NRLF B M bMT 320 UNIVERSITY OF CALIFORNIA AN ELEMENTARY TREATISE ON * THE PLANETARY THEORY. ELEMENTARY TREATISE THE PLANETARY THEORY, WITH A COLLECTION OF PROBLEMS. BY C. H. H. CHEYNE, B.A. SCHOLAR OF 8T JOHN'S COLLEGE, CAMBRIDGE. MACMILLAN AND CO. AND 23, HENRIETTA STREET, COVENT GARDEN, 1862. * {The Eight of Translation is Reserved.'} Camirttfge : PRINTED BY C. J. CLAY, M.A AT THE UNIVERSITY PRESS. C-5 ASTRONOMY LIBRARY PREFACE. IN this volume, an attempt has been made to produce a Treatise on the Planetary Theory, which, being elementary in character, should be so far complete, as to contain all that is usually required by students in this University. But it is not without diffidence that I submit my volume to their notice. In the earlier part of it, the methods which have been adopted are to some extent original*, and the general arrangement of the second chapter will, it is believed, be found to be new. Through the kindness of the Publishers, a portion of Pratt' s Mechanical Philosophy has been placed at my disposal. Of this I have availed myself, particularly in the chapter on the Stability of the Planetary System ; but, on the whole, comparatively little has been reprinted ver- batim from that work. Among other sources of information, my obligations are mainly due to Ponte*coulant's TMorie Analytique du Systeme du Monde, Airy's Mathematical Tracts, and Frost's Planetary Theory in the Quarterly Journal of Mathematics: but I have also referred to the * Some of these have already appeared in Mathematical Journals. M706148 VI PREFACE. Mecanique Celeste, the Mecanique Analytique, Mrs Somer- ville's Mechanism of the Heavens, (a work forming a complete Mathematical treatise on Physical Astronomy,) a Memoir by Prof. Donkin on the Differential Equations of Dynamics, Phil. Trans. 1855, &c. A collection of Problems has been added, taken chiefly from the Smith's Prize and Senate- House Examination Papers of the last twenty years. In conclusion, I would express my sincere thanks to Messrs A. Freeman, P. T. Main, and other friends, of St John's College, for the valuable assistance which they have afforded me, and would venture to hope that the work will be found useful. C. H. H. CHEYNE. ST JOHN'S COLLEGE, October 9, 1862. CONTENTS. CHAPTER I. INTRODUCTION. ART. PAGE 1. Necessity of approximate methods 1 2. Deviations of the planets from elliptic motion 3. Elements of the orbit 2 4. Plane of the orbit 5. The Sun and planets considered to attract as if they were collected into their respective centres of gravity 6. Principle of Superposition of Small Motions, when admis- sible ; 3 7. Difference between the Lunar and Planetary Theories 8. Component of the disturbing force in any direction. Dis- turbing function 9. Meaning of the symbol -=- 5 as 10. Disturbing function independent of any particular system of co-ordinates that may be employed 6 11. Transformation of the expression for R' 7 12. To explain how R may be expressed in terms of the time and the elements of the orbit 8 14. Relations between the partial differential coefficients of R 10 Vlll CONTENTS. CHAPTER II. FORMULAE FOR CALCULATING THE ELEMENTS OF THE ORBIT. ART. PAGB 18. Equations of motion 14 1 9. Definition of the term fixed in the plane of the orbit 16 21. Principle of the method of the Variation of Parameters... 17 22. Application of this method to the equations of motion 18 23. Definition of the instantaneous ellipse 19 24. To obtain formulae for calculating the elements. Process explained, 24. Mean distance, 25, 26. Excentricity, 27, 28. Longitude of perihelion, 29. Node and inclina- tion, 30 35. Epoch, 36, 37. Mean motion, 38 20 39. Recapitulation of formulae for calculating the elements 35 CHAPTER III. DEVELOPMENT OF THE DISTURBING FUNCTION. 42. Expansions of r L , r/, 1? 0/, z, and^ 37 43. Substitution of these in the expression for R 40 46. Form of the terms in the development of R 43 47. Determination of that part of R which is independent of the time explicitly 50. Order of magnitude of the periodical terms 45 52. Proof that (# 2 +#' 2 2aa' cos<)~' can be expanded hi a series of cosines of multiples of 48 53. Calculation of the coefficients 49 58. Simplification of the expression for F 56 CONTENTS. IX CHAPTER IY. SECULAR VARIATIONS OF THE ELEMENTS OF THE ORBIT. STABILITY OF THE PLANETARY SYSTEM. AttT. PAGE 60. Definition of the term secular 60 6 1 . Formulae for calculating the secular variations 62. Approximate method of calculation 61 63. Stability of the planetary system ; 64, of the mean dis- tances ; 65 68, of the excentricities and inclinations 62 CHAPTER Y. SECULAR VARIATIONS OF THE ELEMENTS CONTINUED. INTEGRATION OF THE DIFFERENTIAL EQUATIONS. 72. Integration of the equations for the excentricity and longi- tude of perihelion 69 74. Stability of the excentricities in the case of two planets ... 72 76. Examination of the expression for the longitude of peri- helion 73 77. When the apsidal line oscillates, to find the extent and periods of its oscillations 74 78. Geometrical interpretation of the equations which give the secular variations of the excentricity and longitude of perihelion 75 79. Integration of the equations for the inclination and longi- tude of the node 77 80. Stability of the inclinations in the case of two planets 79 81. Examination of the expression for the longitude of the node.. 80 CONTENTS. AET. FAGB 82. When the line of nodes oscillates, to find the extent and periods of its oscillations 80 83. Inclination of the orbits of two mutually disturbing planets to each other approximately constant 82 84. Geometrical interpretation of the equations which give the secular variations of the node and inclination 85. Integration of the equation for the longitude of the epoch. 85 86. Secular acceleration of the Moon' s mean motion 86 87. Formulae for calculating the node and inclination of the ' plane of a planet's orbit relatively to the true ecliptic. . . CHAPTER YL PERIODIC VARIATIONS OF THE ELEMENTS OF THE ORBIT. 89. Definition of the term Periodical Variations 89 91. Expressions for the periodical variations of the elements 90 92. Long inequalities 91 93. To select such terms in R as will produce the principal inequalities of long period 91 94. Relation between corresponding terms of the long inequa- lities in the mean motions of two mutually disturbing planets 92 95. Variations of elements whose periods are very long 96 96. Distinction between secular and periodic variations 97 98. Periodic variations in radius vector 99 99. Periodic variations in longitude 100. Example of the processes of this Chapter 100 \ CONTENTS. xi CHAPTER VII. DIRECT METHOD OF CALCULATING THE INEQUALITIES IN RADIUS VECTOR, LONGITUDE, AND LATITUDE. ART. PAGE 102. Methods of Lagrange and Laplace 105 103. Equations of motion 105. Equation for the perturbation in radius vector 106 106. Equation for the ]Jterturbation in longitude 108 1 07. Equation for the perturbation in latitude 109 108. Integration of the equation for the perturbation in radius vector 110 109. First approximation to the value of Sr.... Ill 110. Omission of the arbitrary term 113 112. Certain terms to be neglected 114 113. Second approximation to the value of Sr 115 114. Calculation of perturbations in longitude 116 115. Determination of the constant g 117 116. Long inequalities 118 117. Integration of the equation for the perturbation in latitude 119 CHAPTER VIII. ON THE EFFECTS WHICH A RESISTING MEDIUM WOULD PRODUCE IN THE MOTIONS OF THE PLANETS. 118. Possibility of the existence of a very rare medium 121 119. Equations of motion 122 Xll CONTENTS. AET. PAGE 120. Formula for calculating the mean distance 122 121. Formula for calculating the excentricity 123 1 22. Formula for calculating the longitude of perihelion 124 123. Transformation of the above formulae 125 124. Formula for calculating the epoch 126 125. Assumed form of the density 127 126. Effect of the medium upon the elements, the orbit being supposed nearly circular 128 127. The medium, though insensible to the planets, may yet affect the motions of comets 129 PROBLEMS .. 130 APPENDIX. ON THE FORM OF THE EQUATIONS OF ART. 39 140 ON THE GEOMETRICAL INTERPRETATION OF THE FORMULAS FOR THE SECULAR VARIATIONS OF THE NODE AND INCLINATION 145 ON THE METHODS OF CALCULATING THE MASSES OF THE PLANETS . 146 EERATA. Page 80, line 9, for excentricities read inclinations. 2?r , 2?r THE PLANETARY THEORY. CHAPTER I. INTRODUCTION. 1. To determine the motion of a system of bodies sub- ject only to their mutual attractions, is a problem the mathe- matical difficulties of which have not yet been overcome: hence in the particular cases of this problem which the Lunar and Planetary Theories present, recourse must be had to methods of approximation. Happily the arrangement of the Solar System renders approximate methods possible, and in the skilful hands of the Mathematicians of the last century, they have been brought to a high state of perfection. 2. If the Sun were the only attracting body, the planets would describe exact ellipses, agreeably to Kepler's first law ; but in consequence of the attractions of the planets them- selves, slight deviations from elliptic motion are produced. The method of calculating these deviations, to which our attention will chiefly be directed, is due to Euler ; it consists in supposing the planets to move in ellipses, the elements (or arbitrary constants) of which are continually though slowly changing*. * The legitimacy of this hypothesis will appear when we come to treat of the equations of motion. See Arts. 21 and 22. C. P. T. 1 2 PLANETARY THEORY. 3. Now the elements of an elliptic orbit are (i) the mean distance, or semi-axis major, (ii) the excentricity , (iii) the longitude of perihelion, i. e. of the point of the orbit nearest to the Sun, (iv) the longitude of the epoch*, or mean longitude at the epoch from which the time is reckoned, (v) the inclination of the plane in which the orbit lies to some fixed plane of refer- ence, (vi) the longitude of the ascending node. Of these (i) and (ii) determine the magnitude of the orbit, (iii) determines its position in its own plane, (v) and (vi) determine the posi- tion of this plane, and (iv) has reference to the position of the body itself in its orbit. If the planets moved accurately in ellipses, these would be constants : we must however be prepared to consider them as variable quantities, which it will be the object of the problem to determine. They are tejmed the elements of the orbit. 4. But further, not only is it found that the true orbit of a planet is not an ellipse, but that it is not even a plane curve, although the departure of the planet from the plane in which it is at any instant moving is extremely slow. We define as the plane of the orbit the plane containing the radius vector and direction of motion of the planet at the instant under con- sideration. 5. We shall suppose the Sun and planets so distant from each other that they may be considered to attract as if they were condensed into their respective centres of gravity; a supposition which would be rigorously true if these bodies were exactly spherical, and either of uniform density or composed of concentric spherical shells, the density of each shell being uni- form throughout. The errors, however, thus introduced into the motions of translation are found to be inappreciable for the planets, though not in the case of their satellites. The mo- tions of rotation will not be considered in the present treatise. * Also briefly termed the epoch. INTRODUCTION. 6 6. Moreover, since the masses of the planets are extremely small in comparison of that of the Sun, it follows that in cases where it is not necessary to carry the approximation beyond the first order of these masses, we are permitted to avail our- selves of the Principle of the Superposition of Small Motions, and thus to reduce the problem to a case of that of the Three Bodies. 7. So far the Theory of the Planets resembles that of the Moon, and the same method of treatment might be employed in both cases. But they differ in- this respect : the ratio of the distances of the disturbed and disturbing bodies from the central one* is much smaller in the Lunar than in the Planetary Theory, so that if in the latter theory the approximation were made by means of series proceeding by powers of this ratio, it would be necessary to retain many more terms than are re- quired in the former. For this reason a different method of development is employed. The perturbations of the Moon, however, are far larger than those of the planets, since in the former case the Sun, of which the mass is enormous, and the distance not proportionately great, is 'one of the disturbing bodies. 8. To find an expression for the component in any di- rection of the force which disturbs the motion of a given planet relatively to the Sun. Let M denote the mass of the Sun, m, m', m", &c., those of the planets, and suppose the relative motion of m re- quired. Let x, y, z, x, y', z', x", y", z", &c., be the co-ordinates of m, m', m", &c., referred to any system of rectangular axes * By the central body is meant that whose attraction exercises the greatest influence on the body whose motion is required ; the Sun for instance in the Theory of the Planets, and the Earth in that of the Moon. All the other attracting bodies are called disturbing bodies. 1-2 PLANETARY THEORY. originating in the centre of gravity of the Sun ; r, r', r", &c., their distances from the origin ; p', p", &c., the distances of m, m", &c., from m. Now if to every body of the system we apply forces equal and opposite to those which act upon the Sun, we shall reduce the latter to rest without affecting the relative motion. Hence, considering the action of only one disturbing planet m', the forces acting upon m will be M+m. ,. 2 m direction mm, m . ,. . , -is m direction mm , P m . ,. . ,,, -jj in direction m M, of which the last two constitute the disturbing force. ; 77? Let F= , : then on the hypothesis of Art. 5, Fwill be the potential of m', and the components parallel to the axes of the disturbing force due to the action of m, will be dV mx INTRODUCTION. 5 dV my dy r ' dV mz dz r* ' Let s denote the length of the arc of any curve measured from some fixed point up to m* : then the resolved part of the disturbing force parallel to the tangent at m to this curve will be dV /m'x' dx m'y 1 dy m'z dz" ~ds ~~ (~^ ~ds~ or restoring to V its value d which may be written -^ , if m -n, , t i f\ R = j--(xx+yy +zz). If we express in like manner the disturbing forces due to the actions of m", m", &c., we shall have for the whole component in this direction ds " ds '"' or ^, if B = R + K' + .... ds The function E is called the disturbing function. 77-> 9. From the manner in which -^- has been introduced, it appears that E is supposed to be expressed in terms of s and quantities which do not vary with s. It must however * This arbitrary curve we shall term the carve of reference. 6 PLANETARY THEORY. be borne in mind that in -7- the variation is purely hypo- thetical, and has nothing whatever to do with the actual variation of R due to the motion of the planet. For example, suppose the curve of reference a straight line parallel to the axis of x, and let R be expressed in terms of x, y and z then in this case x only will vary, and the disturbing force parallel to the axis of x will be denoted by -j , y and z being considered constant in the differentiation. Similarly, the disturbing forces parallel to the axes of y and z will be expressed by -7- and -7 respec- tively, the differential coefficients being strictly partial. Again, suppose the curve of reference a circle with its plane parallel to that of xy, and its centre in the axis of z, and let R be expressed in terms of the polar co-ordinates (r 1? 0J of the projection of the planet on the plane of xy, and its distance (z) from this plane ; then in this case t only will vary, and the disturbing force perpendicular to the projected /77? radius vector will be expressed by ,- , r^ and z being con- sidered constant in the differentiation. Similarly, the forces parallel to the projected radius vector and to the axis of z, /77? will be expressed by the partial differential coefficients -= , dR ~j~ respectively. 10. The disturbing function, like the potential, is inde- pendent of any particular system of co-ordinates that may be employed. For jy mm., , ,. 7 ~ ^ XX yy ZZ ^ INTRODUCTION. m in r fx x y y z z p r* \r r' r ' r r m mr ,. 2 - cos ft>, p r if co denote the inclination of r' to r. 11. To express B/ in terms of the polar co-ordinates of the projections of m and m' on a fixed plane, and of their distances from it. Take the fixed plane for that of xy : let r v r/ be the pro- jections of r, r upon it, and 0^ 6^ the inclinations of r l9 r^ to the axis of x ; then x = r x cos 15 y=r^ sin 6 V x = r/ cos 0/, y = r/ sin 0/ ; therefore cca?' + yy + ' = r^ cos (^ ^/) + zz, p '*=( X - X >y+(y-y>Y + ( Z -z'y = r * + r 2 - 2' cos - + - Hence by substitution, R' = , 2 + < 2 - 2V/ COS (^ - 0/) + (0 - ^T} 12. In a subsequent chapter we shall consider the de- velopment of R in terms of the time and the elements of the orbit, in a series ascending by powers and products of the excentricities and inclinations, which in the Planetary Theory 8 PLANETARY THEORY. are very small. At present we shall content ourselves with shewing how R may be expressed in terms of these quantities. We shall assume that the equations connecting the co-ordi- nates, the time, and the elements in an elliptic orbit, hold also in the case of a disturbed planet. 13. To explain how R may be expressed in terms of the time and the elements of the orbit. Let r, denote the radius vector and longitude of the disturbed planet, the latter being measured on a fixed plane of reference as far as the node, and thence on the plane of the orbit : let the elements be a the mean distance, e the excen- tricity, TX the longitude of perihelion, e the longitude of the epoch, (the last two being measured in the same way as 6,) fl the longitude of the node measured on the plane of refer- ence, and i the inclination of the plane of the orbit to the plane of reference. Our object is to express R in terms of t and these elements. Again, let # , T O , e , O denote the longitudes of the planet, of perihelion, of the epoch, and of the node, measured entirely on the plane of the orbit. Let a. sphere be described with its centre coinciding with that of the Sun, and its radius of any magnitude: let the planes of reference and of the orbit cut it in the great circles NM, NP, then the line of nodes will cut it in N] let the radius vector of the planet cut it in P, the projection of this radius on the plane of reference in M, and the lines from which 6, are measured in L, respectively. We shall suppose L to be the same origin as that from which X is measured in Art. 11. Then in the figure LM=d v LN + NP = 0, OP = Q , LN= ft, the angle PNM= i, and PM= the latitude of the planet which we shall denote by X. INTRODUCTION. Hence from the right-angled triangle PNM, tan (0 t - H) = cost tan (0-H) (1), sin X = sin i sin (0 - ft) (2): also r 1 = rcosX (3), z = r sinX (4). Again, from the formulas of elliptic motion*, r = a{l + %e z - e cos (nt + e - OT O ) - Je 2 cos 2 (nt + e - -BJ O ) - ...}, = w<+ e + 2e sin (w + e - tsr ) + Je 2 sin 2 (w + e - T O ) + . .. t but 0-6 = LN- ON=e-e = , %, ^ are symbols of functionality. It follows that dr dr_ + ~ Now since by Art. 11, R is a function of r lt l9 and dR_dB L dr l dR dd dR dz^ d6~dr^ ~dd + W ~dd + dz dd ' dRdRdr dB dO dR dz therefore, by addition, dR dR dR 12 PLANETARY THEORY. whence, by the last article, dR dR dR dR W^~fa + lfa + ~d5' , ' 4 . dR . , dE , dR 17. To obtain -3 in terms of -3 and -^ . de J dr d# If u denote the excentric anomaly, we have* r = a (1 e cos u) ..................... (1) , nt + e TX u e sin u ..................... (3), from which r and 6 may be expressed in terms of t and the elements by eliminating u. Assuming r arid 6 so expressed, , , . dr . dO we proceed to obtain ^- and -r- . -r /-,\ dr. f . du \ Irom (1), -j- a le sin u -j cos uj , and from (3), -7- (1 ecos w) sin u = ............. (4) ; eliminating -j- , we have de du de dr ( esmu -r- = a\ de [1 ecosw ( e cos u } \le cos u) ' 1 e cos u (1 e 2 ) 6 (1 ^ See Tait and SteeWs Dynamics, Arts, no, in, and 122. INTRODUCTION. 13 = a cos (6 VT), by the polar equation to the ellipse. Again, differentiating the logarithms of equation (2), 1 dO I 1 1 \ 1 du _ I f 1 1 ~ 2 \1 + e + f sin (0 - r) de ~ 2 \1 + e f^e/ sin u de ' eliminating -y- by means of (4), 1 1 sn ^el e I e cos if h 2 fjia (1 e 2 ) ; therefore Now since R is a function of e only because it is a function of r and 0, de dr de d6 de 7 . dR /> 1\ . ta . dB r= 6t cos ( tf ~~ ffS] 1- cn \ ~j 1 sin ((/ ~~ iff ) ^ . df \ti T / dfj Since 6 r = 03 about their instantaneous positions : let X,, Y, Z be the accelerations due to the impressed forces in the directions of the axes. Then (Eouth's Rigid Dynamics, Art. 114), dx , -. ds w= Tt-^ and the equations of motion are FOEMUL^E FOR CALCULATING THE ELEMENTS. 15 duo In these equations^, < 2 , < 3 are arbitrary; we propose so to determine them that the axis of x may coincide with the radius vector of the planet, and the plane of xy with the plane of the orbit. In order that the axis of x may coincide with the radius vector of the planet, we must have always ; and therefore dx _ dr dy _ dz ~di ~~ dt ' ~dt ~ ' di ~ and in order that the plane of xy may coincide with the plane of the orbit, we must have always ; and therefore dw _ Hence equations (1) give - and equations (2) become ,72 x =w~^' 16 PLANETARY THEORY. In order to reduce the first two of these equations to the forms they would take if the motion were in one plane, let * It is necessary to define this term, since the definition of Art. 4 is not sufficient completely to regulate the motion of the plane of the orbit, though affording it a distinct geometrical position. FORMULA FOR CALCULATING THE ELEMENTS. 17 since 0, 1 and i do not vary with s. Hence the disturbing force in direction of the radius vector = -, . dr Again, let us take as the curve of reference a circle in the plane of the orbit, with its centre coinciding with that of the sun ; then $s = r0, and we have dR _ldR ~3a"~r'd0' since r, H, and i do not vary with s. Hence the disturbing force perpendicular to the radius vector = - - . We have then Y jjb dR v 1 dR ~/ + ^' = rd8' and the equations become df dt ~ r* d f d8\ dR 21. These equations do not admit of rigorous integration, but we may reduce them by the method of the Variation of Parameters to a system of differentia! equations of the first order. The principle of this method may be explained as follows. Suppose it required to integrate the equations dx dv d*x d' dx dy where P v P 2 are functions of t. The solution of these equa- tions can be made to depend upon that of the equations C. p. T. 2 18 PLANETARY THEORY. y> *' di> ft) t dx dy\ dx where c 1? c 2 , c 3 , c 4 are arbitrary constants or parameters. The method of the Variation of Parameters consists in so determining c x , c 2 , c 3 and c 4 as functions of 2, that these inte- grals (and therefore the two final integrals of the equations 2 = 0, which can be obtained from equations (ii) by ft rp // 7/\ eliminating r- and -^J shall satisfy equations (i). That c r c 2 , c 3 , and c 4 can be so determined, may be seen as follows : by the solution of equations (i), values of x and y and there- fore of -V and -i can be found as functions of t and constant dt dt quantities ; if these be substituted in equations (ii) the requi- site values of c x , c 2 , c 3 and c 4 will be obtained. For an ex- ample of the application of this method, see Boole's Differential Equations, Chap. IX. Art. 11. 22. If in equations (1) and (2) of Art. 20 we put ^ = 0, and then integrate them, we obtain i = {l + e cos (0,-flr,)} .................. (3), FORMULA FOR CALCULATING THE ELEMENTS. 19 where h, e, OT O are the constants of integration. Equation (3) indicates motion in an ellipse, of which e is the excentricity, -sr the longitude of perihelion, and h twice the area described in an unit of time. If the mean distance in this ellipse be denoted by a, we have in addition h* = tM(l- ds ds ds ds r sm (0 O) JT> i JT> therefore r sn On substituting this value for ^ in equation (3), we obtain , . .dl dE h sm % -j- = -JT , dt di 9 28 or since PLANETARY THEORY. = pa (1 e 2 ), and ri 2 a? /JL, da dt na dR ~~ sm 32. Again, suppose the plane on which r and 6 are mea- sured, instead of passing through 8N 9 to pass through a line 8G in the plane of the orbit perpendicular to SN; and take for the curve of reference a circle AP with its centre C in SO. Through 8G draw a plane inclined at a small angle to the plane of the orbit, cutting the sphere in the great circle nmp 9 and the circle AP in p. Draw Nm perpendicular to np. Then Nm which measures the inclination of the two planes = SO sin i, and hence if Pp = Ss, s r cos (0 H) 811 sin i ; therefore Again, cfe r cos (0 O) sin i " * = Ln + np-(LN+NP) = nm nN FORMULAE FOR CALCULATING THE ELEMENTS. 29 therefore -7- = (1 cos i) -y- : ds 'as dr di also -y- = 0, -y- = 0. ds ds dR dR de dR dfl Hence -j- = -TTT -y- 4- ^7^ - /- as de/ ds ail as fd^ . N d^) 1 rcos (0 H)sin 2" On substituting this value for ^in equation (2), we have di 1 dR - cos ) jTT f du) r j A sin * (ail x du I (dR . 2 * /^ dR\ . . -T-S bo-Msin 1 - hr+^")f ( Art A sm ^ c/ll 2 33. The only remaining element is the epoch, but before proceeding to obtain a formula for its calculation, we shall give another method of obtaining the results of the last two articles. 34. To obtain a formula for calculating the inclination. (Second method.) If the motion of the planet be referred to the polar co-ordi- nates of its projection on the fixed plane of reference and its distance from this plane, we have the equation 1 d ( 2 dO\ 1 d r^^J-r d_ / 2 d\ _ dR cft'V! ~dtJ~dOC Now if 8^4 , S^j denote the vectorial areas swept out in 30 PLANETARY THEORY. the time St on the plane of the orbit and the plane of reference respectively, we have 8^= $A cos i: but M- :'*** Li- 2 2 - therefore r^ ^ = r z -?- tos ^ = h cos ^. Hence our equation of motion becomes d , dR , . .di .dh dR -Asn^ + ccm^^- _ . .di dR .dh therefore Asnu - - cos ^ - __ "" na ( 1 dR i (dR d. .. -.-777 + tan- -T- + -T- r- jj, V(l e 2 ) [sin tdQ, 2 \ 35. To obtain a formula for calculating the longitude of the node. (Second method.) Since the velocity of the planet at any time can be ex- pressed in terms of the co-ordinates and elements of the instan- taneous ellipse constructed for that time, in the same form as FOKMUL.E FOR CALCULATING THE ELEMENTS. 31 if it moved iii this ellipse, its component in any direction can also be so expressed. Hence, considering R as a function of r 1 , B v z (see Art. 11), the values of -~ , -^ , -7- , and there- fore of , (where ^ denotes that R is to be differentiated at ^ at with respect to t only so far as it involves t through involving the co-ordinates of the disturbed planet) may be expressed in the same forms as if the elements were invariable. Now we have seen (Art. 13) that R may be thus ex- pressed : #-/(*,*, o>t)i or since evidently, 6 H = Q I1 , We have then, considering the elements variable, d (R) _ dR dr dRd(6 Q +Q,-l ) dRdti dR di dt. ~d dd ~ ~3T and, considering them invariable, dRdO ^ ~ + ^ dt ~ dr dtd0 Q dt " * Equating the two values of , we obtain ow _ da~di + ~dJdt~ dR dR * We have made this transformation, because, although the value of -^ is the same in form as if the elements were invariable, this is not the case with dd dt' 32 PLANETARY THEORY. , , . 4 _ N d& dl and (see Art. 29), -^- = -=- cos i ; (dR - fdB dB\\ dl , dR di therefore \-j^ + (1 - cos i) -j- +-=- }\ -j- +-rr ^=0. \dl ' V de d^i] dt di dt Substituting for -r- its value dl na dR dt ~~ fj, V(l e 2 ) sin i di " 36. Jb obtain a formula for calculating the longitude of the epoch. If R be expressed in terms of t and the elements (see Art. 13), since nt + e always occurs as one symbol, we may write Rf(nt + e, a, e } TV, O, i). Differentiating, the elements being considered variable^ we have d(R) _dRd(nt+e) dRda dRde dRd^ dt de dt da dt de dt d^ dt djRdQ dRdi + ~d& dt + ~di dt' and differentiating as if the elements were invariable, which is permissible for the reason explained in the last Article, d(B) _ dR dt ~ n de' Equating the two values of dt dR L _dR L f dn de\ dRda dR de dR ~ n ' t+ dRda dRdi d& dt + di dt' FORMULAE FOR CALCULATING THE ELEMENTS. 33 Substituting for -y- , -7- , &c., their values (it Out _L na ~ _ de dt dt p de da pe de de -e 2 ) fdR dR\dR na V '(I - e*) dE dR \de d^ff) de fjue de dix nqtan 2 dRdR na dR dR -e < c?^ 2a at fj, de ' d*% Ztfa dR or -r = -- - -- . 39. We will here recapitulate the formulae which have been obtained for calculating the elements of the orbit. da _ 2na? dR dt** ft ~3e' de _na(l- e 2 ) dR na V(l - e 2 ) fdR dR\ ^ ' dt~ fie de ^ \de dvrj' ~dt~ ie de + i^/l-e 2 di' de Zna^dR ^V(l-e 2 ) ,, -.dR dR i(dRi + Ln+ We have also (vii) the equation d?%__ ZtfadR df ' ~T de' 36 PLANETARY THEORY. but this forms no new relation, since it has been deduced from (i). 40. When the elements have been calculated by means of the above formulae, the position of the planet will be given by the equations = f + e + 26 sin (f + e - r) + e 2 cos 2 (f + e - r) + CHAPTER III. DEVELOPMENT OF THE DISTURBING FUNCTION; 41. IN the first chapter we have obtained equations by means of which R may be expressed in terms of the time and the elements of the orbit ; we now proceed to shew how the actual development may be effected in a series ascending by powers and products of the excentricities and the tangents of the inclinations. In the Planetary Theory these are ex- tremely small, and the series will converge rapidly. Ac- cordingly in the present treatise small quantities of orders higher than the second will be neglected*. 42. If we recur to Art. 11, it will be seen that, consider- ing only one disturbing planet, == 1 2 + r* - 2r^ cos (0, - 0/) + (z - The first step towards the required development will be the expansion of r v r^, 6 V 0/, z and z in terms of the time * We may remark that to this order of approximation the inclinations, their sines, and tangents will be equal. 38 PLANETARY THEOKY. and the elements of the orbit. For this purpose we may employ the equations which have already been obtained in Art. 13, viz. : tan (0 X - O) = cos i tan (9 - O) , sin X == sin i sin (0 Q), r x = r cos X, s = r sin X, OT) -e 2 cos2 (n + e OT) ...k = nt + e + 2e sin (nt + e *r) + - e 2 sin 2 (ntf + e ) + ..., with similar equations involving the co-ordinates and elements of the disturbing planet. (i) To expand r v . We have r t = r cos X = r (1 sin 2 X)^ = r ji _ 1 sin't sin* (0 - H) + ...i = r |l - itan t sin* (0 - fl) + ...I to the same order of approximation, or substituting the expansions for r and 0, r x = a ! 1+ - e 2 -tan 2 ^ e cos (nt+ e r) -e 2 cos 2 (w^+e suppose. DEVELOPMENT OP THE DISTURBING FUNCTION. 39 Similarly, r^ = a (1 + u). (ii) To expand t . We have tan (0, - 0) = tan {(0 t - U) - (0 - U)} tan (^-H)- tan (0-11) " 1 + tan (6 l - U) tan (0 - 11) _ (cos^-1) tan (0- 11) " l + cositan 2 (0-H) -2 sin 2 1 tan (0-11) = M-- -.-.= ->.- -. 1 + tan 2 (0 - to) - 2 sin 2 | tan 2 (0 - H) = - sin 2 | sin 2(0-11) - ... ; therefore X - = - sin 2 1 sin 2 (0 - U) - ... = - tan 2 | sin 2 (0 - U) - ..., to the same order of approximation; or, substituting the expansion for 0, 5 i = nt + e + 2e sin (w + e &) +-r& sin 2 (nt + e ty) 4 - tan 2 | sin 2 (nt + e - U) + = nt + + v, suppose. Similarly, 0/ = n't + e + v'. 40 PLANETARY THEORY. (iii) To expand z. We have z r sin X = r sin i sin (6 O) = r tan i sin (^ O) ... to the second order; or, substituting the expansions for r and0, z = a (tan /sin (TZ + e O) -f ...]. A similar expression may be found for z. 43. Having obtained the expansions of r t , r', 6 t , 0', z, z' we must now substitute them in the expression for E. This may be effected as follows. Let R' be the value of R when u, u, v, v are severally zero : then, writing for nt 4- e (nt + e') we have R' = m ' [{a 2 + a' 2 - 2aa' cos < + (z - z'Y}^ = m' [(a 2 + a' 2 - 2aa cos <)"* - -^- 2 cos m' (a 2 + a' 2 - 2aa' cos ^>)- ( - z') z , - 1 3 a " AT T> Tlf / r / IN ow R = R + -j- au + - r - 7 au + -=7- (v t? ) v aa , , , , + -77-7 aa uu +. -,,au (vv)+ , , , , aw (v - v ) s ' da d DEVELOPMENT OF THE DISTURBING FUNCTION. 41 44. It will be shewn in a subsequent article that (ft 2 + a/ 2 -2ftft' cos <)-', can be expanded in a series of the form Assume then "* = (a 2 + a' 2 -2a' cos <)"* = - <7 + (^cos^-f- <7 2 cos2< + ... Thus m au ldC Q fdC. l\ dC 2 } - -j-*- + [-J- 1 - -J cos + -j- 2 - cos 26 + .... X 2 rfa \da a J da J 42 PLANETARY THEORY. - m'au (v - ') j(^-* - -^ sin + 2 -j sin - W (, - O jgj + g sin * + 2 g* sin 2* + ...} , (zz 3 a 45. By Art. 42, 1.1 1 w = - e j tan i e cos (nt + e w) e 2 cos 2 (w + e -sr) ^5 4 2 1 +*rtan cos 2 (?i-{-e fl) + ... 4 v = 2e sin (w + e r) -f - e 2 sin 2 (n + e - w) 4 - tan 2 1 sin 2 (n* + e - fl) + . . . , 2 = (tan i sin (n + e H)+ . . .}, with similar expressions for u , v', 2'. Hence w 2 = e 2 cos 2 (w^ + e r) + . . (v - v ') 2 = 4e 2 sin 2 (n< + e - r) + 4e' 2 sin 2 (w' + e' - w 1 ) See' sin (nt + e -or) sin (n'< + e' r') + . . . = 2 (e 2 + e' 2 ) - 2e 2 cos 2 (n* + e - w) - 2e' 2 cos 2 (w'*+e r - r') - 4ee' cos ((/> - r + ') + 4ee' cos {(/i + n') -fe + e'or sr'} +. .., ww' = ee f cos (n^ + e -sr) COS(TI^ + e ') + ... DEVELOPMENT OF THE DISTURBING FUNCTION. 43 2 u ( v _ #') = e 2 sin 2 (nt + e vr) + 2ee cos (nt -f e OT) sin (rc' + e' -BJ') + . .. = e 2 sin 2 (w$ + e r) ee' sin (^ w + r') + ee sin {(w + w') t + e + e' tar *r'} + . . ., >. 9 FOj 9*^ 2 ' tan ^ a tan ^ a A 1 * tan** 1 cos2(^+6 f -Q y )-^ tan ^ tan ^ cos (^~ Q +^ / ) &c. 46. If these values be substituted in Art. 44, it will be seen that cosines will be multiplied only by cosines, and sines by sines. Hence the series will consist of two parts, one inde- pendent of t explicitly, and the other consisting of periodical terms of the form Pcos{(pnqn') t+ Q}, where p and q are any positive integers or zero, P is a function of the mean distances, excentricities, and inclinations ; and Q a function of the longitudes of perihelia, nodes, and epochs. The former part is denoted by the symbol F: we proceed to determine its value as far as the second order of small quan- tities. 47. To determine that part of R which is independent of the time explicitly. If those terms only be written down which either are, or after reduction will become, independent of t, we have ,(C adC fe* tan 2 1\ a' dC (e 2 tan 2 "~ ~~~"~~ 44 PLANETARY THEORY. a z d*C e* , ^^o^_A/altanV o^tanV\ 4 da 2 2 + 4 da' 2 24^2 2 / + - ( Cj -- - 2 cos (j) 4ee' cos (^ -sr + ') f -7^ -- ^ J sin <^> ee sin (0 w + ') f -5 sn > ee sn a + - cos ^ ad tan z tan ^' cos ( 11 + fl') + ...!-. Now cos <^> cos ((j) & + iff') and sin sin (< vr -BT') con- tain the term J cos (w '), cos < cos (^ O + fl') contains the term J cos (fl II') ; hence i 2 4 4- j aa' D x tan i tan i' cos (H ft') + . . . r . We shall hereafter be able to simplify this expression. 48. We have seen that the remaining terms of E are of the form Pcos {(pnqn) t+ Q}: if then values of p and DEVELOPMENT OF THE DISTURBING FUNCTION. 45 q could be found such that pn qn 0, this term, being in- dependent of t explicitly, would form an additional term in F. No instance of this, however, occurs among the planets. 49. In consequence of the extreme smallness of the ex- centricities and inclinations of the planetary orbits, terms in R of orders higher than the second may in general be neglected : but it sometimes happens, as in the Lunar Theory, that higher terms become sensible through the process of integration. This we shall consider in a subsequent chapter, but the following proposition has an important bearing on the subject. 50. The principal part of the coefficient of a term in II of the form P cos {(pn qn) t + Q] is of the order p ~ q. DEF. By the principal part of the coefficient is meant that part of P which is of lowest dimensions in e, e', tan i, tan i'. If we return to the expression for R in Art. 44, it will be seen that in order to obtain the general term it will be necessary to multiply the product of the general terms of the expansions for u a , u'P, v"*, v' s , 2 e , z' by cos k (f> or sin k . Now (1) in the expansions of u, u, v, v ', z, z' the follow- ing law is observed to hold : The number which multiplies nt + e or n't + e' in the argument of any term represents the order of the principal part of the coefficient of that term. (2) The same holds good in any power of w, u', v, v', z, or z'. For, consider a term P cos (put + q) in u z . It can only have arisen in the following ways; partly from the multiplication of two terms in u of which the arguments are Int + X and mnt + y^, where I + m p ; and partly from such as have the arguments Int + V and mnt+ftj where I ~m=p. In the former case the order of the coefficient will be l + m, which equals p, in the latter it will be I' + m, and this is 46 PLANETAEY THEORY. greater than p. Hence the principal part of the coefficient of a term P cos (pnt + g) in w 2 , will be of the order p. Since then the law holds in w 2 , it may be shewn in like manner to hold in the product of u z and u, i.e. in u s . Thus it may be proved for any power of u. In like manner it may be shewn to hold for any powers of u, v, v', z, or z. (3) The same law is true for the product of any powers of u, v, z ; and likewise for the product of any powers of u, v, z . This may be proved by a method similar to that of (2). (4) In the product of any powers of u, u, v, v', z and z', the order of the principal part of the coefficient is the arithmetical sum of the multipliers of nt and n't. For let us consider a term If cos {(In I'ri) t + N}. Now this must evidently have arisen from the multiplication of L cos (Int + X) with L' cos (I 1 n't + X'), or of L sin (Int + X) with L' sin (l'rit + \ r ), where by (3) L is of the order I and L' of the order I'. Hence H will be of the order I + I'. Now atiy term in the development of R of the form Pcos {(pn qn] t+ Q] must have arisen partly from the cos multiplication of P t . &(/>, or as it may be written S1U with P 2 ' {[(p -Jc}n-(^-Tc}n'}t + and its multiples, we shall now give a proof of this and shew how the coeffi- cients may be calculated. 52. To shew that (a 2 + a' 2 2aa' cos $)~* can be expanded in a series of cosines of multiples of<$>. Suppose a greater than a', and for write a ; then Ob ( a + a ' 2 _ 2aa cos ft' 8 = + ... +^4fc cos &< + ... , where A Q , A , &c., are functions of a and a. The series which they represent will be always convergent provided a is less than unity, or a greater than a. If a be less than a', we have only to interchange a and a in the above, so that a will then denote the ratio of a to a. 53. To calculate C and C t . In the preceding article, let s = - ; then C.P.T. 50 PLANETARY THEORY. Unless a be small, these series will converge too slowly to be practically useful. More convergent series might be obtained, but according to Pontecoulant (Systeme du Monde, Tome III. p. 81), it is more convenient to employ elliptic integrals for the purpose, in the manner we proceed to ex- plain. We have , ft * \ ,-*. ^-^n v-wt* f * -k 1 i (a + a 2aa cos 0) ' 2 2 2 + - (7 2 (cos 30 + cos 0) + . Integrating both sides of these equations with respect to between the limits and 2-Tr, we obtain r 2-rr 7 -t "if* %1T 7 -L , I d(b 1 I am 7rC =l z ; 7T4 = aJ 7 a ^1 ' PO^ TrO, _ r 2 "- cos ^ d$ 1 ~J ( a * + d*-2aa o ^w. 1- tt zio/t* cue ^>y ~ " o (1 + &. 20C COS 0J ' J These integrals may be reduced to the standard forms of elliptic functions by assuming sin (0-0) = a sin 6 (1), whence tan# = ^ (2). cos (p a From (1) cos (0 - (jf , ,, c?0 cos (9 0) oc cos therefore - = 7:5 -, du cos (c/ a) + sin sin < /cos 2 /A t. /) = V(l - a 2 sin 2 6} - a cos Q ; therefore 1 + a 2 2a cos ( = 1 - a 2 sin 2 + a 2 cos 2 - 2a cos 19 V(l - a 2 sin 2 0), 2a cos < = 2a 2 sin 2 + 2a cos 6 V(l - a 2 sin 2 0), or cos < = a sin 2 + cos V(! 2 sin 2 ^) Now as < increases from up to 2 TT, also increases from to 2-7T ; hence 2acos =- f *' 4-2 52 PLANETARY THEORY. 1 [ 2 asin 2 <9dfl 1 = -- -77^ - a */i\ H --- a^Jo V(l- sm'0) a7r i ff 27r do c = *7 . aa?r V V(l -a" sm 2 0) J COS Hence with the usual notation for elliptic integrals (see Todhunter's Integral Calculus, Art. 222), The numerical values of F[a 9 ^} and E[a, ^) may "be \ / \ *J found from Legendre's tables of elliptic functions. 54. Given C k and C^ to obtain C k+1 . We have . (a 2 + a' 2 - 2aa cos 0)-i = 1 (7 + C7 X cos ^ + . . . + (7, cos ^ + . . . ; differentiating with respect to , aa sin (a 2 + a' 2 - 2aa f cos 0)" 1 = Q sin ^> + 2 C a sin 2(^> + . . . therefore aa 1 sin (- C Q + <7 X cos + . . . J = (a 2 + a' 2 - 2aa'cos ^) (^ sin ^+ 2C 2 sin equating coefficients of sin 7c (a 2 4- a" 2 - 2aa' cos <)~ f = (7, sin $ + 2 <7 2 sin 2< + . . . + fc O k sin 1c$> + . . therefore aa' sin ^ (-D Q + D^ cos ^ + D z cos 2^> + . . .1 = 0,8111^ + 2^ sin 2 + ... ; equating coefficients of sin /(/>, tkC> = *) ............... (2). Again, (a 2 + a' 2 - 2aa cos *)"* = 5 A + A c os + D 2 cos and (a 9 + a' 2 - 2aa cos ^)~^ = | (7 + (7 X cos ^> + (7 2 cos therefore - <7 + Ci cos 4S - 2aa'cos ^) + A c equating coefficients of cos &0, 0,= (a? + a'*) D t -aa' (D^ + flJ .......... (3), writing Tc + I for &, G M = (a" + a' 2 ) D^ - aa' (D t + D^} ......... (4). Eliminating D k _ : between (1) and (3), 1) C k = (a 2 + a' 2 ) D H - 2aa'D tfl ......... (5). 54 PLANETARY THEORY. Eliminating JD^ between (2) and (4), (6). Finally, eliminating D k+l between (5) and (6), (a 2 + a'*) C k - or D t = , {(a 2 + a") ft - 56. To calculate the successive differential coefficients of C k and D k with respect to a aw^ a'. We have (a 2 + a' 2 - 2aa' cos 0)~* = -C + C l cos ^ + (7 2 cos 2^> + ... 2 4- Ci cos ^ + . . . : differentiating with respect to a, -(a -a' cos <) (a 2 + a' 2 - 2aa' cos $"* = * + cos * + substituting for (a 2 + a** 2aa cos ^>}~^ its expression in series, - (a- a'cos<) (-Dt + Dt cos ^+ ... -f D k cos ^0+ ...) 2 c?a ^a equating coefficients of cos k,' By giving to A; in succession the values 1, 2, 3, &c., those of -~, ~~j^> & c * ma 7 ^ e f lin d? the right-hand member DEVELOPMENT OF THE DISTURBING FUNCTION. 55 being calculated by the formula of the last article. By equat- ing the parts independent of , we obtain The value of -j-? may be found by differentiating the expression for D k in Art. 55, and substituting for ^ and da ^. k+l their values as given by the present article. The successive differential coefficients of C k and D k with respect to a may be obtained from the expressions for -~- and ~ by simple differentiation and substitution. 57. We might determine in the same way the successive differential coefficients of C k and D k with respect to a ; but when those with respect to a have been found, the former may be derived from them, as we proceed to shew. On ex- amining the expansion of (a 2 + a'* 2aa cos )"* in Art. 52, it will be seen that A k is a homogeneous function of a and a of 2s dimensions. Hence C k and D k are homogeneous functions of a and a', the former of 1, the latter of 3 dimensions. It follows that ~ , -^ will be homogeneous, functions of 2 and 4 dimensions respectively ; and so on. Now by a known property of such functions dC k , dC k a -T- + a ~ da da which determines -^ : da 56 PLANETARY THEORY. which determines , dor da da' da , * : da da -, da 2 da da da ' d 2 C which determines , , 2 * : and thus all the differential coeffi- da cients of C k may be determined. In like manner all the successive differential coefficients of D k may be calculated. We are now in a position to simplify the expression for F. We have (Art. 47.) ?K da'* + j aa'D^ tan / tan i' cos (O - fl f ) + . . .! . The following proposition will be found useful. 58. To shew that -> , = - D., an^ *Aa* y ^ = - D . da da da da We have - (7 + (7,008^)+ <7 2 cos2<+ ... = (a 2 + a' 2 - 2aa' cos 0)~* ; therefore - -^ + -^ cos + -^ cos 26 + . . . 2 da c?a da = (a a' cos <) (a 2 + a' 2 2aa' cos )~^J DEVELOPMENT OF THE DISTURBING FUNCTION. 57 therefore + 3 (a a cos <) (a' a cos 0) (a 2 + a' 2 2aa' cos <)" = cos (f) (a 2 -f a' 2 2aa' cos <)"* 3 [ad (1 + cos 2 <) - (a 2 + a' 2 ) cos 0} (a 2 + a' 2 - 2aa' cos = cos (a 2 + d z 2aa cos )^ 3 {aa sin 2 cos (a 2 + a' 2 2aa' cos ^>)} (a 2 + a' 2 - 2aa cos )~ = 2 cos (a 2 + a' 2 2aa cos )"^ sin 2 ^> (a 2 + a' 2 - 2aa' cos c^)- 1 - Now* (a 2 + a' 2 - 2aa' cos ^)"l = 1 J) + D^ cos ^ + ... ; differentiating with respect to < 3 ad sin (a 2 + a' 2 2aa' cos <)~^ = D sin <> + 2D sin 2> + ... therefore -j ^ + -= f-, cos 6 + ... o .7,. j,.' fa da 1 = 2 cos ^ [- D + DI cos -f sin (Z) x sin < + 2Z> 2 sin 2<^> + ..), whence, equating the parts independent of ^>, and also the coefficients of cos 2dada'~ 1 2 " 2 ' o or , /, da da and , *, da da 58 PLANETARY THEORY. 59. Since -y- 5 - is a homogeneous function of a and a of da 2 dimensions, da 2 da da da ' ^ f dC , a*d*C aa' therefore a r-^ + - - 7 2 = da 2 oU 2 2 a , Sim July, a 5 ^ + -^= 5 a'l> 1 . Hence the coefficients of e 2 and e z in the expression for F are each equal to - aa'D . 8 Again, since C t is a homogeneous function of a and a of 1 dimensions, ^ ,i ^ a -T- 1 + a -T- ; * = - (7. ; aa aa hence the coefficient of ee f cos ( -cr') 4 but (Art. 55) 2k C k = aa' (D^ - D k+l ) ; therefore, making k=l, . hence the coefficient of ee f cos (r -nr') - - aa 2 . Again, (Art. 56) DEVELOPMENT OF THE DISTURBING FUNCTION. 59 therefore a?D Q + a -j- = aa D^ . Similarly, a' 2 D +*'jf** aa ' A Hence the coefficients of tan 2 * and tanV are each equal to -\aa'D^ 8 Finally, the expression for F becomes F= m' { + 1 aa D, (e 2 + e' 2 ) - \ aa D^ ee cos (tr - w') [28 4 - aa'Dj (tan 2 i + tan 2 1") + aa ^ tan t tan i' cos (1 - flt') CHAPTER IV. SECULAR VARIATIONS OF THE ELEMENTS OF THE ORBIT. STABILITY OF THE PLANETARY SYSTEM. 60. WE have seen in the preceding Chapter, that the disturbing function, when developed, consists of two parts ; the one independent of the time explicitly, the other in- volving it under a periodical form: we shall consider sepa- rately the effects of these two parts. In the present Chapter our attention will be directed to the first or non-periodical part of jR, which we have denoted by F. The inequalities thus produced in the elements of the orbit are termed secular, in consequence of their very slow variation. 61. By differentiating the expression for F in Art. 59, with respect to the elements, we obtain ^-0 d~ ' dF m ,_. , . , , N --aaD^ee sm (sr OT ), dF m ,- m ,-T. , , i- = aaD.e aa 2J Z e cos (or or ), de 4 4 }}> di m'ncfd m'ncf . -^ fl . .. ,~ ^ /N -, tan z -T- = -- Z\ (tan ^ tan % cos (H - H )} wt ^/^ ^ = A + A^ (e* - tan 2 ) + ^ 8 (e' 2 - tan 2 * ') e' cos (CT txr') + A 4 tan ^tan i" cos (H H'), where in the last expression, A, A 19 &c., have been written to denote certain functions of a and a. 62. To calculate approximately the secular variations of the elements of a planet's orbit , in a given time. Let a 09 e , tsr , &c., be the values of the elements at some given epoch ; a + Sa*, e + Se, OT O + SOT, &c., their values after an interval t : then Sa, e, SCT, &c., are the required variations. By Maclaurin's Theorem, * It will be shewn in Art. 64 that Sa is always zero. 62 PLANETARY THEORY. u> + * which may be carried to any required degree of accuracy, but in practice the first two terms will generally be sufficient. We have supposed the variations of the elements required at a time t after the epoch ; if they be required at a time t 'be- fore the epoch, we have only to change the sign of t in the above. We may remark that ( y ) , (~y~) > & Ct are f * ne order of the disturbing force, since they involve the first power of m ; fd 2 e\ fd z ^\ -I i ? i I -Ta ) , -7^ ) , &c. will involve in 2 and be of the second \at / \ at / order ; and so on. In the short period of one year all terms after the first may be neglected, so that putting t=I, we have Hence the coefficient of t in the above formulae is called the annual variation. 63. Since the elements of the planetary orbits are con- tinually changing, it will be interesting to shew that the dimensions of these orbits, and their inclinations to the ecliptic, nevertheless fluctuate between very narrow limits. This constitutes what is termed the Stability of the Planetary System : in order to establish it, it will be necessary to prove the stability (i) of the mean distances, (ii) of the excentri^ cities, (iii) of the inclinations. SECULAR VARIATIONS. 63 64. To prove the stability of the mean distance of the j- planets from the Sun, and of their mean motions. By Art. 61 = 0, so that a is constant. .Now it will be shewn in a subsequent chapter, (see Art. 91), that to the first order of the disturbing force, the periodical terms of R can produce only periodical variations*; consequently to this order, the mean distance is susceptible of no permanent change. The same is true of the mean motion n, since Jk it = i , and p does not alter. We are hereby assured of the a' 2 impossibility of any of the bodies of our system ever leaving it, in consequence of the disturbances it may experience from the other bodies ; and this secures the general permanence of the whole, by keeping the mean distances and periodic times per- petually fluctuating between certain limits (very restricted ones) which they can never exceed or fall short of. This result may easily be extended to all orders of the ex- centricities and inclinations : for since nt + e always occurs in R as one symbol, e cannot occur in F because t does not, so , dF f da. that , and therefore -r- is zero. de dt 65. To prove the stability of the excentricities of the pla- netary orbits. We will first consider the case of two planets only. By Art. Gl, de m'na z a . . * This result is also true when the square of the disturbing force is included: for the demonstration the reader is referred to Ponte'coulant's Systems du Monde, or to Laplace's Mecanigue Celeste. 64 PLANETARY THEORY. . ., , dv mria'*a ^ , . , < Similarly, -j- -- - - Z> 2 e sin (r w). ut ^/^ Now since Z> 2 is' the coefficient of cos 2< in the develop- ment of (a 2 + a' 2 2aa' cos <)"*, an expression in which a and a' are similarly involved, it follows that A' = A- Hence, multiplying the above equations by e, ?-, e, na n a respectively, and adding, we have m de m , de _____ _ P _ I _ __ p _ _ _ r\ na dt rid dt ~ therefore, since a experiences no secular variation, na na A similar equation holds for any number of planets. Ee- placing for convenience - by (a, a'), we have dp - = m'na (a, a') e sin (r -cr') Cut m'na (a, a") e" sin (-sr -BT") ... ju t/c/ f / / f \ * / ' \ _ _ = ^^ a (a j a) e sin fr w) a n r t / t n\ tt ' ft ti\ mna (a, a ) e sin ^r -cr ) . . . = mri'a" (a", a) e sin (" in-) etc t n n r it i\ t * r tt t\ mna (a , a ) e sin (& ty ) . . . Since D 2 ' = J9 2 , it follows that (a, a') = (a f , a) : hence mul- ying 1 we obtain tiplying these equations by e, -r-, e, &c., and adding, na n a SECULAR VARIATIONS. 65 m de m , de m" , de" whence by integration \na Now observation shews that all the planets revolve round the sun in the same direction, so that the mean motions n, n ', n", &c. are of uniform sign. Hence all the terms of the left- hand member of the above equation are positive. We learn also from observation that the excentricities of the planetary orbits are at present very small indeed, with the exception of the Asteroids, the masses of which are very small. Hence the constant must be small. Since, then, all the terms of the first side of the equation are positive, and their sum always equals a small constant; it follows that every term is small, and therefore that the excentricities are always small*. 66. To prove the stability of the inclinations of the planes of the planetary orbits. By Art. 61 . n tan f sin (H - fl'). at 4/* a . ., , di' mricPa n , . . /rv _. Similarly, -^ = D; tan i sin (fl O). As in the last article, it may be shewn tnat D' t = D t . Hence, multiplying the above equations by m m' ., - tan i , tan i , na na * It should be noticed that the above is satisfactory only for those planets whose masses are considerable, which is the case with Jupiter and Saturn ; but the stability of the excentricities is not confined to these planets. For a com- plete discussion of the subject the reader is referred to Pontecoulant's Syst&me du Monde, or to Laplace's Mecanique Celeste. C.P.T 5 66 PLANETAEY THEORY. respectively, and adding, we have m .di m' ., di' tan i -j + -r-, tan i -j- = 0, na dt na dt or to the same order of approximation, m . d (tan {] m , .. d (tan i'} - tan i --3 - + tan i -^j- - = ; na dt na at i therefore tan 2 i + -j, tanV = C. na na A similar equation would (as in the case of the excentri- cities) be true for any number of planets. Now the inclina- tions of the planetary orbits to the ecliptic are at present very small : hence if we take for our fixed plane of reference a plane coinciding with the present position of the ecliptic, it follows, as in Art. 65, that their inclinations to this plane must always remain very small*. 67. The stability of the excentricities and inclinations may also be established as follows. By conservation of areas 2 (mh cos i} = const., or since A 2 = fM (1 - e 2 ) = (M + m} a(l- e 2 ), if M denote the mass of the sun, we have m 2 Since a is constant, if we neglect -7-77, and the fourth >y Jj-L powers of the excentricities and inclinations, this may be written * We may remark that the above demonstration, like that of the preceding article, is applicable only to the case of planets of considerable mass. SECULAR VARIATIONS. 67 / or to the same order of approximation 2 (m Ve 2 ) + 2 (m Ja tan 2 i) = C. Since we know from observation that all the planets re- volve round the sun in the same direction, all the radicals in this equation must be taken with the same sign. Also, since the excentricities and inclinations are at present very small, the constant must be small. Hence it follows, as in Arts. 65 and 66, that the excentricities and inclinations must always remain very small. 68. It may be observed that the result of the preceding article proves the stability of the excentricities and inclina- tions as far as the third order of small quantities, while in Arts. 65 and 66 it was only established to the second order. We will now shew that if small quantities of orders higher than the second be neglected, the equation of the preceding article includes those of Arts. 65 and 66. On referring to Art. 61, it will be seen that to the second order, the excentricities and inclinations are given by equa- tions independent the one of the other. Each must therefore be the same as if the other did not exist. Hence in the equation of the preceding article, if we make successively i = and e = 0, we have 2 (m yW 2 ) = 0, 2(m V tan 2 *) = X - .Zty). In like manner for the planet m, writing a' for , we nave The forms of these equations suggest the following par- ticular integrals : u=Msm(gt + y), v = M cos (gt + 7), u = Jf 'sin (# + 7) , v M' cos (# + 7) . Substituting these in the differential equations, we obtain from either of the first two and from either of the last two gM 1 = a! (D^M 1 - DJM) -, eliminating the ratio M : M' or - and the roots of this equation will be real and unequal, real and equal, or impossible, according as (a + a')' A' -*'(*' -A 1 ) SECULAR VARIATIONS. is positive, zero, or negative. Now (a + a') 2 A 2 ~ *' ( D ? ~ A 2 ) = ( ~ a? A* a positive quantity, since n, ri and therefore a, a' are of like sign. Hence the values of g will be real arid unequal : denote them by ft, # 2 , and let %, 7 2 5 ^i> ^J ^', M* 5 he tlie cor- responding values of 7, M, M' respectively. Then the com- plete solution of the differential equations will be u = M l sin (ft* + %) + M 2 sin (gf + 7 2 ), v = J/ x cos (ft* + 7 X ) + M z cos (^/ + 7 2 ) 5 u = 7^' sin (^ + yj + Jf,' sin (^ + 7,), ' = If/cos (gj + 7j + Jf 2 ' cos (g z t + 7,). Of the constants in these equations, four are arbitrary and must be determined from observation. We have cos {( ffl -g z )t + 7l - tan = u = , cos (^* + 7,) + M % cos (^ + 7 2 ) ' with similar equations for e and tzr'. 73. Had we considered a system of several planets, we should have obtained by a similar process cos r - = M sin< Jsin with similar equations for each of the other planets. 74 From the form of the expression for e in Art. 72, the stability of the excentricities, in the case of two planets, may ]be inferred. We have 72 PLANETARY THEORY. Consequently the excentricity fluctuates between the limits M^ + Mz and M^-M^, and since we know from observation that MI and M 2 are very small*, it follows that the excentri- city must always remain very small. The period of the changes in the excentricities = 2?r , and is the same for each planet. In the case of Jupiter and Saturn this amounts to 70414 years ! The greatest and least excentricities which Jupiter's orbit can attain are '06036 and 02606, those of Saturn '08409 and '01345; the maximum of each excentricity taking place at the time of the minimum of the other. This follows from the equation na na which has been obtained in Art. 65. 75. It appears from the preceding article that the stabi- lity of the excentricities is a consequence of the periodical form of the solution of the differential equations, a result which depends upon the fact that ff l and # 2 are real and un- equal. Now we have seen that in order that this may be the case, it is only necessary that shall be positive, a condition which might be satisfied if the signs of n, ri, and therefore of a, a' were different. In this case, then, the stability would still subsist. Let us however con- sider what would be the effect of equal or impossible roots to the quadratic from which g is found. In the former case a term would be introduced into u, u, v, and v proportional to * In the case of Jupiter and Saturn, Sir John Herschel finds that M v = -'01715, M a ='04321, for Jupiter; MI= "04877, J)if 2 ' = -03532, for Saturn.- the year 1700 being taken as the epoch. See Article, Physical Astronomy in the Encyclopaedia Metropolitans. SECULAR VARIATIONS. 73 the time, and in the latter the periodical terms would be re- placed by exponentials. Consequently the excentricities would increase indefinitely with the time, and the stability would no longer subsist. 76. We now proceed to examine the expression which has been obtained in Art. 72, for the longitude of perihe- lion, viz. tan w = MI sin (gf + 7,) + M 9 sin M, cos (gf + %) + MI cos (gf + f 2^1/2 cos {(g l ff 2 ) t -f 7 X 7 2 J The maxima and minima values of w, if such exist, will be found by equating -=- to zero. Thus cut If this (disregarding sign) be not greater than unity, the perihelion will oscillate, the period of a complete oscillation being the same as that of the excentricities, viz. - ; but n ~ n ffi fft if, as is the case with Jupiter and Saturn, this be greater than unity, the longitude of perihelion has no maximum or minimum, and the perihelion moves constantly in one direc- tion. Again, , 2 2e 2 2 Hence when e is a maximum or minimum, j- will be at either a maximum or minimum, and the apsidal line will be 74 PLANETARY THEORY. moving most rapidly or most slowly, different cases occur- ring according to the signs and magnitudes of the quantities involved. . 77. When the apsidal line oscillates, to find the extent and periods of its oscillations. We have (Art. 72) tantsr sin (9f + %) + ^ am (gf + 7.) . i cos (&* + 7i) + K cos (9f + 7.) ' , r , s tan-zzr tan (g.t + y.) therefore tan [m gj 7 ,) = - ^^ x - 1 + tan -57 tan (gf + 7^ sn x + 2 COS if = - Also by the last article, if T be the least positive angle , . . < 2 2 whose cosine is -f ~dt ' l l2 ~ OS T Different cases will occur according to the signs of J/J, M^ &c. Suppose M v M z to have different signs, g^ and ^ positive, and g^ greater than g 2 . Then ty increases as t in- creases, and -J- will be negative, or the apsidal line will regrede, while cos ty cos r is positive, i. e. so long as ^ is between 2mr T and 2n7r + T : T- will be positive, or the Cut apsidal line will progrede, while ^ is between 2n7r + r and 2 (fl + 1) 7T-T. To find the angle through which the apsidal line regredes and the period of the regression. Let t', t" be the values SECULAR VARIATIONS. 75 of t, r', -cr" the values of r corresponding to the values 27Z7T - T and 2w-7r + r of r : then ~ 7 2 = sn T . 9 tan (w ?J %) = -=7 ^7 - Ifj + M, 2 cos T / fi \ tan ( 37 q.t yj = ^ - ~ . MI + Jf 2 cos r From these equations the values of t', t", vr', vr" may be found, and thus -57' w" the amount of regression will be known. The period of regression In like manner the amount and period of the progression may be obtained. The latter will be found to be - The period of a complete oscillation will be the sum of 27T the periods of the regression and progression, that is > 9\ 9z which agrees with the preceding article. 78. The motion of the centre of the instantaneous ellipse in consequence of the secular variations of e and -cr may be exhibited geometrically as follows. We have, by Art. 72, e cos r = Jfj cos (gf + y t ) + M z cos (gj + y a ), e sin tar = M^ sin (gf + yj + ^T 2 sin (^ a < + y 2 ) . Let a circle be described in the plane of the orbit with its centre 8 coinciding with that of the sun, and its radius equal to J^a, where a is the mean distance. Let a point P describe this circle uniformly with a velocity g^ starting from 76 PLANETARY THEORY. 0. Again, with centre P and radius equal to M z a let another circle be described, and let a point Q describe this circle uniformly with a velocity g^ starting from C. Let SL be the line from which longitudes are reckoned, and draw PK parallel to it: then if the angle OSN be equal to 7 1? and CPK to 7 2 , the angle PSN will be equal to ffj + v^ and QPK to g 2 t + 7 2 . Produce QP to meet the circle again in E, and draw QN perpendicular to SN. Then, supposing M t and M a to be both positive, we have = SPcos PSN+ PQ cos QPK (gj + 7 X ) + M^a cos (gj + 7 2 ) = ae cos iy. Similarly, it may be shewn that QN = ae sin -or. Hence, the apse being supposed to move from L in the direction contrary to that of the hands of a watch, Q will be the centre of the instantaneous ellipse. SECULAR VARIATIONS. 77 If M l be positive and M z negative, it may be shewn in like manner 'that the centre of the ellipse will be E. If M lt M 2 be both negative, join QS and produce it to Q' so that SQ' = SQ: then the centre of the ellipse will be Q'. A similar construction will of course apply for the motion of the further focus. 79. To integrate the equations for the inclination and lon- gitude of the node. We have (Art. 61) for the planet m 7 vn/nn^n j- = D : tan i' sin (ft - ft'), tan i-jjj- = D^ (tan i - tan i' cos (ft ft')} ; with similar equations for the planet m'. To integrate these, assume p = tan i sin ft, q = tan i cos ft, p = tan i' sin ft', q = tan i' cos ft' ; therefore -f- = tan i cos ft - T - + sin ft (1 + tan 2 i} -y . at at at Substituting the expressions for -T- and -j , and writing cit a/t m na a . ,di , . , i.-i -i a lor , since tan i -y being of the third order may be omitted, we have -y- = ccZ^ (tan i' cos ft' tan i cos ft) Similarly, 78 PLANETAKY THEORY. . , r , , , ... , f Also for the planet w, writing a for - 5 , The forms of these equations suggest the following par- ticular integrals : (Jit + 8), / = JV'sin (^ + 8),' q=N'cos (ht + 8). Substituting these in the differential equations, we obtain from either of the first two, and from either of the last two, hN' = v!D, eliminating the ratio N : N' 9 or therefore h = - (a + a') Z\, or A = 0. Denote the former by ^, and let 8 1? 8 2 , ^, J^ 2 , ^', JV 2 ', be the values of 8, JV, ^V' corresponding to h = \ and A = 0. Then NJ = N 2 , and the complete solution of the differential equations will be p = I sn t + + 2 sn q=N l cos (^< + 8J + ^ s cos ' = JV sin t + S + ^ sin 2 , = N cos SECULAR VARIATIONS. 79 Of the constants in these equations, four are arbitrary, and must be determined from the known values of i and t at some given epoch. We have then tan 2 i =/ + f = A? + ^ 2 2 + 2AyV 2 cos (JiJ _ p N, sin (hj + B t ) + N a sin 8 2 tan 11 = -^7 - it. ^~~\ AT s> q N t cos (^ + oj + jy, cos 2 with similar equations for i' and H'. Had we considered a system of several planets, we should have obtained a result precisely similar to that of Art. 73. a* 80. From the form of the expression for tan i, the sta- . bility of v the inclinations, in the case of two planets, may be inferred. We have tan 2 * = N? + N* + ZN^ cos ( Consequently tan i fluctuates between the limits N^ + ^V 2 and NI ~ N z ; and since we know from observation that N^ and j\ T 2 are very small, it follows that the inclination must always remain very small. Further, the periods of the changes in the inclinations of the orbits of the two planets are the same, being - ; and as appears from the equation of Art. 66, the maximum of each inclination will take place at the time of the minimum of the other. In the case of Jupiter and Saturn, the period is 50673 years ; the maximum and minimum inclinations of Jupiter's orbit to the ecliptic are 2 2' 30" and 1 17' 10", those of Saturn's orbit 2 32' 40" and 47'. 81. We now proceed to examine the expression which 80 PLANETARY THEORY. has been obtained in Art. 79 for the longitude of the node. We have n _ ffi sin (\t + 8 t ) + JV 2 sin 8 2 ' The maxima and minima values of O, if such exist, will j Out be found by equating -j- to zero. Thus . 2 If this (disregarding sign) be not greater than unity, the node will oscillate, the period of a complete oscillation being the same as that of the excentricities, viz. r . But if it be h greater than unity, there cannot be any stationary positions, and the node will move continually in one direction. It may be shewn, as in Art. 76, that the motion of the node will be fastest or slowest whenever the inclination is either a maximum or minimum. 82. When the line of nodes oscillates, to find the extent and periods of its oscillations. It may be shewn as in Art. 77, that if -\Jr be written for j^-j.^ _ S 2) and T denote the least positive angle whose . . N, cosine is Tn? , cos T sn ~~ 1 COS T COS A|T ' and tan 2 i y- = h^N^ (cos ^r cos T). Different cases will occur according to the signs of N^ , N t and h t . Suppose N 19 N z of like sign, and \ negative: SECULAR VARIATIONS. 81 then ^ decreases as t increases, and the line of nodes re- gredes so long as ty is between 2/iTr + T and 2w7r T, and progredes so long as ty is between 2nir r and 2 (n 1) TT + T. Let ft', ft" be the values of ft corresponding to the values 2n7T + T and %nir T of ty ; then tan (ft' - S 2 ) = - cot T, tan(ft"-S 2 )=cotT; 7T therefore ft' S 2 = WTT + T , therefore ft' - ft" = 2r - TT, which is the angle through which the line of nodes regredes. Also the period of this regression may be shewn as in Art. 77 2r to be j- . Similarly, the angle through w T hich the line of nodes progredes may be shewn to be 2r TT, and the period of the progression 7 - . ~~ "i The period of a complete oscillation will be the sum of rt__ the periods of the regression and progression, that is =- , ~~"i which agrees with the preceding Article. The remaining cases corresponding to different arrange- ments of the signs of N l9 N z and \ may be treated in like manner. The mean value of fl is imr + B^, (this will be found to be the case whatever be the signs of N t , N^ and \ ;) and the mean value coincides with the true whenever sin ty = 0. Since then ty is the same both for the disturbed and disturb- ing planet, the nodes of both orbits will arrive simultaneously at their mean positions. C.P.T. 6 82 PLANETARY THEORY. In the case of Jupiter and Saturn N 2 is for each planet numerically less than N 19 so that the node oscillates; the extent of oscillation being 13 9' 40" in Jupiter's orbit, and 31 56' 20" in that of Saturn on either side of their mean position, the ecliptic being taken for the plane of reference, and supposed immoveable. 83. To shew that the inclination of the orbits of two mutually disturbing planets to each other is approximately constant. If 7 denote this inclination, we have by Spherical Trigo- nometry, cos 7 = cos i cos i' + sin i sin i' cos (XI XI') = cos i cos i' {1 + tan i tan i' cos (XI XI')} = (1 + tan 2 1')~* (1 + tan 2 /')"* (1 + tarn tarn" cos (XI - XI')} = 1 - i {tan 2 i + tan 2 i' - 2 tan /tan { cos (XI - XI')}, if we neglect small quantities of orders higher than the second. Now tan 2 i + tan 2 i' 2 tan i tan i' cos (XI XI') = (P-P'}*+(1-CL}" therefore 1 - cos 7 = - (^ - A 7 /) 2 , A or sn = ~; whence it follows that 7 is constant. 84. The equations which give the secular variations SECULAR VARIATIONS. 83 of the node and inclination may be interpreted geometrically as follows*. The equations to be interpreted are p = N l sin (hj + 8J 4- N z sin S 2 , q = JV; cos (hjt + 8J + JV; cos S 2 , where p tan i sin fl, t q = tan i cos H, Since in the differential equations from which these have been obtained, small quantities of the third order have been neglected, we have to the same order of approximation N t sin (hj + 8J sn sn sn sin i cos & = N t cos (hf 4- 3J + ^ 2 cos . . Let a sphere be described with its centre coinciding with that of the sun, and its radius of any magnitude: let the fixed plane of reference cut it in the great circle AL, L being the origin from which longitudes are measured, and LA a * For this elegant geometrical interpretation, the Author is indebted to Mr Freeman, of St John's College. See Appendix, Art. 7. 62 84 PLANETARY THEORY. quadrant. Let another fixed plane inclined at a small angle I to the former cut the sphere in NM, and let LN= a). Let K, P, Q be the poles of AL, NM, and the plane of the orbit respectively: join KA, KL, PA, PL, QA, QL, PQ, KP, and produce the latter to meet AL in 0. Then N will be the pole of OK, and ON a quadrant ; therefore Let PQ= p, and the angle QPO=6. Then from the right-angled triangles A OP, LOP, we have cos AP sin /cos w, cos LP= sin Jsin co, sin AP&in APO sin o>, sin ^IPcos APO = cos I cos w. Similarly cos A Q = sin i cos O, cos L Q = sin /sin H. Now cos J. = cos AP cos P$ + sin AP sin P$ cos APQ = cos JUPcos P + sin ^LPsin P (cos APO cos 0P<2 - sin APtfsin OPQ], or sin i cos O = sin /cos o> cos p + sin p (cos / cos &> cos sin &> sin 6) = sin /cos co cos p + sin p cos (0 + o>), neglecting sin 2 - /sin p. A Similarly, sin i sin fl = sin /sin w cos p + sin p sin (0 + co). Now these equations will be identical with equations (1), if we suppose sin p=*N l9 sin / cos p = N z , SECULAB VARIATIONS. 85 We have then the following interpretation : the normal to the, plane of the orbit moves uniformly so as to generate in space a fixed right circular cone. v^x 85. To integrate the equation for the longitude of the epoch. We have (Art. 61) ^ = A+A, (e* - tan 2 i) + A^ (e' z - tan 2 t ') + A s ee cos (OT -or') + A 4 tan i tan i' cos (fl t') . Now from the formulae of Art. 72, we obtain e" = M? + M' + 2^, cos {( 9l -g,} t+y.-y,}, ee cos w - = + (MM + MJ cos ((g. - g) t + 7, - In like manner, from the formulae of Art. 79, tan 2 i = ^ 2 + JV 2 2 + 2JV; N 2 cos fat + S t - 8 tan 2 1 '= AT' 2 + ^ 2 /2 + 2NJN t cos {^ + S x - tan i tan t ' cos (O - fl') = J^^' + ^" 2 2 If these values be substituted in the expression for ~ dt it takes the form de dt cos ((&-&) * + 7, - yj + B 2 cos (^ + S, - 8 86 PLANETAEY THEORY. where Bn, B t , and B 2 denote certain constants. Integrating, we have A e ^ + + _^. " > '^* ^ ^^ We may omit the term -Zfotf, if we consider it as furnishing a correction on the mean motion n, which thus becomes (1 + B) n. With this understanding B, , jD,, If this expression be developed, we may again omit the term involving the first power of t, and consider it as affording a further correction to the mean motion*. Thus we shall ob- tain a series of the form 86. In the Theory of the Planets this inequality is insen- sible, but in that of the Moon it amounts to upwards of 10 seconds in a century, forming what is termed the secular ac- celeration of the Moons mean motion. Thus it appears that this inequality does not, as its name would seem to imply, contradict the general theorem of the invariability of the mean motions, since it is due to a variation not of the mean motion, (as we have employed the term in the preceding pages,) but of the epoch. If however, as in the Lunar Theory, the epoch be omitted, any variation in the mean longitude will of neces- sity be thrown upon the mean motion ; only in this case, n will not be given by the equation n*a* = fi. 87. We have hitherto supposed the planetary motions to be referred to a fixed plane, but have left the particular plane * The advantage of thus disposing of these terms arises from the fact that the mean motion as determined by observation is the complete coefficient of t in the expression of the mean longitude. SECULAR VARIATIONS. 87 undetermined. In practice it is usual to take the position of the ecliptic at some given epoch, as for instance the year 1800; but since it is to the true ecliptic that astronomers refer the celestial motions, we will now obtain formulas for deter- mining relatively to the plane of the earth's orbit, the position of that of any other planet. Let then m, m denote the masses of the earth and the planet considered, and suppose the orbits of r& and m but little inclined to each other and to the fixed plane of reference. Let X, X' denote the latitudes of points in these orbits corre- sponding to the same longitude l then (see fig. to Art. 13) tan X = tan i sin (0 X - H), tan X' = tan i' sin (0 X - O'). Now since i and i' are very small, we may replace tan X, tan X' by X, X' respectively : thus X' - X = tan i' sin (^ - H') - tan t sin (0 t - 11) = (tan i' cos H' tan i cos H) sin t (tan i' sin O' tan t sin 11) cos t , or, with the notation' of Art. 79, V-X=( 2 '-2)sin0 1 -(|/- d p)cos0 1 (1). Now let 7 denote the inclination, v the longitude of the node of the orbit of m relatively to that of m ; then approxi- mately X' X = tan 7 sin (6^ v) = tan 7 cos v sin 9^ tan 7 sin v cos (2). Hence equating coefficients of sin A and cos X in equations (1) and (2), tan 7 cos v CiQi tan 7 sin v = p' p ; whence tan 2 7 = (p' p}* + (q #) 2 , 88 .PLANETARY THEORY. and io^v , -. 2-2 These expressions determine the position of the orbit of m relatively to that of ???,, when the values of p, p, q, and q are known. Differentiating them, and neglecting small quan- tities of orders higher than the second, we obtain dv (dp dp\ . fdq dq\ -31 = (j* ~~fl} Sm v + \~jl J* COS v > , dt \dt dtj \dt dtj dv _ (dp dp\ cos v fdq dq\ sin v di ~ \di ~~ ~di) tan 7 " \dt ~ dt) tan 7 * If the values of -~ , -^- , &c. be substituted, these equations give the variations of 7 and v. 88. For the theory of the invariable plane of the solar system, the reader is referred to Pontecoulant's Systeme du Monde. CHAPTER VI. PERIODIC VARIATIONS OF THE ELEMENTS OF THE ORBIT. 89. WE come now to consider the variations produced by the periodical terms of R. These are called Periodical Variations, as opposed to the Secular Variations produced by the non-periodic terms. We have seen indeed that the latter are for the most part periodical in form, but in the Planetary Theory, the term Periodical Variations is restricted to those we are about to consider in the present Chapter. 90. We have seen (Art. 46) that the general type of a periodical term is P cos {(pn qri) t + Q], where P is a func- tion of a, e, i; and Q is a function of r, e, fl. Now such a . dR dR , dR , term will produce a similar term in -3 , -T- , and -^r ; out a term of the form P sin {(pn qri) t+ Q} in ^ , -^ , /77? an( j . If then these be substituted in the equations all of Art. 39, they will take the forms 90 PLANETAEY THEORY. -&=?* cos ((P n ?') * + <2)' + G}, where P 1? P 2 , &c. are functions of the elements of the dis- turbed and disturbing planets, and involve the first power of the disturbing mass. 91. In integrating these equations, we may in general consider the elements which enter in the right-hand members as constant and equal to their values at the epoch from which the time is reckoned*. Let then a, e, -or, &c. denote the values of the elements at epoch, Sa, Be, SOT, &c. their periodical variations after an interval t : then integrating the above equations p Ba ; -- 1 7 cos {(pn qri) t + Q}, pn Hr CTfl p Be -* r cos {(pn qri] t + Q}, p SOT = -? , sin {(pn qri} t + Q}, pn qn * This is equivalent to neglecting the square of the disturbing force : see Art. 95. PERIODIC VARIATIONS. 91 p '' = " 7^+W C S {( P U *** * + ' Hence it appears that the variations produced by the periodical terms of It are all periodical in form. 92. It will be seen that all the expressions of the last Article involve the divisor pn qri , while Sf involves the divisor (pn qnf. If then it should happen that either pn + qn or pn - qn' is very small, a term in R containing (pn qn) t in its argument, though of a high order, may have a sensible effect on the elements of the orbit. Now since p and q are either positive integers or zero, pn + qn cannot be small unless n and n are small, a case which does not occur with any of the planets: but we have instances in which pn ~ qn is small*. Since the period of such inequalities is very great (being , ) , they are called inequalities of long period, or long pn "** qn / inequalities. 93. To select such terms in R as will produce the prin- cipal inequalities of long period. * If in any case the mean motions of two planets were exactly commen- surable and in the ratio of p to q, the corresponding term of R, as we have already remarked (Art. 48), would cease to be periodical and would form a part of F, but no instance of this occurs among the planets. 92 PLANETARY THEORY. We have seen that the dimension of the principal part of the coefficient of a term containing (pn ~ qri) t in its argu- ment is p ~ q (Art. 50) ; hence if we can find two integers p and q nearly in the ratio of n to n, and having a small difference, the corresponding term of E will produce an im- portant long inequality in the elements of each planet. In the case of Jupiter and Saturn n : n :: 5 : 2 nearly, and 5 2 = 3; hence there is a long inequality arising from a term in R of the form P cos{(2/i 6ri) t + }, the prin- cipal part of P being of the third order. This inequality is interesting in an historical point of view, having long baffled the labours of mathematicians and appeared inexplicable on the hypothesis of gravitation. It was at last successfully explained by Laplace. For the Earth and Venus, n : n :: 8 : 13 nearly, so that there is a long inequality arising from a term in R of the fifth order. The discovery of this inequality is due to the Astronomer Royal. Finally, in the case of Neptune and Uranus, n : n :: I : 2 nearly, hence there is a long inequality arising from a term in R which is of the first order. 94. Between corresponding terms of the long inequali- ties in the mean motions of two planets, arising from the near commensurability of n and n', there is a simple approx- imate relation. Let m, m be the masses of the two planets, -5, R' their disturbing functions: then by Art. 8, considering only the mutual action of m and m, we have PEKIODIC YAKIATIONS. 93 We shall distinguish the first and second terms of R and R as the symmetrical and unsymmetrical parts respectively, since the co-ordinates of m and m are involved symmetrically in the former but not in the latter. Since then the symmetrical parts of R and R' differ only in having m and m' interchanged, if m'M cos {(pn qn} t + Q] be any term in the symmetrical part of jft, that of R will con- tain the term mMcos {(pn qn} t+ Q}. Confining our attention to these terms, we have (Art. 39) d*$_ 3n*adR_ Znad(R) dt* fj.de jj, dt ^ m'M sin {(pn qn} t + Q} ; ,T r r^ %ri*ap m'M therefore 8f = - _ sm {(pn - qn') t+Q}. a- -i i **> 3n' 2 dq mM . ,, ,. ^ Similarly, S = ^ > sm {(pn-q.n}t+ Q}. TJ > Hence ~ i = -- of w/t/i a q approximately, since qn is nearly equal to pn ; therefore or since ^' differs from ^ by a quantity of the order of the 94 PLANETARY THEORY. disturbing force, the square of which we are neglecting, we have 5? m'ija' S~ ~ m V ' the required relation. The same relation is also approximately true in the case of terms arising from the unsjmmetrical parts of E and R'*. For denoting these by J5 X and E{ respectively, we have x' Now the equations of motion of the planet m referred to rectangular axes are = _ * ' dt? r* dx Hence, the differential coefficients being taken as if the elements were constant f _ df ? r' 3 ~ 1 dt* m therefore B * For the demonstration of this we are mainly indebted to The Theoi'y of the Long Inequality of Uranus and Neptune: an essay which obtained the Adams Prize for the year 1850. By E. Pierson, M.A. j* This is simply an analytical artifice : we merely assert that if the differen- x' tial coefficients be so taken, then -^, &c., and therefore RI and JRi will take the above forms. PERIODIC VARIATIONS. 95 , mf,d*x t ,d*y ,d* , mf,x t ,y Similarly, B, = - (x -^ + y w Now any term in E^ containing (pn qn'} t in its argu- ment can arise only from the combination of terms in #, y y and z, containmg^??^ in their arguments with terms in -- , ~ , and -^ containing qrit. Suppose then x and x when dt cit developed in terms of t and the elements to contain respec- tively the terms L cos (pnt + Z), L' cos (qnt + I'). d z x Hence the product x -^ will contain the term - i LL'fn* cos {(pn - qn} t + 1-1'}, d z x and the product x --, 2 the term - LL' V cos {(jw - ^n') * + 1 - Z'} : A the coefficients are in the ratio (fri* to ^?V. Similarly, the coefficients of the same cosine in y j- and z -7-5- are to at at those in y' - and z -^ in the same ratio. Hence if MqV cos {(pn qn} t + Q} be any term in R^ then E^ will contain the term - J^V cos {Q?/i - 2/0 < + Cl- 96 PLANETARY THEOEY. Confining our attention to these terms, we have s., Srfap m'MqV . ~ s ~^ : ma m'na TT , Hence ^ = -- -? = -- nearly, eg map mn a m the square of the disturbing force being neglected. By means of this relation, when one of the long inequali- ties is known, the other may be calculated : it may be used as a formula of verification. 95. We have remarked that in integrating the equations of Art. 90, we may in general consider the elements which enter in the right-hand members as constant and equal to their values at the epoch from which the time is measured. In the case, however, of inequalities whose periods are very long, the secular variations of the elements in the interval produce a sensible effect. In order to take account of these, we may integrate our equations by parts, considering the elements variable ; and then substitute their values as calcu- lated by the method of Art. 62. For example, consider the equation = Psin X, suppose. Integrating by parts, and remembering that n is constant with regard to secular variations, we have I PERIODIC VARIATIONS. 97 P 1 dP . . 7 COS X + 7 7V2 -JT sm * pnqn (pn qn) dt I d*P (pn-qnf dt 2 i * P t -'; therefore o? = 7 ^2 sin X ^ r^ -,- (^>n - qn ) 2 (^>^ ^ ) dt + -T- cos X + (pn-qrif dt^ T (pn-qn')* df 1 (Art. 22), and - therefore = + (1 - cos i) ~ (see Art. 29) 72 100 PLANETARY THEORY. since (1 cos i) T- being of the order of the square of the disturbing force may be neglected. Let Sr, SO, and Sh be corresponding variations in r, and h ; then d( dt JO dW h / Bh or :77 + -^r == - dt dt r neglecting the square of the disturbing force ; therefore dt ~ r* r* > which gives the variations in longitude. The value of Bh may be found from the formula dh = dE dR dt de dix ' For the periodic variations in latitude, we refer to Ponte- coulant's Systime du Monde, Tome I. p. 492. 100. As an example of the processes of this chapter, we will calculate the variations in radius vector and longitude due to the term m'Me cos {(n 2ri) t + e 2e' + -cr} in JR. Considering this term only, we have E = m'Me cos {(n - 2ri) t + e - 2e' + ^} = m'Me cos X, suppose. dR , dM dR Hence r =m e cos X, -j~ =m Jf cos X, da da de dR , _ . . dR -7-= wiJfesinX, -j = m Me sm X, dR dR PERIODIC VARIATIONS. 101 Substituting these in the formulae of Art. 39, and neg- lecting small quantities of orders higher than the first, we have da 2na* tlt , . ~-Ti~ m Me sin X, de na t ^ r . 3J- wJfBinX, dm na ,,, e -T- = m .M cos X, de 2nd* ,dM I na ,,, -77 = m -j-ecos\ + - mJfcosX, at p da 2 fj, rr = m Me sin X. of p By integration we have ., 2m M na*e oa = , cos X, m'M na ---- r-7 cos X, p n 2n m'M na n m'M 2m'a dM\ nae . = I ---- -; - r-/ sin X, \2 fju fjudajn-2n *,, 3m M . . 6f = --- 7 - TTj smX. p (n 2n) z Also r = a l + e 2 - e cos (f+ e - -or) - e 2 cos 2 (f+ e - r) - 102 .PLANETAEY THEORY. therefore, small quantities of orders higher than the first being neglected, aS-l-co B (?+ e -,r), - dr = a [e cos (f + e r) e cos 2 (f + e - r)}, - r - = a(esin(f + WOT -T- =oesm XT , Jr o, dr ., <7r -, dr ^^ dr ~ Now Sr = -j- 3a + -j- Se + -j- SCT + -^ Sf + -7- Se 2m' M na?e 7 cos ii -w 9 DIRECT METHOD OF CALCULATION. 113 110. We may, however, omit the last term: for, con- sidering this only, the radius vector of the planet becomes a \ 1 e cos (nt + e tzr) + ^ cos (nt B) + ...[ A = a [1 - {e cos (e w) cos B} cos w A + {e sin (e -or) + z sin B} sin n + . . .] a [1 e 1 cos> (nt-\- e wj +...}, if e x cos (e -cj-J = e cos (e -cr) ^ cos B, ^ sin (e -crj = e sin (e ur) + - 2 sin i?, from which e 1 and ^ may be determined. Now since the ellipse upon which our approximations are based, has been obtained by neglecting the disturbing force, we may in the elliptic formulae replace e and CT by e^ and ^ respectively, since they differ by quantities of the order of the disturbing force. If this be done, our first approximation becomes Sr = 4-(2 M > + a^) n'a \ 2daJ p H r~2 r- 1 ^77 cos {(P n qn) I + Q}- a \n (pn qn) \ 111. In order to obtain a second approximation, this value must be substituted for &r in the right-hand member of the equation of Art. 108. Also since the square of the dis- turbing force is neglected, we may write e^ and r x for e and ix in this equation. We will write for brevity Sr = L + P 2 cos {(pn - qri] t+Q}. C. P. T. 8 114 PLANETARY THEORY. Substituting this in the equation of Art. 108, and omitting those terms which have produced the first approximation*, we have = - 3tt\ cos (nt + 6-vr 1 )[L + P 2 cos {(pn - qri) t + Q}] = 3n^e 1 L cos (nt + e -GrJ W^PgCOS 3 112. On the form of this equation, we have an important remark to make. In consequence of the term 3n^e : L cos (nt + e -crj, its integral will contain the term - e^ntL cm^nt + e wj. Here, then, we are met by a difficulty: our equations Irave been formed on the hypothesis that the square of Sr is small enough to be omitted, whereas here, we have a term capable of indefinite increase. This term then, if retained, would ultimately vitiate the whole approximation. The diffi- culty might, as in the Lunar Theory, be obviated by writing en for n in the elliptic formula, which amounts to supposing the perihelion to be in motion. Its motion is however better found by the method of the variation of elements. Indeed it * These terms are omitted for the sake of brevity: in order, therefore, to obtain the complete second approximation, we must add to the integral of the above equation the result of the first approximation. DIRECT METHOD OF CALCULATION. 115 may be shewn that such terms as those we are considering, lead to the formulae which have already been obtained, for the secular variation of the elements*. We shall accordingly altogether neglect such terms, and suppose the elements of the ellipse on which our approximations are based, to have been previously corrected for their secular variations. 113. With this understanding, the complete integral of the equation of Art. Ill, will be 3 * T> rBr = - 2 *-{(/? + !) n-qn'}* 2 n* - {(p -l)n- qri} cos [{(p-l)n- qri} t + Q - e + rj f A cos (nt B). If this be added to the result of the first approximation, we obtain for a second approximation 1 / , m JOA P. rSr=-i(2mg + -a cos {(pn qn) t + Q] cos [{(p + l)n- qn cos [{(p - 1) n - qri} t+Q-e + vr^ + A cos (nt B). * It is thus that the Secular Variations are first obtained in the Mecanique Celeste. See Ponte'coulant's Systdme du Monde, Supplement au Livre II. 82 116 PLANETARY THEORY. The arbitrary constants might Ibe disposed of as in the first approximation, but it is more convenient in practice to determine them otherwise. In order to obtain a second ap- proximation to the value of Sr, it is only necessary to multiply the right-hand member of the above equation by - {1 + e t cos (nt -f e vrj}, neglecting e*. 114. To calculate the perturbations in longitude. We have (Art. 106) z*a n d.r$r jj dr fdR , f [dR _. JiW = 2 = -- Br -y 2a -7- dt %n y- dr. dt dt J da JJ de Taking for simplicity, the first approximation to the value of rSr, which has been obtained by neglecting the first power of the excentricity, we have 1 / m dC\ P. rSr = - 2 277i'^ + a-7- +^ -. s -- 7^ n \ 2 da / n 2 (p/i gw ) cos {(pn gn] t + Q] ; therefore d.rSr 2Pnn . also, neglecting the first power of the excentricity, = . dt By Art. 108, writing - m'C for F, we have j dR 1 , cZC rfP DIRECT METHOD OF CALCULATION. 117 dQ nP PMMQOi pn gn therefore 2a I ~- ^ and n\--r-dt= - ^r 3nP pn qn y da pn gn where/ is an arbitrary constant. Hence by substitution, we have =/- (m'a ^ + 3m' (jw + D, cos + ... + A cos or, putting z equal to 0, and substituting a' tan H sin (n't + e Of) for z (see Art. 42), - ma tan i' sin (n't + e' - Q') j- D + -, 3 + A cos + ...i . This expression, after reduction, consists of terms of the form Q], where p and q are positive integers, and either may be zero. Considering one such term, our equation becomes - + rSx = Psin 120 PLANETARY THEORY. Now as in Art. 108, - 3 = ft 2 (1 + 3e cos (nt + e r) + ...}; hence, neglecting the product eS\, The integral of this equation is p If instead of taking for the fixed plane of reference, the plane of the orbit of the disturbed planet at epoch, we take a plane slightly inclined to this, we may omit the arbitrary term. For, denoting the planet's latitude with respect to this plane by X, we have approximately X = tan i sin (nt + e 11), and it may be shewn as in Art. 110, that omitting the term in question is only equivalent to changing slightly the values of i and H. CHAPTER, VIII. ON THE EFFECTS WHICH A RESISTING MEDIUM WOULD PRODUCE IN THE MOTIONS OF THE PLANETS. 118. IN the preceding chapters, we have supposed the planetary motions to take place in free space, and the results of calculations based upon this hypothesis manifest a very close agreement with observation. There is, however, a remarkable circumstance connected with Encke's comet which seems to indicate the possibility of the existence of a very rare medium, too rare indeed to cause any sensible resistance to the motions of the planets, but which, as we shall presently see, may yet influence the motions of comets, in consequence of the extreme smallness of the masses of these bodies. It has been observed that the comet above referred to (which describes an elliptic orbit in a period of about 3j years,) has since its appearance in 1786, been moving round the Sun with an increasing mean motion. Encke attributes this to the resistance of a medium pervading space. We shall therefore proceed to examine the effects which such a medium would produce upon the elements of a planet's orbit, assuming the resistance to vary as the product of the density of the medium and the square of the velocity of the planet. We shall neglect, in the present investigation, all forces except the Sun's attraction and the resistance of the medium ; conse- quently the planet may be supposed to move wholly in one plane. 122 PLANETARY THEOEY. 119. Let r, 6 be the radius vector and longitude of the planet, s the length of an arc of its actual orbit measured from some fixed point to its position at time , and p the density of the medium. Then if Jc be a constant, we may represent the resistance on the planet by kp (-J- J , and the equations of mo- tion will be d'r dO\ z ju ds z dr 7/1 If r 2 -j- = hj these may be written .d'r de\ z L , dsdr d ( 2 d&\ 7 7 ds (r*-j-) =-7cph-r ....................... (2). dt\ dt) r dt These equations are the . same in form with those of Art. 20, and may be treated in a similar manner, kp-^- -j taking the place of -7- , and Jcph -j- that of -^ . We have from equation (2) dh .. ds 120. To obtain a formula for calculating the mean dis- tance. We might proceed as in Art. 25, but we shall here employ the method of Art. 26. We have d*s i dr ds RESISTING MEDIUM. 123 and by a known formula of elliptic motion dt r a' Differentiating the latter, we obtain ds d*s _ 2yt6 dr fi da 2 diW~~fdi + a*~di ds and multiplying the former by 2 j- , dsd 2 s 2/4 (di) > -da = 121. To obtain a formula for calculating the excentricity. We have, as in Art. 27, Differentiating as if were constant, and -writing k--r , r for - = - kp dtdt) + 7 124 PLANETARY THEORY. Now from equation (3) 2 ) /g \ ds (r~ l )dt' This result may also be obtained by differentiating the formula h? = fj,a(l e 2 ), and substituting the expressions for dh , da j- and -y- , as in Art. 28. 122. To obtain a formula for calculating the longitude of perihelion. We have, as in Art. 29, Differentiating as if r and 9 were constant, and writing 7 dr ds r d*r kp -T. -T. tor -T-O , dr 2 . - . J-nr j dr ds , fi ^ fl t p \ dh dh , ds or since 7t = ~ dt > EESISTINQ MEDIUM. 125 dr v d'uf , a but from equation (4) of Art. 22, dr fQ . fM _cosec(0 )-=?; therefore -7- = " sm (0 ") T dt e J dt 123. Before proceeding to obtain a formula for calcu- lating the epoch, we shall express the results of the preceding articles in a form convenient for application. If u denote the excentric anomaly, we have r OL (1 e cos u) (1), nt + e tar = M e sin u (2). Hence ( -y \dt a \\ e cos u _ fjb I + e cos u $ a 1 ecosw' 6?s //I + e cos tA . a therefore -7- = wa A / - , since no? = v w. a^ V V 1 ~- e cos w / From (1), by differentiation as if the elements were inva- riable, we have dr du 126 PLANETARY THEORY. and from the equation dr dr equating the two values of -j- , (jit . //i v ha . du From (2), by differentiation as if the elements were inva- riable, we have dt \ .. . -j- = - U & cos du n ^ Hence, by substitution, the formulas of the preceding articles become da , 2 , -j- = - 2&pa 2 (1 + e cos u du N //I 4- e cos u\ ) A / ( - - , ' y \l - e cos u) ' //1 + ecostA A / - - , V Vl-ecosw/' -j Arfft/fsiA/ v * " v J COS cv du dix _ 2/jo \/(l e 2 ) . //I + c cos iA rtw e \/ \1 e cos M/ " 124. To obtain a formula for calculating the epoch. By differentiating the formulae r = a (1 ecos w), -sr = u e sin w, considering the elements, first variable, and then invariable, and comparing the results, we obtain du de . . da ae sin u -=- a cos u -, (1 e cos w) -y- , ndt + e. Thus, on substituting the values of -^ , ~ , and RESISTING MEDIUM. 127 dn de d^ du de t -j- + -T- -j- = (1 e cos u) -j sin u -r : dt dt dt v ' dt dt' whence, eliminating -y- efo, o?e c?-cr __ cos u ede (\ e cos w) 2 da dt dt dt e sin u dt ae sin u dt ' fl 77 As in Art. 37, we may omit the term t -7- if we bear in mind that the mean longitude will then be denoted by de dt j- , and reducing, we obtain de %Jcpa , *\ a //I + -y- = jl VU - e) e cos Mj sin w A / ( - , . du e V U ~ e cos w / 125. The formulae of the preceding articles are sufficient to determine the elements of the orbit at any time, and being perfectly general, are applicable as well to the motion of comets, as to that of planets, but before we can integrate them, we shall require a knowledge of the form of p. Now the analogy of the terrestrial atmosphere would lead us to suppose that if the sun be surrounded by an ethereal medium, its density decreases as the distance from the sun increases. Moreover, the researches of Professor Encke on the comet which bears his name, seem to indicate the law of the in- verse square. We will, however, assume p to be such a function of r. that when multiplied by A /(r - -) , and J V V 1 ~~ e cos U J developed in a series of cosines of u and its multiples, it takes the form A + Be cos u + Ce z cos 2w + ... 128 PLANETARY THEORY, Thus our formulas become da -:- = 2Jca? {A + (A + B) e cos u +...}, de ( . Be 1 -7- = 2&a \A cos u H (1 + cos 2w) + ... k dm n7 ( A Be . 1 e ~T~ 2&a -j -4 sin w H sm 2u + ,...> , > / -T- = ka (Ae sin w + ...). aw 126. Supposing the orbit nearly circular, to examine the effect of the medium upon the elements of the orbit. Since the orbit is nearly circular, we shall neglect squares and higher powers of e ; thus the preceding formulas give on integration a = const. 2&a 2 [Au + (A + B) e sin u}, 7 f A Be f sin 2 e = const. - 2ka ^Aewu + (u+ {Be A cos u H 4 e = const. lea Ae cos u. Hence in an entire revolution of the planet, the mean distance is diminished by Trlca z A, and the excentricity by %7rkaBe, while the longitudes of perihelion, and of the epoch remain unchanged. Also from the formula n = ~- , it ap- pears that the mean motion is, in an entire revolution, in- creased by RESISTING MEDIUM. 129 127. We have already remarked that no traces of a resisting medium have yet been discovered in the motion of the planets : but, since k varies inversely as the mass of the body acted upon, the formulas of Art. 123 shew that such a medium, though too rare to influence the planets, might yet sensibly affect the motions of comets, in consequence of the extreme smallness of their masses. C P. T. PROBLEMS. 1. SUPPOSING in the Problem of the Three Bodies the relative orbit of two of the bodies to be a circle described uniformly, obtain equations for determining the motion of the third body ; and transform the system of co-ordinates, so that the plane of the circular orbit being that of xy, the axis of x shall always pass through the two bodies in that plane. 2. Shew that the plane of the orbit of a planet revolves about the planet's radius vector as an instantaneous axis*. 3. A particle is describing an orbit round a centre of force which is any function of the distance, and is acted upon by a disturbing force which is always perpendicular to the plane of the instantaneous orbit, and inversely proportional to the distance of the body from the centre of the principal force. Prove that the plane of the instantaneous orbit re- volves uniformly round its instantaneous axis. 4. Find when the curvature of the instantaneous orbit of a body, acted on by disturbing forces, is the same as that of the actual orbit ; and shew that this is always the case when the only disturbing force arises from the action of a resisting medium. * In this and the following problem, the plane of the orbit must be supposed to have no angular velocity about a normal to itself. See note to Art. 19. PROBLEMS. 131 5. If R be expressed on the one hand as a function of r x , l9 and z (Art. 11), and on the other as a function of r, 0, t, and 1, being measured on the plane of the orbit from the node, prove that and obtain a formula for calculating the inclination. 6. If R be expressed as a function of t and the usual elements, obtain the formulae dt h sin i di ' di_cot(0-n)dR dt~ h ~ di ' where is measured on the plane of reference as far as the node, and thence on that of the orbit, and h = r 2 [ -7- - 2 sin 2 - =- ) . \dt 2 dtj i 7. The central force being - a + ~ , obtain the following equation for the apsidal motion d _ Vq(l -e a ) /A' cos (0 - dt ~ a, e and & being elements of the instantaneous ellipse. 8. A body revolving about a centre of force, which varies inversely as the square of the distance, is constantly retarded by a small constant force; find the alteration of the major axis, excentricity, and apse, in one revolution. 92 132 PLANETARY THEORY. 9. When the disturbing function E is independent of 6, find expressions for -=- and -7-. m' If ^ = , these expressions give variable values for e and -cr, whereas the motion of the body actually takes place in a fixed ellipse: shew this, and explain the apparent paradox. 10. A planet describes an orbit under the action of a force ^ tending to the Sun, p not being quite constant: obtain the following equations for the variations of the ex- centricity and longitude of perihelion ; = cos (6 OT), 'CT . fn . lie -j = sin (u CT). If djjb be always positive, what in a whole revolution is the nature of its effects upon the excentricity and position of the major axis ? 11. If the equation of the Moon's orbit be reduced to the form shew that the excentricity and longitude of perihelion may be found from the equations de /. //i \ dnr /. //i \ ^= -/sin (0-*r), e =/cos ((9 - r). Apply these equations to find e and -or, when / is a small disturbing force, depending only upon the Moon's distance from the Earth. PROBLEMS. 133 12. Assuming the differential equation for s in the Lunar Theory to be d*s - (3 3 rt x/1 + s = - w 2 5 - + - cos 2 (0 - shew that if 7 be the longitude of the Moon's node, ^ = - | m 2 {1 - COS 2 (010 - 7) - COS 2 ((9-7) + cos2(0-ro0)J. From the above expression for -Jr , find the ratio of the mean motion of the node to that of the Moon, taking into account terms of the order m 4 . 13. If two planets disturbing one another were revolving in periods of 350 and 201 days, what form of terms in the disturbing function would demand examination ? 14. The periods of Venus and the Earth are 224*7 and 365*256 days respectively; find approximately the period of the long inequality arising from their mutual perturbations, the important term in the disturbing function R being of the form PeV 2 cos (13 (nt + e) - 8 (rit + e') - 3*7 - 2w'}. 15. The radius vector of a planet is affected with a small periodical inequality ; shew that its effect may be represented by continued and periodical alterations of the excentricity and PT longitude of perihelion, the period of either being ^^ ^ , where P is the period of the planet and T that of the in- equality. 134 PLANETARY THEORY. 16. If in addition to the force of the Sun on a planet there be a small force tending towards the Sun, and varying inversely as the w th power of the distance of the planet from the Sun, prove that the perihelion of the orbit will have a progressive or regressive motion according as m is greater or less than 2. Can you explain this result by reasoning similar to that used in Airy's Gravitation ? 17. It has been found by comparing theory with ob- servation that the perihelion of Mercury progresses at a rate greater by a than that due to the attraction of known bodies : shew that this increment would be accounted for if the law of / i force tending to the Sun were ~ + ^ , and if tf = ac 4 A /- , the T T y' C orbit being supposed to be nearly a circle, and the mean dis- tance to be c. 18. The central force acting on a body being shew to terms inclusive of p' and the square of the excen- tricity, that the motion is in an ellipse revolving uniformly about the focus. 19. Shew by means of the formula da _ 2na* dR dt IJL de that the chief perturbation of the axis major of the Moon's orbit may be expressed by the equation = a 1 + ^~T - ~\ cos 2 ( nt + ~ n 't ~ 2n (n n ) PROBLEMS. 135 where n and ri are the mean motions of the Moon and Sun respectively. 20. A satellite revolving in an ellipse of small excen- tricity is disturbed by another satellite revolving about the same primary ; find approximately the variation of the mean distance and the motion of the apse, corresponding to the terms ^ r* [1 + 3 cos J2 (n - ri) t + e - e}] in the function R, having given na - e 2 dR _ dt fi, de ' dt fjLe de * 21. Prove that, neglecting periodical variations, the ex- centricity of any orbit can always be represented by the diagonal of a parallelogram, whose sides are constant, and angle varies uniformly. 22. Given the equations tan 2 1 = A? + N* + 2N,N S cos (hj + B l - S 2 ), fan o _ " explain the nature of the motion of the node, when the mini- mum inclination is zero. 23. Prove that as far as secular variations only are con- cerned the function F is constant. 24. Considering only secular variations, obtain the fol- lowing equations : / ^N 2 / tan .^ = \na dt) \na dt J 136 PLANETARY THEORY. 25. If the squares of the masses of two mutually dis- turbing planets were to each other inversely as their mean distances, shew that the nodes would oscillate through equal angles. 26. If Mj m, m be the masses of three bodies mutually attracting according to the law of gravity, M being- much larger than m or m', and if v, v be the velocities of m, m at distances r, r' from the centre of Jf, supposed fixed, shew that the equation of vis viva for this case may be assumed to be 2 t '2 c*n / rf m m m> m '\ * mv* + mv* + 2if - --- + , -- ; = 0, \2a r 2a r J %a and 2a being the major axes of the instantaneous ellipses of m and m. 27. Infer from the foregoing equation by the method of the variation of parameters the ratio of simultaneous changes in the mean distances and mean motions of two planets mutu- ally disturbing. 28. If r be the true radius vector, 6^ the projected longi- tude, and X the latitude of a planet, obtain the following equation of motion : d\\ z i dR 29. Obtain the following equation between the pertur- bations of a planet in longitude and radius vector, whatever be the law of force, provided it be central and a function of the distance only, and provided such a function as E can be found : dt j dt dr -4^Sr-2r dr . PKOBLEMS. 137 where F denotes the central force, and Ji twice the sectorial area described by the undisturbed planet round the Sun. 30. If the orbits of two planets which disturb each other be very nearly circular, shew that the inequalities of the radius vector may be immediately deduced from those of the longitude by means of the equation d.W Tt 31. Integrate the equation -4r^ + n* r$r = 2 {Pcos (put + Q)}, determining the arbitrary constants so that Sr = 0, and 7 r\ -4 = 0, when t = : and shew that for small values of t t Pt z the case of p = 1 being included. 32. A planet moves in a resisting medium of which the resistance apply the equation to obtain the following, in which e 2 is neglected : df + n*.rSr + n*. rSr . 3e cos (nt + e -or) + 2/^ 2 a [nt-+ e sin (nt + - r)} = 0. 138 PLANETARY THEORY. 33. The co-ordinates of the position at any time t of a disturbed planet being x + &e, y + y, z + $z, reckoned from the Sun's centre as a fixed origin, and referred to the plane of motion at a given epoch; and r being the heliocentric distance, x, y the co-ordinates of the position which the planet would have had at the time t, if the disturbance had ceased at the given epoch ; obtain the following equations for determining Sx, Sy, 8z to the first order of the disturbing force : ^() ,/ in which JJL is the sum of the masses of the Sun and planet, and R' is put for f* u* dR W + ^ = d^ (3)> APPENDIX. 141 Let a, b, c, d, e, f be the six elements introduced by in- tegrating these equations when R 0, and for -v- , -M- , -T- write a?', y', z: then a;', ?/', and z can be expressed as functions of t and the elements ; hence dx' da dx db where in f-T-J the elements are supposed constant. If in equation (1) we put R equal to 0, we have ] does not contain the time explicitly. The same is of course true of [a, c], [b, c], &c. It follows, then, that whatever system of elements be adopted, we can always express their differential coefficients in terms of the partial dif- ferential coefficients of E with respect to them, multiplied by coefficients which do not involve the time explicitly. 6. From the formula of Art. 3 of this Appendix, which is due to Lagrange, those of Chapter II. may be deduced : for this we refer to Pontecoulant's Systeme du Monde, Tom. I. p. 542. APPENDIX. 145 ON THE GEOMETRICAL INTERPRETATION OF THE FORMULA FOR THE SECULAR VARIATIONS OF THE NODE AND INCLINATION. 7. In Art. 84 we have shewn that in consequence of the secular variations of the node and inclination, the normal to the plane of the orbit moves uniformly, so as to generate in space a fixed right circular cone. This result may also be obtained somewhat differently: we will here indicate the process *. The equations to be interpreted are p = tan i sin ft = N^ sin (hjt + 8 X ) + JV 2 sin S 2 , q = tan i cos ft = ^ cos (hjk + SJ + N 2 cos S 2 . Employing the figure and construction of Art. 84, it will be found that cot QA = tan * sin ft, cot QL = tan i cos ft. These are the geometric representations of p and q. Now it may be shewn by Spherical Trigonometry that n A n tan /sin o> + tan p (sin co cos 6 + sec /cos o> sin 0} COt ty A. U = 7 T~- -7; , 1 tan 1 tan p cos u QJ _ tan /cos 6) + tan p (cos co cos 6 sec /sin + tan p sin (6 + o>), cot QLO tan /cos &> + tan p cos (0 + ). * This method as well as that of Art. 84, is due to Mr Freeman, of St John's College. C. P. T. 10 146 PLANETAEY THEORY. From these equations the interpretation follows as in Art. 84. The advantage of this method is that it affords a geo- metric representation of p and q: on the other hand, in the method of Art. 84, the trigonometrical reduction is simpler. ON THE METHODS OF CALCULATING THE MASSES OF THE PLANETS. 8. There are in general two methods of determining the masses of the planets ; either by observations on a satellite, when the planet is accompanied by a satellite ; or by compar- ing the inequalities produced in their motion by their mutual action, as deduced from observation, with the same ine- qualities calculated from theory. The secular variations are best adapted to give the most exact results ; but these are not yet known with sufficient accuracy to allow of this use. We are therefore obliged to recur to the periodic variations, and, by combining a vast number of observations, gather from them the most probable results*. 9. When the planet is accompanied by a satellite the formula for calculating its mass may be obtained as follows : Let M, m, m be the masses of the Sun, the planet, and the satellite : P, P' the periodic times of the planet about the Sun, and the satellite about the planet ; a, a the mean distances of the planet from the Sun, and the satellite from the planet. Then we have o__4 r. p '=- m + m f P z a n therefore * Ponte"coulant's Systime du Monde, Tome m. p. 340. APPENDIX. 147 or approximately, This equation gives the mass of the planet when that of its satellite is known. If the latter be neglected, the formula becomes m P*a* 10. In the case of the Earth, this method is not suf- ficiently exact, but the following may be employed. The attraction of the Earth on a body at its surface, in the parallel of which the square of the sine of the latitude is - , is very o nearly the same as if the Earth were condensed into its centre. (See Pratt's Figure of the Earth, Art. 89.) Let then sin 2 1 - , g = the Earth's attraction on a body at its surface in o latitude I, b the mean radius of the Earth, E the mass of the Earth, M the mass of the Sun, P the length of the year, and a the mean radius of the Earth's orbit. Then _E 2* - = E gVP* gP* /b\ 5 - = = _ - /b\ 5 therefore -^ = , , , = ?_ ( - ) , M 4wV 4?r 2 6 \a) ' where - = sine of Sun's parallax = sin 8".57. 11. For the methods of calculating the mass of the Moon we refer to Ponte'coulant's Systeme du Monde, Tome IV. p. 651. 148 PLANETARY THEORY. Tables of the numerical values of the masses of the planets, and of the elements of their orbits, will be found in Herschel's Outlines of Astronomy, pp. 693 et seq. THE END. CAMBEIDGE: PRINTED AT THE UNIVERSITY PRESS. MACMILLAN AND CO.'S $500ks -^ FOR COLLEGES AND SCHOOLS. The Series of CAMBRIDGE CLASS-BOOKS FOR * THE USE OF COLLEGES AND SCHOOLS, which have been issued at intervals during the last ten years by HACMILLAN AND Co., is intended to embrace all branches of Mathematics, from the most elementary to the most advanced, and to keep pace with the latest discoveries in Mathematical science. 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It is of importance that those who wish to culti- vate any subject jnay be able to ascertain what results have already been obtained, and thus reserve their strength for difficulties which have not yet been conquered. And those who merely desire to ascertain the present state of a subject without any purpose of original investigation will often find that the study of the past history of that subject assists them materially in obtaining a sound and extensive knowledge of the condition which it has attained. The Author has en- deavoured in this work to ascertain distinctly what has been effected in the Progress of the Calculus, and to form some estimate of the manner in which it has been effected : accordingly, unless the contrary is distinctly stated, it may be assumed that any treatise or memoir relating to the Cal- culus of Variations which is described in this work has undergone thorough examination and study. By J. H. 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