 195, 107, 561, 569. 208 TRUE LONGITUDE AS ARGUMENT [OH. vn This equation may be regarded as expressing a 'mean anomaly* g P in terms of the true anomaly/. The formula 3'11 (6) which gives the true anomaly in terms of the mean may there- fore be used. It gives g - P + 2e sin (g - P) + |e 2 sin 2 (g - P) + . . . .., ............ (6) by Taylor's theorem. The terms dependent on P 2 will be very small and higher powers of P may be neglected. The perturbations Pare expressed in terms of/,/i, and their expression in terms of t by continued approximation constitutes the second step. The first approximation consists in putting in them f=g+E g , fi = g' + n ^E g , ............... (7) where g' = n't -f e' sr'. Consider any term of P : where B, Si are constants. With the use of (7) this term becomes B sin (jg +J 1 g t + BI) cos j(j + ~^J E^ + B cos (jg +j l g' + B l ) sin The second factors of these are expressible as Fourier series with argument g, either by harmonic analysis or by the respective formulae l-t(j+Ji^)X' + ..., (j+fi%)E,-..., ...(8) for expansions of sines and cosines in terms of the angles. An important point to notice is the fact that for long period terms j + ji ft'/ 71 i s small, so that the effect of the transformation in changing the terms which usually have the largest coefficients is small. The transformed value of P is substituted in (6), which then gives / in terms of t to the first order of the disturbing forces. 36, 37] TRANSFORMATION FROM v TO t 209 If this first approximation be denoted by f=*g + Eg + f, a second approximation is obtained by replacing E g by E g -f Sf in (8), and adding the second order term in (6) with 8/=0. For the great majority of the terms in P, powers of the eccentricity beyond the first in the transformation may be neglected. For all such cases, a perturbation in n^t becomes a perturbation in v, where s = j ' -f jfi . n'/n. For the final step we have where the last term contains the perturbations due to the trans- formation from an origin in the osculating plane to one in the fixed plane. These, having the square of the inclination as well as the disturbing mass as factors, are very small, so that the values (5), (6) for/,/i with E g = 2esmg will serve. The remaining coordinates 1/r, 9, T are transformed in a similar manner, that of 1/r being found with the aid of 2*2 (2). The value just found for / in terms of t is used in the expression for 1/r, while the values (5), (6) will be sufficiently accurate for substitution in the expression for 6, T. 7'37. The method of this chapter is closely allied, as far as its final form is concerned, with that of Hansen*. Substantially, his method requires the expression of the true longitude in the form v=g+m + tff + 2eam(g+tg)+ e' 2 sin2 (# + fy) + ..., so that all the perturbations are expressed by adding 8g to the mean anomaly #. Equations 7 '36 (1), (2) show that the same thing is done hero, with the difference, however, that while Hansen calculated dg in terms of t y it is here calculated in terms of v. That the theory is more simple than that of Hansen is due to the fact that it can be expressed by means of equations which follow forms well * The original theory is given in a volume, Fundamenta Nova, etc., Gotha, 1838. 210 TRUE LONGITUDE AS ARGUMENT [OH. vn known in other dynamical problems. The two principal objections which may be urged are, first, the necessity for expressing the true longitude of the disturbing planet as a function of that of the disturbed planet, and second, the possible need for the final transformation given in 7 '36. As we have seen, the latter requires comparatively little additional calculation, while it is doubtful whether the former transformation, which is needed only in the development of the disturbing forces, requires more labour than that in terms of t. As pointed out in Chap, iv, each requires substantially four operations, two of which are more simple in the present method than in that of Hansen, while one of the remaining operations is more complicated. There is, however, a special advantage possessed by the present method when we are dealing with the perturbations produced by an exterior planet on an interior one having a considerable eccentricity. In the transformation giving v' as a function of /, the principal elliptic term, 2e sin /, enters only with the factor n'/n which in most cases is not much greater than 1/2. Thus the powers of this elliptic term have a maximum effect nearer to those of e than to those of 20. To some extent this is compensated by the factor r 2 which accompanies R in the developments, but the convergence is much more easily controlled with the method of this chapter than with the usual independent variable. F. APPROXIMATE FORMULAE FOR THE PERTURBATIONS 7*38. It is often useful to get an idea of the order of magnitude of the perturbations in a given problem. This is particularly the case when extensive calculations are to be undertaken to obtain the general perturbations accurately; an approximate preliminary calculation may save much unnecessary labour. Approximate formulae can also be utilised when the interval of time during which the results are needed is short or when the constants of the orbit are not well known. In obtaining such formulae, we shall neglect the inclination, so that v = v, and attention can be confined to the equations 7*1 (3), (4), (5). If in these equations we put e 2 ), q = (1 + Sq) q 0j u = (l+e cos/+ Su) q , | = f*> n Q t + = (!-#)% (1 -f e cos/)~ 2 + n St, J (1) 17-39] APPROXIMATE FORMULAE 211 ,hey may be written .............................. (2) (3) d *,_ -">*_ f ift o SM \ ~ + e cos/y/ = JSg -28w + (68^-8^)6 cos/ + ....... (4) Consider a term with argument <r in the disturbing function md suppose that this term gives rise to terms A si no- in ^-(r*Rqo), Bcoscr in r z -^- . ...... (5), (6) [n general, A> B will be of the same order of magnitude. Substituting (5) in (2) and integrating, we obtain * 2^1 da- _ gg, -- COS CT, S T- ................ (7) * s dv ^ With the help of (5), (6), (7), equation (3) becomes (& + i) Su = - f + J8) cos a- + \eA cos (<r +/) \rtw / \ s / \eA cos(ov /), }he integral of which furnishes 2A \coso- eA fcos(<r+/) . cos(cr-/)] ~"" + "~' ...... (B) jince l-(* l) 2 = + 2s-s 2 . The substitution of (7), (8) in (4) and a subsequent integration jive n Q St. 7*39. Let us first neglect the terms in 7*38 (8), (4) which have tihe explicit factor e. The remaining terms give /2A cosr tt--f S /2A D \ coso- -f +JB)= - - a , ..-(1), \ o / -I- o 212 TRUE LONGITUDE AS ARGUMENT [OH. vir The transformation to the time as independent variable ia immediately made, since with e = 0, we have Sv= n^St, and /=#,/! = / in <r. These results hold for all terms in the disturbing function whether they contain e or not; we have neglected e only where it appears explicitly in 7*38 (3), (4) and in the transformation to the time as independent variable. Denote the former by the suffix zero, and the additional parts factored by the first power of e by the suffix unity. To obtain the results to the first power of e, it is sufficient to substitute the terms with suffix zero in the previously neglected terms. Hence n Q j-Sit- 2SiU -h (68 Q u S Q q) e cos/. ............... (5) To the same order, the formulae for the transformation to the time as independent variable give 2e sin g -j-nQ$ Q t - n Q Sit, ...... (6) with/=<7,/i g f in all the formulae. The terms of chief importance are usually those in which s or s 1 is small and the order of magnitude of these is given by S Q v = n S Q t. When s 2 is small the additional terms with argument a / must be considered. 7'40. Solution to the first powers of the eccentricities. The development of a'/ A a'r cos S/r' z as far as this order is * Si {c^ cos i^r e (D 2mi) a f cos (i^r +/) + e' (D + 1 - 2i) di cos (fy +/i)J + a{cos^r + i$(l -f 2rn)cos(^-/) + i* (1 - 2m) cos (t +/) - 2e' cos (^ -/,)}, where a = a/a', -^ =/ 4- w ~/i *', D = d/a 7 a, m = w'/w, * The development to the second order with respect to the eccentricities and inclination is given by C. A. Shook, Mon. Not. R.A.S. vol. 91, p. 558. 39-41] DETERMINATION OF CONSTANTS 213 and , t = 0, 1, 2,.... CN- - j /- /. x bince r-^- a-^ ) r a (I ecos/), or da we have r 2 ,r = a 2i }Da $ - cos i\Jr e (Z) 4- 1 2rm) /) cos (ti/r 4-/) 4- e' (D + 1 - 20 Dcti cos ( 7?2/ 4- a 2 - -- { cos -v|r + e (1 -f m) cos (^ /) 4- e (1 - m) cos (^ -f-/) - 2e' cos ( - a - Sii {a 4 sin ity e(D + 2 2nn) a< sin (i- - a, sn {- sin yfr -f ie(3 + 2m) sin (^ -/) + \ e (3 - 2m) sin (^ +/) - 2e' sin 0/r - Substitutions from these formulae in 7*39 (1), (2), (3) give SO<?K &o u > nS^t, and from the first term in each parenthesis the additional terms in 7*39 (4), (5). In most cases values of i beyond 4 will not be needed. G. FINAL DEFINITIONS AND DETERMINATION OF THE CONSTANTS 7*41. The method of this chapter suggests the following definitions of the constants. The mean motion n and the epoch e are such that when all periodic and secular terms are suppressed, the true longitude shall be represented by nt 4- e. If terms dependent on the second powers of the masses be neglected there is no difference in the values of n, e whether t or v be used as the independent variable, and the additions depending on these second powers will usually be insensible to observation, B&SPT 14 214 TRUE LONGITUDE AS ARGUMENT [OH. vn The constants e$, OTO have been defined to be such that the principal elliptic term in OO/T, where n 2 Oo 8 = /i, shall be repre- sented by 2 cos (v cj ). Since any definition depends on the specification of some particular coefficient in a particular coordinate, a change to another definition can always be made. In numerical work it is usually sufficient to make any small correction due to an altered definition in the elliptic terms only. The constants 7 , Q are defined above by making the principal term in the latitude equal to sin /o sin (v # ) Here / is the inclination of the two orbital planes. The change to i y the inclination to any other plane of reference, is made by the formulae of 1*32, and the change to i$ is made to correspond. The slight difference when t is used as the independent variable will be sufficiently accounted for in the determination of the constants from observation. 7*42. A process of approximation is used in the determination of the values of the constants from observation. The perturbations may be calculated with the osculating elements at some given date unless previous work has given elements more nearly approximating to the constants of the theory. Thus constructed, the theory is compared with the observations, or with a selection from them. The differences are assumed to be due to erroneous values of the elements and are analysed so as to determine their corrections. While the formulae for the perturbations should be examined to see whether these corrections make any sensible difference in them, it will usually be found sufficiently accurate to correct the elliptic terms only. 7*43. The detailed work connected with the determination of the constants, as well as their correct definitions, has to be carried out what- ever method be used to calculate the perturbations. The difficulty of avoiding error in performing the work can to some extent be lessened by 41-43] DETERMINATION OF CONSTANTS 215 carrying it out as far as possible in a systematic way, since few checks of its accuracy are available. Most of the work of developing the disturbing forces can be done by harmonic analysis in the manner explained in 7 '7, and this work has the advantages of being easily systematised and of carrying its own checks. The integration of the equations cannot be done in this manner, but the steps with the method of this chapter are easy and simple. The final step of comparing the calculated results with observation, although dismissed here in a few sentences, is or may be as laborious as that of calculating the general perturbations, but it is necessary if good values of the constants are to be obtained. 14-2 CHAPTEE VIII RESONANCE 8*1. Resonance is usually defined as a case of motion in which a particle or body, moving or capable of moving with periodic motion, is acted on by an external force whose period is the same as that of the motion of the body. This definition, while it describes the apparent character of the phenomenon, implies the existence of certain conditions which are not present in actual mechanical systems. Let us take the usual illustration, namely the equation t ,, -r 4- n*x = m sin n t. at* When n =f n r , we have the solution . / , x in IA x = c sin (nt -f a) + - -75 &m n t. / a But when n n f , the solution is x = c sin (nt + a) \ mn't cos n't, where, in both cases, c, a are arbitrary constants. The illustration is defective because such an equation does not arise in any actual mechanical system except as an approxi- mation, and because the approximation is valid only when x is small. The solution, therefore, breaks down as soon as n n f becomes too small. In actual mechanical problems, either the left-hand member which, equated to zero, gives the undisturbed motion, is not a linear function of x y or else the variable x is present in the expression for the disturbing forces, or both of these conditions may be present. 8*2. In the previous chapters we have based our procedure on the plan of continued approximation with respect to the disturbing mass. In the elliptic approximation this mass was neglected. In the first approximation to the disturbance the 1-3] FAILURE OF PREVIOUS METHODS 217 elliptic values were substituted in the expressions for the disturbing forces, and the equations were again integrated. In the second approximation, the new values were substituted for the coordinates in the disturbing forces and the equations were again integrated. This procedure carried the implication that it was possible to develop the perturbations in positive integral powers of the disturbing mass, and that the coordinates would be expressed as sums of periodic terms. It is true that terms with coefficients increasing with the time were admitted, but it was seen that this was merely a convenient device adopted in order to abbreviate the calculations when the results were needed for a limited interval of time only. The terms so treated had periods which were long in comparison with the interval during which the expressions were to be used for comparison with observation. In cases of resonance, this procedure fails. The reasons for its failure may be exhibited in several ways. That which is most fundamental in the mathematical development is due to the fact that expansions in powers of the disturbing mass have to be replaced by expansions in powers of the square root of of some other fractional power of this mass. Further, there is a fundamental discontinuity in the passage from non-resonance to resonance, which cannot be bridged by any mathematical device, since it is a physical characteristic of the motion. The principal features of certain of the resonance problems in celestial mechanics can be illustrated by the motion of a pen- dulum which can make complete revolutions about a horizontal axis as well as oscillate about the vertical, and a following section (8*5) contains an analysis of these motions made from the point of view needed later. 8*3. We shall be concerned mainly with those cases of resonance which occur in the present configuration of the solar system. A certain number of such cases are present in the satellite systems of Jupiter and Saturn, where the periods of revolution round the planets appear to be very nearly in the ratio of two small integers. In the planetary system, we have the Trojan group of asteroids whose members circulate round the sun with 218 RESONANCE [OH. vin the same period as Jupiter. The motions of this group are treated in the following chapter. The most difficult problem is, however, to find out why, amongst the numerous asteroids which circulate between the orbits of Mars and Jupiter, none are known having periods exactly twice or three times that of Jupiter, or periods in the ratio to that of Jupiter of two small integers. A discussion of certain features of this problem will be given in this chapter. 8*4. In the actual cases of observed motions in the solar system, so far as they have been developed, we know of no case in which the discontinuity referred to in 8*2 is present in an observable form. We have referred to resonance as a set of cases in which the periods of revolution are in the ratio of two small integers. Since the final expressions for the co- ordinates contain all multiples of the frequencies, each pair of these can be regarded as a possibility for resonance conditions. But these frequencies are observed quantities, namely, those of the mean periods of revolution, and since such a pair of observed quantities can always be expressed as the ratio of two integers, it would seem that resonance must always be present in any three body problem. The question goes further than this. It will appear below that the phenomena of resonance occur not only when the observed periods are exactly in the ratio of two integers but also when these periods are nearly in such a ratio. In other words, resonance occurs not only for a pair of special values of the periods but also for a range of values and this range is finite. One difficulty, namely, the question of the accuracy of our measures of the periods, disappears to some extent, but it is replaced by another, namely, the consideration of the infinite number of periodic terms which must have the resonance property. The discontinuity referred to is not a place where either the coordinates or the velocities are discontinuous in a physical sense, but is one in which an infinitesimal change in one or more of the constants will ultimately produce a different type of motion. Thus the computer arrives at a situation where he needs a considerable increase in the accuracy with which the constants obtained from observation must be known in order to choose between two possible routes. And this process appears to continue as the approximations follow one another. From his point of view, there can be no general solution of the problem of three bodies, that is, there cannot exist one. set of formulae giving the coordinates in terms of the time and the initial conditions which will serve for more than one set of such initial conditions, which will be valid for all time, and which can be used for calculation of the position. This conclusion may be a result of the mathe- matical devices which he adopts, but is more probably due to an inherent difficulty, namely, that of finding expressions which shall be continuous 3-5] MOTION OF A PENDULUM 219 functions of the constants which can be determined from observation. Any proper solution of the problem requires also the consideration of the limi- tations placed on the observer ; it is not solely a mathematical problem. 8*5. The motion of a pendulum. The fundamental equation in resonance problems appears to be K*smx = (1) This is the same as the equation of motion of a simple pendulum of length , if /c 2 = g/l, and if x be the angle which it makes with a vertical line drawn downwards at time t. Since the substitu- tion, x -f- TT for x, changes the sign attached to # 2 , the equation with /c 2 replacing /c 2 gives the same motion as (1). The equation has the integral Y-C'-f 2* 2 cos#, (2) where G is an arbitrary constant : for the motion to be real it is necessary that G 4- 2/c 2 5 0. There are three types of motion depending on C > 2# 2 , C < 2/c 2 and the intermediate case C 2* 2 . (i) (7>2/e 2 . As dxfdt never vanishes in this case, it is always either positive or negative, and the pendulum is making complete revolutions in one sense or the other. We have dx f const. as the integral. If we put dx 1 _ J^ f 2 " dx^ n ~ 2^ Jo (C + 2* 2 c n can replace as the arbitrary constant, and the solution can be expanded in the form K? /e 4 x nt 4- e H 5 sin (nt -f e) -f 5 ^ sin 2 (nt -f e) + (3) n on The periodic portion of this series can be regarded as an oscilla- tion about the mean state of motion which is revolution with a period 27T/W. The half-amplitude of this oscillation is evidently 220 RESONANCE [OH. vra less than TT and it decreases as n increases. It is convenient to consider n, e as the arbitrary constants of the motion to be determined from the initial conditions. (ii) C < 2/e 2 . Here dx/dt = when x = a, where The integral can be written in the form sin 2 |a -sin 2 |#), and x is a periodic function of t oscillating between the values a, where a < TT. The solution can be expanded into the series C 3 x = c sm (pt -f /?) + - T^T sin 3 (pt + ft) -f . . . , (4) where p K (1 -^c 2 + ...). It is convenient here to consider c, /3 as the arbitrary constants, since the limit of p as c approaches zero is tc> a quantity inde- pendent of the arbitrary constants. (iii) (7= 2# 2 . Here dx/dt %K cos -|#, the solution of which gives x -f TT = 4 tan" 1 cxp. (fct 4- c/o), (5) where a<> is one arbitrary constant, the other having a particular value. When t= <x> , x= TT: at both places dx/dt = 0, d 2 x/dt 2 = 0, and it follows by differentiation of (1) that all higher derivatives of x vanish. Near this point, while x approaches one of the limits TT, t tends to become an indeterminate function of x. It should be noted also that x is a discontinuous function of the arbitrary constant (7, since the motion is of a quite different type according as <7-^2* a from C - 2* 2 > or from C- 2** < 0. This result is of course characteristic of unstable equilibrium, but the point of view stated here is required in the applications to be made below. Attention is drawn to the following facts which are obvious consequences, but which are needed for the interpretation of resonance equations. 5-7] MOTION OF A PENDULUM 221 (a) The mean value of dx/dt in (i) is n and in (ii) it is zero. (6) As n passes from positive to negative through zero the solution given under type (i) is a discontinuous function of n at n = 0. With certain initial conditions, there is a range of solutions (independently of the time constant) corresponding to the case n = 0. This range of solutions constitutes type (ii) and is characterised by the constant c or a which is related to it, and can have any value between TT. (c) In case (i) the series giving the solution proceeds along powers of * 2 ; in case (ii) it depends on V/c 2 . There is no analytical continuity between the two types of solution, and they cannot be represented by one and the same analytic function of t. (d) In case (i), the adopted arbitrary constants are the period of revolution and the time of passage through the vertical. In case (ii) they are the amplitude of the oscillation and time at which this oscillation vanishes. 8'6. A more general type of motion is exhibited by the equation with its integral (~\ = C- where/(#) is assumed to have an upper limit/(K). We get the same three types of motion according as <7>2/(K), C< 2/(fc), C =%/(*). In the first case x can take all its possible values ; in the second case it is limited by the value given to (7. In the first case also dx\dt never vanishes and it has a mean value different from zero ; in the second case x is a periodic function of t and the mean value of dx/dt is zero. When #=2/(K), CZf(x) is divisible by (x - /O 2 since K is the value of oo which makes f(x) a maximum, so that dx/dty d 2 x/dt' z and consequently all higher derivatives of x vanish. 8'7. The disturbed pendulum. The characteristics of resonance phenomena can be well exhibited by considering the equation x-72/yi -y-2 4- /c 2 sin x = mic 2 sin (x n't e'), ......... (1) (it which may be regarded as the equation of motion of a pendulum disturbed by a periodic force. We shall suppose that m, n', e' are 222 RESONANCE [OH. vra given constants and that m is small compared with unity. We shall further suppose that when ra = 0, the pendulum is oscillating with a small amplitude, so that only the type (ii) with the solution 8*5 (4) is under consideration for the undisturbed motion. To solve the equation (1) conveniently, it is advisable to use the method of the ' variation of arbitraries ' The method is given in the following article for a more general type of equation than (1), as it serves to illustrate in detail the plan to be followed in cases where the undisturbed motion is periodic, and also the nature of the change of variables useful when resonance problems in celestial mechanics have to be considered. 8*8. It is proposed to find the solution of the equation when that of d? 4 "^'^" ^ is periodic and is known. Suppose bhat the solution of (2) has been obtained in the form x~x(l,c\ l nt+e, n = func. c, (3) where x (I, c) is a Fourier series with argument I and with coefficients depending on c, the arbitrary constants being c, e. The solution (3) has the following properties. Let be regarded as a function of two independent variables I, c and let us form ?i 2 3 2 #/9J 2 , substituting the result in (4) If n be the function of c defined by (3), the variables /, c disappear from (4) and the substitution reduces (4) to zero. The disappear- ance of I, c is not dependent on their values : they may be any functions of t or any variables whatever or constants. Let us suppose that they are variable. We have dx dx dl ex dc 7, 8] CHANGE OP VARIABLES 223 We are about to replace x in (1) by two new variables I, c which are related to x by equations (3). Since we are replacing one variable by two others, a relation between the new variables is at our disposal. Let us so choose it that dx dl dx dc _ dx ,~^ '~ w ' .................. ( } . i dx dx then di n m Whence, since t is present in (6) only through I, c, n, d*x d*x dl 9 / dx\ dc Substituting this in (1) and making use of the fact that (4) is zero, we have dl 9 / dx dc x ( \ , / x\ c ,, /(7N n W'(dr n ) + dc( n dl)'dt- m( t > .......... (7 > The equations (5) and (7) may be regarded as linear equations to find dl/dt n, dc/dt. Their solution gives dc m dx , dl m dx , . , Tj r 9 / dx\ dx (Px dx /r . x where K = - (n -^ }. ~r - n ^ . 5- ................ (9) dc\ dl/ dl dl 2 dc ^ / It is easy to prove that K is a function of c only. For since the expression (4) vanishes identically for all values of l y c } its partial derivatives with respect to them will do so also. We thus obtain two equations between which df'(x)/dx can be eliminated and it is found that the result can be expressed in the form dK/dl 0, showing that K is independent of I and is therefore a function of c only. If <' = d<f>/dx, where <f> is a function of x, t, we can express (8) in the form dc m d6 dl m 9<f> /irkX _ __ _ _ _ ij _ _ r / 1 1 1 \ dt'Kdr dt~ Kdc' ............ v ' where <f> has been expressed in terms of /, c, t by inserting for x its value (3) in terms of I, c. 224 RESONANCE [CH. vra Since n t K are functions of c only, we can put (10) into canonical form with new variables Ci, JB, defined by dci = Kdc, dB = n dc\ = nKdc, and the equations then become i /r>, \ /D, i\ /-i\ ~ = -( B + m <t>), ^ -(B + Kf) ....... (11) 89, Solution of the equations for I, c. When m is small, the usual method of approximation is the substitution of constant values of j? , Q, Co for n, e, c in the terms factored by m which then become functions of t and can be integrated. If we put I = / -f /! = n Q t -f f + hj o = Co -f Ci, and neglect powers of / l5 Ci beyond the first, we have dn\ from which Ci and then li are immediately found. In cases of resonance this procedure breaks down, and it is necessary to proceed as follows. Differentiate the equation 8*8 (9) for dl/dt and substitute the expressions for dc/dt, dl/dt in the result. We obtain ^ - A ( - ty\ *? _ -^ ^ - ^. -? 2 ^ dt 2 ~dc\ ~K^c) t dt i Kdldc'Tt Kdcdi _ m fdn d^ _ _8^> _ 9 2 j> \ ~'"~ ?l "" Since the last line of (2) has the factor m 2 it may, in general, be neglected in a first approximation. In the applications, I, t are present in </> only as a sum of periodic terms with arguments il j (n't -h e'), where ri y e' are given constants. When this is the case cty^jV cty dt i dl ' 8,9] THE RESONANCE EQUATION 225 and the first approximation to (2) can be written d * 1 , v/- /\a m 9 / * 9< M /Q , -T7 2 -f 5 (in - w ) ^ 5- ~ - =-> ai ) - ....... w) ar J 7 iKoc \in-jn oil The standard type is that in which < has the form < = c^ cos ^-f&, li ilj(n't+e), ......... (4) where a it b are functions of c only. The equation for l t is then If, in a first approximation, we put c = c , n = n Q) K = K Q all constants (5) takes the form of the equation for the pen- dulum. [If the coefficient of sin l t be negative, we put Z. = i7-j(?i' + e') + 7r instead of the value (4).] There are therefore the types of solution considered in 8*5 . Type (i) is that in which dli/dt is never zero so that iri^jri does not vanish. Type (ii) is that in which li oscillates about the value [or TT]. With type (i), we put li = i(nQt + o)-j(n't + e') = l i0 in the second term of (5) and deduce 9 / cti \ . , /a . as a first approximation. With type (ii), we choose n , e to be such that iiiQ jn' -=* 0, i Q je f = or TT, (7) and li is an oscillating function. If the oscillations be small we can put sinli = liy n n^> c = Co, K = !Q in order to find a first approximation. This gives 7 / \ <> \ ma i fi n \ /o\ I- = X sin (pt 4- Xo), p "If" 13") > () X, X<> being arbitrary constants. With similar limitations, the equation for c gives dc f t Whence c = c mt\ - cos (^ + Xo), (9) 226 RESONANCE [OH. vni where c is determined from WQ =jri \ since T? O is a known function of CQ. The coefficient of the periodic term in (9) is X, ..................... (10) v ' and we thus have the first term of an expansion in powers of m^. If the coefficient of ra^ is not large, the assumption that we can put c Co in the coefficient of sin I in (5) is justified. The difficult cases in celestial mechanics are those which depend on the value of c . If (10) becomes infinite as c tends to zero, and if the coefficient of the periodic term in (9) is com- parable with c , this method of approximation breaks down. The analogy of (5) with the pendulum equation no longer exists and special devices have to be employed in order to find out whether resonance is possible. A case of this kind in which i = l,j = 2 is treated below. In general, the solution (6) corresponds to the case of an ordinary perturbation and (8) to a case of resonance. The various features noted in 8*5 as peculiar to the two types of solution are present and can be interpreted in the light of our knowledge of the motion of the pendulum*. 8*10. The general case of resonance in the perturbation problem. We recall the method of integrating the equations S (dc t . Swi - dw t . 8^) = dt.S ( Ai~ 2 + mR), . . .(1) where mR now denotes the disturbing function, ra being the disturbing mass with that of the sun as unit. We had, with a slightly different notation, E = RQ + 2Aco*jiN, .................. (2) where A was a function of the c t and JI'MI +J2W, ...... (3) 3 ' = n't 4- e', WZ^VT'. * A more elementary treatment of resonance with applications to the motions of one and two pendulums is given by E. W. Brown, Rice Institute Pamphlets, vol. xix, No. 1. Also reprinted separately and issued by the Cambridge University Press. 9, 10] GENERAL CASE 227 In a first approximation we put dw^dt = WQ and obtained integrals for the values of c it f w i which contained the divisors j\n*+ji!n'. It was assumed that no one of these divisors vanished. Let us now suppose that there is one term for which this condition does not hold, or rather, in order not to limit the argument too much, let us assume that there is one term in which neither ji nor j\ is zero but in which j\n + j\n f is so small that the approximation is no longer valid, but that we can approximate with all the remaining terms. We shall see later on that this latter condition cannot hold, but that an approach to the solution can be made by supposing that it does hold. All these remaining periodic terms can be eliminated by changes of variables in the manner explained in 6 '6. We can therefore suppose that the equations (1) refer to the new variables after such terms have been eliminated and that R^Ro + AcosfrN, .................. (4) where jiN has the value (3) and R consists of those parts of R which are independent of MI, Wjf. Let us change the variable Wi to W\ where ji Wi =jiwi +ji'wi +J 2 W, ............... (5) so that* ji8Wi= i /iSwi, jidWi^jidw^+tfridt ....... (6) It is easily seen that the left-hand member of (1) merely requires the substitution of Wi for Wi if we replace the right- hand member by c i ji Next, replace Ci by a new variable GH defined by (8) Previously, n$ was defined for the case m = by the relation ^2 Clo -3 = ft 0) so that the second relation (8) is the same as ?i o + ji'n' = 0. This definition does not demand that rc shall be * The symbols d and 8 have the same signification as in 5 '3. 228 RESONANCE [CH. vm the final mean value of n, since there may be a constant portion in en which prevents this. The only condition needed at this stage is that CH/CIO shall be small so that the expansion of Ci~ a in powers of this ratio shall be possible. We then have j^ The first two terms of the right-hand member of (9) do not contain the variables and may therefore be omitted from (7) ; the coefficient of Cn vanishes in virtue of (8). Hence, inserting flo = /^io~ 3 =-ji / ?* / /ji> ............... (10) we obtain for (7) the expression n cos ...... (11) The last expression is the characteristic form of the Hamil- tonian function for cases of resonance. It is to be remembered that no, CIQ are, by definition, functions of n' only and are there- fore independent of c,-, Wi, w { . 8*11. The equations for c n , Wi become -^ = --mjiABUij l N', ....................................... (1) dQ Cu a c n 2 3^0 d -^ - ~ ~ -7- - - ~ """ 5~~ - 5 dt ecu CIQ Ci<f den oc n ...... (2) The right-hand members of the equations for c 2 , c 3 , w 2 , w 3 all contain the factor m. If then we replace the variables Cu, c 2 , 3 by Ci, (7 2 , G z , where c u = mi(7i, 02=020 + ^^2, c 3 = Cso + m^Cs, ...(3) and ^ by m""*! 7 , the equations can be written iv, nj GENERAL CASE where TF 2 > TF 8 are written for w 2 , w$ to preserve symmetry of form. The quantities Cao, CM are now constants which are at our disposal. It will be noticed that the factor m has disappeared from the coefficient of cosjiN. In order to apply these results to actual problems, we need to know what the new variables mean in relation to the dis- turbed elliptic orbit. We have d = (/ia)* c 2 = ci {(1 - e 2 )* - 1}, C B = ci (1 - e a ) } (cos i - 1). The replacement of d by Ci -f m^Ci, with the expansion in powers of wJCi/Cio, implies that we assume an initial major axis 2a and that its variations are small compared with 2oo. The factor wi would seem to imply that Ci is not infinite when m = 0. Mathematically this is correct, but as the whole problem demands that in shall not be zero, we can at present be content with the previous statement. Next, since in the problems considered e< 1, we have The replacement of c% by Cw+ndCt implies that there is a value CQ such that (e e } -f- m% is not very great. But care is necessary if we contemplate expansions in powers of m^C^Cm. For perturbations by Jupiter, m//j, is of the order 10~ 3 so that mi is of order '03/xi. Thus expansions in powers of (<? 2 ^20)^ will converge too slowly for useful numerical computation if e is much less than 1. (See the last paragraph of 8*24.) The same difficulty does not occur in the case of c 3 ; for the expansion is made in powers of (2 sin Jt) a , so that it involves positive integral powers only of Ca CSQ. However, if we contemplate a develop- ment of Li in powers of m*, which these changes imply, we may be in danger of not obtaining a real approximation if (2 sin Ji) 2 is comparable with wi. These difficulties are actually present in the consideration of the motions of the asteroids circulating between the orbits of Mars and Jupiter. They play a much smaller part in the resonance cases amongst the satellites of Jupiter and Saturn, mainly because the disturbing mass-ratios are much smaller. B&SPT 15 230 RESONANCE [OH. vra 8*12. Let us suppose that expansions in powers of m^ OS/CM are possible, and let us further suppose that, in a first approximation to the solution of the equations 8*11 (4), we can neglect mi The coefficient A then becomes a constant, A Q , and R is a function of w 2 , w 3 only ; thus C 2 , 0$ disappear from Q and w 2f w 3 are therefore constant. The remaining equations are rt GI /i\ /n\ -3?! -\ ............ (1), (2) ...... (3), (4) The first two equations give l 710 *r rx -,-^T - 3w o ' a smji tf = 0, U-JL CIQ or, since w a , ^3 are constant, so that dN/dT=dWi/dT, This is the pendulum equation previously discussed. In the type of solution where jiN makes complete revolutions so that dN/dt never vanishes, we have an ordinary perturbation; this was expressly excluded from the definition of N. In the second type jiN oscillates about the value or TT according as A Q is negative or positive. This oscillation is known as a libration. In general, therefore, it appears that, under the stated con- ditions, such oscillations are possible. If the amplitude of the oscillation is small so that we may replace sin^i^V" by j^N or by jiN -f TT, we have, after the replacement of dT by its value m^dt, tf=Xsin(p* + Xo), if = a>ji*\A* ~m, ...(6) CIQ where X, Xo are arbitrary constants. The frequency p is proportional to the square root of the dis- turbing force, while the coefficient and phase are to be determined from observation. In all cases except that of the Trojan group 12] GENERAL CASE 231 of asteroids in which ji = /, some power of e, e f , F will be present in A and it is therefore necessary to consider the possibilities of expansions in powers of e^ > e'&, T$ as well as those in powers of ra*. The value of Ci is given by (2), (6). We find Ci = CIQ + m*(7i = Cio - t ^-r p\ cos (pt + X ). . . .(7) ^oji The small factor p in the coefficient of the periodic part of d is consistent with the assumption, made earlier, that expansions in powers of m^Ci/Cio are possible. It shows further that while the libration of N, that is of the angular position of the body, may have a finite amplitude, that of GI and therefore of the major axis is small. Since R Q) w 2 , w 3 are constants, the integrals of (3), (4) are 2 ~Ci + const., ............ (8) C 8 = w**!^+-i 8 ft + const ............. (9) dw 3 fr ^ ' Now R Q contains w z , w 3 , w 2 ' only in the form of cosines of multiples of w a w 2 ', ^2 4- w 2 ' 2w 3 . In order therefore that (7 2 , C 9 shall not increase continually with the time, it is necessary that w 2 = w 2 =^3 ...................... (10) Since jiN = j 1 w l + J 2 w 2 + J 3 iv 3 +ji'wi +J 2 W, where the sum of all thej iy ji is zero, the condition (10) gives jlN = ji(Wi-W2)+ji'(Wi-W2) .......... (11) If, however, e f = 0, RQ is a function of the c t only, so that the condition (10) is not needed and as w% disappears we have where, as usual, jl+jl'+J2 + J3 = 0. Since the value of Q in 8'10 (11) does not contain the time explicitly, the integral Q = const, exists. We have made no direct use of this integral in the investigation just given. This 15-2 232 RESONANCE [CH. vni omission corresponds to that in the case of the pendulum making small oscillations where it is more convenient to solve the equation directly than through the medium of its first integral. 8*13. The constants. The general solution of the equations of motion requires the presence of six arbitrary constants. When the libration ofjiN is zero, and e r = 0, the constants present in wi, wi, w 3 are all determinate since w z , w a are given by 812 (10) and that in w\ by the condition that j\N must be zero or TT. The constant Cu is determined by 8*10 (8). Thus the constants CM> Cao on ty are a k our disposal. But as the libration in general will exist and as its presence introduces two new arbitrary constants, the loss of four arbitrary constants is reduced to a loss of two. Since the two conditions 812 (10) disappear when e' = 0, there is no loss in this case. Thus when e r = 0, there is a finite range of values for each of the arbitrary constants: in other words, the resonance cases are not particular solutions, but are merely types of solutions in which all the arbitrary constants have finite ranges. When e 1 4= 0, the question of the ranges of the constants cannot be settled by the approximation used above: this involved the neglect of terms factored by m but the retention of those factored by m%. The conditions 812 (10) may be merely limiting values about which oscillations can exist in the same manner that N = is a limiting value about which librations are possible. The treatment of this case for the Trojan group will be found in Chap. ix. 814. It is evident that the change of variable, Wi to W\, eliminates t from all the angles for which the ratio ji/ji is the same: all these terms have in fact the resonance property and should properly be included with the single term chosen above. After the change of variable, the Hamiltonian function does not contain the time explicitly and there is an integral of the equations, namely, M 2 7i V 2 1 4/1 = const. 12-14] GENERAL CASE 233 The succeeding change from ci to c n gives, by 8*1 (9), . c 10 V Cio 2 c 10 3 This equation may be regarded as determining Cn in terms of the remaining variables. A comparison with 8*5 (2) will show that Cn plays a role similar to that of dxjdt in the integral for the motion of the pendulum, and that the presence of resonance depends on the value attributed to the constant. It should be pointed out that the investigation given in the preceding articles does not prove the existence of resonance; it merely shows that so far no condition preventing resonance has appeared. The illustration afforded by the motion of the pendulum must be regarded as showing only the general nature of the problem. Difficulties from which it is free appear as soon as we begin to consider even the simplest case of actual resonance in the solar system. Some of these arise from the fact that the consideration of a single resonance term is not sufficient. For example, in the case of the 2 : 1 ratio, the principal terms present in the disturbing function are A l e cos ( \GI 2w x ' + w 2 ), A x V cos (w\ - 2wj' + 10%). In the ordinary planetary theory, the variation of w%' and especially its secular part can be neglected in a first approximation and the result may be later corrected sufficiently to satisfy the needs of observation. If, how- ever, the former angle is oscillating about a mean value, it is necessary to consider the nature of the motion of the latter according as it oscillates or makes complete revolutions. Another difficulty not exhibited by the pendulum is the existence of types of motion in which small oscillations do not exist but in which oscillations of finite amplitude can exist. In certain simple cases these types may be dealt with by the use of the periodic orbit and of variations from this orbit. But these methods have heretofore given little or no in- formation as to the range of the oscillations and this range may be of importance in actual problems. If, for example, the eccentricity of an asteroid can become so large under the influence of Jupiter's attraction that its orbit can intersect that of Mars, it is only a question of time until a close approach to that planet will occur and such a close approach may alter the orbit so fundamentally that a completely new investigation of its further motion will be needed. A method of approach to the investigation of such cases is given below. 234 RESONANCE [OH. vm GENERAL METHOD FOR RESONANCE CASES 8*15. Certain features of resonance problems have been developed in the previous sections of this chapter. In this and the following sections a method of procedure applicable to certain of the actual cases of resonance in the solar system will be given. The integers j,j f , for which j^ jiV is very small or zero, are usually less than 5. Here n Q , n r are observed mean motions whose ratio can be expanded into a continued fraction. If the successive convergents be formed, the practical cases of resonance are those in which a convergent with small numbers is so close to the ratio that the next convergent is a fraction with large integers. Since the order of the coefficient with respect to the eccentricities and inclination is l^i ^i'l (cf. 4*15), it follows that the coefficients corresponding to the higher convergents will be very small, and it will be assumed that their effects can be neglected in the limited intervals during which it is desired to obtain an approximation to the motion. The terms for which jin Q j\n' is not very small or zero can be eliminated by the method of 6 '6 and the resulting function there- fore contains Wi, Wi only in the combinations p(jiWiji'wi), where p is a integer. Further, since the new terms produced by the elimination of the short period terms have the factor m 2 , they may, in general, be neglected. Thus we can take as the Hamiltonian function mR = ~- + rafi + m2A cos pjiN, ...... (1) where R Q contains the terms in the elliptic development of R independent of Wi, Wi' y and ji N contains these variables only in the combination j\fw\ j\w\) where ji, ji are given integers. Any multiples of w^^ WQ, w^ f may also be present in the angles. The substitution of W for w\ defined by similar to that of 8*10 (5), is made. The equations still remain canonical if to (1) we add the term ji'n'ci/ji. 15, 16] GENERAL METHOD 235 In view of the relations Ci = (^a)*, n 2 a 3 = p, it is convenient to put Ci = C ( 1 + z)~l, c = (^a )*, wo 8 (to 3 = /*, NO = ji n'/ji , ......... (2) so that ft = w (l -M), ........................ (3) and is a variable which in stable motion must oscillate between limits which are small compared with unity. With the notation (2), the function (1) with the additional termji'n'cj/ji can be written rc Co {i (1 + *)* 4- (1 + *)-*} + 01 JR, or, on expansion in powers of #, and with 2 replacing Ci we have As ^ is not present explicitly in R, the expression (4) equated to a constant is an integral of the equations. It may be written * 2 -* 3 +... +6mjR/w Ctt=(7. ............ (5) The symbol z corresponds to m^Ci used in 8*11 and therefore has the factor m*: it follows that C can be regarded as having the factor m. This equation is analogous to the first integral in the motion of the pendulum. In a first approximation, it is assumed that constant values can be given in R to all the elements except W. Retaining only the lowest power of z, we have 2 r = (C r -6mJfZ/7? Co) i ................... (6) This will be expected to furnish at least two types of motion depending on the value assigned to C. It will be shown in 8*17 that we can go a step further and include in the fundamental equation, which is that for W, terms depending on m$. 8'16. Although we are concerned in this chapter with resonance cases, it is of some interest to apply 8'15 (6) to cases in which s is small but never zero for any value of t. 236 RESONANCE [OH. vin Suppose that mR contains a single periodic term denoted by An c Q cos/! TV and that we include the non-periodic portion mR$ in C. As z does not vanish, we can expand 8*15 (6) in the form If #00 be the observed value of the mean motion, the definition of z gives for its mean value ^oo i ./i^oo-/!^' j/ 1 36mM a v A ~ ~~ . \J * 1 A -r-r- ~~~77i> 1~ I .......... I & ) n ftitQ \ 16 6 Y2 / Since n w is nearly equal to n , they can be interchanged in the coefficients of periodic terms. By hypothesis, the value of C given by (2) is small compared with unity. Thus the coefficient of cosj\N in (1) receives the small divisor C-. Hence the periodic term in z is large compared with the term having the same argument in R. The canonical equation for w\ , with the definitions of W 9 z, n Q in 8'15, givea d W dR ~dt= n Z - m ^ the second term of which can be neglected in comparison with the first. Integrating, and making use of (1), we have rrr / x , . . , r . TF= (9^ - tt ) t + const. - - - sin ^ ^- ^ \~~c~~) J Thus the principal perturbation produced in the longitude by a term of long period is and there is also another long period term with argument 2ji N having a coefficient - feji (coef. of sinjiN) 2 , a result in accordance with that obtained in 6'18 and also in 7-32. 817. The equation for W. With the definitions in 815, the canonical equation for W becomes dW dR Differentiating with respect to t, we obtain dz &R ( 9R\ ~Tt ~ m ^~^nr \n z-m 5-}-m %-, -jf at acidW \ 3cj/ 9ci a at _ v (*IL ** 4. & R *?A dt)' 16, 17] FUNDAMENTAL EQUATION 237 where, in the last term, i has the values 2, 3. Since the derivatives of Ci, <?a, c 3 , Wz, w s contain m as a factor, we obtain, on neglecting terms in this equation which are factored by m a , dz But the first of equations 8-10 (1) with 8*10 (7) gives, on substituting for Ci its value 815 (2), or, neglecting terms of order m 2 , dz 3 , t Now K is a function of Ci. If we put therein CI = CQ (1 and expand .R in powers of z, we obtain where the notation ( ) implies that c has been substituted for c. Hence, to the order m', PR On combining this result with (2), (4), and noticing that (d*R/dcidW)o disappears, we obtain in which, to the order of the terms retained, we can put ^^ ~ n Q dt Thus the variable Ci has been eliminated as far as the order % and we can write (7) in the form 3 ( + ~ m ~ ....... It is to be remembered that the variables c a , 03, w z , w 3 are still present in the last term of (8). 238 RESONANCE [CH. vm 818. Another integral can be obtained when e' = 0. Since the disturbing function is a function only of the differences of the angles Wi,W2,w 9> W> W> w e have dR dR , 3-R dR dR _ A m T" ^ i ?\ T > T ^ > " ............. (*) dwi dw 2 But by hypothesis the part of R which we are using contains wi t MI only in the combinations jiWi ji'wi, so that On changing the variable Wi to W y the new disturbing function has the same properties. Hence, from (1) and (2), I ^ /*_ | j j - t \ ji/dW dwz dwz dwj' Thence, with the help of the canonical equations 810 (1), we obbain by integration /, ji\ , f 3 R n / \ 1 J -r ci -h c 2 -f c 3 = const. m ~ > dt (6) \ ji/ J ow 2 If, in accordance with a previous notation, we put and make use of 815 (2), we obtain When e' =Q, R is independent of ^2' = ^ ' The last term o disappears, and the equation becomes an integral. THE CASE <?' = r = 819. The variables c 3 , w 3 disappear and the canonical system reduces to one with four variables. The differences of the angles Wi, Wi, w* are present in R and the ratio of the multiples of Wi t W is fixed by the resonance condition. Hence a single angle N and its multiples are alone present in R. The system admits of the two integrals 815 (5), 818 (4). The latter enables us to eliminate the variable c. From the former, 18, 19] THE RESTRICTED CASE 239 with the equation for dz/dt y we can eliminate N and thus obtain an equation giving dz/dt in terms of z. After the integration of this equation, giving z in terms of tf, the remaining variables may be found without difficulty. The process thus described will be followed below but will be simplified by the omission of terms known to be small in comparison with those retained. We assume that z, e 2 , m are small compared with unity. The omission of higher powers gives e\=* e 2 , and from 8*18 (4), --z = const. or # = E-l\l- J 4Jz % (1) where E is a constant. Next, if R = RQ -f %A P cospjiN, R 0y A p are functions of c\ or z and c 2 only, and by the use of (1) can be expressed as functions of z. Since RQ contains only even powers of e, it can be expressed as a positive power series in z, and the constant part is all that need be retained, although the retention of z, z 2 creates little additional labour. The coefficient A p has the form where Oo> <#2?-" catl be expressed in series of positive powers of z\ the same limitations as those made in the case of RQ permit us to retain the constant term only. Finally, on the same basis, we put p = 1 and thus reduce R to a single periodic term and a constant portion. With these limitations, we can put R = const. + riQCo Ae J cosjiN. (2) On substitution of this in 8*15 (5), we can suppose that the constant part is included in C and thus obtain 2* = G-6mAe J co*jiN. (3) With the same limitations, 8*17 (4) becomes ~~ 7fi = 3? 1 m Ae J sin ji N. (4) 770 Cut 240 RESONANCE [CH. vni The elimination of jiN between (3), (4) with the help of (1) gives 4 / d? \ 2 ...... (5) This equation has the form dzldt={f(z)}^ and gives t as a function of z. For values of / less than 5, the integral is of the elliptic type and the discussion of (5) or of its integral gives the chief characteristics of the motion. For the cases of chief interest in the solar system, J is, in fact, less than 5, and the equation includes all these cases. The most serious limitation is that introduced by the assumption e r = 0. 8*20. Particular cases. These are classified according to the values of /. For J = 0, we have jt=ji = 1. This case, that of the Trojan group of asteroids, is treated in detail in Chap. ix. It permits of numerous simplifications, but the development ofR takes a quite different form. For /=!, the ratio ji/ji has the values 1/2, 2/8, 3/4,..., in the cases of exterior bodies disturbing interior ones, and their inverses when interior are disturbing exterior bodies. For these ratios, the mean values of a/a' are '64, *76, '82, It is doubtful whether the expansions are sufficiently convergent for numerical calculations beyond the ratio 4/5. The case 1/2 is discussed in detail in the following paragraphs. The case J==2*, corresponding to the ratios 1/3, 3/5, 5/7,..., is rather more simple than the case J= 1, owing to the fact that only even powers of e are present in the formulae. This case also arises when we take into account the inclination of the orbit. This and the higher values of J are chiefly of interest in the applications to asteroids disturbed by Jupiter. * See Charlier, Mech. des Himmels, Absch. (1) ; D B. Ames, Mon. Not. R.A.S. vol. 92, p. 542. 19-22] CASE OF THE 2 : 1 RATIO 241 THE CASE ji = l, ji' = 2 8*21. Change of scale. If we put a)z, - t, ( ~ } e, for z, t, e respectively, ??o&> \o/ where a> = (12A 2 m 2 )%, the equations given in the preceding sections become, with the given limitations and with appropriate changes in E, C, d W dr * -~> , N= Wl - 2^' + OT = W + <*, Cvv Clu For an asteroid disturbed by Jupiter with n Q /n f = 2, we have* = -000716. For the change of scale we have o> = *0183, so that an actual eccentricity '1 has the value *78 in the new scale. The values of the variables in the new scale are thus comparable with unity. 8*22. There are two problems. One, that dealing with the conditions under which N is an oscillating angle (resonance) or a revolving angle (non-resonance). The other, the conditions under which z can pass through the value zero. The latter is not, in the limited case here treated, strictly a resonance problem, but it becomes one when e' ^ and it is applicable to the cases of the apparent absence of asteroids for which the osculating mean motion is exactly twice that of Jupiter. Conditions to be satisfied. (i) Since z measures the deviation of the major axis from a mean value, it must be an oscillating function and must there- fore lie between finite limits which will be denoted by s d: we shall choose d to be a positive number so that s + d is the maxi- mum value of z and s d the minimum. * Mon. Not. R.A.S. vol. 72, p. 619. 242 RESONANCE [OH. vm (ii) The limiting values of z are given by dzjdt = 0, so that s d are two of the roots of -(z*-C)* = ................... (1) (iii) The left-hand member of (1) must be > for all other values of z. (iv) The convergence of the developments is doubtful if the actual eccentricity is greater than about '3; this gives a limit 2*4 to the variable in the new scale. (v) The conditions s = d separate the cases in which z can or cannot take the value zero according as s > or s < 0. (vi) The equation e 2 = E+2z gives de dz d 2 e /de _ e dt~dt' dt 2 \dt """3?" It is necessary to have d 2 z/dt 2 =f= when dzfdt = 0, in order that equation 8*21 (1) shall give a determinate value of z for all values of t. It follows from the equations just written that e y de/dt cannot be zero simultaneously. Since e is not negative, it can vanish only if de/dt vanishes simultaneously: hence e is never zero. Since z = s d are two of the roots of (1), it is easily deduced that CW + CP--JJ-, E = 4*ffi + -28, ...(2), (3) and that 8*21 (1) may be written -4(^) These results give ............... (5) so that the maximum and minimum of e are given by I ...................... (6) ZZ-Z4J RlSti'LKLVLXilJ Z:l VADXi '416 8-23. The identities compared with 8*22 (6) and with z 2 = s 2 -I- d 2 - n 6 cos N, ZiS show that if 5 > 0, d > l/4s 2 , the extreme values of N are TT, 0, but that if d< 1/4$*, jV" takes the value TT at both extremes, Hence, the relation d = l/4s a separates oscillating from revolving angles of N, that is, the resonance from the non-resonance case. It is easily seen that the same statement is true if s < 0, but that in the latter case N takes the value zero at both extremes when 8*24. The four values of z given by 8*22 (4) when dz/dt are z l = s -f d, z% = s d, By hypothesis 2 is to lie between z\, z. This condition demands that or that 3 > 4 > #1 > ^2- (a) For 5 > 0, d 2 > 1/5, the descending order of magnitude is zi, z%, z$, 24. The condition z 3 < z z gives d<2s, d<s-f^-2. fr<S (6) For 5 > 0, d 2 < 1/5, the roots #3, # 4 are imaginary and dz/dt is always real between z = z, z%. Hence for s > the boundary is d = s + l/4s 2 , when d < 2s, d 2 =l/s, when d> 25. If we regard 5, d as the rectangular coordinates of a point on a curve, the two conditions are the equations of two bounding curves which meet and have the same tangent at d = 2s = 2^. 244 RESONANCE [OH. vm (c) If s < 0, all four roots are real, and the ascending order of magnitude of the roots is z Z) zi, #4, z%. This demands that 2s + d > 0, d<-s- l/4s a . Since d > 0, s < it is easily seen that the latter condition includes the former. It also requires that s + d < 0, and as we have s d < 0, it follows that z does not change sign. Hence, z cannot vanish for s < 0, that is, if z can vanish its middle value is positive. -1-5 -1-0 -0-5 1-6 The conditions that z may vanish are therefore quite compli- cated. The boundary of the region consists of the four curves, and d, s are both positive. On the other hand, the resonance regions for N are simply those portions included between the two curves for s < and s > 0, where z is real. In Fig. 2, z is imaginary in the regions with inclined shading; z can be zero in the region with horizontal shading; N oscillates 24, 25] RESTRICTED 2 : 1 CASE 245 in the region with vertical shading; the two latter regions overlap as shown. Passage from s > to s < is effected when the minimum in the former case is the same as the maximum in the latter, that is, along the boundary d = s 4- l/4s 2 for s > to the boundary d = s l/4s a for s < 0. Along these two curves and also along d?=l/s, two of the values of z are equal and W jzdt is indeterminate near t= oo in the same sense as in the case of the pendulum near its highest position. The only other case of equal roots for z is that given by d 0. When s > and d is small, the second factor of 8'22 (4) is approximately 4s 2 -f l/ which is constant. The solution is then z = s + d cos (qt -f <?o), <f = s 2 -f l/4s. The same solution is available for s < provided s is not too near the value given by s 2 -f l/4s = 0. In these cases the mean value of e is 1/2 \s\, so that s must not be too small. The case d = in which z oscillates about the value s is the resonance case for N TT when the libration is zero. From 8-22 (2), (3), we deduce C 2 - E=(s*-d so that C 2 E changes sign at the boundary separating the region in which z can be zero from that in which it is never zero. But C 2 also when d 2 =8 2 +l/s, a relation which does not enter into the discussion given above. It follows that the condition O L =E is not the necessary and sufficient condition that z shall take the value zero*. In vol. 4, chap. 25 of his Mfaanique Ce'leste, Tisserand treats the resonance case by supposing that the eccentricity is equal to e -f Se, where e Q is a constant, and he expands in powers of 8e/e Q . With a proper choice of e Q this is theoretically possible, since e is essentially positive, but it gives very slow convergence in the most important cases those in which e Q is small. 8*25. Applications. The discussion in the previous sections is applicable to the cases of asteroids whose mean motions are nearly twice that of Jupiter. The statistical discussions f show * For a different and less complete discussion see E. W. Brown, Mon. Not. R.A.S. vol. 72, pp. 609-630. f These have been numerous. Fairly complete lists are given by S. G. Barton, A.J. 702, 838; A. Klose, Mitteil. Univ. Riga, 1928. B&SPT 16 246 RESONANCE [OH. vnr that while there are numerous asteroids with mean motions somewhat greater and somewhat less than twice that of Jupiter, there is none which can be stated with certainty to have the relation satisfied within a certain range. This result refers to osculating elements. If we omit the short period terms, the variable z may be regarded as an element of this nature and the vanishing of z corresponds to the exact relation. Now we have seen that the limiting case in which z can be zero is given by s = d and the maximum value of z is then 2s. According to 8*22 (6) the maximum value of e is then 2s -f 1/2$ and the least value which this expression can have is given by 5 = J. The least maximum of e is, therefore, 1'5. On referring back to the scale relation in 8'21, we find that this gives a least maximum for the eccentricity of *13. So far, therefore, nothing has been proved which prevents the existence of asteroids which can have an osculating mean motion exactly twice that of Jupiter. But it has been shown that if such asteroids can exist, the elements, in particular the eccentricity, must oscillate through a considerable range of values ; small oscillations are impossible. The existence of asteroids or satellites in which the angle N oscillates is a quite different question. What has been shown is that if such orbits exist, the middle value s of z must be different from zero. Small oscillations or librations about this value are possible. We have, for example, the case of Titan and Hyperion, satellites of Saturn, where the ratio is nearly 3 : 4, a case similar to that of 1 : 2. THE CASES e'4=0 8*26. These cases are much more difficult, mainly because the integral e 2 = E f (1 ji/ji) z no longer exists. But in the cases of the ratios j\ j\ + 1, where the principal terms are of the first order with respect to the eccentricities, it is possible, in a first approximation, to utilise the results obtained above by a change of variables. For simplicity let us consider the case ji= 1. The additional first order term in R has the form n Q CQe'A' cos N' 9 where 25, 26] EXTENDED 2 : 1 CASE 247 JV's= W + vr'. Arguments similar to those used above give the equations for W, z : - *r / At ' *T/\ -- , _ __ sm N + e A sm N) 9 n at n^ at z* = C-6mAe cosN- 6m A' e' cos N'. Instead of the variables 02, w 2 , let us transform to the variables 2> (72 defined by sin w 2 , #2 = 0i Ci* cos w 2 , where, as before, 2 = isi^i 2 - According to 5'14, the equations for Ci, W, p 2) <?2 are still canonical. Let us change from p 2 , q* to new variables defined by sn cr, ^2 = ^2 + ^ cos where X is a function of Ci only. We have with similar equations for dq^'/dt, Sq%'. Now dp2/dt dR/dq z , and JK is a linear function of p 2 , q 2 for the only terms we have under consideration. It follows that dp 2 /dt does not contain either eccentricity as a factor while dci/dt does contain them. The second term in the equation for dp 2 /dt is therefore two orders, with respect to the eccentricities, higher than the first term and maybe neglected. The canonical equations may therefore be written dt . $H=dci . 8W-dW. Sci + dp 2 . Sq 2 - dq 2 . Sp 2 p z ' . Sq 2 ' - dq 2 f . Sp 2 ' 3 (e f cos OT' dp 2 e' sin cr' dq%) Sci. C/Ci But the approximation z = dW/dt to the equation dW/dt=dH/dci involved the neglect of all parts of It in this equation and this is the only way in which the coefficient of 8ci in the canonical set just given arises. It follows that the equations for Ci, TF, PZ> q* are still canonical. 16-2 248 RESONANCE [CH. vm These results suggest that we can put ,A r e cos -or = e cos *& + e ? cos -& , e sin ts = e sin *r + e -~r sin IB , so that __ e A sin N -f e'A ' sin N' = 4 sin N, } ^ _ e^. cos JV^-f e'^1 ' cos N' = e^. cos N, ] and that when we do so, e, N will have the same properties that e, N had in the case e' 0. In particular, we shall have and the limits previously given for e will now apply to e. In the cases of the asteroids disturbed by Jupiter we have b'/b s=s -36, e' = '048 (loc. cit. 8'24), so that unless e is small the additional terms will not give large corrections to the results previously obtained as far as the vanishing of z is concerned. In the cases of the small oscillations of N or JV, it appears that these must take place about the values or TT and that & - tar' must oscillate in a similar manner. But the argument, based on the assumption that TS' is constant, is not necessarily valid if -or' has a mean motion. 8*27. The methods of this chapter are constructed mainly for the treat- ment of those cases of resonance which arise in the solar system. The theory of periodic orbits is applicable as a first approximation in certain problems : the asteroids which form the Trojan group are examples. In general, however, this theory fails, either because the numerical applications are too remote or because the restrictions under which the theory is developed avoid the very difficulties which the actual problems present. The methods given above apply to cases of resonance in which both periods of revolution are present. The perihelia and nodes are angles which in general revolve and there are possibilities of resonance relations between their periods of revolution. In the comparatively short interval of time during which observations have been made, such relations are unimportant because, with the very long periods involved, expansions in powers of the 26, 27] APPLICATIONS TO COSMOGONY 249 time give the required degree of accuracy. Comparable with these are the new periods introduced by the librations, and there are, therefore, further possibilities for resonance relations. So long as the past history of the solar system was supposed to be confined within an interval of 10 8 years, deductions as to its initial configuration from its present configuration ap- peared to have some degree of value ; the extension of this interval to 10* years or longer makes these deductions quite doubtful. The doubt appears not so much in the ranges of values possible for the mean distances as in the ranges of the eccentricities and inclinations. The indications furnished by the theory of resonance as applied to the solar system point towards the possibility of occasional large osculating eccentricities and inclinations at some time in the future. On the other hand, statistical evidence appears to indicate that these elements will tend to be confined within narrow limits. A discussion of these and other difficulties involved in the attempts to apply the theory to the solar system will be found elsewhere*. * E. W. Brown, Bull. Amer. Math. Soc. May-June, 1928; PuU. Astro. Soc. Pac. Jan. 1932. CHAPTEB IX THE TROJAN GROUP OF ASTEROIDS 9*1. The triangular solutions of the problem of three bodies. The problem of three bodies does not, in general, admit a finite solution in terms of known functions. Laplace, however, has shown that there is a solution in which the three bodies always occupy the vertices of an equilateral triangle. The plane of the triangle is fixed and any two of the bodies describe ellipses having the same eccentricity about the third body which lies in a focus. Further, if n, a be the mean motion and semi- axis major of any one of these ellipses, the relation tt 2 a 3 = sum of the masses is found to be a necessary consequence of the solution. Other sets of finite solutions, in which the bodies are collinear, are known but they will not be considered here. Small changes from the triangular configuration or in the appropriate velocities, or perturbations by other planets, cause oscillations about the triangular configuration, provided the masses satisfy a certain limiting configuration ; the study of these oscillations and the applications of the theory are the objects of this chapter. Ten asteroids, each of which, with Jupiter and the Sun, apparently satisfies the given conditions, have been dis- covered, the first in 1901, more than a century after Laplace gave the solution, and the last in the year 1932. They have re- ceived names taken from the Iliad of Homer and from this cir- cumstance constitute what is usually called the Trojan group. We shall first prove the existence of the triangular solutions and of small oscillations of a certain kind about this solution ; these will indicate some of the characteristic features of the motion. A general theory for the motion of an asteroid of the Trojan group will then be based on the methods used in Chaps. VI and VIII. 1,2] THE TRIANGULAR SOLUTIONS 251 The problem differs from that of the ordinary planetary theory in several respects. In the first place, the development of the disturbing function given in Chap. IV cannot be used because the ratio of the mean distances of Jupiter and the asteroid is very near unity and that development ultimately depends on series in powers of this ratio which do not converge when the ratio is unity. In the second place, the motion is a case of resonance, since the ratio of the mean motions of the asteroid and Jupiter oscillates about the value unity. Thirdly, these oscillations, instead of being small, may have very considerable amplitudes and require special methods if an accuracy com- parable with that of observation is to be secured. Another point, brought out in Chap, vm, is the development in powers of the square root of the ratio of the mass of Jupiter to that of the Sun, instead of in integral powers of this ratio as in the ordinary planetary theory ; since the square root of a small fraction is much greater than the fraction, the rate of numerical con- vergence may be much diminished in consequence. Still another peculiar feature is the theory of the long period terms produced by other planets, and notably by Saturn. A first approximation to their coefficients cannot be obtained by neglecting the action of Jupiter, and these coefficients tend to become greatest, not when the periods are longest, but when these periods approach most nearly to that of the principal libration. 9*2. Existence of the triangular solutions. Since the motion takes place in a fixed plane, the latter may be used as the plane of reference. With the use of the equations of 1*23, those numbered (5), (6) disappear and v = v. Let us take one of the bodies, mass m , as origin and let the coordinates and masses of the other two bodies be r, v, mi and r', v' t m'. It is then necessary to show that the equations *-W d( dv Jt) ~dr' 5V 5 _ dt* \dt ~ 9/ ' dt \ dt 252 THE TROJAN GROUP OF ASTEROIDS [CH. ix where, according to 1'9, 1*10, with the inclinations zero, ^ mo + mi , , fl r cos (v v')) 7 _, H ' __ * I AV) ' J .. V / I I ^ I J.' ~ ~ in i A ~"~ 7o i j \ *-* / r (A r 2 J ^ ' , _ 7/? + w' (1 r' cos (v' - v)\ /r . jp j_ ^j j v 10) r (A r 2 j A 2 = r 2 -f r' a - 2rr' cos (v - v'), are simultaneously satisfied by r = r' = A, v t/ = 60, with elliptic motion for each of the bodies. According to 3'2(1), (2), these conditions demand that --_ -- dr ~ dr' ~ r 2 ' dv "~ fo' ~ From (5) we have dF _ w Q + m t , (r - r' cos (v-v') cos (v t;' -' dr ^ = _ dv ~ with similar expressions for dF'/dr', dF'/dv'. It is at once evident that the equations (7) are satisfied by the given relations provided /u = m Q -f mi 4- rti'. The elements n> a, are evidently the same for the two ellipses with n 2 a 8 = 7W -f m\ 4- w'. For the remaining elements we have e 6 / =cr t*r'=4 60. 9'3. 2%^ equations of variations. These equations are defined by giving to the coordinates in the general equations of motion small additions to their elliptic values, the squares, products and higher powers of these additions being neglected. This procedure is not sufficient for the calculation of the general perturbations, but it serves to indicate their nature to some extent. The actual calculation of the perturbations is more easily carried out by quite different methods. The problem will be limited now and throughout the remainder of this chapter by supposing that the mass mi of one body (the 2, 3] EQUATIONS OF VARIATIONS 253 asteroid) is so small compared with either of the masses of the other two bodies (the Sun and Jupiter), that it can be neglected in the equations of motion. We then have F' = (m Q 4- w')/r' and the motion of m r relative to mo is elliptic with mo in one focus. In the present section, two further limitations will be made. The motion of m' relative to mo will be supposed to be circular and to receive no disturbance, and the disturbance of mi will be supposed to take place within the plane of motion of m', so that the problem of the motion of mi is still two-dimensional. According to these assumptions, the undisturbed motion of 77*1 will be circular. Denote this motion by the suffix zero and the disturbed values by r = r + Sr, v = VQ -f Sv. Substitute these values in 9'2(1), (2) and expand in powers ot Sr, Sv and of their derivatives, neglecting powers and products of these quantities above the first. Since the equations are satisfied when 8r, Sv and their derivatives are zero, this pro- cedure gives , -Ji ~J7 ji ^~2 ^-^- Sv, dt ) dt dt 2 d ( 2 d * , O dv <> 5^ \ f & F \ S> , / 92 A 55 -r. r 2 -j-. Sv + 2r ~-~ Sr = 5-=- Sr + ( 5-0 ) ov. dt\ dt dt ) \drdv Jo \9^ 2 / These are the 'equations of variations.' The second derivatives of .fare formed from 9'2 (5), (6). They are ') m' 3m' , , x)2 1 8 2 f /I IN , . , /x 7 5-^-= ~ ( Xa /a ) r sm (v - v ) m 9r3v \A 3 r' 3 / v The limitations imposed above give r = a = r / = A, VQ v' = 60, 254 THE TROJAN GROUP OF ASTEROIDS [OH. nc The substitution of these values in the second derivatives of F and in the equations of variations gives for the latter, s . 2 m -m ,, or zarz ~r A ov n*or = j-= o?* -f -=-- ra ov, dt 4a 3 "" 4a 2 d * .3 A/3 ,, 9m' ^ ^ r = + "T~2- m ^ r + T 8v. dt ~ 4a 2 4a These equations being linear with constant coefficients, their solution is obtained by assuming where A, B, X are constants. The substitution of these values gives, after division by e xt , the conditions The elimination of the ratio A : between these equations gives X* + X' 8 n - 2 *- + + T- = 0. V a 3 / 4a 8 \ a 3 ; The use of the relation ?i 2 a 8 = r??o + ^i / and the introduction of m, where reduce this to X 4 + X 2 /i 2 + ^-w 4 /^ (1 - m) = 0. If 27//i(l-wi)<l, or m < '04 approximately, the roots are all pure imaginary and the motion is oscillatory. Since m < *001 in the case of the Trojan group, the condition is easily satisfied. If powers of m beyond the first be neglected, the roots are so that the periods are 2-rr -r n V^w, 2w -T- n (1 - 3] PERIOD OF LIBRATION 255 With 27T/n = 11*86 years, l/m=1047, the former period is 148 years and the latter nearly the same as that of revolution of the asteroid or of Jupiter round the Sun. The oscillation having a long period is a first approximation to the effect usually known as the 'libration.' The short period oscillation will be seen below to correspond to the principal elliptic term in the motion of the asteroid, so that the principal part of the motion of the perihelion is %g-mn. The ratio B:A for the long period oscillation is given by either of the equations (1), (2) with \ = m (27ra/4)i From (2) we find aB 3m'-ani-m _ 1 t, A ~~ -27a 3 ?i 2 m-9ra' ~~ + T * \/3m' with the aid of the relations a?n 2 = m Q + m' = m' jm> Since the second of the two terms is large compared with the first, the approximate ratio of \aB\ to | A\ is 1 : V3?Ai or 18'7 : 1. As | A\ is the amplitude of the oscillation along the radius vector and \aB\ that perpendicular to it, it follows that the former is small compared with the latter, the ratio being nearly as V8m : 1. The features of the motion brought out in this investigation, namely, the long period of the libration, the small disturbance along the radius vector as compared with that perpendicular to it, and the presence of Vm, will be utilised in the general theory which follows*. Incidentally, it may be pointed out that they are present in all resonance problems occurring in connection with planetary motion, as can be shown from the results of Chap. vin. We now proceed to the general theory of the motion. * The small oscillations were first fully treated by E. J. Eouth, Proc. Lond Math. Soc. vol. 6 (1875) ; see also his Dynamics of Rigid Bodies, Part n, Chap, in . A special case of them is treated by Charlier, Himmels-Mech. vol. 2, Chap. ix. 256 THE TROJAN GROUP OF ASTEROIDS [OH. DC GENERAL THEOKY OF PERTURBATIONS DUE TO JUPITER 9*4. The disturbing function. According to 1*10, the force-function for the action of a planet, mass m' , on one of mass nil when the sun, mass mo, is taken as the origin of coordinates, is r ,- x (1) where A 2 = r 2 + r' 2 - 2rr' cos 5, ............... (2) /S being the angle between the radii r, r'. If 8 be eliminated between (1), (2), F can be written in the form F= mo + mi + m ' + R^ + R, ............ (3) r r x 7 , D ,/l 1 1 A 2 1 v*\ ... where Brsm U~r + 27*~ 2?j' ............ (4) since the term | m'/r', thus introduced, can play no part in the equations of motion which depend only on the derivatives of F with respect to the coordinates of mi. In the form (4), .72 = when r r' = A. It can also be written a form which at once shows that the first derivative of R with respect to any coordinate of mi vanishes when A = r = r'. The mass mi of the asteroid will be neglected in comparison with mo-fw'. The osculating mean motions n t n' and mean distances a, a! will then be connected by the relations 9'5. The equations of motion. We start with the variables c^, Wi (defined in 5*13 and used in Chap, vi) which satisfy the canonical equations, i+R .......... (1) 4, 6] GENERAL THEORY 257 Define a new variable r by the relation Wi = n' -f e' + r = Wi + T, ............... (2) and in (1) replace the variable Wi by r. Since dw-i^n'dt + dr, 8^ 1= =8r, the equations for the variables c^, T, ^2, ^3 will still remain canonical if we add H'CI to R in the right-hand member of (1) so that the Hamiltonian function becomes (3) Since all the d have the dimension \V, arid since a/a' is near unity, let us put d = Ci (1 4- X) = \Va' (1 + #), C 2 = Ci'^2 > C3 = Ci'# 3 . . . .(4) The equations for the variables #, # 2 , # 3 , T, w 2t w 9 still remain canonical and can be written dx . ST dr . Since JK is a homogeneous function of a, a' of degree 1 and contains m' as a factor, the Hamiltonian function is now a pure ratio like the variables x> # 2 > #3> T, w 2 , w^, and ofR/p has the factor m = m'//^. It will be seen below that x consists of portions which have either the factor m or the factor TI Vm, where TI is the coefficient of the principal term in the libration. No case of an asteroid is yet known* in which \x\ exceeds *01, and as eccentricities and inclinations are in general of the order *1, x will be treated as of the same order as the square of the eccentricities. We shall carry R as far as the fourth powers of the eccentricities and inclination for the long period terms and on this basis it is advisable to retain terms of orders x*, mx 2 . In such terms the eccentricities and inclination can be neglected, and then the retention of terms of order x*, mx 2 presents no serious difficulty, but as it will much simplify the exposition to neglect them, they will not be retained in the developments which follow. In any * Except possibly Anchises discovered in 1931. 258 THE TEOJAN GROUP OF ASTEROIDS [OH. ix case we are going to neglect terms in the Hamiltonian function which have the factor w 2 , provided such terms have no small divisors, so that the neglect here of the terms of orders # 4 , ma? follows a general rule. With these limitations, the right-hand member of (5) can be expanded in powers of x, and becomes n'dt . S (I a* - W + + 2x ~] to order , . . .(6) \2 p fJb OCLJ where R is the value of R when a = a' or x = and similarly for the definition of 3R/9a. The approximation a = a! (1 -f 2#) has been used in the expansion of R. 9*6. Form of the expansion of R. As in Chap. IV, it is assumed that R is developable in powers of the eccentricities and mutual inclination. The development there given is available also here as far as the stage where it is reduced to odd powers of 1/A (cf. 4'13), with a (JL and to derivatives of these powers with respect to a, wi. With WI WI=T, A becomes a function of a, F, r, and r is an oscil- lating angle. The angles present are g, g' , Wi + Wi - 20 or r, w Z) WB, Wi, -cr'. Thus R consists of a sum of terms of the form where K> K' are functions of x, x%> ^3, r and N of r, w^, w 3> t. The particular point to be remembered is that r is present both in coefficient and angle because K, K' contain functions of A . We shall distinguish between the terms containing w\ after the substitutions, g tVi Wz, g' = Wi' w^, WI = T + WI, and those independent of Wi. Our preliminary investigation showed that dr/dt was small so that the terms in the former class have short periods and those in the latter have long periods or are constant. The latter class is also distinguished by being inde- pendent of t explicitly when the orbit of Jupiter is an ellipse. 5-7] THE SHORT PERIOD TERMS 259 These considerations give us the form of the development, but the actual development will be carried out by a method quite different from that of Chap. IV, chiefly because very considerable abbreviations of the work are possible with the use of the special properties which R possesses in the case of the Trojan group. In general, it appears that we can secure sufficient accuracy for observational needs by taking the short period terms to the second order with respect to the eccentricities and inclination, and the terms independent of iv\ to the fourth order. Classified with respect to the arguments g, g' ', w\ 4- iv\ 26 used in Chap. IV, the latter are Arg. 0*, orders 0, e 2 , e' 2 , e\ e 2 e' 2 , e'\ T 2 ; 9~9*> orders ee' , e z e f , ee'*\ >} 2#-2<7', order eV 2 ; 2g-(wi + wi'-20), order (?T \ 9 + o' ~ ( w * + w i ~~ 2#)> or der 00' r 5 2g' - (w l + wi - 2(9), order e f * T. 9'7. Elimination of the short period terms. The plan adopted is that used in Chap, vi, namely, a change of variables which leaves the equations canonical. Owing, how- ever, to the fact that the variable r is contained in both coefficients and angles, the form of the transformation function S has to be modified. Further, use can be made of the fact that with the neglect of m 2 , one variable x appears in the differential equations only in a linear form as shown in 9*5. Let the short period terms be denoted by R t and put where, in accordance with 9'5 (6), K, L are independent of x, but are functions of # 2 > #3, T, and N^j'n't -f multiples of T, w 2> w 9 + const., as shown in 9*6. * The term of order T is included in A . 260 THE TROJAN GROUP OF ASTEROIDS [CH. n: Let us transform to new variables #o, #20 > w> T o> WM, iu& by means of the transformation function /, where = sn sin jy + cos jy jn \jn j In this last expression, Ki t LI are the values of K, L when TO is put for T ; MI, Qi are functions of #, # 2 > #3> T o to be determined; NQ is the value of N when T O , 1%) , WSG are put for T, ^2, ^3* According to the general theory, the relations connectingTthe new and old variables are _as = _d_s_ \ dx> XQ " X aV ...(2) 3 Cf Ci Q ^ ' . __ == __^_ ' 9 S and the equations for the new variables will still be canonical provided we add d$/dt to the Hamiltonian function. From (1), and SS/dt = 2 (^i + #ii) cos A 7 + ^2 (Mi cos A r o - (?i sin N ). . . .(3) In performing fche transformation to the new variables, we shall neglect terms factored by w 2 . We recall that x has the factor m* while 8 has the factor m. Hence, to the order m$, from (2), ) , ^ 3 = *o 3 , ...(4) where SQ is the value of 8 when # , ^2o> ^30 are substituted for #> ^2> #3 therein. Next consider the portion of R independent of t explicitly, denoting it by R c . Since the new variables differ from the old by terms having the factor m y we can put ~z cO i "O o oa /JL /it da 7,8] CHANGE OF VARIABLE 261 where terms factored by m 2 are neglected. (Incidentally, it may be noticed that the terms of order m? are all of short period, so that even if we were calculating the long period terms with terms of order m 2 retained in the Hamil toman function, these could still be neglected.) Similarly, we can replace E, t by R^o, and dS/dt by dS /dt. If then in (4) we put for S Q its value deduced from (1), we obtain for the new Hamiltonian function expressed in terms of the new variables, f * a - 2* 3 + -' R.,0 + 2/r. 8 ^ /JL /A da - x Q 2 (Mo cos N Q - Q Q sin JV ), . . .(5) where, in all cases, the suffix zero denotes that the old variables are replaced by the new in the corresponding functions. The terms arising from R to have been cancelled by the same terms present in dSo/dt, and terms factored by xgm have been omitted. Finally, if we determine the coefficients MI, Qi by the relations ZKrfN Q 3 a^ ^ 1 = jV9V ^ = ~JV8^' ............ (6) relations which still hold to the required order when the suffix zero is inserted, the terms of order m% will disappear from (5) and it will be reduced to its first line. The remaining portion of this chapter will be devoted to the determination of the new variables in terms of the time. After this work is done, the old variables will be obtained in terms of the time with sufficient accuracy if we substitute #o> #20, #30 for x, #2> #3 i n S and its derivatives in equations (2). 9*8. The expansion of R in powers of e, e', F. We have, as in 41 (2) and 110 (2), cos 8 = (1 - JP) cos (v - v') + |F cos (v + v'~ 20). If we put sCt, -<;')> B&SPT 17 262 THE TEOJAN GROUP OF ASTEROIDS [CH. ix the expansion of R given by 9'4 (4) in powers of the second term of cos S as far as the order F 2 can be put in the form ' cos 2 ( + '- 20)!, ...(1) 1 16 r' 3 "A^ where RI== ^ + 2 ^*~ 'r ~ 2?* (2) The long period terms defined in 9 7 will be calculated as far as the fourth order with respect to e, e' y F*, and the short period terms to the second order. For this purpose, put a(l+/3), ~ (4) so that RQ, A are the values of Z?i, AI when the eccentricities vanish. Taylor's theorem then gives a'Ri_a'lt a u'p ?R a'dlto 1 a'p* 2 S 2 .Ro r' ~ ~7~ + 7- r " 8 + ? ? IT + 2~7~ a 3 8 + -' ...... (5) which is to be continued to the fourth powers of p, f for the long period terms. For the calculation of the coefficients of the derivatives of R Q in (5), we have, from 316, = 1 + (e - |e 3 ) cos# + e 2 cos 2 .& = 2 (e - ^e 8 ) sin^r + |e 2 sin 2#, with similar expressions for a'/r', E'. Since gr = Wi + isr, gr' == wi 4- ^r', the long period terms will be those whose argu- 8, 9] EXPANSION OF B 263 ments are multiples of g g'. The expressions (6) are sufficient to obtain such terms to the fourth order, in spite of the omission of terms of the forms e 4 cos 2g, e 3 cos 3g, e* cos 4*g in a/r, and of similar forms in E, a'/r', E'. For a term with argument 2g must be combined with one with arguments 2g, g + g', or 2g' to give terms of the required form and these have coefficients of the second order, so that the combination is of the sixth order. Similarly for the other terms omitted. Finally, RQ and its derivatives have no short period terms, so that it is sufficient to omit such terms in the expansions of powers of p, The advantage of this mode of development is seen by a reference to the results given in 9*9. Four of the coefficients are zero, three others are the same except for the numerical factors, jwo others have the same property, and two more differ only in ;he fourth order parts. 9*9. The coefficients of the derivatives of JK * To obtain these expressions put a/r = 1 4- u\ , a'/r' = 1 4- MI, so that of a of , p -j = --- 7 = Ui Ui r r ' r T ' has no long period term. The functions needed have the form where i -f j 4. The calculations appear to be most easily carried out by expressing each such product as a sum of terms, each of the form PQ', where P is a function of ui, E, and Q' of ui, E',r'\ from these products the terms independent of g, g' ', and those with arguments g g' , 2g 2g', are easily selected. The positive and negative powers of r, r' which are needed and the positive powers of E, E 1 can be read off from Cayley's tables*. Use can be made of the fact that pa'/r', E E' both change sign when e, g are interchanged with e' y g', so that terms of the forms ee' sin (g g'), e 2 e' 2 sin 2 (g g') cannot be present in * Mem. Roy. Astr. Soc. vol. 29, pp. 191-306. 264 THE TROJAN GROUP OP ASTEROIDS [OH. ix products of p/r f , p 3 /r' 3 with f, f 8 (in the fourth order terms the divisor r' takes the value a'). The remaining terms have the form (eV e' 9 e) sin (g #'), and these disappear on account of the relation between the coefficients of cos g, sin g in a/r, E, respectively. The following results for the terms independent of g, g' and for those dependent on g g' and its multiples, to the fourth order with respect to e, e', have been obtained : - (2ee' + f e a e' - f ee' 8 ) cos (g - #') - f eV 2 cos 2 (g - g'), - (Zee' - feV - ee' 8 ) cos (g - g') - ^eV 2 cos Z(g- g'), - (3e 3 e' + ee' 3 ) cos (0 - 0') + eV 2 cos 2(g-g'), 4 ^ a = ( e 4 _ e '4) _ ( 2e 3 e ' _ 2ee '3) cos (^ _ g ') t = 3 = (e3e ' + ee/3) sin (sf ~ ^' } ~ e2e ' a sin == 3 r r 6 r - (4e 8 e' 4- 4ee' 8 ) cos (g - ^r') + 2eV 2 cos These are ready for substitution in 9*8 (5) which is the develop- ment of the first term of d'Efn in 9'8 (1). To the second order, we have . ...(1) 9, 10] EXPANSION OF R 265 The calculation of the short period terms to the second order presents no difficulty. For the portion m (of IT') jRi, we obtain afpjr' = ecosg e' cos g' + e 2 cos 2g e' 2 cos 2#', a' f/r' = 2e sin g 2e' sin #' -f f e 2 sin 2$r - ee' sin (# + g') - e' 2 sin 2#', r' = - a'^/r' = e 2 cos 2g - 2ee' cos (gr -f g') + e' 2 cos 20', r' = e 2 sin 2g - 2ee' sin ( + ') + e' 2 sin 2'. Up to this point, no use has been made of the fact that a' /a is near unity, so that the development just given is quite general, at least as far as the second order. In order to deduce the results of 4'32, the expansions of the derivatives of R Q in terms of the coefficients A t and cosines of multiples of Wi w{ are to be substituted. In making the comparison, the difference in the definitions of the symbol a should be remembered. However, the expansion to the second order is not difficult whatever the method used ; it is in the calculation of the terms of higher orders that the expressions become long and complicated, so that for them the method should be suited to the problem. 9'10. Calculation of the derivatives of R Q . These derivatives can all be reduced to the calculation of derivatives with a = 1. For, according to the definition in 9*5, and \ oa a==1 since it has been pointed out (9*7) that the first power of cc is sufficient in the expansion of R. The last result still holds if we substitute for R Q any one of its derivatives. Since we can neglect the fourth order terras in the coefficient of #, this coefficient will not need derivatives of R Q beyond the third and the latter are already required in the calculation of the term independent of x. For the calculation of the derivatives put so that by 9'8 (3), (4), -o a 1 q - 266 THE TROJAN GROUP OF ASTEROIDS [CH. ix Whence, when a = ] , Q 2 = 2 (1 - q), T> 1 1 3 dRo 1 1 1 ___ - g 8 4 Q ' da 3 ~ 2 g 8 8 Q 1_1_161 9a 4 ~Q 6 2Q 3 16Q 9 1 31 - 9T 2 _ ,3 ~ U ' 9'11. TAe additional portions of a'Rj/ji depending on F. The second term of 9'8 (1) has the factor T and the long period terms which it produces are of the fourth order at least, since the argument v + v' 20 must be combined with the arguments 2g, g + g', or 2g' to produce multiples of Wi - Wi, and these terms have the respective factors e 2 , ee', e' 2 . 10, 11] EXPANSION OF It 267 In order to expand it, write v + v f 20 in the form v - v' 4- 2t/ - 26. It is then easy to prove that '- 20) where, as before, a'R\fr' is expressed as a function of r, #, #'. But, with the help of 9'8 (6), cos (2v' - 2(9) = cos (2wi' - 2(9) cos 2" - sin (2w/ - 2(9) sin 2" = cos (2 Wl ' - 2(9) . (1 - 4e' 2 + 4e' 2 cos 20') - sin (2wi' - 2(9) . (4e' sin #' + f e /2 sin 2^'), and the only portions of this which will give long period terms as far as the fourth order including those with factor F are cos(2^i'-2(9)- 2e / cos(2M; l / -0 / - 20) + f e' 2 cos(2w/ - 2#' - 2(9). ...... (1) For sin (2v f 2^), these cosines are changed to sines. The first of the three terms of (1) gives long period terms by combination with the short period terms of a'Ri/r' having arguments 2#, g + g' ', 2g', the second with those having arguments g, g', and the last with RQ. The derivatives with respect to F, T present no difficulties, since they can be formed directly from the results in 9*10. For the short period part which is taken to the second order only, we put v = Wi,v' w\ , r = r' = a', A x = A = Q. The third term of a'Rjp, in 9*8 (1), having the fourth order factor F a , gives the single term 3F 2 /16Q 5 . The portions due to the factor 1 JF have been retained throughout on account of the large numerical multipliers which accompany them. As in Chap. IV, their presence causes but little additional calculation, and adds considerably to the numerical convergence for large values of F. But it is easy if desirable to 268 THE TROJAN GROUP OF ASTEROIDS [OH. ix expand the derivatives of R given in 9*10 in powers of F. For, when a = 1, F = 0, 3Q __ 1 cos r d 2 Q __ I cos 2 r ~ ~ ,, , s\ ~ r COS T , -no COS 2 T ,~. so that (^--Hr*-..., ......... (2) where Q 2 = 2 - 2 cos r. 9*12. Transformation to the canonical elements. The element # 2 is given (cf. 9*5 (4) and 513 (1)) by * 2 =(l+tf)f(l-* 2 )*-l}, ............... (1) from which e can be expressed in terms of ( x$ and x. But the canonical elements # 2 , w 2 will be replaced later by |? 2 , #2> where j9 2 = 61 sin w 2 (ci/c/)*, # 2 = ^i cos w; 2 (ci/Ci')i, ...... (2) These correspond to the elements in 5'14. As w 2 = -cr, and as we neglect powers of x beyond the first, these give esin tzr = a (l i^ 2 #). ^ costxr = which, with the relation g g' = r + tv' tsr, permit us to express a'Rijr' in terms of these canonical elements with but little additional calculation. For the terms containing F, 0, the substitutions ps = (2F)i (1 - \# -f J0) sin 0, g 3 = (2F)* (1 - \# -h |a;) cos 0, ...... (3) can be further abbreviated by putting unity for 1 except in the factor 1 ^F, where we put with 6 a = ^2 2 + 9 r 2 2 - These changes are not necessary in the solution of the equation for T, provided, in forming xdR/da we remember that a is present 11-13] EQUATION FOR THE LIBRATION 269 in R through e, F when we transform to the canonical elements. The coefficient of x in the expansion given above can be obtained with sufficient accuracy from / 3 i 9 ^ 9 \ a'R -2a 5 ---!e 5 --r= , V da * de oTJ JJL and in this expression, R can be limited to the terms of the second order. 913. The equation for T. It has been shown in 97 that, by a suitable change of variables, the short period terms can be eliminated from the disturbing function, and that the equations for the new variables, which are distinguished by the suffix zero, have the same form as those for the old variables if we simply omit from the latter the short period terms. The suffix is unnecessary during the solution of the equations for the new variables and will be omitted. The Hamiltoniau function, namely, Q /->< ^ Tt . C\ ^ V\> C An& *Iff& i VI \ 'in* _ ft-iX/ t\Aj ~\ JLVg f ^O/ "7T , /& IJL oa will be denoted by f ,-E 2 2#3 -f (7 + 2# F, to order m*. According to 97, with its reference to 9'5, <7, Fare independent of x, t. The equations for #, r are then dx dU , 3F , s /1X r^- = - 4-2^ -^ , to order wi* (1) 71 Cl6 OT OT ~ r =. 3# + 6^ 2F, to order m (2) n eft v 7 Differentiate the last equation with respect to t. Since the derivatives of oc, x 2 , X B> w%, w& have the factor m and that of r has the factor m^, we obtain ;7T,- 2 ^ tt> to order mi... (3) 270 THE TEOJAN GROUP OF ASTEROIDS [OH. ix On substituting for dx/dt, dr/dt from (1), (2), we see that the portions depending on V disappear and that the equation becomes 0/ , . . ss - 8 < 1 - 4a? ) , to order m* .......... (4) The disappearance of V from the equation for r, the solution of which is the principal part of the problem, is fortunate because it enables us to find T with high accuracy without transforming the disturbing function to canonical variables. It is true that V is needed in the determination of x, but this is a comparatively simple problem when T has been obtained in terms of the time. Since x = ^ -, -y- , to order m^ t o n at the transference of the factor 1 4# to the opposite side of (4), where it becomes a factor 1 -f- 4#, gives the somewhat more con- venient form of (4): 1 d 2 r 2 d (I dr\* dU ^ , 3 "2 Ti2 ~ o -TIT 1 -/ T^ I = - 3 ^~ , to order m*. ?i 2 dt 2 3 n r/^ \?i di/ dr Finally, if we put ^ = ^ -f- ^ (r -f- const.), it is easily seen that the equation reduces to -S-2 + 3 ?T =. to order m*. (5) /Yl'^x-/-^ A x-1 ' ' ^ ' which is the fundamental equation for the determination of the libration. 914. First approximation to r. We shall obtain this approximation on the assumption that #2> #3> W2> Wa are constants: it will appear later that their variable parts are divisible by m%. With this assumption, U becomes a function of a single variable r and the equation 9*13 (5) admits the integral const. = C- 6U, 13, 14] CALCULATION OF THE LIBRATION 271 so that n' fc -f const. - J (0 - 6 U)~*dr gives the solution. A reversion will give r in terms of fo. From a theoretical point of view, this is sufficient to determine r, but the process is inconvenient for calculation, and it is better to find r as follows. Since we know that r oscillates about one of the values 60, let us put T = 60 + ST, and expand the second term of 9*13 (5) in powers of ST. If J7- be the ith derivative of U with respect to r with r = 60 inserted, the equation becomes _ ...... (i) When e e' F = 0, the formulae of 9*3 give In this case, when powers of Sr beyond the first are neglected, (1) reduces to the solution of which is r = bcos(vn'ti + i>o) = bcos <f>, v* = 2?-m, ...... (2) where 6, VQ are arbitrary constants. The result agrees with that obtained in 9*10. In the further approximations, we are assuming that 6 is a parameter which is small enough for expansions in powers of b to be possible. The largest known value for b is that in the case of Hector for which it is near *3. As z> = *079, we have Ji/& = *008 in this case. The statement in 9*5 that x is less than '01 in all known (see footnote to 9*5) cases is thus justified. In general, when constant values of e, e', F are used, we have, for the first approximation, = & cos </>, i> 2 =3tf 2 ....... (2') 272 THE TROJAN GROUP OF ASTEROIDS [OH. ix For the next approximation, the value (2') is substituted for Sr in the term f ^(Sr) 2 of (1) and the term 3Ci is included. The equation becomes 1 _7X T = -3/i-f t/3& 2 -7 3 & 2 cos2<. ...(3) The addition to 8r is the particular integral corresponding to the right-hand member. It is TT 1 7T t TT O JL 7y A TT A TT- A, _ "1 ^ ' For the next approximation, we substitute the sum of (2), (4) in the term f f/3(8r) 2 , and (2) in the term Jt/VSr) 3 . The ad- ditional terms on the right of (3) are - 3C/3 b cos $ ( - -J - ^ yj? b 2 + Y 2 7j or - The particular integral corresponding to the term with argument 30 is A * fr 3 ^ 2 3(/ 2 '9-l 8Vi/2 In the term with argument </> we can put b cos </> = Sr, since it has 6 3 as a factor. On combining it with the first approxima- tion we have This shows that instead of the value 3C/2 for v 2 , previously used, we must put in order that ST may still remain periodic. 14, 15] FURTHER APPROXIMATION 273 The process is continued as far as may be necessary. Each alternate approximation requires an addition to the value of v. The process of approximation followed above is that which is usual in the case of equations of the type (1). It is to be remembered that the value 3C/2 is merely an approximation to v 2 and that the latter must contain other terms if Sr is to remain periodic. The equation may be solved also by putting in (1) and equating to zero the coefficients of cost<. A series of equations of condition are obtained which have to be solved by continued approxi- mation on the basis that 6, b Q and 6 2 , Z> 3 , ... are quantities of the first, second, third, ... orders. The coefficient of cos (f> determines v, and b is an arbitrary constant. The arbitrary constants 6, V Q replace the usual arbitrary constants %, f which become fixed as soon as the triangular solution is adopted. 9*15. The equations for the remaining variables. The development of the disturbing function in 9 f 8 shows that the long period terms containing the argument 6 are of the fourth order. If we neglect terms of this order, the canonical equations give dx^fdt 0, or #3 = const. Further, since # 3 = T (1 - e 2 )* d/d' = T (1 - e 2 )* (1 + x\ we have, to the same degree of accuracy, F = ,r 3 = const. The equation for 6 is found with the aid of 9'12 (2) and is integrated after this expression has been developed as a Fourier series with argument </> with the solution given in 9*14. To the same degree of accuracy, 9*12 (1) shows that # 2 = i^ 2 . The variables p 2 > <?a, defined in 912(2), reduce to p 2 = esin'cr, g^ ecos tzr ................ (1) The limitation enables us to neglect the part of R which has a? as a factor. Hence, with the Hamiltonian function in 9*13, the equations for p t) g 2 become - __ n'dt~ 9c ' ri dt~ 8s ' with the briefer notation s for p% and c for q%. 274 THE TROJAN GROUP OF ASTEROIDS [CH. ix The development of a'Rj^ which contains c, s to the second order, is given by 9*9 (1). With g g' = T ta- -f TO-', the substi- tution (1), and the notation this development may be written T {- se' cos (r + r') + ce' sin (T + r')} + JP (s 2 + c 2 -f e' 2 - 2e' s sin (T + -sr') - 2e'c cos (T+ r')}. The equations for s, c are therefore 1 d* -, ^ = Te' sin (T + w') - Pe' cos (T + w') + PC, n af 1 r/r- -, ^ = 2V cos (T + w') + Pe' sin (T + w') - Ps. ?l C6t The only variable present in T 7 , P is T and we suppose that T has been expressed as a function of t by means of the solution obtained in 914. Further, as T = 60 + Sr, it is supposed that any of the functions of r present can be expanded into Fourier series with argument </>. 9*16. Solution of the equations for s, c. As is usual with linear equations, we first find the comple- mentary function, which is the solution of 0, ............ (1) ^ ' -- , ~ n dt n dt where P = P + PI cos <j> -f P 2 cos 2< + . . with </> = vn't -\-VQ\P has the factor m. It is at once seen that the solution is s = sin (-CT! 4- o)> c = ^ cos (*TI -f where d^ildt = P, so that p '' (2) where eo> WD are arbitrary constants. We thus have (P- \ P^n't -f TQ + S ~^ cos *0 ) > lv ' , p. \ C = Q COS ( PfiU't + Wo + S ~ COS !</> J . 15, 16] EFFECT OF JOVIAN ECCENTRICITY 275 In the usual language of celestial mechanics P n' is the mean motion of the perihelion. It appears then that this part of the motion gives a constant ' eccentricity ' and a variable motion to the perihelion. Since the coefficients Pi/iv contain the factor bm^, the solution to this order can be expressed in the form s = Q sin (P^nt -f- WQ) p + 2 ~ {sin (P Q n't + ^ + i<f>) - sin (P Q n't + CT O - i<f>)}, tv with a similar form for c. The mean motion of the perihelion is divisible by m, the periodic part being divisible by 6i?A The particular integral corresponding to the terms factored by e' in 9*15(3) is required. These portions are functions of r only and can therefore be expressed by Fourier series with argument <. Suppose the solution to be of the form s = e r 2 (Si cos {</> + s^ sin if) = e' (s c -f s,), ,.. c = e' S (Ci cos icf> + c^ sin i</>) = e' (c c + c,), so that the suffixes c, s denote expansions in cosines and sines of multiples of <j> respectively. These are to be substituted in 9'15(3). The coefficients of cosi</>, sini^> equated to zero will give the coefficients s it /, Ci, c/. In a first approximation we retain only the terms of lowest order with respect to m in r. According to 9'13, fa is then equal to t, and r = 60 -f 56* cos i<. The terms factored by e' in 9*15 (3) are then expressible by cosines of multiples of <. The only sines which will be present will be those arising from ds c /dt, dcc/dt f Sj, c 8 . We must therefore have '), -(5) Since dcj>/dt has the factor m*, the required conditions can be satisfied only if s c , c c = const. + cosines of multiples of <, factor m, s, c, = sines of multiples of </>, factor m*. 276 THE TROJAN GROUP OF ASTEROIDS [CH. ix If, then, Po be the constant term in the expansion of P as a Fourier series with argument <>, the equations (5) and (6), with the notation (4), give P Co = const, term in the expansion of P cos (T + r') - Tsiu (r + w'), PO SQ = const, term in the expansion of - P sin (r + vr')-T cos (r + vr') 9 to the order m. Since the coefficients c iy c/, Si, s/ have at least the factor mi, the terms PC,, Ps, in (5), (6) can be neglected in finding a first approximation to these coefficients. Hence s*, c* are given by f Sj = {_ p cos (r + or') + Tsin (r 4- tar')} dt, and they will have the factor m*. When these have been found, the first of (5), (6) give s c , c c , the variable parts of which have the factor m. The same plan is followed when terms factored by ra$ are retained in 9*15(3). The work is simplified by remembering that in these expansions the coefficients of cosines of multiples of have even powers of m% as factors, while those of sines have odd powers as factors. The latter also have & as a factor. It follows that the errors of c , S Q as determined in the approxi- mation just given, have the factor m, those of c it s t the factor m 2 , and those of c/, s t f the factor m*. When b = we have T = + 60, T = 0, and P is reduced to a constant. The particular integral is then s = 6 / sin(60 + w'), c = e'cos(60+'cr / ), ...(7) and the complete solution is obtained by adding these to the complementary function which is (8) 16, 17] HIGHER APPROXIMATIONS 277 Since the mean longitudes of the asteroid and Jupiter differ only by the constant 60, the terms (7) can be interpreted by the statement that Jupiter impresses its elliptic terms on the motion of the asteroid. This result might have been anticipated from the triangular solution given in 9 '2, for the mass of Jupiter is so large compared with that of the asteroid that in this solution its elliptic motion will be dominant, In the ordinary planetary theory, the terms (8) would be expanded in powers of t Thus, neglecting terms factored by m 2 , we have s = eo sin vr Q -f P^te^ cos 1^0, c = # cos W Q The constants (7) might therefore have been supposed to be in- cluded in the arbitraries sin TOO, e Q cos tsr , which are to be determined from observation. Thus although these constants are affected by an error of order m in the first approximation, most of the error will be absorbed in the determination of e , T O from observation. Owing to the need for further approxima- tion, however, they must be kept separated. The fact thab s, c differ from constants by terms having the factor w* at least, justifies the assumption made in 9*14 that e, CT may be treated as constants in finding a first approximation to r. 9'17. Higher approximations and final results. The equations for T, x have been taken to the order m% and solved with s, c, p 3 , qz constant. Since the variable parts of the latter have the factor 6m*, it follows that the errors of the equations for r, x have the factor 6m* and also a factor of the . order of the squares of the eccentricities and inclination. The error of r has therefore this latter factor and also the factor 6ra*. For the next approximation, the variable values of s, c, p Q , q$ are inserted in U. Since the resulting addition to T will be small, it will in general be sufficient to find it from the following equation, deduced from 9*13 (5), /I d* .A* .,. , , dU ~/2 j^s + v *\OT = additional terms m -% - ; \n * air ) or B&SPT iS 278 THE TROJAN GROUP OF ASTEROIDS [OH. ix the additional part of x can then be obtained. If further approximations to the values of the remaining variables are needed, they can be obtained in a similar manner. The final results give the values of T O , #o> #20, #30 > ^20* ^30> r of the variables which replace them, where the suffix zero which was dropped according to the statement in the first paragraph of 9'13 is now replaced. The values of the original variables are then found by substituting these results in 9*7 (2). Now the portions dependent on the derivatives of S in these latter equations all have the factor m, so that the effect of substituting variable for constant values of the variables with suffix zero will be very small, with one exception, that of r now called T O , and for the latter the portion independent of m will serve. 9'18. Numerical developments. A literal theory in which the expansions are made in powers of b, e , r , e' y m* can be formed which will give a close approximation without an excessive amount of calculation. Even with so large a value of b as *3, the series for r converges rapidly owing to the numerical divisors which the integration produces. In this respect the theory of the Trojan group differs from the ordinary planetary theory where expansions in powers of a con- verge so slowly that they are useless for numerical calculation. The work can, however, he greatly simplified when the numerical values of the parameters are known. The chief part of r is a Fourier series with argument <f> and most of the further calculations consist in the calculation of various functions of cos r, sin r. The functions are rapidly calculated if harmonic analysis be used ; five, or at most seven, special values of < will be sufficient. Analyses of the special values of the functions are needed only when they have to be integrated or differentiated. Harmonic analysis can also be used conveniently to complete the solution of 9'13 (5) when an approximate value of r has been obtained by the method of 9*14. Suppose that such an approximate value is r = 60 + 2^008 ifa d(t>ldt v, (1) and lot the required correction be Sr= SbbiCoaiQ-tfoZbii sin fy, 66 1 = 0, (2) where we neglect squares of the correction to v. As the arbitrary constant can be left unchanged, we can put 56 1 = 66 = 0. With the value (1) of r, the function dU/dr is computed by harmonic analysis and compared with dt*. Let the sum of these be denoted by c 4- *i cos <f> + eg cos 2 < 4- . . . . 17-19] CONSTANTS OF THE ORBIT 279 Then a further approximation will be obtained by solving the equation in which 3 2 U/dr 2 is computed like 3 U/dr. On substitution of the expression (2) for 5r, it will be found that the coefficient of t disappears and that the values of 8b t , dv can bo obtained by equating to zero the coefficients of cos 1$. The process can be repeated if necessary. Since the principal part of 33 2 (7/d r 2 is i> 2 , the principal part of Sr is found at once. 9'19. Determination of the constants from observation. The nature of the orbit is usually set forth by giving the values of the osculating elements at some given date : these are found from the observa- tions by methods which are outside the scope of this volume. A procedure for finding the values of the constants used in the theory from these elements is contained in the following plan. An approximation to the short period terms can be obtained by substitut- ing for the elements with suffix zero in the terms arising from S, their osculating values. The same procedure is followed with the terms due to the action of Saturn determined below. We then obtain an approximation to the elements with suffix zero by the use of tho equations which connect them with the actual osculating elements. By comparison of the elements with suffix zero with the literal series for r, x in powers of 6, values of 6, v are obtained. With these the short period terms, particularly those dependent on the angle $, can be calculated again and the same procedure repeated. At this stage the values of the constants attached to the remaining elements can be found with high accuracy. If necessary, the whole procedure may be repeated, but it will rarely be necessary to do so except perhaps for the constants 6, qf> , which are sensitive to small changes in the elements. The process does not differ essentially from that which would be followed in the ordinary planetary theory if the methods of Chap, vi be used to determine the perturbations. In the latter, however, the elements with suffix zero contain the long period and secular terms only, and if desired we can treat these like the short period terms, using the observed osculating elements to find a first approximation to n , c , etc. Thus while the methods of Chap, vi have certain disadvantages which have been pointed out in 6*25, the custom of defining an orbit by giving the values of its osculating elements at a given date, makes the determination of the constants of the orbit from these values a simple problem. In the case of the Trojan group, the following modification gives the constants more rapidly. The values of the osculating elements are found at several dates by carrying them forward or backward by the method of 18-2 280 THE TROJAN GROUP OF ASTEROIDS [OH. ix special perturbations. Since all the short period terms have periods ap- proximating to that of revolution round the sun or sub-multiples of this period, a mean value of an element with suffix zero can be found by analysing its values at the various dates into a Fourier series with argument 2rr/w-' and choosing the constant term as the first approximation to its value at a mean date. PERTURBATIONS BY SATURN 9*20. The calculation of the perturbations of an asteroid of the Trojan group by Saturn is difficult because the procedure of the ordinary planetary theory cannot be followed. This procedure consists in finding the perturbations due to each planet separately, then those due to their combined actions, and adding the results. Here, in finding the principal perturbations produced by Saturn, we cannot; neglect the action of Jupiter, even in a first approxi- mation. Thus the problem is one of four bodies rather than of three, and in this respect it is similar to that of the action of the planets on the moon. The disturbing function for the direct attraction of Saturn is that given by 1*10 (1). An indirect effect is also produced by the action of Saturn in causing Jupiter to deviate from elliptic motion, so that, in the disturbing function due to the direct action of Jupiter, it is necessary to add to the elliptic elements of that planet the perturbations caused by Saturn. An indirect effect of the action of Jupiter is also present in the perturbations this planet produces on the motion of Saturn. It is assumed that the mutual perturbations of Jupiter and Saturn on one another are completely known. The largest term in the action of Saturn on Jupiter has a coefficient in the longitude of Jupiter of nearly 1200" and a period of some 870 years, this long period being due to the fact that the period of revolution of Saturn is nearly 2 times that of Jupiter. Since the period of revolution of the asteroid is the same as that of Jupiter, we might expect to find a term of similar magnitude in the motion of the asteroid. It will be shown, however, that the action of Jupiter fundamentally alters the direct effect of the action of Saturn, and that the indirect effect produced by Saturn is the largest part of the action of that planet. 19-21] PERTURBATIONS BY SATURN 281 It will be assumed that these effects are, in general, small compared with those which we have been considering in the first part of this chapter, and that, in developing the disturbing functions due to Saturn, we can put for the coordinates of the asteroid their values in terms of the time. These additional portions can then be separated into long and short period terms. The latter may be eliminated as before by a change of variables which will give additional portions to the function 8. We have then to consider the effect of small additive terms in the function U on the variables with suffix zero; as before this suffix will be dropped until the final results for the long period terms have been obtained. 9*21. The equation for Sr and its solution. The equation for r given in 9 '18, namely, ' is still true when we add to U the portions due to the actions of other planets, provided the conditions laid down remain satisfied. These conditions demanded that the principal part of x should be given by oo^ ^drfn'dti, and that the terms present in x should be large compared with the corresponding terms in U. With the conditions laid down in the last paragraph of 9*20, the inclusion of the action of Saturn will require the solution of equations of the form /n . (2) where t/"has the meaning previously given and A,p,p Q are known constants. To simplify the exposition, we shall put ti t', this amounts to the neglect of terms of order higher than those retained. Let T = TO be the solution of (1) and r = T O -f Sr that of (2). If squares and higher powers of Sr be neglected, we have where in U and its derivatives the value r = T O is inserted. 282 THE TROJAN GROUP OF ASTEROIDS [CH. ix The solution of (1) gave T O as a function of t and of two arbitrary constants 6, v$\ if this solution be substituted in (1) the constants b y VQ disappear identically. We can therefore differentiate (1) with respect to b and VQ and obtain . *.. _ n'* dt* \9fc / 9T 2 ' db ~ ' n'* dt* \fa) + drf ' fa ~ ...... (4) Hence Sr=|? g T = ^<> .................. (5 ) ob di/o are particular solutions of (3) when A = 0. It follows from a well-known theorem that a particular solution of (3), corresponding to A =f 0, is (6) vy i/J'O J Ul/ where Equation (6) may be tested by substitution in (3), and equation (7) by eliminating d 2 U/dT<? from (4). Since T O has been obtained as a Fourier series with argument <j> = vn't+ z^o, the derivatives of T O with respect to i/ , b will still be Fourier series provided v be independent of b. Actually, v is a function of b and dr/db contributes a non-periodic portion n't(dv/db)(dT/dvo)', it is easily seen, however, that this non- periodic part disappears from (6). The principal term in T O T 60 is 6cos(im' + j>o)> where z> 2 = 27m/4. With this value of T O we obtain a result which might be deduced directly from (3) since in this case 33 2 7/3r 2 = v 2 . With the complete value of r, the divisor v 2 ~p 2 will be replaced by divisors of the form i*v*-p 2 . 21, 22] INDIRECT EFFECT OF SATURN 283 9*22. Indirect perturbations. These arise through the substitution for the elements of Jupiter their disturbed instead of their elliptic values. We shall suppose that the perturbations of the plane of Jupiter's orbit can be neglected so that we can still use it as a fixed plane of reference. The perturbations to be considered are therefore those of a', e', Wi = w', wz = w'. These perturbations, chiefly due to Saturn, have the mass of Saturn as a factor and are substituted in an expression which has the mass of Jupiter as a factor. The resulting short period terms will be very small and, in any case, they are supposed to have been eliminated by the change of variables. It is found that the only terms likely to produce sensible effects are those of long period in the motion of Jupiter, producing terms of long period in the motion of the asteroid. For their calculation, we return to the point in the original canonical equations where the transformation r = w w' was made (9*5). This transformation still holds if w' is any function of t independent of the elements of the asteroid provided that the expression Cidw'/dt be added to the Hamiltonian function. As before, w' is then no longer present explicitly in the dis- turbing function. If we denote by 8a f , Se' y SOT' the perturbations of a', e', -or', the equations for Ci, T become dd dR d*R , . a 2 /? , , d*R . , /1X -rr =^- + n ^T/S +;Ta-/Be +3-5- -/ or , .................. (1) at dr drda drde dr ffjf , d , A dR -T7= r 3 -tto ~T* W } "5 -- dt \Ci 3 dt J dci PR , , PR , , d*R a a' > ~^i"a ' -" -> , dcida ocide BciPtsr ...... (2) where n$ is the constant term in dw'/dt. We have seen in Chap, vi that the variations Sa', 80', STST' contain the first powers only of the small divisor which is present in the case of a long period term, and these variations, when multiplied by the second derivatives of R, will give very small terms which can be neglected, at least in a first approximation. 284 THE TROJAN GROUP OP ASTEROIDS [OH. ix If the procedure followed in 9*6 and 911 be then adopted, it is easily seen to lead to the equation an equation which replaces 9*13 (5). To find the principal term we put ti=t. Suppose Sw' = B sin (n'pt + p Q ), and that we substitute this in (3). According to 9*21 (8) the principal part of Sr is given by Bp 2 8r = -%-_ 2 sin (n'pt 4- p Q ). But Sw = Sr -f w'. Hence J ft (4) The effect of a perturbation in w' therefore depends on the relative magnitudes of v, p. If v is large compared with p, (4) gives approximately Sw = B sin (n'pt + p Q ) = $w'. The result Sw = $w' applies also to a secular term since such a term can be expressed as one of very long period. Hence the general proposition: If the period of a perturbation of Jupiter by Saturn or by any other planet is long compared with the period of libration of the asteroid (about 150 years), the perturbation of the longitude of Jupiter is directly impressed on the longitude of the asteroid. The principal perturbation of Jupiter by Saturn has a period of some 870 years, so that the indirect perturbation of the asteroid differs from the direct perturbation in the motion of Jupiter by less than three per cent., although this indirect effect is one of the second order with respect to the masses while the direct effect on Jupiter is one of the first order. Terms in which p is large compared with v have been treated as short period effects and therefore do not enter into the dis- cussion. 22, 23] INDIRECT EFFECT OF SATURN 285 Terms in which p 2 is nearly equal to v* would give rise to much larger perturbations in the motion of the asteroid than those present in the motion of Jupiter. There are no such terms having sensible coefficients in the perturbations produced by Saturn, Neptune, which has a period of 164 years, comes nearest to pro- ducing such a term. 9*23. To obtain the perturbations of #, the equation 9*13 (1) is used. When pjv is small, Sr is small compared with Sw' and therefore (d/dt) ST with (d/dt) Sw'. The latter is, however, large compared with dR/dc^ which has the factor ra and has no small divisor. Hence, since /^ 2 /Ci 3 = (1 + #)~ 3 n'> or, since S% = JSa/a', this gives (d/dt) Sw' = f 8n'. Hence 8a = S&'. The long period terms in the major axis of Jupiter are therefore impressed on that of the asteroid. For the perturbations of e, r, the equations 9'15 (3), which are still true if ef ', BT' are variable, are used. Let e', or' receive long period variations 80', SOT'. Then, as before, it may be shown that PC, Ps may be neglected. When the libration vanishes, the equations for 8s, 8c reduce to ^8s-P S{e'cos(60 4V)} f ^8c = P S {e'sin ( 60 + w% If 27r/n'p be the period of the variations of &e', S-sr', the corre- sponding coefficients in Ss, Sc are diminished in the ratio P /p which is usually small, so that &, 8c will be negligible. For the terms depending on <f> when b =j= 0, the divisor is approximately iv if p is small compared with v and the coefficients of such terms in 8s, 8c will be still smaller than those just treated. Thus the long period variations of e', tzr' are not impressed on the asteroid but produce effects which are much smaller than those in the motion of Jupiter. 286 THE TROJAN GROUP OF ASTEROIDS [CH. ix 9*24. The direct action of Saturn. The disturbing function for this action is given by 1'10(1), and it can be expanded in terms of the elements of the asteroid and of Saturn by one of the methods used in the ordinary planetary theory. The expansion, if made in a literal form, is available whether the elements be constant or variable. The disturbing function, denoted by R' y adds a term a'-R'/A 6 to the Hamiltonian function in 9*5 (5). The variable x which measures the deviation of Ci/Cj' from unity is still small, so that 9*5 (6) still holds when we add R f to R. Let the elimination of the short period terms, as made in 9*7, include those arising from Saturn. For simplicity, we shall retain the lowest power of x only, so that the additional terms in S are similar to those found in Chap. vi. The Hamiltonian function used in 9*13 therefore becomes f a? + -(R e + R,') = f x* + U+ U', P with the notation adopted there. The equations 9*13 (1), (2) then become l^'-^J 7 ?E1 I L^ T -__<} _' n' dt ~ dr + dr ' n' dt and equation 9*13 (3) reduces to 1 ^-_ql d_d_(dU n*d&~~ n'dt dt\dx On substituting for dx/dt, this gives JL rf2r - ( d JL + dU '\ _ 1 fiV'\ n' 2 dt 2 ~ \ dr ~dr) dt \dx ) * For a term of long period, the last term of this equation is small compared with dU /dr, and may be neglected in a first approxi- mation. The equation then reduces to J ^j.Q^-.q^ '" 8 rfi 8 + *ST~ dr The right-hand member of this equation has the mass of Saturn as a factor, and we can substitute for the elements of the 24] DIRECT EFFECT OF SATURN 287 asteroid the values obtained from the action of Jupiter alone. In accordance with the previous work, it will be sufficiently accurate to use the series for r and to take all the other elements constant. Let any term in the right-hand member be denoted by A sin(pn't-\-po). The method of 9*21 is now available for the solution of the equation. The principal part of the addition to r will be given by A - 2 - ^ v p The principal long period term due to the action of Saturn will have nearly the same period as it produces in the motion of Jupiter, namely, about 870 years, and therefore p is small com- pared with v. Hence the small divisor due to a long period term is much larger than in the ordinary planetary theory and the resulting effect much smaller. Thus Jupiter not only impresses on the asteroid its own inequalities of period long in comparison with that of the libration, but it also prevents the asteroid from having any very large terms of this nature arising from the action of another planet. The substitution of the series for r will also produce terms with arguments (p iv)n't + p iv Q . When i = l we shall have divisors v 2 (p p) 2 or 2pv approximately. Such terms will have the factor 6, and if b is large they may be sensible. The theory of the perturbations of Jupiter by Saturn given by Leverrier* can be utilised for calculating the perturbations of the asteroid by Saturn, since the numerical value of the ratio of the mean distances is the same, and since Leverrier gives the contribution of each separate power of the eccentricities and inclination, so that the change to those of the asteroid can be easily computed. But the convergence along powers of the inclina- tion is so slow when the inclination is large, as in the cases of certain members of the group, that the value of the coefficient of the principal long period term obtained in this way is doubtful. Another difficulty arises from the fact that the mean motion of the perihelion of the asteroid is comparable with that of the argument of this term so that it cannot be * Paris Obt. Mtm. vol. x. 288 THE TKOJAN GROUP OF ASTEROIDS [OH. nc neglected, even in a first approximation : see "Theory of the Trojan Group " referred to in 9 '25. 9 '25. The literature connected with the triangular solutions of the pro- blem of three bodies is extensive : much of it is concerned with the possible orbits which may be described under different conditions but which have no present applications in the solar system. Amongst the earlier develop- ments arriving at a more general theory for the asteroids of the Trojan group may be mentioned those of L. J. Linders (Stockholm Vet. Ak. Arkiv y Bd. 4, No. 20) and W. M. Smart (Mem. Roy. Astron. Soc. vol. 62, pp. 79- 112, 1918) ; the latter used the method adopted by Delaunay for the development of the lunar theory. In a paper by E. W. Brown (Mon. Not. Roy. Astron. Soc. vol. 71 (1911), pp. 438-454), the particular periodic solution which constitutes the principal part of the libration is shown to be a linkage between the orbits of planets outside and inside that of Jupiter and those of satellites of Jupiter, the passages between them going through the collinear solutions. The linkage bears some resemblance to that which joins the two sets of solutions of the equation for the motion of a pendulum. A literal development sufficiently complete to give the position of an asteroid of the group within a few seconds of arc has been made by E. W. Brown ("Theory of the Trojan Group of Asteroids," Trans, of Yale Obs. vol. 3, pp. 1-47, 87-133). The method of Chap, vn is used for the development of the action of Jupiter : the various problems which arise in finding this action are closely analogous to those set forth in this chapter. The problems and theorems connected with the action of Saturn are dealt with in detail. The theory was applied to the asteroid Achilles. This numerical application has been revised by D. Brouwer (Trans, of Yale Obs. vol. 6, pt. vn) who has added tables for finding its position at any time. W. J. Echert has applied the same theory to Hector (Ib. vol. 6, pt. vi), the libration of which runs up to over 20. A. APPENDIX ON NUMERICAL HARMONIC ANALYSIS A'l. Let F c (x) be a periodic function of x, period 2?r, which is expansible in the form F c (x) = CQ + Cicos x + C2cos 2# + . . . + c n cos nx + ....... (1) The problem under consideration is the rapid numerical cal- culation of the coefficients c^. It is supposed that F c (x) con- tains numerical constants only, and that after some term c n cos nx the remainder of the series may be neglected. Under the latter condition (1) may be regarded as an identity satisfied for all values of x. If then we choose n -f 1 numerical values of x and calculate the corresponding values of F c (x) and of the cosines of the multiples of #, we obtain Ti + 1 relations which may be regarded as n + 1 linear equations, giving the n + 1 unknowns c , Ci, ... , c n . The effectiveness of the method depends on the ease with which the special values of F e (as) can be computed and on that with which the linear equations may be solved. It is found that the questions principally to be considered are the choice of the special values of x and the best arrangement of the work for finding the c-. Similar remarks apply to the development of an odd func- tion of x in the form F 8 (x)-Sismx + s%sm2x+ ... -f s n smnx+ ..., ...... (2) except that n values of x only are needed since there is no con- stant term. If F(x) contains both even and odd functions of x and if we stop at the nth harmonic, 2n + 1 special values of x are needed In cases where the calculation of special values of F e (x) and of F 9 (x) is needed and where much of the work is the same in each case, the same special values of x should be used. 290 APPENDIX [APP. A'2. Choices of the special values of x. If F(x) contains both even and odd functions of x, they can be separated by choos- ing the values of x in pairs , 2?r . Since cos ia = cos i (2?r a), sin ia = sin i (2zr a), we have F(a) + F(%7r -a) = cosine series, F(a)F(27ra) = sine series. In future it will be supposed that this separation has been effected so that the forms (1) and (2) of A'l can be considered separately. Next, since F e (d) + F c (7r-a), F 8 (a)-F s (7r-a) contain even multiples of x only, while F c (a)-F c (7r-a), F 8 (a) + F B (ir-a) contain odd multiples only, the choice of pairs of supplementary values enables us to separate the equations giving the c t - or the Sj into two sets, one containing the coefficients with odd suffixes and the other those with even suffixes. Finally, choices of a which are multiples of TT/?I, where n is an integer, have obvious advantages. The special cases n = 3, 4 or 6 will suffice for most of the requirements in the plans for develop- ing the disturbing function and the disturbing forces outlined here. For the rare cases in which eight harmonics are needed, it is advisable to add the values # = 45, 135 to those given by n = 6, since the work done with six harmonics only can be fully utilised and repetition of it is not needed. Schedules for this last case are to be found in Trans, of Yale Obs. vol. 6, part 4, pp. 61-65. A'3. Determination of the nth harmonic in F s . Since F 8 vanishes for x = 0, TT, these values are useless for the computa- tion of the S{. Moreover, since sin nx vanishes when x is any multiple of TT/W, it is the coefficient of the rath harmonic which is undetermined with this set of values of x. It is sometimes possible to estimate its coefficient with sufficient accuracy; where this is not the case, one of the following devices may be adopted. A-2-A-4] HARMONIC ANALYSIS 291 For n = 3, the chosen values of a? are 0, 60, 120, 180. The value # = 90 may be added to find 53. If this value be used also in F C) we can find c 4 and then get higher accuracy for c 2 (see A*4). For n 4, 6 we may proceed as follows : In most cases the calculation of dF s jdx for x = 0, TT will be found to be easy. These give the values of % 2s 2 + 3s 3 4s 4 -f . . . which, combined with the relation furnished by the other values of x inserted in F 8t will give the value of s n . Only one of the two values is necessary, but the solution of the equations is simplified by using both : they give also the (n + l)th harmonic and higher accuracy to the (n l)th harmonic. A*4. Errors of the coefficients. When multiples of irfn are used for the special values of x, the solution of the linear equations for the cosine series actually gives, instead of c t -, i < n, the value of d + C 2n -i + Ctn+i + C 4n -i + , if we include all the terms of the series. Thus the principal part of the error of c t - is the rejected coefficient c 2n -i When i = n, we actually determine the value of so that the error of c n is c 3/l . Hence the lower the harmonic the higher the accuracy with which it is found, with the exception of the nth which has an error equal to the coefficient of the 3nth harmonic, approximately. In the case of F s we determine the value of Si S^ni 4" fyn+i &4ni "^ instead of Si, for i < /?, so that the principal part of the error of s f is $2n-i, an d tne same rule with respect to the errors of the coefficients of the lower harmonics holds. When the harmonic with coefficient s n (which vanishes when x is a multiple of TT/U) is found for n odd by using the additional value x = 90, it is easily seen that we actually determine 292 APPENDIX [APP. for without its use we find s n _ 2 - s n +2 + , and with its use we find s n _ 2 s n + s n + g . . . , so that the principal part of the error of $ n is 2s n+2 . If x 90 be also used in finding F c , we determine c n _! with an error c n +s and c n +i with an error c n +a, the errors of the other coefficients being unaffected. When n is even, the use of the additional values dF/dx with # = 0, TT gives us s n _i + s n+ s + . . . , s n + 2s n + 2 + . . . , s n +i + s n +3 4- . . . , so that the principal part of the error of s n _i is s n + 3 and that of s n is 2s n+2 , the remaining earlier harmonics being unaffected. Thus in the case of F 9 the nth harmonic has the greatest error exactly opposite to the case of F c where it had the smallest error. A'6. The schedules made out below are given in detail so that they may be used without preparation for harmonic analysis either for a single function or for many functions. In the latter case, the work should be carried out in parallel columns, any one step being the performance of the same operation in all the columns; since most of the operations are simple additions and subtractions, much time can be saved by this reduction to routine computation. In the cases where the coefficients are known to diminish with some degree of regularity along values of i the results themselves furnish a test of the accuracy of the work; in any case, one or two of the special values may be reproduced from the final results with but little labour. A'6. Notation for the schedules. The function to be analysed is denoted by F and the argument by x. For an even function, we put F = C + Ci COS X + 02 COS 2x + . . . , and for an odd function F = SI sin x 4- $2 sin 2x + .... For the special values of x y 0, 180; 90; 45, 135; 30, 150; 60, 120; A-4-A-6] HARMONIC ANALYSIS 2 the corresponding values of F will be denoted respectively by ET I7 ' . I7 I7 37 '. XT' JP * . IT* I?'. TO, ^0 > ^9; ^4, ^4 3 ^3> ^3 5 ^6, ^6 3 and, in the case of an odd function, The letters A,B,C>... used in the schedules are defined therein The first column of each schedule gives the symbols; the second column gives their numerical values in the examples the third, omitted in the computing forms, is explanatory. The error of any coefficient is shown in the final values. Thus c 3 (+ c 9 ] indicates that if CQ were known it would be subtracted in order to find Cs. Since the same function has been used in all the examples for the cosine analyses and its derivative in the sine analyses : direct comparison of the errors of the various results is possible. B&SPT 19 294 APPENDIX [APP. A'7. SCHEDULE FOR 3 HARMONICS. COSINE SERIES. Values of x = 0, 180, 60, 120. JP = (l-- FQ 63246 Addition for FQ 1-26491 extra value # = 90 FQ 83666 FQ 1-14018 \A -94869 A ~F Q +F ' 1-89737 2c + 2c 2 FQ I'OOOOO B =FQ+FQ' 1-97684 2(? -c 2 J4-/i = 2c 2 - '05131 n A B - -07947 C 2 - -02566 D =$C=ct - -02649 D-c 2 =c 4 - -00083 2cQ=B + J) 1-95035 C o( + c 6) '97518 A f ^FQ-FQ -63245 2^ + 2^ ^( + c 5 ) --31 199 B 1 =FQ~FQ - -30353 C!-2c 3 c 2 ( + c 6 ) - '02566 Sc^A' + B' - -93598 6' 3 ( + c 9 ) - '00423 c\ -31199 c 4 ( - <? 6 ) - -00083 Zc^c.-B' - -00846 c, ( + * 6 ) 97518 Ci(+<? 6 ) - -31199 C 2 (+C 4 ) - -02649 C3 (+c 9 ) - -00423 A'8. SCHEDULE FOR 3 HARMONICS. SINE SERIES. Values of x = 60, 120, 90. F = '3 sin x (1 - '6 cos a?)-*. F Q ' 31053 22787 30000 53840 31085 01085 08266 04772 31085 04772 01085 A-7-A-10] HARMONIC ANALYSIS 295 A'9. SCHEDULE FOR 4 HARMONICS. COSINE SERIES. Values of # = 0, 180, 90, 45, 135. .F = (l - -6cos#)*. ^0 63246 A' = Ft - F{ - -43466 V2 (^ - c 3 ) FQ 1-26491 B'^JZA' - -61470 2c!-2c 3 FI 75877 C' = FQ - Fd - -63245 2ci + 2c 3 Ft 1-19343 4e 3 =C'-B' - -01775 F Q 1-00000 4c 1 = C" + 5 / -1-24715 A ^FQ+FJ 1-89737 2c +2c 2 +2c 4 Co(+c 8 ) -97522 B -J4 94869 Cj( + c 7 ) --31 179 2c 2 =i-F -05131 C 2 ( + C 6 ) - '02566 C = B+F 9 1-94869 2c +2c 4 C 3 ( + C 5 ) - -00444 D =FI+FI 1-95220 2c -2o 4 c 4 (+c 12 ) - -00088 4c =<7+ D 3-90089 4ct = C-V - -00351 A10. SCHEDULE FOR 4 OR 5 HARMONICS, SINE SERIES. Values of x = 0, 180, 90, 45, 135. F = '3 sin x (1 - '6 cos ar)~*. ^ 47434 A'=F d -F d f -23717 4* 2 +85 4 ^V 23717 B' =%A' -11859 2* 2 + 45 4 ^9 30000 25 2 =/4-^ 4 ' -10182 ^4 27957 454=^'- 25 2 -01677 Fl 17775 * *i(-*7) -31169 A = Ft + FJ 45732 >J2(8i+8 3 -8 6 ) * 2 ( - *e) -05091 ^ =.4W2 'oJjijoi S\ ~\~ 3g SQ s 3 ( + 5 7 ) -01281 2! =^ + ^9 62337 5 4 ( - 25 6 ) -00419 (7 =5-^, 02337 25 3 -25 6 5 5 ( + * 7 ) -00113 D =F d +F d ' 71151 2,^ + 653 + 1055 2^9 60000 ^ =Z)-2/9 11151 853 + 855 # =i^ 02788 2*3 + 2*5 4*3 =6^+0 05125 4,-?5 -0-G 00451 19-2 APPENDIX [APP. All. SCHEDULE FOR 6 HARMONICS. COSINE SERIES. Values of = 0, 180, 90, 30, 150, 60, 120. jF(l--6cos0)*- F, A = B = C = =(7+1) 4c =^+2c 4 63246 1-26491 1-00000 69311 1-23272 83666 1-14018 1-89737 2-00000 1-92583 1-97684 3-89737 3*90267 4c -2c 4 - -00530 - '00177 3-90090 - -10263 - -05101 -15364 - -05121 - -00020 2 - 4c e 2c 3 - -63245 - -30352 - -53961 - -93597 -93463 -1-87060 - -00134 -31199 ^+5 - '00847 97523 - -31177 - -02561 - -00423 -00089 - -00022 -00005 A-11,A*12] HARMONIC ANALYSIS A12. SCHEDULE FOR 6 OR 7 HARMONICS. SINE SERIES. Values of x = 0, 180, 90, 30, 150, 60, 120. F= '3 sin x (1 - '6 cos )- F d 47434 A 1 =F 3 -F,' 09474 V3(* 2 +* 4 ) Fd 23717 B 1 =FQ-FQ 08266 V3(5 2 -5 4 ) F 9 30000 C' =24' W3 10940 2* a +2*4 F, 21642 D' =25'+V3 09545 2* 2 -^*4 Fj 12168 4s 4 =c'-jy 01395 FQ 31053 4s 2 =C' + D' 20485 FQ 22787 E 1 =F d -F d ' 23717 4*2+8*4+12*0 0' ='-4*2 03232 8*4+12 6 FZ+FS 33810 Si + 253 + 55-57 8*4 02790 3*3 ^FS+FS'-I ^ '03810 12*6 = 0' -8*4 00442 * 3 01270 4 , A =F^+FQ 53840 v/3(*!-* 5 + * 7 ) *l(-*ll) 31179 JV B =^ + V 31085 *!-* 5 + *7 2 ( - 5 lo) 05121 C =* 3 + ^o 31270 * 1 +* 6 -*7 a(-*) 01270 2! =B+C 62355 i(-*) 00349 D = C-B 00185 2$ 5 -2*7 s 5 ( + * 9 ) 00103 E =F d +F d ' 71151 2! + 6*3 + 10*5 +14 r * 6 ( + 2* R ) 00037 -E- 2*! 08796 6*3 + 10*5+14*7 r( + ^) 00010 H --6*3 -07620 -6*3 K = -5Z) - -00925 -10*5+10*7 24*7 =G+H+K 00251 2*7 00021 2* 6 =7) + 2* r 00206 298 APPENDIX [APP. J\'13. Double Harmonic Analysis. When a function of two angles can be expressed in the form F (a>, y) = 2 (A jtj > cosjx cos j'y + B jtf sin j# sin j'y ^, > + G *,f cos j >aj sin ?'y + A*,/ sin ^ cos /2/)> where J,j' = 0, 1, 2, ..., the numerical calculation of the co- efficients can be reduced to a double application of single harmonic analyses. First, the choice of pairs of values a, 2?r a separates the terms containing cosines of multiples' of x from those contain- ipg sines by addition and subtraction. A similar choice for y makes a similar separation in each case. Thus the analysis is reduced to that of each of the four groups, and values for x, y equal to or less than 180 are to be used. A*14. Consider the first group given by = 42J.4 jj' cos joe cos j'y. The special values of A XtV are arranged in a block, each line containing those corresponding to a special value of x, and each column those corresponding to a special value of y. The special values in each column are analysed by one of the schedules for cosine analysis and give series 2-4^ cos j#, in each of which y has a special value. The results are rearranged in a block in which all the numbers Aj t y corresponding to a given j are placed in a column, the successive columns thus containing the special values of the coefficients of cos Ox, cos x, cos 2x, . . . for the special values of y. The special values in each column are then analysed by one of the schedules for cosine analysis and give the coefficients AJJ>. The process for finding the BJJ is the same with the exception that the cosine analyses are replaced by sine analyses. For Cjj>, the first block of analyses is that for cosines, and the second that for sines. A-13-A-16] DQUBLE HARMONIC ANALYSIS ^299 For Djj' t the first block of analyses is that for sines, and the second that for cosines. The choices of special values of x, y are made on the same plans as those developed for single harmonic analysis. It is not necessary that^the same choices of values be adopted .for so as for y. A*15. The derivatives are to be used in the sine analyses instead of the zero values of the functions F. This is possible because the derivatives of the cos jx terms in F disappear from dF/dx when x = 0, TT ; and similarly those of the cosj'y terms from dF/dy when 2/ = 0, TT. A*16. In the method of development of the disturbing function and disturbing forces outlined in 4*15, the separation into the four sets of terms is made at the outset. Each function which vanishes for g 0, TT or for g' = 0, TT, is replaced by a derivative as shown above. When the development is made in terms of the angles f>fi (4*19), the special values of the first two sets are found together and must be separated by addition and subtraction: similarly for the third and fourth sets. An example in which the calculations are shown in detail will be found in the Tables for the Development of The Disturb- ing Function, by Brown and Brouwer*. * Cambridge University Press, 1933. INDEX (The numbers refer to pages.) Anomaly, true, mean, eccentric, 64 Approximations in terms of time first, 146 second, 156 in terms of true longitude elliptic, 176 first, 178 second, 198 principal part of second, 163, 203 in Trojan group theory first, 270 second, 277 Asteroids, Trojan, 250 resonance effect on, 245 Astronomical measurements, 5 Astronomical unit of mass, 7 Attraction Newtonian law of, 7 proportional to distance, 136 Bessel's functions, 55 Canonical differential equations, 117 Canonical equations of motion, 24 Canonical set of variables, 123 Canonical set, Delaunay's, 131 Canonical set, Poincare's, 132 Complete integral, 123 Constants of integration in true longitude theory, 213 in case of resonance, 232 in Trojan group theory, 279 Contact transformation, 118 d'Alembert series, defined, 66 discussed, 141 Declination, 5 Delaunay canonical variables, 131 Delaunay modified set, 132 Departure point, 23 Determining function , 118 Development, of disturbing function (see Contents, Chap. IV) properties of development, 95, 139, 142 in terms of true longitude, 178 numerical development, 98, 181, 278 for Trojan asteroid, 261 Differential equations canonical, 117 Jacobi's partial, 121 (see Equations of motion) Disturbing function, 10 for double system, 11 for satellite problem, 13 for Trojan asteroid, 256, 258 (see Development) Earth-Moon system, 11 Eccentric anomaly, 64 as independent variable, 30 elliptic expansions in terms of, 68 disturbing function in terms of, 99 Elements of ellipse, 16 (see Elliptic variables) Elliptic expansions in terms of eccentric anomaly, 68 in terms of true anomaly, 71 in terms of mean anomaly, 72 literal, to seventh order, 79 by harmonic analysis, 80 Elliptic motion, 62 fundamental relations, 64, 65 Jacobi's method, 127 true longitude theory, 176 Elliptic variables, 16 Delaunay's, 131 Delaunay's modified, 132 Poincare's, 132 non-canonical, 133-136 Encke and Newcomb equations, 26 Epoch, 64 Equations of motion rectangular coordinates, 8 planetary form, 9 satellite form, 11 polar coordinates, 22 canonical form, 24 using as independent variable the true longitude, 28, 174 the eccentric anomaly, 30 the true longitude of disturbing planet, 31 for Trojan asteroids, 256 Expansions Lagrange's theorem, 37 by symbolic operators, 45, 87 of functions of major axes, 102-112 (see Elliptic expansions and Con- tents, Chaps. II and IV) Force function, 8 Fourier expansions (see Contents, Chap. II) 302! INDEX Gravitation, Newtonian law of, 7 Harmonic analysis, ^289 * for developing disturbing function/ 98, 180 for elliptic expansions, 80 in Trojan group theory, 278 Hypergeometric series, 56 * Jacobi's method for elliptic motion, 127 Jacob! 's partial differential equation, 121 Jacobi's transformation theorem, 119 Jupiter's effect on Saturn, 205 on certain asteroids, 245 on Trojan asteroids, 284, 287 Kepler's equation first used, equation (16), 64 numerical solution of, 81 Kepler's laws, 66 third law discussed, 7 Lagrange's expansion theorem, 37 Latitude, 6 Latitude equation, 29, 193 Law of gravitation, Newtonian, 7 Laws, Kepler's, 7, 66 Libration, 230, 255 Long period terms denned, 153 second approximation to, 161, 202 of disturbing planet, 165 case of a single, 171 Longitude, 6 true, as independent variable, 28, 174 mean ,64 , Major axes, functions of, 102-112 stability of, 198, 202 Mass, determination of, fy astronomical unit of, 7 * Mean distance, 66 Mean motion, anomaly, longitude, 64 Measurements, astronomical, 5 Osculating ellipse, 16, 126 Osculating orbit, 125 Osculating plane, 15 Pendulum, motion of, 219 disturbed motion of, 221 Perturbations general and special, 2 of coordinates, 150 mutual, of Jupiter and Saturn, 205 Fertprbations (wntinued) approximate formulae for, 210 "' effect of resonance on, 226 of Trojan asteroids by Saturn, 280 Planetary form of equations, 9 Planetary problem*' 2 Poincar6 canonical variables, 132 Polar coordinate equations of motion, 22 Potential function, 8 , / Power series, numerical devices, 59, 60 Reference frames, 15 Bqsonance, defined, 4, 2J.6 (see Con- tents, Chap. VIII) Bight ascension, 5 Satellite form of equations, 11 Satellite problem, 2 Saturn's effect on Jupiter, 205 on Trojan asteroids, 284, 287 Secular terms, 148 second apprdximation to, 161, 167 effect of, on second approximation, 200 Short period terms elimination of, 143, 259 * effect of, on second approximation 159 Small divisors source of, 84 discussed, 154 in true longitude theory, 193 Solution of canonical equations, ,*138 (see Contents, Chap. VI) of true longitude equations, 185 (see Contents, Chap. VII) of equations of variation, 151, 197, 253 Time, method of measuring, 5 Transformation contact, 118 theorem, Jacobi's, 119 to elliptic elements, 149, 194 to time as independent variable, 207 Triangular solution, 250 Trojan group, 250 (see Contents, Chap. IX) True longitude as independent variable, 174 (see Contents, Chap. VII) Variation of arbitrary constants, 17, 125 equations of, 149, 194, 252 CAMBRIDGE : PRINTED BY WALTER LEWIS, M.A. AT THE UNIVERSITY PRESS