AN INTRODUCTORY TREATISE ON DYNAMICAL ASTRONOMY CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, MANAGER LONDON : FETTER LANE, B.C. 4 NEW YORK : G. P. PUTNAM'S SONS BOMBAY, CALCUTTA, MADRAS : MACMILLAN AND CO., LTD. TORONTO : J. M. DENT AND SONS, LTD. TOKYO : THE MARUZEN-KABUSHIKI-KAISHA All rights reserved AN INTRODUCTORY TREATISE ON DYNAMICAL ASTRONOMY BY H. C. PLUMMER, M.A. \ ANDREWS PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF DUBLIN AND ROYAL ASTRONOMER OF IRELAND I- PEEFACE JL HIS book is intended to provide an introduction to those parts of Astronomy which require dynamical treatment. To cover the whole of this wide sub- ject, even in a preliminary way, within the limits of a single volume of moderate size would be manifestly impossible. Thus the treatment of bodies of definite shape and of deformable bodies is entirely excluded, and hence no reference will be found to problems of geodesy or the many aspects of tidal theory, Already the study of stellar motions is bringing the methods of statistical mechanics into use for astronomical purposes, but this development is both too recent and too distinct in its subject-matter to find a place here. Nevertheless the book covers a wider range of subject than has been usual in works of the kind. Thereby two advantages may be gained. For the reader is spared the repetition of very much the same introductory matter which would be necessary if the different branches of the subject were taken up separately. But in the second place, and this is more important, he will see these branches in due relation to one another and will realize better that he is dealing not with several distinct problems but with different parts of what is essentially a single problem. In an introductory work it therefore seemed desirable to make the scope as wide as was compatible with a reason- able unity of method, the more so on account of the almost complete absence of similar works in the English language. The first six chapters are devoted to preliminary matters, chiefly connected with the undisturbed motion of two bodies. These are followed by five chapters VII to XI dealing with the determination of orbits. This section is intended to familiarize the reader with the properties of undisturbed motion by explaining in general terms the most important and interesting applica- tions. It is in no sense complete and is not intended to replace those works which are entirely devoted to this subject. Otherwise it would have been necessary to describe in detail such admirably effective methods as Professor Leuschner's and to include fully worked numerical examples. Here, as else- where, the aim has been to give such an account of principles as will be vi Preface instructive to the reader whose studies in this branch go no further, and at the same time one which will help the student to understand more easily the technical details to be met with in more special treatises. Though the actual details of practical computation are entirely excluded, the fact that all such methods end in numerical application has by no means been overlooked. A distinct effort has been made to leave no formulae in a shape unsuitable for translation into numbers. The student who feels the need will have no difficulty in finding forms of computation in other works. At the same time the reader who will take the trouble to work out such forms for himself will be rewarded with a much truer mastery of the subject, though he should not disdain what is to be learnt from the tradition of practical computers. An outline of the Planetary Theory is given in the seven chapters XII to XVIII. The first of these deals exclusively with the abstract dynamical principles which are subsequently employed. It is hoped that this synopsis will prove useful in avoiding the necessity for frequent reference to works on theoretical mechanics. The reader to whom the methods are unfamiliar and who wishes to become more fully acquainted with them may be referred to Professor Whittaker's Analytical Dynamics, where he will also find an intro- duction to those more purely theoretical aspects of the Problem of Three Bodies which find no place here. To those who are familiar with these principles in their abstract form only the concrete applications in the follow- ing chapters may prove interesting. A chapter on special perturbations is included. Here, as in the determination of orbits, the need for numerical examples may be felt. To have inserted them would have interfered too much with the general plan of the book, and they will be found in the more special treatises. But it was felt that the subject could not be omitted altogether, and a concise and fairly complete account of the theory has there- fore been given. It may seem curious that with the development of analytical resources the need for these mechanical methods becomes greater rather than less, but so it is. Chapter XIX on the restricted problem of three bodies is intended as an introduction to the Lunar Theory contained in Chapters XX and XXI. The division of these two chapters is partly arbitrary, for the sake of preserving a fair uniformity of length, but it coincides roughly with the distinction between Hill's researches and the subsequent development by Professor Brown. In the second a low order of approximation is worked out, and it is hoped that this will serve to some extent the double purpose of making the Preface vii whole method clearer and of pointing out the nature of the principal terms, which are apt to be entirely hidden by the complicated machinery of the systematic development. The rotation of the Earth and Moon is discussed in Chapters XXII and XXIII. The treatment of precession and nutation is meant to be simple and practical, and the opportunity is taken to add an account of the astro- nomical methods of reckoning time in actual use. In the final chapter of the book the theory of the ordinary methods of numerical calculation is explained. This is necessary for the proper understanding of Chapter XVIII, but it also bears on various points which occur elsewhere. Numerical applications find no place in this work. But let the mathematical reader be under no mis- apprehension. The ultimate aim of all theory in Astronomy is seldom attained without comparison with the results of observation, and the medium of comparison is numerical. Hence few parts of the theory can be regarded as complete till they are reduced to a numerical form. .This is a process which often demands immense labour and in itself a quite special kind of skill. It is just as essential as the manipulation of analytical forms. Originality in the wider sense is not to be expected and indeed would defeat the object of the book, which aims at making it easier for the student to read with profit the larger and more technical treatises and to proceed to the original memoirs. A certain freshness in the manner of treatment is possible and, it is hoped, will not be found altogether wanting. Few direct references have been given as a guide to further reading, and this may be regretted. But the opinion may be expressed that for the reader who is qualified to profit by a work like the present, and who wishes to go further, the time has come when he should acquire, if he has not done so already, the faculty of consulting the library for what he wants without an apparatus of special directions. Sign-posts have their uses, and the experienced traveller is the last to despise them, but they are not conducive to a spirit of original adventure. Since the main object in view has been to cover a wide extent of ground in a tolerably adequate way rather than to delay over critical details, the absence of mathematical rigour may sometimes be noticed. Very little attention is given to such questions as the convergence of series. It is not to be inferred that these points are unimportant or that the modern astronomer can afford to disregard them. But apart from a few simple cases where the viii Preface reader will either be able to supply what is necessary for himself, or would not benefit even if a critical discussion were added, such questions are extremely difficult and have not always found a solution as yet. It is pre- cisely one of the aims of this book to increase the number of those who can appreciate this side of the subject and will contribute to its elucidation. The reader who wishes to proceed further in any parts of the subject to which he is introduced in this book will soon find that the number of systematic treatises available in all languages is by no means large. He must turn at an early stage to the study of original memoirs. It is not difficult to find assistance in such sources as the articles in the Encyklopoidie der Mathematischen Wissenschaften, which render it unnecessary to give a bibliography. The subject is one which has received the attention of the majority of the greatest mathematicians during the last two centuries and in which they have found a constant source of inspiration. Their works are generally accessible in a convenient collected form. For the benefit of any student who wishes to supplement his reading and has no means of obtaining personal advice, the following works may be specially mentioned : Determination of Orbits and Special Perturbations. 1. J. Bauschinger, Bahnbestimmung der Himmelskorper. (A source of fully worked numerical applications.) 2. Publications of the Lick Observatory, Vol. VII. (Contains an exposition of A. 0. Leuschner's methods.) Planetary and Lunar Theories. 3. F. Tisserand, TraM de mecanique ce'leste. (The most complete account of the classical theories.) 4. H. Poincare, Lemons de mecanique celeste. 5. H. Poincare, Methodes nouvelles de mecanique celeste. 6. C. V. L. Charlier, Die Mechanik des Himmels. 7. E. W. Brown, An introductory treatise on the lunar theory. (Gives full references to all the earlier work on the subject.) The great examples of the classical methods in the form of practical application to the theories of the planets are to be found in the works of Le Verrier (Annales de VObservatoire de Paris), Newcomb (Astronomical Preface ix Papers of the American Ephemeris) and Hill (Collected Works). The most suggestive developments, apart from the researches of Poincare, are contained in the work of H. Gyldn (Traite analytique des orbites absolues des huit planetes principales) arid P. A. Hansen. All these works will repay careful study, but the suggestions are not to be taken in any restrictive sense. The author of the present book has the best of reasons for acknowledging his debt to most of the writers mentioned above and to others who are not mentioned. Some of the proof sheets have been very kindly read by the Rev. P. J. Kirkby, D.Sc., late fellow of New College, Oxford. Acknowledge- ment is also due to the staff of the Cambridge University Press for their care in the printing. It is not to be hoped, in spite of every care, that no errors have escaped detection, and the author will be glad to have such as are found brought to his notice. H. C. PLUMMER. DUNSINK OBSERVATORY, Co. DUBLIN, 20 February 1918. CONTENTS CHAPTER I THE LAW OF GRAVITATION SECT. PAGE 1, 2 Kepler's laws 1 3, 4 The field of force central 2 5 Acceleration to a fixed point for elliptic motion .... 3 6 More general case .......... 4 7 Laws of attraction for elliptic motion. Bertrand's problem . . 5 8 The apsidal angle 6 9 Condition for constant apsidal angle 7 10 Bertrand's theorem on closed orbits 8 11 Summary of results 8 12 Newton's law 9 13 Gravity and the Moon's motion 10 14 Dimensions and absolute value of the constant of gravitation . . 10 CHAPTER II INTRODUCTORY PROPOSITIONS 15 Forces due to a gravitational system . . . . . . 11 16 Potential of spherical shell 12 17 Attraction of a sphere 12 18 Potential of a body at a distant .point 13 19 Equations of motion and general integrals 14 20 The same referred to the centre of mass 15 21 A theorem of Jacobi 16 22 The invariable plane 16 23 Relative coordinates and the disturbing function . " . . 17 24 Astronomical units . 19 CHAPTER III MOTION UNDER A CENTRAL ATTRACTION 25, 26 Integration in polar coordinates 21 27 The elliptic anomalies ......... 23 28 Solution of Kepler's equation (tig. 1) . . . . . . 24 xii Contents SECT. PAGE 29 Parabolic motion 26 30 Hyperbolic motion .......... 26 31, 32 Hyperbolic motion (repulsive force) 27 33 The hodograph (fig. 2) 30 34 Special treatment of nearly parabolic motion 30 CHAPTER IV EXPANSIONS IN ELLIPTIC MOTION 35 Relations between the anomalies ....... 33 36 True and eccentric anomalies ..'...... 34 37 Bessel's coefficients 35 38 Recurrence formulae .......... 36 39-41 Expansions in terms of mean anomaly 37 42 Transformation from expansion in eccentric to mean anomaly . 40 43 Cauchy's numbers 41 44 An example 43 45 Hansen's coefficients ... . - 44 46 Convergency of expansions in powers of e ..... 46 47 Expansion of Bessel's coefficients 47 CHAPTER V RELATIONS BETWEEN TWO OR MORE POSITIONS IN AN ORBIT AND THE TIME 48 Determinateness of orbit, given mean distance and two points . 49 49 Lambert's theorem 50 50 Examination of the ambiguity 51 51 Euler's theorem 53 52 Encke's transformation 53 53, 54 Lambert's theorem for hyperbolic motion 54 55 Ratio of focal triangle to elliptic sector 57 56 Ratio to parabolic sector 58 57,58 Ratio to hyperbolic sector 59 59 A general theorem in approximate forms ....'.. 61 60 Two applications. Formulae of Gibbs 62 61,62 Approximate ratios of focal triangles ..... 63 CHAPTER VI THE ORBIT IN SPACE 63, 64 Definition of elements (55 65 Ecliptic coordinates 57 66 Equatorial coordinates 68 67 Change in the plane of reference .... 69 68 Effect of precession on the elements ....... 70 69 The lociis fetus 71 Contents xiii CHAPTER VII CONDITIONS FOR THE DETERMINATION OF AN ELLIPTIC ORBIT SECT. PAGK 70 Geocentric distance and its derivatives 73 71 Derivatives of direction-cosines - 74 72 Deduction of heliocentric coordinates and components of velocity . 75 73 The elements determined . . ' 75 74 The equation in the heliocentric distance 76 75 The limiting curve (fig. 3) 77 76 The singular curve 77 The apparent orbit. Theorem of Lambert . . . . . . 81 78 Theorem of Klinkerfues 82 79 The small circle of closest contact 80 Geometrical interpretation of method . CHAPTER VIII DETERMINATION OF AN ORBIT. METHOD OF GAUSS 81 Data of the problem 85 82 Condition of motion in a plane 85 83 The middle geocentric distance 86 84 The fundamental equation of Gauss . . ( . 87 85 First and last geocentric distances 89 86 First approximation 90 87 Treatment of aberration 91 88 True ratios of sectors and triangles 91 89 The solution completed 93 CHAPTER IX 90 Data for a parabolic orbit 94 91 Condition of motion in a plane 94 92 Use of Euler's equation 95 93 Deduction of parabolic elements 96 94 The second place as a test 97 95 Method for circular orbit . 98 96 Method of Gauss 100 97 Circular elements derived 101 xiv Contents CHAPTER X ORBITS OF DOUBLE STARS SECT. PAGE 98 Nature of the apparent orbit 103 99 Application of projective geometry (fig. 4) . 104 100 Five-point constructions (fig. 5) 106 101 Other graphical methods 107 102 Alternative method . . 107 103 Use of equation of the apparent orbit 108 104 Elements depending on the time ....... 110 105 Special cases ......... 110 106 Differential corrections ......... 112 107 113 108 Use of absolute observations . 113 CHAPTER XI ORBITS OF SPECTROSCOPIC BINARIES 109 Doppler's principle . . ...... 115 110 Corrections to the observations 116 111 Nature of spectroscopic binaries 118 112 The velocity curve (fig. 6, a and b) . 118 113 Special points on the curve 120 114 Analytical solution for elements 121 115 Properties of focal chords . - . . . . 122 116 Properties of diameters 123 117 Integral properties of velocity curve ...... 124 118 Differential properties 125 119 Differential corrections to elements 126 120 Dimensions and mass functions of system ..... 126 121 Application to visual double stars 127 CHAPTER XII DYNAMICAL PRINCIPLES 122 Lagrange's equations 129 123 The integral of energy . . 130 124 Canonical equations * . 131 125 Contact transformation 132 126 The Hamilton- Jacobi equation . . 132 127 Variation of arbitrary constants 133 128 Hamilton's principle 134 129 Principle of least action . 135 130 Lagrange's and Poisson's brackets 136 131 Conditions satisfied by contact transformation 138 132 Infinitesimal contact transformation 139 133 Disturbed motion related to an integral . . . . . . 140 134 Theorem of Poisson . 140 Contents xv CHAPTER XIII VARIATION OF ELEMENTS SECT. PAGE 135 Hamilton-Jacob! form of solution for undisturbed motion . . . 142 136 Interpretation of constants . ........ 143 137 Lagrange's brackets 144 138 Poisson's brackets 145 139 Equations for the variations .- . 146 140 Modified definition of mean longitude 147 141 Alternative form of equations for the variations 148 142 Form involving tangential system of components . . . 149 143 Systems of canonical variables 152 144 Delaunay's method of integration 153 145 Subsequent transformations 155 146 Effect of the process 157 CHAPTER XIV THE DISTURBING FUNCTION 147 Laplace's coefficients .......... 158 148 Formulae of recurrence 159 149 Newcomb's method of calculating coefficients 160 150 Direct calculations required . . . . .. . . 161 151 Continued fraction formula 162 152 Jacobi's coefficients . . . 163 153 Partial differential equation for coefficients ...... 164 154 Hansen's development 166 155 Tisserand's polynomials 167 156 Determination of constant factors 169 157 Symbolic form of complete development 170 158 Newcomb's operators 172 159 Indirect part of disturbing function . . . . . . . 173 160 Alternative order of development 174 161 Explicit form of disturbing function 175 CHAPTER XV ABSOLUTE PERTURBATIONS 162 Orbit in a resisting medium 177 163 Nature of the perturbations 178 164 Perturbations of the first order ........ 179 165 Secular and long period inequalities . . . . . . 180 166 Perturbations of higher orders 181 167 Classification of inequalities 182 168 Jacobi's coordinates . . 184 169 The areal integrals. Elimination of the nodes 185 170 Equations of motion 186 171 Equations for disturbed motion 187 172 Poisson's theorem 188 173 Effect of cornmensurability of mean motions . . . . . 190 xvi Contents CHAPTER XVI SECULAR PERTURBATIONS SECT. PAGE 174 The disturbing function modified 192 175 Form of expansion 193 176 Effect of symmetry 195 177,178 Explicit form of secular terms 195 179 Orthogonal transformation of variables 199 180 Solution for eccentric variables 200 181 Solution for oblique variables 202 182 Other forms of the integrals 203 183 Upper limit to eccentricities and inclinations .... 204 CHAPTER XVII SECULAR INEQUALITIES. METHOD OF GAUSS 184 Statement of the problem 185 Attraction of a loaded ring 186 Geometrical relations between the orbits 187 Equation of the cone 188 The final quadrature 1^9 Introduction of elliptic functions ..... 190 Integrals expressed by hypergeometric series 191 The potential in terms of invariants .... 1 92 Transformation of coordinates CHAPTER XVIII SPECIAL PERTURBATIONS 193 Nature of special perturbations 218 194 The difference table 219 195 Formulae of quadratures 220 196 Application to a differential equation 221 197 An example 221 198 Method of rectangular coordinates 222 199 Equations of motion in cylindrical coordinates .... 224 200 Treatment of the equations 225 201 Perturbations in polar coordinates deduced 226 202 Equations for variations in the elements 227 203 Calculation of disturbing forces 228 204 Perturbations in the elements 229 205 Case of parabolic orbits 230 206 Necessary modification of coefficients 231 207 Sphere of influence of a planet 234 Contents xvii CHAPTER XIX THE RESTRICTED PROBLEM OF THREE BODIES SECT. PAGE 208 Jacobi's integral 236 209 ' Tisserand's criterion . . 236 210 Curves of zero velocity (fig. 7) 237 211 Points of relative equilibrium , . 239 212 Motion in the neighbourhood 241 213 Stability of the motion 242 214 The varied orbit 243 215 Elementary theory of the differential equation 245 216 Variation of the action 247 217 Whittaker's theorems 248 218 Use of conjugate functions 250 219 Applications 252 CHAPTER XX LUNAR THEORY I 220 Choice of method 254 221 Motion of Sun denned . . 254 222 Force function for the Moon . . . . . . . . 256 223 Equations of motion 257 224 Hill's transformation 258 225 Further transformation 259 226 Variational curve defined 261 227 Equations for coefficients' 262 228 More symmetrical form . 263 229 Mode of solution 263 230 Polar coordinates deduced 265 231 Another treatment of problem ........ 265 232 Equation of varied orbit 267 233 Hill's determinant 268 234 Properties of roots 269 235 Development of associated determinant ...... 270 236 Adams' determination of g ' . . 272 CHAPTER XXI LUNAR THEORY II 237 Small displacements from variational curve 273 238 Finite displacements 274 239 Terms of the first order . . . 275 240 The variation 276 241 First terms calculated . . 277 242 Motion of the perigee . . . 278 243 Principal elliptic term. The Evection 279 244 Terms depending on solar eccentricity . . . . . . . 280 xviii Contents SECT. PAGE 245 The Annual Equation . ........ 281 246 The Parallactic Inequality 283 247 The third coordinate 284 248 Motion of the node .285 249 Further development . . . 286 250 Mode of treatment 287 251 Consistency of equations 287 252 Higher parts of motion of perigee 288 253 Definitions of arbitrary constants ........ 289 254 Remaining factors in the lunar problem 291 CHAPTER XXII PRECESSION, NUTATION AND TIME 255 Euler's equations 292 256 Mutual potential of two distant masses 293 257 The moments calculated 294 258 Steady state of rotation 294 259 Equations of motion for the axis 295 260 Change of axes for the Moon 296 261 Expansions for elliptic motion introduced ....... 298 262 Mode of solution 299 263 Luni-solar precession . . . . . . . . . . 299 264 General precession (fig. 8) 300 265 Nutation 302 266 Nutational ellipse 303 267 Numerical values for precession 304 268 Results for nutation. Moon's mass 305 269 Annual precessions in R.A. and declination 306 270 Sidereal time 307 271 Mean time 308 272 Tropical year 310 273 General remark . 310 CHAPTER XXIII LIBRATION OF THE MOON 274 Cassini's laws > . . . . 312 275 Optical libration 312 276 Equations of motion .... 313 277 First condition of stability . 314 278 Libration in longitude 315 279 Equations for the pole 316 280 Second condition of stability 318 281 Third condition for moments of inertia 319 282 Second order terms 320 283 Axis of rotation 321 Contents xix CHAPTER XXIV FORMULAE OF NUMERICAL CALCULATION SECT. PAOB 284 Representation of a function 323 285 The operators A, 8 . . . . . 324 286 Stirling's formula 325 287 Formula of Gauss 326 288 Bessel's formula 327 289 Lagrange's formula 328 290 Mechanical differentiation 329 291 Inverse operations 330 292 The first integral 332 293 The second integral .... . 333 294 Properties of Fourier's series 333 295 Mode of solution for coefficients . . 334 296 Fundamental formulae . . . . . . '. . 335 297 Simplifications . 335 298 Special case (s= 12) 337 299 Property of least squares 338 300 Periodic function of two variables 339 INDEX . . . . . . . 341 THE LAW OF GRAVITATION 1. The foundations of dynamical Astronomy were laid by Johann Kepler at the beginning of the seventeenth century. His most important work, Astronomia Nova (De Motibus Stellae Martis), published in 1609, contains a profound discussion of the motion of the planet Mars, based on the obser- vations of Tycho Brahe. In this work a real approximation to the true kinematical relations of the solar system is for the first time revealed. Kepler's main results may be summarized thus : (a) The heliocentric motions of the planets (i.e. their motions relative to the Sun) take place in fixed planes passing through the actual position of the Sun. (b) The area of the sector traced by the radius vector from the Sun, between any two positions of a planet in its orbit, is proportional to the time occupied in passing from one position to the other. (c) The form of a planetary orbit is an ellipse, of which the Sun occupies one focus. These laws, which were found in the first instance to hold for the Earth and for Mars, apply to the individual planets. In a later work, Harmonices Mundi, published in 1619, another law is given which connects the motions of the different planets together. This is : (d) The square of the periodic time is proportional to the cube of the mean distance (i.e. the semi-axis major). These deductions from observation are given here in the order in which they were discovered. The third (c) is generally known as Kepler's first law, the second (6) as his second law, and the fourth (d) as his third law. But the first statement is of equal importance. In the Ptolemaic system the " first inequality " of a planet, which represents its heliocentric motion, was assigned to a plane passing through the mean position of the Sun. Even in the Copernican system this " mean position " becomes the centre of the Earth's orbit, not the actual eccentric position of the Sun. In consequence no astronomer before Kepler had succeeded in representing the latitudes of the planets with even tolerable success. p. D. A. 1 2 The Law of Gravitation [CH. i 2. It is undeniable that in making his discoveries Kepler was aided by a certain measure of good fortune. Thus his law of areas was in reality founded on a lucky combination of errors. In the first place it was based on the hypothesis of an eccentric circular orbit and was later adopted in the elliptic theory. In the second place Kepler supposed (a) that the time in a small arc was proportional to the radius vector, (b) that the time in a finite arc was therefore proportional to the sum of the radii vectores to all the points of the arc, (c) that this sum is represented by the area of the sector. Both (a) and (c) are erroneous, and indeed Kepler was aware that (c) was not strictly accurate. Mathematically expressed, the argument would appear thus: hdt = rds, ht = Irds 2 (area of sector). Both the supposed fact and the method of deduction are wrong, yet the result is right. But if it should be supposed that Kepler owed his success to good fortune it must be remembered that this fortune was simply the reward of unparalleled industry in exhausting the possibilities of every hypothesis that presented itself and in checking the value of any new principle by direct comparison with good observations. It must also be remarked that Tycho Brahe's observations were of the proper order of accuracy for Kepler's purpose, being sufficiently accurate to discriminate between true and false hypotheses and yet not so refined as to involve the problem in a maze of unmanageable detail. Another factor in Kepler's success was his knowledge of the Greek mathematicians, in particular of the works of Apollonius. 3. Kepler had no conception of the property of inertia and he was therefore unable to make any progress towards a correct dynamical view of planetary motion. It is interesting to analyze his results and to see what is implied by each of the above statements taken by itself. According to the first statement the planets move in a field of force which is such that every trajectory is a plane curve. If we suppose that the acceleration at each point is a function of the coordinates of the point, an immediate deduction can be made as to the nature of the field of force. For let A, B be two points on a certain trajectory, and let P be a third point not in the plane of this curve. Then P can be joined to A and to B by plane trajectories. The planes in which AB, PA and PB lie meet in one point (which may be at infinity). The acceleration at A is in the plane OAB and also in the plane OAP. Hence it' is along AO. Similarly the acceleration at B is along BO, and the acceleration at P is along PO. But the point is determined by the two points A and B. Therefore the acceleration at every point of the field is directed towards the fixed point 0, and the field of force is central (or parallel). Now the planes of the orbits all pass through the Sun. Hence the Sun is the centre of the field of force in which the 2-5] The Law of Gravitation 3 planets move. For an analytical proof of the general theorem see Halphen (Comptes Rendus, LXXXIV, p. 944). 4. To this the second statement adds nothing with regard to the nature of the forces, and might indeed have been deduced from the first. For it tells us that f f \r' i de=\(xdy-ydx) = J J . the Sun being the origin of coordinates and h being a constant. By differen- tiation we have xy yx h or xy yx = 0. Thus yl'x = yjx, which proves that the. acceleration is towards the Sun at every point, i.e. the field of force is central. Clearly the argument might be reversed, and the law of areas deduced from the fact that the accelerations are directed towards a fixed centre, which has already been obtained from the first statement. Both this theorem and its converse are given in Newton's Principia, Book I, Props. I and II. 5. We shall now investigate the law of acceleration towards a fixed point under which elliptic motion is possible. In the first instance it will not be* assumed that the fixed point is the focus of the ellipse. Apart from the interest of the more general result, this is the more desirable because many pairs of stars are known in the sky the components of which are observed to revolve around one another in apparent ellipses ; but the plane of the motion being unknown it is only a matter of inference that either star is in the focus of the relative orbit of the other. For it is the projection of the motion on a plane perpendicular to the line of sight which is observed. Let then the ellipse ' ^ 2 2 a 2+ P~ be described freely under an acceleration to the fixed point (/, g). Any point on the ellipse can be represented by (a cos E, b sin E). The angle E which is known in analytical geometry as the eccentric angle is called in Astronomy the eccentric anomaly of the point. The accelerations being -asinE.E-acosE.E 2 , 6 cos E. E - b sin E. E* along the two axes, we have - a sin E. E - a cos E . E* _ b cos E . E - b sin E . E* acosEf bsinE-g whence E__ ag^cosJE 6/sin E p ,_, E ab ag sin E bfcosE' 12 4 The Law of Gravitation [OH. I This is an integrable form, giving immediately E = h(ab-agsmE-bfcosE)~ l ..................... (2) or abE + ag cos E bfsin E = h(t Q where h and t are constants of integration. If we put h = aba, E-smE + !j-cosE=n(t-t ) ..................... (3) Ct and this may be considered a generalized form of what is known as Kepler's equation. By adding 2-Tr to E it is evident that 2?r/M = T is the period of a whole revolution. Kepler's form applies when the motion is about a focus of the ellipse, and can be obtained by putting /= ae, g = 0, so that E-esmE=n(t-Q ........................... (4) This equation is of fundamental importance. The point for which E = is the nearest point on the orbit to the attracting focus and is sometimes called the pe'ricentre. The corresponding time is t and n is called the mean motion. By (1) and (2) the components of the acceleration become - . ab(f-acoBE)h* .E-acosE.E 2 = 7 1 ---- ^. ^ o ----- =- (ab ag sin a oj cos h) 3 ri 7 rr aft ( MS * , ' I I I ,* V I !. (O) a b where r is the distance of the point on the orbit from (f, g). 6. Before examining this result more closely, it may be noticed that the method is quite general and may be applied to any central orbit. For if the coordinates of a point (x, y} on the curve be expressed in terms of a single parameter a, we have similarly x'u + x"a 2 _ y''d + y"d 2 or = ^" (y - 9} - y" ( x -f) & where x', y' . . . denote derivatives with respect to a, and a, a derivatives with respect to the time. Hence on integration, a. = - h {x' (y-g)- y' (x -f)}~ 1 J (xdy - y doc) -fy + gx = h(t- 1 ). 5-7] The Law of Gravitation 5 By taking the last integration over one revolution in a closed orbit it is seen that h represents twice the area divided by the periodic time. The components of the acceleration become /) and the total acceleration is therefore R = h*r (x'y" - x"y') [x (y - g) - y' (co -/))- . = h?r/p 3 p where p is the radius of curvature at the point and p is the perpendicular from (f, g) to the tangent at the point. This of course is the well-known expression for the acceleration towards the centre of attraction. The same orbit will be described in the same periodic time under the central attraction R' to another point (/', g') if R' = Kr'lp' 3 p that is, if R'IR=p 3 r'/p' 3 r. This result is equivalent to Principia, Book I, Prop, vii, Cor. 3. 7. We now return to equation (5) which may be written .................. (6) where q and q n are the perpendiculars on the polar of (f, g) from the point (x, y} on the orbit and the centre of the ellipse respectively. Hence the ellipse represented by the general equation ax 2 + 2hxy + bf + 2gx + 2fy + 1 = .................. (7) can be described under an acceleration directed towards the origin if the acceleration follows the law R = m 2 r(l + gx+fy)- 3 , m* = n'&*/C 3 .................. (8) where A and C have their usual meaning for the conic (7). Conversely, if the law (8) is given, the trajectory is always a conic whatever the initial conditions may be. For (7) is a possible orbit, and / and g are determined by the law, while a, b and h are three arbitrary constants which can be chosen so as to satisfy any given conditions, such as the initial velocity given in magnitude and direction at a particular point. There now. arises the interesting question whether any other form of law besides (8) exists, for which the trajectories are always conies (Bertrand's problem). Let R = ro"r//(' tv*o . or _d_u _du _d_u doc ' dy ' "dz where U = Gmm' jr. If m is attracted not by a single particle mf but by any number typified by m { at (x i} y iy Zi) the components of the total force are similarly _dU _d_U _dU doc ' dy ' dz where U= It is evident that U is the work which the system of attracting particles will do if the particle m is moved from its actual position by any path to some standard position, except for a constant ; it is the potential energy of m due to its position relative to the attracting system. If we put , U=-mV V is called the potential of the attracting system at the point P. When the potential is known it is evident that the components of the attraction can be easily calculated. 12 Introductory Propositions [CH. II 16. The case of a homogeneous spherical shell is of elementary im- portance. Let m be the mass per unit area, a the radius and r the distance of the point P from the centre. If is the centre of the sphere, two cones with semi-vertical angles and + d(f>, each having its vertex at and OP as its axis, will contain between them an annulus on the surface of the sphere. The potential of this annulus at P is d V = Gm . 2-Tra sin . ad/p where p* = r 2 + a 2 - 2?-a cos < or pdp = ra sin . dcf> so that dV= Gm . Hence V = where p 2 and p^ are the values of p at the ends of the diameter through P. These values are p 2 = r + a, p 1 = \r a. If r > a, P! = r a and p 2 p 1 = 2a ; if r < a, p l = a r and p 2 p\ = 2r. Also the whole mass of the shell is M = 4>7rma z . Hence when P is a point outside the shell F= GM/r or the potential and the forces derived from it are the same as if the whole mass of the shell were concentrated at the centre. On the other hand, when P is a point inside the shell, V=GM/a or the potential is constant and the forces derived from it are zero. 17. From this elementary proposition follow immediately two corollaries : (1) A sphere of uniform density, or one composed of concentric strata of uniform density, may be treated, so far as its action at an external point is concerned, as equivalent to a single particle of equal mass placed at its centre. (2) For a point within such a sphere, the sphere may be divided into two parts by the concentric sphere passing through the point. The outer part is inoperative and may be ignored, while the inner may be replaced by a particle of equal mass situated at the centre. The heavenly bodies are for the most part approximately spherical in shape, and though not uniform in density their concentric strata are in general fairly homogeneous. They may therefore be treated in most cases, as regards their action on other bodies, as simple particles. The motion of a body within a sphere may be illustrated by the motion of a meteor within a spherical swarm, or of a star in a spherical cluster. If 16-is] Introductory Propositions 13 the swarm fills a sphere uniformly the mass operative at any point varies as the cube of the distance from the centre. Hence the effective force towards the centre varies directly as the distance. As another example it may be proved that if the density of a globular cluster varies as (1 + r-) ~ *, r being the distance from the centre, each star moves under a central attraction varying /I i O\ y as r (1 + r 2 ) '. 18. An approximate expression can be found for the potential of a body of any shape at a distant point. Let the origin of coordinates, 0, be taken at the centre of gravity of the body and the axis of x be drawn through the point P, the distance OP being r. Let dm be an element of mass at the point (x, y, z}. The corresponding element of the potential at P is Gdm Gdm {(r xf + y 2 + z*Y (>' 2 2ra? + r p r r \p \r \p where P 1} P 2 , ... are the functions known as Legendre's polynomials. The first terms are easily obtained by expansion in the ordinary way, and ix _ ' Hence if the expansion is not carried to terms beyond the second order, } r\ ^r" 1 " 2r 2 / But if A, B, G are the principal moments of inertia at 0, and / is the moment of inertia about Ox, since p 2 has been written for a? + y 2 + z' 2 , 2 - # 2 ) dm J and since is the centre of gravity, \xdm = 0. j Hence and we see that the potential of the body at P differs from the potential of a particle of equal total mass placed at the centre of gravity by a quantity depending only on l/r 3 . Except in a few cases this quantity is negligible 14 Introductory Propositions [CH. n in astronomical problems not only by reason of the great distances which separate the heavenly bodies in comparison with their linear dimensions, but because they possess in general a symmetry of form which makes A + B + G 3/ itself a small quantity. 19. We see then that in general a system of n bodies of finite dimen- sions can be replaced by a system of n small particles of equal masses occupying the positions of their centres of gravity. The total potential energy of the system is rr /- / where m iy mj are two of the ma'sses and r^ their distance apart. For if we start with any one of the particles this sum, which consists of ^n(n 1) terms, represents the potential energy of a second in the presence of the first, of a third in the presence of these two, and so on. The equations of motion are 3w in number and, according to 15, of the form 317" dU dU Now a/7 T- T- X. = ^num- 3 = (i j\ ^ -/* .. f*Jf V * r / Hence or 2 Wf Xi = a 1} 2 miifi = a 2 , 2 rriiZi = a 3 and where (x, y, z~) is the centre of gravity of the system. Thus we have the six integrals corresponding to the fact that the centre of gravity moves with uniform velocity in a certain direction. Again, we have dU dU 1V . . *-, * ^ t j ' ij Hence or and similarly = c 2 i8-2o] Introductory Propositions 15 These are called the three integrals of area and express the fact that the sum of the areas described by the radius vector to each mass, each multiplied by that mass and projected on any given plane, is constant. They also show that the total angular momentum of the system about any fixed axis is constant. Finally we have ^ ^/ . dU . dU . dU\ Znii (xiXi + yiyi + z t * t ) = - 2, (xi + y { + z { =- ) = -dUJdt whence, on integration, ^ mi (x? + yl + z?) = h-U i where h is constant. This is the integral of energy. There are then in all ten general integrals for the motion of a system of particles moving under their mutual attractions : and it is known that no others exist linder certain limitations of analytical form (Bruns and Poincare). They are in fact simply those which apply in virtue of the absence of external forces acting oil the system. 20. Let the centre of gravity (x, y, z) of the system be now taken as the origin of coordinates. If (ft, 77^, &) are the new coordinates of mi, Xi = x + &, yi = y + rj iy z i = z+ i and = 2mii)i = 2 m^i = 0. The equations of motion become g__atf dU u dU aft' mir}i ~ dn' m ^-~a where U is the same as before, but r^ is now given by rj - (ft - & + (it - vj) 2 + (6 - ) 2 - For the integrals of area we have {(y + i, t ) (I + &) -(z+ (ijiti - &) 4 (yz - ~zy) (since 'S.mtrn = *S,mgi Sm^j = 2m<{j = 0) or Smi (r)i%i- tity) = d + (a 2 b 3 - a^l^nii = c and similarly - ft{<) = c 2 + (ttsfej - a 1 b i )/'2 16 Introductory Propositions [on. n The integral of energy becomes h-U= &mt {(x + & + (y + vtY + (* + *)"} = 2 mi & + W + fc) + i Oi 2 + 2 2 + o^/Swi or where = h-\ (a? + a 2 2 + a 3 2 )/2 ra { . 21. An interesting equation involving the mutual distances of the masses can be deduced. We have , = 2 Wi& 2 . 2w.j + 2wi . Swjf/ - 22m; ; . 'Zmj = 22m t -.2wi|r with similar equations for the other coordinates. Hence 2 Wfrn^ry 2 = 2,mi. 2ra t - (^i 2 + T// + ^ 2 ). It follows that flimriflR = 2 2^i i f + - 22 since ?7 is a homogeneous function of the coordinates of degree 1. The form of the result is due to Jacobi. Now U is essentially negative. Hence if K be positive the second derivative of 2 w t - //>, r^- 2 will be always positive and the first derivative will increase indefinitely with the time. Thus the first derivative, even if negative initially, will become positive after a certain time and therefore 2mjWjry 2 will increase without limit. This means that at least one of the distances will tend to become infinite. We see therefore that a necessary (but not sufficient) condition for the stability of the system is that h' must be negative. 22. The angular momenta whose constant values are c l} c 2 , c s are the projections on the coordinate planes of a single quantity. They are there- fore the components of a vector which represents the resultant angular momentum about the axis For this axis, which is fixed in space, the angular momentum is a maximum. The plane through the origin which is perpendicular to this axis and therefore fixed is called the invariable plane at 0. About any line through in this plane the angular momentum is zero, and about any line through 20-23] Introductory Propositions 17 making an angle with the invariable axis (1) the angular momentum is \J(Ci + c.2 + c 3 2 ) cos 0. The position of the invariable plane is dependent on the position of the chosen origin of reference. Here we have considered the angular momentum as arising purely from the translational motions of the bodies treated as particles. In reality the total angular momentum of the system includes also that part which arises from the rotations of the bodies about their axes. This part itself is constant if the system consists of unconnected, rigid, spherical bodies whose concentric layers are homogeneous. Under these conditions the invariable plane at a point, as determined by the translational motions of the system alone, remains permanently fixed. The conditions hold very approximately in a planetary system. But precessional movements and the effects of tidal friction cause an interchange between the rotational and translational parts of the angular momentum, without disturbing the total amount, and to this extent affect the position of the astronomical invariable plane as defined above. The centre of gravity of the system may be taken instead of an origin fixed in space. The invariable plane is then c/f+c/17 + c.^O .............................. (2) and this is the invariable plane of Laplace. Its permanent fixity is subject to the qualifications just mentioned. A simple proposition applies to the motion of two bodies, namely that the planes through a fixed point and containing the tangents to the paths of the two bodies intersect the invariable plane at in one line. This is easily seen to be true. For the first plane passes through the origin, the position of the first body (x ly y lt z^) and the consecutive point on its path (x l + x 1 dt, y \ + y\dt, z l -\- ^dt). Hence its equation is x (2/1*1 - 2/1*1) + y (*i#i - *i#i) + z (x.y, - x, yO = 0. Similarly the equation of the second plane is (2/2*2 - 2/2*2) + y (Mg - *2# 2 ) + Z (# 2 / 2 - # 2 2/2) = " The equations of these planes together with that of the invariable plane may therefore be written j = 0, a 2 = 0, m l a l + m 2 2 = and these evidently meet in a common line of intersection. 23. When we deal with the motions in the solar system it is convenient to refer them to the centre of the Sun as origin. Let M be the mass of the Sun, m the mass of the planet specially considered and let there be n other p. D. A. 2 18 Introductory Propositions [CH. n planets, of which the typical mass is m { . Then the total potential energy of the system is where pi is the distance of rnt from the Sun, A; the distance of w f from m and r the distance of m from the Sun, so that rj = (Xi - ajT + (y t - ytf + (* - z-f pf = (xi - X? + ( Vi - F) 2 + (* - ^) 2 Ar = ^ - xf + ( yi - 7/) 2 + (Zi - zj r* =(x-X)* + (y-Yy + (z-Z)*. The equations of motion of the Sun are MX- dU MY-- dU MZ-- dU ~dX' W 3Z and of the planet considered dU -dU W mx = -^ . my = -^ , mz ^ . dx dy oz If (, i), ^) are the relative coordinates of the planet, Hence, if (fi, m, f<) are the coordinates of w^ relative to the Sun, fc_ i?? 1 m dx + M j(x- Xj) M(x-X} A Ai 3 r 3 f (m + ~ ^i-i r 2w m(X-ac)\ - - r Lr r 3 v If then we put (3) "i P< we have for the equations of relative motion . 3 + ........................ (4) and similarly Ji = -(m + ^)(?.5+~ ........................ (5) { (m + JOe.I + ........................ (6) 23, 24] Introductory Propositions 19 The function R is called the disturbing function. When, as in the solar system, the masses of the planets are small in comparison with that of the central body, M, we see that the forces derived from this function are small . in comparison with the attraction of M. Indeed a first approximation to the motion of the planet considered, which may now be called the disturbed planet, is obtained by putting R 0. 24. A double star, or system of two stars physically connected and at the same time isolated from external influences, may be considered to nresent a case of the problem of two bodies. In the solar system the disturbing effect of the other planets is always operating. Since, however, this effect is small in comparison with the attraction of the Sun it is useful to neglect R and to consider the orbit which a particular planet would have if at a given instant the disturbing forces were removed and the planet continued to move. as part of the system formed by itself and the Sun alone, its velocity in direction and amount at the given instant being that which it actually possesses. Such an orbit is called the osculating orbit corresponding to the given instant. The actual orbit from the beginning will depart more and more from the osculating orbit, but for a short interval of time the divergence between the two will be so small that an accurate ephemeris can be calculated from the elements of the osculating orbit. The usefulness of the conception of the osculating orbit goes much deeper than this, as will appear later. Now the equations (4) to (6) show that in the problem of two bodies, since R = 0, the relative motion is that which is determined by an acceleration (m + M)G/r* towards the body M which is considered fixed. But by 11 (13) a law of this form leads to an elliptic orbit with mean distance a and periodic time T, where nT = 2-7T, n 2 a 3 = (m + M) G. We can now introduce the usual system of astronomical units. Provision- ally they are taken to be : Unit of time : one mean solar day. Unit of length : the Earth's mean distance from the Sun. Unit of mass : the Sun's mass. Corresponding to this system G is replaced by the constant k 2 , so that which differs little from the Earth's mean motion. Here T is the sidereal year expressed in mean solar days and m is the mass of the Earth expressed as a fraction of that of the Sun. The numerical values adopted by Gauss were : T = 365- 256 3835 m = 1/354 710 22 20 Introductory Propositions [CH. n which lead to k = 0-01 7 202 098 95, log k = 8-235 581 4414 - 10. It may be useful to add that 180 . k/ir = 3548"-18761, log (180 . kjir) = 3-550 006 5746 which differs little from the Earth's daily mean motion expressed in seconds. The number k is called the Gaussian constant. The numerical values of ra an = r - + r 2 2 = 2fi/r + c ........................... (2) The geometrical meaning of the constant c has yet to be found. 22 Motion under a Central Attraction [CH. m 26. From the second equation of motion d -hu*~ dt~ u de where u = 1/r. Hence the first equation of motion becomes +-$- the integral of which is -7)} ........................... (3) where e and 7 are the two constants of integration. But this is the polar equation of a conic section of which the eccentricity is e and the focus is at the origin. The semi-latus rectum in this connexion is more usually called the parameter and denoting it by p we have p = h?lp, or h = Also . , du i But by (1) and (3) r 2 = ^ {1 - e 2 cos 2 (6 - 7)} + c. Hence or c = Thus if 2a is the transverse axis of the orbit, c - p/a for an ellipse, c = for a parabola and c = + /j,/a for an hyperbola. The equation of energy (2) becomes therefore Again, i/r being the angle which the direction of motion at (r, 6) makes with the radius vector (drawn towards the origin), fig v cos i/r = r - sin (6 7) ft v$u\"ty- = rQ = hu = ^ {1 + e cos (0 7)} fv are the components of the velocity along the radius vector (inwards) and perpendicular to it. The form of these expressions is to be noted. For they evidently represent (a) a constant velocity V= p/h ^(p/p) perpendicular to 26, 27] Motion under a Central Attraction 23 the radius vector, and (6) a constant velocity eV in a direction making an angle |TT + ^ 7 with the radius vector, that is, perpendicular to the transverse axis. Thus at perihelion the velocity is 7(1 4- e} and at aphelion (in the case of elliptic motion) the velocity is V(\ e). Since h = vr sin -vjr, the preceding equations may be written fj,e sin (0 7) = v*r sin ty cos i/r fjue cos (# 7) = w 2 r sin 2 ^ /"- giving e and 7 when v and ty are given at (r, 6). Thus /u 2 (e 2 - 1) = v*r (v 2 r - 2/z) sin 2 \/r. 27. In rinding the relations which subsist between positions in an orbit and the time it is necessary to consider separately the three kinds of conic section. The closed orbit, or ellipse, will be discussed first. The line 9 = 7 is drawn from the pole (the Sun) in the direction of peri- helion. The angle 6 7 is measured from this line and is called the true anomaly. Let it be denoted by w. Then, if t is the time at perihelion, t - t = y h 3 f dw 1 + ecosw) 2 ' The corresponding result in terms of the eccentric anomaly E has already been found ( 5). It will be convenient to write down the relations between the radius vector and the true and eccentric anomalies in the forms which are most frequently required. We have Hence x = r cos w = a (cos E e} y = r sin w = a V(l e 2 ) sin E. = a(l-ecosE) ..................... (5) 1 + e cos w r cos 2 %w = a(le) cos 2 \E r sin 2 $w = a(l+ e) sin 2 \E i- (6) This last equation may be regarded as the standard form of the relation between w and E. If we write e = sin (0 < < < 90), as is commonly done, then tan \w = tan (45 + $) tan \E tan \E= tan (45 - |c/>) tan \w 24 Motion under a Central Attraction [CH. in where ^w and ^E are always in the same quadrant. Also cos E e e + cos w cos w = p. . cos L = ' 1 e cos & I + e cos w - e sn - e 2 sn sin w = ~ =;- , sin E = 1 e cos A and it readily follows that 1 e cos E ' 1 + e cos w If now we employ (5) and (7) we obtain _h? f dw t t n ; /j? Jo (1 +ecosw) 2 'p 3 \ [ dE 1 - ecosE But /A=w 2 a 3 where w is the mean motion; the angle n(t t ) is called the mean anomaly and may be denoted by M. We have therefore once more obtained Kepler's equation M = n(t-t ) = E-esmE .. ...................... (8) the angles M and E being expressed in circular measure ; or if M and E are expressed in degrees, e must also be converted to the same form by the factor 180/7r. 28. The complete solution of the problem of elliptic motion is contained in the equations given above. No difficulty in numerical solution arises except in the case of Kepler's equation when E is to be found for given values of e and M. The general method applicable in such cases may be illustrated here. By some means an approximate solution E is found. Let E + &E be the exact solution, and M = E e sin E . Then M = M + (1 - e cos E ) A.E'o + ... when E e sin E is expanded in a power series in &E by Taylor's theorem. Neglecting higher powers of &E we have = (M - M )/(l - e cos E ) and hence a second approximation E 1 = E + &E . If this value is not sufficiently accurate the process may be repeated until a satisfactory result is obtained. 27, 28] Motion under a Central Attraction 25 In order to obtain a good approximate solution at the outset a great variety of methods have been devised. These depend upon (a) the use of special tables, (6) an approximate formula or a series, or (c) a graphical method. Thus to the first order in e, E Q = M + e sin M and to the second order in e where tan E = sec tan 2^ tan ^ = tan (45 + ^) tan the verification of which may be left as an exercise. Among graphical methods we can refer only to one, given by Newton (Prindpia, Book I, Prop. xxxi). Consider a circle of unit radius and centre C rolling on a straight line OX. Let E be the point of contact and A the point on the circumference initially coinciding with 0. Let P be a point on the radius CA such that CP = e and M and N the feet of the perpendiculars from P on OX and CE. Then if E = Z A CE = arc A E = OE, Hence if the circle is rolled (without slipping) along OX until the point P is on the ordinate PM where OM=M, the point of contact gives OE= E, which can therefore be read off when M is given. The locus of P is evidently a trochoid. It may also be noted that the ordinate PM= CE-CN = 1-ecosE which is the corresponding value of rfa or of dM/dE, and so gives the factor required for the improvement of an approximate value E . For references to practical applications of the above principle see Monthly Notices, R. A. S., LXVII, p. 67. 26 Motion under a Central Attraction [CH. in 29. In the case of parabolic motion dw = A /( ) f H 1 + tan2 W d (tan and therefore a quantity M may be denned by the relation ............... (9) A table, known as Barker's Table, gives M (or M multiplied by a certain numerical factor) with the argument w. An inverse table giving w with the argument M will be found in Bauschinger's Tafeln (No. xv). Or w may be deduced when t t is given thus. The equation (9) may be compared with the identity Hence tan \w = X - X if X 3 ' Let X = tan 7, X a = tan Then flif O / I 1 /V f \ r>nt 9 /3 .iu o / I I \ "ot mju ^-AJ V \jpv tan $ = tan 3 7 and tan ^w = 2 cot 27. By these equations w can be calculated directly when t is given. 30. Hyperbolic motion along the concave branch of the curve under attraction to the focus may be treated in an analogous way to elliptic motion by using hyperbolic functions instead of circular functions of the eccentric anomaly. Thus we have x r cos w = a (e cosh F) y = r sin w = a *J(e 2 1 ) sinh F so that r= a(e^ 1) =a (ecoshF-l) (10) 1 + e cos w 29-31 ] Motion under a Central Attraction 27 r cos 2 %w = a (e 1) cosh 2 %F r sin 2 \w = a (e + 1) sinh 2 $F F (11) e cosh F , ,-, e + cos w cos w = T ^ = , cosh f = 3 - e cosh JP 1 1 + e cos w sin w = 1. sinh F = e co.sh ^ 1 1 + e cos - 1 1 + By employing (10) and (12) we now obtain ^ 2 Jo(l = V \/rJ J \/(e 2 -l) ' e'-l 7 -,P) (13) which is the analogue of Kepler's equation for this case. Analogy suggests the use of hyperbolic functions, but full and accurate tables of these functions are not always available. Hence it is convenient to introduce /, the Gudermannian function of F, where (Log denoting natural- logarithm) F= Log tan (45 + /) or sinh F= tan/, cosh F = sec/, tanh ^ = tan /! We may also put e = seci/r. The principal formulae (10), (11) and (13) then become r = a(esecf 1) (14) tan w = cot^i/r tan^-/ (15) and V(/^a~ 3 ) (t -t )=e tan/- Log tan (45 + /) (16) The last equation may also be written VO^cr 3 ) \(t- - = tan iw, + 4 tan 3 Aw, y 20 / J5 32 Motion under a Central Attraction [CH. in the analogy of which with (9) of 29 is evident. Here B is unknown, but the supposition that B = 1 will lead to a good first approximation to tan ^w l and hence to A, and a nearer value for log B can then be taken from the table. This in turn will lead to a second approximation to tan^w l5 and so on until the correct value is. reached. Now let T = tan 2 E = or where G is a function of the second order in A, i.e. a small quantity of the fourth order in E, which like log B can be tabulated with the argument A. Hence /- //l+e\ //1 + e A tan iw = V T . A / ) =* A / 1 1 \\l-e) V \1 - e - Finally, by 27, r cos 2 fyv - a (1 - e) cos 2 J^^ = ^/(l + T) or r = so that the problem of finding w and r is solved by the aid of the tables giving log B and C with the argument A without introducing E explicitly into the calculation. The method with very little change is adapted equally to hyperbolic orbits. The tables will be found in the Theoria Motus of Gauss, or in an equivalent form in Bauschinger's Tafeln, Nos. xvn and xvin. CHAPTER IV EXPANSIONS IN ELLIPTIC MOTION 35. The fundamental equations of elliptic motion found in the last chapter, namely M=E-esinE, e = sin< (1) (2) '1 + e\ tan ^ w = . I ( I tan \ E = tan (|< + 1 TT) tan v VI 6/ ~ tan iJ^, /3 = tan ^ {(/8 + #)-' - (8 + x' 1 )- 1 }] . . .(5) P. D. A. 3 34 Expansions in Elliptic Motion [CH. iv The equation (3) gives /I _ /02\2 It is evident that some expansions will be made more simply in terms of ft than of e. Hence it will be useful to have the development of any positive power of ft in terms of e. Now ft + ft~ l = tan $(f> + cot ^ = 2 cosec = 2e -1 or = + & Hence by Lagrange's theorem " for the only terms which survive arise when q = 2p + m. Hence ... ...(7) and it is readily seen that this series is absolutely convergent. 36. Since * = (2/-/3)(l -/%)- it follows that log x = log y + log (1 - /fy- 1 ) - log (1 - Hence .) ......... (8) But x and y can be interchanged if the sign of ft is changed at the same time. Therefore / sn w + sn w .... It is also easy to express M in terms of w. For, by (5), log z = log a + log (1 + ftx- 1 } - log (1 + ftx) + ft cos 6 {(x + /S)- 1 - (or 1 + /3)- 1 } = log x - ft (x - x~ l ) + ^ft- (x 2 - #- 2 ) - |/3 3 (ar 5 - x~ 3 ) + ... + ft cos {-(>- a:- 1 ) + ft (x- - x~*) - ft- (X A - x~ s ) + ... j = log x - 13 ( I + cos 0) (as - x~ l ) + /3 2 ($ + cos 35-37] Expansions in Elliptic Motion 35 and therefore M = w 2 {/3(1 + cos ^>) sin w /3 2 ( + cos <)sin 2w + /3 3 (^+cos<)sin3w ...}. By this expansion the equation of the centre, w M, is expressed as a series in terms of the true anomaly. 37. We have now to consider the expansions in terms of M~, which are of the greatest importance because they are required in order to express the coordinates as periodic functions of the time. And first we take the case of r" 1 . Now a dE - = (1 ecosA) * = -pjri . r dm This is an even periodic function of E and consequently of M. Hence a 1 ' w 2 f 71 " (l-ecos E)- 1 dM + 2 - cos pM (l-e cos E)~ l cos pMdM >' If. 7T Jo 2 f rr ScosjxMI cos(pE pesinE)dE it J o 00 = 1 + 2 2J 7j, (^e) cospM (9) where 1 [* J (pe) = cos ( pE pe sin E ) dE. TTJo J p (pe) is called the Bessel's coefficient of order p and argument |?e. We shall briefly study the properties of these coefficients so far as they are required for our immediate purpose. Let F(t) = exp. [\x (t - r 1 )} = 2a p V>. CO For t write exp. ( ti/r). Then 4- exp. ( ix sin ty) = "2, a p exp. (- ipty). This is a Fourier expansion, showing that a p = ^ exp. i (p-^r x sin -\Jr) rfi|r and combining the parts of the integral which are due to i/r and 2-Tr i/r we have 1 a - C os ?-r a; sn 32 36 Expansions in Elliptic Motion [CH. iv Thus the coefficients in the expansion of F(t) are precisely the coefficients which we have to study. Now F(t) = exp. (%xt) exp. ( Hence J p (x} is the coefficient of those terms for which a = @+p, or If p is positive, takes the values 0, 1, 2,... and the expansion becomes \_ xP (i p(a = If j9 is negative, /3 takes the values p, p + 1, . . . , because a cannot be negative. 38. The effect of changing the signs of x and t is to leave F(t) unaltered. HGHCG j p (x} = (-iyj p (-x) ....................... (12) Similarly F(t) is unchanged if 1~* is substituted for t. Hence J p (u) = (-VrJ^(x) ........................... (13) Again, the result of differentiating F(t) with respect to t, gives Ja? (1 + t~ z \ 2 J p (x) tP = 2pJ p (^ tP~\ Equating the coefficients of t p ~ l we have &{Jp-i(a)+J p +i(*j\=pJp(ab ..................... (14) On the other hand, if we differentiate F(t) with respect to x, we have or, equating the coefficients of t p , %{J p ^(x)-J p+l (x)}=J p '(x) ..................... (15) These simple recurrence formulae show that, with any given argument, Bessel's coefficients of any order, and their derivatives, can be expressed as linear functions of the coefficients of any two particular orders, or of any one coefficient and its derivative, e.g. 38 Expansions in Elliptic Motion [CH. iv (integrating by parts as before) . = a + - S cos pM - {cos (p mE pe sin E} ' cos (p + mE pe sin E)} dE {J p - m (pe)-J p+m (pe)} .................. (20) The constant term has not been determined. It is 1 [* a = I cos mE dM wJo 1 f" = - cos m" (1 - e cos J) dE 7T Jo 1 f "" = - {cos mE - \e cos (m + 1) 2? \e cos (w 1) .#} d# T Jo and thus a = 1 if m = = \e if m= 1 = 0. if m>l. The particular case of m = 1 is simplified by (15), so that .................. (21) 40. From the last expansion it follows that ......... (22) Any positive power of r can be expanded by means of (20). For example 7 .2 - = (1 - e cos E)- \Ju = l + ^-2e cos E + |e 2 cos 2E _, cos pM T . ,, cos pM , , = 1 + |e 2 + e 2 - 4,e 2 Z J p ' (pe) + e z ^ * [J p ^ (pe) -J p+2 (pe)}. Now, by (14) and (15), 2(jt>-l) , 2(p + l) T JP-* (pe) - Jp+* (pe) = - J Jp-, (pe) - - J p+l (pe) _/^ f^ 4 4 = - J p '(pe)-~ J p (pe). Hence (23) p 39-4.1 ] Expansions in Elliptic Motion 39 The expansions of the rectangular coordinates can be written down at once by means of (18) and (21). Thus, if x, y have this meaning and not as in 35, x = a cos E ae nnc 11 n/f \ (24) (25) and e 2 } a sin E Other important expansions can be derived from those already obtained by differentiation or integration. For instance, the equations of motion give directly d?x a?x = whence ^=-,2pJp(pe)cospM ..... . .................. (26) y 2 (27) 41. The expansion of functions of the true anomaly in terms of the mean anomaly is in general more difficult. But sin w and cos w are readily found. For ( 27) - e 2 ) sin E sin w = 1 e cos E . d ,, dE (28) by (22). And cos E e cos lu ^ 1 - e cos & -i 1 ~ e2 e ' r 2(l-e 2 ) v by (9). 40 Expansions in Elliptic Motion [OH. iv 1 e- in (w -M) = e sin M '- - 2 J p (pe) [sin (p + 1) M - sin (p - 1) M] Hence also for the equation of the centre, - e- 2 J p (pe) {sin + V(l - e 2 ) 2 Jp' (pe) {sin (p + l)M+ sin (p - 1) M} 1 e 2 i * = ^e+- J 2 (2e) + \/(l e 2 )J 2 '(2e)\sin.M+ 2 a^sinp....(30) where a p = - - - { Jp_! (p - 1 . e) - J p+l (p + l. e)} e 2 ) {J'p-, (p-l.e) + J' p+l (p + l. e\}. This expansion for the equation of the centre in terms of the mean anomaly is important, although the coefficients are rather complicated. Hence, as far as e a , sin (w - M) = e (2 - fe 2 ) sin M + fe 2 sin 2M + $e 3 sin 3M w - M = e (2 - |e 2 ) sin M + \& sin 2M + |f e 3 sin 3M as can easily be verified. *42. For some purposes 1 Laurent series in the exponentials x, y, z of 35 are more convenient than Fourier series in w, E, M. Clearly x~ l dx = i dw, y~ L dy = i dE, z~ l dz = I dM. Let S = a + 2 (a p cospd + b p sin pd} = a + 2 { (ap - ib p } T*>+$ (a p + ib p ) T-P] where log T = id. By Fourier's theorem r 2ff r 2tr Trttp = ScospddQ, Trip = I SsinpOdQ Jo Jo /27T r27r (Op - ibp) = ST-P dd, TT (ctp + ib p ) = ST? d0. J .'0 7T Hence S = where /*? rzn- Jo This well-known form, intermediate between Fourier's and Laurent's, is general and includes the case p = 0. It has been used already in 37. Formulae have been found which make it possible to pass from any Fourier's expansion in E to one in M. The general result may be expressed in a slightly different way. For, since y has the same period as z, * The reading of 4246 can quite conveniently be deferred till after Chapter XIII. 41-43] Expansions in Elliptic Motion 41 where = I " y*z~ m dM = im~ l | y?d ($-"*) r2n = pin' 1 exp. {ipE - tm (E-e sin E)} dE Jo = ^7rpm~ l J m ^ p (me) (m ^ 0). But when ra = 0, 2vA = r yv dM = j * V 0--e cos E) dE Jo Jo Hence generally, for any function of y, p ffl=l p = B - \e (B l + B^) + S 2pm- l B p J m , p (me) z m . m- :1 p 43. There is another form of calculation, due to Cauchy, in which Bessel's coefficients do not appear explicitly. Let 8 be any periodic function, such that Here, by (4), rt Sy-'*> exp. \_\pe (y- y~ l )~\ (1-ecosE) dE Jo ! Jo Jo "Sy^> (1 -$e(y + y~ 1 )} exp -p dE Jo where (31) the coefficient B p of U expanded in powers of y l being thus identical with the coefficient A p of S expanded in powers of z l . 42 Expansions in Elliptic Motion [CH. iv Again, /2n (] . t"2w ,]?-p = -il 8z-f-*-~dM=ip-*l S-~-fdM Jo dM Jo dM r 2jr r/iSf r 2 *- r/s = -ip~ l z-P^dM=-ip-* z-p^dE Jo dM Jo dE dy l-Zn 1 (JQ Jo P ^ ^ I^V Jo where I JO F = p Ty the coefficient 5^ of F expanded in powers of y l being thus identical with the coefficient A p of 8 expanded in powers of z l . The form (32) becomes illusory when p = 0. Now the exponential function occurring in (31), (32) can be expanded in a series with Bessel's coefficients having the argument pe. That returns to the methods already considered. But another process is possible and has advantages if S is of suitable form. This consists in developing first in powers of y y~ l . Let p= oo where j and q are integers (not negative). The numerical coefficients N are called Cauchy's numbers and it is evident that a knowledge of them will be required in this method. By comparing coefficients of t p in the identity (t + t-y +i (t - 1- 1 )? = r 1 (t + r j y (t - t- i )i + 1 (t + t~ i y (t - 1-^ it is evident that N - AT" 4- N iv P> J+ii 1 ~ a fliJi 9 T P+i,J, 1' From a double-entry table giving N_ p> 0> q with the arguments p, q, therefore, similar tables giving N_ ptltg , N_ p> ^ q , ... can be readily constructed. The effect of interchanging t and t~ l shows that The expansion is either even or odd and the highest term is V +q . Hence j + q p is a positive even integer, and if p =j + q, N=l. 43, 44] Expansions in Elliptic Motion 43 It is now only necessary to consider the construction of the table for W-P, o, q when p is positive. But this is indicated by (t - t-y = 2, N_ p , , g * = 2 . 7 q ! - T -. t r (- ry- r rl(q r)l whence p = 2r q, and a * W (~\ ^i)] iftte-jW- The tabulation of Cauchy's numbers, which are all positive or negative integers, is therefore an extremely simple matter. 44. To consider an example, let S = (J - 1 Y" = (-e cos E) m = (- \e) m (y + y~ l ) m . Then U = {(- \e) m (y + y~ l ) m + (~ i*) m+1 (V + 2/" 1 ) m+1 l exp. \_\pe (y - y- 1 )] = {(- &) m (y + y~T + (- ^) m+l (y + r l ) m+l } 2 (W 1 (y - y~ l } q l! [jv - ? AT ^ V 2 e ^ * Tl 1V ~P, >, 9 ~ IV -P, *+!, 9-1 , -p, 7)1-1, P 9 ^ 44 Expansions in Elliptic Motion [OH. iv is the coefficient of y p ~ l in V and therefore also the coefficient of z p in S. Comparison with the previous result shows that A7 A7^ A7 //c-XV p^ ffi j^ q-\-i ~~~~ Y'-^-V p^ j/j, ( q 1^-i.Y p^ ?/3,-}-;i ( q i is an identity. From this the recurrence formula (m p + q + 2) N_ p+2t m! g-2(m q) N_ p< m< q + (m + p + q + 2) N- p - z , m , q can be easily deduced. 45. The development in terms of M or z of the functions fr\ n sin fr\ n mw, x m \aj cos \a/ is of special importance. Here n is any positive or negative integer, and if m is also a positive or negative integer it is only necessary to consider the second form. This involves Hansens coefficients X"' m , where Now r /r\ 2 1 -I- /9 2 /r\ 2 dM=-dE=(-1 secd>dw=-^~(-} dw a \aj 1 - y8' 2 \a/ of which the last form follows from the areal property of elliptic motion, r 2 dw = hdt = n~ } hdM = ab . dM = a 2 cos dM. Also and therefore X' m can be expressed by a definite integral involving y and E, or by one involving x and w, by means of (4), (5), (6), thus exp. [$ie (y - y~ 1 )] dE and O (I- @ 2 ) M+S (1 + /3*)--i -< (1 + 00)-"-** (1 + /3^- 1 )- w ~ 2 - i o exp. [i/9 cos <^> {(/3 + a-- 1 )- 1 - (y3 4- a)" 1 }] dw. The first of these forms shows that (1.+ /3 2 ) n+1 Z"' w is the coefficient of y*'- in the expanded product F]F 2 , where F! = (1 - j3y)> l+l - m exp. (^tey) F 2 = (1 - ^y-i^+i+m exp. (- Itey- 1 )- Similarly the second form shows that (1+ /S 8 )"* 1 (1 - $*)--3X n .' m is the coefficient of ae i ~ m in the expanded product X^X Z , where X, = (1 + /3#)- n - 2+i exp. [i cos . /3x (1 + /Sa?)- 1 ] JT 2 = (1 + yS^- 1 )- 71 " 2 ^ exp. [- i cos . /&;- 1 (1 + /Sar 1 )" 1 ]. 44, 45] Expansions in Elliptic Motion 45 The deduction of Hansen's formulae in this way is not difficult, and has been given by Tisserand (Mec. Gel., I, ch. xv). An obvious method consists in expanding the exponential function oc- curring in the first of the two integral forms in a series with Bessel's coefficients. Thus '" = (1 + fl 2 )-"- 1 2 J v (ie) I * y>+~* (1 - /3y) n+1 ~ m (1 -^u- l ) n+1+m dE ' p where X n ' m is clearly the coefficient of y i ~P~ m in the expansion of r 1 (0) = (1 - fly)**-" (1 - /3y~r +l+m and therefore equally the coefficient of y-i+p+ m in the expansion of Y n _ m () = (!- /3y-i)n+i- m (1 - (3y) n+l+m . Now - , - 2 r- IV #** VP i-V-P-k + V j- ~ where h=p + k, and if ^ is positive the coefficient of y p is / R\P i-~(i-P + V * (i-p)...(i-p-k + \) j...(j-k + l) p\ (p + l)...(p + k) kl in the ordinary notation for a hypergeometric series. Hence there are two possible forms for X' : F(ip n l, mn l, i pm + l, /3 2 ) p ml '( i + p n I, m n l, i + p + m + l, /3 2 ) of which the first is available if i p m > and the second if ip m<0, for then the third argument of the series is positive and the binomial coeffi- cient has a meaning. If i p = m both forms become X n ' m = F (m n I, m n - I, I, /3 2 ). 1 9 p When n is assumed to be positive, at least one of the first two arguments of the series is always negative, and therefore the series is a polynomial in $ 2 . For in the first form with i p m > 0, the second argument is certainly 46 Expansions in Elliptic Motion [CH. iv negative if m is positive ; if m is negative, n + 1 m > and the binomial coefficient shows that i p m < n + 1 m, so that the first argument is negative. Similarly when the second form is valid it also is a terminating series. When n is negative one of the known transformations of the hypergeometric series may be necessary to give a finite form. Hence Hansen's coefficients are reduced to the form where X"' m represents, with a simple factor, a hypergeometric polynomial in /3 2 . This form was first given by Hill. 46. The periodic series in M found above are evidently legitimate Fourier expansions, satisfying the necessary conditions with e < 1, and as such are convergent. The Bessel's coefficients are given in explicit form by the series (11) which also is at once seen to be absolutely convergent for all values of e. But in practical applications the expansions are generally ordered not as Fourier series in M but as power series in e. Under these circumstances the question of convergence is altered and needs a special investigation. Now E = M + e sin E considered as an equation in E has one root in the interior of a given contour, and any regular function of this root can be expanded by Lagrange's theorem as a power series in e, provided that \esrn E\<\E-M at all points of the given contour*. We have then to find a contour with the required property, and to examine its limits. We are to regard e and M as given real constants. The equation E = M + p cos % + t>p sin ^ where p is constant, defines a circular contour. At any point on it sin E = sin (M + p cos %) cosh (/> sin %) 4- 1 cos ( M + p cos %) sinh (p sin ^) so that | sin E j 2 = sin 2 (M + p cos ^) cosh 2 (p sin ^) + cos 2 ( M + p cos ^) sinh 2 (p sin %) = cosh 2 (p sin ^) cos 2 ( M + p cos ^) while \E-M * Cf. Whittaker's Modern Analysis, p. 106 ; Whittaker and Watson, p. 133. 45-47 1 Expansions in Elliptic Motion 47 The most unfavourable point on the contour for the required condition is that at which [ sin E is greatest. And our series is to be valid for all real values of M. Hence the condition is always fulfilled if it is fulfilled when sin % = + 1, cos (M + p cos ^) or % = fa, M = ^7T in which case | sin E = cosh p. Thus the required condition becomes e < p/cosh p. The greatest value of e is therefore limited by the maximum value of p/cosh p, which is given by cosh p = p sinh p, Inspection of a table of hyperbolic cosines shows at once that p/cosh p is greatest when p is about 1'20 and that its value is then about f. With ordinary logarithmic tables an accurate value can be obtained without difficulty thus. Let tan a be the greatest possible value of e, so that tan a = p/cosh. p = 1/sinh p. It easily follows that exp. p = cot \OL, coth p = sec a whence, by the equation giving p, cos a Log cot | a = 1 or, using common logarithms and taking logarithms once more, log cos a + log log coti a + 0*362 21569 = 0. In this form it is easily verified that a = 33 32' 3"-0, tan a = 0'662 7434 . . . . This last number is then the limiting value of e, within which the expansion of any regular function of E in powers of e is valid for all values of M. The orbits of the members of the solar system have eccentricities which are much below this limit, with the exception of some, but not all, of the periodic comets. 47. In the form in which Bessel's coefficients occur most frequently in astronomical expansions, 2 . 4 . (2j + 2) (2j~+ 4) 48 Expansions in Elliptic Motion [CH. iv It may be convenient for reference to give the following table : e 4 e s 8 192 9216 -J 2 (2e) = e & f . _ . t/G e 8 V 16 640 2 _ 4e 3 / 4e 2 4e 4 e J 4 (4 =_^i ._ + -_ 2 , ^_625e 4 / 25e 2 625e 4 Js ^ = "~ -\ T" / / \ ~t *^^ ^^ ' ^ SJ '' W ' ~8 + 192- 9216 (Op2 P 4 ,,6 1 -T + 8-f + 625e 4 /, 35e 2 375e 4 These can easily be carried further if necessary, but they are often enough for practical purposes. Bessel's Coefficients occur naturally in several physical problems discussed by Euler and D. Bernoulli from 1732 onwards. In 1771 Lagrange* gave the expression of the eccentric anomaly in terms of the mean anomaly, the result (19) above, and found the expansions of the coefficients as power series, thus anticipating Bessel's work (1824) of more than half a century later. * Oeuvres, in, p. 130. This reference, which seems to have been overlooked, is due to Prof. Whittaker. RELATIONS BETWEEN TWO OR MORE POSITIONS IN AN ORBIT AND THE TIME 48. Since a conic section can be chosen to satisfy any five conditions it is evident that when the focus is given, and two points on the curve, an infinite number of orbits will pass through them. The orbit becomes determinate when the length of the transverse axis is given, though in general the solution is not unique. For let the points be P 1; P 2 and the focal distances r 1} r 2 . In the first place we take an elliptic orbit with major axis 2a. The second focus lies on the circle with centre Pj and radius 2a r^ ; it also lies on the circle with radius P 2 and radius 2a r. 2 . These two circles intersect in two points provided (c being the length of the chord PjPg) 2a 7-j + 2a i\ > c or 4a> 7*1 + - 2 + c (1) If this inequality be satisfied two orbits fulfil the given conditions; if not, no such orbit exists. We notice that the two intersections lie on opposite sides of the chord PiP 2 , so that in the one case the two foci lie on the same side of the chord, in the other on opposite sides. In other words, in one orbit the chord intersects the axis at some point between the foci, while in the other orbit it does not. Only when 4>a = r 1 + r. 2 + c the two circles mentioned touch one another in a single point on PjP 2 and the two orbits coincide. In this case the chord passes through the second focus. When the orbit is the concave branch of an hyperbola the second focus lies on the circle with centre Pj and radius ^ + 2 and also on the circle with centre P 2 and radius r 2 + 2a. These circles always intersect in two distinct real points since r l + 2a + r 2 + 2a> c always. There are therefore always two hyperbolas which satisfy the con- ditions. The second foci lie on opposite sides of the chord and hence in the one case the chord intersects the axis between the two foci and the difference p. D. A. 4 50 Relations between two or more Positions [CH. v between the true anomalies at the points P 15 P 2 is less than 180, while in the other case the chord intersects the axis beyond the attracting focus and the difference between the anomalies is greater than 180. Under a repulsive force varying inversely as the square of the distance the convex branch of an hyperbola can be described. The position of the second focus is again given by the intersection of two circles, the one with centre Pj and radius i\ 2a and the other with centre P 2 and radius r 2 2a. These circles intersect in two points provided i\ 2a + r 2 2a > c or + r 2 -c ................................. (2) There are then two hyperbolas and in the one case the chord intersects the axis at a point between the two foci while in the other it cuts the axis at a point beyond the second focus. * It is easy to see similarly that it is always possible to draw four hyper- bolas such that one branch passes through Pj while the other branch passes through P 2 . These have no interest from the kinematical point of view since it is impossible for a particle to pass from one branch to the other. The case of parabolic solutions, two of which always exist, can be inferred from the foregoing by the principle of continuity. But it is otherwise clear that the directrix touches the circles with centres PI, P 2 and radii r 1} r 2 . These circles, which intersect in the focus, have two real common tangents either of which may be the directrix. The corresponding axes are the perpendiculars from the focus to these tangents. In the case of the nearer tangent it is evident that the part of the axis beyond the focus intersects the chord PjP 2 and the difference of the anomalies is greater than 180. In the case of the opposite tangent, on the other hand, it is the part of the axis towards the directrix which cuts the chord and the difference of the anomalies is less than 180. These simple geometrical considerations show that, when the transverse axis is given, two points on an orbit may be joined in general by four elliptic arcs (of two ellipses), by two concave hyperbolic arcs, by two convex hyper- bolic arcs ; and in particular by two parabolic arcs. This conclusion is qualified by the conditions (1) and (2) which of course cannot be satisfied simul- taneously. All these different cases must present themselves when we seek the time occupied in passing from one given point to another, as we shall at once see. 49. Let E lf E z be the eccentric anomalies at two points P 1} P 2 on an ellipse, and let 2G=E 2 +E ly Zg=E,-E,. Then ?"i = a (1 e cos E } ), r 2 = a (1 e cos E 2 ) 48-50] in an Orbit and the Time 51 and ri + r 2 = 2a{l-e cos \ (E a + EJ c = 2a (1 e cos 6? cos #). Again, c being the chord P 1 P 2 , c 2 = a 2 (cos # 2 - cos Etf + a 2 (1 - e 2 ) (sin jB" 2 - sin Etf = 4a 2 sin 2 G sin 2 g + 4a a (1 - e 2 ) cos 2 6r sin 2 #. Hence if we put cos h = e cos G then c 2 = 4a 2 sin 2 g (1 - cos 2 h) or c = 2a sin # sin h and r-j + r 2 = 2a (1 cos g cos A). If further we now put e = h + g, 8 = h g or e-8=E 2 -E l , coslt(e+8) = ecos(E 2 + E 1 )....: ....... (3) we have n + r. 2 + c = 2 E l and //. = &*(! +m) = w 2 tt 3 the time of describing the arc PiP 2 is given by nt = E 2 - E 1 - e (sin E 2 sin #,) = e - 8 - 2 sin (e - 8) cos (e + 8) = (e 8) (sin e sin 8) ................................. (6) where e and B are given by (4) and (5) in terms of ^ + r. 2 , c and a ; and this is Lambert's theorem for elliptic motion. 50. It is evident that (4) and (5) do not give e and 8 without ambiguity, and this point must be examined. We suppose always that E 2 E l < 360, i.e. that the arc described is less than a single circuit of the orbit ; and we assume that the eccentric anomaly is reckoned from the pericentre in the direction of motion. Now it is consistent with (3) to take (e + 8) between and TT and we also have (e 8) between the same limits. Hence \e lies between and TT and ^8 lies between \TT and + \ir. But the equation of the chord P 1 P 2 referred to the centre of the ellipse shows that it cuts the axis of x in the point x = a cos (E 2 - E^/cos $ (E* + EJ, y = Q 42 52 Relations between tivo or more Positions [CH. v so that, if Q is this point, A the pericentre and F 1 F 2 the foci, x ae _ cos I (e 8)- cos J, (e + 8) ' sin |e sin |8 aT^tt ~ cos (E. 2 - Ej - cos \ (E z + &\) ~ sin %Ei sin i#2 x + ae _ cos (e 8) + cos (e + 8) cos |e cos |8 AQ ~ x a ~ cos (E 2 L\) - cos \ (JE., + A^ T sin \E y sin i A', ' Now sin |e and cos ^8 are always positive. We may also take E 1 less than 2-Tr and sin ^E l positive ; then sin \E Z is negative or positive according as the arc includes or does not include the pericentre. In the first equation the left-hand side is negative when the chord intersects the axis between the pericentre and the first (attracting) focus ; in the second when the intersection falls between the pericentre and the second focus. Otherwise both members are positive. Hence we see that sin |8 is positive if (1) the arc contains the pericentre and the chord intersects F^, or (2) the arc does not contain the pericentre and the chord does not intersect F^A ; and that cos |e is positive if (3) the arc contains the pericentre and the chord inter- sects F Z A, or (4) the arc does not contain the pericentre and the chord does not intersect F 2 A. In other words, sin ^8 is positive when the segment formed by the arc and the chord does not contain the first focus, and cos |e is positive when the segment does not contain the second focus. Let ej and Sj be the smallest positive angles which satisfy (4) and (5). The other possible values are 2?r ej and Sj . If we put ntz = e 1 sin ej , nt^ = Si sin 8^ there are four cases to be distinguished, namely: (a) t = t 3 -t l when the segment contains neither focus; (6) , t = t. 2 + 1! when the segment contains the attracting, but not the other focus ; (c) t = 2-rr/n -L-t, when the segment contains the second, but not the attracting focus; (d) t = 27T/M -ti + t when the segment contains both foci. It is easy to see from 48 that when the extreme points of the arc alone are. given these four cases are always presented by the geometrical conditions and can only be distinguished by further knowledge of the circumstances. Usually it is known that the arc is comparatively short and hence that the solution (a) is the right one. 50-52] in an Orbit and the Time 53 51. The corresponding theorem for parabolic motion is easily deduced as a limiting case. For when a is very large e and S are very small. Hence (4) and (5) become ae- = T! + r z -f c, S 2 = i\ + n c. 1 3 At the same time, if we replace n by /* /a , (6) becomes = iO'i + r 2 + c) f + $ (r, + r 2 -cf. As this applies to the motion of a comet, and the mass of a comet may be considered negligible, we may therefore write 6kt = fc + r 2 + cf + (r, + r 2 - cf ..................... (7) which is the required equation. It was first found by Euler. As regards the ambiguous sign, the second focus is at an infinite distance and does not come into consideration. But 8 is negative or positive according as the,. segment formed by the arc described and the chord contains or does not contain the focus of the parabola. Hence the lower (+) sign is to be used when the angle described by the radius vector exceeds 180, and the upper ( ) sign is to be used when this angle is less than 180, as it almost always is in actual problems. 52. The solution of (7) as an equation in c is facilitated by a trans- formation due to Encke. We put c = (n + r 2 ) sin 7, < 7 < 90 and r) = 2fo/(r 1 + r 2 ) f . Then (7) becomes 3?7 = (1 + sin 7) f + (1 - sin 7)* = (cos 7 + sin |7) 3 + (cos 7 sin %y) 3 ............... (8) First we take the upper sign, in which case 877 = 6 sin 1 7 cos 2 7+2 sin 3 ^7 = 6 sin ^7 4 sin 3 ^7. If we put sin fy = \/2 sin 0, < J@ < 30 then 377 = 2V2sin@, 0< @ < 90 .............................. (9) and sin 7 = 2 V2 sin V(cos 0). Hence c =(r l + r a )vifj, ............................................. (10) where /JL = siny/rj = 3sin J V( cos 0)/sin .................. (11) 54 Relations between two or more Positions [OH. v Since /A and 77 are both functions of (a), fj, can be tabulated with the argument 77. When such a table is available (cf. Bauschinger's Tafeln, No. xxu) and 77 is known, c is immediately given by (10). In the second place we take the lower sign in (8), so that 877 = 2 cos 3 7 + 6 sin 2 ^7 cos ^7 = 6 cos 7 4 cos 3 7. If now we put cos 7 = \/2 sin J0, 30 < J < 45 then 377 = 2V2sin, 90 < <135 .................... ....... (12) and sin 7 = 2 \/2 sin J V(cos ) as before. Hence (10) and (11) apply equally to this case, with the difference 'that @ as given by (12) is an angle in the second quadrant instead of the first. Except for this the solution is formally the same in both cases, but different tables would be necessary. The case of angular motion exceeding 180, however, seldom demands consideration in practice. 53. For motion along the concave branch of an hyperbola under attraction to the focus we' have ( 30) r-i = a (e cosh E l 1), r z = a(e cosh E 2 1) and we may suppose E 2 > E^. Hence r, + r 2 = 2a {e cosh (E 2 - E,} cosh \ (E 2 + E,) - 1 } = 2a {cosh (e - B) cosh $ (e + B) - 1} where E 1 ) ...... (13) Again, the chord c is given by c 2 = a? (cosh E 2 - cosh E^f + a? (e> - 1) (sinh E 2 - sinh E,) 2 = 4a 2 sinh 2 (E 2 - E,) sinh 2 \ (E 2 + E,} + 4a 2 (e 2 - 1) sinh 2 \ (E 2 - E 1 } cosh 2 (E 2 + EJ = 4a 2 sinh 2 1 (e - 5) {- 1 + cosh 2 (e + 5)} or c = 2a sinh | (e - 8) sinh (e + 3). Hence r, + r 2 + c = 2a (cosh e - 1) = 4a sinh 2 e ............... (14) t r 1 + r 2 -c=2a(coshS- l) = 4asinh 2 S ............... (15) 52-54] in an Orbit and the Time 55 But on the other hand if /j, = & 2 (1 + m) = ri>a s nt = e sinh E 2 E 2 (e sinh E l E^) = 2e sinh (E 2 - E\) cosh \ (E 2 + E,) - (E 2 - EJ = 2 sinh 1 (e - 8) cosh (e + 8) - (e - 8) = sinh e sinh 8 (e 8) ........................ 77.. 7 ....... (16) where e and 8 are given by (14) and (15). This is the form which Lambert's theorem takes in this case. We may take ^ (e + 8) as defined by (13) positive ; and (e 8) is positive since A' 2 > E t . Hence e is positive. Now the equation of the chord referred to the centre of the hyperbola gives for the intercept on the axis x = - a cosh | (E 2 - #,)/cosh (E 2 + E 1 ), y = or, ( ae, 0) being the attracting focus within this branch, ac+ae=-a {cosh (e - 8) - cosh (e + 8)}/cosh (E 2 + EJ = + 2a sinh e sinh 8/cosh (#2 + ^1) ..................... (17) The left-hand side is negative or positive according as the intersection falls beyond the focus or on the side of the focus towards the centre. Hence sinh ^8 is positive when the angular motion about the focus is less than 180, and negative when it exceeds 180. Thus the sign of 8 is determined. If we put mi 2 = fa + r 2 + c)/4a, m 2 * = fa + r 2 - c)/4a then sinh |e = + m l} sinh ^8 = m 2 or _ exp. | e = + ?>*! + \/m^ + 1, exp. %8 = + m 2 + Vm 2 2 + 1 sinh e = 2mj Vw/ + 1, sinh 8 = + 2?% VW + 1- Hence (16) can be written (Log denoting natural logarithm) nt = 2m l ra^ + 1 + 2m 2 m 2 2 + 1 - 2 Log (TO! + Vmj 2 + 1) 2 Log (m 2 + Vw 2 2 +1) where the upper or the lower sign is to be taken according as the angular motion about the attracting focus is less or greater than 180. ' 54. The corresponding theorem for motion along the convex branch of an hyperbola under a repulsive force from the focus can be proved similarly. In this case ( 32) ?*! = a (e cosh E l -}- 1), r., = a(e cosh E 2 + I). Hence r, + r 2 = 2a (cosh \ (e + 8) cosh \ (e - 8) + 1} 56 Relations between two or more Positions [CH. v where e-S = E 2 -L\, cosh (e + 8) = e cosh (^ 2 + #,) ......... (18) and as in 53 o = 2a sinh \ (e B) sinh ^ (e + 8). We have therefore r! + r 2 + c = 2a(coshe + I) = 4acosh 2 |e ............... (19) r, + r 2 c = 2a (cosh 8 + 1) = 4acosh s |8 ............ (20) Then by 32 (22), if // = w 2 a 3 , wi = e sinh E 2 + E 2 (e sinh ^ + ^j) = 2e sinh (# 2 - #,) cosh (# 2 + #,) + E 2 - E, = 2 sinh \ (e - 8) cosh (e + 8) + e - 8 . = sinh e sinh 8 + e - 8 .................................... (21) where e and 8 are given by (19) and (20). This is analogous to the other forms of Lambert's equation. Putting as before Wj 2 = (T-L + r 2 + c)/4a, m 2 2 = (n + r 2 c)/4a we have of necessity cosh \ e = + m l , cosh ^ 8 = + m 2 but there is again an ambiguity in the values of e and S. Now we may take E z > E l and \ (e 8) positive ; and we may define ^ (e + 8) as the positive value which satisfies (18). Hence e is positive and exp. (^'e) > 1. To the equation (17) now corresponds x ae = - 2a sinh ^e sinh |S/cosh | (E 2 + E^) showing that 8 is positive if the chord intersects the axis at a point on the side of the focus towards the centre. It must be noticed that this focus is, as before, the focus within the branch and not the centre of force. Hence exp. ^8 > or < 1 according as the angular motion about this focus < or > 180. It follows that exp. (e) = + in! + Vwj 2 - 1, exp. (|S) = + w 2 Vm 2 2 - 1 sinh e = 2wj Vm^ - 1, sinh 8 = + 2w 2 Vm 2 2 - 1 and hence that + 2 Log (w^ + Vraa 2 - 1) + 2 Log (m 2 4- Vw 2 2 - 1) where Log denotes natural logarithm and the upper or the lower sign is to be taken according as the motion about the internal focus (not the centre of force) is less or greater than 180. In all cases, whether the motion is along a parabola or either branch of an hyperbola, when two focal distances are given in position and nothing 54, 55] in an Orbit and the Time 57 more is known about the circumstances, the discussion of | 48 shows that the ambiguities in the expressions for the time of describing the arc corre- spond to the distinct solutions of the geometrical problem. Hence they cannot be decided without further information. In practice, however, it rarely happens that the angular motion about a focus exceeds 180 and this limitation, by which the upper sign can be taken, will be generally understood. 55. A quantity of great importance in the determination of orbits is the ratio, denoted by y, of the sector to the triangle. The case of elliptic motion is taken first. Since n = h/ab, where h is the constant of areas, twice the area of the sector is, by (6), ht = ab {e 8 (sin e sin 8)j. But if (#], i/j), (x 2 , 7/ 2 ) are the extremities of the arc, twice the area of the triangle is 2A = O'j 2/ 2 -#22/0 = ab {sin E z (cos E l e) sin E l (cos E z - e)} = ab {sin (j a - EJ - 2e cos (E 2 + E,} sin (E 2 - &\)} = ab {sin (e 8) (sin e sin 8)} by (3). Hence = 6-8-(sine-sm8) J sin (e 8) (sin e sin 8) This expression contains a implicitly and this quantity is to be eliminated. Let 2f be the angle between r z and r 2 and let g, h have the meaning assigned to them in 49. Then 1 6a 2 sin 2 ^ e sin 2 8 = (r^ + r 2 + c) (^ + r 2 c) = (n + r.tf - r* - r* + 2r,r 2 cos 2/ = 4^7*2 COS 2 / whence 2a (cos g cos h) = 2 cos/ vVjr 2 . Also by (4) and (5) r, + r. 2 - 2a (sin 2 |e + sin 2 8) = 2a (1 cos g cos A) and therefore ?*i + r 2 2 cos/cos (/ vrjra = 2a sin 2 ^. Again, by (22), nt V - -; -- -. - sin 2, f 1 - ^l] - X,r. 2 (l + ^pj + Z 3 r 3 (l - ^- 3 ) ...(38) 62 Relations betiveen two or more Positions [CH. v 60. Now in the case of the motion of two bodies in a plane we have x = /JLX I r 3 , y Hence substituting x and y successively for X in the formula just obtained we have, to the fourth order in the intervals of time, . = a? lTl (1 + Mi/ r i 3 ) - #2 T 2 (1 - pA 2 /r 2 *) + x z r. A (1 = y l r 1 (l+ At^j/n 3 ) - y 2 T 2 (1 - pA a jrf) + y a r a (l + pA a /r a f ) The solution of these equations in the ordinary form gives ^i') T 8 (1 - /*4 a /r, 8 ) - r 3 (1 + nA a /r a ) . ^22/S ^'32/2 ''32/1 ~T **a2/3 "l2/2 ^22/1 But the denominators are respectively double the areas of the triangles whose sides are pairs of r lt r Z) r 3 . Hence we have the formulae of Gibbs, ' ' ' r, (1 + pAJrf) r 2 (1 - ^,/r,") T, (1 where, according to the customary notation, tr 2 r a ~] denotes double the area of the triangle whose 'sides are r 2 , r s , and A l} A 2 , A 3 have the values found above (37). This expresses the ratio of the triangles correctly to the third order of the time intervals. A second interesting example is provided if we take X = r 2 . In this case we have ( 25 and 26) Hence the formula (38) gives = ~ (T! (T 2 r 3 - T! 2 ) + T 2 (TIT,, 4- r 2 2 ) + T 3 (T^ - r 3 2 )} = - (3T!T 2 T 3 - T!* + T 2 3 - T :i 3 ) = {3TJT2T.J -f STjT^T, + = -yLtTjTaTs/O, ................................................... (40) The form (40) applies to an ellipse and gives the means of calculating an approximate value of a when r 1; r 2 , r 3 are known. It must be adapted to the hyperbola by changing the sign of a. For the parabola the right-hand side vanishes and we have the relation between the three radii vectores /r,. + A. 2 r 2 /r 2 + A a r 3 /r 3 ) which holds provided we may neglect terms of the fifth order in the time. 60, 61 ] in an Orbit and the Time 63 61. Returning to the formula^ of Gibbs (39), in which the denominators are correct to the fourth order, we have 3 r s r 8 ] [r a rj _ 1 T,[r,r,] l-^ 2 r 2 to the third order. But to the first order = l r /M A 1 '2 ' Hence TiJ^vyl , T S [r a r,] TaCrirJ - r -^-i T s- 9* I 9* 4* I ->* ** i** * For the coefficients we easily find from (37) 12 (4, + A 3 ) = T lT3 + r 2 2 + Tl r 2 - r s 2 = 2 (r 2 2 - r 3 2 ) 12 (A, + A 2 ) = T lT3 + T 2 2 + T 2 T 3 - T! 2 = 2 (T 2 2 - TV) 12 (^j Tl + ^!T 3 ) = T! (T lT2 - T 3 2 ) + T 3 (T 2 T 3 - Tj 2 ) = T/ + T 3 3 and therefore r*y 7^ I T] fi'j* ^ 4*?^ ^ These formulae are correct to the third order and if the' terms involving r 2 be omitted they express the ratios of the triangles in terms of the single distance r 2 to the second order. Hence their value for the determination of orbits. 64 Relations between two or more Positions [CH. v 62. Without loss of accuracy the ratio? can be expressed in terms of the two distances r x and r 3 instead of r 2 and r 2 . The forms found by Encke may be derived thus : we have to the first order whence and therefore 7*3 7"j To, Tw , 7*i 1 1 , = 2r 2 + r. 2 (T! - T S ) T^T4 ( T I - T 3) or 8 , 24(r,-r i ; - T, r 2 s (r : + r 3 ) 3 (r x + r..Y T, In the terms of the third order we have simply W* _L iff* . * \ '2 "\'3 '!/ T . \ A 4/* (7* 1 o* )4 Hence the ratios of the triangles to the required order become I>i?y] T 3 _ (,a_ T a\_y*V' '^ T \s vi T S / 7I~TTTS T I [ r, ;Q ) r 1 T 3 Vr 2 V ... (42) 3 (rT r./~T 1 2 )+ (^ + ^4 T ^ T /' where, if t 1} t%, t 3 are the times corresponding to the distances r u r 2 , r 3 , Equivalent but rather simpler expressions in terms of the extreme distances may be obtained by observing that 1 _Ji ^ Ji--JL_?^ whence r JL _r 1 T, n T-B I..l 3 3 .3' 4^ ^3 3* r 3 7^1 T 3 r 2 7'j 7^ 3 By substitution in (41) it is easily found that (43) '_ T 3 From the method by which all the expressions of this kind have been derived it is clear that the results apply equally to all undisturbed orbits, elliptic or hyperbolic. CHAPTER VI THE ORBIT IN SPACE 63. Hitherto we have considered the relative motion of two bodies only as referred to axes in the plane in which the motion takes place. It is now necessary to specify the manner . in which the motion in space is usually expressed. We take a sphere of arbitrary unit radius with the Sun at its centre. The ecliptic for a given date is a great circle on this sphere. That hemi- sphere which contains the North Pole of the Equator may be called the northern hemisphere. On the ecliptic is a fixed point 7 which represents the equinoctial point for the given date and from which longitudes are reckoned in a certain direction. The plane of the orbit is also represented by a great circle which intersects the ecliptic in two points. One of these fl corresponds to the passage of the moving body from the southern to the northern hemisphere and is called the ascending node ; the other node is called the descending node. The longitude of fl, or 7!!, may be denoted also by n : it is an angle which may have any value between and 360. The angle between the direction of increasing longitudes along the ecliptic and the direction of increasing true anomaly along the orbit is called the in- clination and may be denoted by i. It is an angle which may lie between and 180. Let P be the point on the great circle of the orbit which represents the radius vector through the perihelion and Q any other point on the same great circle representing a radius vector with the true anomaly w, so that PQ = w. We may denote the arc HP lying between and 360 by , so that OQ = w + w. This angle, reckoned from the ascending node to any point on the plane of the orbit, is called the argument of the latitude. It is possible to regard w as an element of the orbit, but it has been more usual to define the element -or, which is called the longitude of perihelion, as the sum of the two angles fl + w although only one of these is measured along the ecliptic. The angle CT + w or D 4- w +- w is called the longitude in the orbit. We have thus defined the three elements, the longitude of the p. i>. A. 5 66 The Orbit in Space [CH. VI ascending node, the inclination of the orbit and the longitude of perihelion, required to fix the position of the orbit in space, and with these it is necessary to mention the date of the ecliptic and equinox to which they are referred. 64. The motion must now be definitely related to the time. Let t be an epoch arbitrarily chosen and T the time of perihelion passage. Then, n being the mean motion, the mean anomaly corresponding to the epoch is M = n(t -T). Either M or T might be regarded as an element of the orbit, but in the case of a planetary orbit it is more usual to employ the mean longitude at the epoch, e, which is defined as the sum CT + M . Thus at any time t, if u = -GT + w is the longitude in the orbit and E the eccentric anomaly, the position of the planet is given by where -esmE = M = n(t- T) = n (t t ) + e - -or. The mean motion and the mean distance are connected by the relation ( 24) no? = /jf = k" (1 + m) a where m is the mass of the planet (negligible in the case of minor planets). The complete elements can now be enumerated and illustrated by the case of the planet Mars : Mars (m = 1/3 093 500) to 1900 Jan. 0, O h G.M.T. e 293 C 334 Epoch Mean longitude Longitude of perihelion Longitude of node Inclination Eccentricity Mean motion Log of mean distance 44' 51"-36 13 6 -88 n 48 47 9 -36 i 1 51 1 -32 e 0-093 308 95 n 1886"-51862 log a 0-182897033 Equinox 1900.0 The number of independent elements is six, corresponding to the six con- stants of integration which enter into the solution of the equations of motion, these being in their general form three in number and of the second order. When the orbit is parabolic the eccentricity is 1 and the mean distance is infinite. The scale of the orbit is indicated by the perihelion distance q and the time of perihelion passage T is given instead of the mean longitude 63-65] The Orbit in Space 67 at a chosen epoch. Thus preliminary parabolic elements of Comet a 1906 (Brooks) are shown as follows : T 1905 Dec. 22-29263 G.M.T. a, 89 51' 53"-7 ] ft 286 24 22 -1 [ 1906.0 i 126 26 7 -3 J q 1-296318. 65. If axes (x 1} y 1} z-^) be taken such that Ox-^ passes through the node, Oy 1 lies in the plane of the orbit, and Qz is in the direction of the N. pole of the orbit, the coordinates of the planet (or comet) are ac 1 = r cos (&> + w), y^ r sin (to + w), z^ = when its true anomaly is w. Let the axes be turned about Ox l so that Oy^ takes the position Oy 2 in the plane of the ecliptic and Oz 2 is directed towards the N. pole of the ecliptic. Then #2 = x \, 2/2 = y\ cos i z 1 sin i, z 2 = z l cos i + yi sin i. Next let the axes be turned about Oz 2 so that Ox 3 passes through the equi- noctial point and Oy 3 is in longitude 90. Then x 3 x 2 cos ft 2/2 sin ft, y 3 = y 2 cos ft + x 2 sin ft, z 3 = z 2 . Hence the relations between (a? 3 , y s , z 3 ) and (a? 1} t/j, ^) are given by x l y 1 z^ x 3 cos n cos i sin H sin t sin H y 3 sin n cos i cos fi sin i cos fl 2^3 sin i cos i. This scheme will give the heliocentric ecliptic coordinates of the planet. It is convenient to write sin a sin A = cos O, sin a cos A = cos i sin ft sin 6" sin B' = sin ft, sin b' cos .5'= cos i cos ft for then x s = r sin a sin (A + w + w) y 3 = r sin V sin (5' + w + w) z 3 = r sin t sin (&> + w). Hence, if R, LI, B l are the geocentric distance, longitude and latitude (the last always a very small angle) of the Sun, which may be taken from the Nautical Almanac, and A, \, /3 are the geocentric distance, longitude and latitude of the planet, A cos X cos /3 = R cos L^ cos BI + r sin a sin ( A' + o + w) A sin A, cos /3 = R sin j cos J5 a + r sin 6 sin (B f + + ?#) whence the geocentric ecliptic coordinates of the planet. 52 68 The Orbit in Space [OH. vi 66. Were the elements given with reference to the equator instead of the ecliptic, and this is sometimes done (though not often), the same formulae would give equatorial coordinates with the substitution of R.A. and declination for longitude and latitude. To obtain equatorial coordinates from ecliptic elements another transformation is necessary. Let the last system of axes be turned about Ox 3 so that Oy s comes into the plane of the equator and the new axis Qz is directed towards the N. pole of the equator. Then the obliquity of the ecliptic being denoted by e , y 3 sne . From the above relations between (x 3 , y 3) z 3 ) and (x l , y 1} z^ it follows that (# 4 , 2/4, z t ) and (x 1} y lt Zj) are related by the scheme : x l y l z, x 4 sin a sin A sin a cos A cos^a 2/ 4 sin b sin B sin b cos B cos b z 4 sin c sin C sin c cos G cos c where it is easily seen that sin a sin A = cos ft sin a cos A = cos i sin ft cos a = sin i sin II sin b sin B = cos e sin ft sin b cos B = cos e cos i cos ft sin e sin i cos b = cos e sin i cos ft sin e cos t sin c sin G = sin e sin II sin c cos C = sin e cos i cos ft + cos e sin i cos c = sin e sin i cos ft + cos e cos i. The heliocentric equatorial coordinates of the planet now become # 4 = r sin a sin ( A + co + w) 2/ 4 = r sin 6 sin (B + to + w) z t =r sin c sin (C + to + w). Thus, for example, the above elements for Comet a 1906 lead to x, = r [9-803389] sin (243 29' 42"'3 + w) 2/ 4 = r [9-999830] sin (331 33 15'1+w) z 4 = r [9-887772] sin ( 60 14 19 '5 + w) referred to the equator of 1906'0. Let (ac, y, z) be the geocentric equatorial coordinates of the planet and (X, Y, Z) the corresponding geocentric coordinates of the Sun, which may be taken directly from the Nautical Almanac or other ephemeris. Thus 66, 67] The Orbit in Space 69 But ar = A cos a cos 8, y = A sin a cos S, z = A sin & where A, a, S are the geocentric distance, right ascension and declination of the planet. These coordinates can therefore be calculated from the equations A cos a cos 8 = X + r sin a sin ( A -f a> + w) A sin a cos 8 = Y + r sin b sin (B + w + w) A sin 8 = Z + r sin c sin ((7 + a> + w). This form of equations, introduced by Gauss, is very convenient for the systematic calculation of positions in an orbit. 67. The direct transformation of the elements from one plane of refer- ence to any other may be made as follows. Let yAB represent the first plane of reference, ^AC the second plane and BCP the plane of the orbit. The first set of elements are yB = ft, BP = w and 180 -B = i. The new elements are 7^= ft', CP=w, and C=i'. Also the position of the new plane of reference relative to the old may be defined by yA = fl 1} A =^ and the arbitrary origin 71 by ^ A = ft . Hence the sides and angles of the triangle ABC are a = &> &>', b = ft' ft , c=fl Hj A = i 1} 5 = 180-;, - &)') sin \i! = sin J (ft - ft^ sin (i + i^ cos | (fl' fl 9 4 w ') sin i' = cos ^ (H ftj) sin ^ (t i,) sin (O' fi = a> (a, and (by Napier's analogy involving B + C and A) ... cos^(n + n / -2n 1 -^ 1 )^ tan %(i-i'} = - - V- tan A 7^. cos i (Q lr 4- ^i) 68. When the interval t is moderately short, however, these rigorous equations for the effect of precession are not required and it is more con- venient to use differential formulae. We now consider ' and i') and define the position of the ecliptic and equinox at T + t relative to those at T by flj = II, ^ = TT and O = II + ty, so that a = G) O &>, b n n -/r, c = n ii 4 = 7r, 5= 180 -to, (7=*. Hence by substitution in (1), (2) and (3) cfo' = cos (ft II i/r) C^TT sin (&> o) sin i . dll sin t . d (H II -^r) = cos t sin (ft II i/r) cfor cos (&> &>) sin z' . dH sin i.dw= sin (ft II -^r) dir cos (H II ->/r) sin TT . dll. 67-69] The Orbit in Space 71 But in the coefficients of dYl we may put i = i a , w = &> and TT = 0, this being the mutual inclination of the fixed and moving ecliptic. Hence we have simply di fdt = - cos (O - II - i/r) dirfdt dl/dt = d-^r/dt + cot i sin (1 II i/r) dir/dt dw /dt = cosec i sin (H II i|r) dTr/dt. These are to be integrated between t = ^ and t = t 2 , and the" coefficients of d^/dt are variable with the time. Provided the interval is no more than a few years, it is sufficiently accurate to proceed thus. Writing 4 = *i (t'2 O cos (O II i/r) dir/dt O 2 = H, + (t. 2 - <,) {d-fr/dt 4- cot i sin (fl - II - " 2 = &>! (f 2 1) cosec i sin (O IT i/r) dirjdt we take II + i/r, dTr/dt and dty/dt from appropriate tables (e.g. Bauschinger's Tafeki, No. xxx) with the argument T + ^ (4 + ^). With = 11! and i = ^ approximate values of H 2 , i 2 can be obtained and the calculation is then repeated with the corresponding values | (f^ + fl 2 ), | (^ + 1' 2 ) substituted for O and i. 69. It is impossible to correct the first observations of a moving body for parallax in the ordinary way because its distance is unknown. But the line of observation intersects the plane of the ecliptic in a certain point, called by Gauss the locus /ictus, the position of which can be calculated. If the observation is then treated as though made from this point the effect of parallax is allowed for and also the latitude of the Sun. Let the observation be made at sidereal time T at a place whose geo- centric latitude is sin T, p sin <), p being the Earth's radius at the place, and the corresponding ecliptic coordinates (ph lt ph 2 , ph 3 ), where Aj = cos I cos b = cos < cos T h 2 = sin I cos b = cos sin T cos e + sin < sin e h z = sin b = sin cos e cos (f> sin T sin e e being the obliquity of the ecliptic and I, b the longitude and latitude of the Zenith. Similarly H! = cos X cos ft = cos 8 cos a H 2 = sin A, cos /3 = cos 8 sin a cos e + sin B sin e H 3 = sin ^ = sin 8 cos e cos B sin a sin e are the direction cosines of the line of observation, A,, /3 being the geocentric longitude and latitude of the observed object. The Nautical Almanac gives R!, LI and B 1 the geocentric radius vector, longitude and latitude of the Sun. 72 The Orbit in Space [CH. vi Hence in heliocentric ecliptic coordinates the equation of the line of obser- vation is x + R! cos Zj cos 5j A a p _ y 4- RI sin L^ cos ^ A 2 p &i H 2 z + -Ri sin I?! A 3 p _ . ~ffT where A is the distance from the place of observation to the point (x, y, z) positively in the direction away from the object. If then this line intersects the plane of the ecliptic in the point (the locus fictus) x = R cos L, y = R sin L, z = A = (h 3 p - R, sin B l )IH 3 R cos L = R! cos L! cos B l + p^ (h s p R! sin B^ H^jH^ - R sin L = R! sin L^ cos B 1 + ph 2 (h 3 p R l sin B^ H 2 /H 3 . But these exact equations can be simplified, regard being had to the small quantities involved. For B l < 1" in general, so that sin B l = B 1> cos B^ = 1. Also we may put p = pR l where p is the solar parallax, 8"'80. Hence writing R = R 1 + dR l} L=L l -{- dL 1} we have cos Zj . d-Rj + R l sin L^ . dL l ^pR^ (h 3 p B^ R 1 H 1 /H S sin Zj . dRi R! cos L^ . dL l =pR 1 h 2 (h 3 p J5J R^^jH^ whence dR 1 /R l =p (Aj cos Zj + h 2 sin L^ (h 3 p BJ (H l cos L^ + H 2 sin Z x )/ H s dL^ =p (Aj sin Zj A 2 cos L^ (h 3 p Bj) (H l sin L^ H 2 cos Lj)/ H 3 or again dR l jR l =p cos b cos (J^ i) ( p sin 6 B^ cos (Z x X) cot ft dLi = p cos b sin (7^ 1) ( p sin b BJ sin (L^ - X) cot /3 A / R, = ( p sin b - 50/sin /3. Here both jp and B l are naturally expressed in seconds of arc. Thus dj , the additive correction to the Sun's longitude, is appropriately expressed in the same unit. The Nautical Almanac gives log^, to which the additive correction is j , -r> dR l Iog 10 e dR^ r . ono . d . log R, = -- . .,, = -- [4-3234 - !0]. Finally, had the observation actually been made from the locus fictus it would have been made later in time by the interval required for light to travel the distance A. But the light equation, or the time over the mean distance from the Sun to the Earth, is 498 8- 5. Hence the additive correction to the time of observation is (in seconds) A 498-5 A --'- The reduction to the locus fictus is a refinement rarely employed in practice. CHAPTER VII CONDITIONS FOR THE DETERMINATION OF AN ELLIPTIC ORBIT 70. There are certain properties of the apparent motion of a planet or comet on the celestial sphere which bear on the problem of determining the true orbit and which can be considered with advantage apart from the details of numerical calculation which are necessary for a practical solution. They are closely connected with the direct method of solution devised by Laplace, but they equally contain principles which are fundamental to all methods. Let (x, y, z) be the heliocentric coordinates of the planet, (X, Y, Z) the heliocentric coordinates of the Earth. Then x = m and m being the masses of the planet and the Earth. Let (a, b, c) be the corresponding geocentric direction cosines of the planet, so that x = X + ap, y=Y + bp, z = Z + cp .................. (1) p being the geocentric distance of the planet. The observed position of the planet is given in right ascension and declination (a, 8), and if the equatorial system of axes be chosen, a = cos a cos 8, b = sin a cos S, c = sin 8. Since x = X + dp + Zap + ap fjuK/r 3 /j, X/R 3 + dp + 2dp + ap = or X and similarly 74 Conditions for the Determination [CH. VII These are three equations in p, p and p + f^p/r 3 , the solution of which can be written down at once in the form -P 2/5 /*/"- /Mo/IP a d X b b Y c c Z a d X b b Y c c Z add b b b c c c (2) the value of p not being required. 71. The determinants in (2) can be calculated when the first and 'second derivatives of the three direction cosines are known. Now d sin a cos 8 . d cos a sin B . 8 d = sin a cos 8 . d cos a cos 8 . d 2 + 2sin a sin 8 . dS cos a cos 8 . 8 2 - cos a sin 8 . 8 c = cos 8.8 sin 8 . 8 2 . The derivatives d, 'd, 8, 8 are most simply calculated from a series of observed values by Lagrange's interpolation formulae. If the number of observations is three, made at the times ti, t 2 , t s , we have according to this rule, whence (t-t 2 }(t-t 3 ) \vj " tg/ \ 1 ^~ ^3/ 2i-L-t, a l T , "2 r 2 "T 2o, 2a - t,) (t s - t,) (t s - t,) or, if we choose t = t 2 , the time of the middle observation, TjTaTs . d = - T 1 r 2 T 3 .d= where + T 2 (TJ - T 3 ) . 2 + T 3 2 . 3 = 2r 2 . 2 + 2r 3 . a 3 = Tj 2 ( - a,) + T, 2 ( 3 - 2 ) == '2 "1 These formulae, which apply equally to the declinations, mutatis mutandis, are only correct if the observations are made at very short intervals of time and are ideally accurate. Since the accuracy of observations has practical limitations, moderately long intervals must be used and a greater number of observed places is necessary for satisfactory results. Our immediate concern, however, is rather with general principles than practical methods of calculation. 70-73] of an Elliptic Orbit 72. It is now possible to calculate the quantity I given by 75 add -& 2 a a X b b b b b Y c c c c c Z and we then have by (2) lp = (l + m )/R 3 - (1 + mVr 3 (3) The mass of the planet, m, must be neglected in a first approximation to the orbit and this is one relation between p and r. In essence it is fundamental in all general methods of finding an approximate orbit. A second relation is available because we know the angle ^r between R and p, namely r 2 = R z + p* + 2Rp cos -^ (4) while the projection of R as a vector in the direction of p gives R cos i/r = aX + bY + cZ, (0 < ^ < 180). If r be eliminated between (3) and (4) an equation of the eighth degree in p results, and it will be necessary to examine the nature of the possible roots. For the moment we suppose that the appropriate value of p has been found. Then the corresponding value of p is given by (2) and the components of the velocity can be calculated, since by (1) x = X + dp + ap, y=Y+bp (5) where X, Y, Z must be found from the solar ephemeris by mechanical differentiation. Thus when p and p are known, (1) and (5) give the three heliocentric coordinates of the planet and the three corresponding components of velocity at a given time t. From these data the elements of the planet's orbit, assumed for the present purpose to be elliptic, can be calculated without difficulty. 73. Since equatorial coordinates have been used hitherto, the elliptic elements of the orbit will also be referred to the equatorial plane. If new coordinates (, tj, ) be taken so that the axis of passes through the node and the axis of through the N. pole of the orbit, the transformation scheme is (cf. 65) : x y z cos ft' sin ft' - sin ft' cos i' cos ft' cos i' sin i' sin ft' sin i' cos ft' sin i' cos i' 76 Conditions for the Determination [OH. vn Hence in the plane of the orbit, = x sin H' sin i' y cos ft' sin i' + z cos i' = = x sin fl' sin t' y cos H' sin i' + cos t' = giving for the determination of H' and i' sin H' sin i' cos O' sin i' cos i ' (6) yz yz xz xz xy xy Also, if u is the argument of latitude (or rather of declination), | = r cos u = x cos H' + y sin H' .............................. (7) and rj = x sin O' cos i' + y cos H' cos i' + z sin i' or r sin w = z cosec t' ................................................... (8) by the above equation for . Similarly, if V is the velocity and % the angle between F and the radius vector produced, = F cos ( w + x) = ar cos ft ' + y sin ft ' .................. (9) i) = Fsin(w + %) = z cosec i' ........................... (10) Thus F and ^, as well as r and w, are determined. Now if w is the true anomaly at the point, the polar equation of the orbit gives p = r (1 +ecosw) ........................ (11) pcotx = re sin w ..... . ........................... (12) since tan % = rdw/dr. But the constant of areas is h = Vr sin % = V(/^p) = k \/p ..................... (13) giving p and hence e and w. The mean distance a can be deduced from the known values of p and e, or directly from the relation F 2 = 2/z/r - fju/a .............................. (14) and the mean motion n from the equation //. = k 2 = n 2 a 3 . Also the element vr' is given by r' = H' + w w. Finally the epoch of perihelion passage is deter- mined by the two equations n(t-T) = -esinE .............................. (15) E being the eccentric anomaly at the point of the orbit observed. 74. We now return to the consideration of the solution of equations (3) and (4), following the method of Charlier, which gives the clearest view of the geometrical conditions of the problem. The first of these equations is based on the assumption that the point of observation is moving under gravity about the Sun. The point which so moves is in reality the centre 73-75] of an Elliptic Orbit 77 of gravity of the Earth-Moon system and, strictly speaking, the observations should be reduced to this point and not the centre of the Earth. But this is a matter of detail which our immediate purpose does not require us to stop and consider. Similarly we may neglect the mass of the Earth as well as that of the planet and put R 1 . Then the equations become simply lp = \-\lr* ....................................... (16) r 2 = 1 + 2/> cos ^ + p 2 ........................... (17) where I and ty are known. The position of the planet becomes known when either p or r has been found, and it is simpler to eliminate p. Thus pj* = pr s + 2lr 3 (r 3 - 1) cos ^ + (r 3 - I) 2 or l* r *_(l<> + 2Zcos-f + l)r 6 +2( cos ^ + 1)^-1 = ...... (18) Now the coefficient of r 3 is 2 (I cos ^ + 1) = {(1 - 1/r 3 ) (r 2 - 1 - p 2 ) + which is obviously positive, whether r is greater or less than 1. And the coefficient of r 8 is essentially negative. Hence, by Descartes' rule of signs, there are at most three positive roots and one negative root. The latter certainly exists because the last term is negative (the equation being of even degree), and two positive roots must satisfy the equation, namely +1 (corresponding to the Earth's orbit) and the root required. There must be a fourth real root, and therefore in all three real and positive roots, one real and negative root and four imaginary roots. But the third positive root may or may not satisfy the problem. Now by (16) r is greater or less than 1 according as I is positive or negative. If then the two roots which are in question lie on opposite sides of 1, the spurious root can be detected and a unique solution of the problem can be found. But if they lie on the same side, they cannot be discriminated between in this way, and an ambiguity exists. If we divide (18) by (r 1), we obtain /(r) = l*r 6 (r + 1) - (2r 3 cos ^ + r 5 - 1) (r 2 + r + 1) = 0. Thus /(0) = + 1, /(+ 1) = 2^-3 cos T/T) so that the roots are separated by +1, and a unique solution exists, if 1(1 3 cos i/r) is negative. 75. The geometrical interpretation is instructive. The equation (16) for different values of the parameter I represents a family of curves in bipolar coordinates, the poles being E (the Earth) for p and 8 (the Sun) for r. The planet lies at the, intersection of one of these curves with a straight line 78 Conditions for the Determination [CH. vn 75] of an Elliptic Orbit 79 drawn through E in a given direction. But there may be two intersections, and this will happen if /(+ 1) or pH (1-3 cos i/r) = (1 - 1/r 3 ) {1 - 1/r 3 + f (1 +/> 2 -r ! )} is positive. This expression changes sign when we cross the circle r = 1 and again when we cross the curve l-l/r 3 + f (l+p 2 -r 2 ) = 0. Putting p 2 = 1 + r 2 2r cos < we get for the polar equation of this curve with the origin at $ 4 - 3r cos = 1/r 3 ........................... (19) or in rectangular coordinates, showing that the curve has an asymptote 3# = 4. Moving the origin to E we find at once that E is a node, the tangents being y= 2x. The whole curve consists of a loop crossing the SE axis at the point r = '5604, < = TT, and an asymptotic branch, and is shown as the " limiting " curve in the figure. The plane of the figure is that containing S, E and P (the planet); it is only necessary to show the curves on one side of the axis because this is one of symmetry. A few curves of the family (16) are also shown in the figure, for values of I which indicate sufficiently the different forms. When I = we have the circle r = l, called here the "zero" circle. It is evident that when I is negative r < 1 and the curve lies entirely within the zero circle, while when I is positive r > 1 and the curve lies entirely outside this circle. When I has a large negative value, the curve consists of a simple loop surrounding S and an isolated conjugate point at E. As I decreases from oo the loop increases in size until, when l = 3, the loop extends to E, where there is a cusp. Afterwards as I approaches the loop, still passing through E, approximates more and more closely to the zero circle. When I is positive the form of the curves is rather more complicated. It must be remarked that I cannot be greater than + 3. For I = (r 3 - l)/rp = (r~ l + r~ 2 + r~ s ) (r - I)/ p. But r > 1 and r 1 < p. Hence the limit is established and we have only to follow the values of I from + 3 to 0. At first the curve consists of a small loop passing through E. As the value of I falls the loop expands, tending to enfold the zero circle. Finally, when I = + 0'2959, it reaches the axis again and forms a node on the further side of S. As the value of I falls still further the curve breaks up into two distinct loops. The larger continues to expand outwards at all points and recedes to infinity, while the inner, always passing through E, contracts until finally it becomes the zero circle. These features in the development of the family of curves will be evident in the figure. 80 Conditions for the Determination [CH. vn It will now be apparent that the limiting curve and the zero circle divide space into certain regions and that the solution of the problem of determining an orbit by the method indicated is unique or not according to the region in which the planet happens to be. Thus we distinguish four cases : (1) If the planet is within the loop of the limiting curve there are two solutions. (2) In the space between the loop and the zero circle the solution is unique. (3) Outside the zero circle and to the left of the asymptotic branch of the limiting curve there are again two solutions. (4) If the planet lies to the right of the asymptotic branch of the limiting curve only one solution is possible. It happens that newly dis- covered minor planets are usually observed near opposition and therefore this is the case which most commonly occurs. 76. There is another curve which has considerable importance in the problem of determining an orbit by a method of approximation and to which Charlier has given the name of the " singular " curve. We may find it thus. If we eliminate r between the equations (16) and'(17) we have lp = 1 - (1 + '2p cos i/r + p 2 ) ~ ' which is an equation giving the values of p for a line drawn through E in the direction i|r. Two of the values become equal and the line touches the curve (16) if I = 3 (cos $> + p)(l + 2p cos i/r + p n -)~% = 3 (cos i/r + p)/r\ Hence the locus of the points of contact of the tangents from E to the family of curves (16) is (l-l/r 3 )/> = 3(cos^ + p)//- 3 or 2r 2 (r 3 -l) = 3(p 2 + r 2 -l) or again 3p 2 =2r 5 - 5r 2 + 3 ........................ (20) This is the equation of the singular curve. If we change from bipolar coordinates to the polar equation with the origin at 8, we obtain 3 (1 - 2r cos + r 2 ) = Zr 5 - 5r 2 + 3 or r 3 = 4 - 3 cos /r .............................. (21) Comparison of this form with the equation (19) of the limiting curve shows at once that these two curves are the inverse of one another with respect to 75-77] of an Elliptic Orbit 81 the zero circle. From this relation the form of the singular curve, which is shown in figure 3, becomes apparent. The importance of the singular curve arises thus. In general a line through E meets a curve of the family (16) either in one point (besides E) or in two distinct ppints. In the latter case the coordinates of the planet are regular functions of the time and can be expanded in powers of the time, but each is expressed by two distinct series between which it is impossible to discriminate. When, however, the planet is situated at a point on the singular curve, the two distinct series coalesce and each point of the singular curve corresponds to a branch point where we may expect the coordinates of the planet to be no longer regular functions of the time. This is in fact the case. Charlier obtained the equation of the singular curve by noticing that along this curve expansion of the coordinates as power series in the time ceases to be possible. 77. If the masses of the Earth and of the planet be neglected, (2) may be written in the form , 2 3 where A^ A.,, A 3 represent three determinants and l=,& t /k?A l . It is clear, as we have already noticed, that rR if I is positive. Now the equation of the plane of the great circle tangent to the ;ipparent orbit at (a, b, c) is a a x b b = (23) The coordinates of the Sun on the celestial sphere are ( X/R, Y/R, Z/R) and of a neighbouring point to (a, b, c) on the apparent orbit (a + at + ^dtf, b+ ..., c + ...). Hence the ratio of the perpendiculars from these points to the above plane is - A,/JE * | 2 A :S = - 2/lk 2 t*R. Thus I is negative if the Sun and the arc of the planet's orbit lie on the same side of the great circle touching the orbit, and positive if the Sun and the arc are on opposite sides. In the first case r < R, in the second r>R. Hence we have the theorem due to Lambert, which may be expressed by saying that an arc of the orbit of an inferior planet appears concave to the corresponding position of the Sun, but the arc described by a superior planet appears convex. This test makes it immediately apparent whether a planet or the Earth is the nearer to the Sun. It may happen that A 3 vanishes. It is then necessary to express the coordinates of neighbouring points on the orbit to the third order p. D. A. 6 82 Conditions for the Determination [CH. vn (a dt + ^tit 2 ^i'it s , b ..., c ...). The result of substituting in the left- hand side of (23) is a a a b b b and the double sign shows that the curve crosses the tangent great circle. In the language of plane geometry there is a point of inflexion on the apparent orbit. Now if A 3 vanishes either r = R or Aj = 0. Thus such a point of inflexion occurs either when a comet reaches the same distance from the Sun as the Earth or when the great circle which touches the orbit of a planet passes through the position of the Sun. 78. When the apparent orbit of a planet reaches a stationary point the curve either crosses itself and forms a loop, or without crossing itself it pursues a twisted path, passing through a point of inflexion. At such a point, as we have just seen, the tangent in general passes through the Sun. There is a related theorem, due to Klinkerfues, which applies to the case of a loop. Let P 1} P 2> P 3 be three positions of the planet in space, E l , E. 2> E 3 the corre- sponding positions of the Earth and S the position of the Sun. If the first and third positions correspond to the double point on the loop, E 1 P l and E X P 3 are parallel and lie in one plane. Let SP 2 meet the chord PjP 3 in p 2 and SE 2 meet the chord E 1 E 3 in e 2 . If ^ is the time taken to describe PjP 2 or E 1 E 2 and t 2 the time along P 2 P 3 or E 2 E 3 ,t 1 : t 2 is the ratio of the sectors SP 1 P Z , SP. 2 P 3 or very nearly the ratio of the triangles SP^, Sp 2 P 3 , that is P\P?. ' pzPs- But similarly ^ : t 2 is nearly equal to the ratio E^ : e 2 E 3 . Hence PxP 3 and E 1 E 3 are divided by p 2 and e 2 in approximately the same ratio and therefore e 2 p 2 is parallel to E 1 P 1 and E S P 3 . Consequently the three planes E 1 SP 1 , J 2 e 2 Sp 2 P 2 , E 3 SP 3 have a common line of intersection, namely the line through 8 parallel to E 1 P 1 and E 3 P 3 . But on the geocentric sphere these three planes correspond to three intersecting great circles. The first and third intersect in P, the double point on the apparent orbit. Hence the great circle joining any intermediate point on the loop to the corre- sponding position of the Sun also passes through the double point, at least very approximately. It may be inferred then that if any three points on such a loop be joined to the corresponding positions of the Sun, the three great circles will meet in one point which is also a point on the apparent orbit. 79. There is some interest in finding the geometrical meaning of the three determinants A 1} A 2 , A 3 in (2) or (22). Bruns has noticed that A 3 = V 3 k, where k is the geodetic curvature of the apparent orbit on the sphere and V the velocity in this orbit at the point (a, b, c), so that V 2 = a? + b 2 + c 2 . 77-so] of an Elliptic, Orbit 83 But we shall now express these determinants in terms of the small circle of closest contact or circle of curvature. This passes through the points (a, b, c), (a + dt, b + bt,c+ct) and (a + at' +^dt'*, b + ..., c +...), and the equation of its plane is x y z 1 =0 a b c 1 a b c d b c or x (be -be) -\- y (cd - cd) + z (db db) = A 3 ............... (24) Now a? + b- + c 2 =1 ad + bb + cc = ad + bb + cc = V- by successive differentiation. Solving these as linear equations in a, b, c, we obtain aA 3 = be be V" (be be) arid two similar equations. But (a/F, b/V, c/F) are the direction cosines of the point PI on the tangent 90 from (a, b, c), and the pole of the tangent is (a , 6 , c ) where Fa = be - be, F6 = ca ca, Fc = ab - db so that be be = aA 3 4- F 3 a , . . . and S (be - 6c) a = A 3 2 + F 6 . The equation of the circle of curvature (24) becomes then ( A s + a F 3 ) x + (6A 3 + 6 F 3 ) y + (cA 3 + c F 3 ) z = A 3 . Hence, if &> is the angular radius of this circle, cos 2 w = A 3 2 /(A 3 2 + F 6 ) and therefore A 3 = F 3 cot 6. This then is the geometrical meaning of the third determinant. 80. Next we take A 2 . If (A, B, G) are the geocentric direction cosines oftheSuri, X = -AR, Y = -BR, Z= - CR and A, = - R [A (be - be) + B (cd - ca) + C (ab - db)} ~r = - R ~r {A (be - be) + B (cd -ca) + C (ab - db)} "dt = - RV(Aa + Bb + Cc ) - RV (Ad, + Bb + Cc \ 62 84 Conditions/or the Determination of an Elliptic Orbit [CH. vu Here A, B, C are of course constants. Now (cr , 6 , c ) is the pole P of the tangent at P, (a, b, c). The arc PP passes through the centre of the circle of curvature and while P is initially describing a circle of angular radius o> about this centre P is describing a circle of radius 90 eo about the same centre. If the velocity of P , which is in the direction of the pole of PP opposite PI, is V, F'/cos o> = F/ sin o>, / V = - / F, &/ V' = -b/ F, c / F' = - c/ F. Hence A 2 = A, F/ F + P F cot w (Ad + Bb + Cc). Again S being the position of the Sun on the sphere, and r the perpendicular arc from S to the tangent PP, at P to the apparent orbit (positive if drawn from the same side of PP X as P or the centre of curvature). Also Aa + Bb + Cc = Fcos SP, = Fsin v where v is the perpendicular arc from S to the normal PP to the apparent orbit at P (positive if drawn from the same side of PP as PI). Hence A 2 = RVsmr + jRF 2 coto>sini>. Thus the geometrical significance of the three determinants has been determined and we may write (2) in the form R Fsin r R ( V 2 cot o> sin v V sin r) F 3 cot to which shows in the clearest way how this method of determining the orbit depends on a knowledge of the simple quantities F, F, r, v and a>, which can be specified without reference to any particular axes. To these must be joined the equation (4), which enjoys the same property. It has been remarked ( 75) that I cannot be greater, than + 3. Now I = AS/^A! = - F 2 cot w/k-R sin r. Hence for a superior planet, F 2 < 3k*R | tan w sin r \ which sets a limit to the apparent velocity when w and r are known, or to the curvature of the path when F and T are known. CHAPTER VIII DETERMINATION OF AN ORBIT. METHOD OF GAUSS 81. Since a planetary orbit requires for its complete specification six elements, it is to be expected that three positions of the planet, i.e.. three pairs of coordinates, observed at known times, will suffice to determine its path. And this is in general true, though there are exceptional circumstances in which further observations may be necessary. The formulae are a little simpler when ecliptic coordinates are employed, and though this is not essential we shall take as the data of the problem : the times of observation t lt 2 , 3 the longitudes of the planet \ 1} X 2 , X 3 the latitudes of the planet &, $3, /3s the longitudes of the Earth L lt L. 2 , L 3 the Earth's radii vectores R 1} R 2 , R 3 . The angular coordinates are referred to a fixed equinox which will apply to the resulting elements. The Earth's longitude (which differs by 180 from the Sun's longitude) and radius vector can be derived from the Nautical Almanac or other national ephemeris : the Earth's latitude can be neglected, or, if desired, allowed for by using the method of the locus fictus ( 69). At the time ti let r t be the heliocentric distance of the planet and p f its geocentric distance. Referred to a fixed system of rectangular axes through the Sun let (x i} y { , z { ) be the coordinates of the planet, (A { , B i} (7 t -) the direction cosines of RI and (a t , b i} Ci) the direction cosines of p i} so that 82. Since the three positions of the planet lie in a plane passing through the Sun y\ y (y\ z * l/s^i) and (y^z.^ y^z^ are the projections on the yz plane of the areas [r 2 r 3 ], [r :j rj] and for 2 ]. Hence i for s ] - as t [r,r s ] -f x, [r^-,] = or = 0...(1) = 0. . .(2) = 0. . .(3) [r 2 r 3 ](a l p l + A l R 1 )-[r 1 r 3 ] And similarly [?- 2 r C 3 }; and these are the representative points of the directions of R l} p 2 , R 2 , R 3 on the sphere of unit radius. The perpendiculars to the plane are therefore the sines of the perpendicular arcs to the great circle through (a 1} 6 1} GJ), (a 3 , b z , c 3 ) and if these arcs are B-!, y8 2 ', B 2 , B 3 respectively (due regard being paid to sign) our equation becomes for ;! ] p 2 sin y3 2 ' = [r 2 r 3 ] R l sin B{ fo r s ] R 2 sin B 2 + forj R 3 sin B 3 . . .(4) 83. The points on the sphere just named are E J} E 2 , E 3 , representing the heliocentric directions of the Earth and lying on the ecliptic, and P 1} P 2 , P 3 , representing the geocentric directions of the planet. The great circle men- tioned is PjP 3 . Let this circle intersect the ecliptic in longitude H 2 and at the inclination r) 2 . Then we have the same relation between any one of the perpendicular arcs and the longitude (reckoned from H. 2 ) and latitude of the point from which it is drawn as exists between the latitude of a point and its 82-84] Method of Gauss 87 right ascension and declination, the obliquity of the ecliptic being replaced by 77 2 . That is to say, sin fa' = cos 772 sin fa sin 7/ 2 cos /3 2 sin (X 2 ~ ^2) sin B-i' = sin r}. 2 sin (L^ H 2 ) sin B 2 ' = sin r} 2 sin (L 2 H 2 ) sin B 3 = sin 7/ 2 sin (Z 3 7/ 2 ) and as regards the points P 1( P 3 = cos 7; 2 sin & - sin rj z cos fa sin (Xj ^ 2 ) = cos 77 2 sin fa sin 7; 2 cos /3 3 sin (\ 3 jEQ. The latter give, by addition and subtraction, 2 tan 772 sin { (Xj + X 2 ) H.^} = sin (& -f /3 3 )/cos /3j cos /3 3 cos ^ (X 3 Xj) 2 tan 773 cos { (Xj + X 2 ) // 2 } = sin (/3 : , /3j)/cos /Sj cos /9 3 sin ^- (X 3 Xj) and determine rj. 2 and //.j. We now put 2 r 3 ]/[rv 3 ], w s = [n^Mnni]. The equation (4) then takes the simple form PZ = Cj^j + c 2 c 3 n 3 . Now this is a purely geometrical relation involving the intersections of any plane through the Sun with three lines drawn in given directions through the positions of the Earth. If we imagine the plane to move into coincidence with the ecliptic, c 1; c 2 , c s remain unaltered while in the limit p lf p 2 , p 3 vanish and r 1} r 2 , r 3 become coincident with R ly R 2 , R 3 . Hence if we put N, = [R.R.^/IR.R,] = R 2 sin (L a - L^jR, sin (L, - LJ N 3 = [R,R. 2 ]/[R, R,] = R, sin (L 2 - LJJR, sin (L, - L,} the equation Q = c l N 1 + c 2 c 3 N 3 must be an identity, and this can be verified. Hence by the elimination of c 2 p 2 = c, (JV, - ?h) + c 3 (N a - O (5) which is the required equation for p 2 . 84. Since fa' is the perpendicular arc from P 2 to P^P 3 it is geometrically evident that if the observed arcs of the planet's orbit are of the first order of small quantities (and we assume them to be small) /3./ is a quantity of the second order. Hence the equation (4) shows that if we are to obtain a value of p. 2 which is a real approximation and not merely illusory we must at the outset employ values of the ratios of the triangles which are correct to the 88 Determination of an Orbit [OH. viu second order in the time intervals. Accordingly we use (41) of 61 and neglect the terms of higher order than the second ; that is to say, where It is necessary to neglect the mass of the planet and put p = k~: this can safely be done in calculating a preliminary orbit, for which the perturbations are entirely neglected. The equation (5) for /o, therefore becomes = fco-W .............................................. , ..... (8) where k , 1 are completely determined quantities. But if S 2 is the angle (< 180) between p 2 and R z produced, r.? = R 2 2 + pJ + 2R. 2 p 2 cosS 2 ........................ (9) where cos S 2 = cos P. 2 E 2 = cos /3 2 cos (\g L 2 ). If now /> 2 be eliminated from (8), which corresponds to the definite form of Lambert's theorem ( 77), and (9), an equation of the eighth degree in r 2 results. The nature of the roots of this form of equation has already been discussed in 74. But Gauss replaced the eliminant by a much simpler equation which is easily found. We have r 2 R 2 p. 2 - K ---- r . ^ -- - sm o., sin z sm (8 2 z) where z is the angle subtended by R 2 at the planet in its intermediate observed position. Hence by (8) RZ sin (8 2 - z) ' 1 sin 3 z " = " or / sin 4 z/R 2 3 sin 3 B 2 = -R 2 sin (8 S z) + k sin z and therefore if we put m cos q = k + R 2 cos & 2 ??i n sin q = R 2 sin , mm = IQJR^ sin 3 S 2 where m is given the same sign as 1 , we have the simple form m sin 4 z sin (z q) .............................. (11) 84, 85] Method of Gauss 89 and this is the equation of Gauss. This form of equation does not avoid the possibility of an ambiguity arising from two distinct roots, which is inherent in the problem. But when only one appropriate root exists, it is easily found by successive approximation. In the most common case, that of a minor planet observed near opposition, z q is small and a first approximate value is given by z = q + m sin 4 q. When z is found the corresponding first approximations to p 2 and n are given by (10). 85. We have now to find the corresponding values of p l and p 3 . For this purpose we return to the equations (1), (2) and (3), and eliminate p s and R... The result can be written down at once in the form or where the determinants as before are represented by their first lines, the other rows being obtained by change of letters without change of suffixes. Since the same form of equation must remain true, the directions of p lf p 2 , p s being preserved, when the plane of the orbit is made to coincide with the ecliptic, in which case p l = p. 2 and n becomes N lt the equation N 1 R l \A l ,a 3 ,A 3 \ = must be an identity. Hence 'iPi i,f'3,A 3l = p, a 2 ,a 3 ,A Now '!,:>, A* '= cos/SjCosXj cos/3 3 cosX;> cosL 3 cos fii sin Xj cos /3 :! sin X 3 sin L 3 sin & sin /3 3 = cos /3 X cos $ 3 { tan & sin (X 3 L 3 ) + tan /3 3 sin (Xj L 3 )} the axis of z being drawn towards the pole of the ecliptic and the axis of x towards the First Point of Aries. Similarly a 2 , a 3> A s | = cos /3 2 cos /3 3 { tan /S 2 sin (X 3 L s ) + tan /3 S sin (X 2 L^)} and | A 1} a s , A 3 1 = sin /3 3 sin (L^ L s ). Hence where ,, _ tan /3 2 sin (X 3 Z 3 ) tan # 3 sin (X 2 Z 3 ) tan^ sin (X 3 L 3 ) tan yS 3 sin (X, L 3 ) j\f , _ RI tan /3 3 sin (L s L^) tan /8] sin (\. L^) tan /8 3 sin (Xj L 3 ) ' 90 Determination of an Orbit [CH. vni Similarly the result of eliminating p l and ^ from the original equations is to give (interchanging the suffixes 1 and 3) n s p 3 cos ft s = M 3 p 2 cosj3 2 + (N 3 -n s )M 3 ............... (13) where ,, _ tan # 2 sin (Xj Zj) tan fa sin (X 2 L\} tan # 3 sin (Xj L^ tan fa sin (X 3 L^) , _ R 3 tan fa sin (L r - L 3 ) tan fa sin (Xj L^ tan & sin (X 3 LJ ' The coefficients M l} M^, M 3 , M 3 as well as N lf N 3 are constants throughout the process of approximation, but n l} n 3 must be taken at this stage from the approximate forms (6) and (7). Then (12) and (13) give values of p l and p 3 corresponding to the approximate value of p 2 already obtained. 86. The heliocentric distances, longitudes and latitudes of the planet are next deduced by the formulae Ti cos bi cos (k L^ = pi cos & cos (\i - Li) + RI Ti cos bi sin (l t - L t ) = pi cos fa sin (\ Li)' ......... (14) Ti sin bi = pi sin fa (i=l, 2, 3), which are at once found by taking the axis of x successively along R 1} R z and R 3 , the axis of z being always directed towards the pole of the ecliptic. But these coordinates give the position of the plane of the orbit, for tan i sin (4 H) = tan b l tan i sin (l s II) = tan b 3 where i is the inclination and O the longitude of the node ; or in a form more suitable for calculation 2 tan i sin (^ (^ + 1 3 ) li} = sin (6j + 6 3 )/cos h cos b 3 cos \ (1 3 h) 2 tan i cos [ (^ + /) flj = sin (6 3 6j)/cos 6j cos 6 3 sin ^ (/ 3 ^) And now the three arguments of latitude Uj, giving the differences of the true anomalies, can be calculated, for tan Uj = tan (lj fl) sec i ........................ (16) (j = 1, 2, 3). In the case of a comet, it is the practice to take Uj < or > 180 according as the latitude is positive or negative ; in the case of a planet, Uj is placed in the same quadrant as lj O. If we calculate n 1} n s from _ r 2 sin (u s w 2 ) _ r 2 sin(w 2 w,) rjsin^Mg ?/,)' r 3 sin(w 3 M : ) we shall not obtain improved values of these ratios, because these equations have a purely geometrical basis and merely serve as a useful control on the accuracy of the calculation ; the values already obtained should be reproduced. 85-8s] Method of Gauss 91 87. We have now arrived at preliminary approximations to the values of the geocentric distances p l , p 2 , p 3 , the heliocentric distances r lf r 2 , r s and the arguments of latitude u-^, u 2 , u 3 . From these quantities we might proceed to deduce a complete set of elements. But our results are not accurate for two reasons : (1) the effect of aberration has been ignored, and (2) the expressions (6) and (7) employed for ^ and n 3 were of necessity only approximate. The effect of aberration may be stated thus. The light observed at-time t left the source whose distance is p at the time t A, where Ai = 498 8 -5 p/l day = [776116] p in days, 498 s- 5 being the light-time for unit astronomical distance. Had the source moved in the interval A uniformly with the velocity of the observer at time t, its position at time t would be correctly inferred from the observa- tion, without correction, since in that case there is no relative motion between the source and the observer. If now we correct the observation for stellar aberration according to tfie ordinary rule the observer's motion attributed to the source is eliminated and we have the direction of the observed body at time t A from the observer's position at time t. This is the most convenient procedure in the present case, because it enables us to retain the Earth's coordinates (R, L) at the times of observation t throughout the calculation and to make no subsequent change in the planet's observed coordinates (A,, j3) supposing them to be corrected for stellar aberration at the outset. This avoids many changes which would otherwise be necessary in the calculation of subsidiary quantities. It only remains when approximate values of p become known to correct the time t by subtracting A in so far as these relate to actual positions in the orbit. In particular, the corresponding corrections must be applied to the. time intervals T I} r 2 , r 3 . * 88. A better approximation to the values of n l9 n s might now be made by using the formulae of Gibbs or those of 62 and with these values the whole calculation might be repeated. But we proceed at once to introduce the accurate formulae for the ratio of the sector to the triangle, (25) and (26) of 55 in the case of an elliptic orbit. The sectors are i -y i [r. r s ] , \y* [n r 3 ] , %y s |Y, r 2 ] and are proportional to T I} r 2 , T,, (now corrected for aberration). Hence Here y 2 2 = w 2 2 /( 2 + si y? ~ V* = m.i (2g. 2 - sin 2#>)/sin 3 by the formulae quoted, and in the present notation jr.,, cos (u 3 - u^}, m,? = fe^T| 8 / 92 Determination of an Orbit [CH. vin The corresponding equations for y lt y 3 can be written down by a symmetrical interchange of suffixes. Various methods have been devised for the convenient solution of these equations, generally involving the use of special tables. In the absence of such tables, and they are not necessary, we may proceed thus. Writing the cubic equation in the form f _ y * _ ^ Q (2g) = 0, Q (2g) = 3 (2g - sin 2$r)/4 sin 3 ,7 where Q (2g) approaches the value 1 as g approaches the value 0, we compare it with the identity (X 3 - X- 3 ) - 3 (X - AT 1 ) - (X - X- 1 ) 3 = 0. Thus y = c/(\ - X- 1 ) if c 3 _ c 2 ^ 4m 2 Q X 3 -X- 3 ~3~ 3 that is, if c = 2m VQ = H*- 3 - ^~ 3 )- Hence if X 3 = cot /9, 3raV# = cot /3 and if X = cot ^7, y = m\jQ tan x = q (y), p and q being any functions. These correspond to two curves P and Q. Starting with the approximate value x l we find yi=p(i) and hence (#,, y^ 88, 89] Method of Gauss 93 the point P l on P. Next we find similarly (x 2 , y^ the point Q l on Q. This gives the new value x. 2 of x and with this we find successively (# 2 , y 2 ) the point P 2 on P and (x 3 , y 2 ) the point Q. 2 on Q. But if the successive values #], # 2 , # 3 do not differ greatly, the chords PiP 2 , QiQ? lie close to the curves P and Q and their intersection nearly coincides with the intersection of the curves. In this way we find for the correction to the third value x. A X X 3 = \X 2 X 3 ) I \\X% Xi) \X 3 -^2/i In the above case two solutions of (19) with application of the correction just indicated will generally suffice for the accurate determination of g and y. 89. When the values of y 1} y 2 , y-j have been thus obtained we have new values of n l and n 2 by (17). The next step is to recalculate p 2 by (5) and Pi, ps by (12) and (13). Hence r 1; r 2 , r 3 and 1 1} 1 2> 1 3 by (14), new values of ft and i by (15) and finally u lt u 2) u 3 by (16). This brings us back once more to the equations (18) in y. If the result of solving them with the improved values introduced is to leave n^ and n 3 practically unaltered, our object is attained. Otherwise it is necessary to repeat the above steps until a satis- factory agreement is reached. When this stage has been arrived at the problem has been solved, and it only remains to calculate the other elements of the orbit, ft and i having been obtained in the last approximation. The three equations p = rj{l+ecoB (v } - to)}, ( j = 1, 2, 3) are linear in p, e cos &> and e sin to. The symmetrical solution gives p = T^TS 2 sin (u s w 2 )/2 r 2 r 3 sin (u 3 u 2 ) e cos &) = 2 r 2 r 3 (sin u 3 sin w 2 )/2 r 2 r s sin (u 3 u 2 ) e sin &) = X r 2 r 3 (cos u 3 cos u^fSt r 2 r 3 sin (u-j u. 2 ) whence e = sin (/>, &> = CT O and a =p see 3 . This, however, is not the simplest solution. The areal velocity h = fc\/p ( 26) and hence A-r, Vjo = [r, r 3 ] 7/ 2 = y, r,r, sin (M - j) .................. (20) Thus, p being known, we have f) f) \ - +- 2 = 2e cos ^ (u, + u 3 2&)) cos ^ (u 3 u^ Tl Ts V ......... (21) IP f) = 2e sin (wj + u 3 - 2 + l + Ej-e"'BmEj-n(t i -t ) .................. (22) where taji i EJ = ^j tan | ( Wj - ), (j = 1 , 2 or 3). The times tj are here corrected for aberration ( 87). CHAPTER IX DETERMINATION OF PARABOLIC AND CIRCULAR ORBITS 90. The method explained in principle in the last chapter requires no assumption as to the eccentricity of the orbit. Its practical convenience is greatest, however, when the eccentricity is comparatively small. On the other hand the majority of comets move in orbits almost strictly parabolic. For these it is important to have approximate elements after the first observations have been secured, in order that an ephemeris may be calculated to guide observers as to the position of the object. For this purpose the method of Olbers (published in 1797), which depends on the assumption of a parabolic orbit, has continued in use to the present time. Although only five elements have in this case to be determined we still use three complete observations of the comet giving the longitude and latitude (Xj, /3j) at the three times tj. We again take (Rj, Lj) as the corresponding radius vector and longitude of the Earth and pj the geocentric distance of the comet, so that as before x i = a JPj + A J R j> Vi = b jPi + B J R J> Z J = c )Pj + G jRj- Here (xj, y/;, Zj) are the heliocentric coordinates of the comet, (dj, bj, GJ) the direction cosines of pj and (Aj, Bj, Cj) the direction cosines of Rj. In the ecliptic system of axes adopted, a,- = cos \j cos /3j, bj = sin Xj cos ySj, GJ = sin fy. We shall express p 3 in terms of p l and for this purpose it is possible to eliminate p. 2 and R 2 from (1), (2) and (3) in 82. The same result may, however, be deduced from the condition that the orbit is plane in another way. 91. If 8 is the Sun, E lt E 2 , E s the three positions of the Earth, and Cj, C 2 , C s the three positions of the comet, S, C lt C. 2 , C 3 are coplanar. Hence [7*1^2] tetrahedron SE 2 dC 2 [r 2 r 3 ] tetrahedron 2 C. A B,R, C 2 R, a 2 p.,+A 2 R 2 , c. 2 p 2 +C 2 R 2 , G 2 R.^ c 2 p 2 +C 2 R 2> b 3 p s +B 3 R s , 90-92] Determination of Parabolic and Circular Orbits 95 A 2 B 2 C. 2 -T- A B. i-ttj, bipi~\-B 1 RI, c\pi-\-C>iRi a 2 b 2 c. 2 a 3 p 3 +A 3 R 3 , b 3 p 3 +B 3 R 3> c 3 p 3 +C 3 R 3 or, representing determinants by single rows, [i\f-2]{p 3 \a3,A 2 ,a 2 +R 3 A 3 ,A 2 , a. 2 ]} 4-[r 2 r 3 ] {/^ } a 1} A 2 ,a 2 i + R, ^,^1.,, a 2 j} = 0. But if, leaving the directions of p l , p 2 , p 3 unaltered, we move the- plane of the orbit into coincidence with the ecliptic, we see that in the limit [RiR 2 ]R 3 \ A 3 , A 2 , a 2 \ + [R 2 R 3 ] R l \ A lt A 2 , 2 j = must be an identity. Hence [r 2 r 3 ] Ich. A 2> a 2 \ _ i \[R 2 R 3 ] [r a r s ]| j A lt A 2 , a z a 3 , AS, a 2 Now = Mp l 4- m. ct l} A 2 , c.j = b 1} B 2 , b. 2 c Co c I cos \! cos ft, cos L 2 , cos X 2 cos ft sin Xj cos ft, sin 1/2, sin X 2 cos ft sin ft ' , , sin ft = sin ft cos ft sin (X. 2 Z 2 ) sin ft cos ft sin (Xj Z 2 ) and the other determinants can be written down by simple substitutions. Thus M = [r 2 r 3 ] sin /Si cos /3 2 sin (Xa L 2 ) sin /8 2 cos & sin (X, i? % 2 ] ' si n ft cos ft sin (X 3 L 2 ) sin ft cos ft sin (X 2 L 2 ) sin ft sin (i/j L. 2 ) (1) and wi = JR ([^! R 2 ] [rjTa] j sin ft cos ft sin (X 3 L 2 ) - sin ft cos ft sin (X 2 L 2 ) ' In the practical problem the time intervals are usually small and it is possible to substitute the ratio of the sectors for the ratio of the triangles, both for the comet and the Earth, so that I 2 *S I I -^'2 -^"3 I tJ "~" ^2 Thus m = and with sufficient accuracy we may write (3) where M has the value given by (1) and (2), unless the comet is near the Sun and describes large arcs in comparatively short intervals. The effects of parallax and aberration are entirely neglected. 92. The next step is to express r lt r 3 and the chord c joining the extremities of these radii in terms of p 1 . We have rf = 2 (a,^ + A.R,)- = p l 2 + R 1 2 + ^p.R, cos ft cos^ - LJ ......... (4) r 3 2 = 2 (Ma 3 p 1 + A 3 R 3 y- = J/ V + R./ + 2M PI R 3 cos ft cos (X, -,).. .(5) 96 Determination of Parabolic and Circular Orbits [CH. ix and = A 2 pi 2 + (T 2 + 2pj /t<7 cos ..................... .............................. (6) where / A 2 = 2 (Ma,. a^f = M* + 1- ZM {sin ft sin ft + cos & cos ft cos (X 3 - X,)} g* = 2 (-4 S Us - -4,-R,) 2 = # 3 2 + Uj 3 - 2U,U 3 cos (L s - Z,) hgcos = R 3 \M^a 3 A s "S.a 1 A s \ R 1 {M'^a s A l ^a^A-^ = J/cos ft {U 3 cos (\ 3 L 3 ) - R l cos (X 3 Zj)} COS ft {#3 COS (Xj jL 3 ) .Kj COS (Xj Zj)}. If jE'jC' is drawn equal and parallel to E 3 C 3 it is clear that CC 3 = E 1 E 3 = g, GC^hp,, (?! (7 3 = c and (7,00, = 180 -. But Euler's equation gives 6A; (t, - t,) = (r, + r s + of - (r, + r a - cf and this must be satisfied by the appropriate value of p l in (4), (5) and (6). This value must be found by a process of approximation and for a suitable starting point we may consider c small in comparison with r^ + rg, r 1 = r s and R, = 1. Then . 6lc (t 3 - t,) = (n + r s ) f . Sc/O-, + r,) = 3 V2 . c Vn or 2& 2 (, - ^) 2 A 2 = (Pi 2 + 2p, cos /> . (///n- ^ 2 ) (Pi 2 + 2 pi c s & cos (A! - A) + I} 4 . With approximate values of the numbers which occur in this equation it is easy to find by trial a value of p 1 which is correct at least to one decimal place. Then with this value of p l it is possible to calculate c in two ways : (i) directly by (6), (ii) through r 1} r 3 given by (4) and (5) and inserted in Euler's equation, which may be written ( 52) in the form 3fc (, - O/V2 ( r i + r *f = ' sin e ' c = 2 V 2 ( r i + ') sin o @ V cos I - -( 7 > or solved by special tables. Two values of c thus correspond to a hypo- thetical value of pi, and the latter must be varied until the discrepancy between the former is made to disappear. A rule analogous to that given in 88 leads quickly to the desired value of pj. For if the values p/, p" lead successively to the differences AjC, A 2 c in c, it is easy to see that the value of P! to be inferred is given by Pi = Pi" + (pi" - pi) A 2 c/(A lC - Age). In ordinary cases the correct result is quickly obtained in this way. 93. When p l and p 3 = Mp^ have been obtained it only remains to de- termine the elements of the orbit. The formulae of 86 arc again appropriate, namely TJ cos bj cos (lj LJ) pj cos fa cos (\j LJ) + Rj TJ cos bj sin (lj LJ) = pj cos fy sin (\j - LJ) TJ sin bj = PJ sin $,- 92-94] Determination of Parabolic and Circular Orbits 97 (j = 1, 3), for the heliocentric distances, longitudes and latitude of the comet. Here r 1} r z should reproduce the values finally arrived at in the course of determining p^. Also 2 tan i sin {i (^ + 1 3 ) HI = sin (6j -I- 6 3 )/cos 6 X cos 6 3 cos ^ (1 3 Z x ). . .(8) 2 tan i cos { (^ + 4) O} = sin (6 3 fcj/cos &! cos 6 3 sin (Z 3 - Zj). . .(9) (0 < i < 90 if ^ > /!, 90 < i < 180 if 1 3 < IJ give H and i. The Arguments of latitude are given by tan Uj = tan (lj fi) sec i (jl, 3), where in this case < Uj < 180 if 6; > 0. By the equation of the parabola yV/ = \fi\ cos ^ ( Uj ) = \/r 3 cos ^ (w 3 CD) ............ (10) whence 0*3 V^i sin | (i -f M 3 - 2o>) sin ^ (M 3 u t ) V^s + V r i C S i (z*j + W 3 2ft)) COS i (w 3 UT) or tan i ( ttl + u 3 - 2ft>) = ^ 3 ~ ^ cot { (u, - 1^) ............ (11) yr, -4- v ? 'i which gives &> = w H and also 5-, the perihelion distance. Finally, T being the time of perihelion passage, we have ( 29) T = tj- q* {tan ^ (M,- - o>) + i tan 3 ^ (w,- - co)} A/2/& ......... (12) (j i = 1, 3). This completes the determination of the five elements. 94. It is to be noticed that while the first and third observations have been completely used, the second observation has only entered partially into the calculation. In fact the five elements have been determined from six given coordinates in a unique way because X 2 , ft. 2 have not been used independently but only in the form cot /3 2 sin (X 2 X 2 ) in the equation (1) for M. Consequently it cannot be expected that the elements will satisfy the second place exactly and the magnitude of the discordance is an im- mediate test of the derived orbit. The second place is therefore calculated by finding ( 29) w 2 = u 2 + n 2 Z 2 B! = l^X z + Wj Y 2 + w^a, B 2 = 1 2 X^ + m 2 Y l + n 2 Z t we have a 2 = Pl * - 2A lPl a~ cos nr = p^p 2 cos 6 B 1 p l B 2 p 2 + X^X 2 + Y l Y 2 + Z 1 Z 2 . Hence 4 2 sin 2 $nr = p^ + p 2 - 2pip 2 cos 6 2 (A 1 BJ p^ Z (A 2 B 2 ) p., + (X. - Xtf + (Y 2 - F,) 2 + (Z, - Ztf cos 2 \e {p 2 - Pl -$(Ai-A l -B a + BJ sec 2 %ey + sin 2 1 6 [p 2 + p l -$(A*+A 1 - B. 2 - B,} cosec 2 & O} 2 sec^ - \(A 2 + A l -B a - 5,) 2 cosec 2 0. The equations, which must be solved by trial, can therefore be reduced to the form sini/r^ MJa, p t = a cos ^ + A l ~\ sin i/r, = M 2 /a, p 2 = a cos fa + A 2 I... (13) 4a 3 sin 2 |nr = cos 2 \Q (p 2 p l 6j) 2 + sin' J (p 2 + p l 6 2 ) 2 + cj where (without the transformations -appropriate to the coordinate system) M? = X:- + I? + Z* - A?, MJ = X.? + F 2 S + Z* - A./ b, = (A.,-A 1 - B 2 + A)/2 cos 2 \e 6 3 = (A 2 + A l -B 2 - 5^/2 sin 2 c = (X, - XJ- + (Y 2 - Y,r~ + (Z 2 - Zrf -(A 2 -B 2 -A, + 5,) 2 /4 cos 2 ^ - (^ 2 - B 2 + A, - B^/4, sin 2 \ 0. A trial value of a gives, by (13), fa, fa and hence p l} p 2 ; these lead to a value of n and the process is continued until values are obtained consistent with the relation w 2 a 3 = & 2 . In the case of a minor planet log a = 0'4 is indicated as the appropriate initial value. With the above formulae the calculation can be performed directly in equatorial coordinates, and little will be gained by introducing the ecliptic system. When a and n have been 72 TOO Determination of Parabolic and Circular Orbits [CH. ix found, p-i, p 2 are also known by (13) and hence the heliocentric coordinates of the planet Vi = lipi-Xi, yi = m 1 p 1 -Y l , z l = n 1 p 1 -Z l %2 = 4/t>2 - X*, 2/2 = intfz - YZ, z z = KZP* - %z- 96. Gauss has given a method for finding a circular orbit, based on ecliptic coordinates. Let (JR^, L^, (R 2 , L 2 ) be the heliocentric distances and longitudes of the Earth at the times t lt t 2 and (\, /3j), (X 2 , /3 2 ) the cor- responding observed longitudes and latitudes of the planet. If in the plane triangle SE 1 P l the angle at Pj is denoted by z l and the exterior angle at E, by B lt P 1 SE 1 = B 1 -z 1 and a sin z l RI sin ^ (14) Similarly in the triangle SE 2 P 2 , with similar notation, asin^ 2 = R 2 sin 8 2 (15) The directions of the sides of the two triangles are now represented on a sphere of unit radius, SE 1} 8E 2 being represented by E l , E 2 on the ecliptic, SP l} SP 2 by two points P 1} P 2 . If G l} G 2 represent E^P^, E 2 P 2 , these points lie respectively on the great circles E- 1 P 1 , E 2 P 2 and the arcs E 1 G 1> E 2 G 2 are S x and 8 2 . Let the circles E l G l , E 2 G 2 cut the ecliptic at the angles f y l , y 2 . Then the projections of the radius through GI on the radius through E ly the radius through the point on the ecliptic 90 in advance of -fc'j and the radius through the pole of the ecliptic give cos y3j cos (A-i jLj) = cos B 1 cos & sin (X x Lj) = sin S l cos ^ sin ySj = sin 8 1 sin ^ and similarly cos /3 2 cos (X 2 L 2 ) = cos 8. 2 cos /8 2 sin (X 2 Z/ 2 ) = sin & 2 cos 7^ sin /3 2 = sin S 2 sin j 2 whence S 1} S 2 and 7^. 7 2 . Let the circles E l P l) E 2 P 2 meet in D at an angle 77. If DE l = (f> l and DE^ = <^. 2 , the analogies of Delambre applied to the triangle DE 1 E 2 in which the side E^E Z is Z 2 L^ and the adjacent angles are 71, TT 70, give ^ + <^> 2 \) (TT _ fir TT 7 2 - " 8 Jf S ^_-j2)}' cos [44 2 /J (4 or more explicitly sin ^77 sin ^ (^ + 2 ) = sin ^ (Z 2 - 1^} sin (7, + 7,) sin ^77 cos % (! + <^ 3 ) = cos (Z a - A) sin \ (y 2 - 7, cos ?; sin | (0! - ^) = sin | (X a - LI) cos | (7.3 + 7 cos \T) cos |(^! - < 2 ) = cos | (L 2 - L^ cos ^ (73 - 7 95-97] Determination of Parabolic and Circular Orbits 101 whence < 1; < and 77. But since the arc E 1 P 1 = B 1 z 1 and DP l = (f>i 8i+Zi and DP 2 = while P l P 2 = n (t. 2 ^), n being the mean motion. Hence cosn(t 2 t l )=cos( 2 8 2 +^ 2 ) + sin(^) 1 S 1 +2' 1 )sin(^) 2 S 2 + or better, since n ( 2 1) is a small angle, sin 2 1 re ( 2 - j) = cos 2 77 sin 2 i (^ + .sv, - s,) + sin 2 \T\ sin 2 (x 2 + z^ + zj.. .(16) where The solution is conducted in the usual way. Since S lt S 2 are known an assumed value of a gives z ly z by (14) and (15). Then ^,, ^ 2 an d ? being known, the value of n is deduced from (16), and the process is continued until values are found which satisfy the relation W 2 a 3 = & 2 . When this has been done, the values of z 1} z have also been found, and hence the geo- centric distances are given by p l sin z-i = .Rj sin (Si - z^), p 2 sin z z = R 2 sin (S 2 z 2 ) but these distances are not actually required. Since the arc E l P l on the sphere is 8 l z l and makes the angle ^ with the ecliptic, we have the heliocentric longitude and latitude of P 1 (as in the case of G^) given by cos 6j cos (^ Z,) = cos (B 1 z^) cos 6 t sin (l-i Zj) = sin (Sj ^) cos ^ l sin b l = sin (Sj z^) sin 71 with similar formulae for (1%, b 2 ) the heliocentric longitude and latitude of the planet in its second position. 97. If (/i, &]), (1 2 , 1 2 ) have been thus obtained the remaining elements are easily found. For by (15) of 86" the node and inclination are given by 2 tan i sin (i (^ + 2 ) H} = sin (6j + b 2 )/cos h cos 6 2 cos ^ (1 2 ^) 2 tan i cos { (/, + 1 2 ) n j = sin (b. 2 6j)/cos 6j cos 6 2 sin (/ 2 /j) and then the arguments of latitude by tan MJ = tan (^ II) sec i, tan u 2 = tan (Z 2 H) sec i with the check u 2 MI= n. (^ 2 ^). As the fourth element the argument of latitude i.i at a chosen epoch t may be taken, and this is simply u = MJ + w (< - tj) = u 2 + n (t t 2 ) where t l} t 2 may be antedated for planetary aberration. If, on the other hand, the heliocentric coordinates (x ly y l} z^ and (x 2 , y 2 , z^) have been found as in 95, and i' is the inclination of the orbit to the 102 Determination of Parabolic and Circular Orbits [OH. ix plane z = and ft' is reckoned in this plane from the axis of x towards the axis of y, the plane of the orbit is x sin ft' sin i' y cos ft' sin i' + z cos i' = and as this is satisfied by the two points on the orbit we have sin ft' sin i' _ cos ft' sin i' _ cos i' y\z z y%Zi x\z<2 (K^ZI Xi y z x z y\ The solution can then be completed as before, the arguments- u being now reckoned in the plane of the orbit from the node in the plane z=Q. The meaning of the quantities 6j, b 2 and c in 95 may be seen thus. Let an axis of z be taken perpendicular to p 1 and p, and an axis of x midway between the directions of p l and p 2 , so that (l lt w,, HI) become, (cos ^6, sin |#, 0), ( 2 , w 2 , n. 2 ) become (cos|#, sin ^ft, 0), and (X lt F,, Z^, (X.,, Y, Z. 2 ) become (i/, F/, *,'), W, F/, /). Then 6, = (X/ - Z,') sec |^ If the difficulties of reducing this apparently simple problem to a practical form of calculation are carefully considered, in view of the small quantities which occur, the merit of the method in 96 will be better understood. The reader must realize that the general problem of determining orbits from observations close together in time is essentially a question of arithmetical technique, and not of any particular mathematical difficulty. This is well illustrated in the history of the problem, especially in the eighteenth century. It is to be remarked that the problem of finding a circular orbit to satisfy the given observations cannot always be solved. That a solution is not necessarily to be expected with arbitrary data can be readily seen, though the equations, not being algebraic, are too complicated to make a general discussion of the conditions feasible. It is enough to say that cases have occurred in practice in which a circular approximation to the orbit has proved impossible. The number of minor planets already discovered is approaching a thousand, and the most frequent eccentricity is in the neigh- bourhood of 012. CHAPTER X ORBITS OF DOUBLE STARS 98. There exist in the sky pairs of stars the components of which are separated by no more than a few seconds of arc, and frequently by less than one second. So close are they that they can only be seen distinctly in powerful telescopes, if indeed they can be clearly resolved at all. Such pairs are so numerous that probability forbids the idea that the contiguity of the stars can be explained by chance distribution in space. They must be physically connected systems for the most part and it is to be expected that the relative motion of the stars will reveal the effect of mutual gravitation. That this is actually true was discovered by Sir W. Herschel. The motion is referred to the brighter component as a fixed, point. The relative motion of the fainter component takes place in an ellipse of which the principal star occupies the focus ( 24), unless there are other bodies in the system, or there proves to be no physical connexion between the pair. The apparent orbit which is observed js the projection of the actual orbit on the tangent plane to the celestial sphere, to which the line of sight to the principal star is normal, and since the point of observation is very distant compared with the dimensions of the orbit the projection can be considered orthogonal. Hence the law of areas holds also in the apparent orbit, which is equally an ellipse. But in this orbit the brighter star does not occupy the focus : its position gives the means of determining the relative situation of the true orbit. The observations give the polar coordinates, p, 8, of the companion, the principal star being at the origin. The distance p is expressed in seconds of arc and the linear scale remains unknown unless the parallax of the system has been determined. The position angle 6 is reckoned from the North direction through 360 in the order N., E. or following, S., W. or preceding. The planes of the actual and apparent orbits intersect in a line called the line of nodes and passing through the principal star. The position angle of that node which lies between and 180 will be designated by H. Thus if the line of nodes is taken as the axis of , 104 Orbits of Double Stars [CH. x On the other hand, in the plane of the actual orbit, the longitude of periastron X is the angle measured from this node to periastron in the direction of orbital motion. Hence in this plane, if the line of nodes is taken as the axis of x, x = r cos (w + X), y = r sin (w + X) where r is the radius vector and w the true anomaly of the companion. But if i is the inclination of the two planes to one another, = x and v) = y cos i, so that . p cos (6 O) = r cos (w + X) p sin (9 H) = r sin (w + X) cos *'. Here the limits contemplated for i are and 180. If < i < 90, 6 and w increase together with the time and the motion is direct. If 90 < i < 180, B decreases with the time and the motion is retrograde. This is a departure from the more usual convention according to which i is always less than 90. It is then necessary to state whether the motion is direct or retrograde, and in the latter case to reverse the sign of cos i. Ordinary visual observations of double stars, however, must leave the position of the orbital plane in one respect ambiguous, since there is nothing to indicate whether the node as defined is the approaching or receding node. The two possible planes intersect in the line of nodes and are the images of one another in the tangent plane to the celestial sphere. In addition to the three elements, fl, X, i, now defined, four other elements are required. These are a, the mean distance in the true orbit, expressed like-p in seconds of arc; e, the eccentricity of the true orbit; T, the time of periastron passage ; and P, the period (or n = 2?r/P, the mean motion) ex- pressed in years. 99. The measurement of double stars is difficult and the early measures were very rough indeed. As the accuracy of the observations is not high refined methods of treatment are seldom justified and graphical processes have been largely employed. The observed coordinates may be plotted on paper and the apparent ellipse drawn through the points as well as may be. Let C be the centre and S the position of the principal star. The problem consists in finding the orthogonal projection by which the actual orbit is projected into this ellipse and the focus F into the point S. The direction of the line of nodes can be determined by the principles of projective geometry. Conjugate lines through the focus F form an orthogonal involution. They project into an overlapping involution of conjugate lines through S. Of this involution one pair is at right angles and as in this case a right angle projects into a right angle it is clear that the line of nodes is parallel to one of the pair. Let SA, SA' ; SB, SB' be two pairs of conjugate lines through 8. When the apparent ellipse has been drawn these can be 98, 99] Orbits of Double Stars 105 found by drawing tangents at the extremities of chords through S; or by inscribing quadrangles in the ellipse, for each of which S is a harmonic point. On CS as diameter describe a circle, centre K. Let A l} AJ ; B lt B^ be the points in which the conjugate lines intersect th"is circle and let A 1 A l ', B l B l ' intersect in 0. Corresponding points of the same involution on the circle are obtained by drawing chords through 0, and if OK meets the circle in N, N', SN, SN' are the orthogonal pair of the involution pencil required. Let CABNA'B' be a transversal of the pencil drawn parallel to SN' so that A A', BB' subtend obtuse angles at 8. This is an involution range of which N, since it corresponds to the point at infinity, is the centre, so that AN. NA'=BN . NB'. On NS take the point F such that NF* is equal to this constant product. Then F is the intersection of circles on the diameters A A', BB' and A FA', BFB' are right angles. Hence if NF be rotated about Fig. 4. CN until FS is perpendicular to the plane CNS (the plane of the apparent orbit) right angles at F will be orthogonally projected into the involution of conjugate lines at S. The position of the focus F of the actual orbit has therefore been found, and the orthogonal projection by which the true and the apparent orbits are related. The true orbit may be plotted point by point on the plane of the paper, with its centre C and focus F. For if P' is a point on the apparent orbit and P the corresponding point on the true orbit PP' is perpendicular to CN and PF, P'S meet on CN. In particular, if X' (fig. 5) is a point where OS meets the apparent orbit, the corresponding point X in which the perpendicular through X' to CN meets CF is a vertex of the true orbit and CX = a. The eccentricity is given by CS CF CX'~ CX = e Orbits of Double Sf - 2Hx;y + Ay 2 + c 2 = represents the projection of the circle on the latus rectum ( 101), or an ellipse with axes 2p and 2p cos i and its transverse axis coinciding with the line of nodes. It is therefore identical with the equation (x cos ft 4- y sin ft) 2 (y cos ft x sin ft) 2 _ p 2 p" cos 2 i and thus - B/c 2 = p~ 2 cos 2 ft + p~* sec 2 i sin 2 ft H/c* = ( p~* p~ z sec 2 i) sin ft cos ft - ^1/c 2 = ^>- 2 sin 2 ft + p~ z sec 2 1 cos 2 ft or p- 2 tan 2 1 sin 2ft = - 2#/c 2 ~ 2 tan 2 i cos 2ft = 5 - 2p~ 2 + p~ 2 tan 2 1 = - (B + A)/c 2 which determine ft, p and i. Again, the perpendicular from the focus on the directrix is a (e~ l e) = pe~*. Hence the intercepts on the line of nodes and on the line perpendicular to it between the focus and the directrix are p/e cos X, p/e sin A,. The projections of these intercepts, also at right angles, are p/e cos X, p cos i/e sin X. But the projection of the directrix is the polar of the origin, or the line gx +fy + c = 0. . Hence (g cos fl +/sin H) p/e cos X + c = ( g sin n +/cos fl) p cos i/e sin X + c = so that e and X are given by the equations e sin \ = p cos i (/cos H g sin ft)/c e cos X = p (/sin fl +$r cos fl)/c. Equations for the five geometrical elements in the above form were first given by Kowalsky. The form of the equation which represents the projection of a circle is defined by the fact that the asymptotes of the projected ellipse are parallel to the projection of the circular lines and therefore to the tangents from S to the apparent orbit. It will be found that the projection of the auxiliary circle, referred to its centre, is in the usual notation C' 2 (Bx- - ZHxy + Ay-) + A 2 = 110 Orbits of Double Star* [CH. x and that of the director circle C* (Bx> - ZHxy + Ay 2 ) + A (A + Cc) = while the eccentricity of the true orbit is given by l-e* = G'c/A. 104. In some few cases a double star has been observed over more than one complete revolution. The period P is then known approximately and the date T of periastron passage, when the companion is situated on the diameter of the apparent orbit through 8. Otherwise, when the geometrical elements have been determined, two dated observations suffice to determine these two additional elements. For two observed position angles 6 l , 2 give the corresponding true anomalies w lt w. 2 and hence the eccentric anomalies EI, E 2 , since /n e \ tan (6 - H) = tan (w + \) cos i, tan %E L- tan ^w. v \1 ~t~ 6/ Then n (t, -T) = E l -e sin E lt n (t, -T) = E 2 -e sin E* determine n = 2-7T/P and T. In practice a larger number of such equations will be employed in order to reduce the effect of errors in the observations. The law of areas can also be applied directly to the apparent orbit, for if a^ is the area described by the radius vector between the dates t lt t. z , and A l is the area of the ellipse, P = ( 2 O ^i/ff-i, and similarly T can be determined. A primitive method which has been used for measuring the areas consists in cutting out the areas in cardboard and weighing them. When the parallax w of a double star is known, a/vr is the mean distance in the system expressed in terms of the astronomical unit. Hence ( 24), if m, in' are the masses of the components, k* (m + m') = 47r s a 8 /w 3 V J8 while A; 2 = 4-Tr 2 if the mass of the Sun-Earth system and the sidereal year are taken as units. For this purpose the mass of the Earth is negligible and thus, P being expressed in years, ,m -f m = a?/tv 3 P- is the combined mass of the system, compared with that of the Sun. 105. The apparent orbit can be reconstructed, on an arbitrary scale, from observed position angles alone. This course was advocated by Sir J. Herschel, who considered the measured distances of his day very inferior in accuracy. With this object the position angles are plotted as ordinates with the time as abscissa. Owing to inaccuracies the points will not lie exactly on a smooth curve, but such a curve must be drawn through them as well as possible. Let i/r be the angle which the tangent to the curve at the point 103-105] Orbits of Double Stars 111 (t, 6} makes with the axis of t y so that d0/dt = tan ty. But since Kepler's law of areas is preserved in the apparent orbit, p 2 6 = h, an undetermined constant. Hence p = \/(h cot \|r) and the apparent orbit can thus be derived graphically from the (t, 6} curve. The elements with the exception of a can then be obtained and finally a is determined by the measured distances, of which no other use is made in the calculation. The opposite case may arise, and is illustrated by the star 42 Comae Berenices, in which the determination of the elements must be based on the distances. Here the plane of the orbib passes through the point of observa- tion, i = 90 (or practically so) and the position angles serve only to determine O. If the star has been observed over more than one revolution the period P may be considered known. Corresponding to the point (a cos E, b sin E) on the orbit, the observed distance is while p = a cos E cos X b sin E sin X - ae cos X = R cos (E + /3) ae cos X If the observations are plotted for a single period, from maximum to maximum, the result is to give the curve a; = nt = nT+ E e sin E y = p = R cos (E + /3) ae cos X which is a distorted cosine curve. Maximum and minimum correspond to E = /3, TT /3 and give nti = nT /3 + e sin /3, y^ = R ae cos X nt 2 nT + TT /3 e sin /3, y 2 = R ae cos X whence R and ae cos X, while in addition n (t 2 - ti) = TT 2e sin /3. These equations may be supplemented by a simple device. Taking the origin of a; at the first maximum let the curve y R cos x ae cos X also be drawn. Let P be a point on this curve and Q the corresponding point on the first curve such that the ordinates at P and Q are equal. Then at P, x = E + (3, so that Hence the curve y = esin(x {3) + ft n(T ,) can be constructed by laying off on each ordinate through P a length equal to QP. This is a simple sine curve, the form of which will serve to show 112 Orbits of Double Star* [CH. x any irregularities in the (nt, p) curve from which it is derived. The ampli- tude is 2e, represented on the scale by which 2?r corresponds to the period in x. The value of e being thus known gives /3 from (t. 2 x ) and hence a and X, since a cos \ = R cos /3, a sin \ = R sin /3/\/(l e 2 ). T is then given by the maximum and minimum of the original curve. But the sine curve has its maximum at x = /3 + |TT and its central line is y = ft n(T tj). These conditions must also be fairly satisfied by the adopted solution. 106. Graphical methods, such as those sketched above, only provide a first approximation to the solution of a problem. Here in general the obser- vations are too rough to make a closer approximation feasible. But if it is necessary to improve the elements thus found, each observation gives one equation in the following way. Let da, dfl, ... be the required corrections to the approximate elements, a, H, .... For the time t of an observation 6 (or p) can be calculated. Its value is e c =f(t,a, n, ...). But the observed value is e o =f(t, a + da, n + dfl, . . .). If then the elements have been found with such an accuracy that squares, products and higher powers of da, dfl, . . . can be neglected, 6 - 6 C = ^~ . da + ^. . dl + . . . oa oil a linear equation in da, dl, .... And similarly with p. The coefficients are = ? = da da a - - an an~ .~r = ^ sin 2 (6 H) tan i, ^. = -p sin- (6 - H) tan i ^- = cos i, = p sin 2 (6 H) sin i tan i o\ p- d\ 80 no* . dp na 2 ( . . dp ^= - cos i V(l - e*), ^T= - pepsin E + V(l - e'-) __ dn n 'dT' dn~ n ' dT d0 r" fa 1 \ . . dp dp fa 1 \ ap 5- * -; I - + 1 , sm w cos i, ^ = (- + - - sin w - cos w de p-\r \-e-J de d\\r 1 - e-J r the verification of which may be left as an exercise. 105-108] Orbits of Double Stars 113 107. In some cases the position of a binary system has been measured relatively to some neighbouring star C which is independent of the system. Let A be the principal star, m^ its mass, (ac l} y^ its coordinates at the time t', and similarly let B be the companion, m 2 its. mass, (x z , y 2 ) its coordinates. A series of measures of AB gives x. 2 x 1 = p cos 0, 2/2 2/i P sm $ while the measures of AC give x 3 x l} y 3 yi, (# :i , y 3 ) being the position of C. Let (f, 77) be the c.G. of AB, so that (mi + m 2 ) = m l x l + m 2 x 2 , (r^ + w 2 ) <>7 = m^ + 7n 2 i/ 2 . But the motions of C and of the C.G. of AB are uniform and independent. Hence | = x 3 + a + fit, i] = 2/3 + a' + fit where ft, ft' are the proper motions of the C.G. relative to C, and (a, a') is its position relative to C at the chosen epoch to which t refers. Thus (m l + m. 2 ) (x s + a + fit) = m l x l + m^x z or a + fit f(cc z #j) + % 3 x l = and a' + fft -f(y, -y 1 ) + y 3 -y 1 = similarly, where /= TWa/CWj + m a ). From a series of such equations a, a', /3, /3' and f can be determined and therefore the ratio of the masses of A and B. But if a is the mean distance, P the period and to- the parallax of the system AB, and the masses of the individual stars, expressed in terms of the Sun, become known. 108. In certain cases the absolute coordinates of stars apparently single have exhibited a variable proper motion. It is then assumed that the varia- tion is periodic and due to orbital motion in conjunction with an undetected body. The motion to be investigated is relative to the C.G. of the system, which itself is supposed to move uniformly. In the plane of the orbit the coordinates are a' (cos E e), b' sin E, and therefore in the plane of projection, when referred to the line of nodes and the line at right angles, they become x = a' (cos E e) cos A, b' sin E sin \ y={af (cos E e) sin X + b' sin E cos X} cos *'. Hence the orbital displacement in the direction of the position angle Q is = ayos (tl-Q)-y sin (ft - Q) = g cos E + h sin E ge p. D. A. 8 114 Orbits of Double Stars [CH. x where g a {cos X cos (11 Q) sin X sin (O Q) cos i] h = b' {sin A, cos (O Q) + cos A, sin (H Q) cos i] and Q = 90 for displacements in R.A., Q = for displacements in declination. The observations of one coordinate, say 8, therefore give a series of equations of the form S = B + fj>&t + g cos E + h sin E ge with E - e sin E = n (t - T). From these e, n (or P), T, /AS, & > g an d h can be determined. Since g and h are functions of a, H, A, and i, these four elements cannot be derived from observations of one coordinate alone. But from observations of the other coordinate, say a, the corresponding quantities g and h' can be found and the elements of the motion are then completely determinate, including a', the mean distance from the C.G. of the system. In the two notable examples of this kind, Sirius and Procyon, the companion was discovered afterwards. It thus became possible to find the relative mean distance a and hence the ratio of the masses, since Hence, the parallax being known, the individual masses of the components have been determined. It is to be noticed that, when the companion cannot be observed, the function of the masses which can be found is mj' (n^ + m. 2 ) For this is equal to of'/v^P*. ~ 2 CHAPTEE XI ORBITS OF SPECTROSCOPIC BINARIES 109. Another class of orbits which are based on pure elliptic motion is presented by those systems which are known as spectroscopic binaries. It is now possible to determine the radial velocities of the stars in absolute measure with high accuracy. This follows from the application of Doppler's principle to the interpretation of stellar spectra. On the simple wave theory of light this principle is easily explained. A light disturbance travels out- wards from its source in a spherical wave front which expands in the free ether of space with the uniform velocity U. Let a fixed set of rectangular axes be taken in this space, and let (x l , y lf Zj) be the position of the source at the origin of time. Let (u lf v lt w^ be the velocity components of the source, supposed to be in uniform motion, and t the time at which a light disturbance is emitted. Similarly let (# 2 > 2fe> ^2) be the position of the observer, also supposed to be moving uniformly, ( 2 , v 2 , iu 2 ) the velocity components, and r the time at which the specified disturbance reaches him. For simplicity the motions have been considered uniform, but obviously they are immaterial except as regards the source at the instant t and the observer at the instant T. Let the corresponding positions be A, B respectively and let the distance AB = R. Then dR _-, / dr \ tr dr where (a, ft, 7) are the direction cosines of AB and V lt F 2 are the projections of the velocities (u v , v lt w^), (u 2 , v 2 , w 2 ) on this line. But since the wave reaches B from A in the time (r t), Hence ,_ , .-> , dt U-V., U U(U-V Z ) ' 82 116 Orbits of Spectroseopic Binaries [CH. xi Now ( V 2 Vi) is the component of relative velocity of A and B, measured in the direction of separation of the two points. This is a definite quantity. But F 2 is a component of the observer's absolute motion in free ether, and this is unknown. Presumably it is small in comparison with U, and the last term can be rejected as a negligible effect of the second order. Or, on the theory of relativity, V z is not only unknown but unknowable, and the effect is completely compensated by a transformation of the ideal coordinates of space and time into another set which is the subject of observation. All this has its counterpart in the theory of aberration, with which it is intimately related. Whether the limitation is imposed by the imperfection of practical observations or by the ultimate nature of things, it is necessary to be content with the effect of the first order. If the light emitted at A has the wave length \, the frequency of a particular phase in the wave train at A is U/\. But the number of waves emitted in a time dt is received at B in the time dr. If then the apparent wave length of the light received at B is X' and the apparent frequency U/\', UX' 1 dt = UX'- 1 dr and therefore V_rfr_ V \ ~ dt ~ f U where V is the relative radial velocity of A from B. Thus the application of Doppler's principle gives where AX is the increase of wave length (or displacement measured positively towards the red end of the spectrum) of a spectral line, of which the natural 'wave length in the star is supposed known. Further details on the practical methods of reduction would be out of place here, and this explanation must suffice. It is usual to express V in km. /sec., and the velocity of light maybe taken to be U= 299860 km. /sec. 110. From the measured radial velocity must be deduced the radial velocity of the star relative to the Sun, or rather relative to the centre of gravity of the solar system. This requires the calculation of certain corrections, of which the most important are due to (1) the diurnal rotation of the observer, and (2) the annual elliptic motion of the Earth relative to the Sun. The effects of perturbations of the Earth and Sun are compara- tively small. An observer situated on the equator is carried by the Earth's rotation over 40,000 km. in a sidereal day. This means a velocity of 0'46 km. /sec. Hence the velocity of an observer in latitude < is 0'46 cos km. /sec. always directed towards the E. point. If is the angular distance of the star from this point at the time of observation, cos 9 = cos 8 cos (h + 90), where B is the 109, no] Orbits of Spectroscoptc Binaries 117 declination and // the W. hour angle of the star. Hence the additive correction corresponding to (1) is v d = + - 46 cos < cos Q-- 0*46 cos cos 8 sin h. Again, the Earth's elliptic velocity is compounded ( 26) of one constant velocity V l perpendicular to the radius vector and another eV l perpendicular to the major axis, e being the eccentricity of the orbit. These .vectors are directed to points in the ecliptic of which the longitudes are 90 and F 90, where 8 is the longitude of the Sun and F the longitude of the solar perigee. Let (I, /3) be the star's longitude and latitude. Hence the required correction for the Earth's orbital motion is Va = +V l cos /3 [cos (I - B + 90) + e cos (1-T + 90)}. Now Fj is precisely that vector on which the constant of stellar aberration depends, so that if k" is this constant, V 1 = k"Uj 206265" = 29-76 km./sec. when the standard value of k, 20"'47, is adopted with the value of U given above. Hence the correction for (2) is v a = + 2976 cos (sin ( - 1) + e sin (T - I)}. It is evident that the process might be reversed and the value of k deter- mined by observing the apparent radial motion of one or more stars at different times of year. This has been done at the Cape Observatory, with the result that the standard value of k was reproduced very exactly, an excellent test of the theory. Indeed this is probably the best available method of finding the constant of aberration : it will be noticed that the adopted value of U, being a factor of both V l and V, will scarcely affect the resulting value of k. When the necessary corrections have been applied to the apparent radial velocity of a star, the star's radial velocity is obtained relative to the solar system. This is affected by the motion of the latter relative to the stellar system as a whole. Hence conversely when the radial velocities of a number of stars scattered over the sky are known, it becomes possible to deduce the motion of the solar system relative to the average of those stars in absolute measure. If, further, CT is the parallax of a star, and /* its total annual proper motion, its transverse velocity is p/nr when expressed in astronomical units per year. Now with the solar parallax 8"'80 and the Earth's equatorial radius 6378'249 km., the astronomical unit (or Earth's mean distance from the Sun) is 149,500,000 km. Hence this unit of velocity is equivalent to 4*737 km./sec. and the star's transverse velocity is 4737 /H/OT km./sec. Thus the velocity of a star relative to the Sun can be completely determined in absolute measure. This concerns questions of stellar kinematics which are now entering the region of dynamics but lie outside our present scope. 118 Orbits of Spectroscopic Binaries [CH. xi 111. Repeated determinations of the radial velocity of a star yield values which in the majority of cases are constant within the errors of observation. The motion of the star is apparently uniform. But in other cases, perhaps a third of all the brighter stars, changes are observed which prove to be regular and periodic. These are attributed plausibly to the motion of one component in a binary system. Such spectroscopic binaries differ from visual doubles only in the scale of their orbits, which prevents them from being resolved even in the most powerful telescopes, while their periods are to be reckoned in days instead of years or even centuries. It may appear that the spectrum of the second component should also be seen. When the components are fairly equal in brightness, as in ft Aurigae, this is so ; the lines of the spectrum are seen periodically doubled. But with other stars, and this is the more common type, the companion is relatively so faint that only one spectrum is shown : it is quite unnecessary to suppose that the companion is then an absolutely dark body. Even when both spectra are visible the secondary spectrum is often difficult to detect and usually difficult to measure. As a particularly interesting example Castor (a Geminorum) may be quoted. The telescope reveals this star as a visual double, and the spectroscope shows that both components are themselves binary systems. More complex systems can be inferred from spectroscopic measures alone. Thus Polaris, which appears in the telescope as a single star, has been shown to be a triple system, consisting of a close pair revolving round a more distant third body. Here the motion will be considered in the first instance of one component of a binary system about the common centre of gravity, and it will be seen how far the elements of an elliptic orbit can be deduced from the measured radial velocities, these being based on the comparison of the star's spectrum with that from a terrestrial source (usually the spark spectrum of iron or titanium). 112. Since the period is generally short, the observations extend over several revolutions and the period P is determined by obvious considerations with fair exactness. This being known, the observed velocities can be referred to a single period with arbitrary epoch and plotted as ordinates with the time as abscissa in a diagram called the radial velocity curve. Such a curve is illustrated in fig. , while the relative orbit is shown in fig. b, corresponding points being indicated by the same letters. The focus of this orbit is G, the centre of gravity of the system. The line of nodes AGB, passing through A the receding node and B the approaching node, is the line drawn through G in the plane of the orbit at right angles to the line of sight. The points P 1} P 2 mark the position of periastron and apastron, and the angle from G A to GP^ , measured in the direction of motion, is the longi- tude of periastron, w. The true anomaly at any point of the orbit being iv, the longitude of this point from A is u = o> + w. Let i- (0 < i < 90) be the Ill, 112] Orbits of Spectroscopic Binaries 119 inclination of the orbit, this being the angle between its plane and the plane which is normal to the line of sight, and let e be the eccentricity. 150 200 250 300 days Fig. 6 : (a) upper, (b) lower. The orbital velocity of the star is compounded ( 26) of one constant velocity V 2 transverse to the radius vector and another eV 2 perpendicular to the major axis. These may be resolved along and perpendicular to the line of nodes. The former components contribute nothing to the radial velocity. The latter are + V 2 cos u and + e F 2 cos &> in the direction GE which is 120 Orbits of Spectroscopic Binaries [CH. xi drawn at right angles to GA. This line makes the angle (90 - i) with the line of sight, and hence the radial velocity which is measured is V = 7 4- (cos u + e cos &>) V 2 sin i where 7 is the radial velocity of the point G, that is, of the system relative to the Sun. It is at once evident that V 2 and i cannot be determined inde- pendently from the radial velocities alone, and the equation may be written V 7 + K(cos u + ecosco), K=V z sini or again, V = 7' + K cos u, 7' = 7+ Ke cos &> where K, 7 and 7' are to be taken as constant. 113. When the velocity curve has been drawn the maximum and mini- mum ordinates are approximately known. These are y = 7' + K, y = 7' K, which require u = 0, u = 180. The maximum and minimum points, A, B, therefore correspond with the receding and approaching nodes. The line y = y can then be drawn in the diagram, intersecting the velocity curve in E, F. These points require u = 90, 270 and the corresponding points in the orbit are the extremities of the focal chord at right angles to the line of nodes. The velocity curve is thus divided at A, E, B, F into four parts corresponding to four focal quadrants, each bounded on one side by the line of nodes. The part which contains the periastron passage will be described in the shortest time and that which contains the apastron passage will require the longest time. The opposite extremities of any focal chord give equal and opposite values to (Vy)- In particular, the periastron and apastron points, P 1} P 2 , are located on the velocity curve by the further condition that their abscissae differ by P, the half period, and the points LI, L 2 corresponding to the ends of the latus rectum by the condition that they are equidistant in time from P l or P 2 . The four points P lf P 2 , L 1} L 2 on the velocity curve are easily found graphically by trial and error. Again, let be the centre of the orbit and COD the diameter which is conjugate to the diameter parallel to the line of nodes, so that the tangents to the orbit at C and D are also parallel to this line. Hence V=y at C and D on the velocity curve. Let an axis of z be taken parallel to GE in the plane of the orbit, so that T7 dz . t, Now the integral represents the area of the velocity curve measured from the line y = 7. Hence by taking the limits at A, C, B, D it follows that the positive area of the velocity curve from A to C is equal to the negative area from C to B, and the negative area from B to D is equal to the positive area 112-H4] Orbits of Spectroscopic Binaries 121 from D to A. These conditions, which can be tested by a planimeter or some equivalent method, make it possible to draw the line y 7 in the diagram. At K lf K 2 , the extremities of the minor axis, the radial velocities relative to G are equal and opposite. Hence on the velocity curve K^ and K 2 are at equal and opposite distances from the line y = 7 and equidistant in time from P 1 or P 2 . Thus these points can also be found graphically without difficulty. 114. It is supposed that the period P is known, and this gives the mean daily motion, yu,= 27r/P. The other quantities which can be derived from the velocity curve are five in number, namely T the time of periastron passage, K = V 2 sin i, y the radial velocity of the system, &> the longitude of the node, and e = sin the eccentricity of the orbit. The most satisfactory direct method of finding these elements is based on the representation of the curve (see Chapter XXIV) by a harmonic series in the form V= Fo + 2r,-sinO>* + $) where t is reckoned from some arbitrary epoch. This is always possible by Fourier's theorem. But V = 7 + K cos G) (e + cos w)K sin &> sin w = 7 + *2K cos D cos 2 . e~ l 2 J,- (je) cosjM 2K sin w cos . ^J/ (je) sinjM by 41, (28) and (29). Now M = p(t-T) and therefore F = 7 and rj sin (jfiT + ft) = 2K, . er* Jj (je) where K^ = K cos to cos 2 , K 2 = K sin &> cos ............ (1) There are now only four quantities to be determined, which may be taken to be K-i, K 2 , T and e. Thus the four equations corresponding to j = 1, 2 are alone required : those of a higher order are useful only when there is reason to suspect that the motion is not purely elliptic. Now these give ( 47) (2) ( l- 122 Orbits of Spectroscopic Binaries [OH. xi showing that r z jr-i is of the order of e. Hence, by division, 5e 2 e 4 r," sinOT+ V 24 "96 rg cos iT + &) / _ Te 3 _ e^ _ \ T+A) = \ 24 96 "V and, by subtraction and addition, r, sin Q7 7 +&,-&) e 3 e sn sin the last equation containing no term in e 5 . Eccentricities as high as 075 are met with occasionally, but even so it is evident that (pT + /& ft) is a very small angle which can scarcely exceed 2 and is generally negligible. If then it is possible to neglect a 2 and the last equations become T -i . (1 + a cot (4/3, - 2&)} - e - j whence From this equation e is easily found by trial and error, and then a, which gives T, is found from (3). The equations (2) give K l and K 2 , whence finally K and = 7' 7. A number of interesting properties have been used for the purpose. Among these are the properties connected with a focal chord of the orbit. Let ^ be the time at a certain point of the orbit and w and E l the iu-116] Orbits of Spectroscopic Binaries 123 corresponding true and eccentric anomalies. Let t 2 be the time at the other end of the focal chord through the point and 180 + w and E 2 the true and eccentric anomalies. Then (1 - efi tan ; = (!+ e) 4 tan \E lt ^ (t, -T) = E 1 -e sin E l (1 e) cot \w = (1 + e) tan ^E 2 , /j,(t 2 T) = E 2 e sin E z . Hence - (1 _ e ) = (1 + e) tan ^E 1 tan |# 2 or and therefore = (E 2 -EJ- sin (Es-EJ. Also tan (# 2 - #0 = - J (1 - e 2 )* e- 1 (cot w + tan | = cot cosec w. Hence, if 2?? = E. 2 -E lt p (t 2 ti) = 2?; sin 2?7, tan < sin w = cot 77. Similarly, if 3 , 4 are the times at the ends of the perpendicular chord, where the true anomalies are 90 + w, 270 + w, P (t\ ts) = ST/' sin 2?/, tan < cos w = cot rj '. The angles 17, 77' are easily found, especially with the help of a suitable table of the function (x sin #), and hence or e and w = u w. But the ordinate at the point ^ gives y with the major axis. Hence in this case tan w = cos $ tan E and therefore $/* (t, -t 1 -$P) = e(l+ tan 2 &> sec 2 <) " - = e cos ) ~ - cos which gives e = sin < when e cos &> = (7' j)/ K is known. Also - e = /& (*,-,- P) sec I/A (, + ^ - 2T 7 ) which gives a relation between e and T. Another pair of such points is K lt K 2 , corresponding to the ends of the minor axis. Since E = in this case, Let Uj, u 2 be the longitudes at K lt K 2 . Then the radial velocities at these points, relative to G, are + %K (cos Wj cos w 2 ) = + K sin | (w ? HI) sin ^ (z^ + u 1 )= K cos sin o>. This quantity is therefore given by the ordinates at K 1} K 2 on the velocity curve, relative to the line y = 7. 117. The velocity curve also possesses interesting integral and differential properties which may be useful. It is necessary to have a consistent system of units, and since those of time and velocity have already been adopted, the unit of length is fixed and the natural system is: Unit of time = 1 mean solar day = 86400 mean sees., Unit of length = 86400 km. = 0'0005779 astronomical units, Unit of velocity = 1 km. per second, Unit of mass = that of the Sun. Now the constant of areal velocity in the orbit is p V 2 = lirabjP pa 2 cos < so that a sin i = V 2 /j.~ 1 cos sin i = K/jr 1 cos . The argument relative to the areas -of the velocity curve in 113 can now be made more precise. For the* tangents to the orbit at C and D, referred to the principal axes of the ellipse, are x sin a) + y cos a> = V( 2 sin 2 &> + 6 2 cos 2 to) 116-iis] Orbits of Spectroscopic Binaries 125 and the perpendiculars on them from the focus G are z\ , 2 2 = ae sin &> + a \/(l e~ cos 2 &>). Measured from the line y = y let AI be the area of the velocity curve from A to C, A l from C to B, A. 2 from B to D, and ^3 from D to A Then cos < - e cos &) cos . e sin &> A ^2 = K*pr- cos 4 <. When A lt A 2 have been measured in the proper units these equations deter- mine (j> (or e) and &>. 118. If the tangent to the velocity curve makes an angle -\/r with the axis of time, dV ,, . dw tan \/r = -TT JL sin t* -rr cfa at and r being the radius vector in the orbit, the constant areal velocity is pa? cos sin u (1 + e cos w) 3 and at special points on the curve tan i/r has these values : A,B : ti = 0, 180 : tan i|r = ^, F : it =90, 270 : tan) 2 P l} P 2 :w = 0, 180 : tan ^ = + pK sec 3 <^> sin w (1 e) 2 Xj , Z., : w = 90, 270 : tan ty = + //,/f sec 3 ^> cos &> K l} K. 2 : w = + (90 + ^>) : tan i/r = + //.X" cos ^> cos (&> + ^>). If tan i/r is found graphically at any of these points, attention must be paid to the scales in which ordinates and abscissae are represented. These expressions can then be used in order to find o and . Since r oc (sin u cot ty] r, w = w &> and u at any point on the velocity curve is given by the ordinate measured from the axis y = j', it is possible theoretically to plot the actual orbit to an arbitrary scale, point by point. This is scarcely a practical method, but deserves mention as the counterpart of Sir John Herschel's method for double star orbits ( 105). 126 Orbits of Spectroscopic Binaries [CH. xi 119. The values of the elements found by any of these graphical methods are approximate only. They can be improved by the addition of differential corrections, SX to K, Be to e, &a> to <, ST to T ,and S/j, to /A. Thus each observation gives an equation of condition of the form V - V c =8y + cosu.SK and it is easily found that dw -^- = sin w(2 + e cos w) sec 2

. It is more usual to give 7, the radial velocity of the system, than 7', but this quantity can be derived finally from the relation 7 = 7' Ke cos &>. 120. When the elements of an orbit specified above have been obtained, by whatever method, some information can be gained as to the dimensions and mass of the system. An equation already found in 117 gives a sin i = Kfir 1 cos < . 86400 km. when the unit of length there adopted is explicitly introduced. Let m be the mass of the star whose spectrum is observed, and m' the mass of the other star. Then u?a? (l + ,Y = (m + m) C \ m / where C is a constant depending on the units employed. These being as stated in 117, the special case when m' = 1, m = 0, gives 47T 2 1 = (0-0005779)*' lo S C It follows that m' 3 (m + m')- 2 sin 3 i = [3'81443 - 10] K 3 fjL~ l cos 3 = [3-01625 - 10] K 3 P cos 3 and it is only this function of the masses, involving the unknown inclination of the orbit, which can be determined when only one spectrum can be observed. If, however, the radial velocity V of the second component of the system can be measured at the same time, which is possible when the two superposed spectra are of comparable intensity, ii9-i2i] Orbits of Spectroscopic Binaries 127 One such equation will give the ratio m : m when 7 is known and two will give 7 in addition without any knowledge of the orbit. It has been supposed that the radial velocities have been determined by referring the stellar spectrum to a comparison spectrum from a terrestrial source, as mentioned in 111. When there is no comparison spectrum, as when an objective prism is used, and the stellar spectrum shows double lines, it is still possible to deduce the orbit of the system from the relative displacements of corre- sponding lines. But the orbit is then the relative orbit, a is the mean distance of the components from one another, and it is easily seen that (m + m'} sin 3 i must be substituted for the above function of the masses. 121. The true spectroscopic binary cannot be resolved in the telescope. But one or both components of a visual double can, when bright enough, be observed with the spectrograph, and very interesting results can be gained in this way. Let a, a' be the mean distances of the components relative to the centre of mass, expressed in terms of the linear unit 86400 km. The astronomical unit contains 1730 such units. Let a" be the visual mean distance and -a" the parallax of the system, both expressed in seconds of arc. Then , , mm , ,. > ma = ma = - , (a + a ) m + m rs m + m and therefore V = 7 -i- K (cos u -f e cos (a) 7 -f p) a" = 7-1- 1730 a sm i sec (cos u + e cos V = 7 1730 u, sin i sec d> (cos u + e cos (cos u + e cos o>) a" ITS" the parallax can be found, and hence the total mass of the system. One question remains in the determination of the true orientation of a double star orbit in space, which can only be decided by radial velocity 128 Orbits of Spectroscopic Binaries [en. xi observations. For the spectroscopic binary i has been defined so that < i < ^TT, while for the visual double 0) have the same sign, where is the longitude of periastron of this star, reckoned from its receding node in the direction of motion. But X is the longitude of periastron of the companion at its first node O (< TT). Hence if the expressions V l V 2 , cos (w x + X) cos (w 2 + X) have the same sign, X = &>. This means that the principal star is receding and the companion is approaching when the latter is at its node fi. If on the other hand the expressions are of opposite signs, X = to + TT and the companion is receding at fl. Otherwise it. may be possible to determine the velocities V, V of the principal star and the companion respectively at the same time. Then the expressions V V' f cos (w + o>) + e cos o> have the same sign, and therefore if the expressions V V, cos (w + X) + e cos X have the same sign, X = to, while if they have opposite signs, X = to + IT. The same consequences follow as before. Thus a knowledge of either V 1 V 2 or V V removes the ambiguity with regard to the true position of the orbital plane, which remains after the elements of a double star have been deter- mined from visual observations alone. CHAPTER XII DYNAMICAL PRINCIPLES 122. It will be convenient in this chapter to recall some of the salient features of dynamical theory and to consider as briefly as possible the form of those transformations which are of the greatest importance in astronomical applications. We shall start from Lagrange's equations. Let the system consist of a number of particles whose coordinates can be expressed in terms of n quantities q l} q 2 ,...,q n and possibly of the time t. Let m be the mass of a typical particle situated at the point (x, y, z). Then doc dx dx x = 3- + 5 . QI + . . . + 5 . q n dt dq, dq n so that dx __ doc dq r dq r ' Hence d (. dd?\ d f. dx\ d (. dd?\ d f. dx\ -j- *m =-r- ) = m -j- (x ~ 1 dt \ dqJ dt \ dq r J = ~ -- - a* dq r dq r where X is the component of the force acting on m. If T is the kinetic energy of* the whole system, Hence adding all the equations of the preceding type for the three co- ordinates and all the particles, ~r. ^rr- - ^ -- h J. T ^ ^ + ~ . dt \oq r j \ dq r dq r dq r / dq r Now the forces which occur in astronomical problems are in general con- servative, and we can write 2 (Xdx + Ydy + Zdz) = -dU P. D. A. 9 130 Dynamical Principles [CH. xn where dll is a perfect differential. U represents the work done by the forces in a change from the actual configuration to some standard configu- ration and is called the potential energy. We therefore have . d_ /dT\ = d(T-U) dt d) d dt \dq r J dq r But U does not contain q r , and hence, if we write T = U + L, this becomes ^ (dL\ = dL 9 which is the standard form of Lagrange's equations. The function L is often called the Kinetic Potential. In the absence of moving constraints (or some analogous feature) within the system = ... = 0. ot Then T is a homogeneous (positive definite) quadratic form in q 1} ..., q n . 123. If L does not contain explicitly, the equations admit an integral called the Integral of Energy. For in this case dL = 2( PD PZ>---> Pn- Hence L and H can be expressed either in terms of (q r , q r ) or of (q r , p r ). Thus ^ T ^ dL 5, ^ dL ~ . oL = 2t ~ . oq r + 2t ^-7- . oq r r oq r r eq r re . 3l ,. *dL o2^q r ? - = 2i q r op r -f- 2i< ^rr- od r dq r r r dq r and therefore BH = 2 (q r 8p r p r Bq r ) r since dt \dq r j dq r ' It follows that a IT 3IT ' ' = f =1 2 ^ (1} dp r ' dq r ' and this is the form of the equations which is called canonical. When L has its natural form, H= T+ U. If L does not contain t ex- plicitly, neither does H, and the integral of energy (2) becomes simply H= k. 92 132 Dynamical Principles [CH. xn 125. Let us consider the differential form dO = 2 p r dq r Hdt r or d(^p r q r 0) = ^ q r dp r + H dt. r r If dd is a perfect differential, the right-hand side of both equations must also be perfect differentials, and this -requires that dp r _ dH dq r _ dH dt dq r ' dt dp r or the canonical equations must be satisfied. Let us suppose now a trans- formation from the variables (q r , p r ) to the variables (Q r , P r ) such that . ....... (4) where dW is a perfect differential and W is expressible either in terms of (qr, Pr) or of (Q r , P r ). Such a transformation is called a contact transforma- tion, or in the particular case when (q r ) can be expressed in terms of (Q r ) alone [by relations not involving (p r ) or (P r )] an extended point transformation. If W contains t in addition we may write dW dW 2 P r dQ r ^p r dq r 3 . dt = d W -~ . dt j* T Ov Ov so that when dd is introduced Each_ side of this equation is a perfect differential provided d6 is a perfect differential, and in this case where dK * _ dK ~dQr' ^~dPr' Since these equations equally with the form (3) express the conditions required if dd is to be a perfect differential, they must be equivalent to (3). Thus we see that any transformation of variables satisfying the condition (4) leaves the equations of motion in the canonical form. 126. In consequence of (4) dW _dW ~Wr' Pr ~3qr' Hence K will vanish in virtue of (6) provided dW + = .................. (8) 125-127] Dynamical Principles 133 This equation is known as the Hamilton- Jacobi equation. But when K = 0, -Pr = &, & = * where a r and (3 r , by (5), are arbitrary constants. Hence if any function W can be found which satisfies (8) and contains n arbitrary constants (a r ) in addition to (q r ) and , the solution of the problem is completely expressed by the 2n equations (7) written in the form SW 3W a^'-A. ft-aj- <> where (/3 r ) are n additional arbitrary constants. If H does not contain t explicitly we may write W=-ct n t+W where W is a solution, containing (n 1) constants (a r ) apart from a n but not t, of the equation The solution (9) is therefore replaced by (H) 127. In the set of equations (7) W is an arbitrary function of (Q r , q r ). Instead of making W a solution of (8) let it satisfy the equation where H is the Hamiltoniau function of another problem also presenting n degrees of freedom. Hence as before P-r ~ @ r , Q r = <*r where (ce r , {3 r ) are the 2n arbitrary constants of the problem defined by H . Hence the equations (5) and (6) become where 3F dt 0> Thus if the H problem has been solved and the constants of a solution of the corresponding Hamilton-Jacobi equation are known, the same form of solution applies to the H problem with the difference that the quantities which remain constant in the first problem undergo variations in the second 134 Dynamical Principles [OH. xn problem which are defined by (12). This is the foundation of Lagrange's method of the variation of arbitrary constants. The simple form of (12) depends essentially on the function K being expressed in terms of the constants which occur in a solution of a Hamilton-Jacobi equation and which may be called a set of canonical constants. If we suppose that the problem defined by H has been solved by some other method than through the medium of a Hamilton-Jacobi equation, a different set of constants will be obtained. Let A m be a typical member of such a set. Then A m is some function of (a r , f3 r ). Hence dA -^ o/~> "i ,-. ' ~ da r d/3 r d/3 r da r j s - da r 'dAg'd/3 r 3/3 r 'dA s 'da. r - 1 ( A A } ^ K - Z [A m , A.J ^ where K = H H as before, and a form of expression which will be defined later ( 130) as a Poisson's bracket. 128. Let us consider the integral J=f t> Ldt= ( (T- U)dt Jto J t = ! f> (-H+2p r q r )dt ........................ (13) J t by the first set of equations in 124. We have therefore - SH + 2 r S where S denotes a change in (q r , p r ) but leaves t at each point unaltered. Hence 8J = if 8q r = at the limits and if the canonical equations are satisfied. And this proves Hamilton's principle that in the passage from one fixed configuration to another the integral J has a stationary value for the actual motion as compared with any other neighbouring motion in which the time at corresponding points is the same. 127-129] Dynamical Principles 135 If however 8 denotes a change in t, 8J = _ 8 t ' Hdt + 8 I %p r dq r J t .'0 o Hence when two neighbouring forms of motion, each compatible with the canonical equations, are compared, the complete variation between two positions and 1 is 8J= IZprSq^ \H8t\\ Accordingly, if the initial time is taken as fixed and ( far~ * and But this is the Hamilton- Jacobi equation. Hence the integral J" is a par- ticular solution of this equation. And further, since we have reproduced the equations (8) and (9) of 126 except that J is written in the place of W, we see that J is that solution which contains the initial values of the coordinates as its n arbitrary constants. 129. Let us suppose now that H does not contain t explicitly, so that the integral of energy H = h exists. Then if = r J t ........................ (14) t i 1 r* 1 ' r + (Zq r 8p r - 2,pr8q r ) dt. JO Jtv But 2q r 8p r - 2p r 8q r = 2 JT- 8p r + 2 ^ 8q r op r oq r = 8h and therefore Sh . dt. . This is the complete variation of J and it vanishes between fixed terminal points if 8h = in each intermediate position, i.e. if the time is assigned to each displaced position in such a way that the equation H = h is satisfied in the varied motion. Under these conditions the integral f'(T- U+h)dt J t 136 Dynamical Principles [OH. xn has a stationary value in the course of the actual motion as compared with motion in any neighbouring paths. This integral is called the action and the proposition established is known as the principle of least action. When T is a quadratic function of the velocities h T+U and the integral becomes .............................. (15) t and in problems which involve only one material particle this is simply rt, ri /== v 2 dt=\ vds ........................... (16) J to JO where v is the velocity of the particle (of unit mass). The integrals which we have found to be stationary are not necessarily minima. The necessary conditions in order that an integral rti J = f(lr, qr)dt * shall be an actual minimum are : (1) The first variation 8J vanishes between fixed terminal points. (2) The function of (e r ) rif M =f(q r , q r + e r )-^ r ^~ is a minimum. (3) Between the terminal positions and 1 no intermediate position P exists such that and P can be joined by a neighbouring path which satisfies the dynamical conditions and is other than the path considered. The nearest point to on the path which does not satisfy this condition is called the kinetic focus of the point 0. 130. It is necessary to study the properties of certain expressions connected with the transformations which are frequently employed. Let u l} HZ, ... , u 2n be 2n distinct functions of (q r , p r ). The first expression is du m ' du which is called a Lagranges bracket and is denoted by [HI, M M ]. The second expression is du m du m duA ^ 8 (MJ, u m ) . = -- -r - .- \ = . . ,. dq r dp r dq r dp r j r (q r , p r ) This is called a Poissons bracket and will be denoted here by the symbol [ui> u m }. It is evident that we have [MI, u m ] = - [u m , ui], (1 4= m) . {ui, u m ] = -{u m , ui}, (I 4= m) 129, 130] Dynamical Principles 137 There are also relations between the two types of expression, and these we shall now investigate. Let two linear substitutions be defined by and z m , where r can have all values l,...,n and I and m can have all values 1, ... , The result of eliminating y r , y n+r is to give o du m j 2n 2 [HI, Um] Z 1t .(19) But the substitutions can be reversed by writing z m = TWr " a^r y- 2 y n +r The equivalence of these forms is easily verified since au, ag r i _ " ra^ a^i _ "5 ^\ *' * I 5 'N "> 8< ?r *ij ? L 8 ?*- 9w d When y r , 2/ n+r are eliminated, these give ^ " /3w z 9w m 8w w 3 ^ tn = 2 act 2 U- . -5 -5 . a .(20) The resultant substitutions (19) and (20) must therefore be equivalent, and accordingly their determinants, written in the forms and {*,, Wi}, {M!, Ma},..., {MU {^2, Wa}, (W 2 , Ma},..., {Ma, (21) are reciprocal. This means that any constituent of either determinant is equal to the co-factor of the corresponding constituent in the other determinant divided by that determinant. Any Lagrange's bracket is thus expressible in terms of Poisson's brackets, and vice versa. 138 Dynamical Principles [CH. xn 131. Let us now consider the explicit conditions for a contact trans- formation. We have in this case r r r r I a perfect differential. Hence ap always, and unless I = m, in which case It is at once evident that these conditions may be written [P,,P m ] = 0, [Q f , Q M ] = for all values of / and w, [Qi, ^m] = for all unequal values of I and m, and [Qi, PI] = I for all values of I. In other words, in the case of a contact transformation all the Lagrange's brackets vanish with the exception of those which are of the form [Q t , PI], and these are all unity. Let us now put Ur=Qr, U n+r = P r , (r=l, 2, ..., W). Then the substitution (19) becomes simply X r = Z n ^. r , ^n-^r = %r- But this shows that all the Poisson's brackets occurring in (20) vanish except those which are of the form [HI, ui* n ], and these may be written {Qt, P t \ = 1 or {P lt Q t ] = - I.' The conditions for a contact transformation are therefore of the same simple form whether expressed in terms of Lagrange's or of Poisson's brackets. Again, the substitutions of 130, 131, 132] Dynamical Principles 139 become identical when m = n + I, since z n+ i = xi. Hence dqr^djj. d_Pr = _ Pr o dp r dq r we must have d ^ , WJ^L x d * H s \ T (Sq r ) = 2, 5 5- . bq g + 5 = . 6p s W V Vdpr^s 3/>r9p / dt with similar equations for Bp r . Now let us suppose that the new variables are those given by (22). These will lead to a particular solution of the varied motion provided d dt\dp r j s \dp r dq s 'dp s dp r dp s ' dq s j -A v f^[ ?^_^" ?K\ dp r s \dq g 'dp g dp g 'dq s ) ,-, /dH (PK dH cPK 2t I . . __a_ s /_ . dK_ . dK\ + = !_/^_^ A@x\~lL@K} \dt dt ) dt \dpj dt \dpj - dp r \ dt J with a similar set of conditions arising from the equations for Bp r . But it is evident that all these conditions will be satisfied if K is an integral of the system, for then K = 0. We thus see that if K is an integral, the equations (22) are a particular solution of the equations for the disturbed motion. 134. Let u be another integral of the undisturbed system. Then u + AM must also have a constant value in the disturbed motion. But by (23) AM = e {u, K} when the disturbed motion is that obtained by the infinitesimal contact transformation derived from K. Hence {u, K} must be constant, and we have Poisson's theorem : if u and K are two integrals of a" system, the Poisson's bracket {u, K} is also an integral. It might be supposed that a knowledge of two integrals would thus lead to the discovery of all the 133, 134] Dynamical Principles 141 integrals of a problem. This is not so in general. The known integrals are more often of a generic type, particularly in the case of those gravitational problems with which we have to deal, and fall into closed groups. For example, if we start from two integrals of area we obtain by Poisson's theorem the third integral of the same type and no further progress can be made in this way. In order to obtain fresh information it is necessary to start from integrals which are special to the problem considered. Let u 1} u 2 , ... , u^i be 2n distinct integrals of the problem. Then each Poisson's bracket of the type {u r , u s } is constant throughout the motion. But we have seen in 130 that a Lagrange's bracket [u r , iig] can be expressed in terms of all the Poisson's brackets. Hence \u r , Ug] is also constant through- out the motion. But this gives no means of finding additional integrals of the problem, for in order to calculate [u r , u s ~] it is first necessary to express (q r , Pr) in terms of the 2n integrals (u r ). And this presupposes that the problem has been completely solved. CHAPTER XIII VARIATION OF ELEMENTS 135. The Hamilton-Jacobi equation corresponding to elliptic motion about a fixed centre of attraction is very simply solved when the variables are expressed in polar coordinates (r, I, X), so that (I, X having the same relation to one another as longitude and latitude) qi = r, 2 = X, q 3 = I. Then, after suppressing the factor m in the potential energy U and therefore treating the mass factor in the momenta as unity, U = - fir-\ //, = & 2 (1 + m) = ri*a 3 PI ^ PZ = r ^-> PS r " c s 2 X . / H = T + U = $ (p* + r~%* + r~ 2 sec 2 X . p./) - pr~\ The Hamilton-Jacobi equation ( 126) therefore takes the form, since H does not contain t, /8FV 1/8^Y+ L. 5.Y_2a+ 2 ' i (dr ) + rUx ) + r 2 cos 2 xl dl ) ~ "* l + r where W=W' a. l t. Integration by separation of the variables is then easy. For /8FV obviously satisfy the equation. Hence Tir/ f r / 2/i a 2 2 \ 7 " = I ( 2j + -- - I ar + 135,136] Variation of Elements 143 is an integral which contains the three independent constants 1} a. 2 , 3 . Therefore the complete solution of the problem is given by the equations 2,11 n z \~% +- * dW f x _i /3 3 = -^ = Z ., sec 2 X (a,, 2 3 ' 2 sec 2 X) 2 cX o 3 / where /9j, /3 2 , /3 3 are three additional constants. The lower limit r is also arbitrary. It may be identified with the pericentric distance, and then the integrals depending. on r will vanish at the pericentre. 136. We have now to determine the meaning of the six constants of integration. Since the integral in the first equation vanishes at perihelion, & is clearly the time at this point. Also, by the same equation, f..?*-^ r r 2 = 2! (r r x ) (r r 2 )/r 2 . But at an apse, r = and r = a (1 + e). These then are the values of r l} r 2 , and hence /i = 2a !, a./ = 2a 2 (1 e-) otj or ! = - fjL/2a, a 2 = \/{/ia (1 - e 2 )}. Also if we put or 3 /a 2 = cos i the second and third equations become on integration - /3 2 = /! (r) + sin" 1 (sin X/sin i) /3 3 = sin" 1 (tan X/tan i) or sin X = sin i sin {/i (r) /8 2 } tan X = tan i sin ( + /3 3 ). This last equation shows that the motion takes place in a fixed plane making the angle i with the plane X = 0, which may be taken to represent, for example, the ecliptic, with I and X as the longitude and latitude of the planet. Thus the meaning of 3 = o 2 cos i is defined, and /3 3 is simply the longitude of the node. The preceding equation then shows that /, (r) yS 2 is the angle between the radius vector of the planet and the line of nodes, i.e. the argument of latitude. But at perihelion the integral /i (r) vanishes. Hence /3 2 is simply the angle in the orbit from the node to perihelion, or ta O in the ordinary notation. The canonical elements which we 144 Variation of Elements [CH. XIII have introduced can therefore be expressed in terms of the usual elements (T being reckoned from the epoch when the mean longitude is e) thus : e 2 )] cos i, /3 3 = H. The homogeneity of these constants will be increased by introducing a = instead of a,. This makes 2^ = -^ /a? and W= W + /x 2 i/2a 2 . Hence will be replaced by ft, where 9a a 3 V Since the integral vanishes at perihelion, and t = T at this point, /s-e*:. /e..r-r + .. a 3 V a The other constants are easily seen not to be affected by the change in a lf ft, which can accordingly be replaced by where e is the mean longitude of the planet at the time t = 0. 137. The expressions for a, 2 , a 3 , ft, ft, @ 3 in terms of the ordinary elliptic elements which have just been found make it very easy to calculate the Lagrange's brackets r , v /8a d/3 d/3 da\ [u, v] = . u dv .^ dv where u, v are any pair of the six elements a, e, i, O, tn-, e. Since a, a 2 , a., are functions of a, e, i alone and ft, /3 2) /3 3 are functions of U, -nr, e alone, the Lagrange's bracket for any pair of either set of three elements vanishes. It is equally evident on inspection that [e, e], [i, or] and [i, e] also vanish, the two constituents never occurring in a corresponding pair of canonical constants. Hence the complete array of Lagrange's brackets may be set out thus : a e * 11 or 6 a [a, H] [a, tsr] [a, e] e [e, fl] h] i [. ft] n - [a, ft] - [e, n; 1 -[,n; nr [a, r] -!>, 1 o e - [> e ] 136-138] Variation of -Elements 145 where the first constituent of each bracket taken positively is placed in the column on the left and the second constituent in the line at the top. The brackets in the second diagonal really contain only one term and are at once seen to be [a, e] = - V/A/O, [i, ft] = Vytt (1 e*) . sin i while the remaining three brackets contain two terms and are [a, ft ] = i V( 1 e 2 ) /JL/CL (1 cos i) 4 [e, O] = - e *Sjta (1 - cos t)/Vl e 2 . The value of the whole determinant depends simply on the constituents in the second diagonal and is evidently A = [a, e] [e, r] a [i, H] 2 138. It is now easy to form the reciprocal determinant, the constituents of which are the Poisson's brackets of pairs of elements. On account of the large number of zeros in the above determinant a corresponding number of minors vanish and the rest can be calculated without difficulty. It can in fact be verified by simple inspection that the reciprocal determinant takes the form : a e i ft tx e a [a, i\ e {*] {*> 6 1 i {i, ft} {i, r*} (> e l ft - {i, 0} -or -{ r} - {i, v] e - [a, 1 - {, *} -M the first constituent of each bracket (written positively) being indicated in the column on the left and the second ' constituent in the top line as before. It is also clear that the partial substitutions ( 130) #! = [a, li] z 4 + [a, tar] z 5 + [a, e] z 6 # 2 = [e, fi] 4 + [e, -sf] z 5 = [i, O] z 4 P. D. A. 10 146 Variation of Elements [OH. xin and 4 = I*. ft} a ':- Z 5 = {e, -or} # 2 + {*', w} x s z 6 = {a, e} #1 + {e, e} # 2 + ft, ej # 3 must be equivalent, and it readily follows that ' {a, e} = !/[,*] = -2 V5/ e '* dt where A r represents any one of such elements. Since we have found the expressions for all the Poisson's brackets, the equations for the variation of 138-140] Variation of Elements 147 the usual elliptic elements can at once be written down in an explicit form. They are as follows : da i dR dt =2Va/^.^- de cot d> dR tan ^d> cos d> dR dt *J lL(t 9'57 di 1 dR tan|i 'N ~" ^ dt cos d> sin i V/*a 8fl cos d> v yu.a WOT de dn = 1 dR dt cos d> sin i V/ta 9* rfor _ cot d> 9.B tan \i dR ~~TT / - "^ h ~~ ' "/ "jr 7 " rtc v yLia c*e cos d) v/ia 0* rfe 97iJ tan ^d> cos d> dR tan ^i dR dt i r" da v/xa de cosd>.v/Lta- ^* A slight simplification has been- made by writing sin d> in place of e in the coefficients of the partial differentials of R. 140. The above set of equations for the variations of the elements is fundamental. An important point must be noticed in regard to them. The variation of a entails a corresponding variation of n which is determined by the relation n 2 a s = p. Now the disturbing function R is a periodic function of the mean anomaly and is expressed in terms of circular functions of mul- tiples of nt. Hence the derivative of R with respect to a would contain the same circular functions multiplied by t and this introduction of terms not purely periodic would be inconvenient. The difficulty is avoided by an artifice which should be carefully noted. We consider n (as distinct from a) to occur only in the arguments of these periodic terms. Otherwise a is used explicitly or if it is more convenient to use n outside the arguments, n is simply a function of a given by n 2 a 3 = p. Now e enters into R only in the form nt + e through the mean anomaly, so that dR _ 1 /9.R\ 8e 1, \97i/ a=colls t.' , Hence - = -2V^T dt "da 2\ dn-/dR\ dR\ dn dR .9a/ w=cons t. da 8e ^9^\ dn da ~^ * ~j ~T7 -\- . . . < f' a /w=con8t. U /a Ml 102 148 Variation of Elements [CH. xm or de . dn -. / 7 /9jR\ -r + t -j- = 2 va/fjb ( ^- ] (It Ctt . \U(l / )i = const. If then we take e' instead of e, where de dn _ de' dt dt dt or e + nt = e + Indt = e' + I the form of the above equations for the variations of the six elements will be unaltered, since dR = dR de ~ de' but their natural meaning will be so far altered that (1) n in the mean anomaly is not to be varied in forming the derivative with respect to a, and (2) nt in the mean anomaly is to be replaced by Indt. The secular terms which would arise from the cause mentioned are thus avoided. The value of n is deduced directly from the value of a, and we have 4 f = ^ - a - dt. If this integral be denoted by p we have also d*p .,- da 3 dR or which gives the finite variation of this part of the mean longitude in the disturbed orbit. 141. When e (and therefore ) is small, and this is commonly the case, the coefficients in the variations of e and OT which contain cot (f) as a factor become large. This gives rise to a difficulty which can be avoided by intro- ducing the transformation li-i = e sin -or, &j = e cos ta. The result of making this change, which can be verified without difficulty, is to substitute for the corresponding pair of equations dhi cos < 9-R &! tan |t dR h^ cos < dR dt V/ia 9&! cos < V/ia di 2 cos 2 1^> V/^a 9e dki_ cos< dR Ajtan^i 9^ ^ cos dR 'dt Via 9A, cos < x/tta di 2 cos 2 <> Via 9e 140-142] Variation of Elements 149 Similarly, when the angle between the plane of the orbit and the plane of reference is small, a pair of coefficients in the variations of i and fi become large, and the transformation h-z = sin i sin H, k z = sin i cos U is useful. The result, which can be verified with equal ease, is to replace the equations named by the pair rf// 2 cos i dR h 2 cos i /dR dR\ dt cos V/^a dk. 2 2 cos 2 ^' cos \//JM \9w 9e / dk 2 _ cosi dR k 2 cosi fdR dll dt coadtvua dh. 2 2cos 2 Aicosd> \fua V?w 9e 142. Another form of the equations for the variations of the elements, in which the disturbing forces appear explicitly, is of great importance. Let 8, T be the components of these forces in the plane of the orbit along the radius vector and perpendicular to it, and W the component normal to the plane. Let u be the argument of latitude and (X, /u., v) the direction cosines of the radius vector, so that ( 65) A. = cos u cos fl sin u sin II cos i fjb = cos u sin fi + sin u cos fl cos i v = sin u sin i. The direction cosines of the transversal and of the normal to the plane may be written 9X 9/4 dv , 1 9\ 19/4 1 dv du' du' du sin u di ' sin u di ' sin u di which must satisfy the conditions duj ' sin'u If o-'o da- \du da-/ smu \di da 150 Variation of Elements [CH. xm by the conditions mentioned. Now r = a(l ecosE), tan -Jw *= u = iff O 4- iv, E e sin E = nt + e OT. In accordance with 140 we treat n, as it occurs implicitly in u, as inde- pendent of a, and replace nt by Indt. Hence dR 8r' rS ~o~ = > ^5 = da da a di sinw duj sin u di (since X contains O both explicitly and implicitly through u) = rT]2 ( ~- 3-7^ ) 1 [ + - ~^(~~n) rW = rT (cos i 1 ) 4- (- sin u cos u sin i) sm w v = 2rT sin 2 %i r W cos u sin *'. The remaining elements enter into (X, //-, v) only implicitly through w, so that in their case ^ _ sin w + aT sec (1 + e cos w). Since r and w are both functions of e OT, 142] Variation of Elements 151 and finally dR _ Q dr dw ^ -- *3 ^ r I J- ~zr~ de oe oe sn w --- sin E esirfE \ 1 - 1 ( esirfE \ 1 - 1 \ = aS - cos E + - -~ } + rrsra n - =, + .. - = V 1 e cos E) \l-ecosE 1 e 2 / , e cosE fl+ecosw 1 .= , ^ - :, 1 e cos E \l-e- 1 e' = aS cos w; + rT sin w (2 + e cos w) sec 2 0. It only remains to carry the expressions found for the derivatives of R into the equations of 139 for the variations of the elements. The results are as follows : da -^7=2 Va'/A* {& tan sin w + T sec (j> (1 + e cos w)] ^ e _ = v a/ p cos <|> (>S sin w + T(cos w + cos E)} '- = rW cos M/COS rf> Vtta rti c?O . . -^- = r W sin u/cos 9 sin i 2 sin 2 i ~ + 2 cos sin'it. From the first two equations we get for the variation of the parameter ja = a (1 - e-) ^ = cos 2 6 ~ 2a sin -y- = 2?- T cos v'a/u . cU dt at It has been convenient to derive the above important set of equations from those which involve the derivatives of the disturbing function. But their form would be the same if the components of the forces were not such as can be expressed as the differentials of a single function. Thus they hold, for example, in the case of elliptic motion disturbed by a resisting medium. Since w 2 a s =/u, is constant, the equation for "the variation of a maybe replaced by ~ = 3 {$sin . lit 1 52 Variation of Elements [OH. xm Also - (e BT) = 2rSf^/(/jui) cos -- + r W sin w t^an ^i = {(a cos 2 cos w 2r sin <) S rT sin w (2 + e cos w)}/sin which gives the variation of the mean anomaly, dM d , fdn , part of the variation of nt being included in e as explained in 140 and mentioned above. 143. It has been seen in 139 how the canonical solution of the problem of undisturbed elliptic motion leads to the canonical equations appropriate to the form of motion which follows from the introduction of disturbing forces. With a slight change of notation, L = a = V(/K), I - nt 13 = e - r + nt H-% 0= -*-r-U JJ= 3 = Vl/ua (1 - e 2 )} cos i, A = - /3 3 = fl and the canonical equations become dL ^dR dl = _dR dt ~~ dl ' dt~ dL dG _dR dg _ dR ~dt ~dg' dt = ~dG dH_dR dh_ dR dt~dh' 'dt dH' But there is here a change in the meaning of R due to replacing the element /3 by the mean anomaly I. If the disturbing function in the usual form quoted in 139 be denoted by R , the variation of I follows from d ., . _ 9jR dR _ 9^o dt (l ~ ~dL' dL = dL~ and therefore ndL = R - L'L-^dL = R This change in R has no effect in the other equations, and since R is a function of e OT + nt, dR/dl is the same thing as dR/d/3. The above canonical equations are precisely those on which Delaunay's theory of the Moon is based. Without changing L let the transformation L G = } , G H = .?, c /< = &>, h = co 2 , 142-144] Variation of Elements 153 be made. Then \dL + co 1 dp 1 + a) z dp z (IdL + gdG + hdH) = and this expression is therefore a perfect differential. Hence by 125 the transformation from the variables L, G,H; l,g,h to the variables L, /i, /? 2 ; X, ft>!, a> 2 is one which leaves the equations of motion in the canonical form. The angle A, = e + nt is the mean longitude, and o) 1 = - OT, &> 2 = ft are the longi- tudes of perihelion and the node, reversed in sign. Again, consider the transformation % = (2p)^ cos &), ?; = (2p) 2 sin o>. In this case r)dj; wdp 2p sin 2 codco + sin o> coswdp wdp sin 2ft> &) is a perfect differential. Hence the variables L, p l , p. 2 ; \, (o 1} eo 2 can be changed to L, &, % z \ \ i7 lt % and the canonical form of the equations will still be preserved. These variables have been used extensively by Poincare. Since (sin^> = e), | 1} 7?! are of the order of the eccentricity, and are called by him the eccentric variables. Similarly, since sn 2 2 , rj 2 are of the same order as the inclination, and are therefore called the oblique variables. 144. The account which will be given of the lunar theory in later chapters will be based on a method which is quite different from Delaunay's. But the latter is in reality very general and therefore Delaunay's mode of integrating the canonical equations of the previous section will now be indicated. The form of the disturbing function will be taken to be R = B A cos (i-i I + i 2 g + i 3 h + i t rit + q) + R^ = -B-Acos0 + R 1 = R + R 1 where R l represents an aggregate of periodic terms similar to the one written down and n', q are constants. The term B and the coefficients A are functions of L, G, H only and in comparison with B these coefficients are small quantities of definite orders. Let #! = i l I + i*g + iji 6 i 4 n't q. 154 Variation of Element* [CH. xm Then the variables L,G,H-l,g,k can be replaced by L,G',H'; ir^frh. provided is a perfect differential ; and this condition is clearly satisfied if G' = G - ir*i*L, H' = H- ir l i z L for then d W = 0. If now -R a = 0, a solution of the problem can be found. For corresponding to the equation R = - B - A cos (#! + i 4 nt + q) the Hamilton-Jacobi equation takes the form /. dW \ dW -B-Acos(i 1 ^ r + i t n't + gr + - = \ oLi J Ot and a solution involving three constants G, g, h' is W = Ct + ir 1 1 OdL - i~ l L (i t n't + q) + g'G' + h'H' provided -B-A cos + C- i~ l L . i t n' = 0. This equation, which is in fact one integral, may be written C = B 1 +Aco&0, B, = B + i.n , i^L. . The solution, by 126, takes the form (a r = C, g', h' ; & r = c, - G', - H') t+c+ ir 1 JL \OdL = 0, ir 1 O l = ir 1 (O-i.n't-q) du J a G' = const., g = g' + i l - l ^-,\0dL H' = const., h = h' + ir 1 555 f^^- 0/2 J The lower limit of the integral involved is a function of (7, G', H', but the integral is so defined that the integrand 6 vanishes at this limit. The solution can also be written ~ 1 14, 145] Variation of Elements 155 At this point (C, g', h' ' ; c, G' , H'} are absolute constants, resulting from the solution of a Hamilton-Jacobi equation when the Hamiltonian function is R R l . Hence, by 127, the further treatment of the problem depends on taking these constants as new variables, and solving the canonical system dC dR, dGT dR, dt 8c ' dt. dg' ' dt dh' dt = ~dC' ~dt = ~"dG" ~dt = ~d~H'' But circumstances now arise which require further examination. For R l is now a function of the new variables, instead of the old, and the form of the function is important. 145. In the partial solution C = B, + A cos 0, ^ = V {A* -(C- B,Y-} = Asin0 where B l) A are functions of (and the constants C, G', H'), and @, are functions of t to be determined. The forms to be expected may be seen in this way. The above equations give (cos 0) d6 = 0/0 + 2tf r sin r0 when vanishes with t + c. Hence (t + c) is an odd periodic function of and therefore of X = (t + c). Thus, being some constant, = X + 20, sin r\, \ = 6 (t + c) and =/(cos 0) = @ + S@ r cos r\. These forms, which without a critical examination of the conditions have only been made plausible, are actually found in practice. It follows that L = t\ @ + i^&rcos r\, G=G'+i. 2 +i 2 '2 , 90 A sin ,. 9 = 9 + \^7^- a d\ = g +g (t+c) + Zg r sin r\ OLr (7 30 A sin , fj-, . ,d\ = h' + h (t + c} + hr sin r\ /" J and the original variable I is given by 1^1 = 6 iji't q i 2 g iji = \ i^n't-q i i {g'+g Q (t + c)} i 3 {h' + h a (t+c)} 156 Variation of Elements [CH. xin Now, since and () contain G, 0', H', these constants also enter into g , h and therefore into the coefficients of t in the arguments of the terms in R^. Hence t will appear outside the circular functions in the derivatives of R 1 with respect to C, G', H'. This inconvenient circumstance must be avoided by a change of variables. Now d 6 d = 6d -(t + c)dC+(g- g') dG' + (h - h') dH' by the form of the partial solution, and therefore d(ct-jd0\=-de-cdC + (g- g') dG' + (h - h') dH' + Cdt. This is a perfect differential and when each side is expanded in the form of a secular and a periodic part, the same must clearly hold true for each part separately, at least when the number of periodic terms is finite ; and in practice the remainder after a certain number of terms must be treated as negligible. But cos rX) o = A + SA r cos r\, A = + 2r r 0,. Hence, when the periodic terms are omitted, Cdt - A d\ - cdC + g (t + c) dG' + h (t + c} dH' is a perfect differential, to which d (A X) may be added ; and therefore the variables C, G',H'; c,g',h' can be replaced by A , G', H' ; X, K, rj where K = 9' + 9o (t + c), 77 = h' + h (t + c). This follows from 125, which shows that at the eame time RI must be replaced by .Rj C. All is now expressed in terms of the last set of variables, and secular terms are thus removed from the arguments of the terms in R^. It is convenient to make a final simple transformation. Since (ijX' X) dA n 4- ivtcdAo + irfdA.0 = d {A (i 4 nt + q)} + i^n'A n dt if . ijX' = X i 2 K i;?; i 4 n't q the variables A , G', H' ; X, K, r) can be replaced by .A' = 1^0, G" = G' + i 2 A , H" = H' + i,A ; V, /e,iy but at the same time it is necessary to add i t n'A Q to R^ C. Thus finally, if R' = R,- C + i 4 n'A 145, 146] Variation of Elements 157 the system of canonical equations dA' aft d&" d& dH" dR' dt~ ax' ' dt ' dtc ' dt drf d\' == _aB / d* = _dR/ dr, _dR' dt ~ 8A" dt'~ dG"' dt ~~ dH" is obtained. 146. If the value of X' be compared with the expression for I in terms of X it will now be seen that i^l = ^x,' + 2 (0,. i. 2 g r iji r ) sin ?-X and thus X' and I differ only by periodic terms. The same is true of K, g and 77, h. The periodic terms would disappear with A, as also those in and 6, and A would coincide with and . Hence the final variables are the same as the original variables when .4 = 0. The form of R' differs from that of R mainly in the complete removal of the term A cos 6, and naturally the most important term will be first selected for elimination. Periodic terms will be introduced into the arguments of R', but it is easily seen that on expansion they give rise to periodic terms of a higher order than A cos 6. The same process can be repeated indefinitely, until all sensible terms are one by one removed, together with those of a higher order introduced at an earlier stage. It has been assumed that ^ is not zero. If ^ = 0, i^g or i 3 h can take the place of ij 1. There are also terms for which ^ = i 2 = i z = 0. In the lunar problem these depend on the mean longitude of the Sun and are removed by a single preliminary operation analogous to the above. Delaunay's expression for the disturbing function contains over 300 periodic terms, and their removal involves practically 500 operations of the above kind, reduced to the application of a set of formal rules. This immensely laborious task was carried out unaided. But the result is the most perfect analytical solution which has yet been found for the satellite type of motion in the problem of three bodies. The solution is not limited to the actual case of the Moon, since it is expressed in general algebraic terms. The satellite type of motion may indeed be defined as that type for which the Delaunay expansions are valid. It seems an interesting problem of the future whether such satellites as Jupiter VIII and IX will be found to satisfy this definition. Their conditions differ widely from those of the lunar problem, in particular in the fact that the motions are retrograde. CHAPTER XIV THE DISTURBING FUNCTION 147. The development of the disturbing function R in a suitable form gives rise to many difficulties, partly of analysis, partly of practical computa- tion, and is the subject of an extensive literature*. It is possible to deal here only with a few of the more important points. The principal part of the disturbing function for two planets involves the expansion of A" 1 , the reciprocal of their mutual distance. It is therefore important to consider the nature of this expansion, or rather of A~ 2 * in general, where s is half an odd integer. For this more general form will give the derivatives of A" 1 , A 2 being a rational quantity, and these will naturally occur when A" 1 is expanded in terms of any contained parameter. It is convenient to consider first the case of two circular, coplanar orbits. Then, if H is the difference of longitude in the plane, A 2 = aj 2 -f tt 2 2 - 2a : 02 cos H a t , a 2 being the radii of the orbits. Let aj < a 2 , a = i/a 2 , ^H = log z, i? = 1 and therefore tt 2 ~ 2 A 2 = 1 + a 2 - 2a cos H = (1 - az) (1 - a*" 1 ). Hence the function to be examined is = (1 + a 2 - 2a cos H)~ g = $b s + 2 6,* cos iH. i Since the function is unaltered when z and z~ l are interchanged, b s ~ i = bg i , and i may be treated as positive. The coefficients b 8 { are called Laplace's coefficients. By Fourier's theorem, &/ = (1 oz)~ 8 (1 as" 1 )"* ^ -1 dz 7TI J = - I (1 + a 2 - 2a cos t)~' cos td< W.'O * Cf. H. v. Zeipel, Encykl. der Math. Wist., vi, 2, pp. 560-665. 147, us] The Disturbing Function 159 The first (complex) integral is due to Cauchy ; the path of integration is taken round a circle of unit radius. By introducing the Weierstrassian elliptic function p 0) = z - i (a + a- 1 ) Cauchy's integral clearly becomes an elliptic function, and Poincare has shown how this function can be reduced to a calculable form. But another method will be followed here. The coefficients b s i are easily developed as power series in a-. For, with the use of gamma functions, (1 - az}-* (1 - or- 1 )- - 2 r^- +-P).- a?** . 2 r ^+gl_ 0.1 z~i p T( and therefore, when p = q + i, r(<7 + i) a*, 2, <7 But this can be recognized as a hypergeometric series, and when it is expressed in the ordinary notation, bj=^F(s, s + i, t + 1, ^ ............... (2) By the known properties of the hypergeometric series, this expansion is convergent when a< 1. There are many equivalent forms, but (2) is enough for the present purpose. 148. Laplace's coefficients are subject to several formulae of recurrence, which facilitate their calculation. That such exist follows from the known relations between sets of three contiguous hypergeometric functions. Instead of finding them directly, a more general function may be considered, for this reduces to b s i when j = 0. In the integral ) write z = a%, and then It follows that TOO The equivalent forms TTICC I s - (i +j + *'a 2 ) /'' + (i +j + s - 1) a B^ = . . .(4) The identity (i-a^-^a-r 1 )"*^' 1 = (1 -'a a ) '-' (1 - t- 1 )-"- 1 {(I + a*) f'+J- 1 - a 2 f w - ' +j ~ 2 } gives similarly on integration (s + j) B iJ = s(l + a 2 ) 5*'^ - 8CLB i+ *' j - sa B l l\ j ^ J' s ' s+1 s+l s + 1 and after eliminating the last term by means of (4) with s + 1 in the place of s, When j = 0, (4) and (5) give formulae which apply to Laplace's coefficients. Derivatives of the latter with respect to a can then be expressed as linear functions of Bj'J. i 149. Newcomb's method of calculating the coefficients b s i , together with their derivatives in the form subsequently required, can now be explained. Let 2s = n, 8 = -, D = a dy? da and let This is not Newcomb's definition of c^i, but it is the equivalent. Thus D cji = { J ( - 1) + i + 2j} cj'l + c n ^ +1 and therefore J +1 ............ (6) so that these derivatives of a higher order are easily deduced from those of the next lower order. Let 148-iso] The Disturbing Function 161 and then, by (4), where The development is to be carried to a definite order fixed by i = k, say 11. In the first place p n k >J is calculated for the required values of n, j by a direct method. Next p n k ~ l 'i, , p n l ' j are deduced in succession by (7). For i = I, s = -|-, the formula (3) becomes + a d l ' = or The first coefficient d ' is calculated directly. Then (8) gives d ' * ( j = 1, 2, . . . ) in succession. The formula (5), when i = 0, gives \n [in + (j + in) a 2 ] c^ 2 - $n (j + n) a cj or (),j _ w 2 / w c " +! i H + (j + i) i, where n1, 3, ..., j = 0, 1, 2, ..., and k is the highest value of i to which the expansion is carried. Now a complete elliptic integral which can be found in great variety of ways. Newcomb commends for the purpose the arithmetic-geometric mean, which follows from the identity /I"- /* (a n 2 cos 2 + V sin 2 $)~ * <& => (a? n+l cos 2 ^ + 6 2 , l+1 sin 2 ^r)~ * dtyr J o 7o where 2ft)i+i dn + O n> 6"n+i = ft0n< P. D. A. 11 162 The Disturbing Function [CH. xiv This is obtained immediately by the transformation of Gauss . ,_ 2(/ n sin^r (a n + b n ) cos- ^ + 2a w sin 2 ^ and can be extended indefinitely by successive steps. It is obvious that the sequences a n , b n have a common limit A and hence that the value of the integral is 7T/2A. In the present case is easily obtained. The calculation of p n k> i is based on the hypergeometric series (2). It is clear that s + i, i + 1, a 2 ) = ^( + 1, s + i+ 1, i + 2, a 2 ) i + 1 and therefore generally i>+j) r(+t+j) r< + i) Wfo...)- -r^)-- r( , + i) - r(i+j + l) f Hence, by (2), - 9 Q itT/ , ; ,-, ? " , ; i i a a -^" and therefore, since n = 2s, f.j = B 's 3 = ^n + i+j-l F (n +j, jn + i +j, i +j + 1, a 2 ) g F ' j g'~ 1 'J i+j ']^(^n+j, ^w + i+j-1, t'+j, a 2 ) 5 The quotient of .the two hypergeometric series can be converted into a continued fraction by a known theorem* of Gauss, and as it converges rapidly a few terms suffice to give its value. By this method Newcomb determined the required values of p n k 'i. 151. In order to obtain the desired form of the continued fraction it is not necessary to introduce the hypergeometric series. By (3) and the following equation, B i+i,j a B i+l>j+1 ^B l ' j+l 1 >J *' '"' * ' AJ'"" _ g- _ S 8 Pn-2 ~~ ni-l,j + 2 ~ D77+l _J a D*-1.J+1 s 1 s s and by (4), (i -s + 1) afl; +w+1 -(i +j -f- 1 + ia 2 ) ti i ; j+1 + (i +j + s) a B^ 1 '^ 1 = 0. *lChrystal's Algebra, n, p. 495. 150-152] The Disturbing Function 163 These are three linear equations in B s i+1 J +1 t 5/- J ' +1 , Bj-*'J +l , which can be eliminated. The result may be expressed in the form : (i-s+l)a i+j+itf+l (i+j + s)a ; = 0. 2 -t^n - 2 1* + 1, / i + lj.7 + Ci Pn Pn After expansion and division by (1 a 2 ) this gives or 1 ' ^ ~ ia } i<* - * + !) X " - (' + J + 1)1 Therefore (7) gives (2s = n) __ (i+j + * - 1) a i+j- ( S +JXT- *) a 2 (t + j + 1 - (t - (_t + j + * - 1 ) a (s + j) (1 - s) a 2 1 - 1 - * + j + 1 + s-l) (a + j) (1 - s) a 2 (i-s+ !_) (i ' i- i- i- and this is the required form. The relation between the alternate constituents is obvious enough, for the substitution of j + 2 for j and n 2 for n (or s 1 for s) clearly has the effect of increasing each factor by 1 in the numerators and by 2 in the denominators. As i = k is a fairly large number in the direct calculation of p^'i, the" even constituents are small and the calculation is based on an odd number of terms (generally five). With the use of subtraction logarithms the process is rapid. 152. The next step is to consider two circular orbits in planes inclined at an angle /. Let L 1} L 2 be the longitudes in the two planes, reckoned from the common node, and let fji = cos , v = sn , x = L l L z , y LI + L z . Then the angular distance between the planets is given by cos H = cos Zj cos L 2 + sin L l sin L 2 cos J = p cos x + v cos y 112 164 The Disturbing Function [OH. xiv and a 2 A- 1 = (1 + of - 2 cos H) ~ > = &> + 2 2 > cos ix + 2 2 6' > cos JT/ + 422 &'> i cos we e=l j=l i=lj=l where """ 77 1 ) cos ix cosjy dx dy. 1 /""" f 77 = 2 I I When v is small A" 1 can be expanded in powers of v. Thus 0-2 A" 1 = {1 + a 2 2a cos # 2av (cos y cos #)] ~ * =o or where It is only necessary to compare the coefficients of f*'^ in these expressions in order to have &* as a power series in v, the coefficients being functions of a. Thus, for example, as far as v 2 , 26*- = &* - av 6/ +1 + ft*- 1 + a 2 ^ 2 6' +2 + 4&s' + fc-* - ... It is easy to continue these developments further, and this is the method used by Le Verrier and Newcomb. But its validity is limited. The binomial expansion (10) of a 2 A -1 is convergent only when 1 + a 2 2a cos x 2 a (cos y cos x) and since the most unfavourable case, cosx= cosy = 1, must be included sin- J J = v < (1 a) 2 /4a. It has been proved by H. v. Zeipel that the same limit applies to the expansion of Jacobi's coefficients b { 'i. This condition is satisfied in all cases by the small inclinations of the orbital planes of the major planets. 153. Among the orbits of the minor planets, however, are some whose inclinations to the plane of Jupiter exceed the above limit. It is therefore desirable to find a more general form of development. Let F-* = (1 + a 2 - 2ao-)- s = 2(7/a /l . 152, 153] The Disturbing Function 165 The coefficients C s n are polynomials in a, which are in fact Legendre's poly- nomials when s = ^. Differentiation with respect to er and log a gives tfs+2 (1C n V * n n 1 i 2 2sa ~ da- Jfs+2 ,72/7 n _, u/ \J S _ , . 2-7 o " = 2(s+l) a 2sa - 2 wC,"a n = (o- - a) (1 + a 2 - 2acr) 2sa - 2 ?i 2 (7/a n - (o- - 2a) (1 + a 2 - 2a cos ?/) - -^ (/*" cos a? + v" cos y) $0" L IMV . /n = ~i~> ^2 + ;n /* + r - 1 / (^ - + ^ 2 ) /* z; dr a 2 , , f ^ ) I > -^ -- r 5~7 oar fj, v oyj cos x + j 2sv (w" v'*} \ cos y and therefore if yLl V 2s 4 - -, (^ - l\ = 2.s- - x - , fc- - l) = JV the equation takes the required form d 2 C 1 3 . . w (TO + 2s) C = -V-, 1 r- pv dP Vyu,^ o*- 2 yu, v oy 2 154. At present p, and y are any functions of t. Let ^- = (l-p l }(\ -p s ), v z =p l p. 1 . Then it will easily be found that the first condition becomes ipfi'vv'M = ( Pl - p 2 ) 2 /,//,/ = 0. Hence either pj = p 2 or p 2 is independent of t. The first case has the more obvious importance since it gives directly v = PJ = sin 2 J, yu, = 1 - p, = cos 2 1 /. The second condition may be written and the right-hand vanishes because //, + v = 1. Hence the method can only be pursued further when s = , but this happens to be the most important special case. If now t = v, v = // = 1, //," = v" = 0, and the partial differential equation (13) in C becomes v-~. v oy 2 ' dv 153-iss] The Disturbing Function 167 On inserting the series (12) and comparing the coefficients of cosix cosjy this gives n (n + 1) A n i j = - v (1 - i/) d ^' j + (^ + J-] A\ tj + (2i/ - 1) -^ . ' iJ ' rttt~ \ I 11 i / tj \ / y-/i But the direct expansion of F s shows that since cosix cosjy arises from terms of the form (/u,cos# + vcosy), A n it j must contain /u, l V as a factor. It is therefore proper to write 4\ f (i-iOMJ?^^ + ^(*^> 1 )--M : ^+W-")( i +^ 1 +)^W-^ \lll/ \J(jV Now -B\ j is a polynomial in v with a constant term, and this equation gives the law of its coefficients. But the equation is clearly'of the form satisfied by a hypergeometric series. Hence A n ij = c/jL l viF(i+j n, i+j + 1+n t 2j ' + 1, v) (15) where c is a constant depending on i,j, n. This gives the form of Hansen's development in powers of a, namely a. 2 A" 1 = 2 a w . A\ j cos ix cosjy, (n > i + j). n, i, j The determination of the constant c may be deferred. 155. This is the simplest, most obvious application of the method. But its possibilities, though limited, are not exhausted. The first condition for its use is also satisfied by making p 2 a constant. This may be expressed by where J is to be treated initially as constant, though finally it will be identified with ./. The relation /i + v = 1 no longer holds formally, but is replaced by and the result of differentiating this twice with respect to t and eliminating tan ^ J" shows that the right-hand side of the second condition (14) is 1. Therefore s = 1. At first sight this case has no present interest, since s is not half an odd integer, but the reason for considering it further will be seen later. The development will be in powers of sin 2 |J as before, but it will be convenient first to make t = ^J, so that // = sin ^J cos J , v = cos | ./sin J , /j," = v v. 168 The Disturbing Function [on. xiv Then the partial differential equation (13) for C becomes n (n + 2) G = - ' - sec 2 1-^- 9 - cosec 2 1 -- - 2 cot '2t =- . ot 2 das 2 dy 2 dt The form of the solution resembles the previous case, suggesting (7=2 ii i v^T n i t j cos ix cosjy and the comparison of coefficients of cos ix cosjy after the substitution gives Now let the independent variable be changed to r = sin 2 = sin 2 \ J, so that - , 2 dt dr dt 2 and the previous equation becomes 4 (T 2 ~ T) ~ + 4 {(t+ j + 2)r- O'+l)) J + (t+j -n)(i+j+ 2 + n)T\ } = 0. Now T\j is a polynomial in T with a constant term, and this equation determines the formation of its coefficients. But again it is an equation of the type satisfied by a hypergeometric series. Hence where c t is independent of r. But //, and v, and therefore T\j, involve J 9 symmetrically with /, and therefore it is evident that Cj contains as a factor the same polynomial with T replaced by T O = sin 2 1 Jo. Hence T\ j = c 2 F(r )F(r) where c 2 is a constant independent of T and T O . This is clearly general, whatever the values of J and ,/ . A return to the actual problem can now be made by putting J U =J, and then r = v and + n ... ' ^ 2 ' 2 which gives the form of expansion 2 2 A~ 2 = 2 a n . T n u /*V cos ix cosjy n, i, j (i+j< n). The form of proof is essentially that of Stieltjes. The squared (terminating) hypergeometric series is a polynomial of Tisserand. The more general utility of this result will now be easily seen. For a 2 2 A- 2 = (1 + a 2 - 2 cosH)~ l = (1 - OLZ}~ I (1 - az' 1 }^ = {z (1 - az)-> - z~ l (1 - a*- 1 )" 1 } (z - z~ 1 }- 1 = 2 a n (z n+l - z~ n ~ 1 } (z - z~ l )~ l 155, ise] The Disturbing Function 169 Hence, by comparing the coefficients of a", sin (n + 1) H/sin H = 2 T n i j/j. i v j cos ix cosjy. But CO (a. 2 ~ l A)~ s = $b g + 2 &/* cos nH i = ^6 g + 2 %b g n {sin (n + 1) H sin (n - 1) Ity/sin 7/ and therefore (a 2 ~ ] A)-* = li; 1 + 12 b, n 2 (T?_. - r;: 2 ) /*V cos iar cosjy . . .(16) which is Tisserand's development in a series of Laplace's coefficients. 156. To complete the result it is necessary to find the numerical factor c 2 . Now the final term of F(oi, /3, 7, x}, a., /3, 7 being positive integers, is Hence the term containing the highest power of v in T\ j fj, i v j is 2 V*. But a 2 2 A~ 2 = {1 + a 2 2a cos x 2av (cos y cos x)}~ 1 = 2 (2ay) m (cos y - cos #) m (1 + a 2 2a cos a?)-- 1 and the highest power of v associated with a n is given by the terms (cos y - = 2 when The same terms appear in the form 2 T n ; j fj, { vJ cos we cos jy = K^ T\ } fjfvJ^rf where K = 1 when i and j = 0, K = 1 when i or j = 0, and K = | otherwise. The highest power of v has already been found in this form, and comparison of the coefficients of v n %'rf gives finally The development (16) is now completely defined. 170 The Disturbing Function [CH. xiv The numerical factor c in Hansen's development (15) can be found similarly. For the term containing the highest power of v in A\ j is On the other hand the terms associated with a n and the highest power of v in a 2 A -1 are by (10) contained in and these are now known. As before, the coefficients of i/ n f'V in the two forms of aaA"" 1 can be compared, and thus (2n) ! <2y) ! (-l)'(n !) 2 F ( + J) (n + j - i) ! (n + t' + j)\ H {ft (n i .7)] I}' T (n + 1) where II denotes the product of four factorial factors. Now ^(n ij) is an integer, n ij is even, and. the sign is the same on both sides. Also r(n + l) = n\, 2 2W F (n + i) . n I = T () . (2n) !. Hence finally _ (2?) ! [ j which completes the determination of Hansen's development. The results obtained for inclined circular orbits may now be summarized. Since cos ix cosjy = cos i (L^ L 2 ) cosj (Li 4- L 2 } = | cos [(i + j) LI - (i -j) L 2 ] + | cos [(i -j) L l - (i +j) L a ] it is possible to write where log\ 1 = t// J , logX 2 =tZ 2 ; and it has been shown how the coefficient A(p l ,p 2 ) can be developed (1) in powers of j' = sin 2 ^J, (2) in powers of a = Oi/a 2> (3) as a series in Laplace's coefficients. 157. The preceding developments of A" 1 or A~ 2S apply to circular orbits, but they are not on that account to be regarded as mere approximations to the forms actually appropriate to the orbits of the solar system. On the contrary they constitute the essential source from which the latter forms must be generated by the most convenient means. Now quite generally A 2 = 7-j 2 + r 2 2 2rjr 2 cos H and LI, L 2 must be replaced by (a t +Wi, w. 2 + w. 2 , where to l} w.> are the longitudes of perihelion reckoned from the common node, and w lt w 2 are the true anomalies. When the eccentricities e } , e., vanish the radii ?, , r., become 156, is?] The Disturbing Function 17 f l the mean distances a 3 , a 2 , and w l , w. 2 can be identified with the mean anomalies M 1} M 2 . The corresponding value of A may be written A . Taylor's theorem can be expressed in the familiar symbolical form f(x + y) = exp. (y 3-)/(*) = exp. (yD}f(x} \ U,t / which means simply that if the exponential function be exparicfccfas though yD were an algebraic quantity, the result otherwise known to be true is formally reproduced. Thus generally, /(! + y lt ar a + 2/ 2 , ...) = exp. (y l D 1 + y. 2 D 2 + ...)/Oi, a? 2 , ) where D r operates on x r alone. Now when e l = e 2 = 0, Ao" 1 =/(!, 02, A, A) is an expansion of which the form has been completely determined. The more convenient developments refer not to r a but r/a, and the change from the argument a to the argument r is made additive by taking log a as the variable instead of a. Thus in the present case x 1 = \oga 1 , # 2 = log a,, x s =-L 1 = to- i + M 1 , x 4 = L z = w 2 + M 2 y 1 = logr 1 /o 1 , y a =logr a /a s , y 3 = w 1 -M l ,. y 4 = w 2 -M 2 n a , 8 n 8 8 -L'l ^\ i ^l-i > -^2 -~, i ^2 o - j 9 log aj dc^ 9 log a 2 9a 2 j r ) , = A = tX A !'- 8 ,. A. 9//! 9Xj ' 9// 2 " 9^2 Then generally = exp. [log ~ - A + log J . A + (w, - 3fO D, (/, a- 2 But in the notation of Hansen's coefficients ( 45) where log a? = tw, log = iM. Hence in a corresponding symbolic notation, since log ac/z = i(w M), A-i V Y D> '~ ll>3 i V D -'~ lI>i Simplifications are now possible owing to the form of /. In the first place A -1 is homogeneous, and of degree 1, in a l} a 2 . Hence A + D 2 = ! + a. 2 ^- da., 172 The Disturbing Function [OH. xiv But further /has been expanded in the form and so that A> A can be replaced by ip 1} ip. 2 , and A. A do not operate on X a , X 2 . Hence the. symbolic form of the complete expansion becomes Pi , P* i, j where log X a = i (w 1 + M^, log Xo = i (&> 2 + M 2 ), \ogz 1 = iM 1 ,\ogz. 2 = iM 2 , and the symbols X are respectively functions of e, , A and e 2 , A- 158. This leads immediately to Newcomb's operators as defined by Poincare. For the functions X can be expanded in positive powers of e, so that where m 1 - i ,m 2 - \j | = 0, 2, ..., since Xi n > m is of the order fe 1 *- at least. The operators II are combined by Newcomb in the notation but the combined symbols, though tabulated by him over a wide range, seem to present no practical advantage over the constituent operators. The final form of the development of A" 1 can therefore be written and the completion of this part of the problem depends on the practical treatment of Newcomb's operators II, which are polynomials in D, p of degree in, with numerical coefficients. The definition of the symbols is given by 2 n,- (A p) e* Hence in particular 2 n^ (D, 0) e m ^ = (-) , 2 n/ (0, ^) e m , i \O>/ m, i and therefore , 2 n^ (D, p) e m z i = 2 H,-" 1 (D, 0) e w ^ . 2 H / (0, p) e^'. l, i m, i ii. j Comparison of the coefficients of e m z i on both sides then gives n/ (A p) = 2 n/ (A o> n;q w (o, P ) , j where w = 0, 1, ..., m, and j has all the values which make n \j and ? n 1 1 j positive integers (including 0). This formula, due in another 157-159] The Disturbing Function 173 notation to Cowell, makes the calculation of IT/' 1 (D, p) depend on the expansion of r/a and x v . But these are known forms. The first is given by (22) in Chapter IV. Means of deriving the latter have been given in 45. In fact and therefore it is necessary to expand X.'^* in powers of e and the resulting coefficients will represent 11^(0, p). They are purely numerical and can be tabulated for all moderate values of m, i and p. Other methods have been suggested to facilitate the calculation of Newcomb's operators. But the above will suffice to make clear the principles involved. 159. The disturbing function due to the complete action of a single planet can now be considered. By (3) of 23 this is R = Gm \ -r- - . (xx + yy' + zz r \ A T ' where (x, y, z), (x, y', /) are the heliocentric coordinates of the disturbed and disturbing planets ; r' is the radius vector of the latter. The constant G may be reduced to unity by the choice of appropriate units, and the dis- turbing mass m' may be understood as a common factor to be restored ultimately. Thus R = (,-2 + r 's _ 2rr' cos H) ~ * - rr'~* cos H where H has its previous meaning, the mutual elongation of the two planets as seen from the Sun. The principal part, already discussed, is symmetrical in r, r', but the indirect part is not so. Hence a distinction must be drawn, according as the disturbing planet is superior, when r = r 1; r' = r 2 , or the disturbing planet is inferior, when r = r. 2} r' = r l . Now when the eccen- tricities vanish, by 152, (/.A- 1 = 6' + 2b l > n cos x + 26 ' 1 cos y + ... cos H = \ /* cos x + v cos y and R A" 1 = BR = aa'~ 2 (//. cos x + v cos y) is the correction required to change A" 1 into R. This can be effected by giving corrections to b 1 ' and 6 - 1 , thus = a (a' > a) ; or- (a > a') where a < 1 always and a' is the mean distance of the disturbing planet. If these corrections are carried into the expansion in terms of v ( 152), as used in 174 The Disturbing Function [CH. xiv the chief planetary theories, it will affect the Laplace's coefficients only to this extent : 8b = - a, &b % = -2 (a' > a) $b = -a- 2 , S6 4 = -2a- 3 (a > a') for it is easily verified that these changes will give the required corrections to 6 1 ' , & M . In the exponential form they apply equally to b~^, b>~ 1 , and b,~\ Thus the indirect term is very simply incorporated in R Q , in which e 1 =e 2 = 0, and the full expansion of R in terms of the eccentricities can then be deduced in the manner explained for the development of A from A . It is most important to remark that while the indirect part modifies the coefficients of certain elementary periodic terms, it affects in no way the constant term which is independent of the time. 160. Another order of development is possible by expanding A" 1 initially in terms of rj/r 2 . If this ratio is small, as in the case of the solar perturba- tions of the lunar orbit, this method has great advantages. By 153 this expansion takes the form A" 1 = 2 r^ r 2 - n ~ 1 A \ j cos ix cos j y n, i, j where A\j is given by (15) and x, y have their true meanings, Wj + W 2 = co, + Wi + (&> 2 + w 2 ). It is more convenient to use the exponential form, and with a slight change of notation for the coefficients, A- 1 = 2 r^r^-^An ( Pl , p,) tf*pf* n, p t , p 2 where log /^ = i( Wl + w^, log /i. 2 = i (w. 2 + w 2 ), \p 1 -p,\ = 2i, p 1 + p 2 = 2j and n \p 1 , n \p 2 \ are even positive integers. Hence n, p, , p, where logXj^ *(! + MJ, logX 2 =t( Pi ) Ps 9\> + M, L' w + M'. The subsequent process introduces e, e into the coefficient A, which already contains powers of v sirf^J, and adds multiples of M, M' to the argument. In the ordinary notation for the elements, w = OT I) ;Y, to' = tzr' II* % where ^, %' are the distances of the intersection of the orbits from their ecliptic nodes. Hence R takes the form R = 2 AftPtf cos \hM + h'M' + (p + q) O - fl) -(p-q) ( V > - ft') - p ( X - % ') -q( X + x ')]. Now the two orbits with the ecliptic form a spherical triangle ABC in which = X'> b = X> c = n s - flj ^=1, B = 7T-i', C = J where i, i' are the inclinations of the orbits to the ecliptic. Hence, as in 67, if the intersection be taken as the ascending node of the disturbing orbit on the disturbed orbit, sin $(x + %') sin J= sin (H' - II) sin (i' + i) cos ^ (% + %') sin ^ /= cos i (II' O) sin ^ (i' i) sin ^ (% - %') cos | J = sin -| (II' H) cos (*' + 1) cos ^ (% %') cos ^ J= cos | (fl' O) cos \ (i! i) and therefore v exp. ^^ (% + %')= s in \i! cos |i exp. ^ t (ft' H) sin ^i cos \i' exp. \i (II' H) /i^exp. ^t(% %')=cos^i'cos^-iexp. |t(H' O)+sin^ism^''exp. ^i(H' II). It follows that 1/9 cos ^ (% + %') = 2 6 g cos s (H' H), yi sin ? (% + %') ^ ^ g in s (&' ^) /i^ cos p (% - %') = 2 a g cos s (H' H), ^ sin p (^ ^') = 2 a s sin s (H' O) where a s , b s represent simple coefficients involving i, i'. Thus % %' can be eliminated from R, which now takes the form R = S A cos [hM + h'M'+(p + q) (vr-fl)-(p- q) (r' -n')-(s + s') (IT - fl)] 176 The Disturbing Function [CH. xiv where A now contains a, a, e, e', i, i' and also powers of v. But from the above analogies of Delambre, v = sin 2 (IT - n) sin 2 (i' + i) + cos- (IT - H) sin 2 i (*' - i) = ^ (1 cos t cos i) ^ sin i sin i' cos (fi' 1). Hence these powers of v can be removed from the coefficient without altering the form of the arguments, which are only changed by the addition of some multiples of O' - H. Thus finally R = 2 A cos [hM + h'M' +gv + #V +/O +f&] = 2 A cos [h (nt + e) + // (V* + e') + u u -\-u v 2 = fj, au~ 2 -5- dt dt "Vw -, M j \(ju U i C* j . at dt Now let u~*d = H, - t = Hu*^ and the first equation of motion becomes u i d f TT *du\ . ^ TTn , fduV . . rr du Hu 1 -JQ ( H u~ 2 or P. D. A. 12 178 Absolute Perturbations [OH. xv But by the second equation of motion H=h-a0 where h is constant. Hence d z u p _ d0* + (h-a0?~ It is enough to retain the first power of a, so that and the integral is u = ph- 2 {l+ecos(0-y) + 2cth- 1 0} ..................... (1) where e and 7 are constants. 163. The osculating ellipse at the point 6 = 1 is obtained by supposing the resisting medium to disappear at this point and the subsequent motion under the central attraction to be undisturbed. The path is then u = pr 1 {1 + &i cos (6 - 70}- The motion at the instant is the same in the actual trajectory (1) and in this ellipse, and thus = 1} u = Ui, u and 0, and therefore H = H^ and dujdd are the same for both curves. Let /u/r~ 2 = p~ l . Now H l is the constant of areal velocity in the ellipse, and hence pr L To the first order in a. then j Again, by equating the values of u and dujdd, pr 1 {1 + e l cos (0j - 70] = p- 1 [l + e cos (6 l - 7) + 2a^~ 1 6^ Pi~ l { e l sin(0 1 j l )}=p~ l { e sin (6^ 7) + 2a/z~ 1 } and to the first order in a ! cos (0j - 70 = e cos (0, - 7) - 2aA~ 1 e0 1 cos (0! - 7) ej sin (#j 70 = e sin (^ 7) 2a/i~ 1 2a/t" 1 e0 1 sin (0j 7). Hence 0! cos (7! 7) = e 2ah~ l e0 1 2ah~ l sin (0! 7) e x sin (71 7) = 2ah~ l e cos (#1 7) and, still to the first order, [e6 l + sin (#1 7)} 1 cos (0j 7). Between these terms an important practical distinction is at once apparent. That in A^ depending on 0^ will diminish the eccentricity indefinitely until the orbit becomes circular. It is a secular term. The other terms are 162-164] Absolute Perturbations 179 periodic, and when a is small their effect, not being cumulative, is small also. In practical applications, to Encke's comet for example, they can be neglected. Then A7j = and the direction of the apsidal line is unaffected by the resist- ing medium. In a complete revolution the secular effects are given by ei Pi and the corresponding changes in the mean motion and the mean distance are given by A/>! __ 3A!_ 3 A/)! 3e l &e 1 \ + e^ 6-rra % = ~2~aT = ~2"^7 ~ 1 - e? ~ 1 - ef ' ~h since a 1 =p 1 (1 e*)~ l . Thus the most important effects of a resisting medium are a steady increase in the mean motion and a steady decrease in the mean distance, which must ultimately bring the disturbed body into contact with the centre of attraction. 164. This simple example has been chosen, apart from its intrinsic interest, because it illustrates certain important points. There is, in the first place, the osculating or instantaneous ellipse, which is p^u = l + e 1 cos (6 7j) and not pu = 1 + e cos (6 7). The latter is a definite curve which may be called an intermediate orbit and may serve usefully as a curve of reference. Indeed it' has been so used in what precedes. But it is not the osculating orbit at any time. There is also the distinction drawn between periodic and secular disturbances in the motion, of which the former may be relatively unimportant compared with the latter because these, however slow, are cumulative in effect. The general nature of disturbed planetary motion can now be considered. For two planets only, the disturbing function has the form, found in the last chapter, R = 2F(a, a, e, e, i, i') cos T, T= [h (nt + e) + h' (n't + e') + g<& + g'ts' +/Q + /'ft'] where (a, n, e, i, ft, BT, e) are the elements of the disturbed orbit, (a, ri, e, i', fl', CT', e') the elements of the disturbing orbit. The equations of 139 are now available for finding the variations of the elements. In accordance with the artifice explained in 140 the mean longitude e is taken in a special sense there defined, and a in the coefficient and n in the argument of any term are treated as independent in forming the partial differential coefficients of R. Therefore dR dR dR da ' de ' di 122 180 Absolute Perturbations [CH. xv are all of the form 2(7 cos T, and are all of the form 2(7 sin T, where T is the argument of the term. Hence the equations for the variations are themselves of the form ?-24nnr,.. at In the first approximation the right-hand members (which contain the dis- turbing mass as a factor) are calculated with the osculating elements of both orbits for a certain epoch, and these elements are treated as constant. The equations can then be integrated, and in fact 8, a = - 2 C, cos Tj(hn + h'ri), . . . These are the absolute perturbations of the first order. Similarly the pertur- bations of the first order in the masses can be calculated for all the disturbing planets concerned and the results can be combined by addition. 165. Each term in the perturbations represents a distinct inequality in the motion of the disturbed planet. It will now be seen that the inequalities are of two kinds. The multipliers A, A' have all integral values, positive and negative, including 0. When A = A' = the disturbing function R is reduced to that part which does not contain the time. Thus da dl and the inequalities are secular. From the present limited point of view they will increase indefinitely and in the course of time will modify the conditions of the planetary system profoundly, uncompensated by any check. But one remark can be made immediately. The most important element as regards the stability of the system is clearly the mean distance a. Now when A = A' = 0, not only does t disappear from R but also e. Hence sm t 9i and after substituting the above expressions for a,..., e and expanding by Taylor's theorem, The reduction of the right-hand side to a suitable form will be readily understood in general terms, apart from the complexities which will naturally arise in the practical calculation, and a simple integration, requiring the introduction of no arbitrary constant, will give the expression of S 2 ft. Similarly the perturbations of higher orders, so far as they are of sensible magnitude, can be found successively, when those of the lower orders have been deter- mined, for all the elements. 167. The general form of the results will now be apparent. In the first order the inequalities are of the forms A cos (vt + h), At only. In the higher orders the terms obtained by the algebraic composition and subsequent integration of these two forms will clearly belong to one of the three types A cos (vt + h), A t tn , A t m cos (vt + h) which may be called respectively periodic, purely secular and mixed terms. The term order may be retained to denote the degree a of A in the masses. As A is also a function of the eccentricities and inclinations, which are also in general small parameters, it may be limited to a homogeneous function in these parameters. Then the degree of the term is the degree of this function and represents its order in respect to the eccentricities and inclinations. A further classification is used by Poincare. The order of a term being a, the rank of a term is represented by a - m, or by the order less the exponent of t. A term of high order is initially small, but if m is large it will grow rapidly in importance, so that ultimately the terms of the lowest rank will have the greatest significance. The occurrence of long-period terms with small divisors has been noticed. In the higher orders these divisors will be combined and raised to higher powers by the subsequent integrations. Let m' be the sum of the exponents of such divisors in any term. Then the class of that term is defined .by the number o ^ (m + m'). It will now be clear that the value of these different categories depends on the length of time contemplated. For relatively short 166, 167] Absolute Perturbations 183 intervals the most important terms are those of low order. In longer intervals the terms of low class rise into prominence. And finally it is the terms of low rank which have the greatest influence in the ultimate destiny of the system. But here 'a question naturally arises. How far is the form in which the terms present themselves natural to the problem, and how far are they the artificial product of the particular method by which they are obtained ? It is evident that the physical importance of this question is notr quite the same in all cases. Thus a mean motion in the position of the node or perihelion may be admitted without any serious direct consequences to the nature of the system. On the other hand, a purely secular term in the mean distance or the eccentricity, taken by itself without compensating circumstances, must ultimately prove fatal to the stability. The general problem suggested is very difficult and the reader is referred to the first volume of Poincar^'s Lecons de Mecanique Celeste for a thorough discussion. It must, however, be pointed out that the form of the results may be perfectly legitimate, so far as it goes, and at the same time not in any way inconsistent with the stability of the system, though a decision is beyond the range of the above elementary methods. It is impossible to be satisfied with the solution here described as a final representation, and this feeling is ob- viously suggested by considering the mixed terms. Since the corresponding oscillations increase in amplitude indefinitely with the time the departure from the original configuration will become so great that the fundamental assumption of small displacements in forming the equations for the variations will be contravened. Then one of two things will happen. Either the mutual forces will tend to restore the original configuration, and there will be stability, or the forces will tend to magnify the disturbance, and there will be instability. But in either case equally the method adopted breaks down and the funda- mental question remains unanswered. How then are the statements to be reconciled, that the method which is the method on which the existing theories of the major planets are actually based may be perfectly legitimate, and that, while the form of the terms to which it leads obviously suggests instability, complete stability is never- theless entirely possible ? The simple answer is that it is only necessary to imagine that v in the argument of any term is itself a function of the disturbing masses. Now the above method involves a development in powers of the masses, and when the parameters which represent the masses are thus forced out of the circular functions they carry the time t explicitly with them, and the appearance of secular and mixed terms is a natural consequence. Yet the development in terms of the masses may be convergent and entirely legitimate. In this way it will be seen that the occurrence of secular and mixed terms is compatible with stability, though a profound discussion is necessary for a positive conclusion on this point. 184 Absolute Perturbations [CH. xv The case of a planet moving in a resisting medium is quite different. There is then a definite loss of energy and the effect of the secular changes is not doubtful. 168. In the theories of the planets on which the existing tables have been based the coordinates of the planets relative to the Sun have been used and this fact governs the form of the disturbing function, which is distinct for each pair of planets. For practical purposes this choice of coordinates is an obvious one. But for theoretical purposes it is unsuitable, chiefly because, like the common system of elliptic elements, it is ill adapted to the transfor- mations which are an essential feature of the dynamical methods initiated by Hamilton. Another system of coordinates, due to Jacobi, will therefore' now be introduced. Let (gi, rn, &) be the coordinates of the mass TO; in a system of n masses m^mz, ...,m n , the origin being any fixed point. The masses are taken in any fixed order, represented by the suffixes, which is quite independent of any arrangement which may be visible in the system. Let Let (X{, Y it Zi) be the coordinates of the point Gi, which is the centre of mass of the partial system m 1} m 2 , ..., mi, so that /Z;_! Zi_!, & = X 1 . Let (xi, yi, z^ be the coordinates of mi relative to Gi- l} so that Thus (# 2 , 2/2. z -i) are the coordinates of ra 2 relative to m 1} or ( 2 ,, rj^^, 2 i)', (x 3 , 2/3, 23) are the coordinates of m 3 relative to G 2 , the centre of mass of m 1} m 2 ; and so on. There are no coordinates (x lt y lt z^). By the above O; - Mf- Hence on eliminating the product term and on addition of all the equations of this type xflm + fj, n X n 2 . = j= The relations between the coordinates have been- written down for one only. But they are linear and the same for all three coordinates separately. 167-169] Absolute Perturbations 185 Therefore they also apply to the velocities. Hence if T is the kinetic energy of the system, i=2 But (X n , Y n , Z n ) are the coordinates of the centre of mass of the system. They are absent from the potential function and are in fact ignorable coordi- nates. The known integrals for the centre of mass follow immediately and these coordinates can be suppressed. The problem of n bodies is thus reduced to a problem of n 1 fictitious bodies and the total order of the differential equations of motion is reduced by 6. 169. The new form of the areal integrals is easily found. For - (mZi - /H-i^i-i) (pi* 4 (pt - M,--,) 2 (M - gifr) = tf ( Yt - Yt-i and hence The sum of all equations of this type gives . . . 2 mi ((ifcfc - ft 17*) - Pi-ilH~\(giii - *&)} =(*n(Y n Z n - Z n Y n ). i=l But 'it is possible to write X n = Y n = Z n = ; that is equivalent to taking the centre of mass of the system as the origin of the coordinates (, 7j iy ft). Thus the areal integrals now take the form 2 niim^jii 1 zi&i XiZ?) = c 2 O j / \ where (c 1} c 2 , C 3 ) are the angular momenta of the system about fixed axes through the centre of mass. The direction of the axes has remained the same throughout. Let (c a , c 2 , c s ) be considered as the components of a constant vector C, mi fji-t fj,i~ l (&i, yi y Zi) as the components of a vector M it and (#,-, y it z { ) as the 186 Absolute Perturbations [CH. xv components of a vector r;. Then in quaternion notation the above three integrals may be represented by the single equation Hence in the problem of three bodies These three vectors are therefore coplanar. But V(r 2 M%) is normal to the plane of r 2 , M 2 , that is, to the instantaneous orbit of the fictitious planet 2. Similarly V(r s M 3 ) is normal to the instantaneous orbit of the fictitious planet 3, and clearly C is normal to the invariable plane. Hence the nodes of the instan- taneous orbits of the two fictitious planets on the invariable plane coincide. This important property explains the so-called elimination of the nodes, which in an explicit form is due to Jacobi. In the more common system of astronomical coordinates it disappears from view. The reader who is un- acquainted with the elements of quaternions will have no difficulty in finding an alternative form of proof, as in 22. 170. The body denoted by 1 will now be identified with the Sun, and i or j will have the values 2, . . . , n. The potential energy of the system, when the units are chosen so that the constant of gravitation is unity, is TT S m i m i _ V m i> m j ' A M / ' Aij where Also the kinetic energy, when the coordinates (X n , Y n , Z n ) are ignored, is T, where Let 1 pi' 1 ^,..., H=T+ U. Then the equations of motion of the system may be written ( 124) dxi _ dH dx^ _ dH ~dt~d^' ~dt'~ ~a^' to**)- Now (l*i - /Xi_i) & = piXi - /ii_! Z f _! = and therefore Sf+i ~ i = x i+\ Hence by the addition of such equations i69-i7i] Absolute Perturbations 187 which expresses the relative coordinates j , , . . . in terms of the coordinates %i,..., and shows that the latter differ from the former only by quantities of the first order in the small masses. In particular, for the body 2, which may be identified with any one of the planets, there is no difference. Let V be reduced to its terms /, of the lowest order in the small masses, which is the first. Then for Ti differs from A 1( i by a quantity which involves the masses. The equations of motion reduce to _ dHi = T jj = ' dt ~ dxi' ' dt or in more explicit form f J L i _ 1 H i - l x i = - m., Xi/ri 3 , (x,y, z). These are the equations of undisturbed elliptic motion, and in particular x 2 = - (m, + m 2 ) x 2 fr 2 3 , (x, y, z) which agree naturally with the usual equations of a planet relative to the Sun in undisturbed motion, and give a mean distance a 2 with the usual meaning. For the other bodies the equations are of the same form and have precisely similar solutions, but the elements o^ will' differ from the ordinary elements slightly because (set, yi, Zi) are not coordinates relative to the Sun unless i = 2. This is not material to the purpose in view because the body 2 represents any planet and any proposition which is proved for it must be true generally. 171. These equations for the undisturbed motion can now be solved in terms of canonical constants. When the latter are treated as variables, they satisfy canonical equations formed with R=U 1 U. As in 143 this value of R may be modified by adding 2 mfj?/2L' 2 , where m = m;/^//^ and p = mifii/ m-i in view of the explicit form of the undisturbed equations. Then any of the different sets of variables explained in that section can be used, and the last set, now denoted by (L 1 , /, /; X, ^ , ij 2 '), will be chosen. The equations for the perturbations can now be written dt d\i ' pi dt Pi dt df)i ' /if dt where V = - U + U, + m* There are n 1 pairs of equations in (Li, \i) and 2(n 1) pairs in (/, ?;/), but there is no need here to distinguish between the eccentric and oblique 188 Absolute Perturbations [CH. xv variables. From this point the former use of (f t -, iji, &) as the rectangular coordinates of m^ disappears. A little explanation may be necessary to account for the appearance of the mass factors of the momenta x{ in the equations. In 135 giving the Hamilton-Jacobi solution for undisturbed elliptic motion the single factor m, representing the mass of the moving body, was removed consistently from U, Tand H. Similarly in 139 UR was written in the place of U, R being the disturbing function in its common form, whereas the true increment in the potential energy is mR. But here it is not possible to divide the more general function U U^ as a whole by any particular mass, though it is possible to do so as regards the set of equations corresponding to a particular value of i. Hence it was necessary to restore the mass factors in the manner shown. But now they can be removed by the change of variables, N and the equations then become _ dt ~d\i' dt ~ 3L, d!j i= dV dm = _dV dt dr)i' dt dgi, where V= - U + U, + m, 2 2 mffi The terms added to U-^ U depend on the Li only, and affect one type of equation, namely so that \i = n t t + h and n t is the mean motion in the preliminary solution. The first-order perturbations of X t - will require the first-order perturbation of Li to be included in the term from which w t - originates. 172. It is not at present very necessary to consider in detail the form of expansion of U U^. It can in the first place be expanded in powers and products of the small masses m.{ and of the coordinates (#,-, y,-, 2,-). The latter can be expanded in powers of Li, ^, 77$ with purely periodic functions of X,:. Hence UU l can be expanded in the same form, and arranged in orders of the masses, beginning with the second since the first has been removed by U l Thus if the fourth order in V be neglected, V=V 2 +V 3 , where F 2 is of the second order and V s of the third, and V 2 contains at most two, V s at most three, mean longitudes X; in its arguments, the coefficients of the periodic terms being rational and integral functions of Li, ;, 77*. 171, 172] Absolute Perturbations 189 The perturbations of the first order can now be obtained in the usual way by neglecting F 3 and substituting initial values of L i} &, ?/; in F 2 , including i^t 4- \i for A;. This process gives Li = L { + ^Li, \ = mt + \i + 8, v, & = & + 5! &, 77* = < + ^ V where Zf, . . . are constants and S^Lf, . . . are the perturbations of the first order. Owing to the form of F 2 , 9F 2 /9Af is purely periodic and free from any term independent of \{. Hence ^X/ is also periodic and free from a secular term. But the other elements will contain a term multiplied by t, arising from the terms independent of \ { in the partial derivatives of F 2 , together with periodic terms. To the second order let ln-Lf+blf + SiLt. In F 3 , which must now be retained, it suffices to substitute the constant values Li ,... for Li,..., and r^-fA/ for A^; but in F 2 it is necessary to substitute L i + S 1 L i , ... for Li,..., though only the first powers of these perturbations are required. Hence the equation ^ (L? + B, L? + S 2 If) = ~ ( F 2 + F 3 ) gives, when account is taken of the solution for the first order, By the same argument as applied to F 2 in the first approximation the last term gives rise to periodic terms only. Hence a search for secular terms can be confined in the first place to the expression _I, dt+ i 2 ^ [*r* dt - 82 *JL f dLf a( + ax,9f/ J a^ a{ diidrjf J Here the multipliers of the integrals are all purely periodic, owing to differentiation with respect to A f . The integrals themselves contain secular terms in t. Hence on integration the products will give rise to periodic and mixed terms, but not to purely secular terms on this account. The latter must arise, if at all, from a constant term in the products. The only way in which this could happen would be connected with terms in the development of F 2 of the form F 2 = B sin (ki\{ + kj\j) + C cos (ki\i + kj\j) = B sin yfr + C cos i/r. But for these 2 ra_F 2 ,,_JH^ f f J ax,- d\id\j 1 + k^ = 0. 190 Absolute Perturbations [CH. xv In a similar way those terms which might produce constant terms neutralize one another between the other pairs of products and therefore no purely secular part of 8. 2 Li can arise in this way. But the above expression is not complete, because SjX,/ depends on SjZ/ as well as on F 2 . For, by the last equation of 171, dSjX,- 3F 2 37-2^3... dt dLf ^ * 7 o l j so that there is an additional part of 8 2 L i (l n t yet considered. It is given by where J. is a constant. But terms in F 2 of the above type, taken in the form D sin (i/r + h), lead to It D sin (* + A > (kjw D cos <* + A > d Z..J..2 - 2 Therefore this part of 8 2 Li is purely periodic. Hence there are no purely secular terms in 8 2 Li, a proposition which Poincare' has proved in the more general form : there are no purely secular perturbations of Li in any order of rank lower than 2. This applies in particular to L. 2 , But 2 = ML./, where M is a constant mass factor. Hence a 2 + &ia 2 + S. 2 a 2 = M (L z + ^L^ 4- 8 2 L 2 ) 2 B.a, = 2ML 2 8 1 L 2 , $,a. 2 = M {(S.L^ + 2L 2 (S 2 Z a )} the affix being now omitted. But 1 L 2 is purely periodic, and 8 2 Z/ 2 has no purely secular term. Hence to the second order in the masses there is no secular inequality in the mean distance, for it has been remarked that a 2 represents the mean distance of any of the planets. This is Poisson's theorem, an extension of Laplace's corresponding theorem for the first order, and it is the most important elementary result bearing on the stability of the solar system. 173. On the other hand there are evidently mixed terms of order 2 and rank 1 in Li. Hence the existence of purely secular terms of order 3 and rank 2 in a 2 can be anticipated. For even without pushing the approximation further and examining 8 S L 2 it is obvious that ^M^L^. 8 2 L 2 constitutes a part of 8 s a. 2 . Therefore the combination of a term A cosmt in 8 1 L 2 with a term Btcos mt in 8 2 L 2 will give a term MABt in 8 3 a 2 . Such terms were first shown to exist by Spiru-Haretu in 1876. 172, i7s] Absolute Perturbations 191 On one condition true secular inequalities of the first order occur in the mean distances. Since U /! = S A cos (ki\i + kj\j + h) to its lowest order, a V/d\i = 2 Aki sin (&A; + kfc + h). For perturbations of the first order the coefficients are constants and \ - n^t, \j Hjt are also constant. Hence dLi/dt = ^Aki sin (mt -f h'). A constant term results, producing a secular inequality, if m = &iW; + kjitij = 0, which is possible only if n if rij are commensurable. This possibility was considered in the previous form of discussion and excluded. But it is in effect ruled out by its own consequences. For if a body were artificially or fortui- tously projected in such a way as to have a mean motion commensurable (e.g. , i,...) with the mean motion of a disturbing body, its mean distance would be subject to a secular disturbance from the beginning, and therefore the commensurability of its motion would be definitely destroyed. Hence if the minor planets be arranged in order of distance from the Sun, it is to be expected that gaps will be found in the frequency at distances corresponding to mean motions commensurable with that of Jupiter, and it is so. And similarly divisions in the rings of Saturn can be attributed to the secular perturbations of the constituent meteoric bodies, produced by the commen- surable motions of any satellite which may be effective. This also has been verified for the action of Mimas by Lowell and Slipher. Nevertheless among the many minor planets a few are naturally found whose motions are nearly commensurable with Jupiter's mean motion. For these the long-period terms with small divisors are highly important, and the terms of low class play a far larger part than in the theories of the major planets. The special difficulties thus presented require special methods of treatment, and such have been suggested by JJansen, Gylden and others. Poincare has used an application of the principle of Delaunay's method. The proper treatment of this class of minor planets presents perhaps the most interesting problems to be found in dynamical Astronomy at the present time. CHAPTER XVI SECULAR PERTURBATIONS 174. In the preceding chapter it has been shown that the mean distances in the planetary system are free from purely secular inequalities when developed to the second order in the masses. The general nature of the secular perturbations in the other elements will now be examined. It may be convenient to modify slightly the equations obtained in 170, 171. By reducing U to its terms of the lowest order the equations of motion there took the explicit form p-i^^Xi = - m^rf, (x, y, z) which are satisfied by the osculating motion of a planet, according to its ordinary definition, when i = 2, but not otherwise. But if // be substituted for U lt where Ui = - 2 (wj + nn) mmi-i/ Wi a form which will be found to differ from U^ by terms of the third order only, the explicit equations of motion become Xi = - (m, + mi) Xi/n 3 , (as, y, 2) which are the ordinary equations in the undisturbed problem of two bodies, and are satisfied by the osculating elements taken in their usual sense. The mass factors of the momenta are as before m^i^//^, but the constants of attraction are p, = m l + m^ Hence the equations for the variations will now be based on V = - U+ US + 2 (n h = -U+U 1 ' + ^ (m, The relation between Z t and L{ is the same as before, but the meaning of both is changed (except when i = 2) in such a way that Li bears generally the same form of relation to di, the osculating mean distance in its ordinary sense, as L 2 to o^. 174, 175] Secular Perturbations 193 Thus the transformations of 143 give, with those of 171, Li = (m-i + miY at", G t = L{ cos fa, Hi = G{ cos i k = i nTi + nit, g i = iffi Q, i , hi = li p it 1 = 2Z// sin 2 %fa, p iy 2 = 2Z/ cos fa sin 2 tj \ = i + nit, &>i,i = ^i, <>t, 2= ^f ; = w; (w! + w;) V;_! pi" 1 a? , :> j = 2Z f * sin ^fa cos ta^, 77,;, j = - 2,; 2 ~ sin <; sin tsr ^ 2 = 2Z^ cos* L sin 2 fa & + 77 2 2 = 4 cos < sin 2 $ i tan r * i, tanft = it is easily verified that the five quantities can all be expanded in powers of 1, i?i, &, ?2. Also r cos w = a (cos E e), r sin w = a cos sin .fi* P. D. A. 13 194 Secular Perturbations [CH. xvi E being the eccentric anomaly, and therefore X/a = e cos M + cos 2 -< cos (E M) + \ sec 2 %(f) {e" cos 2M cos (E-M)- e* sin 2M sin (# - Jlf )} Yfa =es'mM+ cos 2 ^ sin (E - M) - i sec 2 ^ {e 2 cos 2Jlf sin (E - M) + e 2 sin 2ilf cos (# - M)} which are forms easily verified. Since cos 3 ^<, sec 2 ^ can be 'expanded in terms of e 2 = sin 2 , these forms show that X, Y can be expanded in powers of e sin M, e cos M if this is true of sin (E M), cos (E M). But Kepler's equation may be written 6 x cos 6 y cos 6 = 0, 6 = E M, x = esmM, y = e cos M and can be expanded in powers of x, y. Hence sin (E M), cos (E M) can be expanded in powers of e sin M, e cos M, and therefore also X and F. But this shows that X, T can be expanded in powers of e sin nr, e cos sr with coefficients involving periodic functions of X, since M = X -ST. And e sin OT, e cos w can be expanded in powers of 1? T/ I} as can easily be seen, with coefficients involving L. Hence (x, y, z) can be developed in powers of tri> %> 2, *?2 with coefficients involving L and periodic functions of X. There- fore finally U f/i' can be expanded in powers of i(1 , 17^, fi2 , 77^ 2 with coefficients involving Z; and periodic functions of \ if and the supplementary part of V involves LI only. It is assumed that the inclinations of the orbits are very small. Now there are two ways of regarding retrograde motion in an orbit whose plane differs little from the orbits of planets moving in the opposite sense. It is possible to take the mean motion n t as positive. Then the inclination is near TT and is not small. Or it is possible to take the inclination as small, and to regard n { as negative. Then since n^Lf is a positive mass function, Li is negative and therefore f , T/J are imaginary. All the orbits will therefore be supposed to be described in the same (direct) sense, which is true of the planetary system but not always of the satellite systems. This remark has an obvious bearing on theories of cosmogony. For if high inclinations and in particular retrograde motions were unstable, such forms of motion would not be permanently maintained. Now the nebular hypothesis of Laplace is very largely based on the observed fact that the planetary motions are nearly coplanar. If, however, such a type of motion is alone stable, the observed fact loses its significance in this connexion and no deduction of the kind is to be drawn from it. The question of stability in general, beyond the range of inclinations to be found in the actual planetary system, is therefore important, though beyond the range of this work. 175-177] Secular Perturbations 195 176. When the secular part [- U+ ' 'i, 2 which is free from \ is considered, certain properties of the development are easily seen. For this being independent of the direction of the chosen axes, the substitutions f , 2 > 'i, a are all possible without affecting the result. Thus (a) follows when l it -&i are altered by TT, or when the axes of xy are rotated through TT in their own plane. Similarly (6) follows when this rotation is made through ^TT. Again (c) is produced when fl f (but not -or;) is altered by TT, and this is equivalent to reversing the axis of z. Finally (d) is obtained by changing the signs of all the angles X;, fi t -, w t -, which is equivalent to reversing the axis of y. The change in X, : is of no further importance here since X; is absent from the terms now considered. Certain properties of the exponents in the expansion are now obvious. For 2 (p l + q l +p 2 + q 2 ) must be an even number to satisfy (a), and S (pz + Qz) to satisfy (c). Hence '2 t (p 1 + q l ) is also an even number. Similarly (d) requires S ((ft + #2) to be even, and therefore 2 ( p^ +p z ) must be even. Hence in the second degree there can be no terms of the form 77 or. 2, 17^2- But if terms of the fourth degree be neglected, only terms of the second degree involving , 77 remain. These terms can therefore be written down in the form [-U+ Uf] = %%A it j (&, ! & j + i?*, i "nj, i) + 2 i-Bi, j (&, 2 1>-, 2 + i\i, 2 *)j, 2 ) where the coefficients of &&, rjiVj are taken to be the same, both for the eccentric and the oblique variables, in accordance with the substitution (6), and terms i|y, 17^- are reckoned twice when i,j are different, but A i> j = Aj ii> Bi,j = B/,t- 177. It will be of interest to obtain the explicit values of A { j, S t j for the lowest order in the masses. The principal part of the disturbing function is SmiWjA" 1 and it has been seen in 159 that the complementary part contains periodic terms only. The distances A t - ( j involve coordinates (#;, y^ zj) which themselves contain the masses. But to the lowest order these coordi- nates are identical with the relative coordinates commonly in use, and the methods of Chap. XIV can therefore be employed. Two planets, 1, 2, will be first considered. Then in the notation of 152, when the orbits are circular, a 2 A- 1 = 6' = | b - %avb + . . . 132 Secular Perturbations [CH. xvi with the exclusion of all periodic terms. The triangle formed by the two orbits and the ecliptic gives cos J= cos i-i cos i 2 + sin t\ sin z' 2 cos (1^ fl a ) or to the second order in i 1} i 2 , v = sin 2 ^J= i [i\ + i-? ^i\i z cos (Oj fl 2 )}. Since v is of the second order the Laplace's coefficient bs 1 is derived imme- diately from the circular motion. But &i must be modified to include the eccentricities, the orbits being now treated as coplanar. Let A 2 = j 2 + a./ - 2^02 cos 6, 6 = t^ - nr 2 + M l - M z . Then in the notation of 157, A -i = - exp. \( Wl - M,} A + (w 2 - M 2 ) A} AT' \O%f V^a/ and, by (22) of 40 and (30) of 41, r/a = 1 + |e 2 - e cos M - ^e 2 cos 2M + ... w - M = 2e sin M + f e 2 sin 2M + .... Hence (a-ir)^ = 1 - e cos M . D + \& (1 - cos 2M) D + e 2 j(l + cos 2Jf) . D (D - 1) exp. (w - M} D = 1 + 2e sin M . D + f e 2 sin 2M . D + e 2 (1 - cos 2M) . D 2 . All operating terms which do not combine M l , M 2 in the form M : M 2 will clearly produce periodic terms only. And terms already of the second degree are combined with no others. Therefore, when ineffective terms are omitted, since D l + !Z) 2 = 1, A- 1 = (!-a - 3a KV - ) - ^ a (^t 1 - - i C^ 2 + e?} . ar 1 [a^ 1 + 3a [|6^ - (1 + a 2 ) &5 1 + i (^ + e) . ar l {- c^ 1 + fa 2 (6 f - 6 f 2 )j. To simplify this expression the recurrence formulae (4) and (5) of 148 with j = are available : (i-s+ 1) abl +l -i(l+ a 2 ) 6^ + (i + s - 1) ab^ 1 = (i+ S )bi= S (i+*)bi +1 Thus + f V) - t a V = f( and the last line of the expression disappears. Again = f {(1+ a 2 ) & f > l1 + rf\ ,) - i (\ 2 + 7ft- 2 )|* = const. - 2ft,, (Z, - $ (f V + ift,) - 1 (r, )2 + 7?\ 2 )}* = const. S \Li - % (f V + ift ,) - (p i>2 -f 77\ 2 )} = const. If the plane 01 reference is the invariable plane the first two of these con- stants are zero. In that case, when there are only two planets, f) 2 /J; 2 is the same for both and the nodes coincide, which is the property already noticed in 169 and referred to as the elimination of the nodes. These integrals, being satisfied identically, remain true when developed according to order and rank. Thus the third equation gives + ift! + p il2 + 0 is again of the order mi. The canonical equations give for the infinitesimal planet d& = A - = -A -2A As the new mass is regarded as infinitesimal, the motion of the finite planets will not be influenced, and the former solution fy = So,-, i Ci cos (ait + hi) tjj = ^a>j,iCi sin (a t t + h,) holds good. Hence -^ A 0>0 r) = ^Aoja^tCi sin (a^ + A f ) dfln -3; + A t = 2 A jdj iCi cos (dit + hi), dt ij 206 Secular Perturbations [CH. xvi These are the equations for a natural oscillation, together with a set of forced oscillations, and the solution is = cos 0,0* + - ojdj^i ,o "t" 1 cos 7o = <>sm 04 ,o* + ^o) + 2A 0t jOj t iCi (A 0i0 - a,-)- 1 sin where a fl , h are arbitrary constants. In general this solution shows that the eccentricity (and a similar form applies to the inclination) of the orbit of the infinitesimal mass will remain small. For , ?o contain m^ as a factor, and A 0> j(A 0t0 at;)" 1 is of the order of mfmt~~. An exception occurs when A 0>0 is nearly equal to o^, that is, when the period of the free oscillation nearly agrees with one of the forced periods imposed by the main planetary system. The corresponding amplitude then tends to become infinite. This condition is fulfilled at the mean distance from the Sun 1*95, or near the inner limit of the minor planets (Eros excepted), but for the inclinations only (A 0j0 fit). But before any positive conclusion can be drawn for this case, the extremely limited development of the disturbing function must be remembered*. * Cf. Charlier's Mechanik des Himmels, i. CHAPTER XVII SECULAR INEQUALITIES. METHOD OF GAUSS 184. A beautiful method of calculating the secular perturbations of the first order, due to the action of one planet on another, was proposed by Gauss in 1818. It was this method which was applied by Adams to the path of the Leonid meteors. Further developments have been given by several writers, and references will be found in an article* by H. v. Zeipel. The principle of the method is extremely simple. Equations for the variations of the elements have been found in a suitable form in 142. As an example we may take (/u. = n 2 a s ) di 1 r W cos u dt no? ' cos Here the right-hand side can be developed in terms of M, M', the mean anomalies of the disturbed and disturbing planets, in the form di | . A^ + 1 AJJ cos (jM+j'M 1 + q) and hence, the coefficients being constant in the first approximation, i - i, = A 0>0 t + 2A j>f sin (jM+-j'M' + q)/(jn +j'ri). If therefore the mean motions n, n' are incommensurable, so that (jn +j'ri) can never vanish, A Qi0 t constitutes the secular inequality in i. Now VfJi~\ 1 f 27r f 2ff di l*fr -L U/l 711 ,f- 7Ti,r/ -v- a = 4^ J dt mm |_CfcCJo, ^" JO JO *f* 1 f 27r F 1 f 2ir ~l rcos J - WdM'ldM ...(1) 2-Trna 2 cos .' \^TT J The component W contains as a factor & 2 m' = n 2 a 3 m'/(l + m). We therefore write +m with similar reduced mean values S , T corresponding to S, T. If then a series of values of S , T 0> W Q can be calculated for a number of points * Encyklopddie d. math. Wiss., vi. 2, p. 632. 208 Secular Inequalities [CH. xvii regularly distributed round the disturbed orbit, they can be introduced into the equations for the variations and a simple quadrature will give the secular perturbations of the several elements, that of a being zero. 185. In calculating S , T , W , the disturbed planet occupies a given fixed point P in its orbit. It is clear that S , T , W are components of the mean attraction, with respect to the time, exercised at P by a unit mass describing the disturbing orbit, with unit constant of gravitation. They are the same as would result if the disturbing orbit were permanently loaded so as to constitute a material ring of the same total mass, when the density is proportional to dM' Ids'. Thus it is necessary to calculate the attraction of an elliptic ring of this kind. Let any system of rectangular axes xyz be taken, with origin at P. Let (x , y , z ) be the coordinates of the Sun, (x', y, z') those of a point P' on the disturbing orbit, and let da be the area of an elementary focal sector, dV the volume of the tetrahedron on the base da' with its apex at P. Then 2p . da-' = 6d V = x 6 (y'dz - z'dy') + y (z'dx - x'dz) + z (x'dy - y'dx) where p is the perpendicular from P on the plane of da'. Hence one component of the required attraction at P is 1 Px 27rJo A 3 where a', b' are the semi-axes of the disturbing orbit and A 2 = x' z -f y' 2 + z'*. This takes account of the first (principal) part of the disturbing function only: the second (indirect) part has been left out of consideration because ( 159) it gives rise to no secular terms in the perturbations of the first order. It is now to be observed that x'&~ s dV is a homogeneous function of degree in x', y', z', and can therefore be expressed, since z'dy' y'dz' z' 2 d (y'/z f ), . . . , in terms of x'/z, y'/z', which are connected by some relation f(x'/z', 3/7*') = which is the equation of the cone having its apex at P and the attracting ring as its section. Thus the integral factor of P x (and similarly of P y , P z ) depends only on the form of the cone and not on the particular section. This is true whatever the shape of the ring may be. But in the present case the cone is of the second degree, and the axes may now be identified with its principal axes, P (X, Y, Z). Let PZ be the internal axis and a, ft the semi-axes of the section Z=l. The coordinates of P' can be written Y' = /3sinT, Z'=l where r is the eccentric angle in the section, and A 2 = 1 + a 2 cos 2 T + /3 2 sin 2 T, 6d V = (- ftX, cos T - a F sin r + aftZ,) dr. 184-186] Method of Gauss 209 Hence p = 1__ [ 2w cos T (- ffZ cos r - FQ sin T + afiZ,) dr Zjra'b'p J (i+ a * COS 2 T + S i n 2 T )f - 2/3A" f * cos 2 T i ~/o "A^ and similarly P ~ 2g ff F ( ^ sin2TC?T p = 2a^, r*" dr Tra'&'p J A 3 z ~Tra'b'p] Q A 5 ' These components can now be expressed in terms of the complete elliptic integrals HTvd-l-sin-T)' *- *1-*T>*. For, since sin T cos r cos 2 r sin 2 T + & 2 sin 4 T ** sin 2 rc?T ,! 1 ,, 1 "T" ln-^ Tn/-< *1\ -C'* Hence ( 2 -/3 2 ) V(l + a 2 ) na'b'p ' (a 2 /3 2 ) V(l + fl2 ) OX ~/3 P* = where the modulus k of E and JP is given by 2 _ /02 1 l 0-2 ^ 2 " 1 P p IV - _ . "~" A/ _ ~T 1 + a 2 ' 1 + a 2 186. It is now necessary to consider the geometry of the problem. Let the angular elements of the disturbed orbit be fl, *', <>, and of the disturbing orbit H', i', w'. These are referred to the ecliptic, which it is convenient to eliminate by referring the latter orbit directly to the former. With some change in the notation of 67 the equations there found give sin ^ (ft" + &)' - w") sin |i" = sin (11' - ft) sin \ (i f + i) cos | (ft" + a)' - w") sin t v/ = cos (ft' - ft) sin | (i v - i) sin (ft" - w' + &>") cos ^i" = sin (ft' - ft) cos (i' + i) COS ^ (ft" - 0)' + ft)") COS l'" = COS | (ft' - ft) COS | (t' - l). P. D. A. 14 210 Secular Inequalities [CH. xvn Here H" is the distance of the intersection of the two orbits from the ecliptic node of the disturbed orbit, i" is the mutual inclination of the two planes, and to" is the distance of the perihelion of the disturbing orbit from the intersection. Two sets of rectangular axes, with an arbitrary origin 0, are now to be defined. For (, 77, ) the directions are those of S, T, W, so that 0% is parallel to the radius vector at P, Orj is parallel to the plane of the disturbed orbit and 90 in advance of 0%, and 0% is in the direction of the N. pole of this orbit. For the second set, (x, y, z), Ox is directed towards the peri- helion of the disturbing planet, Oy is parallel to the plane of the disturbing orbit and 90 in advance of Ox, and Oz is directed towards the N. pole of this orbit. Let v be the true anomaly at P, and to + v fl" = v 1 the distance of P from the intersection of the orbits. Then the relations between the two systems of coordinates are given by the scheme : i} (T a; cos to" cos #!+ sin to " sin Vi cosi" cos to" sin Vj+ sin to" cos v l cos i" sino/'sint" y sin to" cos #! + cos to" sin #! cosi" sin to"sinv 1 + costo"cosy 1 cost" cos w" sin t' z sin?^ sin i" cosv^sini" cosi" Thus if r is the radius vector at P, and the origin be taken at the centre of the disturbing orbit, the coordinates of P are (x lt y 1} z^, where ^ = ae + r (cos to" cos v l + sin to" sin v l cos i") yi = r( sin to" cos v + cos to" sin v t cos i"), z l = r sin v 1 sin i" = p and a, e are the mean distance and eccentricity of the disturbing orbit. 187. Consider now the con focal system of quadrics of which the disturbing orbit is the focal ellipse &*&r lm The parameters \,, X 2 , X 3 of the three quadrics passing through the point ( x i> y\ > z i) are given by /. 2 . 2 2 ^ i 3fr if? _i a' 2 + X b'- + X X or as the roots of the cubic X 3 - X 2 (x? + y? + z? -a'*- b' 2 ) + X (tt' 2 &' 2 a' 2 ;z/i 2 fc^a?! 2 a'-Z] 2 6' 2 ^j 2 ) af*V*z* = (2) Now the axes of any tangent cone to a quadric are the normals to the three confocals which can be drawn through the vertex of the cone, and this remains true in the particular case where the focal ellipse is a section of 186, is?] Method of Gauss 211 the cone. Hence the relations between the sets of coordinates ( X, Y, Z) and (#, y, z] are given by the scheme : x y z X Y #,! (a' 2 + X^- 1 p,yi(b' 2 + \ 2 )~ 1 Z p 3 x, (a 2 + A3)- 1 p s ft (I)' 1 + \ 3 )~ l where p l; p 2 , p s are such that Pl * (*, (a 2 + X,)- 8 + y? (6' 2 When combined with the scheme given above for (x, y, z), (f, 77, ), this gives the relations between (X, Y, Z) and (, 77, ). The equation of the cone is (ex,. - xz$ (zy l - yz,)- '- j/2 '*$ for this is" clearly homogeneous and of the second degree in as a; l ,y y l , z Z-L , and its section by the plane z = is the disturbing orbit. Transposed to parallel axes through its vertex (x l} ft, z^) it becomes ____ ' 2 ' 2 * ' 2 /a ___ a' 2 6' 2 z* ' 2 6 /a / &'' ' ^ " a' 2 ' The justification for identifying these two forms is seen on comparing the three functions of the coefficients which remain invariant under a rotation of the axes. It will then be found that the results are equivalent to the relations between the coefficients and roots of (2). It is convenient to write down the equation of the reciprocal cone. The coefficients are the minors of the discriminant of the previous equation ^_! = 0. Hence with due care in choosing the right multiplier the desired equation may be written x 2 (acf a' 2 ) + y' 2 (y^ 6' 2 ) + z'*z? + 2yz y^ + i 2.zxz l x l + ^xyx^ = X a Z 2 + X 2 F 2 + X 3 2 = F l = the invariant relations being identical with those between the coefficients and roots of (2). Also & + * + # = X* + F 2 and it is evident that F_ lt F l can also be readily expressed, by means of the transformation scheme of 186, in terms of , rj, . 142 212 Secular Inequalities [CH. xvn 188. Two of the roots of the cubic (2) are negative and one positive, since two of the corresponding quadrics are hyperboloids and one an ellipsoid. Let Xj < X 2 < < A 3 . The axis of Z is then the internal axis of the cone F^ and it follows that X 3 ' X 3 ' 1 + a 2 X 3 X t The elliptic integrals F, E can therefore be found. The coordinates of the Sun relative to the point P are x = a'e' x 1 , y = y l} z = z 1 in the system (x, y, z) and (X , Y , Z ) can be deduced by the transformation scheme of 187. Hence PX, PY> PZ become known, and the components Pf = S 0> P n = T , Pf = W may be derived by applying the two transformations of 186 and 187. It is unnecessary here to consider the equations for all the inequalities. As a type, (1) now becomes 'di\ nam 1 /" 2ir Tir , ., r = - T 5 ^ cos u . W dM. \dt/ 0t o (1 -f m) cos 2?!- J Suppose that j values i/r g of ty = r cos u . W have been found, corresponding to j points around the disturbed orbit at which M has equidistant values, 0, 2-TT/j, . . . , 2 (j - 1 ) if I j- Then (Chapter XXIV) T|T = a + 2 di cos i M + S bi sin iM where a = - S^g, { = - S^g cos -. , b{ = - 2 ^ g sin ; . J * J s J J s J Hence nam' o (3) (-} = \dtJ 0>0 (1 + m) cos Jo A' 3 ira'b'p ~] ~A^~ ' where X 1 X 2 X 3 = 7<; :) . It is clearly possible to write consistently whence dr (e 3 e-,) (e 2 e 2 ) 2 sin T cos T -7- = )-^ as (e 2 61) (s e s ) 2 and -et) (-)(*-;) 214 Secular Inequalities [OH. xvn But this can be written A'" 1 dr = dv, $>(u) = s where $(u) is the Weierstrassian elliptic function formed with the roots BI, e z , e 3 . When r = 0, $(u) = e 2 , w = o> 2 ; when T=^TT, ^>(u) = e l , U = CD I . Hence d-r /" > M ~ e 3 d = ^-^-u,, * , _ ~ (u) + e 2 u "" _ 77 4- " Jo A" ].fe<*-4)(4:- Jo A /:1 = L, (e 2 - 6l ) (e, - ej du = |_(e 2 - e,) (e a - eO_L = 2 - *i) (e s - ej where The quantities &> and rj will now be found. 190. The reader who is unacquainted with the theory of elliptic functions will notice that nothing beyond the definitions of the functions jjp(w), (w) is here involved, and that these can be easily inferred. In fact, if the variable s be retained, it is easily seen that ds [ e * sds / [4> (s - e,) (s - e 2 ) (s - e s )} where 4 (s - e^ (s e 2 ) (s - e s ) = 4>s 3 - g 2 s - g a , e l #2 = ITT - 7, 3 = y< ^TT since - # 2 s - # 3 = (|0,)* (cos 30 - cos 87). Hence Yl ^-* <"*' shifldfl j/'*' sin20c J ^ V (cos 30 - cos 3 7 ) ' ^ ~ ( M J e , Vjcos 30 - cos 3 7 ) ' Now the Mehler-Dirichlet integral* gives P ( i \ - 1 ~7r where P n denotes Legendre's function of the first kind and order n. Let = 30 2-rr, and then /, &(+H*d0 -_ J^ 2 7T6 ^D - P n (COS 3 7 ) J e , V (cos 30 - cos 87) * Cf. Whittaker's Modern Analysis, p. 219; Whittaker and Watson, p. 308. 189-191] Method of Gauss 215 whence Now put n = \ and + in succession. Thus -i "~ - (C ^ / sm29d0 _i -77 ^rx-~ -fo-q = - 6 * 7T P t (COS 87). ./ 2 V (cos 80 cos 87) F v But the Legendre's functions can be expressed in the form of hypergeometric series* F(n,n + l,l, sin 2 7). Hence finally ,, I,sin"t7) where sin 2 f 7 = (1 <7 ). Thus o> and 77 are expressed in a form not requiring the solution of the cubic equation. These hypergeometric series are not the same as those originally found by H. Bruns as the solution of the problem. But the latter are easily deduced. For P n (z) satisfies the differential equation The result of changing the independent variable to x= 1 z 2 .is which is satisfied by the hypergeometric series F(^n, ^n + ^, 1, x). When z = cos 87, x = sin 2 87 = g~ l (g 1) and since there can be only one convergent series for y in powers of x, this is it. The above series may therefore be replaced by F ( T V, f 1, s in2 37), F(- iV &> 1, sin 2 87) which are the series obtained by Bruns. 191. Let the origin of coordinates now be taken at the Sun, the point P being at (X, Y, Z) or (- X , - Y , - Z ). Then the components P X) PY, PZ (4) can be derived by partial differentiation from the potential = (-k)*[* Tra'b'p Jo ** Z 2 cos 2 T + F 2 sin 2 r-Z z dr a'b'pJo A' 3 rra'b'p ' (e, - e 2 ) (e, - BI) (e 2 - ej Of. Whittaker's Modern Analysis, p. 214 ; Whittaker and Watson, p. 305. 216 where Secular Inequalities #1 = (e 3 - e 2 ) X 2 + (e, - e 3 ) F 2 + (e 2 - e,) Z* G 2 = e 1 (e s - e 2 ) X 2 + e 2 (e, - e 3 ) Y 2 + e 3 (e z - e,) Z 2 Now by ordinary multiplication of determinants and X 2 Y 2 Z 2 A.J A2 /Vg 111 Z 2 F 2 Z 2 "X 1 "X 1 "X 1 *M 'Vj ^3 1 1 1 where 111 L i Xa"" 1 A 3 1 \l Xjj X3 1 1 1 F Q 2V [CH. xvn and e 1} e 2 , e a are the roots in X' corresponding to X,, X 2 , X 3 . The first determinant is clearly Gj, and the determinant below it is The multiplying determinant in both identities is (XJX2X5)"" 1 (X 3 Xj) (X 3 X^) (X 2 Xj) = \KZ~ I (g w-i ai> 6 known and g n is required. As in 197 w 2 can be omitted by substituting w*k* for fc 2 . Then, by (SO), or 1 ' Sx,n + -{* hfqx n (5) 1Z 224 Special Perturbations [OH. xvm Here S Xjn comprises the terms which can be directly calculated, for 2. represents the direct terms, K~*$n follows from the previous stage of the quadrature, and K% n can be extrapolated easily owing to its small multiplier. Also x n =x +l; n is known well enough since it is multiplied by q. But q itself is not accurately known. By combining the three parallel equations of the same type as the last with the above equation for q, it follows that 1 \ / I \ 1 --/ 1 \ where S refers to the three coordinates. Thus, f being easily extrapolated, q can be calculated. The combination of (3) and (5) now gives Sx,n) whence n can be calculated, and therefore n by (4). Thus the quadrature, once started, proceeds step by step. In order to start the quadrature the four dates are taken such that the epoch of osculation coincides with the centre of the middle interval. With = the direct terms in ( are calculated and the difference table is formed. By applying (27) and (34) approximate values of are obtained whereby the indirect terms can be brought in. The process is then repeated until the final approximation is reached. The rest of the calculation, giving the results by means of (30), has already been explained. 199. Special perturbations may also be directly calculated for polar coordinates. Let the cylindrical coordinates of the disturbed mass m be (p, 6, z), the fundamental plane being the plane of the osculating orbit itself at the epoch t , and the initial line passing through the ecliptic node. The rectangular coordinates of the typical disturbing planet, of mass rrij, relative to the Sun are Xj = TJ cos BJ cos Lj , yj = TJ cos Bj sin Lj , Zj = TJ sin Bj. The kinetic energy of m is \m (p 2 + p 2 6 2 + z' 2 \ and therefore the equations of motion are, since r 2 = p 2 + z 2 , d 2 o /dO\ 2 dR /d6\dR d*z dR /\_ *z_ dt\ p dt)~d8' dt*~ where (23) ij { Aj -1 rf 3 [prj cos Bj cos (Lj 0) + zrj sin A/ = p z + z* + rf - 2 [prj cos Bj cos (Lj -0) + zrj sin Bj 198-200] Special Perturbations 225 Hence p - p fr = - k 2 (1 + w) pr~ 3 - k 2 2m, {p&f-* - ( A/- - r/- 3 ) r } cos B 5 cos (L 5 - 6)} d (p 2 6)1 dt = k 2 p 2m,- ( Af 8 - rf 3 ) r, cos , sin (L, - 6) z = -k 2 (l+m) zr~ 3 - k z 2m,- [z&f* - (A,-~ 8 - rf) r,- sin Bj}. Let where / is the same function of q as in (2) and can usually be replaced by 3 simply, because z is merely the perturbation in latitude reckoned from the osculating plane. The equations of motion can now be written : p-p6 2 + k 2 (1 4- m) p~ 2 = pH where H = $& (1 + m)f p ~ 5 z 2 + k 2 2m, {p- 1 (bf* - r/" 3 ) r, cos ^ cos (L t -6)- A,-" 3 } f/ = k 2 p 2m,- (Aj- 3 - r/-) r,- cos _,- sin (J^ - <9) F! = k 2 2m,- (Aj- 3 - r,- 3 ) r 5 sin 5,- + P 2 (1 + m)fp~ 5 z 3 W 2 = k 2 2m* A ; -- 3 + A; 2 (1 + m) /j" 3 . The third equation is now in the required form to determine z. The first two must be transformed in order to obtain p and 6. 200. The second equation gives Udt t where h is the undisturbed constant of areas, so that h - {k 2 (1 + ra) p }* = n a Q 2 cos po> n , a , sin< being the osculating parameter, mean motion, mean distance and eccentricity. Hence , r,r rt p -*dt + \p- 2 Udt dt J J t t L Jt rt Ud J = &> + V + Aft) where is the initial value of 6 and o is the distance of the undisturbed perihelion from the node. The angle A&>, which represents and is defined by the double integral, would vanish in the absence of disturbing forces. In the same circumstances V would be the undisturbed true anomaly. Thus V may be regarded as the disturbed true anomaly and Ao> as a rotation of the apse. In the rotating orbit thus defined, in which the elements p ,a ,e , < keep their osculating values, let p (1 + v)~ l be the radius vector corresponding to the true anomaly F, so that, since V hp~ 2 , 1 + e cos V = p (1 + v) p~ l - e sin V=h~ l p 2 p {- (1 + v) p~ 2 p + vp~ 1 } - e cos V- h~-p 2 p Q {- (1 + v) p + pi/}. P. D. A. 15 226 Special Perturbations [CH. xvm Hence - p) + h- 2 p p 3 v or p = h?p~ 3 + (1 4- v)~*pv - tf (1 + m) But 'o Therefore, by the first equation of motion in the form last found, pH=(l + i/T 1 P V + & 2 (1 + m) (1 + v)- l vp~* - p~ 3 f J which can be written in the form v + H 2 v = H 1 where From this equation, which is of the same form as that in z, v can be found by mechanical integration. Again, instead of finding F by a direct quadrature, the necessary correction N is found to the mean anomaly calculated with the undisturbed mean motion n , so as to reproduce the true anomaly V or the eccentric anomaly E in the rotating orbit. Thus E - e sin E = M + n (t -t ) + N a (1 - e cos E) = p(l + i/)- 1 . Hence, by (7) of 27, ft + HO = ( 1 - e cos E) E = pa ~ l (1 + v)~ l V . dE/d V p h 1 e cos E _ n ~ a (1 + v) ' p 2 ' cos < ~ (T+ i/) 3 and 201. The whole problem is therefore reduced to the mechanical solution of the equations d^v dN_ 2 + v dt*+ ~dt = ' W ^TTi 2 dt When v, N, A&>, z are known, the coordinates r, 6 and the latitude X are given by E e sin E = Jf + n (t t ) + N p sin V = (1 + v) a cos sin E, p cos V=(l+ v)a (cos E e ) SOD-SOS] Special Perturbations 227 Perturbations to the first order will be obtained by calculating the quanti- ties occurring in the differential equations according to the osculating elements, but as they become known in the course of the work their approximate effect on the coordinates of the disturbed planet can be introduced before integra- tion. The integral of U, and also N and A&>, can thus be found by direct quadrature by applying (27) and (28). For v and z, which require exactly similar treatment, the case is slightly different. As before, the time interval w is removed by writing w 2 k' 2 for k 2 , which is equivalent to making this interval the unit of time. Then at any stage n, when z n ^ and K~ l z n are known, y WW? &n " i " 2^n w* - w, (i + 1 IT,)" {(*- - 4 K ) 1, + 1 and this last equation will determine z n with the needful accuracy, and hence z n and K~*z n+l for the next stage. This method is due in principle to Hansen. The perturbations start from zero values and remain small for a considerable length of time. This conduces to accuracy and is an advantage. The method is less simple than that of rectangular coordinates, and for the easier construction of an ephemeris requires the determination of new osculating elements by a process which is itself complicated and is omitted here. Perturbations of the coordinates are recommended by the fact that there are three coordinates as against six elements to be determined by quadratures, and their computation is suitable for practical needs in the case of a body, such as a periodic comet, which can only be observed at relatively long intervals. Otherwise it is preferred to perform the calculation on the elements directly. 202. With slight changes which will be readily understood the equations found in 142 for the perturbations of the elements may be written : dijdt = rW cos u/k^/p dl/dt = rW sin ujk\/p sin i d/dt = [S sin v + T (cos v + cos E)} *Jpjk cos dvr/dt = { pS cos v + (p + r) T sin v + r W sin tan A i sin u] /k\/p sin dnjdt = 3 (rS sin < sin v + pT) cos /pr /t ft n -r.dt .! {dt = [, 1] F 1 + [, 2] F 2 , wdnfdt = [n, 1] F l + [n, 2] F t wdvr/dt = [w, 1] Ft + [or, 2] F 2 + [or, 3] ^3 where [i, 3] = r cos w, [12, 3] = r sin w/sin t [<, l]=p sin v sec (/>, [, 2] = ^ (cos v + cos .') sec [or, 1] = p cos v/sin <, [or, 2] = (p + r) sin v/sin 0, [or, 3] = r sin u tan t' [Jf, 1] = - {[or, 1] + 2?-} cos , [#, 2] = - [w, 2] cos [n, 1] = 3k sin < cos sin w/Vp, [w, 2] = 3k cos Vp/ r - For a minor planet disturbed by Jupiter, 40 days is generally found a suitable value for the interval w. The disturbing function R may be taken in the form found in 199 except that the argument of latitude is now u = v + or H instead of 6. Thus R = k~ ^mj (A," 1 rf 3 [prj cos Bj cos (Lj u) + zrj sin Bj~]} and if the directions of the components S, T, W be recalled, _ dR T _ 1 dR u/ . dR ~^r~ 5 * ~ ~^~ > " "^ op p du oz where after differentiation z 0, because the plane of reference is the plane of the instantaneous orbit. For the same reason p = r. Hence F l = p~ 2 (kwmj) {(Aj~ 3 - rf 3 ) TJ cos Bj cos (Lj u) rA/~ 3 } F z = p ~ * 2 (kwmj) ( Aj~ 3 rj- 3 ) 7*j cos 5j sin (Lj u) - ?)- 3 ) ?) sin Bj and Aj 2 = r 2 + ry 2 2rrj cos Bj cos (Z>j u). 203. Let (,, 6j be the heliocentric longitude and latitude of the disturbing planet, which with log r,- are given in annual tables like the Nautical Almanac. The relations between ecliptic coordinates (x, y, z) and the orbital coordinates (> ^ )> the axis of f passing through the ecliptic node, are shown by x y z cos fl sin n 17 cos i sin U cos i cos 1 sin i sin i sin U sin i cos n cos i 202-204] Special Perturbations 229 which is the scheme derived in 65. Hence = cos Bj cos LJ = cos bj cos (^- H) T) = cos J3j sin LJ = cos fy cos i sin (^ fl) + sin bj sin i = cos Bj = cos bj sin i sin (^- O) + sin bj cos t and thus F 1} F 2 , F s can be calculated, so far as the coordinates of any disturb- ing planet are concerned. But F 1} F z , F s and the coefficients [i, 3], ..., involve also the varying elements and coordinates which depend on them. The elements may be identified with the osculating elements at the initial epoch and the co- ordinates may be calculated as in undisturbed motion. Then the result of mechanical integration will give the perturbations of the first order. When these are known for the several dates covered by the work, the calculation can be repeated with the improved values and a higher approximation can be obtained. The work can be arranged so as to obviate this repetition by including the perturbations to date at each step. 204. The five elements i, fl, $, OT, n require only a single quadrature. The lower limit a + \w is made to coincide with the epoch of osculation and the tables are formed in accordance with (27). The corresponding perturba- tions are then given by (28) or (26) according as a + nw or a + (n + ^) w is preferred for the final date. It is to be noticed that the differential equations for the elements have been reduced to a form in which w occurs explicitly as a coefficient of the derivatives on the left-hand side. It will disappear when the quadratures are effected, its function being to make the unit of time agree with the tabular interval. But the unit of time is not really changed, and with the ordinary Gaussian constant k occurring in the combination kwnij for each disturbing planet remains one mean solar day. Thus the perturbation in n which will be drawn by this process will be the increment in the mean daily motion. Since all the elements are in the form of angles, it is con- venient to express k, so far as it occurs in F l} F z , F 3 through the combination kwnij, by its value in arc (log k" = 3'55...). But in [w, 1], [n, 2] k has its purely numerical value (log k = 8*235. . .). The perturbation in M can be conveniently divided into two parts. The first, 8,M - w~ l | {[M, 1] F, + [M, 2] F a } dt is calculated in precisely the same way as the other five elements. The second is B 2 M= The table having been prepared for the first quadrature on the basis of (27) and (28), the second can be performed by means of (34) and (30). The 230 Special Perturbations [CH. xvin immediate result will give w~"8 2 M, which must therefore be multiplied by w 2 . To avoid this large multiplier it is usual to calculate w&n from w z dn/dt at the first quadrature (giving the increment in the w-day mean motion). This alters the time unit of the acceleration and therefore no multiplier will be required by 8 2 M, a result which can be otherwise seen by noticing that all the tabular entries are multiplied by w, while the integrand is divided by w, being in fact dn/dt instead of w. dn/dt as in the first quadrature actually performed on this plan. 205. In the case of parabolic and nearly parabolic orbits some modifica- tion is necessary. The equations for i, H and ts remain valid, except that it is well to replace by e. The equation for e itself becomes wde/dt = [e, 1] F 1 + [e, 2] F 2 e \r a But the equations for n and M become inconvenient, if not illusory. One suitable substitute is easily obtained by forming the equation for q, the perihelion distance. Since q = a (1 e), do da de 2aw ... . dn de w --^ = (^ i e) w -j aw -j- = ; (1 ? * fi.9n at dt dt 3?i where I V > -^' I """" ^\ \ -* ~~ ^ / I *^) A I ~~" 'v I t. A I = r sin cos ^> (1 e) sin v ap sin w wp 5 = 2a 2 e (1 e) sin v a 2 (1 e 2 ) sin = a 2 (1 - e) 2 sin v = ?-{- ) nr r e \r a) ^ ap* pr v) i er e pr ao 2 fl 2 pr p 3 . _ _ I I __ -t ( I J g) 2 e r|_e 1+eJ e r .e ~~^'4 sin^-y (1 + e 204-206] Special Perturbations 231 Thus a valid form for the perturbation of q is obtained. If F 1} F 2 have been calculated with the angular value of the constant k the results for Se and Sq will require to be multiplied by sin I". Again, an equation can be formed for the variation of T, the time of perihelion passage. Since n(t-T) = M=e-vr+ I ndt dn dT d , dM [dn (t - T) -j- - n-j- = -r (e - or) = -=- - I -j- dt ' dt dt dt^ dt j dt it follows that l(t ~ T} {[n ' 1] Fl + [1I> 2] Fz} ~ n ~ l {[M ' l]Fl + [M ' 2] = [T,l]F l + [T,2']F a where [T, 1] = w- 1 (t - T) [n, 1] - n~ l [M, 1] v\ / -e 2 )sin v(t - T) (1 - e ( ^ p cos -f- ^ n 2 (1 - erf ( p 3ke . , 4 ^J v ^ cos v -- 1 sm v(t- T)> n ( 2e 2 p* J and [T, 2] = ?i-' (t - T) [n, 2] - n~ l [M, 2] (1 - erf (p + r) sin v nr ne (I -erf (I, 8p* But these coefficients are in a form absolutely unsuitable for calculation, especially in the case of a parabola, for which in fact they are required. The difficulty can be, and is best, met for such orbits by calculating special perturbations in rectangular or polar coordinates, instead of directly in the elements. 206. The reduction of [T, 1], [T, 2] to a calculable form is not altogether easy. It can be effected in the following way. The required expressions can be written, since n 2 a 3 = k 2 , p = a (1 e 2 ), 2p*r f- cos v (1 + e cos v) 3k(t-T) } [r,l] jryf- iHl - S -esmn k (1 - e 2 ) 1 2e Z ^r ) Special Perturbations [CH. xvm Now 3 k(t-T)=a?(E-esmE) = -- - {(E - sin E) + (I - e) sm E\ (1 - e 2 )* * E-sinE 2 f = ' tan * v + cos2 * E tan iw (1 + )} tan where - "itan-i^cotf^" - But ( 27) r cos 2 i w = a (1 - e) cos*$E=p(I + e)~ ] cos 2 and therefore Let [T 7 , 1], [T, 2] be written in the form r^ n_ " 3 rm o-i 2p T r (sin v (2 + e cos v) ,,} L-t,/J = , , i\ i~~ ' n "~" * r where Fi = esinv.F, F 2 = (l -f ecos?;) F and therefore Fj cos ^v - F 2 sin $v = (1 e) sin %v . Y Hence F, cos $v - F 2 sin $v = - $ sin $v sin v - - j(r ^ . 2 tan 2 ^t;) -f 3} F, sin * + F 2 cos ^ = - i cos \v sin v -. . ^? VI +e tan 2 ^ J '/l + | cos ^w sin v {'2 (1 + e)- 1 tan 2 \v + 3}. The expressions involving 8 are finite and they are multiplied by l-e, which is a necessary factor. For the other terms, let y, cos v - y z sin v = - 1 sin A y sin v . ^^ I +e y l sn v + y.> cos $v = f cos ^v sin v + (1 + e)- 1 cos |v sin v tan 2 w. 206] Special Perturbations 233 Then yi = % (1 + e)- 1 sin 2 v (3e + tan 2 $v) = % (1 + e)- 1 (1 - cos v) [3e (1 + cos v) + (1 - cos v)} = ^ . r- sin v . g cos G where g sin G ! $., g cos G S 2 tan 4 At; -.2tan 2 iv, ^- -.-- -r^ sin v 1 + e cos ^v 1 + e Hence a cosw which are fairly simple forms, but still require the calculation of g sin G, g cos 6r. In the limiting case of the parabola, S = ^E Z and g sin G = tan 2 ^v sin ^v, g cos (r = tan 4 |v cos |t; which then completes the solution. The more general case of a very eccentric ellipse can be related to the method of 34. In the notation of that section, A A=ri - : FT, T = tan ,- 9E + smE I - A + C Hence 15^ n 1-A + C s=^~ -** ^-j^ + cyi c\ 1-%A' tan 2 A^ l-^ \5 A) 234 Special Perturbations [CH. xvm Now by the method of 34 A (of the order E' 2 ) is found in calculating v, and C (of the order E*) is tabulated with argument A. With the same argument it is possible to tabulate* log and log 77, where Then 2 tan 2 Aw sin A v ~ 2 tan 4 A - and the problem is thus solved in a practical way. Similar treatment can be applied to hyperbolic orbits. 207. It sometimes happens that a comet approaches a planet (generally Jupiter) so closely that the disturbing force due to the planet is actually greater than the force due to the solar attraction. It is then more convenient to refer the motion to the centre of the planet and to treat the solar action as the disturbing force. In the ordinary case the equations of motion of the comet are of the form d*x , , , . as ln (of x x'\ -rr=- k 2 M - + k 2 m - 1 --- I dt 2 r 3 V A 3 p s / where M is the mass of the Sun, m the mass of the planet, and the origin is at the centre of the Sun. If S, P, C are the positions of Sun, planet and comet, CS = r, CP = A, SP = p. The equations involve no assumption as to the relative masses of the Sun and planet, and if they are interchanged the equations of motion of the comet take the form where the origin is at the centre of the planet, so that #=#' + ,...,#' + '=(), The advantage of either form depends on the ratio of the total disturbing force to the corresponding central attraction, and it will rest with the latter if * that is, if fjL = m/M, when Let GPS = 0. Then r cos CSP = p - A cos 6 r 2 = p 2 - 2|oA cos + A 2 . Now in the nature of the case A is small compared with p. Hence r~ 4 = p-* + 4/>~ 5 A cos + 2/>- 6 A 2 (-1 + 6 cos 2 0)+ ... r~ 3 = p~ s + 3/o~ 4 A cos + f /3~ 5 A 2 (1 + 5 cos 2 0) + . . . * Bauschinger's Tafeln, Nos. xxvn, xxvin. 206, 20?] Special Perturbations 235 and therefore r -4 + p -4 _ 2 r - 3 /3 - 2 (p - A cos 0) = p~ 6 A 2 (1 + 3 cos 2 6)+ .... To gain an idea of the planet's sphere of influence the approximation need not go further. On the other side of the inequality the first term preponderates and it can be further simplified by taking r p. Thus the significant terms of the lowest order in A give the inequality p- 6 A 6 (l + 3 cos 2 0) < /*yA- 4 and the polar equation, with coordinates (A, 6) and origin at the centre of the planet, A (1 + 3 cos 2 0) = p*p represents a meridian of the bounding surface, which is one of revolution and differs little from a sphere. Its radius for Jupiter, Saturn and Uranus is about a third, and for Neptune rather more than half, of an astronomical unit. When the comet enters this sphere of influence its relative coordinates (^ - ar/, y l - yj, z l - z^} or (&, tj 1} ) and its relative velocity (&, 7)1, ) are known and its orbit about the planet can be found, with the constant of attraction & 2 m. It remains within the sphere so short a time that the solar perturbation can generally be neglected, and on emergence a return is made to the heliocentric orbit, based on the new position (|f 2 + #./, 772 + y%, > + ^a') or (a? 2 , 7/2, 2 2 ) and the velocity (x 2 , y 2 , 4)- CHAPTER XIX THE RESTRICTED PROBLEM OF THREE BODIES 208. The general problem of three bodies is reduced to a relatively simple and ideal form when two of the masses describe circles in one plane about their common centre of gravity and the third body has a mass so small as not to affect this circular motion in any appreciable degree. Let OXYZ be a set of rectangular axes rotating with angular velocity n about OZ, OX following OF, and let the coordinates of the masses //,, v be ( c l , 0, 0), (c 2 , 0, 0) where pc^ = i/c 2 . The velocity components in space of a small body at P (%, 77, ) are (% nv), rj + n%, ) and hence the kinetic energy of unit mass is The equations of relative motion are therefore where in this case dV dr) 8F a? p l} p 2 being the distances of P from p, v. The result of adding these equations, multiplied respectively by , 17, , gives Jacobi's integral of energy v 2 = p + # + 2 = 2 V + n 2 ( 2 + vf) - C and in accordance with Kepler's law & 2 (p + v) = n* (d + c 2 ) 3 . 209. This integral has a very simple and important practical application. Let us return to fixed axes through //,, so that + Cj = x cos I + y sin I, r) = y cos I a; sin I, = z where I is the longitude of v and / = n. Then 2 + if = (x + ny)* + (y- nx)* %- + rf = x* + y n - 2c a (x cos 1 4- y sin 1} + d 2 . 208-210] The Restricted Problem of Three Bodies 237 Hence Jacobi's integral becomes 2 4. if. _j_ ^2 + 2n (yx ay) = 2 V - 2w 2 Cj (x cosl + y sin 1) + ri*c? C. The special circumstances under which this integral can be usefully employed are these. A periodic comet between two appearances in the neighbourhood of the Sun may pass in close proximity to a large planet, Jupiter for example. In that event the elements may be so altered that at the second return the identity of the comet is doubtful. At times when the perturbations are small and the heliocentric motion is sensibly elliptic, xy yx = *p>p cos the latter being the projection of the areal velocity on the plane of the disturbing planet. Hence - k' 2 /j,/a 2kn V(/^>) cos i = 2k 2 v/p 2 2w a d (x cosl + y sin 1) + n?c? C. It is supposed that the change in the observed osculating elements takes place almost impulsively within the region of the planet's influence. This region is small and nearly spherical. Hence /o 2 is the same at the beginning and end of the encounter, and the changes in x, y and I are small. These can be neglected together with the other planetary perturbations, and therefore approximately /*/a' + '2k~ l n J(np') cos i' = pfa" + 2k- 1 n \'(pp") cos i" where a, a" are the mean distances of the comet, p' y p" the parameters, and i', i" the inclinations of the orbit to the orbit of the disturbing planet, before and after the encounter. For the Sun //.= 1 and A; 2 (I +i/) = w 2 a 8 , where a is the mean distance of the planet, and if v be neglected a'- 1 + 2a ~ * />'* cos i' = a"~ l + 2a ~ * p"** cos i" which is the criterion of identity proposed by Tisserand. It has been assumed that the orbit of the disturbing planet is circular, but some allowance can be made for the eccentricity of the orbit by taking into account the actual motion of the planet at the time of the suspected encounter. 210. Let the problem of 208 be now reduced to two dimensions (=0). Then HP? 4- vpf = H ( + c,) 2 + /in* + v (| - c 2 ) 2 + vrf = (H + v) (f 2 + r)*) + i*c* + vc. 2 *. Let the units be so chosen that k=l and d + c 2 = 1, with the consequence that fju + v = n\ The equations of relative motion may now be written 811 238 The Restricted Problem of Three Bodies [CH. xix where 2H = p. (Zpr 1 + />!) + v (2p*- 1 -f p 2 a ) and the integral of relative energy is v 2 = 2ft - C. These are the equations used by Sir G. H. Darwin, with the masses /* = 10, v = 1, in his researches on periodic orbits. Now it is obvious that v z cannot become negative under any circumstances. Hence the curves of the family given in bipolar coordinates by the equation 2n = (7 are of great importance in the restricted problem of three bodies, because they represent barrier curves which cannot be crossed by trajectories characterized by corresponding values of C. Thus if the barrier curve, or curve of zero velocity, is a simple loop within which a part of the trajectory lies, then the trajectory can never pass outside. If the lunar theory can be compared with this simpler problem it is found that the orbit of the Moon lies within such a closed curve surrounding the Earth, and therefore the Moon cannot recede beyond a certain limiting distance from the Earth. This remark is due to Hill. The simplest view of the general character of the curves of zero velocity is gained by considering them as the contour lines of the surface If the axis of z is taken vertically upwards, and motion for a given value of C is supposed to take place on the actual contour plane z = C, then it is evidently restricted to those parts of the plane which lie underneath the surface, since elsewhere in the plane the velocity becomes imaginary. Now the main features of the surface are easily represented topographically. At the points where the masses /z, v are situated the surface rises to infinity, but in the neighbourhood of these singular points may be treated as two peaks. At any considerable distance from them the terms pp? + vpf are predominant, and the surface rises indefinitely in all directions. Now 2ft may be expressed in the form 20 = 3 O + v) + p ( Pl - I) 2 (1 + Zpr 1 ) + v (p 2 - I) 2 (1 + 2/ir 1 ) and clearly has an absolute minimum value 3 (//, + v) when p l = p 2 = 1, i.e. at the vertices of the equilateral triangle on the line joining the masses //,, v. These points represent the bottom of two valleys, and a simple consideration of the continuity of the surface shows that these valleys must be connected by three passes, one between the two masses and the others on the same line but on opposite sides of the two masses and separating them from the rising surface as it recedes in the distance. If it be added that the highest pass is 210, 2ii] The Restricted Problem of Three Bodies 239 that which lies between the masses and the lowest is on the other side of the greater mass, the general order of development of the contour lines should be sufficiently evident. The critical curves for Darwin's special case, /A = 10, v= 1, are illustrated in fig. 7. The whole is symmetrical about the line SJ. Fig. 7. 211. The points at which the ovals coalesce or disappear evidently correspond to critical values of fl. Take v < /*. The critical values are given by 9fi 911 opi 911 9/3 2 ... 9|r 9/Oj 9^ 9p 2 9g 911 9Ii 9/Oi 9O 9^2 /\ dtj dpi drj 9p 2 877 which show immediately that such points are points of relative equilibrium for the third body. These equations are satisfied in the first place by or p 1 = p 2 = 1. This gives the " equilateral " points mentioned above, where is an absolute minimum. But other solutions are given by = or T? = 0, together with 3fl/3| = 0. This will lead to the three points collinear with the masses. For the first, lying between the masses, so that ^ 240 The Restricted Problem of Three Bodies [CH. xix This is a quintic in p.,, with only one real root. The actual solution in a particular case is easily found by trial and error from the first expression. The second expression, when expanded, gives and to the same order C = p(3+ 3p + 2/> 2 3 ) + v (20T 1 + P2 2 ) = fju (3 + 3a 2 ) + i/cr 1 (2 + fa) = fjt (3 + 9a 2 + 2a 3 ). For the second collinear point, on the further side of the smaller mass i/, _ 1 , 9 Pl _ 8 />2 _ , 1 PI- +P*> af"af- and hence /A p^-p* (l-pj)(l+ pz )* again a quintic in p z with only one real root. For the approximate solution and to the same order (7 = ^(3 + 3p a a - 2p 2 *) + v (2/Dr 1 + p 2 2 ) = /n (3 + So 2 ) + i/a- 1 (2 - I a) = i, (3 + 9a 2 - 2a 3 ). For the third collinear point, on the further side of the larger mass /*,, dp l dp 2 A-l+ft, g| = ^ = and therefore v _ _ pT^^Pi _ (2 + o-) 2 (3(5 + fjo) is a point of relative equilibrium, or as it has been called a point of libration, and as, y are very small, the linear equations x y + 2nd; = l u x + l Q2 y are obtained, and these determine the nature of the equilibrium at ( Vo)- For they are satisfied by the solution x = h cos (mt a), y = k cos (mt /3) provided - 2mnk sin $ = (m 2 + O^) h cos a + Ml n cos /3 2wnA; cos /3 = (m- + n 20 ) /?, sin a + kfl u sin y8 2wm/i sin a = hl u cos a + (m 2 + n o2 ) ^ cos /S 2mnh cos a = hl u sin a + (m 2 + n^) k sin y8. These equations, which result from equating coefficients of cos mt, sin mt, are equivalent to (m 2 + Hgo) A sin ( #)= 2rarc& Ml u sin (a /3) = - 2mn^ cos (a /8) (?n 2 + n o2 ) k sin (a /3) = 2mwA. AH n sin (a - /3) = 2wmA cos ( /3). There are only three independent equations here, and this should be so because the only quantities which can be determined are the ratio of P. D. A. 16 242 The Restricted Problem of Three Bodies [CH.-XIX amplitudes hjk, the difference of phases a /3, and in. The three equations may be written A 2 (m 2 + Ho,,) = k* (m 2 + n o2 ) n n tan (a. /3) = 2mw (m 2 + fla,) (m 2 + n w ) = 4m a a + fl 2 n and these determine a series of infinitesimal elliptic orbits about a point of libration when m has a real value. With certain simple developments such a series can be traced into a family of finite periodic orbits. 213. The third equation, that is the quadratic in m 2 , w 4 - m 2 (4n a - fla, - HOB) + ^20 ^02 - fl a n = decides the question of stability and may be examined more closely. If the roots in m 2 are complex or negative, real exponential functions of the time enter into the disturbed motion and equilibrium is unstable. If the roots are real, but of opposite sign, an unstable mode of motion is associated with a possible elliptic mode and equilibrium is again unstable. Here the point is surrounded by an unstable family of orbits initially elliptic. This is illustrated by the collinear points of libration. For it is easily found that when 77 = n n = o, n^ = ^(2pr 8 + 1) + v (2/or* + 1) so that ftao is positive. Now at the point of libration between the masses 9pi 9p 2 _ -, an _ an P\ + PZ ^> "at + x > ~ 2 f $ Pl P2 and therefore, since rj = 0, i an i an /i i\ / i\ = + I . M f i -- : 1 2 v/oj /> 2 / pi 2 / which is negative since pi < 1. Similarly n o2 is negative at the other collinear points of libration. Hence at these three points the absolute term of the quadratic in m 2 is negative and the roots are real and of opposite sign. Each of the points is therefore surrounded by a family of unstable periodic orbits. It has been suggested by Gylden and by Moulton that the phenomenon known as the Gegenschein is due to sunlight reflected by meteors which, in spite of the instability, are temporarily retained in the neighbourhood of that centre of libration in the Sun-Earth system which is opposite to the Sun and at a distance of about 938,000 miles from the Earth. When both values of m 2 are positive the disturbed motion is the resultant of two elliptic motions, and equilibrium is stable. This may be illustrated by the " equilateral " centres of libration. At one of these an = an = _5 2 n_ = dp! dpz dprfpz api = _ap2 = i jjgi d and 162 244 The Restricted Problem of Three Bodies [OH. xix Also the equation of relative energy gives, when the constant C remains unaltered, dv . 8H vdv _ an ds ds ' dp dp ' Let the undisturbed orbit at P be defined by the quantities s and , and the corresponding point P' on the neighbouring orbit by bs along the undisturbed orbit and 8p normal to it. Then (v + 8v) 2 = (v + \ or to the first order d8s . _ _an 8p an 8s dt dp ' v ds ' v Hence d /Bs\ jr = v ^(- dt dt\v ' Again, let (u, u) be the components of velocity in space of P in directions coinciding with 8s, 8p. Since these lines are rotating with the absolute velocity (0 + n) the kinetic energy of unit mass at P' is dSp . ) 2 - Hence Lagrange's equation for Sp is Now this equation must be satisfied when 8p = 8s = 0, and when the terms which do not vanish have been removed, it becomes Also it must be satisfied when Sp = 0, 8s = v8t, where 8t is constant, for this also represents a point moving on the unvaried orbit. Thus 9 2 F - 2 (6 + n) v a well-known result due to Hill. Again, Lagrange's equation for Bs is which must be satisfied when Bp = Bs = and also when Bp = 0, Bs = vBt. Hence, after removing the terms which are independent of Bp and Bs and then those which contain Bp, fcv 3 2 F This result may be used to give @ another form, namely v dt 2 "W where V 2 = 3 2 /dp 2 + 9 2 /9s 2 = 9 2 /9 2 + 9 2 /d?? 2 . This form may be more convenient than Hill's because V 2 (not to be confounded with the three-dimensional V 2 ) does not depend on any particular direction. For some purposes it is necessary to take the arc s instead of t as the independent variable. Then (3) becomes d ( dSp\ VT- (V-T*-) da \ ds J or again, if Bp = v ~ * Bq, where 215. When the unvaried orbit is periodic, is a periodic function of t with the same period T. The equation (3) is therefore a particular case of a linear differential equation with periodic coefficients. Its general theory may be indicated. Since the equation is unaltered when t is replaced by 1 4- T, g(t + T) is a solution if g (t) is one. But every solution is a linear combination 24(3 The Restricted Problem of Three Bodies [CH. xix of any two others which are independent. Hence if g represents g (t) and G represents g(t+T\ g lt g 2 being any two solutions, #1 = agi + @g 2 , G 2 = 70i + 02 where a, ft, y, 8 are constants, not unrelated. For since g lf g 2 are two solutions of (3) 2 0i=0>0 2 and therefore 2 0i - 0i0 2 = const. = G 2 G,- G& = (020i - 0i 0a) (& - 7). Hence 8 $7 = 1. Let/,,/ 2 be two other independent solutions. Then 0i = a/i +6/J, 2 = c/i + d/ 2 G 1 = a^ + bF 2 , G 2 = c and the result of eliminating g 1} g 2 , G l} G 2 is where (ad be) A = ada + cd/3 aby bc8 (ad - be) B = bd (a -8) + d*/3 - 6 2 7 (ad - be) C = - ac (a -8)- c 2 /3 -f a?y (ad bc)D = bca cdft + aby + adS. Hence A +D = a + 8 is a constant independent of the choice of particular solutions, as well as AD BG = a.8 - fiy = 1. But it is now possible to choose b/d and a/c so that B=C=0. Then F^Afi, F 2 = Df. 2 , AD = l and the functions /-,, / 2 are defined by the property that they are multiplied by constants when the argument is increased by the period T. Hence the general solution of the differential equation may be written 8p = a^e* <, (t) + a 2 e~ kt < 2 (t) where 0,, 2 are periodic functions with the same period as and cosh kT = ^ (a. + 8), a constant which can be derived from any pair of inde- pendent solutions. The quantities + k are what Poincare has called characteristic exponents. If & is a pure imaginary circular functions are involved and 8p has no tendency to increase beyond a certain limit. The periodic orbit is then stable. If on the contrary k is real or complex real exponential functions are involved and Sp will increase indefinitely. The orbit is then unstable. The question of stability therefore involves essentially the determination of k. But this is a matter of great difficulty in general. What is known as Mathieu's equation, generally written in the form - + (a + 16? cos 2z) y = 215, 2ie] The Restricted Problem of Three Bodies 247 of which the solutions are elliptic cylinder functions, is only a particular case of the general type (3) and it is the subject of an extensive literature. On the astronomical side the reader may consult Poincare"'s Methodes Nouvelles, Tome II. See also Whittaker and Watson, Modern Analysis, Ch. xix. 216. The original equations of motion, an , an can also be given a canonical form. Let and then evidently dH dH are equivalent to the above, and they are of the required form. The integral of energy is H = 0. Now consider the integral /= [ (- H + Pl % + p,-r)) dt. Jt Between fixed limits its variation will vanish along a trajectory in virtue of the canonical equations. Therefore it is a minimum (or at least stationary) along a trajectory as compared with its value along any neighbouring path. Let the time along any such path be determined by the equation of energy H=Q. Then the integral becomes = f { Jo from which form, since v 2 = 2O C, the time is absent. Now v^d^ + v^ ds ds 8 !vds= r J J f 1 = l Jo ~ ^r ds) V ds dij I 1 248 The Restricted Problem of Three Bodies [CH. xix and 8 f l n (dr) - r) df ) = n P Jo Jo o Therefore, if 8% = ST; = at the limits, ,-o i \ dsj \ ds, Let the tangent to the orbit make the angle with the axis of , and let < be the normal distance to an outer neighbouring curve, so that d = ds . cos <, dr) = ds . sin , Sf = Sp . sin <, Brj = Sp . cos 0. Then SJ = I [Svds sin d (v cos ^>) 8p + cos d(v sin 0) Sp + 2nSpds} Jo ...(5) q where Jr dv -- op ds J2 being the radius of curvature. Along an orbit K = therefore, and this is a result already expressed in (1). It is further to be noticed that dK = i a z n /idfi _i\dv _ v dR dp ~ v dp 2 U 2 dp R) dp ~ R 2 dp v (dp* v dp ~RJ v dp when K = 0, and since v = 7i(/> comparison with (3) shows that dK @ = -v^. 8/> It follows that the action J round a closed orbit is greater than for any adjacent parallel curve when (H) is positive at every point. In this case the periodic orbit is in general stable. Similarly the action J is a real minimum when is negative at every point. Then, as (3) shows, the periodic orbit is obviously unstable. 217. This remark is due to Prof. Whittaker, who has given another application of equation (5). The quantity K can be calculated for all points on a given curve. Now let K be negative everywhere along a simple closed 216, 2i7] The Restricted Problem of Three Bodies 249 curve A. Then by (5) the value of J will be diminished when taken round another curve adjacent to and surrounding A. Again, let the quantity K be positive everywhere along another simple closed curve B external to A. The value of J will also be diminished when taken round a curve adjacent to and surrounded by B. Now consider the aggregate of all the simple closed curves which can be drawn in the ring-shaped space bounded by A and B. There must exist, if the space contains no singularity of Ii, one of these curves which will give a smaller value of / than any other, and it cannot coincide with A or B for any part of its length. It represents therefore a periodic orbit characterized by the constant of energy C, and thus the existence of such an orbit is established when the two curves A and B can be found which satisfy the conditions stated. The orbit is necessarily unstable. The same author has given another elegant theorem. By Green's theorem (log V ) ddr) = (Jog V)dr)- (log V where the first integral is taken over the area of a closed curve, and the second over its boundary. But if the curve is a trajectory, K = and therefore 3 . d 2n = ^ (log v) ^ + 5- (log v ) a ' + j + ~ 9 op dt) op ds v 3 drj 3 . .dd(f>2n = ^ (log v } -T- - 5~ 0g V ) J + j + d ds di ds ds v Hence = - l This assumes that the enclosed area contains no singularity of the integrand. But this function becomes infinite at the centres of attraction. Surround the mass p at ( c, , 0) with a small circle K^ of radius p. Then since the integral round the circumference becomes a <. a\ , r i / 7 a ,.3 = 7T. 250 The Restricted Problem of Three Bodies [OH. xix Similarly the corresponding integral round a small circle K surrounding the mass v tends to the same limit. Now if the outer boundary contains either of the attracting masses or both, the boundary integral must be diminished by subtracting the integrals taken round ^ or /c 2 as the case may be. Hence the final result is 8^ i JVd 2 c where j= 0, 1 or 2 according as the loop of the orbit contains neither or one or both of the attracting masses, j is the total angle through which the tangent to the orbit turns, and T is the time from one end of the loop to the other. In the case of a periodic orbit in the form of a single closed curve 7= 27T. 218. The equations of relative motion are capable of a transformation which is very useful in some cases. This may be deduced from the intro- duction of conjugate functions in a general form. Let the original equations be dV 77 + 2w n 2 rj = Of) or in the Lagrangian form _d f_ dt Vay " af d (dT\_ + v) where b is the distance of the middle point between the masses from their centre of gravity. For pi* = ( b + c) 2 + rf = 4c 2 cos 2 -| (u + iv) cos 2 \ (u iv) PZ = ( b c) 2 + if 4c 2 sin 2 ^ (u + iv) sin 2 ^(u iv) and hence , r U, V U> V PI PZ c (cosh v + cos u) c (cosh v cos u) ' Also /=//// = c 2 sin (u + iv) sin (u iv) = | c 2 (cosh 2v cos 2a) and 2 + if =/i/a = b 2 + 26c cosh v cos u + ^c 2 (cosh 2w + cos 2w). Hence H' = /*c (cosh v cos u) + vc (cosh v + cos w) + %n 2 bc 3 (cosh 3-y cos u cosh v cos 3u) + y^w 2 c 4 (cosh 4v cos 4u) ^c 2 (h w 2 6 2 ) (cosh 2w cos 2u) and the equations of motion are , dv 9H' -^ + nc* (cosh 2v - cos 2u) ^ = -^- . The time is given by a final integration $ = c 2 1 (cosh 2v - cos 2w) rfT= IfrpidT. These equations are in general very complicated, although they offer essential advantages in studying the motion in the immediate vicinity of 218, 219] The Restricted Problem of Three Bodies 253 one of the masses. Two particular cases may be noticed. In the first the masses are equal, /JL = v and 6 = 0. The equations of motion then become d?u dv -T - nc- (cosh 2v cos 2u) - = . , Here the variables u, v are separated and the equations lead immediately to a solution in elliptic functions. The comparison of this problem with the simplest case of the problem of three bodies is instructive as to the difficulty of the latter. LUNAR THEORY I 220. The theory of the Moon's motion relative to the Earth has been discussed with generally increasing elaboration and completeness by various authors from the time of Newton to the present day. The methods which have been employed also differ considerably, presenting peculiar advantages in different respects, so that it cannot be said definitely that any one method possesses an exclusive claim to consideration. But at the present time three modes of treatment are certainly of outstanding importance, those adopted by Hansen, Delaunay and G. W. Hill respectively. Hansen's theory was reduced to the form of tables by the author ; these tables were published in 1857 and are still in common use, but will shortly be superseded. Delaunay 's work took the form of an entirely algebraic development of the Moon's motion as conditioned by the Earth and Sun alone. His theory has been completed by others and made the basis of tables recently published. Hill's researches, which bear a certain relation to Euler's memoir of 1772, deal only with particular parts of the theory, but the whole work on these lines has now been carried out systematically and completely by E. W. Brown and will form the foundation of a new set of lunar tables now in course of preparation. Here it is only possible to attempt a slight sketch of one method. For this purpose Hill's theory will be chosen, partly because it is destined to receive extensive practical application, and partly because it contains original features of the greatest theoretical interest. The reader who wishes to gain a comparative view of the different methods which have been used in the lunar theory will study Brown's Lunar Theory and may also be referred to the third volume of Tisserand's Mecanique Celeste. 221. Let the mass of the Earth be E, of the Moon M and of the Sun m , the unit being such that the gravitational constant G = 1. Let the origin of rectangular axes be E, (x, y, z) the coordinates of M and (x , y', z'} the co- ordinates of m. Further, let r be the distance EM, r' the distance Em, and A the distance Mm'. Then ( 23) the forces on the Moon per unit mass relative to E can be derived from the force function r, M m m , . F= - + - - - (xx' + yy' + zz'} 220, 221] Lunar Theory I 255 by differentiation with respect to x, y, z ; and similarly the forces on the Sun per unit mass relative to E can be derived from the function E+m M M. , F = -, h -i 7 (xx +yy + zz) r A r 3 by differentiation with respect to x', y', z'. Hence the ^-component of the Sun's acceleration relative to G, the centre of gravity of E and M, is r) W M rl T? 1'' T' V V, u r J.u oju , -r-, /, i// i g i// *** H r ** I Lj ' I A ' \ /l/f ^_ lift rrr I JTJ ^- ft I, t jifj. r ^^ ML fjy rlj \ /L/ H np / f\ 7^ ?L/ ( rv* rr* f flf* + ., ,.; \(E + M} -. - +m'- -+mf E + M \ r* A 3 r- E+M+m (x_ ^x'-x /Q ' A This expression will be derived by differentiating the function M+m' (E M\ Vr / + Ay with respect to a;', or with respect to x l , where (x lt y 1( z^) are the new co- ordinates of m' when parallel axes are taken through G instead of E. Let r l be the distance m'G, 6^ the angle m'GM and 8= cos 0,. Then r'- l = \r^ + -^^ r rr l S + and ( M r M 2 r 2 ) = rr i 1 _ j^n r - P! + 7Jj=^jy -* A - - [ 2# # 2 A 1 ) 2 _ .Cf i L ^ ~t'l T^. >r' / l^-' i^ / .? . i E r E 2 r 2 1 -L P 4- P 4- f ^T^n ' (^ + !/) 2 r 1 2jr2+ -". where P 1} P 2 , ... are Legendre's polynomials Hence, when expanded in terms of r/r 1} E + M+m' L EM Now the Moon's parallax is of the order 1 , the solar parallax is of the order 9" and the ratio M/E is of the order 1/80. It follows that the second term in F^ is of the order 10~ 7 as compared with the first. It can be neglected, at least in the first instance. F^ is therefore reduced simply to the first term, and the meaning of this is that the motion of G about m', or of m' about G, is the same as if the masses E arid M were united at their centre of gravity. 256 Lunar Theory I [OH. xx This motion is elliptic and the coordinates (x 1} y 1} z} can be treated as known functions of the time according to undisturbed elliptic motion. The influence of the other planets is left out of account in the first instance and finally introduced in the form of small corrections. The first task, and the only one considered here, is to find an appropriate solution of the problem of three bodies, the problem being already so far simplified that the relative motion of the Sun and the centre of gravity of the Earth-Moon system is supposed known. 222. The force function F is expressed in terms of (x, y', z') and not the coordinates (x l} y ly z-^) now supposed known. It is necessary to consider the effect of this. The ^-component of the Moon's acceleration is dF ... x ,x x , x =- = - (E + M) - - in - m das r* A 3 r 3 x mf E \ m! ( M ) ~~~ X * fr* ' snce x' = ^ + Mxj(E + M\ x-x' = -x l + ExK E + M). This component is clearly derivable from the force function E + M m'(E + M) m' (E + M) 1- ~V~ ~^A~ Mr' when r and A are expressed in terms of (x lt y 1} z-^) instead of (x ', y', z'}. When A" 1 , r'~ l are expanded in terms of rjr^ this becomes ^E+M m' \(E+My> r^ E 2 - M 2 r 3 E 3 + M 3 r 4 r f r, | EM + r,- + (E + Mf ^ 3 + (E + M) 3 ^ 4 E+M m'r* ( E-M r_ E* ~~ '~ * 3+ * for the term in 1/rj does not contain {x, y, z) and can therefore be suppressed. As a matter of fact the force function which is commonly used for the motion of the Moon is neither F 1 nor the function _ E + M m? _ m'r fi r A r' 2 where is the angle m'EM, but the function _ E+M mf m'r ~ which is derived from F by substituting the coordinates of the Sun relative to G for the coordinates relative to E. Thus 221-223] Lunar Theory I 257 and therefore in the expanded form after suppressing m jr l . This is not the same as F lt but- for practical purposes it can be brought into agreement by a simple device. Let a, a' be the mean values of r, ?v It is found that to a term of the series involving (r/r^ correspond inequalities with the factor (a/ay. If then (E - M) a/(E + M) a' be substituted for a/a' in the results which follow from the use of F 2 , they will be very nearly the same as if they had been derived by using F l . It may be left to the reader to examine the order of the chief outstanding dis- crepancy after this treatment of F^ It is easy to make the adjustment exact. 223. Let the axis Ez be taken normal to the ecliptic and let EX, EY rotate in the ecliptic plane of (xy) with the Sun's mean motion n'. The equations of motion of the Moon are then Now if E + M = /A, since n /s a' 8 = m' (more strictly ra' + '- - (f r*S* * A ^6 the higher terms containing r/rj and therefore the solar parallax as a factor. Let v' be the true longitude of the Sun and let v' = e when t 0. Then the Sun's coordinates are X' = T! cos (v' n't - e'), Y' = r a sin (V w' e'), / = the axis of X being always directed towards the Sun's mean place. When the solar eccentricity is neglected and the Sun's orbit treated as circular, v' n't + e' and TJ = a, so that X' = r, = a\ Y' = z' = 0, rS = (XX' + YY')/^ = X. Hence when the solar parallax and eccentricity are both neglected F 2 = pr- 1 + ri* (f X* - r 2 ) = yu?- 1 + n' 2 (X 2 - $ 7 2 - \z*) P. D. A. 17 258 Lunar Theory I [CH. xx and when, still further, the latitude of the Moon is ignored, the equations of motion become simply X - Zn'Y- 3?i /2 X = - These two-dimensional equations represent the simplest problem bearing any real resemblance to the actual circumstances of the lunar theory. It is the degenerate case of the restricted problem of three bodies when the two finite masses are relatively at a very great distance apart and refers strictly to the motion of a satellite in the immediate neighbourhood of its primary. These equations have great importance in Hill's theory. Again, when the solar parallax alone is neglected, F 2 may be written in the form where the third term, which vanishes with the solar eccentricity, is a quadratic function in X, Y, z. Thus * where A', H', B', G' are functions of t to be derived from the elliptic motion of the Sun. The equations of motion now become *r fy I \T Q / T7" j A f ~\T - TTf T7 17 i O^.' V i ZT' V i Tf' V i + Zn A + li A -f Li I z + n' 2 z +C'z and these are the foundation of the researches of Adams into the principal part of the motion of the lunar node. 224. It is now necessary to give Hill's transformation of the general equations of motion. Let n a m = , , K = - - , v = n - n . n n (n n )- Then, since r 2 = us + z 2 , n being undefined as yet, a' 3 2m 2 (P 2 r 2 + P 3 r 3 /r 1 + ...) M 3 X ^ 3/1* / where H 2 ', Q 3 , ... are homogeneous functions in u, s, z of degree 2, 3, ... and of degree 0, - 1, ... in a'. Let 1' = fl a ' + O 3 + . . . . The kinetic energy of the Moon T is given by = (u + ii'iu) (s n'is) + z 2 . 223-225] Lunar Theory I 259 The equations of motion are therefore O ET ii + 2niii n' 2 u = 2 2 ds s 2n'is n' 2 s = 2 ~ du Let where t , like n, is a constant at present undefined. The previous equations become 3D' D*u + 2mDu + mht, = K u/^ - ds D 2 s - 2mDs + irfs = KS /r s - du --. " dz It is, however, convenient to separate from fl/ (accented for this reason) the part which is independent of the solar eccentricity. This is IV - I2 a = m 2 (3Z 2 - r 2 ) = | in 2 (a + s) 2 - m 2 (us + 2 ). With this change the equations of motion take the form Dhi +2mDu + f m 3 (u + s)- = - ^ r 3 ds IPs -2mDs + 4m 2 ( u + s)-- =- r 3 " dz where ft = fl a + H 8 + . . . . Thus (2) -m^-l ............ (3) ; \i / which vanishes with the solar eccentricity. 225. The next object is to transform the equations in u and s so as to remove the terms involving r~ s . Since ( 123) and ^ 2 contains terms involving explicitly only in H, in this case ws + z* - n'*us = 2,P 2 -^- ot 172 260 Lunar Theory I [CH. xx or in the later notation Du . Ds + (Dzf + | m 2 (u + s) 2 - mW + ^ = - ft + -D r where C is a constant of integration, D~ l is the inverse operator to D, and D t represents the operator D applying to ft only in so far as ft contains t explicitly. This corresponds to the equation of energy. Again, since r 2 = us + z-, the equations of motion (2) give sD 2 u + uD 2 s + 2zD 2 z + 2m (sDu - uDs) + f m 2 (u + s) 2 - 2m 2 z 2 - 2 _/ 8_ft 8_ft 8_ft; \ ds du dz by Euler's theorem, ftp being a homogeneous function of degree p in u, s, z. The result of adding the last two equations is D 2 (us + z 2 ) -Du.Ds- (Dz) 2 + 2m (sDu - uDs) + f m 2 (u + s) 2 - 3m 2 ^ .(4) This is one equation of the required form. The other equations are obtained simply by eliminating the terms with r~ 3 as a factor between different pairs of the equations of motion. Thus from the first pair D (uDs-sDu - 2mus) + f m 2 (u 2 - s- 2 ) = s 8 - - u ^ ...(5) ds du and when the third equation is used, 8f 1 r)fi D (uDz zDu) 2mzDu - im 2 ^ (ou + 3s) = z lu -= ds dz D (sDz - zDs) + 2mzDs - \\tfz (3w + 5s) = z' d ~ - Is ^ Oil OZ These combined give D {(u s) Dz -zD(u s)} - 2mzD (u + s) - m 2 zW an 8n\ ,8n *- 3- - H M s ) o- 8s du/ dz where with the upper sign W = 4<(a + s) and with the lower W = u s. In this more symmetrical form the real and imaginary parts of u and s are clearly separated. Equations in the form of (4) and (5) have two advantages. In the first place the left-hand members are homogeneous in u, s, z of the second degree. Except for the constant C this applies also to the right-hand members when the parallax of the Sun is neglected, and the parallactic terms need rarely be taken beyond the third and fourth degrees. In the second place, whereas X and F can be expressed as trigonometrical series in terms of t, u and s can be expressed as algebraic (Laurent) series in terms of and such series 225, 226] Lunar Theory I 261 can be more easily manipulated. Also if u =/() s =/(-) and therefore when either u or s has been calculated the other can be derived immediately. 226. The general method of the lunar theory, which is common to all forms, consists in choosing an intermediate orbit which bears some re- semblance to the actual path of the Moon and in studying the variations which it must undergo in order that the path may be represented accurately and permanently. This intermediate orbit, since it merely serves as a subject for amendment, will naturally be chosen with a view to simplicity. At the same time, the more closely it represents the permanent features of the actual motion, the less burden will be thrown on the subsequent variations. Thus one might take the osculating elliptic orbit of the Moon about the Earth as the intermediary, neglecting the effect of the Sun altogether. The intermediate orbit adopted by Hill is called the variational curve and this must now be defined. When the solar eccentricity (} = a 2 a* f* -oo = I ft (4*H - ^WO * +1 + 1 (^+1 + ^i WO f-*- 1 ! = a I a-^ 2i+1 -so where -"-21+1 = 3- (&2i ~t~ & 2i a)> " at+i = a ( a 2i ~~ a -2i 2)- As it is necessary to multiply such series together and to exhibit the products as double summations, it is convenient to write or similar equivalent forms, so as to retain always a fixed coefficient a 2i and a fixed power go in the typical constituent. The result of substituting the series in (6) is : ar*C = 22 47*0* a^ j+zi & - 22 (2i + 1) (2j - 2t - 1) a* + 2m + f m 2 2 2 a^- (2a-$ j = 2 2 2j (2j - 4t - 2) a 2l - a_ 2j+2i ^ - 2m 22 2j a2i a 1 3 + f m 2 2 2 a* (a^-af^ - a_ 2? -_ 2i _ 2 ) f 2 -' j where * and j have all positive and negative integral values. The coefficients of every power of f must vanish identically, and therefore a~ 2 C = 2 {(2t + I) 2 + 4m (2i + 1) + f m 2 } a 2 ^ + f m 2 2^ a- 2t - 2 . . .(8) * i when J = 0, and = 2 {4J* + (2i + 1) (2i + 1 - 2J) + 4m (2t + 1 - j) + $ m 2 } + f m 2 2 Oaf (a stf _ 8i _ a + a_ 2j _ 2( -_ 2 ) i = - 2 4j (2i + l-j + m) a 2i a_, ? - +2( ; + fm a 2 a 2l - ( 2 j- 2 i-2 * i whenj/ has any other value. 227-229] Lunar Theory I 263 228. Owing to the introduction of a, one coefficient may be made equal to 1, though retained for the sake of symmetry. Then, if m is a small quantity of the first order, a p is found to be of order \p , being a function of m alone. This fact makes it possible to obtain the coefficients by a process of continued approximation, provided m is sufficiently small. The terms containing a a 2 j, a a_ 2 j in the last equations are obtained when i =j and i = 0, and they are respectively |4j2 + 2 j + l + 4m ( j + 1) + |m 2 } a 2j + {4j 2 - 2j + I - 4m ( j - 1) + f m 2 } a a_ 2j and - 4j (1 + j + m) a a 2j - 4j (1 - j + m) a a_ 2j .............. .(9) Let the two equations be combined so as to eliminate the second of these terms. The result may be written : 2 a 2i |[2j, 2t] a_y +2i + [2j, +] a^_ 2l -_ 2 + [2j, -] a_ 2j -_ 2 i- 2 } = ...(10) i where F9 ' 9 H - _ i fy' 2 -2-4m + m 3 + 4 (t - j) Q'-l-m) j ' '~~8f- 2 - 4m + m 2 3m 2 4j 2 -8j-2-4m(j + 2)-9m 2 W* 4 " T6f ' " 8f - 2 - 4m + m 2 3m 2 20j - 16j + 2 - 4m (5j - 2) + 9m 2 16j 2 ' 8j 2 - 2 - 4m + m 2 the common divisor being chosen so that the coefficient of a a 2 j, [2J, 2j], is - 1, while [2j, 0] = 0. If, on the other hand, the term in a a 2 j be eliminated, the result will be found to be f-a = which can be deduced from the same series of equations (10) by changing the sign of j and then writing i j for i in the first term. This single series is therefore sufficient. The last equation can also be written 2 {[- 2 J> ~ 2 *1 a 2j-2itt-2i + [- 2j, -] a 2j _ 2i _ 2 a 2i + [- 2j, +] a_ 2; -_ 2i _ 2 a 2i } = i and hence the rule for connecting the pair of equations corresponding to j: in terms multiplied by [2j, 2i] change the signs of j and i throughout (both in coefficients and in suffixes) ; in the other terms write [ 2J, ] for [2j, +] and [ 2j, +] for [2j, ], the suffixes being unchanged. 229. Since the coefficients [2j, +] are of the second order in m, the orders of the three terms are respectively 2 i +2 % - j , 2 i +2 i+l-jl + 2, 2 | i \ + 2 i + l+j +2 which are at least 264 Lunar Theory I [CH. xx Let the equations be written down so as to include all quantities of the sixth order (neglecting m 8 ). This requires j = 1, 2, 3. The orders of the terms with the only possible values of i are : j = 1, i = 2 (6, 10, 14), 1 (2, 6, 10), (2, 2, 6), - 1 (6, 6, 6), - 2 (10, 10, 6) j = 2, t = 2 (4, 8, 16), 1 (4, 4, 12), (4, 4, 8) j = 3, i = 3 (6, 10, 22), 2 (6, 6, 18), 1 (6, 6, 14), (6, 6, 10). Hence the required equations are : a 02 = [2, 4] a 2 a 4 + [2, - 2] a_ 2 a_ 4 + [2, +] (2a 2 a_ 2 -f a 2 ) + [2, -] (2a a_ 4 + a 2 _ 2 ) a a_ 2 = [- 2, - 4] a_ 2 a_ 4 + [- 2, 2] 2 a 4 + [- 2, -] (2a 2 a_ 2 + a 2 ) + [-2, +](2a a_ 4 + a 2 _ 2 ) a a 4 = [4, 2] a 2 a_ 2 + [4, +] 2a 2 a a_ 4 = [- 4, - 2] a 2 a_ 2 + [- 4, -] 2a a 2 a 6 = [6, 4] a_ 2 a 4 + [6, 2] a 2 a_ 4 + [6, +] (2a a 4 + a*, 2 ) a a_ 6 = [- 6, - 4] a 2 a_ 4 + [- 6, - 2] a_ 2 a 4 + [- 6, -] (2a a 4 + a 2 2 ). Thus, since a = 1, if m 6 be neglected, a 2 =[2, +], a_ 2 = [-2, -] and then, neglecting m 8 , 4 = [4, 2] [2, +] [- 2, -] + 2 [4, +] [2, +] a_ 4 = [- 4, - 2] [2, +] [- 2, -] + 2 [- 4, -] [2, +]. These values will give a 6 , a_ 6 as far as m 9 , and inserted on the right-hand side of the first pair of equations they give second approximations to 2 , a_ 2 of the same order. It is to be noticed that each stage of further develop- ment carries an equation four orders higher. The ratio of the mean motions of the Sun and Moon, and therefore the numerical value of m, is known with great accuracy from observation. Hill adopted the value m = n '/(n _ w ') = 0'08084 89338 08312. Hence it is practicable to introduce the numerical value of m from the beginning, and the approximation to great accuracy in the calculation of a* 2 , ... is then extremely rapid by the above method. This is the process which has been adopted in the latest form of lunar theory. It is also possible by giving m other values to trace the development of the whole family of periodic orbits of lunar type. These orbits are of great theoretical interest, especially for larger values of m. But it is evident that the effect of the neglected parallactic terms will become more considerable, and such results may differ sensibly from true solutions of the restricted problem of three bodies. Also when m exceeds ^ the question of convergence begins to in- troduce practical difficulties and the method of quadratures, followed by Sir G. H. Darwin and others, becomes necessary. 229-231] Lunar Theory I 265 230. To find the value of a recourse must be had to an equation of motion which has not been reduced to a homogeneous form in u, s. Since fl = z = and r 2 = us, the first of (2) becomes in the present case (D 2 + 2mD + f m 2 ) u + f m"s = KU (us) ~ or a 2 {(2t + I) 2 + 2m (2t + 1) + f m 2 } a^ i+l + f m 2 a 2 o^- 21 '- 1 = w (us) ~ \ This equation must hold for all values of , including f = l. Then w=s= and therefore a 2 {(2i + 1 + m) 2 + 2m 2 } a 2i = /ca~ 2 (S a^)~\ But ( 224) * = /* (w - rc')~ 2 = ft (1 + m) 2 rr\ so that w 2 a a = fi (1 + m) 2 (S a*)-* [S !(2i + 1 + m) 2 + 2m 2 } a*]-' ...... (11) It has been usual to write w 2 a s = /A, a being the mean distance which would correspond to the mean motion n in the absence of solar or other perturba- tions. Thus a = a (1 + powers of m) when the values of a^ are inserted. The precise form of this relation is required only when it is desired to compare two theories expressed in terms of a and a respectively. The con- stant a fixes the scale of the orbit and therefore depends on the parallax, which is observed directly. When the coefficients a^, and a have been determined, (8) gives the value of C, if it be required. For the transformation to polar coordinates, r cos (v nt e) = r cos (v n't e ) = X cos + F sin = \ (u%~ 1 + s%) r sin (v nt - e) = r sin (v n't e ) = Y cos % Xsmi- = % (s u^~ l ) i where e = e' (to n') t 0> since = (n ri) (t t ) and ig = log Hence r cos (v nt e) = a {1 + (a 2 + a_ 2 ) cos 2f + (a 4 + a_ 4 ) cos 4^ + . . .H l**\ / r sin (v nt e) = a { ( 2 a_ 2 ) sin 2 + ( 4 a_ 4 ) sin 4 + ... N which lead to the determination of r and v, the more simply because v nt e is evidently of the second order in m. 231. The use of rectangular coordinates is a distinctive feature of Hill's method. But for some purposes polar coordinates present advantages. By a simple change of units and notation (1) become ? + 2^= -^ d 2 d* r 3 which can be reduced to canonical form by putting (cf. 216) 266 Lunar Theory I [OH. xx The transformation to new variables, r, I ; r', I', defined by p r cos I, p' = r cos I r~ l I' sin I q = r sin I, q' = r' sin I + r" 1 1' cos I will leave the canonical form unchanged, since p'dp + q'dq - (r'dr + I'dl) = and therefore it is an extended point transformation ( 125). Let t be eliminated from the equations by taking I as the independent variable. After writing out the equations in explicit form make the transformation r = l/cr, r' = p/o; T = w/a 2 and finally put e = - 1) ~ = w 2 - p 2 + f cos 2Z + i - e di (to I)-- = 2pco I sin 21 ctl and the integral H = h becomes /> 2 + (&) - If - f cos 2 Z - (Ae f + e) = 0. Assume a solution in the form p = i 2 a 2n e 2Ml / k , w = 2 b 2n e 2inl ' k , e = I c 2n e 2Ml ' k . 00 00 00 For a periodic orbit described always in one direction as regards these series are convergent, and if the coefficients are real, a yn = _<*, b. 2n 6_ 2n , Can = C-2H, and therefore _ 1 dr _ 2nZ p ~rdt~ ^f a ^ sll -y dJ _ S , 2?iZ - i -2 - 3 - @ 4 4'-@ ' 42 _ (S) ' 4 ! -o' 4 2 -0 4 2 -0 " -, (c-2) 2 - -, - 2 - 3 2 2 - 2-e o 2 2 -o' 2 2 - o 2* - -e, -0, c 2 -o - 0, - 2 2 - 2 - O 2 -o' 2 - 2 - ~3 - 2 -! (c + 2) 2 - -, 2 2 - 2 2 - 2 2 -o' 2 2 - ' ' 2 2 - - 4 -3 - 2 -! ( C + 4) 2 - 4 2 - 4 2 - 4 2 -o' 4" -0 4 2 - each row being divided by such a factor that the constituent in the leading diagonal becomes 1 when c = 0. This is Hill's celebrated determinant, which introduced the consideration of the meaning and convergence* of determinants of infinite order into mathematical analysis. 234. The determinant A ( c) = A(c), for the change only reverses the order of the constituents in the leading diagonal. Also A (c + 2j) = A (c), for the displacement of the leading diagonal along itself may be compensated by moving the divisors of the rows. Hence if c is. a root of A (c), + c -f- 2j are also roots. The highest power of c in the development is given by the product of terms in the leading diagonal, and this product is A M- FT It follows that 4J 2 - (2}4-Vo)' = (COS 7TC COS 7T Vo)/(l COS 7T Vo)- A (c) = (COS 7TC COS 7TC )/(1 COS 7T \/o) for this contains the right number of roots, the same as A (c), and the same coefficient of the highest power of c. The roots are those already found, and there are no others. But this equation shows that A (0) = (1 - COS 7TC )/(1 - COS 7T \/o) and therefore c is a root of sin 2 ^7rc = A(0)sin 2 ^7rVo ..................... (16) * Cf. Whittaker's Modern Analysis, p. 35 ; Whittaker and Watson, p. 36. 270 Lunar Theory I [CH. xx The solution of A (c) = is thus reduced to the calculation of A (0). The latter determinant is convergent if 2;* is convergent, and this may be assumed for sufficiently small values of rn. As a matter of fact in the present case A (0) is not only convergent but very rapidly convergent. It may be written in the form A (0V- where Suppose every 0, to be multiplied by 0i. If then the sign of be changed the sign of every alternate constituent in every row and every column is changed. Multiply every alternate row and every alternate column by 1 and the original determinant is restored. This involves multiplication of A (0, 0) by an even power of 1, since the number of rows and columns is equal. Hence A (0, 0) = A (0, 0), and A (0, 0) is an even function of 0. But the power of in any term of the development of A (0, 0) is the sum of the suffixes of the / associated with it. Therefore the sum of the suffixes in any term of the development of A (0) is even. Since ^ is of the order j 2j \ in m, this means that the order of every term is a multiple of 4. It is evident that the determinant A (0) must be developed axially, the term of zero order, 1, coming from the leading diagonal alone. There can be no term in , alone, for , incapacitates by its row and column two units from the leading diagonal as cofactors. Similarly a product ;, incapaci- tates more than two such units unless their rows and columns intersect on the leading diagonal. Thus i =j and the only terms of binary type involve squares. 235. The mode of developing A (0) will be sufficiently understood if m 12 be neglected. The sum of the suffixes can only be 0, 2 or 4. Hence the only possible terms are of the type A (0) = 1 + A* + 2 2 + C! 2 , + Z)! 4 . It is also easy to see how each of these terms arises. Thus 234, 235] Lunar Theory I 271 The next term corresponds to three consecutive diagonal constituents, and c^e, = 2 i Finally, the term in x 4 must correspond to four diagonal constituents only and it is therefore D = 22 &&_! /3//3/-1 = A 2 i j J 3 for, as the two minors must not overlap, i cannot have the values J or j 1. It remains to calculate the values of these coefficients. Let @ = 4a 2 . Then * 32a (2a - 1) Va - j a +j -l) J 32a (2a + 1) (a +j a - j 2, 1 1 1 _ _ _ = _ _ _1 8a (4a 2 - 1 ) ' a + j 8a (4a 2 - 1) (a TT cot ?ra TT cot ^7rV@o 80 4a 2 - 1 = ** . x a 2 - " 2 The other coefficients can be calculated similarly by first reducing to the form of partial fractions. Hill's results include all terms of order less than 16, and with the value of m already given ( 229) he obtained the value c =107158 32774 16012. Without going further than the term of which the form has actually been found here, ............ (17) The argument given above as to the order of the terms refers to x , @ 2 , ... and not to effects arising from . But 1 6 is itself of the first order, and therefore this expression neglects m 7 instead of m 8 . Since m = 0'08 the error in c might be expected to occur at about the seventh decimal place, and in fact it is about 5 units in this place. This simple expression, involving only @o and ] , is therefore very approximate. It may be noticed that + ic (n n) are the characteristic exponents of the variational curve. Since c is real this curve represents a stable orbit for small variations. 272 Lunar Theory I [OH. xx 236. The introduction of the eliminant of infinite order was a bold and original expedient on the part of, Hill, though justified later by analysis. But an analogous method had been used earlier by Adams, whose results were published after the appearance of Hill's. They refer to the integration of the third equation of (2) when H = 0, or D 2 z - z (icr- 3 + m 2 ) = 0. If z be neglected in the coefficient of z, that is in r~ :f , the series already used in 232 may be inserted, and the equation becomes which, since Mi = M_i is of the order j 2i \ in m, is of exactly the same form as (15). A solution is known to be of the type and g must be determined from the infinite set Hence the eliminant is A' (g) = 0, and the solution is given by sin 2 ^7Tg = A' (0) sin 2 \TT \/(2^ ) where A' (0) is the result of replacing ,: by 2M t in A (0). Adams used the value m = n' / n = 0'0748013 exactly, which is not quite the same as Hill's value. He thus obtained the corresponding numbers m = 0-08084 89030 51852, g = 1-08517 13927 46869. CHAPTER XXI LUNAR THEORY II 237. It is now necessary to consider the form of the general solution of the equations (6); in the present chapter equations will receive reference numbers in continuation of those assigned in the previous chapter, so that the latter will suffice without referring specifically to the chapter or section in which they occur. The solution of (15) may now be written 8^=^26^, logft = t(n- ')(*- *0- The arbitrary constant t : makes it possible to assign any required phase to the variation in relation to the periodic solution and as 8N is supposed small (so that SN 2 has been neglected) the coefficients bi may be considered to have a small arbitrary factor. These two arbitraries make the small variation otherwise general. Since c has been determined it would clearly be possible to determine real values of the coefficients (except for the arbitrary factor) by substituting the series in (15), equating coefficients, and proceeding by continued approximation. Again, if So- be the displacement in arc corresponding to 8N, by (2) of 214 adapted to the present notation, or ( 232) B irrt -yr- - -r- + 2 m )8N = - 1 VD - \ Du Ds ] Hence, V being an even function of , i8a has the same form as 8N. But since Ve L * = ivDu, Ve ~ '* = iv Ds and 8N= SX sin i/r - SF cos ^ = fa (8u .-*-&. e 1 *) So- = SX cos^ + 8 Y sin ^ = % ( 8lt e ~^ + gs e ^ it follows that Su = ^ (8N + i8 ^* CJ Qi%j+y&fi T [- J C, Li -f- ZjJ tt 2 -(-2/fi i _j_ 2 [27 c +1 a. .,' 4- 2 r2i c 1 --*+* + e/^ + eL,^) s = ae^ 1 (e~ c + *& + e- z ~ c + ^ 2 ~ c + ^'^ +c + e'-i^ ) to which correspond ( 230) rcos(v - nt - e) = ae {(6 + e ')cos rsin(w - nt- e) = ae{(e - e ')sin where ) r sin (v - nt e) = ae sin 0. Now in elliptic motion (24) and (25) of Chapter IV give, to the first order in e, r cos iv = a ( f e + cos M + ^e cos 2M ) r sin w = a ( sin M + |e sin 2M) r cos (w M) = a (1 e cos Jlf ) r sin (w M} 2ae sin -M. 241-243] Lunar Theory II 279 These can be identified with the former by putting a = a, e = 2e, ) + /i m2e cos r sin (v nt e) = a {^m 2 sin 2 + e sin + (if m + J^m 2 ) e sin (2 - <) + / ff m a e sin The effect of dividing the latter by the former is to add to the second series the terms m' 2 e (cos 2 sin + { sin 2 cos <) = m 2 e {f|- sin (2 + <) - sin (2 - <)}. Hence the longitude is approximately v = nt + e + -^m 2 sin 2 + e sin + (J g B -m + W m2 ) e sin ( 2 - <) + ii m2e sin ( 2 ^ + ) As a constant of integration introduced at one stage of the present method, e may be defined in any suitable way for the later stages. Its value depends on the exact definition adopted and will be found by com- paring the final results with observation. Thus |e as defined by Brown is not to be identified with the e of Delaunay, for example. The difference is not great, however, and its value may be taken to be 0'0549! Thus the co- efficient of the principal elliptic term in longitude, e sin <, is of the order 6'3. The term next in importance has the argument 2 (or 2D I in Delaunay 's notation). The coefficient is right to the order given, though the above derivation left this doubtful, and its value gives (j^ m + ^-m 2 ) e = 73' nearly. The true coefficient, depending on e alone, is 4608". This inequality is the largest true perturbation in the Moon's motion and is known as the Evection. Its discovery from observation is due to Ptolemy. 280 Lunar Theory II ' [CH. xxi The term with the argument 2f -f (f) (or 2Z) + I) is much smaller. The above coefficient gives 157", while the true value is about 175" for the part depending on e alone. It will be noticed that the greater part of it is due not to a true perturbation in the rectangular coordinates but to inter- ference between the variation and the principal elliptic term in deriving the longitude. 244. The terms depending on the first power of the solar eccentricity e' will be next considered. With z = and the solar parallax still neglected, H = f} 2 and (4), (5) become D 2 (us) - Du .Ds + 2m (sDu - uDs) + f m 2 (u + s)*=C- 3H 2 + D~ l (D t O,) D (uDs - sDu - 2mws) + f m 2 (u 2 - s 2 ) = s - u 9s du where (3) gives H 2 = m 2 ^ (3r*S* - r 2 ) - m 2 {3 (u + s) 2 Now rS = (XX' + YY') rr 1 = i (u + s) cos % - 1 1 (u - s) sin %' where ( 223) % = v' n't e' = v' ' is the solar equation of the centre. Hence i*S* = I (u 2 + s 2 ) cos 2% + iws - i* (u 2 - s-) sin 2%' and therefore a' 3 I1 2 = m 2 {f (u- + s 2 ) cos 2% + -us - ft (u- - s*) sin 2%'} - |m 2 (3u- + 3s 2 + 2) *"i where u, s have the values given by the variational curve. The Sun's mean anomaly is $ = ri(t- t s ) = m(n- M') (t-ta) = -i log ^ 3 m . The whole disturbing function must ultimately be developed in powers of f 3 m as far as necessary, the coefficients involving u, s, a'" 1 and e'. But for the immediate purpose it is easily verified that to the first order in e', v W C\ t 1 C\ I I/ Cv *'-\/ A / if = : cos 2y = 1 + 3e cos d> , sm 2v = 4e sin c6 . M 5 /V '/yo / * / Hence n 2 = fmv {u 2 (- Ks m + K 3 - m ) + * 2 (K 3 m - i&- m ) + us ( 3 m + r 3 - m )} A 2 - f mV {u 2 (- ^ 3 m - f &-) + s 2 (ft3 m + if,) + us (^ - r 3 - m )]- Thus the right-hand members of the equations at the beginning of this section will be of the form a * m 2m , a for, as in 238, the suffix of 3 may be suppressed in the calculation with the proper understanding as to the argument corresponding to m in the results. The solution is of the form u = a , ipm *, 8 = a - i p 243-245] Lunar Theory II 281 where A 2f = a.,,, A 2i+m - e'rj,: , A 2 ,-_ m = e'rj' and p has the values 0, + 1 only, until higher powers of e are taken into account. The solution follows the same course as in 239 except that there are now terms on the right-hand side of the equations. The equations of condition corresponding to (20) are thus 2 {[2j + m, 2i + m] a_ 2j+2 ;7?; + [2j + m, 2i + 2j] o^V i + 2 [2/ + m, +] a 2/ -_ 2 ;_ 2 ^ + 2 [2j + m, -] a_ 2j _ 2( ;_ 2 T//} = E\ j+m . This form results from the linear combination of a pair of equations obtained by comparing coefficients of 2 J +m and in these the leading terms by analogy with (9) are respectively . . . + {4/ 2 + 2/ + 1 + 4m (/ + 1) + f m 2 ] a e' Vj + {4/ 2 - 2/ + 1 - 4m (/ - 1) + f m 2 } a u V-; + = *'&*+* ... - 4/ (1 +/ + m] a e'rij - 4/ (1 -/ + m) ao^Y-; + = e'E' 2j+m where / is written for j + -| m. The combination is such that the coefficient of ?;'_; vanishes and that of rjj becomes 1. Hence rv, = (1 -/ + m) K 2j+m + {4,f* - 2j' + I - 4m (f - 1) + f m 2 } E' 2j+m 2 (8/ 2 -2-4m + m 2 ) The divisor, which appears also in the symbolical coefficients [ ], becomes small only through the factor/, whenj = 0, 4/ a = m 2 . 245. The calculation of 77,-, ?// when m is given its numerical value at the outset, proceeds as in the case of e,-, e/ with this difference, that the equations contain definite right-hand members. A particular solution of the differential equations is required, representing a forced disturbance of the steady variational motion. Hence no new constant of integration enters. The machinery is of course absurdly elaborate when only the main parts of the leading terms are sought, but this plan will be pursued. It is easily found that O 2 = f mVa 2 {- \ (^ +ni + - 2 - m ) + \ ( 2 ~ m + - 2 + tt ' + (1 + 6a_ 2 m + - m with the neglect of m in the coefficients of 2 m , but not m . The operator D t applies to " m only and gives a multiplier + m to every term, while the operator D~ l applies to generally and gives divisors 2 + m or + m. Hence to the same order in m Also an 2 an, s -r w = m 2 e 282 Lunar Theory II [CH. xxi Hence # m = E- m = - f m 2 (1 + 6a_. 2 ), E m f = - "_ m = 12m 2 a_ 2 E m " = (- m- 1 + f) E m - ^m--E m ' = f m + ^m 2 E"_ m = (m- 1 - I) E_ m - im- 2 #'_ m = - f m - Sfm*. Thus ?? , 770' must be of the first order in m and give rise to terms of at least the third order in the equations for j = 1. These contain no small divisor and for the lowest order they give immediately : Vi = E" 2+m = | -fi" 2+m = ^ m 2 rji = E" 2 _ W = |" 2 _m = Mm 2 Coefficients of the form [m, y] are of the order 1 in m, but they multiply terms of at least the fourth order in the equations for j = 0. These give therefore to the second order -770 + 2 [m, +] a.rj^ + 2 [m, -] a^'^ = E" m -i) ' + 2 [- m, +] OoV-i + 2 [- m, -] a 77_, = E"_ m where [m, +] = [- m, +] = - f , [m, -] = [- m, -] = f . Accordingly -7; = fm-fm 2 , -770' = -fm + fm 2 . Thus the principal terms depending on the solar eccentricity may be put in the form r cos (v nt e) = ae' {(770 + T7 ') cos <' + (77, + V-O cos (2 -f ') + (^ + 77^) cos (2 - = ae' (fm 2 cos <' + ^m 2 cos (2^ + <') - |m 2 cos (2 - ')} r sin (v nt e) = ae' {(770 - V) sin f + (^ - 77 / _ ] ) sin (2| + f ) + (j?/ - 77^) sin (2f - = ae' {- 3 (m - m 2 ) sin <' - |im 2 sin (2f + <') + ffm 2 sin (2f In deriving the longitude there are no interfering terms of this order, and the last line without a gives the additional terms depending on e'. The term with argument 0' (or Z 7 ) is called the Annual Equation after its period. The value of e is 0'01675 and the coefficient of this part of the term, 3e (m m 2 ), is 770" as compared with the complete value 659". For the argument 2-<' (or W - I') the coefficient ffe'm 2 is + 109", the true value being + 152", and for the argument 2+ 0' (or 2D + I') the coefficient -j^e'rn 2 is 15"'5, the true value being 21"'6. The discrepancies are considerable and show that the parts depending on higher powers of m are large. As series in m the coefficients converge slowly, and hence the great 245, 246] Lunar Theory II 283 advantage of the Hill-Brown method, which by employing an accurate numerical value of m from the beginning avoids expansions in this parameter altogether. 246. In deriving the terms with the characteristic a'" 1 alone, e is neg- lected and therefore fl a = 0, D t l = 0, and H = a, = 2m 2 a'- 1 Par 3 = naV" 1 (or'S 3 - Sr'jS) . = ^m'a'- 1 [5 (M + s) 3 - llus (u + s)} since rS = X = ( + ,9) when e' = 0. The terms on the right-hand side of (4), (5) are thus - 4O :J = - ^m'a'- 1 {5 (u 3 + s 3 ) + 3ws (u + s)} = a? a'- 1 - u 3 = - fmV- 1 (5 - s 3 ) + us (u - s)} = a 3 a'- 1 respectively. The additional terms required in the solution must be of the form n, _ os^'-i r? n j-2i+i (, _ a 2 <7 / ~ 1 / J T J*2*+i M a a c, ^a2i +1 t, , * a, a, c, z ( _ 2t _ 1 1, in order to produce odd powers of Similarly H 4 has the factor a'~ 2 and gives rise to terms with the same arguments as the variational terms. The solution follows the same course as for the terms with characteristic e*, and the relation connecting E" 2j+1 with E 2 j +1 , E' 2j+l is the same as before when y=j+i The principal terms are given by 2j + 1 = + I, 3. The divisor D. 2j > is of the order m when j' = | only. But fl s contains m 2 as .a factor. Hence, when terms of the order rn 3 are neglected in E' 2 j +1 , m 2 can be neglected in m~ 2 n s and the variational coefficients a 2 , a_ 2 are not required. Thus it is enough to write - 40 3 = - $ m'aV- 1 {5 ( + ^ and therefore tt_s = E f 9 = ^r E?, H 1 E'_ .j = -4p- m 2 . Also, to the same order in m, E" = (Zm- 1 4-ii"l E 4- ( 4m" 1 - 2 - 7 -^ E' , = 44m ^4m 2 j-> j i ^ in \^ if/ *^^ 1 *^ V 4 f fi / "^^ 1 ^^ ^^ 1 ^ fi * The equations for a 1( a_j can be adapted from (21) and its correlative by putting c = l, e = / = !. and e ' = _! = _!. To the second order in m these give [1, 2] a,^ -a, + [I, - 1] a_ 2 o_, + 2 [1, +] a a^ = E," [- 1, 1] a.^ - a_ x + [-!,- 2] a_ 2 a_, + 2 [- 1, -] a,*., = ^"^ 284 Lunar Theory II [en. xxi whence ^ = in + i m 2 and therefore - i = M m + ft m2 > -a-i = - ft m - -W- m2 - The additional terms in their elementary form are thus r cos (v - nt e) = a 2 a'~ 3 {(i + a_ x ) cos + (a 3 + a_ 3 ) cos 3f} = a 2 a'- 1 {(|f m + ^m 2 ) cos - ff m 2 cos 3j r sin (v nt e) = a 2 a /-1 {(ctj _,) sin + ( 3 a_ 3 ) sin 3} = a 2 a'- 1 { - (-> m + ^ m 2 ) sin + if m 2 sin 3} and the last line, divided by a, gives the corresponding terms in longitude. The mean parallax of the Sun is 8"*80 and of the Moon 3422 //< 7 ; to the above order a/a'= 0*002571. This gives 114" for the coefficient of the first term (argument or D} and l //- 6 for the coefficient of the second (argument 3 or 3D), whereas the complete values, with the characteristic a/a' alone, are 125" and under 1". The term with argument D is known as the Parallactic Inequality. Its period is one lunation (or synodic month) and the comparison of its theoretical coefficient with observation gave probably the best determination of the solar parallax until the direct geo- metrical method based on the observation of minor planets was adopted. This use of the parallactic inequality is not entirely free from objection because the Moori cannot be observed throughout a complete lunation and systematic error may be suspected, due to the varying illumination of the lunar disc. 247. Hitherto the terms of u, s which are of the first order in the characteristics e, e',aa' -1 have alone been considered. If the third coordinate z be assumed to be of the first order the first two equations of (2) show that u, s contain in addition only terms of the second and higher orders. The third equation of (2) has already been considered in 236, and when O is neglected terms in z of the first order are given by the equation D*z = (22^^) z. Let *) = S ( n ~ n ') (t-t 2 ) = - i log 2 g . Then the general solution is of the form iz = ak Sfcf ( 24 '+ g - -2*-g) where a preliminary value of g has been found in 240 and k, t. 2 represent the two necessary arbitrary constants. As before the suffix of 2 has been suppressed because it does not affect the calculation, though the proper 246-248] Lunar Theory II 285 argument must be retained in the results. The coefficients k t are deter- mined by equating terms in 2 J +e , so that and it is possible to write k = 1. In obtaining k lt k^ to m 2 only it is possible to neglect k 2 , 7c_ 2 and approxi- mate values of M , M l = M^ have been found in 240. Thus the equations are (2 + g) 2 ^ = 2M k\ (2 - g) 2 k., = 2Jtf &_ 1 where (2 + g )s _ 2M Q = 8, (2 - g) 2 - 2M = - 4m - 3m 2 , 2^ = 2^^ = m 2 + J m 3 Hence &! = T 3g m 2 , &_j = m -|| m 2 and to this order in m iZ = ak {* - - s - (f m + || m 2 ^ = 2ak {sin 77 + (f m + f f m 2 ) sin (2f - 77) + T 3 ff m 2 sin (2 -1- 77)}. 248. Here the fundamental term is z = 2ak sin rj = 2ak sin {g (n n') (t t^)} and its general meaning is easily seen, though the exact definition of k must be adapted to the final approximation and then determined (like e) by direct comparison with observation. The maximum value of z is 2ak. But it is also approximately a tan /, a being the mean distance in the orbit projected on the plane of the ecliptic and / being the inclination of the orbit to this plane. Hence k is nearly \ tan /, and differs little from Delaunay's 7=sin \I. Its provisional value may be taken to be 0'0448866 = 9260". At a node 2=0 and the period between successive returns to the same node is 27r/g(w n'). In this time the mean motion in longitude is 27rn/g(n n'). Hence the mean rate of change in the position of the node is {27rn/g (n - n') - 2-Tr} -=- 2?r/g (n - n) = n - g (n - n') = n I 1 ~ g/C 1 + m)} = w (- f m 2 + f|m 3 ) with the approximate value of g found in 240. Since this expression is negative the lunar node has a retrogade motion and completes a circuit in 6890 days or 18 - 9 years, which is reduced by about 100 days when the com- plete value of g is used. These facts have an important bearing on the theory of eclipse cycles. In deriving the elementary terms in latitude with the characteristic k it is enough to take from the variational solution r = a(l - m 2 cos2) and to the order m 2 the latitude is z/r = 2k (sin 77 + (f m + ^|m 2 ) sin (2 - 77) + |m 2 sin (2 + 77)}. 286 Lunar Theory II [CH. xxi The first term, with argument rj (or F in Delaunay 's notation) is the principal term in latitude. Its coefficient is 5 8'. The second term, with argument 2 77 (or 2D F}, has been called the evection in latitude. Its coefficient as found above is 610"'6, the true value being 618"'4. The third term, with argument 2 -f 77 (or 2.Z) + F) has the coefficient 83"'2 as compared with the true value 94" - 5. 249. It is now possible to sketch the whole method of the subsequent development. The greater part of the practical work of calculation has been based not on the homogeneous equations used above, which present advan- tages in special cases (especially the calculation of long-period terms), but on the original equations (2), Dhi + 2mDu + f m 2 (u + s) - -- = - d ~ r* ds KZ 9H - = -!. r 3 dz It is unnecessary to use the equation in s because s =/("~ 1 ) if u =/() ; two real equations are replaced by a single complex one. Also the characteristics entering into u and z are distinct. Hence the treatment of the equations in u and z is also distinct. The order of a characteristic is the sum of the positive powers of the parameters e, e', aa'" 1 , k which compose it : m is a mere number for this purpose, and retains its identity only in the argu- ments. Now suppose that a complete solution u = u 1 , s = s ly z = z 1 to the order /* in the characteristics has been obtained. The next step is to find the solution ic = u 1 + u z , s = s 1 +s. 2 , z z 1 + z 2 , where u 2 , s. 2 , z. 2 represent the terms of order fi+1. Insert these values in the equations, retaining only the first powers of u 2 , s 2 , z 2 . The result is, since r 2 = us + z*, (D + m) 2 (M! + u 2 ) + -|- m 2 (^ + u. 2 + 3^ + 3s a ) - K (u^ + u. 2 ) rr s + ^K'u^r^ 3 (iiiS 2 + u 2 s i + 2^i z 2 ) = ^ OS (D 2 m 2 ) (Z-L + z 2 ) K (z l + z 2 ) rr 3 + ^icz^- 5 (u^ + u^ + 2^^) . oz Now terms of order less than p + 1 must be satisfied identically and therefore terms linear in u lt s l} z l may be omitted. Also terms of order higher than /4+ 1 can be neglected. Hence u ly s 1} z^ may be used in calculating H, and in conjunction with u 2 , s 2 , z 2 it is possible to write U^ UQ, s 1 = s , ^ = 0, ri* = U Q S O = p z=0 is the variational solution of zero order. Hence the equations reduce to (D + m) 2 1/2 + u 2 (|m 2 + ^Kp ~ 3 )+ s 2 (f m 2 + f KU < 2 ./+ T >+ 1 on both sides of the equation gives 'i- j = A j } This series of linear equations, in which the coefficients M i} N f rapidly diminish, must then be solved by successive approximation. When this has been carried out for each series A and every characteristic A, all the terms of order fjL + 1 in u, s have been determined. The treatment of z is precisely similar. 251. But one important question clearly arises. Is the set of linear equations consistent and definite ? If the modulus of the set, which can be written as a symmetrical determinant of infinite order since M t = M_ i} Ni = N-i, is not zero, the solution is certainly definite. This is the general case. But consider the determination of e^, e/ the co-factors of the character- istic e of the first order. By the above method these will be obtained from (23) by putting Aj = A' ^ = and r = c. The consistency of the equations Lunar Theory II [CH. xxi now requires the modulus to vanish. It is obvious that this condition in fact musfc lead to a determination of r which will be identical with the value of c , though the latter was found above in a formally different way. When the equations have thus been made consistent the solution only becomes definite when the arbitrary condition e - e ' = 1 is added, and this condition is equivalent to a definition of e. It is. now evident that the modulus vanishes whenever r = c, or for every series based on the same argument as that of the principal elliptic term. The consistency of the linear equations requires a relation between the coefficients A j} A] which may be expressed by equating the modulus to zero after replacing any column in it by the series A h A-. But owing to the symmetry of the modulus this relation is capable of a much simpler form. Let the equations (23) be multiplied by e/, e'_,- and let the sum be taken for all values of j. Then the coefficient of X,- is (2j + r + 1 + m) 2 e,- + 2Mij +i + 2 #_,+< = because, since ^M i j+i =^M_ i e j ^ i = ^M i ej, i) this is one of the equations of condition. Similarly all the coefficients on the left-hand side vanish, and the required relation appears in the form + A'_je'-j) ........................... (24) j The reason for retaining the terms (D 2 + 2mZ))zi in (22) will now be under- stood. Without them there is no reason why the relation (24) should be satisfied, and in fact it will be contradicted. But let w x contain terms of the form & {[c 2 + 2c (2t + 1 + m)] #^+ c + [c 2 + 2c (2i - 1 - m)] E'-g-*^} where terms obviously of order less than //, + 1 are omitted. Then clearly, if the value of c here be regarded as unknown, it will be possible to adjust its value so as to satisfy the relation (24). 252. The matter is made clearer by considering the actual facts. In the first order there is one such series, with the coefficients e i} e/. In the second order there is no such series and the question does not arise. The primitive value c suffices. In the third order series of this type reappear, associated with the characteristics e 3 , ee' 2 , ek 2 , e (a a'" 1 ) 2 . The contemplated change in c is associated with e through the first order terms. Hence the relation (24) in the third order will give in succession the parts of c which contain e 2 , e' 2 , k 2 and (aa'" 1 ) 2 . Similarly still higher parts of c may be found in con- junction with the inequalities of a higher order. It is natural that the motion of the perigee (and the value of the characteristic exponent) which was determined for highly simplified conditions, should require adjustment 25i-i>53] Lunar Theory II 289 when the conditions are more complicated and the deviation from the periodic orbit is no longer infinitely small. For c let Cj + X'Sc be written, where X'Sc is the part to be determined, its characteristic being X', and let where Bj, B'_j, Dj, D'_, are calculated numbers. With the new value of c the quantities Aj, A' _- t satisfy a certain relation identically as required, and the equations (23) become consistent, but the solution is not definite because any one of the equations can be derived from the rest. An arbitrary condition can be imposed, and the form X ' = X is chosen. The solution is then con- ducted in the following way. The equations for j = are left aside. Three separate solutions are then made of the remaining equations: (1) \j = bj, X'_y = 6'_,- when \ = X ' = and Aj = Bj, A'- } = B'_ } ; (2) X ; = d jt \'_,- = d'_j when X = X '=0 and Aj=Dj, A'_ } = iy_j ; and (3) X,- =f jt X'_; = /'_/ when X = X ' = 1 and A } = A' _ } = 0. The last, which under the different condition X X ' = 1 would have led to j, e'_j, is independent of Aj, A'^ and applies in all cases. The complete solution is therefore \. = bj + dj Sc +fj\, \'-j = b'-j + d'-j Sc +/'_, X . When these are inserted in the equations for j = the result is of the form b, + d Sc +/ X = &' + d 'Sc +/ 'X = and Sc and X are thus determined. The value of Sc must also satisfy the relation (24), so that a check on the accuracy of the work is provided. The solution of the equations (23) for the case when r = c is therefore complete, and the derivation of the higher parts of c has been explained. It may be noted that on the left-hand side of these equations the primitive value c is to be retained for r at every stage, both because it is associated with terms of the full order yu, + 1 and because the theory of the equations depends on the fact that the modulus vanishes. On the other side c will receive its full value so far as it has been determined. When a new part of c comes to be determined in conjunction with inequalities having the characteristic X, 8c is always associated through (D 2 -I- 2mD) (MJ) with the terms in i^ of the first order in e. Hence the new part of c itself always has the characteristic X' = e~ x X, and the numbers dj, d'_j, like/-,/'-/, are the same in all cases. 253. With the equation for z matters follow a precisely similar course, and the exceptional case arises when r = g. The conditions are simpler, because X,- + X'_,- = always, and therefore the arbitrary relation has the form X = V = 0. The terms of the first order with suitable arguments have the characteristic k, and the part of g found in conjunction with inequalities having the characteristic X contains the characteristic k -1 X. P. D. A. 290 Lunar Theory II [CH. xxi The arbitrary condition X = X ' adopted in all cases has an importance beyond that apparent in the actual calculation. The aggregate of the terms considered up to the final stage of approximation gives for the one argument u = ae * = ae The last expression remains unaltered throughout the course of the approxi- mations. Hence the constant e is defined as " the coefficient of a sin I in the final expression of p sin (v nt e) as a sum of periodic terms, where v nt e is the difference of the true and mean longitudes and p is the projection of the Moon's radius vector on the plane of reference." Similarly the terms of the form in the first approximation have no addition made to them subsequently, since X = X/ = 0. Hence the constant k is defined as " the coefficient of 2a sin F in the (final) expression of z as a sum of periodic terms." There is no reason to alter the definition of a, which is based on the variational curve. But it is then to be noticed that the constant of distance in the projection on the z plane will no longer be aa , where a = 1, but will be affected by terms with various characteristics which arise in the course of the approximations as the constant parts of u%~ 1 or s Either m or a, since they are connected by a certain relation (11), maybe regarded as an arbitrary constant of the solution. The remaining three arbitraries have been denoted by t , t lf t 2 . These may be replaced by e, -or, 6, the mean longitudes of the Moon and its perigee and node at the epoch t = 0. Then D = (n n') (t - t ) = (n -n')t+e- e' I = c (n n) (t ti) = c (n ri) t + e OT I' =m(n- n') (t - t 3 ) = n't + e - -er F = g (n - n') (t - t.,) = g (n - n'} t + e-0 where e' is the mean longitude of the Sun at the epoch t = and vr' is the (constant) longitude of the solar perigee. The time t s is not an arbitrary : it depends on the Sun alone and is one of the data of the problem. The formulae for transformation to polar coordinates were given in 230 for two dimensions only. It is necessary to replace r by p, its projection on the plane of the ecliptic, where p 2 = X 2 + Y 2 = us. Then u%~ 1 = p exp. i(v nt e) s = p exp. i (v nt e) z = p tan 253, 254] Lunar Theory II 291 where is the latitude. Hence the true longitude and the latitude are v = nt + e p p The constant of the Moon's horizontal equatorial parallax is based on a, where n 2 a? = E + M. To obtain the parallax at any time this constant must be multiplied by ns a 2 a i = s.' V In these expressions for v, and ar~ l the variational parts u , s are separated from the other terms M I} s lt z, and the expressions are then expanded in terms of the latter. Advantage can thus be taken of the expansions already obtained in the course of the previous work. The conversion to the final form of coordinates therefore entails no great amount of extra labour. 254. This completes in outline the solution of the main part of the problem, in which the Earth, Moon and Sun are treated as centrobaric bodies, and the orbit of the Sun, or the relative orbit of the centre of mass of the Earth-Moon system, is treated as an undisturbed ellipse in a fixed plane. A large number of comparatively small but highly complicated corrections are still necessary in order to represent the gravitational motion of the Moon in actual circumstances. They may be classified thus : (1) The effect of the ellipsoidal figure of the Earth, and possibly of the Moon. (2) The direct action of the planets on the relative motion of the Moon. (3) The indirect action of the planets, which operates by modifying the coordinates of the Sun. These indirect effects are in general larger than the direct effects, and are sometimes sensible in the lunar motion when they are insensible in the relative motion of the Earth and Sun. Among the indirect actions of the planets may be specially mentioned (4) Lunar inequalities produced by the motion of the ecliptic, and (5) The secular acceleration of the Moon's mean motion, which arises from the secular change in the solar eccentricity e under the action of the planets. It is impossible to discuss these matters profitably in a short space. The reader will find references in Professor Brown's Treatise and detailed results in the memoir* which contains his complete and original theory. * Memoirs R. Astr. Soc., MIJ, pp. 39, Ifi3 ; LIV, p. 1 ; LVII, p. 51 ; LIX, p. 1. 192 CHAPTER XXII PRECESSION, NUTATION AND TIME 255. In order to investigate the motion of the Earth about its centre of gravity we take a set of rectangular axes OXYZ fixed in space and a second set Oxyz coinciding with the principal axes of inertia. These are fixed in the Earth and move with it. The two sets are drawn in such a sense that the positive directions of the corresponding axes can be brought into coincidence by a suitable rotation. Their relative situation is defined by the three Eulerian angles 6, , ty, where 6 is the angle betw r een OZ and Oz, < is the angle between the planes OXZ and OZz, and ty is the angle between the planes OZz and Ozx. Then the coordinates are related by the scheme : X Y Z x cos 9 cos cos i/r sin sin -\Jr cos^sin^cos^-f cos cos -fy cos # sin $ si n-\Jr + cos < cosier sin sin ty z sin 6 cos < sin 6 sin cos 6 The result of resolving the angular velocities 6 which is a rotation in the plane OZz, < which is a rotation about OZ, and -^ which is a rotation about Oz, about Ox, Oy, Oz is to give the equivalent angular velocities about these axes, namely twj = 6 sin i/r 2 = cos ^ + sin 6 sin i/r (1) &> 3 = yjr + < cos $ which are Euler's geometrical equations. Let A, B, C be the moments of inertia about the axes Oxyz and L, M, N the moments of the external forces about these axes. Then the dynamical equations may be written in the well-known form : / D /"\ r ~ I O U ) 60.2 61)3 := J 2 -(C-^)w s i = ^(- (2) 255, 256] Precession, Nutation and Time 293 256. The external forces which are here considered are due to the action of the Sun and Moon. An approximate expression for the action of either of these bodies is sufficient and easily found. The potential of the Earth (mass m) at a distant point P has been found ( 18) to be T7 . n ^ dm n (m V = G-2, -- = tr , 1 p \r 2?' 3 where OP = r and / is the moment of inertia of m about OP. This expression is true as regards terms of the second order in the coordinates of points in m relative to the centre of gravity 0. Terms of the third order will clearly vanish in the sum provided that the mass m possesses three rectangular planes of symmetry : and this is sensibly true in the case of the Earth. Terms of the fourth order are small in consequence of the ellipsoidal figure of the Earth and are neglected. Now V is the work done by unit attracting mass at P when the particles of the mass m are brought from infinity to their actual configuration. Hence the work done by a finite mass near a distant point 0' is by similar reasoning, if 0' is the centre of gravity of the attracting mass m, 00' = R, A', B', C' are the principal moments of inertia of m' at 0' and /' is the moment of inertia of m' about 00'. Now since A, B, C and / are of the second order in the linear dimensions of m, terms of the second order in the linear dimensions of m' can be neglected when associated with them. Let the coordinates of 0' relative to be (as, y, z) and of P relative to 0' be 1,- Then 7- 2 / = A (x + f But since 0' is the centre of gravity of the mass m' 2,!; dm' = 2,r)dmf = ^dm = 0. Hence if the expression to be summed be expanded in terms of , ?;, the terms of the first order vanish in the sum and terms of the second order are neglected. To this order of approximation A+B+C 3 (An? + By* + - -- and if / now represents the moment of inertia of m about 00', the complete expression for U becomes f mm' m (A' + B' + C'-M') m' (A +B + G- 31)] r (R 2R 3 2R 3 }' 294 Precession, Nutation and Time [CH. xxn This represents the mutual potential of two masses m, m with sufficient accuracy. In the usual astronomical units ( 24) G = A: 2 . The mass of the Sun is unity and for the masses of the Earth and Moon we take E and//?. Then if the mean distances of the Sun and Moon are a' (= 1 ) and a" and the mean motions n' and n" ' , Gl+E = ri*a' s 257. The moments of the external forces about the axes Oxyz being L, M, N, the work done by them when the Earth receives a small twist defined by the rotations da) 1} da) 2 , da) 3 about the same axes is dU=L dw l + Md(o. 2 + Nd(o s . But U depends on the orientation of the Earth only through the occurrence of / ; and R 2 I = Ax* + By* + Cz* (x, y, z) being the centre of gravity of the attracting body. Hence dU=- 3Gm (Ax dx + Bydy + Czdz)/R 5 . But with due regard to sign, when the axes are rotated, dx = y d(D 3 z da) 2 , dy = zdw l xdw. A , dz = xda) 2 ydw l . Hence, equating the coefficients of d(o l , dw 2 , dw s in the two expressions fordU, L = 3Gm'(C-B}yzlR 5 , M = 3Gm'(A - C)xz\R\ N = 3GW '(B- A)xyj R 5 . These apply to a body possessing three distinct principal axes. But the Earth may be regarded as an ellipsoid of revolution, for which E A and C>A. Under these circumstances L = SGm'(C- A)yz/R 5 , M= - 3GW (G - A) xzjR', N = Q. On the other hand, the term in U which depends on the orientation of the Earth is more generally ' {(20- A -B)z* + (A- B) O 2 - ? / 2 ) + (A+B) R>] /R 5 a useful form for some purposes. The last term on the right, being inde- pendent of the orientation, can always be rejected ; and when the Earth is considered uniaxal, it is possible to use simply U" = -Gm'(C-A)z*IR> ........................ (3) 258. With B = A and N= 0, the third equation of (2) gives &> 3 = 0, ! + fJ?(t)\ d>2 + fJ?W. 2 = where Hence the steady state is given by (o l h cos (p,t + a), o). 2 = h sin (/mt + a). But the instantaneous axis of rotation in the Earth is the line #/o>i = y/e>a = zf(D 3 or ac/h cos (pt + a) = y/h sin (/u.t + a) = z\n which indicates that if h is fairly small the terrestrial pole describes a small circle of radius h/n about the axis of figure in the period 2ir//j,. This is the Eulerian period of A/(C A) (roughly 300) days. Now the angle between the Zenith of a place and the Pole is the co-latitude of the place, an angle which can be constantly observed. Hence the latitude of any place should exhibit a variation with a period of about 10 months. Until a quarter of a century ago no variation of latitude had certainly been detected. Since that time variations (of the order of 0"'3) have been systematically observed and studied and have also been traced in the older observations. But analysis has proved conclusively that these variations contain no part which conforms with the Eulerian period. They cannot therefore be explained by the free motion of the Pole on a rigid Earth. Hence observation justifies the belief that h/n is insensibly small. The variations of latitude observed are always very small and constitute a highly complex phenomenon. The periods of the chief components of the motion of the Pole are about 12 and 14 months. 259. Corresponding to the free movement of the Pole on the Earth's surface we have, by (1), 6 = d) 1 sin T/T + (0. 2 cos ty = h sin (/* + a + i|r) $ sin 6 = coo sin ty a> } cos i/r = h cos (/j,t + a + ty). For the plane OXY we take the plane of the ecliptic which varies but slightly in consequence of planetary perturbations. The value of 6 is about 23. Hence 6 and < are very small in comparison with n, a fact in accord- ance with observation even when the disturbing effects of the Sun and Moon are operative. Hence, further, -^ differs only slightly from n. The rotational energy of the Earth is T, where 2T=^( 2 sin 2 0) + C (ijr + cos 0) 2 . 296 Precession, Nutation and Time [en. xxn Hence the Lagrangian equations of motion are -r (A6) - Aft sin cos + Cty sin (-^ + cos 0) = But since T = JV = 0, T^ + 2 sin 6 cos 6 + Cn 6 sin 6 = 0(7 ft rlTT ~(A(f> sin 2 B + Cn cos 0) = |^ . a c< It has been seen that n is very large compared with and <, and it follows that those terms are of predominant importance which contain n as a factor. Neglecting the other terms on the left the equations become simply . 1 3U 0= (7n sin The complete justification for omitting the terms rejected must be sought by substituting in them the results which follow from the latter simple form of equations, when it will be found that they are practically insensible. The form to be used for U is given by (3), so that a sum of two terms corresponding to the Sun and Moon. For each dis- turbing body it is necessary to find the product of z 2 jR 2 and a 3 /R 3 expressed in appropriate terms and with a suitable degree of approximation. 260. The axes XYZ being fixed in space are defined so that OZ is directed towards the pole of the ecliptic for 1850.0 and OX towards the equinox for the same epoch. By the scheme of transformation z = X sin cos 9, 260] Precession, Nutation and Time 297 (i) Rotate the axes about OZ through the angle fl so as to bring OX to the position OX^. Then X = X, cos O - F a sin fl, Y = Y l cos fl + J^ sin fl, Z = Z l where fl is the node of the ecliptic for epoch t on the ecliptic for 1850.0. (ii) Rotate the axes about OX^ through the angle i so as to bring OY l to the position OF,. Then X 1 = X 2 , Fj = F 2 cos i Z 2 sin i, Z l = Z 2 cos i + F 2 sin i where i is the inclination of the ecliptic for epoch t to the ecliptic for 1850.0. (iii) Rotate the axes about OZ 2 through the angle N fl so as to bring OX. 2 to the position OX 3 . Then X 2 = X 3 cos (N- fl) - F 3 sin (N - fl), F 2 = F 3 cos (# - fl) -f X 3 sin (>V - fl), Z, = Z, where N is the longitude of the Moon's node reckoned through fl in both ecliptic planes. (iv) Rotate the axes about OX 3 through the angle c so as to bring OF 3 to the position OY 4 . Then X 3 = X 4 , Y 3 = F 4 cos c Z 4 sin c, Z 3 = Z 4 cos c + F 4 sin c where c is the inclination of the Moon's orbit to the ecliptic for epoch t. But, if (X 4 , F 4 , Z t ) are the Moon's coordinates, X 4 = r cos (v - N), Y^ = r sin ( v ~ N), Z 4 =0 where r is the radius vector and v is the longitude of the Moon at epoch t reckoned in its orbit; this longitude is the sum of three arcs in the two ecliptic planes and the plane of the lunar orbit. Now i < 1 and, for the Moon, c is of the order 5. Terms of the order t 2 , c 3 and ic are therefore neglected. Then the result of eliminating (X 3 , Y 3 , Z 3 ), (X 4 , Y 4 , Z 4 ) gives X. 2 = r cos (v - fl) + |c 2 r sin (v - N) sin (N - fl) F 2 = r sin (v - fl) - |c 2 r sin (v - N) cos (N - fl) Z 2 = cr sin (v N) arid the result of eliminating (X, F, Z), (X lt F 1} ZJ gives = X 2 sin cos ((/> - fl) + F 2 sin sin (0 fl) + ^ 2 cos + i { Y 2 cos 8 - Z 2 sin sin ( - fl)}. Hence z/r = sin 6 cos (v ) + c cos 6 sin (v JV) |c 2 sin sin (v N) sin (0 JV) + * cos 6 sin (v fl). In squaring this expression terms not involving 6 or (f) can be rejected, because they disappear on differentiation. Also terms involving v with 298 Precession, Nutation and Time [on. xxn coefficients above zero order are found to be negligible in effect. Under these conditions the result becomes ^/r 2 = S i n 2 + S i n 2 Q cos 2 (v - ) + c sin cos 6 sin ( N) + i sin cos # sin ( H) + |c 2 sin 2 0cos2(<-7\0-fc 2 sin 2 (4) 261. Certain expansions in terms of the mean anomaly in undisturbed elliptic motion are now required. When e s is neglected in the formulae of 40, (22), (26) and (27) of Chapter IV become r/a = 1 + |e 2 - e cos M - %e 2 cos 2M tfxlr 3 = (1 - f e 2 ) cos M + 2e cos 23f + % 7 -e 2 cos 3M tfy/r 3 = (1 - f e 2 ) sin . + 2e sin 2J/ + ^ 7 -e 2 sin 3M. The latter give, w being the true anomaly, a 4 sin 2w/r 4 = (1 - e 2 ) sin 2Jlf + 4e sin 3if + -\ 3 -e 2 sin 4 - N) + i sin cos sin (< - fl) + c 2 sin 2 6 cos 2(-N) + % sin 2 6 cos 2 (n"< + n - <) + f e sin 2 cos (" + /* - r) the final term being retained though periodic and not of zero order. For the Sun c = and hence similarly a'V 2 // 8 = (| + f e' 2 ) sin 2 + *' sin 6 cos sin ( - fl) + i sin 2 6 cos 2 (w'$ + // -<#>) + f e' sin 2 cos (n' + // - '}. 260-263] Precession, Nutation and Time 299 262. These expressions give the means of forming U, for For the Moon ( 256) < = GEf = fn,"* and for the Sun a-'"' = G L= jrc' 2 _ a 3 a' 3 1 + E ' Let C-A >"< C-A A8 ~ '~ar-i+/' Kl -*-~CT'\^E (5 Then U _ a?z z a' s z' 2 ~lrv Cn r 5 r 5 = - [K, ( - I c 2 + | e 2 ) + K! ( + f e' 2 )} sin 2 - (^i + K z ) i sin 26* sin (0 - fl - J&LJ { cos 2 (w' + // <) + f e' cos (n' + // OT')} sin 2 - iT 2 {| cos 2 (n"t + p (f>) + f e cos (n'^ + //, TO-)} sin 2 $ -zi r 2 {csin^cos6'sin((/)-^V r ) + ic 2 sin 2 ^cos2(- JV")} (6) The dynamical equations ( 259) sin 8(9 VOn 6 = . f sin ^ 90 VCn which result must be solved by continual approximation. This process, when guided by the facts of observation and limited to practical require- ments for a period of a century or two, is very simple. For it is known that is very nearly constant, while changes progressively but very slowly. Hence it is possible to discuss the secular effects, or precession, and the periodic effects, or nutation, separately. 263. The last three lines in the expression for U/Cn, containing six terms, give rise to periodic terms in 6, <, which can be neglected in the first instance. The secular changes come from the terms in the first line. With sufficient accuracy we may write i sin n = gt, i cos U = g't, e' = e + e : t the quantities e , e 1( g and g being given by the theory of the Sun's motion. The corresponding changes for the Moon are negligible in effect or rather are treated differently. Hence the equations for the secular movements of the Earth's axis are = -{K a (I- f c 2 + fe 2 ) + K, (1 + |e 2 )] cos (K l + K 2 ) ' (g' sin g cos 0) t 3K l e e l . t cos sin c/ 6 = (K l + KZ) cos (g' cos < 4- g sin ) t. 300 Precession, Nutation and Time [CH. xxn When = (1850'0), Q is the mean obliquity of the ecliptic for that date and may be denoted by e . Also $, being the angle between the planes OXZ and OZz ( 255), is 90 by the definition of the axis OX. The periodic effects at the time t = are excluded from consideration here, but their influence is small. Hence initially < = 90 - {K 2 (1 - f c 2 + f e 2 ) + K, (1 + f e 2 )} cos e . t\ ( ' sin f ' 6 j I 6 = e + I (K^ + K 2 ) cos e . gt 2 ) The length of time during which these expressions will be valid depends on the numerical values of the quantities involved. For a short interval from 1850'0 (a century or two) the preceding equations hold good, and may be written _ = 90 at. fit? \ (8) the suffix ra denoting mean values from which periodic changes are excluded. Thus (f> m , 6 m define the position of the mean equator at the time t relative to the fixed ecliptic (1850'0), the coefficients a, /3 and y being now determined by (7). The motion of the mean equator on the fixed ecliptic, measured by 90 m , is called the luni-solar precession in longitude. The angle B m e may be called the luni-solar precession in obliquity. 264. It has been convenient to use a fixed set of axes XYZ, where Z represents the pole of the ecliptic for 1850*0 and X the mean equinox for the same date. It is now necessary to introduce a new set of axes X'Y'Z', where Z' represents the pole of the ecliptic for the epoch t and X' the corresponding mean equinox, i.e. the intersection of the mean equator and ecliptic at the epoch t. Let z represent the N. pole of this mean equator, its position being defined by m , m . The longitude of Z' in the X YZ system is ft - 90 and ZZ' = i, where i sin ft = gt + ht- i cos fi = g't + h'P the terms of the second order being omitted above because they clearly give rise to terms of the third order only in the luni-solar precessions. Let us consider the spherical triangle ZZ'z, of which two sides are ZZ' = i and Zz=6 m . Since XZZ' = Q,-W and XZz = $ m , the angle Z'Zz = (f) m O + 90. The side zZ' , which is the mean obliquity of the ecliptic at t, will be denoted by #/, and the angle ZzZ', which is called the planetary precession, will be denoted by a. Hence cot i sin m = cos 6 m sin (O m ) + cot a cos (H < m ) 263, 264] Precession, Nutation and Time and to the second order i cos (ft m + (gt + hP) sin sin m - f (gt + htf) cos m - (#' + A' 2 ) sin < TO ) cos sin 6 + ^'^ cos e since it is enough to take d m = e and (j> m = 90 at. Hence to the required order at t 2 a= - 4- - - (h + ag' - qq' cot e ) sm e sin e v Fig. 8. Again, in the same triangle, cos m ' = cos i cos m + sin i sin m sin (ft m ) whence, to the second order, (6 m ~ 6m) sin \ (0 m + m f ) = -$i a cos m + sin m (etgtf - g't - To the first order, therefore, m 6m = - g't, sin \ (0 m + m ') sin e + \g't cos e . (9) 302 Precession, Nutation and Time [CH. xxn Hence to the second order ' a - + ff' 2 ) & cos + (g't + lit* - agt 2 ) sin e in "in = ~ i 7. sin e + i%g t cos e = g't + h't- - agt 2 + % g"P cot e ........................ (10) The relations between the various sets of axes are shown in fig. 8. The equator X'y (epoch t) cuts the fixed ecliptic XY in x, where Xx = zZY=QO - m , the luni-solar precession, and xX' = xzX' = ZzZ' = a, the planetary pre- cession. Let ZX' cut X Y in D, so that XD is the negative mean longitude (1850'0) of X', the mean equinox at t. This arc is called the general pre- cession and will be denoted by 90 < m ', so that xD = m ' , n . The angle DxX' = Zz = 9 m and xDX' is a right angle. Hence tan ( m r - <,.) = tan a cos d in and to the second order Thus by (8) and (9) the general precession may be expressed in the form 90 - < m ' = Pt + P'V where P = g cot e P' /3 - cot e (h + ag' gg' cot e ) and by (8) and (10) the mean obliquity of the ecliptic is m ' = e + Qt+Q't? where Q=g' Q' = y + h' - ag + ^g- cot e . 265. To find the periodic effects, or nutation, it is necessary to return to 262 and write =< w + <&, = e m + . Now (f) m and 6 m have been calculated so as to satisfy the secular terms which arise in the equations of motion from the first line of the expression (6) for U/Cn. Hence the six periodic terms of the last three lines alone are now relevant, and the dynamical equations become 4> = - KI {cos 2 (n't + fi-) + 3e' cos (n't + // - BT')J cos 6 KZ {cos 2 (n't + p ) + 3-N) cos ^} B = {K, sin 2 (n't + // - <^>) + ^T 2 sin 2 ("* + ^ - $)} sin ^ + 7T 2 {c cos cos (< - N) - ^c 2 sin sin 2 ( - 7^)|. The Moon's node makes a circuit of the ecliptic in 18 years in the retro- grade direction, so that it is possible to write N=N n -N 1 t. 264-266] Precession, Nutation and Time 303 To the first order in t, which is alone necessary, 6 = e and = 90 at ; the coefficient a can clearly be incorporated with n, n" and N l before integration in those terms in which < occurs, though the change in n', n" is unimportant. Then on integration f 1 Se <& = K l cos e \-=. sin 2 (n't + At/) T sin (n't -f At' *r" (2tt n + K. 2 cos e J 77-77 sin 2 (n"i + xt) r , sin (w" + At t [2n n ( c c 2 ) -f KZ (2w 2?i J f c c 2 ) f , j cos 6 cos ( JV - j\fa) - sn e cos It is unnecessary to add integration constants because these are incorporated in is the nutation of longitude, and 4> being measured in the direction of increasing longitudes. The numerical quantities involved are of such an order of magnitude that a fair standard of accuracy has already been obtained in the formulae. If more precise results were needed, it would be necessary (1) to carry the expansions for the disturbing bodies further, and (2) to continue the process of integration by successive approximation to a higher stage. The latter process would clearly introduce terms of the form at sin (nt + a). Among the terms of the former origin those depending on three times the Sun's mean longitude (n't + //) are the most important, and it may be left as an exercise to the reader to determine them. By far the most important terms in the nutation are those with the argument (N Nj). The other terms being omitted, let ^=K 2 ccose /N l ...... . .................... (11) x = [] sin e = e/Fsin (JV Nj) cos 2e / cos e y = [0] - - J'' cos (N - Nj). Since c/Kis an angle of a few seconds only, x and y may be considered as the rectangular plane coordinates of the Earth's pole relative to the mean pole, x being measured in the direction of increasing longitudes and y upwards towards the pole of the ecliptic. The relative path of the true pole is therefore the small ellipse x 2 cos 2 e + y' 2 cos 2 2e = ^ cos 2 2e 304 Precession, Nutation and Time [en. xxn described in a period of about 18 years. Since cos e > cos 2e the major axis is directed towards the pole of the ecliptic and, since x has the same sign as y, the sense of description is such that the relative longitude of the true pole is increasing when it lies between the mean pole and the pole of the ecliptic, that is, it is clockwise when viewed from a point outside the celestial sphere. The centre of this elliptic motion is carried by precession almost uniformly in the direction of decreasing longitudes round the pole of the ecliptic. 267. Since the manner of the investigation has been controlled by the actual magnitude of the various quantities involved, it is necessary to intro- duce numerical values if the results are to be properly understood. Three quantities are. based on observation, and not derived from theory, namely, the obliquity e at the fundamental epoch 1850'0, the precession constant P and the nutation constant Jf. The values now accepted are e = 23 27' 31"-7, P = 50"-2453, m = ctt + $V = 50"-3684 1 - 0"000 1077 t* e m = 6fl + v? = 23 27' 31"-7 + 0"-000 0066 1 2 while the general precession ( 264) becomes 90 - , and the nutation of longitude ( <3>) is called AX. Comparison shows that no term with coefficient exceeding 0"'05 has been omitted here. Two important astronomical constants are involved implicitly in the constants of nutation and precession, namely the mass of the Moon and the ratio (CA)/C, which has been called the mechanical elliptic! ty of the Earth. For the equations (5) may be written __ 1+/~# 1 '" 2 ' ' C ~3' ' the mass of the Earth, E= 1/333432, being negligible. Here K^ and K 2 , expressed above in seconds of arc, are angular motions in a Julian year, and n, n' and n" are sidereal mean motions in the same unit of time. With sufficient accuracy the above values of n' and n" may be used, and for n the value 2?r x 366^. Hence //(I +/) = 0-012102, /= 1 /81-6 p. D. A. 20 306 Precession, Nutation and Time [CH. xxn for/", the ratio of the mass of the Moon to the mass of the Earth, and C-A JL C * 304-2 for the mechanical ellipticity of the Earth. The mass of the Moon is also obtained as a by-product from the observations of a minor planet in a refined determination of the solar parallax. The value of f found by Hinks in this way was I/ 81 '53. 269. The practical application of the results obtained for precession and nutation belongs to the domain of Spherical Astronomy and will not be pursued in detail here. Nutation is so small that its effects can be, and are, treated independently of those due to precession. Of the latter some- thing more may be said in order to define the two quantities employed in calculating the effects of precession in right ascension and declination. Let a, 8 be the R.A. and declination of a star at the epoch t. These refer to the system of axes X'y'z (fig. 8), which differs by a simple rotation through the angle a about z from the system xyz. Hence the coordinates of the star in the latter system are x = cos 8 cos (a + a), y cos 8 sin (a + a), z = sin 8 whence, by differentiation with respect to t, it easily follows that a 4- a (xy y#)/cos 2 8 8 = z I cos 8. Now the relations between the systems xyz and XYZ are expressed by the scheme : X Y Z x sin (f> cos (f) y cos cos (f> cos sin < sin z sin cos (j> sin sin < cos 0. Here XYZ are constant, and differentiation of the linear formulae for xyz, when XYZ are finally expressed in terms of x, y, z, gives x = (y cos + z sin 0) < y = x cos . (j> z0 z = - x sin . (j> + yd. Hence, when x, y, z are expressed in terms of a, 8, d + d = - cos . - tan 8 sin (a + a) sin . - tan 8 cos (a + a) . 8 = - cos (a 4- a) sin . + sin (a + a) 0. These differential expressions are required to the first order in t, and a0 being of the second order may be rejected at once. Hence (the symbol n being used here in a new sense) 268-270] Precession, Nutation and Time 307 d = m + n sin a tan & p cos a tan B 8 n cos a +p sin a where in = d cos 6 . + $ and # may be replaced by e . With the numerical values given in 267, (9) gives a = + 0"-1342 1 - 0"-000 2380 t 2 a = + 0"-1342 - 0"-000 4760 t and from the luni-solar precessions = - 50"-3684 + 0"-000 2154 1 e= + o"-ooo 0132 1. Hence m = + 46"'07ll + 0"-000 2784 1 n = + 20"'0511 -- 0"'000 0857 t while p = + 0"'000 0002 and is altogether negligible. Thus m and n are the important quantities known as the annual precessions in R.A. and declination. The total precession in R.A. from 1850 for a point on the equator is mdt = mj + m 2 t 2 = 46"'07ll t + 0"'000 1392 V. The expressions found for d, S are the coefficients of the first power of the time and these terms suffice for short intervals only. The further develop- ment of formulae for the transformation of coordinates from one epoch to another according to the methods of astronomical practice must be sought in such works as Newcomb's Compendium of Spherical Astronomy. 270. It is now possible to consider in some detail the astronomical measure of time. The third equation of (1) is a) 3 = ijr + cos 6. 202 308 Precession, Nutation and Time [CH. xxn It is the true equinox which is now involved, affected both by precession and nutation, so that Hence f = &) a (f) m cos 6 m cos 6 + (j> m sin 6 m = w -f- m cos 6 n = co + m 4> cos e with sufficient accuracy, for n can be neglected since -is small and n is about 10~ 4 , and d> being small cos 6 may be replaced by cos e . Hence integration gives for Greenwich sidereal time T = T + (at + mj + m. 2 t 2 cos e (12) where t is measured in Julian years of 365'25 mean days and reckoned from 1850 Jan. 0, Gr. mean noon. The quantity t is an equi- crescent variable in the sense required by the dynamical laws which have been used ; its origin and unit are for the moment of importance only so far as they condition the numerical values of the coefficients. On the other hand the sidereal time T is not uniform, being affected by secular and periodic terms. Hence T is merely an intermediate standard of time. But this in no way affects its practical utility. By far the largest term in cos e is 15"'803 sin & = 1 8 '054 sin of which the period is nearly 19 years, and m. 2 is very small. The irregularities in T are therefore very small and gradual, and far less than the natural irregularities in the rate of the most perfect sidereal clock. Since this instrument shows the hour-angle of the First Point of Aries, it also shows the right ascension of stars on the meridian, and this principle serves both to determine the error of the clock and to measure the apparent positions of the stars. 271. In the next place a mean Sun is defined which moves in the plane of the equator with the uniform sidereal mean motion /u,. Its R.A. at time t, reckoned from the true equinox, is therefore A A + /j,t + nijt + m 2 2 <& cos e and its hour-angle T = T A = T A + (w ft) t is the measure of Greenwich mean time. The constants occurring in A are adjusted as far as possible to secure identity with the mean longitude of the actual Sun affected by aberration. This may be written in the form 7" /-\ I \ + l \ +2\ Z/. I / P/ I f>'tZ\ Li 1 A,n ~T Art* T A. 2 C ) hi -f- ( JLt/ -\- JL v I 270, 271 ] Precession, Nutation and Time 300 where A is the true mean longitude of the Sun when t = 0, X x is the sidereal mean motion, and 2X 2 is the secular acceleration which arises indirectly from the perturbations of the other elements of the Earth's orbit ; k = 20"'47 is the constant of aberration; and (Pt + P't 2 ) is the general precession in longitude. The adjustment of the constants in A and L gives and leaves outstanding between L and A the secular discrepancy (L 2 ?u 2 ) t 2 which would lead ultimately to a departure of the actual Sun, apart from periodic effects, from the meridian at mean noon. This quantity is small and far from certain in amount, and will have no practical effect for many centuries to come. Now at 1850 Jan. 0, Greenwich mean noon, T=t=0, r =A = L and the effect of adding one mean day to T or t is 24 h = 360 - (w - /*)/365'25 whence 6)/365'25 = 24 h + (L, - w^/ 365-25 (a) + wij)/ 365-25 = 24 h + A/ 365-25. Now, according to Newcomb, L, = 279 47' 58"-2 = 18 h 39 m 11 8 '88 L,= 1296027"-6674 = 8640P-84449 L 2 = + 0"'000 1089 = + 8 -000 00726 while in the latter unit (1 s = 15") mi = +3 s -07!41, wi a = + 8 -000 00928 so that A/365-25 = 236 8 -55533, (L, - 7?i 1 )/365'25 = 236 8 '54692. Hence in numbers the equation (12) for Gr. sidereal time becomes T = 18 h 39 m 11 8 '88 + (24 h 3 ra 56 8 '55533) D + 8 '000 00928 1 2 - $ cos e where D = 365'25 t is the number of days reckoned from 1850 Jan. 0. When D is given an integral value this expression gives the sidereal time at Gr. mean noon and its value (less a multiple of 24 h ) is tabulated for every day in the Nautical Almanac. When the nutational term is omitted, AT = (24 h 3 m 56 8 '55533 + 8 '000 00005 1) AD. The secular term is also negligible, and hence 1 mean day _ 86636-55_5 _ 1 sidereal day ~ 86400 s 310 Precession, Nutation and Time [CH. xxn Another period which differs little from the sidereal day, but must not be confounded with it, is the period of the Earth's rotation on its axis, measured by o>. Its ratio to the mean sidereal day is &> + m, 86636-555 = HTvqr-Kk'7 = 1 ' 000 00 97 - &> 86636547 272. A catalogue of astronomical positions gives mean places freed from nutation and reduced to the equinox of a common epoch. Such an epoch is always the beginning of a tropical year and this expression must be defined. It is the moment when the mean longitude of the Sun as above described, L = L + Lj + L.tf is 280 = 18 h 40 m . It follows that the length of a tropical year is 24 h f -ETf~. 365'25 mean days A + zLJ J \ 365-25 1-000 021 3483 + O'OOO 000 000 168 1 = 365-242 20272 - O'OOO 000 0614 1 or 365-242200 mean solar days at the epoch 1900. For the present the secular change is unimportant. Once the beginning of the tropical year is fixed in a particular calendar year, its beginning in any other year may be found by adding so many tropical years. But the details will be better illustrated by a direct example from the year 1900. When = 50, L'= 18 h 40 m 44"123. Now 50 Julian years exceed 50 years of 365 days by 12| days, whereas the calendar inserts 12 leap days between 1850 and 1900. Hence this is the mean longitude for 1900 Jan. 0'5. The mean longitude for 1900 Jan. (Gr. mean noon) is therefore L' - IA/365'25 = 18 h 38 ra 45 S> 845 and must be increased by 74 8 '155 at the daily rate 236 8> 555 in order to become 18 h 40 m . This requires 0'3135 mean days, and the beginning of the tropical year in 1900 is therefore Jan. 0'3135, the fraction of a mean day being reckoned from Greenwich mean noon. This epoch is recorded briefly as 1900*0. It is to the mean equinox of this date that the observations of the year are reduced in the first instance. 273. Such in outline are the main features in the astronomical methods of reckoning time. They involve certain constants which, being based on the comparison of theory with observations, are capable of improvement. But there is no absolute standard of time. Ultimately no doubt the con- tinued comparison of theory with observation according to such a system of time as that described above will bring to light discrepancies in the motions of the heavenly bodies of a kind which cannot be attributed to errors of 271-273] Precession, Nutation and Time 311 observation. Then the question will arise whether these discrepancies can be removed by a mere adjustment of an accepted system of constants in- volved in the measure of time or whether the fault lies in the theory. This is the ordinary experience of practical astronomy. It may, however, prove that what have been regarded as constants are not really constant at all. Thus 6, its square may be neglected. The relations between the coordinates are then given by the scheme : X Y Z X COS (( + \|r) sin ((j) -f ty) COS *fr y sin ( + -v/r) cos ($ + >/r) 8 sin i/r z 9 cos (f> 6 sin < 1 and Euler's geometrical equations become &>! = 6 sin \Jr cos ir o> 2 = cos r + (> sn 314 Libration of the Moon [CH. xxm * The dynamical equations are again of the form A6*! (BC) o) 2 o) 3 = L Ba> 2 - (C - A) 0)3^ = M Cw, - (A - B) Wl a),= N where ( 257) l = 3Gm(C-B)yz/r*, M= 3Gm (A - C) xz\r\ N= 3Gm (B - A} xy/r* m being the mass of the Earth, (oc, y, z) its coordinates and r its distance from the Moon. Let (X, Y, Z) be the ecliptic coordinates of the Earth relative to the Moon. The inclination of the Moon's orbit, c = 5 9', is so small that c 2 will be neglected. Then (cf. 65) - X = r cos (H + + w), - Y = r sin (II + ) the longitude of the Moon's perigee, and w the Moon's true anomaly. But X= fl + w + w is the longitude of the Moon in its orbit. Hence, by the above relations between the two sets of coordinates, x = r cos (X, < -v/r), y = r sin (X $ ^r) z=r6 cos (A. ) + re sin (A, O) the product c6 being neglected in x and y. Let C-B = Aa, A-C=*B/3, B - A = Cy. Then the dynamical equations of motion become &>! + a2<3 = 36rwar- 3 sin (X T/T) {# cos (A, 0) + c sin (A, fl)h 3 sin 2 (X < i/r) As the figure of the Moon is to all appearance sensibly spherical, a, /3 and 7 must be fairly small quantities. And since, further, the instantaneous axis is nearly fixed in the body and very close to the axis of z, o^ and o> 2 must be very small in comparison with &> 3 . 277. It follows that in the last equation the term yw 1 A. Now, if the inclination "of the lunar equator to the lunar orbit be neglected, (0 + -^) measures the displacement of the axis of x from the equinox from which the longitudes are reckoned. Under these simplified conditions the first meridian contains the axis of x and always coincides with the central meridian of the apparent disc. The axis of x is therefore directed approximately towards the Earth and this defines the axis about which the moment A is less than the moment B. This is the first condition of stability. It is also to be inferred that A ^ B. For if A = B, % = and a small disturbance would introduce a secular term in ^ which observation shows to be absent. 278. If 7' = 7(1 +f)~ l the more general equation (4) for ^ becomes % = f ?i'V (a'/r) 3 sin 2 (^ + X n't e). Now (A, n't e) is of the order of the eccentricity of the lunar orbit (055). x is still smaller and a /r differs from 1 also by a quantity of the 316 Libration of the Moon [OH. xxm order of the eccentricity. Hence if the square of the eccentricity be neglected, X = - 3n'y (x + \ - n't - e) or X + 3wV% = - 3n'V 2# sin (ht + hf) where the terms under S represent the equation of the centre and periodic inequalities of the lunar motion. This is the ordinary equation for forced vibrations and the solution may be written in the form x X* + X* wnere %i is a particular solution, corresponding to the forced vibrations, and ^ 2 is the complementary function, corresponding to an arbitrary free vibration. It is easily verified that and Xz = K sin [n't >J(3y) + k'] where K, k' are arbitrary. Terms in Xi can on ty become sensible by reason of H large or h small, and the most promising terms in the lunar theory are consequently the equation of the centre (or principal elliptic term) : ht +h'=g lt H= + 22639"-!, h = 47033"'97 and the annual equation : ht + h' = , H=- 668"-9, h = 3548"-! 6 where g l is the Moon's mean anomaly, is the Sun's mean anomaly, and the unit of time is the mean solar day, so that n' 47435" - 03. The corre- sponding terms in x\ are 377' 11'15 * = 6=3277^7 ' 7 Sm * ~ 6=661865^7 ' 7 8m G It is easily seen that, 7' being certainly very small, it is the second of these terms which is the larger. But the determination of its coefficient from observation has not yet been made with satisfactory certainty. Since the Earth's distance is about 220 times the Moon's radius a geocentric angle of 1" is the equivalent of 4' in selenographic arc near the centre of the lunar disc. As the quantities to be looked for are likely to be of this order, or rather still less, and the observations are very difficult, positive results must be awaited from the study of the large-scale photographs of the Moon which are now available. According to Franz, using the heliometer observations of Schliiter, the coefficient of sin is about 2', giving 7 of the order 0'0003, and the arbitrary libration K, which should have a period of rather more than 2 years, is practically negligible. 279. Since, by (3), &> 3 -I- x = n ' where x ma y now be supposed very small, the first two dynamical equations may be written =L/A\ 278, 279] Libration of the Moon 317 Now let = 6 cos -fy, i) = 6 sin -v/r so that % = cos ty + $0 sin -fy ( + i^) sin \|r = to 2 ta^l r ......... (9) 77 = 6 sin i|r $6 cos ^ + ($ + ty) 6 cos i|r = &>j + 3 may be replaced by n', being multiplied by and 77 which are small. Hence (8) become 77 - (1 - a) w'i + cm' 2 ?? = X / A Expressions for L/A, MjB have been given in (1), and if /= 1/81 be neglected in (6) these are L! A = Saw' 2 (a'/r) 3 sin (\-~ -f) [d cos (X - ) + c sin (X - II)} . / = 3/3w' 2 (a'/r) 2 cos (\ - - -f ) (0 cos (X - 0) 4- c sin (X - H)} and as they are already of the order 6 or c multiplied by a or ft, the other quantities involved are only required to the first order in e, the eccentricity of the orbit. Now g l being the mean anomaly, by Ch. IV (9) and (30) or in a more simple way a IT = 1 + ecosf/j, w g^ = 2esin^! where g 1 = n't + &-, w = \ CT w being the true anomaly and 57 the longitude of perigee. Also ^ is in- significant here, so that by (3) (f> + \lr = 7r + n't-}-e = g l + 'GT + 7r ..................... (10) Hence X ty = lu g l TT = 2e sin g l TT sin (X >/r) = 2e sin g l , cos (X (f> -^) = 1 (a'/r) 3 cos (X - < -v|r) = 1 3e cos ^ J Again, cos (X () = cos (i/r + 2e sin ^rj = cos -fr + 2e sin ^ sin ty 6 cos(X <^>) = 6 cos i/r + e0cos(g l ^) e6cos(g l + i/r) ...(12) and finally X 1 = w + & O = ^ 1 + / nr fl + 2e sin In this way any periodic terms on the right of the equations can be represented by corresponding terms in | and 77. But the coefficients Q, Q' involve P, P' multiplied by the small quantities a or fi, and are therefore extremely small unless A is also very small. Now A=j9 2 (p 2 1) when a and y3 are ignored and therefore, ceteris paribus, sensible terms can be obtained only when p is very near to or +1. Solutions of the same form constitute the complementary function and are determined by (17) when P = P' = 0. Then p is given by A = p* -p (1 - 3y8 - a/3) - 4a/3 = or 2p* = 1 - 3/3 - a/3 + V{(1 - 3/8 - a/3) 2 + 16a/3} It is enough to retain in p the terms of the first order in a, /9, and thus 2p 2 = 1 - 3/3- a/3 (1 - 3/3 - a/3 + 8a/3) so that if ^1, p 2 are the two roots, 279-281] Libmtion of the Moon Thus the periods of the two possible terms are determined with sufficient accuracy, the former being nearly a month, and if the corresponding co- efficients are Q l} Q/, Q 2 , Q 2 ', then by (16) to the lowest order only Hence a solution of (15) when is substituted on the right-hand side is & = Q l sin {(1 - f /3) n't + q,} + Q 2 sin {2 V(- ) t + q 2 } % = - Q l cos {(1 - f ) n'J + ^} + 2 V(- /) & cos (2 V(- and as these expressions contain four arbitrary constants Q 1} Q.,, q 1} q 2 they represent the required complementary functions. These arbitrary terms again appear to be insensible. The important point is that a/8 mast be negative, for otherwise the circular functions would be changed into hyperbolic functions and the motion would be unstable. This means that (C B) (A C) is negative, or again that C is not inter- mediate in magnitude between A and B. This is the second condition of stability which has been found. 281. To terms of the first order only, L/A=Q, M/B = -3firi 2 csin(g 1 + '&-n) where, the secular inequality of the node being taken into account, g 1 + *r = n't + 6, ft = ft - fin't, /* = + O004019. Thus in applying (17), P' = 0, P = c, p = 1 + //,, and therefore (T^-. - (i -.MI +,.) (i + gy^+^+ff^s -< 18 > If a, /3 and p, be regarded as small quantities of the first order and those of the second order be neglected, (19) so that and 77 contain the terms 3/9c 38c These terms contain the explanation of the steady motion of the Moon's axis, which is expressed by Cassini's laws. For the coordinates of the Moon's pole of rotation relative to the pole of the ecliptic may be taken as X = Q cos = | cos ((f> + ^) + V) sin (< -f -v|r) Y= 6 sin (f> j; sin (0 + i/r) 77 cos ( + -\^). Libration of the Moon [CH. xxm Let the free components f a , rj 1 be ignored and also the forced oscillations of the second order which have still to be found. Then X = Q sin (g l + ts - fl - < - ^) 7 = Q cos (g l + OT - H - - i/r). But by (10) (f> + -*lr = y l + '&+Tr and therefore Z = Qsinn, F=-Qcosfl But the longitude of the pole of the lunar orbit is H - ITT, so that its coordinates are similarly Z' = csinfl, F' = -ccosft. Hence these two poles are always exactly on opposite sides of the pole of the ecliptic provided Q is negative. This requires, since Q is given by (19), > /3 > I/A. Hence > A, which is a third condition to be satisfied by the moments of inertia. The resultant of the three places the moments in the order C>B>A where C refers to the axis of rotation and A to that axis which in the mean is directed towards the Earth. It is now clear that the further conditions necessary in order that the second and third laws of Cassini shall remain approximately true are one and the same, namely that those terms which have been neglected in the above argument are really small in comparison with Q. This quantity is the mean value of 6, and its numerical value is 91 ''4 according to Franz. With c = 308'-7 and /* = 0'004019 it follows that -I3 = (C-A)/B = 0-000612 which should be tolerably well determined. It is to be noticed that a, /3, 7 are not independent, but connected by the identity a + + 7 + a/37 = 0. The product is negligible and if 7 = O'OOOS as given above, then a is of exactly the same order as 7. 282. The terms of the second order in e, c, can now be found without difficulty, since here it is legitimate to give 6 and -v/r their values in the steady motion. Thus = # , its constant mean value, and since in the steady motion = H 4- ^TT, ty = g 1 + & O + |TT. Hence without the terms of lower order already treated, the expressions (14) become L/A= 3 3 = n here and neglecting the arbitrary terms 1} %, the existence of which has not been established by observation, and (yu, + v) is relatively unimportant here. One remark is necessary however. For the sake of simplicity and in order to concentrate attention on the main feature of the motion, the coefficients of 2 and ?; 2 in (20) were made numerically equal by the simple expedient of neglecting /u. 2 (= 0'000016) in comparison with fi. Consistently with this p. p. A. 21 322 Libration of the Moon [OH. xxni the factor (1 + /&) has been omitted in finding | 2 > fy, and the result is that >> % do not appear in a> 1; 2 /w' = (! + /*) (2/i 4- /i 2 ) 6 s cos^r. The factor a shows that Ae^ is very small and if yf as well as a be now rejected, Hence in a numerical form the forced rotations are finally given by mjn' = - , = - r-6 sin (w - II) . a> 2 fn' = i) 3 2/j,r) 2 = 1''6 cos (-57 O) 0'*7 cos (g l + CT O) since G = - 91H and /* = 0*004. With the more exact expressions the coefficient in , is numerically greater than that in i] 2 , the difference being //,(!+//, + a) (7 or pG. This amount, 22", may be divided equally between the two coefficients without disturbing the observed mean inclination of the lunar equator to the lunar orbit, and thus 2=- 91''6 sin (g 1 + & - n), 772 = 91''2 cos (g l + o - H). Lastly, by (7), if ^ 2 the free libration in longitude be ignored, / , Oil 0-000242 >= -W- - 073-3 > 7 COB ^ + Q . OQ1865 _ y/ . 7 cos where the coefficients are expressed in circular measure. Thus the position of the instantaneous axis, relative to the principal axes of the Moon, x\ w l = yj 6> 2 = zj o) 3 is determined. It has therefore been seen under what conditions Cassini's laws are approximately true, and how far they must necessarily be modified by disturbing actions. The latest results from observation, by M. Puiseux of Paris, seem to be at variance with the foregoing theory. It is probable that it will be necessary to treat the Moon as a deformable body, as the observed variations of latitude have shown to be requisite in the case of the Earth. The above theory is very largely due to Poisson. CHAPTER XXIV FORMULAE OF NUMERICAL CALCULATION 284. If we consider a function of one variable or argument only, for the sake of definiteness, it can be represented in three distinct ways, namely : (1) By an analytical form, e.g. sin a; or a hypergeometric series F (a, /3, 7, as). The effectiveness of such a form depends on the knowledge of its properties and the facility with which it submits to the ordinary operations of mathe- matics. (2) Graphically, by a curve. This gives a continuous representation. Values of the function corresponding to particular values of the argument can be obtained and the processes of differentiation and integration can be performed mechanically. But the accuracy of the results is limited in practice. (3) Numerically, by a series of isolated values. This gives a discon- tinuous representation, but one capable of very great accuracy. In theory this does not serve to define the function, for it may vary in any manner between the given values. Even in practice the representation does not cover terms in the function with a period of the same order as the intervals between the values. But with due care this limitation causes little in- convenience. Each mode of representation has distinct advantages of its own and to pass from one to another is a problem frequently arising and often attended by great difficulty. The form (1) may be considered the ultimate expression of natural truth, but it has no absolute superiority. Thus integration may be practically impossible in this form and must be replaced by a mechanical quadrature. A function determined by a series of observations or experiments falls generally under the form (3). Now the variable quantities which occur in Astronomy, e.g. the coordinates of the Moon, are in general so complicated, even when an expression in analytical form is available, that for practical purposes it is necessary to use an ephemeris, or a table of values calculated for equal intervals of time (not necessarily one day, as the name would imply). It is therefore necessary to consider how functions represented in 212 324 Formulae of Numerical Calculation [OH. xxiv this way may be manipulated so as to give intermediate values by inter- polation for comparison with the results of observation, and also to render numerical differentiation and integration possible. 285. Let w be the constant interval of the argument and 2/n= = /( + nw) be the function to be considered, the values of y n being given for consecutive integral values of n. A simple difference table can be formed thus : a + (n l)w a + nw a + (n + \)w Now let two operators A, 8 be introduced such that Then it follows that ASy n = A (y n - y n ^) = y n+l - 2y n + y n -i = S (y n +i - y) = Hence the operators A, 8 are commutative, and similarly it is easily seen that they obey all the laws of ordinary algebra. The inverse operators A- 1 , 8~ l may be defined so that A A" 1 = 1, 8B~ l = 1. Then the table of differences may be replaced by a table of operations which, acting on y n , will reproduce the difference table, thus : L^ G O 8 1 AS A AS- 1 A 2 The two operators are not independent, for the position of AS in this table shows that they are connected by the homographic relation Let x be the variable, so that y =/(#), and let D = djdx. Then or 1 + A = e wD . Hence Thus = (1 + wD + i 2 w n -D 2 + . = e lvD ,f(x} .......................................... (2) =f(x) + qwf (x} + Iftff" (x) + . . . = f(x + qw). 284-286] Formulae of Numerical Calculation 325 which is Newton's original formula of interpolation and can be written in the form : ( /~\ \ (3) where j q by a proper choice of n may always be taken < ^, and in any. case should not exceed 1. The coefficients are simple binomial coefficients. 286. The differences A, A 2 ,... are diagonal differences in the table. But the most useful formulae involve central differences, lying on or adjacent to a horizontal line in the table. If the blank spaces in the odd columns are filled by the arithmetic means of the entries immediately above and below, the operators in the complete central line are 1 KA + S) AS (A + S)AS (AS) 2 ... which can also be written, by introducing two new operators K, k, I k K kK K* where .(4) Thus k cannot be expressed rationally in terms of K, and in order to find a formula in terms of central differences it is necessary to expand in terms of K, keeping only the first power of k. Thus ku q + v q ................ -.....(5) where u q = v = It is easily verified that u q (1 + PO + V Q = u q+1 , u q (K + K*) -f v q (1 + %K) = v q+l since *" vr-1 Also - 2 i < - 2r) . + (r + l) (1 + i^)'' (K + = 2 J, !(-/) + (|.;\)} (1 + - (K 326 Formulae of Numerical Calculation [OH. xxiv It is therefore possible to write v q = 1 + q2brK r , u q = q22(r+l) b r+1 K r . Let b r become b r f mv q+1 , u q+1 , and equate the coefficients of K r ~ l in the first, and of K r in the second, recurrence formula. Thus 2rb r ' = 2rb r + (r - 1) &.,_> + qb^ (q + l) b r ' = 2rb r + | (r - 1) b^ + qb r + $qb r ^ and, on eliminating &/, 2r (2r - 1) b r = (q + r - 1) (q - r + 1) &,_,. This shows that where JL is a constant, and since b^ \q, A = 1. Hence -)^ ......... (6 > and the first terms of the complete formula are therefore ... y . ...... (7) This series was found by Newton, but is generally known as Stirling's formula. It is here taken as fundamental, and other results are deduced from it. 287. The formula of Gauss depends on the even central differences and the odd differences of the line below, the operators being therefore 1 K K 2 A These are, in terms of k, K, But (5) may be written in the form (l + ^ = (k + ^)u q where by (6) r, = ,, - $KU, - 1 + s _ 280-288] Formulae of Numerical Calculation 3*27 This gives the coefficients of the even central differences, the coefficients of the odd differences of the adjacent line being still given by u q . The first terms of the complete formula are therefore O ! If the order of the difference table were reversed, 8 would take the place of A and the sign of w would be changed. Hence similarly i , ...... (10) By choosing either (9) or (10) q can always be taken between and + . 288. The formula of Bessel contains the odd differences in the line immediately below the central function, with the mean even differences of the same line, so that the operators are 1+A, A, (1+iA)^, &K, (1 + A)# 2 , .... The odd differences are thus the same as in the formula of Gauss, and therefore = Aw + F = l where, by (6) and (8), - l (ll) This gives the coefficients of the odd differences, and the coefficients of the even (mean) differences are given by F 9 . Hence the first terms of the complete formula are ^ * yn+q Bessel's own form differs from this in the first two terms, being written 328 Formulae of Numerical Calculation [CH. xxiv which is of course equivalent, but is not symmetrical with respect to the middle of the tabular interval. To make this symmetry clearer, let p + ^ be substituted for q in (12), which then becomes + |A) +p . A + tp* . (1 + JA) K +p . P ' _. When the sign of p is reversed, the terms of even order are unchanged and the terms of odd order are simply reversed in sign. If terms of the two orders are computed separately, two interpolations corresponding to I p are obtained at the same time. This is of great advantage in systematic interpolation to regular fractions of the tabular interval, e.g. in reducing the 12-hourly places of the Moon to an hourly ephemeris. Stirling's formula presents a similar advantage. But (13) becomes particularly simple at the middle of an interval, for then q = % or p = 0, and the odd differences dis- appear. Thus -n '+...}? ......... (14) and. this gives intermediate values with great ease and accuracy. 289. When the values of a function y are known only at irregular intervals of the argument x, as in an ordinary series of observations, the function is strictly indeterminate in the absence of other information as to its form. Nevertheless, when n values y lt ...,y n are known, corresponding to as lt ..., x n , a formula y = a +a 1 a;+ ... + a_] x n ~ l can be found which is satisfied by the n values and within the interval a?i to x n will generally resemble the true function closely. The n coefficients can be determined by the linear equations (r = 1, . . . , n). These can be solved in the ordinary way, but it is immediately obvious that the result can be written = \ . . . n (X r Xi) . . . (X r X n ) where the numerator of the fraction written- does not contain (x - #,.). For this equation becomes an identity when x r> y r are substituted for x, y. The expression on the right is a polynomial of degree n 1 in x and the equation, since it is satisfied by every pair (x r , y r ), must be identical with the previous equation, the coefficients in which can be written down by comparison. The formula (15) is due to Lagrange and is directly suitable for interpolation, 288-290] Formulae of Numerical Calculation 329 differentiation and integration. An illustration of its use in a case where n = 3 has been given in 71. When n is large the formula naturally be- comes inconvenient for practical purposes. 290. Returning to the function with known values at regular intervals of the argument, let us consider the process of mechanical differentiation. By (2) wD = log(l + A) = A-iA 2 + iA 3 -... I w 2 Z> 2 = {log(l + A)} 2 = A 2 - A 8 + 1JA 4 -.../' These formulae are suitable only in simple cases where great accuracy is not required. The loss of accuracy is a natural tendency when differentiation is concerned. The forms (16) also apply only to the tabulated value of the argument. But since x = a + (n 4- q) w, wD = wdfdx = d/dq a formula of differentiation can be derived from every formula of interpolation. Thus Bessel's formula (12) gives '- and analogous forms may be derived similarly by differentiating (7) and (9) with respect to q. But there are some particular cases of special simplicity and importance in the formulae of central differences. According to (6) u q is an odd function and v q an even function of q. Now when 9 = 0, d/dq is the coefficient of q and d z /dq 2 is twice the coefficient of < in ku q + v q . These coefficients can easily be taken from ku q and v q respectively, and give, by (6) or (7), 7! ...... ( 18 ) and 21 (19) Both (18) and (19) involve the alternate differences in the central tabular line. Similarly when V q , U q are expressed in terms of p = q + | instead of q as in (8) and (11), V q is an even function and U q is an odd function of p. W T hen q = i,p = and d/dq is the coefficient of p and d*/dq* is twice the 330 Formulae of Numerical Calculation [on. xxiv coefficient of p- in (1 + -1A) V q + A U q . These coefficients can readily be taken from (13), which sufficiently indicates the law of formation, and thus and - (3 2 . 5 2 . 7 2 + I 2 . 5 2 . 7 2 + I 2 . 3 2 . 7 2 + 1 2 . 3 2 . 5 2 ) The distinction between the operators (1 + A) 4 and (1 + A) must be carefully noted. That on the left, (1 + A)", indicates an addition of half the tabular interval to the argument, so as to apply the differentiation at the right point, which is the middle of the interval. That on the right, (1 + ^A), merely denotes the mean of adjacent differences in a vertical column of the difference table. 291. Convenient methods for mechanical integration or quadrature can now be deduced. The formulae for differentiation just found, (18), (19), (20), (21), are of the form wD = kS l (K), w*D* = S 2 (K) wD (1 + A) 4 = AS 3 (K), w*D 2 (1 + A) 4 = (1 + \ A) S, (K) S (K) denoting a power series in K. Hence w- 1 D- 1 = k-^/S, (K), w-* D- 2 = 1 fS a (K) The coefficients of the reciprocals of the K series must be expressed more appropriately, thus : = kite = k(K+ + (1 + A) (1 + ^A)- 1 = (1 + A) {1 + |A 2 (1 + A)- 1 }- 1 = (1 + A) (1 + J It is therefore necessary to multiply -8-^ and $ 4 by (1 + \K) before finding the reciprocals of the series by division in order to have results for D~ l , Z)~ 2 of 290, 291 ] Formulae- of Numerical Calculation 331 exactly the same form as those already found for D, D-. These results are easily found to be (22) (23) ) (24) : +...).. .(25) The development is here carried as far as differences of the fifth order. This is generally sufficient. It is now necessary to examine the meaning of these purely formal results. The operator K, like its components A, 8, is such that KK~ l = 1, and therefore, as K represents a move two places to the right in the table, K~ l represents a move two places to the left. The difference table now requires an extension not hitherto contemplated, and the central line of the table of operators, with the adjacent lines above and below, now becomes : S SK SK* ... [k] K [kK] K* [kK*} ... A ra Here 1 corresponds to the original entry y n in the table. The natural differences as directly formed are expressed simply, while those which are means of the entries immediately above and below are enclosed by [ ]. But while the symbols occurring in the columns to the right of the central column (representing the function itself) will be readily understood, the construction of the columns to the left must now be explained. The numbers in the first column to the left are such that their differences appear in the central column. Thus - Mr-') y n = y n> A*"-' y n = y n + SK~> y n and when one number in this column is fixed, the rest are formed by adding successively (when proceeding downwards) the tabulated values of the function. The entries in this column therefore contain an additive arbitrary constant. The second column to the left is related to this first column in exactly the same way as the first column to the central column, and therefore contains another arbitrary constant, but is otherwise definite. The use of four different operators in the table may seem excessive, since they are all expressible in terms of one. In fact and this suggests another mode of development which has here been de- liberately avoided. But all these operators have simple special meanings 332 Formulae of Numerical Calculation [CH. xxiv and it is important to notice that k8~* and (1 + |A) are equivalent, but quite distinct from AAr 1 , though in the complete table, in which the mean differ- ences are filled in, they all three denote one vertical step downwards. 292. As with A" 1 and the other operators, D" 1 is such that DD~ l = 1, or D, D~ l represent inverse operations. And since D represents differentiation, Dr* represents integration. Thus take the formula (24). The column AAr 1 being formed with an arbitrary constant, the right-hand side of the equation, operating on y n , will produce a function (represented in tabular form) which is w~ l D~* (1 + A)* y n = w~ l D~ l y n+ <- On the application of D or differentia- tion, this becomes w~ 1 y n+ ,. Hence the meaning of the formula is where m is written for n + . The lower limit is arbitrary. But the right- hand side also contains an arbitrary constant, and this constant can now be chosen so as to fix the lower limit of integration. For let this limit be a + ^w. If then ra = , w = in (26) O^Atf-' + ^A-^Atf + Tn/^Aff--...)^ ...... (27) and the value of A/iT" 1 . y is now determined. With it the whole of the corresponding column can be definitely calculated by successive additions of the values of the function. When this is done, (26) represents the definite integral of y between the limits a + \w and a + (n + ^) w. Quite similarly the meaning of (22) is seen to be ra+nw ydx = (kK-* -&k + 7-VV kK - itffa kK' + ...)y n .. .(28) where the lower limit is a when But the latter form is not convenient, because kK~ l y , which is hereby deter- mined, is the mean of two numbers not yet known. Now 2/0 = Atf- 1 y, + 8K-> y , y, - AIT" y, - SK~* y and therefore dfcskK*-...)y t ......... (29) Thus A-K"" 1 . y is determined, and the calculation proceeds as in the previous case. It is to be noticed that, though (27) has been derived from (26) and (29) from (28), (26) can be used in conjunction with (29), giving a and a + (n + ) w as the limits of integration, or (28) with (27), giving a + nw as the upper limit and a + ^w as the lower limit. 291-294] Formulae of Numerical Calculation 333 293. In a similar way (23) and (25) give the second integrals, thus ra+nw r rx K*-...}y n ...... (30) r J b F r [J ............ (31) where m = n+ \ as before. The lower limit c of the subject of the second integration is arbitrary. But if the first summation column, on the left of the function y, has been based on (29), c = a ; if it has been based on (27), c = a + ^w. The lower limit b of the second integration is also arbitrary and corresponds with the additional arbitrary constant in the second summation column K~\ The latter is easily determined by taking the case b = a, n = of (30). Thus = (^ + T V-^o^ + ^V8o^ 2 --.-)2/o ............... (32) This gives K~ l y , and the whole of the second summation column becomes determinate when the first column has been fixed. Or again, if the lower limit b is to be a + \w, (31) gives when b = a + ^w, m = %, n = 0, or This is quite general whatever the value of c, or of A.fif" 1 y , may be. But as c b usually, (27) can be used in this case, and then K- 1 2/o = (A (1+ A) - jti (3 + 2A) K + jrfMn (5 + 3 A) K* - . . .} y . . .(34) When the second summation column is based on (34) and the first on (27) x = a + \w is the common lower limit for the double integration. When (29) and (32) are used in forming these columns, x = a is the common lower limit. In either case (30) and (31) give the values of the double integrals to the upper limits x = a + nw and x = a + (n + -|) w respectively. No attention has been given here to the limitations of the method which are imposed by the conditions of convergence of the expansions employed. In general the question is settled in practice by obvious considerations. But for a critical estimate of the accuracy attainable it is clearly important. 294. There is also a trigonometrical form of interpolation, otherwise known as harmonic analysis, which is of great importance. This is intimately related to Fourier's series, and indeed amounts to the calculation of the coefficients of this expansion. It will be well to recall the principal pro- perties of the series, which may be stated thus : The sum of the infinite series cr. + 2 (a n cos nx + b n sin nx) 334 Formulae of Xnuierical Calculation [CH. xxiy (n a positive integer), where 1 /" 2ir 1 i" 2ir 1 f 2jr a = o~ I y (^) ^j a n = f( x ) cos w # ^> b n = I f(x) sin ?w; e&c An j Q TJO 7! "^o \&f(x) throughout the interval i cos id r + bi sin i0 r ) 294-296] Formulae of Numerical Calculation 335 the coefficients a , a;, 6; can be found in the ordinary way. It is also easy to represent the result .by a formula analogous to Lagrange's formula of inter- polation (15). But when 6 r = 2r?r/(2n + 1) the solution can be effected in a very simple way. It is necessary to consider the sums' of two very simple series. In the first place s-l s-l 2 sin ra = 2 {cos (r |) a cos (r + ) a} /2 sin ^a r=Q = {cos^a cos(s i)a}/2 sin^a = sin ^sa sin |(s 1) a/sin ^a and this is if a = 2pfr/s. Even when p =p's, p and p' being both integers, and therefore sin | a = 0, this remains true, for every term of the series is then zero. Similarly s-l *-l 2 cos ra = 2 {sin (r + ^)a sin (r -) } / 2 sin \ a r=0 = {sin(s ^)a + sin a}/2 sin^a = sin^sacos(s 1) a/sin a and this is also if a = 2pTr/s, unless p =p's. In the latter case each term of the series is 1 and the sum is s. Thus both the series vanish for a = 2pTr/s, except the cosine series when a = Zp'jr. 296. Let u = u r be the value of the function corresponding to the value of the argument 6 = ra.. The series will not now be limited to a finite number of terms. Then s-l S u r cosjra = 2 cosjra + S 2 (a, ; cos jra. cos ira. + hi cosjra sin ira) r=0 r i r = a 2 cosjra + % 2 2 ; {cos (i +j) ra + cos (i j) ra} r=0 u r sin jra = 2 sin jra + 2 2 (a t sin jra cos ira + 6 t - sin jra. sin ira) i r {cos (i j) ra cos (i +j) ra} when a = ZTT/S, for all the sine terms vanish immediately in the sum with respect to r. The cosine terms also vanish in the sum unless j, i+j or i j is a multiple of s (including zero). Thus, j having in succession all values from 1 to (s 1), or .(35) 336 Formulae of Numerical Calculation [CH. XXIY When s equidistant values, u , ..., iig^, (u 8 = ' u o)> ai * e known the operations indicated on the left are easily performed. Then, if the series converges so rapidly that the higher coefficients can be neglected, a, n , a lt b l , ... are deter- mined, as far as aj (,_!>, 6j( g _D if s is odd, and as far as a^ s , 6 ig _ x if s is even. The lower coefficients will naturally be calculated much more accurately than the higher, for there is little reason to suppose a^ s+l small in comparison with a^g-i. But it is well to compute the higher coefficients as a practical test of convergence. 297. It is usually convenient to make s an even number, and indeed a multiple of 4, so as to divide the quadrants symmetrically. Let s = 2n and let the terms of higher order than a n , 6 M _, be neglected. Then (35) become 1 2M ~ 1 1 jr-rr 7 1 v . jr-jr = z, u r . a-, = - 2,u r cos^ , ft,- = - zw r sm- (do) 2w r=0 ' n " n ' n n (j = l, 2, ...,n-l). When j = n, so that a n is determined, but not b n \ and this is natural, for 2n coefficients i addfbion to a cannot be derived from 2n values u r . Let nj be written for j in (36). Then 1 n ^, 1 ( ?VTT\ 1 ^, , ir?r a n _; = - 2, w r cos I rtr } = - 2, ( LY u r cos- n ,. =0 \ w / | w 7 1 < / J r7r \ ! v / J r7r &_, = - S w r sm r?r *- - = z ( 1 )' u r sm - . n \ n ) n n Hence ence 1 f 2iw 2j(n- $ (aj + _)) = - s MO + "2 cos -*- + . . . + u.^_ 2 cos - n ) 1 ( 2?7T 4?7T - n-z) COS -^ - + (g + ?< 2n _ 4 ) COS - - n ( w n I ( JTT 3J7T (2w-l)i7Tl I (tt; tt n _j) ={, COS^ h W 3 COS -+... + M 2n _i COS n ( n n n J 1 f, J7T 3J7T ) = - S(MI + Wan-i) COS 1 ^ h (U 3 + l^jn-s) COS - + ... X n { n n ) I f . in- . 3J7T . (2w-l)jV ^ (o; + 071-;) = - -Ni sml ^ I- u s sm - + ... + w^i sm - ?i ( n n n x . -s) sm ^ / t . . sm - - + M 4 sm - + . . . 4- w 2 - 2 sm - It [ 'ft/ it if/ 1 (, 2/7T 4iV ) - I (Ms - ^2n-2) Sin -A- + ('M 4 ~ U-zn-4) BUI -*j- + ... 296-298] Formulae of Numerical Calculation (j = l, 2, ..., n 1); and 337 n = - "o 71 + U 3 By this arrangement a n _/, &_/ are calculated together with o^, 6; with scarcely more trouble than a/, fy alone. As a practical check on the convergence of the series these higher harmonics should be found. 298. The arrangement can be greatly simplified in special cases. For example, in the case s = 12, n = 6, let the data be arranged thus : Sums : v v^ v 2 v 3 v 4 v- a v Differences : w l w 2 w 3 w 4 w & V V-i V 2 V 3 V 6 V 5 V 4 W l W 2 W s Sums : p PI p 2 PS Differences : q q l q 2 r-y (* 1 '2 '3 a ft Oi Og The equations for the coefficients are (a/ + a 6 _j) = %(v + v 2 cos \jir + v t cos |JTT + v 6 cos JTT) (a, a 6 _/) = ^ (v, cos |JTT + V 3 cos ^-JTT + v s cos f JTT) ^ (6/ + 6 6 _j) = ^ (w! sin ^jir + w 3 sin ^;V + w 6 sin IJI'TT) ^ (bj b 6 -j) = ^ (w 2 sin ^jir + w 4 sin UTT). Hence two cases, according as j is even or odd : j even j odd : (#0 + - ^-j) = i s 2 sin ^'TT and these forms can easily be made more general. p. D. A. 22 338 Formulae of Numerical Calculation [OH. xxiv Then, forj = 2, I (a. 2 + a 4 ) = i 0> - \p 2 \ (6 2 + 6 4 ) = i *i cos 30 (Oa ~ = i *a COS 30 for j i = 1 , (a x + a.) = a 5 ) = 7T f ^ / 2 sm- - ^a + 2 a ;: r=0 S ( t - =1 V ; cos - + ( - sm 298-soo] Formulae of Numerical Calculation 339 But since 2pks are subject to immediate recall. t-JBp^TlH- JUN29 1961 Rwcd UC6 A/M/S AI i r* f\ j-\. J-L j AUG 3 1981 CD37StE7 c lfl UNIVERSITY OF CALIFORNIA LIBRARY -v<