MATHEMATICAL THEORIES PLANETARY MOTIONS BY DR. OTTO DZIOBEK, PRIVATDOCENT IN THE ROYAL TECHNICAL HIGH SCHOOL OF BERLIN-CHARLOTTENBURG. TRANSLATED BY MARK W. HARRINGTON, CHIEF OF THE UNITED STATES WEATHER BUREAU. FORMERLY PROFESSOR OE ASTRONOMY AND DIRECTOR OF THE OBSERVATORY AT THE UNIVERSITY OF MICHIGAN. WILLIAM J. HUSSEY, ASSISTANT PROFESSOR OF ASTRONOMY IN THE LELAND STANFORD JUNIOR UNIVERSITY. Of THE UNIVERSITY ni ANN ARBOSTMlCHIGAN: THE REGISTER PUBLISHING COMPANY. Ube flnlanb ipres0. 1892. Copyright 1892 BY THE REGISTER PUBLISHING COMPANY. PREFACE. The determination of the motions of the heavenly bodies is an import- ant problem in and for itself, and also on account of the influence it has exerted on the development of mathematics. It has engaged the attention of the greatest mathematicians and in the course of their not altogether successful attempts to solve it, they have displayed unsurpassed ingenu- ity. The methods devised by them have proved useful not only in this problem but have also largely determined the course of advance in other branches of mathematics. Analytical mechanics, beginning with Newton and receiving a finished clearness from Lagrange, is especially indebted to this problem, and in turn, analytical mechanics has been so suggestive in method as to determine largely both the direction and rapidity of the advancement of mathematical science. Hence when it is desired to illustrate the abstract theories of analyti- cal mechanics, the profundity of the mathematics of the problem of the motions of the heavenly bodies, its powerful influence on the historical development of this science, and finally the dignity of its object, all point to it as most suitable for this purpose. This work is intended not merely as an introduction to the special study of astronomy, but rather for the student of mathematics who desires an insight into the creations of his masters in this field. The lack of a text-book, giving within moderate limits and in a strictly scientific manner the principles of mathematical astronomy in their present remark- ably simple and lucid form, is undoubtedly the reason why so many mathematicians extend their knowledge of the solar system but little beyond Kepler's laws. The author has endeavored to meet this need and at the same time to produce a book which shall be so near the present state of the science as to include recent investigations and to indicate unsettled questions. The subject of the work is that part of celestial mechanics which treats of the motions of heavenly bodies considered as material points. This is its most important part, and it is fundamental in the theories of rotation, tides and the figures of bodies. The author hopes to treat of these in a separate work. The simplest processes and those which best represent the present state of the science have always been selected. Especial care has been taken to guard against brilliant hypotheses which explorers in this field have so often indulged in but which are not suitable in a text-book. IV PREFACE. References to original sources of information are invariably given. These will be useful to students who desire to study further. Assistance in this direction is also afforded by the historical sketches accompanying each important sub-division of the work. The tables at the end of the book give the numerical values of the elements of the solar system according to Leverrier and Newcomb. For the pecuniary aid which permitted the necessary studies and the publication of this work, the author begs to return his sincere thanks to his Excellency, Dr. von Gossler, Minister of Religious, Educational and Medical affairs. CHARLOTTENBURG, September 1, 1888. DR. DZIOBEK. NOTES BY THE TRANSLATORS. The author kindly consented to read the proof of this translation. Many changes have been introduced by him. M. W. HARRINGTON. The publication of this translation has been delayed in a number of ways. The author kindly offered to assist in reading the proofs and at the same time to revise the work. By reason of his remoteness much time was consumed in sending the proofs back and forth. Moreover, the changes which he introduced in the text were so numerous and of such a charac- ter that it became necessary to reset the type of large portions of the earlier forms. To such an extent was this the case that, after eighty pages had been printed, it was decided to send a type-written copy of the remainder of the translation to the author and to have him revise it before setting the type. This was done with its accompanying delay. In the meantime Professor Harrington had become Chief of the Weather Bureau-, and at this juncture I was asked to revise the remainder of the transla- tion, to incorporate in it the author's corrections and additions and to see it through the press. W. J. HUSSEY. PALO ALTO, CAL., July 8, 1892. TABLE OF CONTENTS. FIRST DIVISION. SOLUTION OF THE PROBLEM OF TWO BODIES. FORMATION OF THE GEN- ERAL INTEGRALS FOR THE PROBLEM OF H BODIES. ALGEBRAICAL TRANSFORMATIONS OF THIS PROBLEM. PAGE. 1. Newton's Law of Gravitation. Motion of Two Points Subject to It 1 2. Elliptic, Parabolic and Hyperbolic Orbits 12 3. The Restilinear Path and the Formula of Lambert and Euler... 19 4. Solution of Kepler's Equation. Development of the Coordi- nates as Functions of the Time 23 5. Historical Notes on the Preceding Sections , 37 6. The Problem of n Bodies. The General Integrals 39 7. The Problem of Three Bodies 50 8. Special Cases of the Problem of Three Bodies 61 9. Historical Notes on the Problem of Three Bodies 74 SECOND DIVISION. THE GENERAL PROPERTIES OF THE INTEGRALS. 10. Poisson's and Lagrange's Formulas 77 11. Development of Poisson's and Lagrange's Formulas for the Elliptic Elements of the Orbit of a Planet 91 12. The Canonical System of Constants of Integration 97 13. The Canonical Constants for the Elliptic Elements of the Orbit of a Planet 104 14. Properties of the Involution Systems..; 107 15. Canonical Transformations of the Canonical System of Differ- ential Equations 115 16. The Partial Differential Equation of Hamilton and Jacobi 120 17. H Not Containing the Time 124 18. The Partial Differential Equation of Hamilton and Jacobi for the Motion of the Planets around the Sun 132 19. Historical Survey for the Second Division 140 VI TABLE OF CONTENTS. THIRD DIVISION. THEORY OF PERTURBATIONS. 20. The Consideration of the Solar System as a System of n Bodies 143 21. The Orbits of the Planets. Theory of Absolute Perturbations.. 148 22. Solution of the Differential Equations for the Absolute Pertur- bations 153 23. Other Formulas of the Absolute Perturbations 161 24. An aly tical Development of the Perturbing Function 167 25. The Development of (a^ 2a 1 2 coBd-\-a^)~^ s in a Trigono- metric Series, 186 26. The Terms of the Perturbing Function of the Degrees 0, 1 and 2 191 27. The Analytical Expressions for the Perturbations 196 28. The Variation of Elements 203 29. Approximate Integration of the Differential Equations for the Variationof the Elements 209 30. The Secular Values of the Elements. Development of the Rig- orous Equations Between Them 211 31. Approximate Calculation of the Secular Values of the Elements 214 32. The Secular Variations of the Mean Longitude 227 33. Periodic Terms in the Elements. A Combination of the Theory of Absolute Perturbations with the Theory of Variation of Elements 229 34. The Stability of the Solar System 233 35. The Effect of Neglected Secular Terms of the Perturbing Func- tion whose Degrees with Respect to the Eccentricities and Inclinations are Higher than the Second 234 36. Terms of Long Period and the Cemmensurability of the Peri- odic Times 241 37. The Exactness of the Formulas for the Variation of the Ele- ments 249 38. The Improvement of the Theory of Variation of Constants by Including Terms Depending on the Second Powers of the Masses 253 39. The Invariability of the Major Axes 256 40. The Form in which the Elements and Coordinates appear as Functions of the Time 262 41. Several General Formulas Relating to the Coefficients in the Development of the Coordinates in Trigonometrical Series 267 42. Brief Historical Review of the Theories of Perturbations 278 43. Notes on the Tables 290 Tables... 292 EEEATA. ^ PAGE LINE FOR 10 -6,-7 motus 11 + 8 t 13 headline PROBLEMS 17 it 21 + 5 1 i 21 + .9 rl ((,-+, 22 + 11 /I 22 -4 to 24 + 10 smallest 27 headline PROBLEMS 31 " " 31 + 6 ()' 40 + 19 m l 42 + 15 = 49 headline TWO 49 + 9 qualities 80 2 = 0. 115 4 (15) 176 1, 2 Sq 212 + 4 W 228 8 Q\ READ Motus. * PROBLEM a smallest value PROBLEM x(x 2a)"\ ()' C = n quantities 15. Bg W g Lines counted from above are + , from below M A.TH E ]VLATIC A.L, THEORIES OF PLANETARY MOTIONS FIRST DIVISION. Solution of the Problem of Two Bodies. Formation of the General Integrals for the Problem of n Bodies. Algebraic Transformations of this Problem. 1. NEWTON'S LAW OF GRAVITATION. MOTION OF Two POINTS SUBJECT TO IT. Newton's law of gravitation is the point of departure in mathematical investigations of the motions of the heavenly bodies. This law reads as follows: Each particle of matter attracts any other particle with a force whose magnitude is directly as the product of their masses and inversely as the square of their distance from each other. Assume that Pj and P 2 are two gravitating particles, the coordinates and mass of the first, referred to stationary rect- angular coordinates, are x lt y lf z lt m lt and those of the second are a? 2 , y^ z 2t w 2 , then the distance between them is (1) r = and this is to be considered as always positive. The total force between the bodies, according to Newton's law, where k is a constant whose magnitude depends on the selected units of mass, distance, and time. 2 THEORIES OF PLANETARY MOTIONS. Since the two particles attract each other, the direction of action is along the line which joins them. The action of P 2 on P 1 has the direction from P v to P 2 and its direction cosines are The direction of the action of P l on P 2 is exactly the reverse, and its direction cosines are #2 2/1 2/2 The components of the first force in the direction of the coordinate axes are therefore m 2 m The components of the second force are, also, m g m 2 z, Consequently the differential equations of the motion of the two points are, when i represents the time, (2) Cv tJO-\ M^l 1 dt 2 and also 2?i ~P ' di 70 = m 1 79 ~ ** # - These differential equations are valid for any system of coordinates. Hence the letters x, y, z, can be cyclically inter- changed. Advantage will be taken of this to simplify the PROBLEM OF TWO BODIES. 3 manner of writing the succeeding equations, in that only one of the three equations will be written, leaving the others to be made from it by exchange of letters. These cases will be indicated by the sign *. Introducing this change, equations (2) become rl^r nc - T e*/l j 2 * / 2 *^\ nil TIT" * wii ^2 s > at r 3 ^* e ^- / 2 7-2 1 ^2 **v9 T 1 o KT 7/vi 7/vo Q " These are six total and simultaneous differential equations and the determination of the motion of the two bodies is reduced to their integration. As they are all of the second order their complete integration will introduce twelve arbitrary constants. That this number is necessary appears directly from the fact that, to make the problem a definite one, twelve condi- tions must be expressed, for instance, six coordinates and six component velocities, for any given instant. To prepare for the integration of equations (2), add the first and fourth. This gives d 2 x d 2 x /0 \ (3) Integrating this once, and calling the constant **, we get dx l dx 2 These equations, on a second integration, give (4) * m l x 1 + m 2 x 2 = a x i -f- &,. Equations (4), in which *a xj $ x are constants, have a simple interpretation. If the coordinates of the center of gravity of the two points, that is the point which divides their distance in the inverse ratio of their masses, are X, Y, Z, then and equations (4) become (6) * (m, + m,)X= aj + p.. From these equations it appears that the center of gravity 4 THEORIES OF PLANETAEY MOTIONS. moves in a straight line with uniform velocity, which is the law of the conservation of the motion of the center of gravity. Eeturning to equations (2), if we refer P 2 to a moving system of rectangular coordinates parallel to the original with the origin constantly at P l , and designate the coordinates of P 2 in the new system by x, y, z, we have (7) * x = x 2 x lf and we observe that these equations are included in the right- hand members of equations (2). Put therefore (8) k 2 (m, + m 2 ) = P., divide the first three equations of (2) by m 19 the second three by m 2) and subtract in pairs, and we get (9) . * ? = - "- These three equations by integration afford the six remain- ing constants and complete the solution of the problem. For, by solving (5) and (7) for *x lt x 2 , we have /1A\ ' -vr m 2 **!*#! (10) * x l = X - - x, , x 2 = X + - - x. m 1 -\- m 2 m l -\- m 2 P l may be considered as representing the sun, P 2 the planet. Equations (9) then determine the (relative) motion of the planet about the sun, and the motion is as if the sun were fixed and the planet attracted by the sum of the masses of the two bodies. To integrate (9), multiply the second by ---*, the third by -f- y, and add. We thus get d' 2 z d' 2 y _ n y ~w~ z ~w~ This is at once integrable, and gives dz dy - where c x is the constant of integration. These are the so-called sectorial integrals, ydz -- zdy is twice the area of the triangle in the plane of yz, whose angles are the origin, the projection of the point x, y, z, and the projection PROBLEM OF TWO BODIES. of the adjacent point x + dx, y + dy, z -f- dz. It is positive or negative, according as the infinitesimal angle at the origin is described by the radius vector in the direction from + y to -f- z, or the reverse. If this triangle is named dS x then * 2d& = c^dt. To fix the coordinate system, we will, once for all, so place it that a rotation from -f y to + z is contra-clockwise when the clock face is in the plane yz and is directed toward + x: and similarly for rotations from -f z to -f x, and from -f- x to 4- y. A sector is then positive when the direction of description is contra-clockwise, and vice versa. Equations (11) show that the areas described by the projec- tions of the radius vector are proportional to the times. For the Qf Qf Qf unit of time these areas are -^ , -^ , -^ . From this it follows A A A that these areas themselves are proportional to the times and p that, for the unit of time, they are -~- , where c is positive, and, A as is always possible, (12) I = c 2 , + c 2 , + : 2 ,. The expression y ~ z -j! is called the moment of velocity about the x axis. It is easily seen that it is equal to the product of the projection of the velocity on the yz plane and the per- pendicular from the origin on its direction. The same is true for the two other analogous expressions. Hence the square root of the sum of the squares of these three expressions, that is, c, equals the moment of velocity about the origin. From (11), by multiplication with x, y, z, and addition, we get a final equation between the co-ordinates, (13) C^ + C y y + C.* - 0, and hence The path of a planet is a plane curve and is so traced that the areas described by the radius vector from the sun are pro- portional to the times. It follows that C*, C y , C z are the direction cosines of a line which we shall call the C-axis and which is normal to the plane of motion. 6 THEORIES OF PLANETARY MOTIONS. For further integration it is best to form an equation in r and i only. r 2 = x 2 + y 2 + z 2 , dr dx . dy . dz r dt = x Tt+ydi +z 7t' and by another differentiation 4 T dt ) d 2 x + y d*y + z d 2 Z + dx 2 H - dv 2 - f d5? 2 dt x dt 2 dt 2 ' dt 2 dt 2 or, by (9), when for brevity (14) r* = f, dr' PL . dx 2 + dy 2 + dz 2 ~ar " 7 ' ~^~ Differentiating again, dr d 2 r f ^ dt Q dxd 2 x + dyd 2 y + dzd 2 z ~W ~P~ dt* or, by (9), This equation has a significant relation with equations (9), in that x, y> z, are replaced by r'. If it is combined with (9) in the same way that the latter were combined with each other to get the sectorial law, and, when the three constants of integra- tion are represented by / , / $ y , f z , we get dr' dx ... Here it will be assumed that / is positive and that * 2 -h V + ^z 1, so that X9 y9 % z become the direction cosines of the ^-axis which lies in the plane of motion and, as we shall soon see, is directed to the perihelion. If we multiply equations (9) by dx, dy, dz, add and integrate, we get the expression for the kinetic energy _!_ dx 2 + dy 2 + dz 2 '/* /, ~ 2 dt 2 ~ r ~ 2a PROBLEM OF TWO BODIES. 7 where the constant of integration, for reasons which will soon appear, is represented by --- ~ -- Between the integrals (11), (17) and (18), exist two identical relations. From (11) and (17) we get at once (19) , C B + C, + . C, - 0. Also, (xdx + ydy -j- zdz V ~dT or, by (18), (20) c 2 = r a (2 - -^-) r' 2 . Also, and then, from (17) and (18), or, since by (15) and (18) /oox dr> ~ (23) vr a a From (20) and (23) follows the second relation between the integrals, (24) c2 A +/2 = ^. In order finally to get an equation between the coordinates, multiply (17) in order by x, y, z, and add. With the help of (20) and (22), we then get (25) c 2 - A* r = f(x$ x + y$ v + *.). This equation represents a surface of revolution of the sec- ond degree which is cut in a conic by the plane (13), which plane by (19) passes through the >y-axis of the surface. Since, 8 THEOEIES OF PLANETABY MOTIONS. also, by (25) it appears that the origin is a focus of the surface, we have The path of the planet about the sun is a conic and the sun is in one of its foci. We have now two lines perpendicular to each other, the - and C-axis, which are fully determined by the motion, and we shall now take a third line, the i?-axis, perpendicular to the others, so that , ^, C form a new system of rectangular coordi- nates. Let y x) i) V9 f) z be the three direction cosines which the ?7-axis makes with the axes of the first system, then the nine quantities ^xt ^yt %s) (26) i) x , ^, 77,, r ? ^X) *t/> *JT| are the coefficients of transformation from the old system into the new. This transformation may be written in either form * 00 = S*. * To find the expressions for y x , ^,, y z we may use the known relations between the nine quantities (26); for instance, or, with (11) and (17) and a slight transformation, /-it? \ dp dx (17a) cfr) x = x-~ p-r, where (14a) P = r 2 ^ + r' 2 = c 2 + pr. dt From equations (17a) a very interesting result can be easily deduced. Writing them dx u. r dr d we get CS ->)'+ (S -/?)'+ ( -/?)- & PBOBLEM OF TWO BODIES. We have also and it appears at once that The hodograph of the motion of a planet is a circle with the radius , and its plane is the plane of motion. c The equations of the orbit are, in the new system, (27) c 2 -^r = /?. Also, (28) $ ^ r) = C. Introducing polar coordinates by putting (29) r* = rcos* L r) = r sin v, we get (27) and (28) in the following forms (30) r = 1 + cos v ~dT = c ' If r and v are found by integration of (30) and (31) we have and p. by (29), and x, y, z by (*26a). As C = 0, we evi- dently have (26b) * x = S x + H TJ X = r (cos v x + sin v rj x ). Assuming, with Gauss, I - A r*r\Q 73 (32) r^-j*cos^, L y x = A x sin B x and we get (32a) * x = r A x cos (v + B x ). The six constants A and B satisfy the three equations ,4 2 _L >4 2 _l_ J2 9 & ~r -oLy ~r A-Z A A x * sin 25, + ^ y a sin 2 5 y + X 8 sin 2 5, = 0, A x * cos 2 B x + ^/ cos 2 B y + ^., 2 COB 2 B a = 0, 10 THEORIES OF PLANETARY MOTIONS. because (32a) must give identically r 2 = x 2 +2/ 2 + z\ If p is the semi-parameter and e the eccentricity of a conic, its equation, when the origin is at the focus and the -f- axis is directed to the vertex nearest it, becomes (30a) r = P 1 + e cos v Hence, in this case, (38) ) Inserting in (33) for ft its value (8), we get (34) e = \f p \f nil ~h m 2 Since c is double the sector described in a unit of time, 2 S if S equals the sector described in the time T, then c = TJFT and hence m and (35) p (111, -[- m 2 ) Hence * For the different planets moving about the sun, the square of the ratio of the sectorial velocity to the time is as the product of the semi-parameter into the sum of the masses of sun and planet. The constant k is Gauss's constant. If the sun's mass is unity, that of the earthf is og^Tn = 0.0000028192. The unit of distance is half the major axis of the earth's orbit, which is also *See Gauss's Werke, (Theoria motus, etc.), Vol. VII, page 12. tThe numerical constants In this translation are from Gauss's Theoria motus. While values more acceptable to astronomers, at present, might be inserted yet, as the object of the book is rather a correct analytical than numerical development of the problems involved, and astronomers are not generally agreed as to the exact numerical values employed, I have preferred to leave these numbers generally unchanged. TBAKSLATOR. PROBLEMS OF TWO BODIES. 11 the mean distance of the earth from the sun. The sector becomes in a sidereal year the entire area of the ellipse, or S = xab = K a V -\l = K o> V a> V p = * V p If the mean solar day is the unit of time, T is the number of such days in a sidereal year, or T = 365.2563835, and by computation k = 0.017202099.t Gauss also proposed so to select the unit of mass that k should equal unity. This is evidently the case when, retaining the above units of distance and time, the sun's mass is assumed as = ( 0.017202099 ) 2 . The unit of mass can then be denned as the mass which, at distance unity, during an interval of unity, gives an acceleration = 1. For if k = 1, then, from (2), OT From this it appears that the unit of mass is fixed when the choice of the units of distance and time has been made. The farther treatment of the problem requires that r and v be expressed in terms of the time t. It follows directly from (30a) and (31), with the aid of (33), that dv (36) dt = = V v- (1 -\- e cos v) If / is the time of passage through the vertex nearest the sun, at which time v = 0, the integration of (36) gives (37) i i- -Pi- f *? - ^J (1 + eeosvY This integral requires different treatment when e equals, is greater than, or is less than, unity. The three cases will be considered separately. * For a more exact value of k, and for its exact signification, see Oppolzer, Lehrbuch der Bahribeatimmungen, Vol. T, page 45. 12 THEORIES OF PLANETARY MOTIONS. 2. ELLIPTIC, PARABOLIC AND HYPERBOLIC ORBITS. Case J, e < 1. Equation (30a), 1, represents in this case an ellipse. If the semi-axis major is represented by [a], then tf_ [a] = ^E-s = * . Hence by (24), 1, ^1 [a] = a, that is, the constant a is positive and equals the semi-axis major of the ellipse. Equation ( 18 ) therefore shows that the velocity of a planet depends only on its major axis and its distance from the sun. The denominator of (37) then never = and v increases continuously, though not uniformly, with t. It can be evaluated directly as a trigonometrical integral, but it is simpler when we introduce an auxiliary angle E/, such that Vj By differentiation this gives _ e = tan E. E increases continuously with v and, if we take E when v = 0, the two angles will be equal for multiples of * and will always be in the same semi-circumference. From (1) it follows that (3) and (2) passes into (4) dv = Vl=?- r ^A^I dE = ^ 1 e-\- tan Js Further, (5) 1 + 6 cos v = 1 e + 2 21 PROBLEMS OF TWO BODIES. and also, from (37), 1, 13 If we make (6) = - (E e sin E). V f* at 1 , , = , and V A* n n(t-t Q ) = M, M = E e sin E. (7) we get (8) Equation (8) is Kepler's equation. It does not directly give what is wanted, that is E as a function of /. To obtain this, the equation must be solved for E, a problem which is discussed in 4. When this is done v can be got from (1), r from (30a), and ^ from (29), and finally x, y, z, by transformations of (26b) or (32a). The three angles v, M and E, are called by astronomers, (v, the true anomaly, M, the mean anomaly, E, the eccentric anomaly. Also, n is the mean daily motion. The anomalies have a very simple geometrical meaning. fc Of THt XJNIVEBSITY 14 THEORIES OF PLANETARY MOTIONS. Let ABCD be the ellipse of a planet, which moves in the direction of the arrow, and let F be the focus occupied by the sun. Let the planet be at P, then angle AFP v, the true anomaly. Extend the perpendicular QP above P to P r where it cuts the circle described on the major axis, then angle QOP' = E, the eccentric anomaly. For nr\ COS V OQ -f- QF = a e = a cos E r cos v = a cos E n , 1 + e cos v' which passes into equation (5). Imagine, finally, on the circle, a point moving with uniform velocity and reaching A and C at the same time as the planet. Let P" be its position when the planet is at P; then angle A OP" = M t the mean anomaly, and n is the daily increase of the mean anomaly. To get the periodic time T, the year of the planet, we note that in this time M becomes 2 ?r, and hence (13) 7r = Tn, and, by (6), 7 751 4-7T 2 a If, as is the fact, m 2 is so small as compared with m 1 that it can be neglected in the first approximation, the right hand member of (13) is the same for all planets, and we reach the celebrated Laws of Kepler: 1. The orbits of the planets are ellipses, in one focus of each of which is the sun. 2. The areas described by the radii vectores from the sun in equal times are equal. 3. The squares of the periodic times of the different planets are as the cubes of their mean distances. The relations between r, v, M, E, e, a, can be given different forms adapted to special requirements. Putting (14) e = sin ? the most important of these, given by Gauss in his Theoria Motus, are the following: PROBLEM OF TWO BODIES. 15 I. p = a cos 2 , II. r = ,3P , 1 -f- e cos v III. r = a (1 e cos E), TTT 17, cos v + e cos E e IV. cos E = = : ! , or cos v = -= - , 1 + e cos v 1 ecosE \r ; i TP A cos E . V. sin $E = +1 ^ = sin i -h e cos . n / r ( 1 e} I ^~ = sin ii _P /T~~L VI cos \E = e cos VII. tan p; = tan Jw tan (45 VIII. sin E = - p a cos ^ IX. r cos v = a (cos E e) = 2a cos (J.E7 + ib + 45) cos (p? ^ ~ 45), X. sin ^ (v ^7) = sin % sin E ^ XL sin $(v + E) = cos }? sin v /~ = cos J^ sin E XII. M = E e sin E. JSL Finally, we will collect and name the six constants which astronomers call the elements of the orbit. They are ' 1. The mean distance or semi-axis major = a, 2. The eccentricity = e, 3. The longitude of the ascending node = , (16) 4. The inclination of the plane of the orbit = t, 5. The longitude of perihelion = TT, 6. The mean longitude of the planet at the selected epoch = e. 16 THEORIES OP PLANETARY MOTIONS. The constants a and e have the same meaning as already given to them. From now on we will so select the xy plane that c C* = ^ ~7, y will always be positive. It divides the plane of the orbit into two parts, of which the one on the + z side is the upper, the other the lower part. The point where the planet passes from the lower to the upper part is the ascend- ing node, and the opposite the descending node. The longitude of the ascending node, , is then the angle between the + x axis and the radius vector directed to the ascending node. The inclination i, is the acute angle which the plane of the orbit makes with the plane of xy. The equation of the plane of the orbit then becomes x sin sin i y cos & sin i -f- z cos i = 0. When i 0, is indeterminate. If ^ is the angle which the perihelion makes with the ascending node, measured on the plane of the orbit, then the longitude of perihelion is measured in part, (0), on the xy plane, in part, (&), on the plane of the orbit. Hence (17) ~ = Q + . This designation appears singular but is convenient, especially when i is very small. When i = 0, * is the angle between the -f x axis and the perihelion. If we take ^ instead of TT, the nine coefficients of transforma- tion become % x cos (t> cos Q sin & sin Q cos i, y = cos & sin Q -f- sin u> cos & cos i, z = sin m sin i, f) x = sin to cos Q cos to sin cos i, (18) < r iy = sin a) sin Q -f- cos to cos cos i, j s = cos u> sin i, x sin & sin i, y cos & sin i, a cos i. The mean anomaly and time have the relation M = nt nL. PROBLEMS OF TWO BODIES. 17 For 4 let us take a new constant s such that (20) -nt = s JT, and (21) M = ni + e -. To w/ + is given the name of mean longitude, (measured from the -f- x axis), and we may make (22) r = nt + e. All the formulas are simplified when e = 0, that is when the orbit is a circle, and the motion is uniform. If, at the same time, the plane of xy is the plane of the orbit, both and TT become arbitrary, and nt -f~ s = becomes the angle made by the planet with the -f- x axis. Introducing polar coordinates into (26b), we have fx = r [cos (v + ~) -f- sin (v + - ) sin & (1 cosi)], y = r [sin (v -f -) sin (v + - fl) cos fl (1 cos t)], 3 = r sin ( v -j- - & ) sin i, in which form they are convenient for computation. \ Case II, e = 1. The orbit is a parabola; a is infinite, and M and E = 0, or lose their significance. Equation (37), 1, becomes easily integrable by substituting tan \v z. The integral then becomes and hence i tan a ^ This equation must be solved for v and the farther solution of the problem then proceeds as in the ellipse. For t = oo, v = TT and r = -\- oo ; for / = + oo, v = -f- -, and r = -j- oo ; hence it appears that when = 0, according to (18), 1, the 18 THEORIES OF PLANETAEY MOTIONS. planet comes from infinity with infinitesimal velocity, reaches its greatest velocity at perihelion, then withdraws again to infinity with a velocity which gradually decreases to 0. Case III, e > 1. Equation (30a), 1, represents a hyperbola. Since r is positive, we must have 1 -(- e cos v > 0, hence cos v > - o Putting cos = , v must vary from 180 -f to 180 (p. 6 At these limits r = GO. If v is taken beyond 180 - 0, equa- tion (30a), 1, gives negative values of r, and we pass to the branch of the hyperbola which does not inclose the sun. This plays no part in the physical problem with which we are engaged, since the planet remains continuously on the first branch. If we call [a] the semi-axis major of the hyperbola, it follows that or, by (33), (34) and (24), 1, [a] = - a, that is, the constant a is negative in the hyperbola, and its absolute value is that of the semi-axis major. M and E become imaginary; that is, they are illusory in the physical problem. Yet an auxiliary angle F can be introduced through the equa- tion (25) tan $ F = rpr tan 4 v = tan i ^ tan J v, and from (37), 1, we get (26) V-^W" =eteuF- loge [tan (J F + 45 )] . Hence, using the expression (27) ti = tan(i^+ 45), *o) ** e (-a)t (28) j 7 ^vs = -?r I u | ioer u. PEOBLEM OF TWO BODIES. 19 When u has been determined by (28), .Fand v can be got from (27) and (25). From (18), 1, it appears that the planet comes from infinity with the velocity -/ , accelerates its velocity to perihelion, and then returns to infinity with a velocity which continuously decreases until it becomes -\l Case I represents the orbits of all the planets moving about the sun. The two others occur with comets and meteorites which move sometimes in ellipses which are usually very long, sometimes in parabolas and sometimes in narrow hyperbolas. 3. THE BECTILINEAR PATH AND THE FORMULA OF LAMBERT AND EULER. In addition to the cases of the preceding paragraphs, there may be another, that in which c x = c* y = c z = 0. In this case equations (11), 1, can be at once integrated, and we get, if p l9 p 2 , p s are the three constants, Pi P% PS or the equations of a straight line passing through the sun. If we take this for the x axis, y and = 0, and r = x. If the planet is on the positive side of the origin, it must pass through the sun to reach the negative side. At the instant it reaches the sun, however, the force becomes infinite, the principles of the Calculus lose their significance, and the dis- cussion closes at this point. We can then always put r = -f- x. The differential equation of the motion becomes < a > iiF = -y ; This can be at once integrated, on multiplying by dx, and gives U. IK. 1 ~^~^ 20 THEORIES OF PLANETARY MOTIONS. where - is again the constant of integration. 2id We will now so select our time that when x OC Q , i = , the velocity, is negative. The planet is then approach- (Jit ing the sun and (3) becomes dx dt = V2/. Designating, finally, by ^ the time at which the planet reaches x ly and reversing the limits of integration, we have (3a) /, _ * = V 1_ J dx "27" / /I 1" / V^ ""2a 1. If a is positive, then must x < 2a. Making, therefore, (3b) x = 2a sin 2 1 ?, or V# (2a a?) = a sin we get 2al /\ . a , at - t = I d

(- g)| //-e_*_ /V/ \ ^ or = ti]* f V/A - log e ((x Ski)J I. If the initial velocity is directed away from the sun, there are also three cases: 1. If a is positive. The velocity = 0, when x = 2 a, and the planet turns back toward the sun at this point. 2. If a = oo. The velocity = when r becomes oo ; that is, the planet goes off to infinity with a velocity which gradually decreases to 0. 3. If a is negative. The velocity continues positive and decreases to -* / , and with this velocity it disappears in \ d infinity. From these considerations it appears that even when the path is rectilinear, it can be considered as an infinitely narrow ellipse, parabola, or hyperbola, according as - = 0. 22 THEOEIES OF PLANETARY MOTIONS. It is now interesting to note that the formula, (3a), is cor- rect for the general motion in any orbit, if x and x l have in the general case other but simple values, becoming in the recti- linear path simple distances from the sun. We shall prove it for the ellipse. It is clear that the time of motion between two points depends only on the distances of these points from the sun, their distances apart, and the semi-axis a. To find its expression in terms of these quantities, represent what depends on one point by the subscript 0, and on the other by the sub- script 1. Then, if we put , We find, by simple transformations, = g M, - M - (A - * ) = 2 (/- e cos<7 sin/), T! -f- r = 2a (1 e cos g cos/), P = 2a (VI e 2 cos 2 g) sin/, r a r = 2a e sin g sin /. To express /j 4 by r , r 19 p, and a, we must eliminate e, /, and g from the above equations, and as in the first three e and g are contained only in the form e cos g, we have the result: The time of motion between two points depends only on a, n + n, and />. If we put e cos g cos a and then we find immediately A t = -^ (

_. . 2 PI 2a ~ Sln "2 PROBLEM OF TWO BODIES. 23 Comparing now (3a) with those which follow, we find, by reversing the process of integration, r\ + n + P (7) r - *o = / V x ~ 2a ' 2 which is Lambert's theorem. We shall find it in 18 by quite another process. 4. SOLUTION OF KEPLER'S EQUATION. DEVELOPMENT OF THE COORDINATES AS FUNCTIONS OF THE TIME. In order to develop the coordinates as functions of the time, it is necessary, first of all, to solve Kepler's equation, (1) M = E e sin E, with reference to E. Numerous methods of solving this cele- brated equation have been devised, when M and e are given, and the simplest of these is that given by Gauss in his Theoria Moius. The later applications require a more general solu- tion, and there are two roads that lead to this, the one through Lagrange's series, (which seems to have been discovered by Lagrange in his study of Kepler's equation), the other through Bessel's functions. Lagrange's series enables us to develop x in ascending pow- ers of e, when the two have the relation expressed by the equation (2) x = y + e ?(x), where ?(#) = -^-i and, if /(a?) = as* , we get _ - 3! 2*+ 6 If now E, sin J, sin 2E... are developed in ascending powers of e and the proper reductions made, the expression for v becomes (11) v = M + e X 2 sinM + e 2 X |- sin 2 Jf + e 3 (]| sin 3 Jf - ^ sin Jf) 11 . ,A sin 4Jf 2j sin 21f J sin 6 Jf - | sin 3 Jf + ^ sin Jf) l03 sn - sn TM sn Before the series (6) to (11) are used, their convergence must be assured. For this purpose the criterion mentioned on page 24 will serve. The equation E = M + esinE has two equal roots if the two differential quotients of both sides are equal. 28 THEORIES OF PLANETARY MOTIONS. 1 = ecosE, cosE = , sin E = A /l r = ~V1 e a 6 \ hence arc cos = .M + i Vl e 2 > <> r > e (12) = cos (M + i VI e 2 ) == cos M cos (i Vl 6 e 8 ) = cos-M sin-M" If Jtf" = 0, the equation becomes the smallest root of which is e 1. e " 2t or, if we put e in, 2 = and from this, solving for [e] v x + a2 , (13) 1 + The smallest root of this equation is = 0.66195...,- e = 0.66195 1. If any other value of M is taken, a root for e is obtained from (12), whose modulus is between 0.66195 and 1. Hence the conclusion: If e < 0.66195, the series (6) to (11) converge in every case. If 1 > e > 0.66195, these series converge only for a part of the path. If e > 1, the series diverge in all cases. For the major planets, moving about the sun, e is always a PEOBLEM OF TWO BODIES. 29 very small fraction, and these series are serviceable, but for many of the asteroids their convergence is tedious. In the latter case another method is in use among astronomers for overcoming the difficulties of Kepler's equation. It is possible, for each value of e between and 1, as shown by Bessel,* to form a progressive series in terms of the sines and cosines of the multiples of M, the coefficients of which are progressive series in terms of the ascending powers of e. Such series can evidently be made from series (6) to (11), of which the law of formation is evident. These series can be looked on as doubly infinite series, and all the terms can be collected together which contain the angle n M. In this way, in (6) for instance, the coefficient of sin M e s e 5 1!2!2 2H 2!3!2 4 To make the discussion general, and especially to show the convergence of the series, it is convenient to begin with Fourier's theorem for the developmeDt of a function in a trigonometrical series. Since E - - M = e sin E remains unchanged when M is increased by 2 x, the value of E M, according to this theorem, must, for all values of E, be equal to a trigonometrical series of the following form E M a + i cos M + a 2 cos 2 M + a 3 cos 3 M -f ... -f &! sin M + & 2 sin 2 M -f 6 3 sin 3 M + ... Then, according to Fourier, o = sr- I (E M)dM=^- I 2 V 2 V 2TT - I e sin E dM = 0, because, in this integral, the elements from M = to M destroy those from M = K to M = 2~. In the same way M ) dM cos n M = 0. * Untersuchungen des Theils der planetarischen Storungen, welcher aus der Bewegung der Sonne entsteht. Abhandlungen, Vol. I, page 84. 30 THEORIES OF PLANETARY MOTIONS. There remain only the coefficients b n . They are 2_r b n = - I (E M) dM sin nM. o Using integration by parts, we get -KIT TI\T coanM /Tn ,, x sin nM dM - - (E M ) n f(E M) si [ - f(dE-dM) coanM. If the integration is from to 2~, the term containing (E M ) vanishes, and we have 27T 27T b n = / (dE dM) cos n M = / dE cos nM nr ) 27T - / dE coan (E e sin E). Making for brevity n e = x, 27T i r n*J b n = ~ I dE cos (nE x sin E) o 1 27T - I cos nE cos (x sin E) dE o 27T -H sin nE sin (x ainE) The terms in the brackets are by Fourier's theorem equal to the coefficients of cos nE, or sin nE, in the development of cos (a; sin.Z), or sin (x siuE) into a trigonometrical series. The development is most easily performed by using exponen- tials. If PROBLEMS OF TWO BODIES. 31 = z, then /_!_ sin E = FT^ , hence Performing the multiplication and arranging by powers of z, a series is obtained of the form . ag __ s , e A a; _ * * 4 re in general, (14) z z z where in general, 2)(2n + 4) 2x4x6 (2n + 2)(2n + 4)(2 + 6) Series (14) is called a Besselian function. It depends on two quantities, x and n, and possesses many noteworthy proper- ties, of which the most important is that for every value of x it is finite, and its value is between 1 and -f- ! This follows directly from the definition 27T (15) J n (x) = ^ / cos (nE x sin E) dE, 32 THEORIES OF PLANETARY MOTIONS. since this integral is always smaller than * 27T I dE = 2-. o Moreover, the roots of J n (x) = are all real, so that the curve y = J n (x) is sinuous, like the curve y = sin x, with the difference that the waves for the first curve are always flatter. Finally, it should be noted that it is always easy to write out Jn + i(%) from J n (x) and J n -i(x), so that, by the transcend- entals J (x) and J^x), the others can be determined. By use of these functions * we get cos (x sinE) = 2 [J (x) + J 2 (x) cos 2E + J,(x) cos IE sin (x sinE) = 2 [Ji(aO smE -f J a (x) sin 3E + J 6 (x) sin 5^ + ...]. 2 2 Hence b n -- J H (x) J n (ne), and (16) E M = 2 [J^e) sin Jf + J 7 2 (2e) sin 2 Jf The quantities r, ^and ^ can also be easily expressed with the help of the Besselian transcendentals. Taking r = a (1 e cosE) a + a a cosJf + 2 cos 2Jf -f- a 3 cos 3Jf + . . . + b, sin Jf + 6 3 sin 2Jlf+ & 3 sin 3 M + and we have at once 6, = 6 2 = &3 = ---- = 0. Farther, 2rr 27r cos E dM a = x- / (a a 6 cos_EJ) dJlf = a ^ / 2 V 2 V 27T = a |^ / 27T dE = a *The Besselian functions are important in other problems. Eeaders who wish farther information about them, will find their principal properties treated of in Todhunter's work, "An Elementary Treatise on Laplace's, Lamp's and Bessel's Func- tions." PROBLEM OF TWO BODIES. 33 and, for n > 0, 27T a (1 e cosE) cos nM dM 27T a (1 e cos E) ae 1 f . I S1 n ,J 2- r/! P I - I sinn (E e sinE) sin jE 1 dE ae n '27T ^ / cos[(n- (i 27T 1 /cos[(n 1) J n 6 sin + 1) -S nesinE'] dE ae n Hence (17) The coordinate >? is derived at once from (16), since In the same way ? can be obtained from (17), since a (1 e 2 ) r C . e The development of i; is more troublesome, and the reader is referred to the treatment of it by Bessel.* *AnalyU#cTie Auflusung der Keplerschen Aufgabe. Abhandlungen. Vol. I, page 27. 3 34 THEORIES OF PLANETARY MOTIONS. The development of is simple. For 1 1 1 dE r a ( 1 - e cos E) a dM ' and, by (16), (18) - = 1 + 2 [J^f?) cosJf + J" 2 (2e) cos + J 3 (3e) cos3Jf + ...]. After developing and >?, as indicated above, x, y and z may be expressed in functions of the time; for M and / are connected by the equation M = nt -\- e n = C JT. Taking C = as the plane of the orbit, the equations (26a) p. 8 become x = Z x -\- yy x , y ?/ -\- I}?) y , and = + >?^ ; the coefficients of , 7? and C being given by (18) p. 16. e and i are generally small; hence , limiting the approximation to their second powers in the values of x and y and to their first powers in that of z, we get "" ( (5 x = a jcos C 4~ -H- [ 3 cos TT -j- cos (2 C ~)] + ^- [3 cos (3C 27r) 4 cos C + cos (2* C)] + - [cos (2fi C) cos C] j- , ^7/ = a|sinC + -|- [ 3 sin TT + sin(2C ^)] H- ~ [3 sin (3 C 2r) 4 sin C + sin (2^r C)] -|- -j- [sin (2i2 C) sin C z ai sin ( C & ) . Finally, from (17) to the same degree of approximation, (20) r = a \l ecos(C TT) + ~ [1 C os2(C w) I ^s and thence (21) r 2 = a 2 1 1 2e cos (C *) + ^ [3 cos 2 (C -)] j. , which will be used later. Formulas (19) can be completed by taking into account the 'higher powers of the excentricity and inclination, but we shall PKOBLEM OF TWO BODIES. 35 content ourselves with deducing a result which is of great importance in what follows. From (8) and (10) it appears that when kae a cos AJKf, or kae a sin A M, is any term in the development of and ^, and A is taken as positive, then we always have A either a positive odd number or = 1. The terms for in which A = 1 are the following: (22) a (cos M + -|- cos 2 M + gf|p 3 cos + 3^3 4 2 cos 4 Jf + j^ 5 3 cos 5 Jf + . . .), and the corresponding ones for >? are (23) a (sin M + -|- sin 2 M + ^p 3 sin By equations (26a) p. 8 and (18) p. 16, the coefficient of in x is % x cos ( TT ) cos 12 sin ( ~ 12 ) sin 12 cos V = cos TT -\- cos (212 TT) sin 2 ^i cos x sin 2 -| i, and that for >?, >y x = sin ( TT 12 ) cos & cos (TT 12 ) cos 12 cos i = sin TT -f- sin (212 TT) sin 2 ^ i -\- sin ?r sin 2 ^ i. Hence, taking for a term of the form kae a cos ^Tlf = 7cae a cos A (c ~), putting it in (26a) p. 8 developing sin 2 ^ i in a series on the ascending powers of i, and using the formula cos a cos ft = J [cos (a /?) + cos ( + /?)], we get, in the expression for x, terms of the form (24) cos r* where in every case A-f- r -f<5=+l, and d = or <5 = + 2, and, also, a -\- p - [A] is odd and positive, except when y? = 0, in which case [A] may also 1. Here [A] is the absolute value of the quantity A. Of UNIVEBSITY 36 THEORIES OF PLANETARY MOTIONS. Terms of the same form can be obtained for iy, so that in x only the form given in (24) can be found. With the aid of (22) and (23), we can at once write out the terms for which [A] = 1. They are (25) [cos C + ~ cos (2: TT) + g~L 3 cos (3: 2*) Exactly the same results can be obtained for y, except that cosine must be replaced by sine. Finally, it follows from the formula for z, (26a) p. 8, that in z there are only terms of the form kae a i? sin (A: + r - + dQ), and for these A + r + s = 0, 3 = 1, and -{- /? [A] is either even and positive or = 0. We will bring these results together. Let the development x be (26) x = ^kae a i& cos (AC -f r x -f dQ). Then the development of y is (27) y = ^kae a i? sin (AC + r- + *) with the same coefficient 7c, and at the same time rA + r + a=:+l, = 0or + 2,so that A + r = 1. (28) < a -f- /5 [A] is odd and positive, or when /9 = 0, in the L limiting case, the first sum. = 1. Let the development of z be (29) z = S/c' e a i? sin (AC -f r - + dQ): then f * + r + ^ = 0, = 1, and \-f/9 [A] is even and positive, or = 0. At the same time in all three cases, [r] even, positive, or 0, p [] even, positive, or 0. The five quantities , /?, A, y, 3 can have all possible integral values, positive or negative, if the conditions (28) to (31) are fulfilled, and it only remains to determine the coefficients k and k' which are pure numbers. This development is the more expeditious, in so far as all the elements , e, i, x, Q, except e PROBLEM OF TWO BODIES. 37 (which, however, appears in C = nt -f- e) are analytically expressed. If, however, the order is that of multiples of C, the development can be expressed as follows: x = S(Z cos AC + ra sin AC), ?/ = S(Z sin AC m cos AC), s = S(Z' sin AC + ra' cos AC). 5. HISTORICAL NOTES ON THE PRECEDING SECTIONS. Newton's law of gravitation is the source of our explanations of the motions of the heavenly bodies. Its correctness is confirmed by attentive and unbiassed observation, while the development of our knowledge of it is due to the inventive audacity and persistence of mathematical analysis. Observation shows that the stars apparently describe circles, their common center being near the pole-star. This is not the place to give the reasons which led Copernicus to conclude that these motions were only the reflex of a single rotation, and, hence to propose the hypothesis that The earth is a body hanging free in space rotating uniformly about an axis which passes through its poles. Aside from this motion, the stars remain at rest, at least for the unaided eye and for the duration of a human life. An exception to this fixity is found in a few bodies, the sun, moon, planets, comets and meteorites. Of these, the first three move witli some uniformity among the fixed stars, while the others show striking irregularities which were noticed in very early times. On the hypothesis of a motionless earth, Ptolemy (Claudius) undertook in the second century after Christ, in his work which was later called the Almagest, to reduce the observed motions to circles, or rather to combinations of circular motions. Though this assumption is now rejected, it must be admitted that it was in accord with the ruder observations o antiquity. As observations continued and irregularities multiplied, they were obliged to combine a large number of circles into cycloids, until at last the entire wheelwork became so complicated that only a few could grasp it all, and to these few was attributed an 38 THEORIES OF PLANETARY MOTIONS. extraordinary degree of learning. This circumstance, together with the religious respect paid in those days to authority, permitted the Almagest to appear as undeniable truth, and any explanation was right or wrong, according as it agreed with that authority or not. The hypothesis of the earth's rotation, which had been already proposed by the Pytha- gorians and Aristarchus, fell into almost entire oblivion, until Copernicus (1473-1543) appeared and shattered a structure which had existed for thirteen centuries. Copernicus's sys- tem was given to the world in a work which appeared just before his death and which was entitled De Orbium Ccelestium RevolutlonibuSj libri VI. By assuming that the earth rotated on its axis and also moved in a circle about the sun, he reached the conclusion that the orbits of the planets were circles, and he explained the observed irregularities by the fact that we observe these circles from a moving station of observation. He adhered to the theory of circular motion, but assumed that the center did not exactly coincide with the sun. The law of gravitation was soon introduced and was gradually brought to a high degree of perfection by the successive labors of a few great men, each one of whom took up the work where it was left by his predecessor. Tycho Brahe (1546-1601) took new observations with perfected instruments, especially on the planet Mars. Kelying on these observations and explaining them by the Copernican theory, Kepler (1571-1630) succeeded gradually in evolving the three laws which are called by his name, and which he published in his two works De motibus stellae Martis and Harmonices MundL He even expressed clear views on universal gravitation, but he was overcome by work and vexations and left the crowning of his labors to a successor who, instead of his method of bold and happy specula- tion, employed a powerful synthesis. Meantime, Kepler's contemporary, Galileo (1564-1642) awoke hanics from the sleep of a thousand years, into which it sank after the death of Archimedes. Galileo's work appeared as Dis- corsi e dimostrazioni Matematiche intorno a due nuove scienze atlenenti alia Meccanica, ed ai movimenti locali. The year of PROBLEM OF H BODIES. 39 his death was that of the birth of Isaac Newton who first con- ceived the idea of gravity in 1666, and scientifically grounded the law of gravitation in 1686 the latter in a work entitled Philosophiae Naturalis Principia Mathematica. It can not lessen Newton's credit that Boulliau had already expressed this law, but only as an assumption without any proof. By the aid of the law of gravitation Newton proved the laws of Kepler and ex- tended their validity to the comets, the bugbears of earlier times. Newton's surpassing genius appears in a clearer light when we consider the state of science at that time. Mechanics was as yet barely outlined, the Infinitesimal Calculus owed its origin largely to him, and that powerful analysis did not yet exist by the help of which our demonstrations attain such extraor- dinary simplicity and transparency. Newton's method was, for the most part, synthetic, yet his results can be reached only with considerable difficulty by the analytical methods which now add so much to our facility in the solution of problems. With the development of the analysis, for which we are especially indebted to Leibnitz and Bernoulli, the subject was simplified and the problem was reduced to the form of three total simultaneous differential equations, the integration of which can be performed in many different ways. The method employed in this book is that invented by Laplace and given in his M6canique c6Uste. The literature on this subject, and especially that of the derivation of the coordinates by Kepler's equation, is remarkably large. The principal contributors were Euler, Gauss, Lagrange and Bessel. 6. THE PROBLEM OF n BODIES: THE GENERAL INTEGRALS. If more than two points attract each other under Newton's law of gravitation, we can, on the principle of composition of forces, get the components of the force acting on any point by adding the corresponding components of the forces exerted by the other points on this point. If we designate the points by P u P 2 , P 3 P n , each point with its proper index, and if we represent the distance between any two points P% and P^ by r so that 40 THEORIES OF PLANETABY MOTIONS. Xi ?/! z l . ,, we get for the components m^ . a , m l '' , m l . a or the first point by (2), 1, and the parallelogram of forces, ,/rf-fcx Cv o(yi o / ^"> *^1 **3 w^l i <2) * m, r^- = 7c 2 ( m 1 m 2 - 3 l - +.WH ra 3 - >* + . . . . at r 12 r 13 . CC n Xi \ -i- m 1 m n - ~ I . 'in ' Similar equations for the other points are obtained by sim- ple exchange of subscripts. An exceptionally elegant form can be given to them," if it is remembered that the right-hand mem- bers of equations (2) are the partial derivatives of a function of the coordinates with respect to those coordinates whose moving force is in the left-hand members. This function, as the differ- entiation at once shows, is /o\ y _ p i 2 i i 3 n - l \ r 19 r lB r r (rt _ 1) This function, the existence of which was first shown by Lagrange, is, after the example of Green (1828) and Gauss .(1839), called the Potential of the forces. It plays an impor- tant part in other branches of Mechanics and Physics. Intro- ducing V, equations (2) take the simple form V satisfies several partial differential equations from which by the assistance of (4), integrable total differential equations can be formed. Since V is a function of the distances, it is a function of the differences of the coordinates, and consequently the sum of its partial derivatives with respect to any coordinate, as x, is equal to zero. Therefore Further, by a rotation of amount

dy l dy 2 dz l dz 2 , ,,. ~aT~ ' ~dT ' "' ~ 7 dT ' ~lf~ ' ' ' s a a( *d in g we & et ^^ f d 2 x dx d 2 y dy d z z dz~\ ^-1 m V dt* ~df ~ dt 2 ~di " dt 2 ~dtJ ^^/ 8F dx dV dy dV dz\ Z ^JV8^TW~ ~fy/~~dt~ ~dz~~dtJ' By integration, this becomes This is the equation of the kinetic energy for this particular case. It expresses the fact that one-half the sum of the pro- ducts of the masses into the velocities differs from the potential by a constant only. PEOBLEM OF n BODIES. 43 It should be noted that the kinetic energy, and therefore, some velocity, can become infinite, according to (14), only when Fis infinite, or when one or more distances of the points = 0. In that case, however, the differential equations (4) lose their significance because some of the right-hand members become infinite. The rules of the Calculus are applicable only up to the instant of collision; hence, in this entire discussion, there is a preliminary assumption that no collision occurs, at least none during the time selected. The integrals (9), (13), (14) are the only ones so far discov- ered for the general case, and it has been lately shown by Professor Brun^ that no other algebraic, nor Abelian integrals exist. While the investigations of the mathematicians have, as will be shown in later paragraphs, brought to light relations of the highest interest, there is, nevertheless, reason to think that the analytical functions so far introduced do not suffice for the solution of the problem. Systems of differential equations are ganerally not submissive to integration; their very generality gives them an obscurity which is not likely to yield to any other than a complete general treatment. As the center of gravity plays so important a part, it is very often advisable to refer the motions to it, by employing a moving system of coordinates, whose origin is at this center and whose axes are parallel to the original axes. If the coordinates in this system are represented by primes, and those of the center of gravity referred to the stationary system by , r h C, then (15) * X = *i' + *> and so forth. If these are introduced into (4), , >?, C, disappear at once from the right-hand members, and they disappear from J %; r/ 2 yi d 2ff the left because r^- = -^m = -777- = 0. In other words: dt* dt 2 dt* The law of gravitation applies also to ihe relative motions about the moving center of gravity. The entire motion of the system, then, consists of two parts: a motion of the system as a whole, represented by that of the center of gravity, and an internal motion about the latter. 44 THEORIES OF PLANETARY MOTIONS. The integrals (9), (13), (14) must remain significant for the primed coordinates, but the constants of integration must have other values, which may be represented by 6's. Naturally, then b x = b x f - b y = bj = b s = bj = 0. Further, from (15), if we put Sra M dz dy~\ ^ ( , dz' ^ dz' ^ dif dr) -+- 7) 2 m -77 C .2j m -^7 -j-r dt dt dt and, since 2 w as' = ^mij = ^mz' = 0, and by ( 11 ), rfC r (1y _ <7/ r/ s a e ' a y r 'dt~ *dt~~ M' 2 we get (16) * Finally, TO ^w dx' 2 + dy' 2 M + dz' 2 Md; 2 - ' 2l 2 d/ 2 + 2 dt 2 This equation shows that the kinetic energy of a system of bodies is equal to the kinetic energy due to the motion of the bodies about their center of gravity, plus the kinetic energy due to a motion, equal to that of the center of gravity, of a body whose mass equals the combined masses of the bodies. From (17) we get, therefore, (18) 2 M where V is the same as V except that the relative coordinates are used. PKOBLEM OF U BODIES. 45 To equation (17) Lagrange gave a form which admits of a highly interesting conclusion concerning the stability of the system. Introducing into (4) the primed coordinates, and multiplying the equations in order by T/, ?//, /, etc., and remembering that V is a homogeneous function of the coordinates of the degree 1 and hence, satisfies the equation we get and, adding twice equation (18) and reducing, (20) This equation can be given a form in which the mutual distances of the planets alone appear. It is V = <"*' - V)' + (*' - 2V') 2 + (^' - V) 2 = x ^ + J&" By multiplication by m% m^ and addition, 2 = - S [fltf (*/ 2 + ^' 2 + ^' 2 ) + 2 A TO/ , (^' * The last sum = (Sm^') 2 + (Sm^') 2 + (Sm^ A ') 2 = 0, and (20) becomes " F' is always positive. If C" is also positive the right-hand member is positive and greater than 2 C". The first derivative of SWJL m^rty will then increase continuously. If it is origi- nally negative, it will at some time = and will afterwards be positive, so that in this case S m% m^ r^ 2 grows with acceler- 46 THEORIES OF PLANETARY MOTIONS. ated velocity to infinity. A distance r% u must eventually become infinite, that is one point must at last separate itself from the others to an infinite distance. If by stability of a system is meant the condition that these distances always remain finite (never and never oo), we can say: In order that a system may be stable, C' must be neither positive nor zero; that is, the kinetic energy must be less than the potential. It does not, however, necessarily follow that when C' is nega- tive the system is stable. Only partial stability can be con- cluded from it since not all the distances can become infinite. For if all points separated infinite distances from each other V would = and the right-hand member of (21) would be negative. Sra^ wr; 2 must finally become smaller, and at least two points approach each other, and this is impossible since at infinity the attraction = 0. The conception of complete stability, as defined above, is of the highest importance for astronomy. The solar system, including only sun and planets, is, in all probability, a stable system. The labors of Laplace and Lagrange and of later astronomers culminate in the problem of stability, and they have shown that this conception can be narrowed down to a great degree when applied to this system. Unfortunately, however, they have not succeeded in giving an entirely rigorous proof, nor, in general, in fixing the conditions of stability. At present it can only be said that our solar system may be stable, and, with great probability, is stable without limit as to time. Finally, we will add a paragraph on the conception of the invariable plane, as represented by Laplace for our solar system. The expressions M_ dtf M dz> dy' dx' y dt~ ' dt 9 ' dt' ' dt' ~dt ~~ y ~dt have the remarkable property that, with a rotation of the system of coordinates, they are transformed in the same manner as the coordinates. Taking the familiar transformation formulas (22) * X" = x'a x + y' fi x + Z > rx) PROBLEM OF n BODIES. 47 and representing the first set by A', B', C', the transformed ones by A", B", C", we get from (22) A" = A' (P w r. r y P*} + B' ( r , *. y r*) + C' (& ft, a.), or, by known relations between the nine coefficients, (23) * A" = A'a x + B'/l x + C' r .. If we represent by ft", C y ", C" the constants which replace" ft', Cy, C z f of equations (16), we get similarly, (24) * ft" = ft' a x + C y ' ft x + ft' /-.. The constants of integration are, then, transformed in the same way as the coordinates. If a new system is selected such that C y " = ft" = 0, then two equations fix the x" axis com- pletely, but not those of y" and z". For from them is derived, ft' : and hence, fix = ty c; 2 + ft' a vcy 2 + cy 2 + o s ' 7 ' = =, and (26) c x " = vcy 2 + cy 2 + ft". The 2/" 2?" plane, thus determined, is Laplace's invariable plane. The sum of the areas projected on it is a maximum, while for every x" plane it is zero. The invariable plane makes but a small angle with the ecliptic, and has the exceptional advantage over the latter, as well as over that of the equator, to which are generally referred the spherical coordinates of the stars, that it is absolutely invariable, while in the course of thousands of years the ecliptic and the equator change their position in space to a notable amount. A proposal to use the invariable plane is, however, for the present illusory, because it can only be fixed after an absolutely exact knowledge of the masses of the planets, and, besides, all astronomical tables now employ ecliptic or equatorial coordinates. If at any moment all points are in one plane and all the velocities in the same plane, they will always remain in this 48 THEOKIES OF PLANETARY MOTIONS. plane, and it becomes the invariable plane of the system. This is approximately the case with the solar system. If C x ' = Cy = C z f = 0, every plane is invariable. This would be the case, for instance, if all the points were at rest at a given instant, or if they were all in one straight line and their motion was confined to it. The motions in the solar system are referred by astronomers, not to the center of gravity, but to the center of the sun. This requires some modification in the formulas which will now be given. Let the Gaussian constant k = 1. Let m 1 be the mass of the first planet, x i9 y lt z l9 its coordinates with reference to the sun, Xi, 2//, #/, these with reference to the center of gravity of the system, etc. Let , 9, C be the coordinates of the sun referred to the center of gravity, and let its mass be M. The potential V has the value (at 1 - ) 2 2' The equations of motion become therefore * M d ^ * rZV _ 1 dt* '' 8* J OTl dt* '' dx,' ..... But, (27) x l = x,' Z, y, = yS TJ, Zl = zj C, etc., and if we put ' V (xi - x^ + fa -y^-i-^- z \* M we get PROBLEM OF TWO BODIES. IV d^ x, 1 dV 1 rt/V A "2 3 . 3 3 Introducing finally, i 8a?i etc. where /j. may be any number not A, the equations of motion take, the form : 2 ' ~ r, 8 + 8^! ' d/ 2 * ~ + The qualities R l , J5 2 are called perturbing functions, for reasons which appear later. They depend only on the masses of the planets and on their distances from each other and from the sun. When oc lt y 19 z lt etc., have been determined by (27), S, y, C are found by the equation MS + *2mx f = M S + 2m (x + ) = 0. Hence, (29) mx and consequently x,' = x l .etc. The integral equations (16) and (17) also take on a some- what different form. They become (30) dx dz dt d ). 7 f^ ^ dx + k(^mz^m Tt 50 (30) and THEORIES OF PLANETARY MOTIONS dz - mx dx = Co', dy dx 7. PROBLEM or THREE BODIES. Three bodies only will now be assumed, not because the results obtained for three can be extended to a larger number, but because the formulas obtained with this limitation present an extraordinary symmetry. No new integrals will be afforded, as has already been shown. It will appear, however, that by a skillful and elegant use of the integrals already found, the pro- blem can be notably simplified. This is especially the case ivhen the development is reduced to the parts which are inde- pendent of the system of coordinates, making what Hesse calls ihe reduced problem of three bodies. The system of nine differential equations are, in this case, d' 2 X t - dV . . n r> \ ~ ~~ ~' ^ == ' ' )' in which (2) 2 2_3 m This is a simultaneous system and requires for its solution eighteen integrations, of which ten have already been performed, six for the law of conservation of motion of the center of PROBLEM OF THREE BODIES. 51 gravity, three for the areal law, and one for the law of conserva- tion of energy. Aside from these results, the system can be reduced to one with only two variables x, and /, for instance. If the first of the equations (1) is differentiated with t as the independent variable, we get d*x t 8 2 F 8 2 F 8 2 F 8 2 F 8 2 F - O - V 2 H~ O - O 8 2 F 8 2 F 8 2 F " Ms 82/3 / { where t,- = 777 ' v =: dr Wi ~ The right-hand member of (4-) is a function of the coordi- nates and their first derivatives. If (4) is again differentiated with respect to /, and the second derivatives are eliminated by d*x- (1), rrr is a l so a function of the coordinates and their first d n x- derivatives, and it becomes evident that -^ is a function of the same quantities. Fixing the attention only on x i9 -rf , -W d*x- TJT ) etc., it appears that in these equations there are only sixteen elements, (the eight other coordinates and their first derivatives), which have a perturbing action, and, to eliminate these we must have seventeen equations. The proper way to d. . . . . a/ Xj, d X{ a X; a . . , i o this is to express rry- ) rrr ' ' * .ii 8 as functions or the Ctt" (Jit CLT nine coordinates and their first derivatives and, by means of the seventeen resulting equations, to eliminate the eight remaining coordinates and their derivatives. The final result is an equa- tion of the form 52 THEORIES OF PLANETARY MOTIONS. that is, a differential equation of the eighteenth order between Xi and t. When (5) is completely integrated, the problem becomes one of pure elimination, and the previous expressions for d 2 x d^x- - |8 * are more than enough to calculate the remaining Cit Qjl coordinates and their derivatives. In the place of a coordinate, any function of the coordinates and their first derivatives may be employed. This will in gen- eral lead to a differential equation of the eighteenth order between the function and the time, but in special cases it may lead to an equation of a lower order. Indeed, it may be the case that the equation is of the first order and takes the simple form ^ - dt ~ ' and this is always the case when

1 t>t/2 - & y >L/2 ^3 - v& ) ^3 CC^ - CC ) X, X, = X'" 9 X, X 3 = X r , Xt X^X", then any such function depends only on the primed quantities. The number of these is eighteen, but it immediately reduces to twelve, since * x' + x" + x'" = o * X' + X" + X'" = 0. This reduction would destroy the symmetry and we will there- fore retain all the eighteen x', X', etc. If we define the twenty-four expressions [A, /-A], [/, 7^], [^i A*], by the formulas of Hesse, = 05*0?" + y*y* + z*z* = [/-/-, A], (12) [A, /,] = &x* -f i/ A F^ - r t PROBLEM OF THEEE BODIES. 55 they will permit the transformation required and it is clear that they may be expressed linearly, by (11), but for symmetry we shall rather take, [l,l],[2,2],[3,3],orr 2 3 2 ,r 31 W, (13) [l,l'J,[2,2'],[3,3'], ,, = [1,2'] -[2,1'], and (14) or, by (11), P = [1,2'] - [2,1'] = [2,3'] - [3,2'] = [3,1'] - [1,3']. Considering the results of (11), (15) V .' we find by simple transformations [1, 2 ] = 4 ([1, 1 ] + [2, 2 ] - [3, 3']), [!', 2'] = - 4 ([!', 1'] + [2', 2'] - [3', 3']), (16) , etc. There are also other simple expressions which permit the transformation (9). These are determinants of the following type: W> II V x ) y > z (17) D = x" , y", z" , etc. X', Y', Z' If two such determinants are multiplied by rows, expressions of the form [^, M], etc., will be obtained. For instance [1, 1 ], [2, 1 ], [!', 1 ] D-= [1,2], [2,2], [l',2] [1, 1'], [2, 1'], [!', 1'] As nine independent functions, which admit (8), are suf- ficient for expressing all the others by them, it follows that there must exist one equation containing the nine quantities (13) and p. To find this equation, let us take the vanishing determinant 56 THEORIES OF PLANETARY MOTIONS. ' , y' , z' , " , 2/" , *" , T, F', Z',0 ", F", Z", and form the squares by rows. The square is [1,1], [1,2], [1, 1'], [1, 2'] [1, 2 ], [2, 2 ], [2, 1'], [2, 2'] [1,1'], [2,1'], [1',1'J, [2',!'] [1, 2'], [2, 2'], [!', 2'], [2', 2'] and if we introduce therein expressions (16), we immediately obtain the equation sought, but in an unsymmetrical form. From inspection of (18) we get an equation of the form (19) A P * + B P * + C = 0, and (18) = 0, (20) ' P = V* + lf 2 ' 4AC ' in which p is expressed in terms of the quantities of (13). We now pass to the formation of the differential equations between the nine quantities of (13) and the time, or rather, since we retain p, between (13), p l and t. To this end we first form the differential equations between * x' t x", x"', and X, X", X", introducing, for brevity, is to be substituted in (25). The inter- change of indices will then give the two other differential equa- tions of the third order. From these an integrable equation can be formed for, from (26) & + & + & = _2^ m l m 2 m 3 dt Hence, from (25) and its analgues, we get 3 r 23 2 -f m 3 m l r 31 2 + m l m 2 r 12 2 ) d 2 (27) | (m 2 m 3 r^ + m^m^ -f m l m 2 r l ^) = MV Then, by integration, (27) |^( 2 3 r 23 ' or equation (21), 6. If equations (25) are multiplied, in order, by , and the products added, the p disappears. It follows that ~~df df PEOBLEM OP THREE BODIES. ndwi dft -i i df dt ~ V2 J m,r 31 3 n dWl ^JL .7/3 f !4 (28) Besides the integral (27), the equations (25) have another, a combination of the surface integrals, found by Lagrange. (29) (7, Q! m : fC,?)i + )'2 , (^, C), r - m + ^ + + %^ + where, for brevity, (3) >?A -^ C^ -^JT = (^> ^)A? etc. The combination given by the new integral is -the sum of the squares of the areal integrals. For , o (^ Oi (^ Q 2 + (^ g) t (C, s^) 2 + (g, vy), (I-, r y ) mj rw 2 But, by a known proposition of determinants, (n, 0.' + (:, ?).' + (?,*),' = [i, i] [r, r] - [i, r] (* 0, (7, 0, + = [1,2] [l',2'] - [1,2'] [l', = [1,2J etc. Consequently + 2 -((tfl, 2]') 2 - m. 4- .... And (31) is the integral sought, when for p its value in (20) is 60 THEORIES OF PLANETARY MOTIONS. taken. The system is reduced to the seventh order and con- sists of 1. A differential equation of the third order (28), 2. " " " second " (29), 3. " " " " second " (31). If / is eliminated from these equations, the order is reduced to the sixth, and this is the extreme of reduction reached to the present time. When the reduced problem shall have been completely integrated, the final solution of the entire problem will, as shown by Lagrange in his celebrated memoir, depend on a single quadrature. If equations (29) are multiplied in order by j, ^, Z lt we get, by appropriate addition, (32) e*, 3 m 2 dt m 2 dt w 3 dt The left-hand member is a function depending on the dis- tances. If it is squared, by compounding the columns, and the previous formulas are used, the desired form is at once obtained. In the same way the expressions o? 2 + 6^2 ~1~ c '3' an( ^ a ^ ~+~ ^3 + c^ 3 may be determined. If , y are taken in the invariable plane, a = 0, and 6 0, and from (32) and the corresponding equations Cj , C 2 , C 3 can be obtained at once, that is, the position of the plane in which the three bodies move, in its relation to the invariable plane. In order to get the six remaining coordinates, the five fol- lowing equations are to be used. I j r\ (33) f 4+ t-^* -<.'=*>.'. / + r t > = iv :* = r,,', PROBLEM OF THREE BODIES. 61 and only one more relation is needed. Putting tan

^ 3 : 2 , B = C 2 3 -3 2 , O c 2 1?3 3 >? 2 . By equations (12), 7, these expressions are unchanged when the subscripts 1, 2, 3 are cyclically interchanged. The other determinants therefore become l w J w '-%/ Calling the three determinants of (1) p l9 p 2 , _p 3 , we get and further, identically, Pi + P* + PS = 0. Designating the common value of the preceding fractions by /, PJ = lm ly p 2 = lm. 2 , _p 3 = ^^3, hence A (nh + wi 2 + ? 3 ) = 0, and as w^ -(- w 2 H- w 3 cannot vanish, (6) A = 0, and hence p l = p z = p 3 = 0. The quantities A, B, C, have a very simple geometrical mean- ing; they equal twice the three projections of the triangle whose UNIVERSITY 64 THEORIES OF PLANETARY MOTIONS. angles are at the three points. By their ratios, they therefore determine the position of the plane of these points. Equations (6) show that the relative motion of the points is in this plane, and the above proposition follows from it. This case is, consequently, included in case I and affords no new reductions. III. In which the three areal integrals and the constant of the kinetic energy are equal to zero. In this case the system of the fifth order, (found after the elimination of /), can be reduced to an order lower by unity. This follows from a principle of great simplicity and import- ance, which may be viewed as an extension of Kepler's third law to the general problem of n bodies. The differential equations of this problem in their original form are # d*x t dV_ df 2 ' dx t ' The second members of these equations are homogeneous functions of the coordinates, of the degree 2. If we write kx {1 ky f1 kZi for x i9 y { , z f and tk% for t, k~ 2 will appear as a factor in both members, and when it is suppressed, the equations have their original forms. From this it follows that from any solu- tion x it y f , ZH another solution kx it ky { , kz i} with a constant factor k can be obtained, if tk% be substituted for / in the expressions for these coordinates as functions of t. This can be expressed as a proposition. From any solution of the problem of 11 bodies another can be deduced in which all the linear magnitudes are increased or decreased in a constant ratio. For such systems the squares of the times, from the first to 1he last places respectively, must be proportional to the cubes of the linear magnitudes. When attention is turned from the position of the orbits in space and time, and fixed only on their positions with reference to each other, the system can be reduced to the eighth order, as has been shown, without the introduction of a single constant of integration. If again the linear dimensions are not consid- ered and only the form of the system is taken into account, PROBLEM OF THREE BODIES. 65 another arbitrary constant (the k above) is eliminated and the order is reduced by unity. But the two given integrals (that of the kinetic energy and that of the sum of the squares of the areas) determine but one of the two integrals in the second case. If the two constants of integration are given, they must, after the above substitution, be multiplied by determined powers of k (the first power + or ), and the product of the two is the" only general integral for this case. It is otherwise when these constants each = 0, for they do not then change on this substitution, but remain = 0. In this special case the last system of the seventh order is reduced by two units. So long as we confine ourselves to real quantities, this case can occur only when the three areal integrals are separately zero. Then there is a further reduction of two units, and hence, When the three areal integrals and the constant of the kinetic energy vanish, the given system can be reduced to the third order. I will not delay farther on this case, as it is a special one for which the mathematical formulas can be at once written out after the above explanations. IV. In which the three distances of the three gravitating points remain constant. It must first be shown that this is possible. For a given instant, assume that r 12 , r 23 , r 31 are given and that their first and second time derivatives vanish. Then, for this instant, the quantities Q and the three derivatives of (25),, 7, also vanish, and we have from these equations, if the third derivatives of the time also vanish for the same instant, 12 ' therefore either (a) r 12 3 = r w 8 = r 31 3 , and, since they are real, (a) r ]2 = r 23 = r sl , or 5 66 THEORIES OP PLANETARY MOTIONS. (b) /> = 0. These two cases are to be considered separately. (a) 7" 12 9*23 T SI . The triangle formed by joining the three points is, for any given instant, equilateral and the first and second time deriva- iives of the distances vanish. Under this assumption, derivatives higher than the third also vanish, that is the distances remain >constant. For differentiating the first of equations (25), we get dQ l l r 13 In which the right hand member vanishes for the given instant .since ~ contains no term free from the first or second deriva- dt tives. In the , same way it can be shown that derivatives of Tiigher orders also vanish. If r is the common value of the three distances, then by (26), *a -t 3 ? and (18) becomes Jr 2 . 0, p ' r IT" ' U J ~ ~nT o, " 2 + 1 , o Jf 1 M 7"' ~ 2 r 1 J/ if T7" ) ?* = 0, -or ^-Y - a Af r 4 ' ^ = 3 Mr. The first of the determinants in (2), this section, becomes PROBLEM OF THREE BODIES. 67 f) p A ~1T ~ T' " > and the two others likewise vanish, and this case becomes a special form of Case II. Equation (34) gives for the angular velocity with which the triangle turns in its plane If all the motions are referred to the center of gravity, it appears that they are as if the invariable triangle turned in its plane around its center of gravity with a velocity -t / Or inversely, if the three gravitating points form the vertices of an equilateral triangle and if each has a velocity, in the direction perpendicular to the line joining it to the center of gravity, such that in the time dt, the angle through which this V' ~\T jpdt, then the triangle rotates as if the system were rigid. (b)

v = s , to ==-- 3 '31 '12 Equation (V) can, with the aid of (I), be brought into a very elegant form. The transformation, which is not an easy one, is most simply performed as follows: If, for brevity, A = w[l,l] B =21 then pip _ p The expression - can, from (I), be expressed ^ in two forms, namely, + P, p 8 ) = WlJ B JfM[l,2], + P 2 P 3 ) = 7^ 2 JL Mv[l, 2], and (Y) becomes - Jf [w,v (^[2,2] B [1,2]) + m 2 M (B [1,1] - But = (M- to) ([1,1] [2,2]- [1,2 J), Hence, = M ([1,1] [2,2] ^^(m.vw + mzwu + m t uv), or = 70 THEORIES OF PLANETARY MOTIONS. This equation divides into three. The first and last factors can not vanish, as long as only positive masses are considered. There remains only the middle one, and that brings us back to the previous case where the three points lie in a straight line. If negative masses are admitted into the problem of three bodies, the first or third factors might = 0. Then the planes of the three orbits no longer constantly coincide, but they rotate about an axis which is generally inclined to the original plane. The rotative orbits of two of the gravitating points are circles about the third, but their planes do not pass through it. Moreover, it is worthy of note that the case in which the sum of the masses vanishes represents a remarkable limiting case of the problem of three bodies. There is then no center of gravity, the expressions 2>mx, 2m?/, 2m depend only on the relative coordinates, and the six integrals of the center of gravity relate only to the relative motion. If there are only two bodies, the path of the one relative to the other is a straight line described with uniform velocity, and the relative coordinates are linear functions of the time. To get the actual path, these functions are to be inserted in the original differential equations (!),!, when they become determinable by direct quadrature. A special case is that in which the relative coordinates are constant, when each point describes a parabola in space. If there are three bodies for which m l -\- m 2 -f m 3 0, the order of the reduced problem, in general the seventh after use of the areal integrals and the proposition of kinetic energy, can be reduced by two, because equation (27), 7, is at once inte- grable and gives m 2 m ?i TK -f- m A m l r 31 2 -f- m l m 2 r 12 2 = Ct* + CJ -f- C 2 . But this can not be farther reduced by elimination of the time, for this is contained explicitly in the preceding expression. V. In which the angles of the triangle, formed by the three bodies, remain constant. In this case put r 23 = ur r 3] = ur r = wr PROBLEM OF THREE BODIES. 71 iii which r u r 2 , r 3 are constant and u alone is variable. Intro- ducing these quantities into (25), 7, we get du Here P/ is what P in (26) becomes when r lt r 2 , r 3 are intro- duced for r 23 , r 31 ', r 12 and (VI) In /> remains yet u and its first and second derivatives in a somewhat complicated form, as is shown by (20), 7. To get an equation in u, which shall contain not p but only ~ , the i Cit x above equation must be differentiated again, and we get du ! dp Multiplying the first equation by 3 ^- and adding, it appears / ...Cudu\ 72 r dii\ / ^ 51- I o 7 d 2 \u-rr l a j V d/ -/ 3 du \ dtJ (Ylla) rr \ *~ ~ in which p' is the value of from (24), 7, when r l9 r a , r 3 are inserted in the place of r 23 , r 31 , r 12 . Prom (VII) can be formed two similar differential equations with the coefficients r/, P/, A 2 p f and r 3 3 , P 3 ', A 3 p f respectively. If rr does not = 0, that is, if the distances are not constant, then, as easily appears, must the coefficients of the equations be in proportion, and (VIII) n 2 :n 2 :n 2 = P/:P 2 ':P 3 ' = A lP ':A 2 p':A 9 p'. 72 THEOEIES OF PLANETARY MOTIONS. These make four mutually consistent equations between the three quantities r u r 2 , r s . Taking the first and third sets of ratios and assuming that p' does not vanish, or, if it vanishes, nevertheless, ,p 2. p 2. ^ 2 __ ^ . 'A it follows that r?: r}: A A A 1 ' ^^ + ^ + ^ = A,: A,: A,: - 1 + -' + -' m 2 m 3 m^ m 2 m 3 But, by (VI), M 2 A. 2 A 2 Since -\ - -\ - cannot = 0, it follows that m l m 2 m s A l = A 2 = A B 0, and (IX) r, = n = r,. We will assume that this = 1, then and the triangle is again equilateral. The remaining conditions of (VIII) are likewise fulfilled, since it follows from (IX) that pi p f p. f = ]\f Equation (VII) now takes the form du du (X) d 2 , . ., = M 3-- It is easy to show that, in this case also, the motion is in one plane. The differential equations (11), 7, become now JfC, PROBLEM OF THREE BODIES. 73 which are the differential equations for the relative motions of two gravitating points the sum of whose masses = M. Hence The relative motion of any two bodies about a third is exactly the same as if the masses were united in the third, leaving the mass of the other two = 0. There remains the case when p = 0, but not A l A 2 = A 3 = 0. We then get from (VIII) r/Ps'-r/P/ = 0, r 2 2 P 3 ' r 3 2 P 2 ' = 0, rs'P/ rVPa' = 0. If p' is written out as it is found directly from (24), 7, we get r 2 p / _ r" P ' _ />' f 1 2] /[2,2][1,1]-[1,2J [3,2][l,l]-[l,2][l,3;h rf~ r, 1 /-[2,2][1,1]-[1,2J [3,l][2,2]-[2,3][l,3n TOI V~ ~^\~ ~ - M But, = A, t when A is an abbreviation, as follows A = (TI + r 2 + ^(r, + r 2 ^(r, r a + r 8 )( r, + r, + r 3 ). The preceding equation, since p' = 0, then becomes 2 '-r/P/ = A - - . - = - A (4, + 4 2 - 4. . v/ 2 '1 In the same manner r 2 2 P 3 ' r 3 2 P 2 ' = - A (A, A, + A,) , r 3 2 P/ n 2 P 3 ' = A ( A, + A, + A,} , or, if it is not true that A l = A 2 = A 9 , then A = 0, that is, points must be in one straight line. 74 THEORIES OF PLANETARY MOTIONS. It appears that in this case also the orbits must be in a fixed plane and that the paths of two points about a third must be as if they were attracted by the third only. Bat the attraction of the third is not that of the sum of the masses, but a somewhat more complicated relation exists. VI. In which one mass, m l , or two masses, m l and m 2 , = 0.. Let m l 0, then the first of (25), 7, passes over into dt 2 ~ ( 2 + "" 3/ r, 3 23 which can be at once integrated, so that we get r 23 as a function of t. After substitution of r 23 in the two other equations of (25), they become differential equations of the third order between r 13 , r 13 and t, of which there is no known integral except when, as Jacobi has shown, r 23 is constant. VI is nearly the case in the solar system for the three bodies, sun, earth and moon. The relative motion of sun and earth, or better of sun and the center of gravity of earth and moon, is nearly independent of the moon's position, and the latter is determined after the former. It is historically of interest that this idea was the first incentive to Lagrange's investigations. If m l and m. 2 = 0, r 13 and r 23 , and generally the relative motions of the first and second points about the third, may be directly determined by 1. The motions of the first relative to the second then follow at once, and it appears that the third equation of (25), 7, can also be integrated. This case is approximately that of several bodies with very small masses with reference to their relative motions about a single body with large mass, and these are closely the conditions for the sun and planets. 9. HISTORICAL NOTES ON THE PROBLEM OF THREE BODIES, The investigations of the preceding paragraphs are espe- cially due to Lagrange, who, in 1772, published his classical treatise entitled, JEssai d'une nouvelle methode pour resoudre le probleme des irois corps. His most noteworthy advance was PROBLEM OF THREE BODIES. 75 in the formation of equation (34) for the determination of /?; this was especially creditable to him because the theory of determinants was then in its infancy, Beyond this point he turned his investigation in the direction of the system formed by the sun, earth and moon, expressed in the symmetry which was always so sought by him. His greatest contribution to the problem was in its reduction to the seventh order, a reduction which is equivalent to the production of a new integral. Jacobi gave another kind of reduction in his memoir pub- lished in 1843, and entitled, Sur V elimination des noeuds dans le probleme des trois corps (CreUe's Journal, p. 115). He introduced into the differential equations the relative coordi- nates about the center oE gravity and then considered two new fictitious gravitating points such that the coordinates of the three points, relative to the center of gravity, become linear functions of the coordinates of these fictitious points, and the conditions ^mx = ^my = ^mz = are identically fulfilled. When these new coordinates are introduced into the expressions for the potential and the kinetic energy, the latter take on a somewhat different form. The system thus formed is of the twelfth order, but is reduced by four by the aid of the areal integrals, and the law of the kinetic energy, and it then appears that, by making the invariable plane the plane of reference and introducing polar coordinates, the problem can be so trans- formed that the nodes of the two variable planes are determin- able by a simple quadrature and there remains a system of the seventh order. This method is remarkably elegant, yet its tal- ented author did not succeed in reducing the problem beyond the point reached by Lagrange. The reduction by one order in the differential equations, accomplished by Lagrange and Jacobi in such different ways, can be performed on the equations in their original form, as has been shown by Eadau, Sur ^elimination directe du noeud dans le probleme des trois corps, (Comptes rendus, LXVII, 1868 p., 841) and by Allegret in his Memoire sur le probleme des trois corps (Journal de Mathematiques, 1875, p. 277). The labors of later mathematicians have resulted in many 76 THEORIES OE PLANETARY MOTIONS. noteworthy transformations of the problem of three bodies. Bertrand and Bour (Memoire sur le problems des trois corps, Journal de I'ecole polytechnique, 1856, p. 35) have brought the reduced problem of the eighth order, by ten known integrals, to a canonical form in which the function H does not explicitly contain the time. This will receive attention in the following sections. Bour has succeeded in putting the expression for kinetic energy into such a form that it separates into two dis- tinct parts. One part depends only on the form and position of the triangle made by the bodies, and on its position and changes of position with reference to the invariable plane. By using this form Mathieu (Memoire sur le probleme des trois corps, Journal de Mathematiques, 1876, p. 345), by introducing a system of eight variables, which have a very simple geometrical meaning, has succeeded in obtaining a canonical group of differ- ential equations. However interesting these studies may be, they do not reduce the order of the final system. This has been done, so far, only in the two cases where either the motions are in one plane, or all three areal integrals = 0. The second case is, as has been shown, only a special case of the first (only for three bodies, however), because the motion must be in one plane when the three areal integrals = 0. The order can then be reduced by two, as shown by Allegret in the memoir cited above. Allegret thought that by special transformations this simplification could be made general, but he fell into an error which Mathieu has pointed out. Another simplification which was introduced by Hesse in his memoir, Ueber das Problem der drei Korper, (Crelle's Journal, 1887, p. 47), also proves illusory, as J. A. Serret has shown, in a note to Lagrange's memoir. This celebrated problem still stands where it was left by Lagrange, a century ago, and though a multitude of versions of the problem have been developed, Bruns shows (Ada Mathe- matica XI, p. 43), that they give no prospect of its more com- plete algebraic solution. SECOND DIVISION. The General Properties of the Integrals. 10. POISSON'S AND LAGRANGE'S FORMULAS. While it is true that only the integrals already given have been so far obtained, yet they do not include every result of value in the problem of n bodies. Certain highly important properties have been discovered, both of the known integrals and of the unknown. While these properties were at first deduced by Poisson and Lagrange from the system which has been discussed, they have been shown by Jacobi to be funda- mental and to belong to a much more general system. His profound investigations, the framework of which will be given in what follows, have been generalized and simplified by. Pro- fessor A. Mayer and Professor Sophus Lie. To arrive at them, it is convenient to bring into another form the differential equations of 6. If the components of the velocity of the bodies be intro- duced as new independent variables by means of the equations / 1 \ d x i difr dzi / _ i o =u " -or ~- v " ~M - w " ==... the equations of motion become du t 6V dv< 6V dw< 6V (2 ) m-i n- -o i Wf n- = ; > nii 77- = -~ dt dxi dt diji dt dzi when, as before, By the introduction of u i9 v it u\ as new variables, their number is doubled, and also that of the differential equations (1) and (2) form a closed system of simultaneous differential 78 THEORIES OF PLANETARY MOTIONS. equations. At the same time, all the differential equations are now of the first order. Equations (1) can be put in the same form as equations (2), when, for the kinetic energy, we put (4) T = $2m t (u? + v? + wf). For then dT and (1) become , dXi T dl Jj dT < lz * T ' dt diii l dt dVi ' ' dt dwi ' This can be still farther simplified. For, if in the place of w,-, V;, w f , other new variables be introduced by the equations we get (6) T =-- ^^i ( U! + V;2 + Wf )> dT dT -~ = rrifU; = rr du { Finally, putting and noting x,-, y ; , z ( occur only in V, and U,, V f , W { only in T, we get the equations of motion in the following form xt dH dH dz dH _ _ _ _ dt ~ dU< ' dt : 8F, ' dt - _ ~ dt ~ dx, ' dt ' dy, ' dt ~ dz; ' For the same of simplicity employ p t for x i9 y it z i9 and q t for Ui, Vi, W it and n for 3n, and the equations (8) become POISSON'S AND LAGRANGE'S FORMULAS. 79 To pass from (9) to (8), we have only to substitute 3n for n; #,-, y it Zi for pr, Ui, V f , W, for q { . The differential equations of motion have thus been brought to the form (9). This form can be given to very many prob- lems of mechanics, and since, when once found, they recur after the most various transformations, and from them are most easily developed the dependent formulas, they are said to be in canonical form. In order to give all possible generality, we will assume that the function H of (9) is any given function of p t and q f , and also of the time t. With this entirely general assumption for H, the equations (9) can be treated as follows: f) FT c) TT Multiply equations (9) by -^ and -755 and we get, by addition, "> If H does not contain /, this passes at once into (ID **=, and hence (12) H=C, when C is the constant of integration. If H contains /, equa- tion (10) becomes 8H a non-integrable equation. The integration of equations (9) introduces 2/i arbitrary constants, a l5 <7 2 , r/ 3 , . . . a 2n . To Lagrange is due the concep- tion, that these constants may be regarded as independent analytical magnitudes, an idea of the greatest importance in very different ways. The object of the integration is to obtain PI and q { as functions of / and of the 2w constants. The result- ing expressions must be of the form a 2n , t) 80 THEORIES OF PLANETAEY MOTIONS where, in the second members, the parentheses contain the functions which are to determine the quantities placed before the parentheses. (This notation is commendable for its clear- ness and simplicity. ) If equations (14) are solved for the a's, the results are functions of p t) q if t, of the form (15) a { = di(p lt . .. p n , q lt . . . q n , t). Equations (14) are the inverse of equations (15), and con- versely. For equations (14) maybe obtained from equations (15) in the same way that the latter were obtained from the former, i merely playing the part of a parameter. It is evident that, in the choice of constants a,-, there is much arbitrariness. For if a system a lt . . a 2n has been selected, %n independent, constant functions of them may be introduced into (14) and (15), and thus we may pass from the given system to a new. one, a/, 2 ', . . . a 2w '. This is a circumstance which will later prove of great usefulness. As a result of this, we can consider the integration of the canonical equations as dependent on the %n equations (15); that is, we can form 2?i functions of p i9 q it t, which, with the help of (9), have the property of becoming constants. Jacobi, in this narrower sense, calls such functions integrals of (9). The integrals are therefore functions of p { , qi,t, which contain no arbitrary constants, but which with the assistance of (9) become themselves arbitrary constants. These functions satisfy a linear partial differential equation of the first order, which may be regarded as their defining equation. For if a (p l9 p 2 , . . . p H , q lt q 2 , . . . q n , t) is such a function, then /-IP- \ (15a) or by total differentiation, M6) - da + ^ d i 4- i - ' W + ' Zlt ~d ^ ~ This equation by (9) is transformed into POISSON'S AND LAGRANGE'S FOEMULAS. 81 6a = di an equation which must be identical, since it contains no arbi- trary constant. It is the partial differential equation mentioned above. Conversely, if the function a satisfies the equation (17), we can obtain (16) and (15 a) again from this; therefore a is an integral. The discovery of %n integrals of (9 ) coincides with the deter- mination of %n particular solutions of (17), and thus the prob- lem is again reduced to the consideration of this partial differ- ential equation. The identical integral, a = constant, is not here considered. We shall now introduce a much simpler method. If / and

,(/> are functions of Pi and q f , and they may also contain /. Hence in which another summation index i' has been selected. Substi- tuting the value of (/, p ) from (18), gives ((f _\ &? df d4> d qi dp/ dp, ' dqs 82 THEORIES OF PLANETAEY MOTIONS. i' = n i n i' = dpi dq/ dq f ' dps dq t dqs ' dp- l i' = 1 i = li' = l n i' = n i = n i' = n , v* ])-(?,[/, ]} Adding to this the similar expressions for ((^ 5 / I and ( (^ >/) > ^ ) gives the identical equation, (20) by means of which Poisson's formula may be easily obtained. Differentiating (19) with respect to >/ and g/, which maybe done by assuming that it becomes identical on the substitution of a from (15), we get a ^ dp,' d(a,H) Q a 6(a,H) dt , Multiplying the first of these equations by - - and the second ft k ' by , where b denotes any function of p and q } and taking the sum with respect to i', gives , H>, 6) ==Q. ' a/ ' ' POISSON'S AND LAGRANGE'S FORMULAS. 83 Exchanging a and b on the assumption that b is also an integral, and subtracting the two equations gives a!L\ , da tigs r 'dps dt / , H), ) = (-(#, 6),o) = -((#, 6),a), and consequently equation (23 ), by the use of (20), becomes (24) ' But this equation may be obtained from (19) by writing (a, b) for a. This establishes Poisson's proposition: (25) If a and b are two integrals of the system (9), then (a, b) is also an integral of this system. By applying this proposition, we see that If a and b are two integrals of the system (8), then "^_ , da db da db , da db \ f \ ) / /, .,,. dXi' du f d iii ' dx { d y { ' ( t = i da db da db da d ~d Vi dy t d z.i d w t d w,- d is also an integral of the same system. Hence it follows that, by pure differentiation, Poisson's propo- sition furnishes a new integral from two given integrals. The in- tegral thus obtained may be entirely new, or it maybe a function of the two already known, or it may even reduce to a constant or zero. If it is a new integral, it may be combined with the original ones, giving two more integrals, and this process may be continued until the integrals obtained are merely combina- tions of the earlier one's. This will occur when 2n integrals have been obtained, or the cycle may close earlier. The follow- ing investigations illustrate these interesting conditions. 84 THEORIES OF PLANETARY MOTIONS. As an application of Poisson's proposition, we will obtain the third areal integral from the other two. We have a 2 vrii (yi Wi z t v t ), b = 2 m f (Zi iii Xi Wi). And, according to (26), these give (a, b) ^*? ( m f O . z f + m { . w f + niiWi - w f -- . ^ fiii) = 2 m f ( ow -^ ), which is the third areal integral. With this the cycle is closed. The application of Poisson's proposition to all the previous inte- grals furnishes no new integrals. The areal integrals, the integrals relating to the motion of the center of gravity and the integrals relating to the kinetic energy, each form a closed system of integrals. We shall next consider Lagrange's proposition. After sub- stituting equations (14) in (9), the latter must be identically satisfied for each value of a l , a 2 , . . .a 2n , t. Hence, equations (9) may be identically differentiated with respect to a\. This gives dH &H dp, ldqidps 'da i' = l and similarly, Now let a^ denote a new constant, and multiply the first equa- tion by ~ and the second by - and add the products. The sum with respect to i is a\dt dap 9 a\ d t POISSON'S AND LAGRANGE'S FORMULAS. 85 t h> dq, dqtdps 8a A 8a M dq t = 1 i' =1 8g, *"8^6^ ' ~8 A 8^ dptdq The second member is symmetrical with respect to a^ aed a M , consequently da* ~ da^di .9 da^ dt l/|U, *J \*\ \S V V/ '. f'jK. i-1 or 1 = This equation proves Lagrange's proposition; viz: The expression (28) d a\ d a^ d a\ d c ing independent of t, is only a function of the constants a. Moreover the expressions [<7 A , a^] have also the property, that (29) [cv, A ] = [a A , a^] and [a A , a A ] = 0. For the system (8) /OAX r T .- < f < , (30) ra A) au]= > mJ 7=r- - ' -^~ ' 7=r- - -^~ V8a A 8a M 8a A 8a M ' 8a 8^ 0y< 0g< 8a A da^ 8a A d Lagrange's formula will not, like Poisson's, furnish new re- sults from certain conditions, for the formation of the expres- sions [a A , a^ ] already requires a knowledge of all the equations (14) and hence of the solution of the problem. Nevertheless the two formulas are so closely related that each may be regarded 86 THEORIES OF PLANETARY MOTIONS. as the converse of the other. Lagrange, although he mentions Poissoii's formula, seems to have overlooked this relationship. In order to determine this relationship, we shall collect some proportions in determinants for use in what follows. Let it be assumed that the determinant (31) A = a ljl9 a lj2 , C/2 j 1 5 ^2 J 2 J of the 2* order is not equal to zero. The minor belonging to any element a\, ^ will, after having been divided by ^J, be deno- ted ^4u,A, so that (32) A - A - Likewise let a determinant be formed of the A's, ^i,,, 4i,i, . . . (33) On account of the conjugate relation between 4 and Dj we shall call these conjugate determinants. From the second determi- nant (34) ai>) , =: ^._g_ and at the same time (35) J/)=l. From the known relations among the elements of J and Z), we have (36) r/ A)1 ^ (37) axM-d and likewise (38) (39) + . . . -f = 0, / POISSON'S AND LAGRANftE'S FORMULAS. 87 If we form the %n equations (A = 1, ... 2w), x 2n as unknown (40) a^ x x l -ha 2 ,A# 2 + . + a 2n ^x 2n ^= and solve them with reference to x l9 x 2t quantities, the results are of the form (41) A 19 A 2/1 + -4 2 , A #2 + + kn, A & = ^A and, in the same manner, the solution of the 2n equations (42) a A ,ii + aA, 2 #2+ + a A , 2 ^ = gives (43) ' ^ A ,l2/l + -^A, 2 2/2+ +^A,2*#2 = #A. These are the most important properties of the conjugate determinants A and D, and they are also valid when the determi- nants are of an odd order. The developments which follow are valid for determinants of an even order only. If the square of ^ is formed by combining the rows or col- umns according to the rule for multiplication of determinants, the result is a symmetrical determinant. By a slight modifica- tion, as^ so far as I know was first shown by Brioschi, this may be made a skew symmetrical determinant, that is, one in which the constituents of the principal diagonal are all zeros and those symetrically placed with respect to this diagonal, are equal but with opposite signs. This may be shown by taking the 1** row of (31) with the negative of the (n-\- 1)* row, the % nd row with the negative of the (n + 2)* row, and so on. In order to make this plainer, replace the a's in the last n rows by 6's, so that the determinant becomes (44) J = a,, , 2 , . . . Cti, 2n , 2 > C?2 ) 2 The transformed determinant then becomes THEORIES OF PLANETARY MOTIONS. 1 , 1 , - i , 2 , ,, , j , O n , 2 , . . . O n , (45) Further, if the columns are compounded, it becomes 1 , Ci , 2 ) Ci , 2 ? 2 > gn ) 2* where, in general, (47) CA, M = a 1 ,A&liM+ 02A02,/* + . 6j , A Oi , /u, - ^>2 J A ^2 > /u, - - and hence, as should be the case Cx , M + C M , A = . Further, denoting the constituents of the last n columns of (33) by the letters B , it becomes "111? -"-1 J 2> "! -*1 > 2> '* -^1 > (48) Z>= 's "' V- ' V ....... -"-2H > 1 j -"-2w > 2 j * ^-SM j i -^W )! -Ojw > By writing D in the new form ~ -"1 > 1 J -^*l> 2 > "T -"! j 1 ) ~T <*1 , (49) D'= M5 2 ) ~T" -^ 2w j 1 ) l~ " Jd > n and combining its horizontal rows, it becomes 1 , i , GI , 2> VI j 2*1 (50) F = DD' = D*= where > 1 2n > 2 ) POISSON'S AND LAGRANGE'S FORMULAS. 89 (51) Op, \ = A\, j BV. ,!+..+ A\, n B,j, , n J5 A , 1 ^L/n, 1 ... B\, n Ap, n , and also C\ , M + Cp. , A = . It follows at once that But a still closer relation exists between the skew determi- nants E and F. Like (44) and (48), or J and D , they are con- jugate. For, in accordance with the definition of conjugate determinants /KO\ A Z? ^ {04) jf\ )f t.- . p> -OAjM A OL ,,ov _I_ _8-D_ 1 8 \ **" / ax ' ** n /i A ^ n z? In order to derive the corresponding expressions for E and F, we write (47), with the aid of (53), in the form 1 r dP d P dP dD dD dD or, also with regard to (49), d D dD' d D d D' di) dD r -L d A O / T> N I ' ' * I /" Consequently, C A , ^ is equal to -FT times the sum of the prod- ucts of the minors of the p. th row of (48) with the corresponding minors of the /* 7i row of (49). According to an extension of the theorem for the multiplication of determinants, c ^^ = f times the minors of the product of (48) and (49) and therefore of F which contains the term OA, M . Therefore 'iiA\ l QF (54) C ^ = 90 THEORIES OF PLANETARY MOTIONS. and likewise (55) C ^=S^' and consequently E and F are conjugate determinants. Therefore, if a?j , . . . , x, 2n is a system of 2 n variables and if a second system y lt . . . , y 2M is determined by the linear substitu- tions (56) C^A^I + C^A^+.-.+^A^H^A G = l,2, ...,2w), then are the solutions of these equations. Still other relations exist among the four determinants -D, J,. E, F. By allowing the index /;. in (47) to take all values from 1 to 2 n and solving the resulting 2 n equations for a, , A , - , a n9 A, &u A, . . . , &, A, we obtain (58) These equations show that the determinant J' may be ob- tained identically by combining the rows of E with the columns of D. By proceeding in the same manner with (51), we get f -4A,M=Cl,A&, t. -5 A j /u, C'! , A df. These developments of the determinants can be directly ap- plied to Poisson's and to Lagrange's Theorems. The equations (15) are the solutions of (14). If they are substituted in (14) r the equations become identical, and in this sense the differentia- tion of (14), therefore, gives (60) / gg \ ^-A j /a /! , A /A j 1 ~T T~ ^2w } A M , 2 n > f\ OP\ OCli . . Op\ v\A} A _ Qp 1 dq, dp, 8g 2 O O I o , d I or simply, (64) C^,A = [A,^]. Therefore, it appears, that if Poisson's n 2 expressions and Lagrange's n 2 expressions are arranged in determinants, the two determinants are conjugate and in this consists the remark- able relation between the two theorems. 11. DEVELOPMENT OF POISSON'S AND LAGRANGE'S FORMULAS FOR THE ELLIPTIC ELEMENTS OF THE ORBIT OF A PLANET. The formulas of the preceding paragraphs can here be applied at once. If we put 92 THEORIES OF PLANETAEY MOTIONS. dx dy dz df, =W ' ft =V ' M then (2) and the differential equations of the motion become dy dH dz ^ = dt~ du' dt~ dv ' dt ' dw' _. - __ 'dx ' dt ~ ~dy ' dt ~ dz ' In this case, equations (26) and (30) of 10 become /A . , da x da dci^da^ 9a x 8a M (4) (x,a M )=- - + ----- + -- . ~ , r -, _ dx d u . dy dv dz die da\ da/j. da^ da^ da\ 8a M ' The six constants of integration are the elements of the orbit. In order, they are t = a = the mean distance, a 2 e = eccentricity. a 3 = Q = longitude of the ascending node, a 4 = i ' inclination of the plane of the orbit, a 5 = - = longitude of perihelion, o 6 = s = mean longitude of the planet for the time / = 0. In this case it is best first to compute Lagrange's quantites [A, M ]. Then, by substituting for -3" and H their values in terms of /, and remembering that Z is zero, the formulas (26 a) 1 can be at once used. The component velocities may be immediately obtained by writing -^ or 5' instead of S and TV- or //' instead of //, in equations (26 a). POISONS' AND LAGRANGE'S FORMULAS. 93 Therefore, putting (7) * X=Z x +Ur lx in which Z x ,y w , etc., denote the six coefficients of (26 a), we get (8) * tfetfV+^ir- The derivatives with respect to & and i are the easiest to obtain since they alone appear in the C^-'s and ^'s. Hence their determination presents no difficulty. The element a ap- pears twice, explicitly as a factor, and implicitly in the angle Denoting the entire derivatives by enclosing them within parentheses and leaving the derivatives, which arise from the explicit appearance of a, without them, we have d_x _ 3 8xtn_ _x_3iu da~2~&~a ~ a~ ~%~ii' etc. Remembering that, in u -^ ~^jyr A/s ' ^ ie e ^ emen t a a P- pears only in the power , it likewise follows that f dn "\ u 3 / dn u , 3 fx i i . _i_ pfp \daJ 2a 2a(// 2a- Sar* The element appears only in Mnt-\-s -. Hence Qx 1 _ u Qfn~n -x- ^T = 3, etc. os ot n nr The element - appears twice, once in Z and //, and again explicitly in the C^'s and r jx 's. Denoting the entire derivatives with respect to - by enclos- ing them within parentheses and the parts of the same, which are obtained by the differentiation of !\ f and r ljn without them, and remembering that - appears in M only in the relation -, we have 94 THEORIES OF PLANETARY MOTIONS. dx ~\ dx dx du \ du du It is somewhat more difficult to form the derivatives with respect to e. From the equations f c = a(cosE e) ( H) s \ 7> = ttVi & sin.E we obtain, sinE ' = an 1 ecosE 1 ecosE In order to form the derivatives of r, r h |', // with respect to e, it is only necessary to know - . By differentiating we get, W = l ecosE' With these preparations it is not difficult to compute the thirty-six derivatives of the coordinates and the component velocities with respect to the elements. In ordgr to form the expressions [a A ,a M ], the number of which in this case is essentially fifteen, a specinl value of t can be at once used in the thirty-six derivatives. We can, for exam- ple, put M=E = and thereby much reduce the necessary computations. Moreover, five of the combinations can be immediately obtained in another way. . % The equation for the kinetic energy, in this case, is (14) H= ~Wi ) which, by the substitution of the coordinates and velocities in terms of the time and elements, must become identical. In POISSON'S AND LAGRANGE'S FORMULAS. 95 this sense, the differentiation of equation (14) with respect to any element k, (a being excepted), gives (15) Using equations (3), and remembering* that tf appears only in nt-^s, we have dx _ ,dx . dn _ Qu . and it follows immediately from (15), that [e,fc] =0. Hence But if k is the mean distance a, the second member of (15) is not 0, but ~-. 2 , and _ j- -. _ /J- hence (17) The element is thereby fully exhausted. With the help of the above preparations, the ten remaining combinations can be obtained without difficulty. The results are as follows: (18) [,<] = I -(I =r sn The remaining Lagrange's combinations are all = 0. Conse- quently, the determinant F, (50) 10, becomes 96 THEOEIES OF PLANETARY MOTIONS. (19) F = a 0, 0, [<5, *], -[a, a], -[e, fl], 0, -[fi, t ], . 0, 0, 0, -[fi, t], .0, 0, -[,'<]> [>,*], 0, 0, 0, -[a, e ], 0, 0, 0, 0, All the elements of this determinant to the right of the sec- ond diagonal are zeros, consequently it is equal to the negative product of the elements of this diagonal, or (20) F= ([a,*] [fl, i] [e, ^]) 2 = J/, 3 ae 2 sin 2 i. Since so many of the elements are zeros, the calculation of the minor determinants is easy. Dividing them by F gives (21) > sni / ->)- I ' yt / \ . . rt (1 008^ . . , - ^ -- ^- , sin i \f /j.a(I e 2 ) / \ V 1 cos i ' sn All the remaining Poison's combinatioDs are = 0. The deter minal E, (46) 10, accordingly becomes a e & i TT e a o, 0, o, o, o, -(/) e o, o, o, o, -(e,*), (e, e) & (22VE7 o, o, o, (,i) , o, i ( ) - o, o, (0, i}, 0, (?; TT), (t,) - o, (*V*)i o, (i, - ), o, e (a, e) , too, o, -ft) , o, CANONICAL CONSTANTS OF INTEGRATION. 97 These tables, (19) and (22), were given simultaneously by Lagrange and Laplace. They show that nine of the combina- tions [r/ A , M ] and (a*, M ) are equal to zero and that they do not contain the elemeiits e, ~ and &. By a proper selection of elements which are functions of these, it is possible to carry the reductions further. This will not be done, but in the following paragraphs a general theory will be developed which can easily be applied to these special cases. 12. THE CANONICAL SYSTEM OF CONSTANTS OF INTEGRATION. The developments of 10 will now be continued with the use of the same notation. If a and 6 are any two given functions of PI and q it a third function of them is defined by Poisson's expression (a,b). Con- sequently, if a and b are two integrals, which may also contain the time, (a, 6) represents a third integral. Again, if a^c^, . . . a 2n is a complete system of integrals, there are, neglecting the sign, \n(n 1) such combinations, and we shall show that they all take very simple values when a proper selection of the integrals is made. In order to reach the fundamental idea of the investigation, we shall next consider Poisson's expressions (a, b) for themselves, without reference to their use in the theory of the integrals of our differential equations. Let a lt 2 , . . . a m be any number of given functions of p f and q it and, further, let b be a given function of a lt 2 , . . . a m , which does not explicitely contain p t and q t . Then dp, V. da, dq { + da 2 8g, + ' " ) * I \ I ? I O ^v, 1 O O ^. I O ^ O ^. I * * * 98 THEOBIES OF PLANETARY MOTIONS. Consequently, f^h s$i\ (1) (a ) , 9 b) = (a^a l ) / x b = ,2, (*>*) 85. Further, if &i is a function of a, satisfying the same conditions as a, it follows that w (MO= 22 (---)-;. A=l /x=l After these preparations, which show that Poisson's combi- nation of "functions of functions" can be at once traced back to the first functions, we assume that ^ is any given function of Pi and q im Then let another function ft of p t and g, be so de- termined that (3) (,,ft)=l. Then, with respect to ft, this is a linear partial differential equation of the first order, with p t and q as independent varia- bles; and ft may, therefore, be selected with all the arbitrariness allowed by the theory of these equations. Now let ft be deter- mined in any fixed way. The partial differential equation (4) (o l ,6)=0 for a third function 6 has (2w 1) independent integrals and the general integral is an arbitrary function of them. As one of these integrals, ,we may take ^ , since it gives the identity (,, aj)=:0. Let 6 n 6 2 , &2 M -2 be the remaining (2w 2) inde- pendent integrals of (4). If we substitute a lt ft, 6 for/, 9?, 0, in the fundamental relation (20), page 82, [10], we immediately obtain, by reason of (3) and (4), (5) (,,(ft,&))=0, that is, (ft, b) is also an integral of the partial differential equation (4) in 6, and consequently the (2n 2) expressions (u &i)> (ft> &2-2) depend entirely upon a l9 b lt . .. 6 2M _ 2 . Let CANONICAL CONSTANTS OF INTEGRATION. 9 c be any selected function of these 2ra 1 elements. Then, from (1), (6) Since the coefficients are functions of ^ and 6j, . . . b n only, it followsfthat (7) (A,o) = a This is a partial differential equation in c, with a lt 6 lv .. 6 2w _ 2 as independent variables. It has 2?i 2 independent integrals, which may be denoted by c 19 C 2 , . . . c 2w _ 2 . Since they are func- tions of a l9 6j, . . . 6 2w _ 2 , th e y a l so satisfy the equation (8) (,,c)=0. The (2w 2) functions c 1} c 2 ', . . . c 2w _ 2 have now the property that any combination (CA,CV) depends on them alone. They form, according to Lie, a group. This may easily be shown as follows : Since a lt fa, c 1$ . . . c 2ro _ 2 are 2n functions of p d) depend only upon a 2 , d l9 . . . d 2w _ 4 . Letting e denote a function of 2 , d lt . . . c 2w _ 4 , (11) ( / 3 1 ,e)=- C/ is a partial differential equation in e with 2 , d, , . . . d 2n _ i as independent variables, and has (2n 4) independent* integrals 0i, &2W-4- Since 2 , &0i e 2w _ 4 are functions of c l9 ...Cz n _ tt they likewise satisfy the equations (7) and (8), and, conse- quently, te,^) = (A, &) = (i, &) = (2,A) = and The functions e form a group, as did the functions c; there- fore the combination (e\, e^) is a pure function of e. Now, con- sider this as a function of p { and q t , and restore the 2n functions ei,...e 2 n-4. Since it appears that it does not contain ,, 2 , /5 U /5 2 . It is evident that this process which we have followed through two steps, can be continued indefinitely. New partial differen- tial equations are successively obtained containing always a decreasing number of independent variables, and no restriction is made in the selection of the integrals. In the end, the fol- lowing result is reached: If ! be any given function of p t and q t , then, in an indefi- nite number of ways, (2n 1) other functions 2 , ... w , A .../?, may be determined, such that, (*,&)=! It is the fundamental idea of this development, that the equa- tions (12) formn(2tt 1) simultaneous partial differential equa- tions between 2n functions and /?, and the 2w independent variables p t and # f . Now, in a system of more than (2rc 1) CANONICAL CONSTANTS OF INTEGRATION. 101 partial differential equations between 2n functions of 2n vari- ables, certain fixed conditions must be satisfied. These condi- tions are here satisfied and in such a manner that the solution possesses the greatest generality possible for a system of n(2n 1) differential equations. For, we are limited in no way by the integration of the successively appearing partial differential equations, in which the given variables are successively replaced by new ones. This principle, which is also of importance in other branches of mathematics, e. g., in the theory of those dif- ferential equations which are fundamental in the theory of inva- riants,* deserves the closest attention. Moreover, one solution of the equations (12) is at once obtained by putting A = p A , ^ = q\, (*> 1, 2, . . . n). These results can be at once applied to the integrals of the differential equations (9), 10, after introducing a new variable r, making (2w + 2) variables in all, and putting H' = H+r, so that H' contains only p^qi and t. Further, for uniformity, introduce the expressions t = q , t=p , and consider H' as the function of p and q which is taken at will in (3). Then (3) becomes i dq t dq t dp, i = l in which p and q are independent variables and A, a depend- ant variable. It is at once satisfied by A t = q Q , and we shall take this quantity for ft in equation (3). The equation (4), which may be written, has (2n + l) independent solutions. One of them is H' itself See Arnohold, Ueber eine fundamentale Begrundung der Invariantentheorie. Berlin, 1863. 102 THEORIES OF PLANETARY MOTIONS. and the others can be so selected as to be independent of r. For, if we put -^- =0, the differential equation 0= (a, H'} be- comes (13) Q=(a,H)+^. This equation is consistent, since its cofficients do not con- tain r. It is indeed equation (19), 10, the one which defined the 2tt integrals of equations (9). For the given value of A, ( A = t = QO), we also have In order to show clearly the fundamental principle of the method, we will, for a moment, put H' = , A i /? , so that (o #>) = 1, and in general (a , ) = (A, ) = 0. From this follows, finally: If we take a system of 2(w + l) variables p it q { , (i = Q,...n), and determine 0? a 1? . . . MJ /? , /5 lf . . . /? M in such a way that they satisfy the conditions (12); then as a special case, we can put = H' H-\-p , Po = qo t where H is independent of p Q and q Q , but is an arbitrary function of the quantities p and q. The equation ( , /5 ) 1 is then immediately satisfied. The remain- ing functions !,..., &,...&,, can be so selected as to be independent of /S . In the combinations (a x ,/9 A ) of these 2w functions, there will be no derivatives with respect to p Q and go- Finally, if we put p Q = r, q = /, the functions !,...*,&&, satisfy equations (19), 10, for a. They are therefore integrals of (9), 10, which satisfy the conditions (12) of this section. By reason of (54), (55), (63) and (64) of 10, Lagrange's com- binations [x, M The 2n integrals ,, ... ,&, .../?, which satisfy the con- ditions (12) will hereafter be called a system of canonical inte- errals. o CANONICAL CONSTANTS OF INTEGEATION. 103 grals, or also, since they do not change with the time t, a system of canonical constants of integration. Among these there is one of extraordinary simplicity. It is obtained by putting a. p/j I3f = qi' 9 where p,-/ and qs denote the initial values of Pi and q< for a fixed epoch f t. The equations (14), 10, here become 15 f \ and their solutions for p t r and g/, (15), 10 Pi' = P<'(pi - : -Pm qi . . q n , t', t) q>' = q In Poisson's expression (a, 6), we now substitute for a and 6, the two functions which form the second members of (16). Since they are independent of / and since no derivatives with respect to t occur in (a, 6), / may receive any value /' before differentiation. But then the functions _p/,g fri might at once be taken, but instead we shall select the moment of velocity which VV(1 e*)= V/-*( hence (3) 2 VVa(l 6 2 ) 1 By reason of (19), 11, Therefore we might put & = TT, but we shall take (4) A = fl-, .that is, the angle between the ascending node and perihelion. Then we have COS I COS smt V /a(l ej 2 sin* If c is a function of a 2 , i2, i,- 8c cos z 8c (/?2,C)=- The partial differential equation (/5 2 ,c) =0 in c, with 2 >^ 5 * as independent variables, has the two independent integrals, d = , and c 2 = 2 cos i = /s/ /> ^ a ( 1 e 2 ) cos i. Further ( otj Oa, Oa, O a, D = da 8ft 8ft 8ft 8ft 108 THEOKIES OF PLANETAEY MOTIONS. If we consider the (2n) 2 derivatives as independent ele- ments, without reference to their meaning as derivatives, they satisfy the n(%n 1) equations of condition (1), (2), (3), and these equations of condition furnish a series of noteworthy properties of this determinant which possess an unmistakable similarity to the properties of the determinant of an orthogo- nal substitution. The determinant of an orthogonal substitu- tion has the property that the combination of any two rows or columns gives or 1, according as they are different or not, and there is a somewhat similar relation between D and the determinant __ dq n ' dpi"" dp (6) 80, da. _da _da n da^ 8 8^" d~q n ' 8^'" 8 which is derived from D by exchanging the last n rows and columns with the corresponding first rows and columns, and taking the signs as indicated. The combination of the rows of (5) gives, by (2) and (3), 1 when they correspond, and when they do not correspond. Hence and consequently (6) D=l. The analogy may be carried further. If we solve the %n equa- tions \ : / * a o ( ~~ ~\ for the p's and g's, we get PROPERTIES OF THE INVOLUTION SYSTEMS. 109 (8) and, if the functional determinants of (8) take the two forms (9) (10) 8/V"' 8/V'" _i_ Gqi ^ Opi r 8ft '"' 8ft'" then, according to 12, ^J and J' are conjugate determinants with the same symbols as D and D'. It can now be shown that D and J' are identical and that D' and J are identical. That is, that (11) 80 This follows immediately from equations (59), 10, by substi- tuting in them the expressions given by (62), (63) and (64), and at the same time putting a and ft in place of a and 6, and then using the equations (1), (2) and (3). 110 THEORIES OF PLANETARY MOTIONS. Determinants like D, whose elements satisfy the equations (1), (2) and (3), are called canonical determinants.* A conclusion of great importance can be obtained from the equations (11). For, if P l and P 2 represent any two of the p's and #'s, the expression is equal to if Pj and P 2 are different, and to 1 if they are alike. If in this, we substitute for the derivatives of and ft , their values from (11), we obtain at once x,x , , > . (12) (PA, 0,0=0, in which the Poisson's expressions are obtained from the earlier ones by replacing and /? by p and q. Hence, the important result: The solution of an involution system for the primitive vari- ables, gives again an involution system. Finally we will deduce the following proposition: If a tt fa is an involution system of the original variables p i9 q {) and we form a second involution system A i9 B t of ,., /?,, then the latter is an involution system of p t - and q t when the expres- sions , fa are replaced by p i} q { . Denoting Poisson's expressions by (A A , B^ a ^ and (A A , B^) PJ q in which the subscripts a, p, in one case, and p, q in the other indicate the independent variables, we have at once from equa- tion (2), 12, (13) (A^B lt ) a9ft = (A^B lt ) f9qt which proves the proposition. And, further, (13) gives the fol- lowing general result: Poison's expression 2( "Canonical Determinants also appear in other problems. See Clebseh and Gor- dan's Abel'sche Functionen, p. 300. PKOPERTIES OF THE INVOLUTION SYSTEMS. Ill of any two functions a, , a 2 of p { , q { remains unchanged in form, when p^ q { are subjected to a canonical substitution. What precedes is a preparation for our problem, the com- plete integration of the partial differential equations (1), (2) and (3), and consequently, the determination of the most gen- eral form of the canonical system. We shall proceed to this problem, by first considering n func- tions (14) a { = ai ( Pl) p 2 ...p n , q l} g 2 , . . . q n ), (i=l, '2, ... n), and the ^n(n 1) partial differential equations between them (15) (A,M) = (>. We may assume that these equations represent the condition that the product of the two determinants (16) da, obtained by combining the rows, shall be a symmetrical determi nant. If, for brevity, we put the determinants become, PDD - -Pi>i (17) D.,,= / M 'Ki Pnj 1) Pn) and equation (15) becomes (18) fflll, ! [MJ 1> *!> t i We will now form the minors of (17) and put 112 THEORIES OF PLANETARY MOTIONS. D a,* Up M , A and arrange them as determinants. PP 1,1,... -t 1) M (19) *,= P.,,,... P., Multiplying (18) by P r , A , (? P , M , where r and /> are indices selected at will between 1 and n, and letting A and /* take all possible values, and summing the resulting equations, gives , A Each three-fold sum can be combined into a single sum. The first member, being summed for A, gives But S px,*Pr,A is either 1 or 0, according as * is or is not A equal to r. Consequently the first member of the above equa- tion becomes By summing the second member, in the same manner, for ^, we get it A Finally, substituting i for A and P., A for r, and P. for />, and we obtain and, hence the following interesting proposition of determi- nants: If a determinant (#,) has the property, that the product, formed by combining its rows with those of a second, is a sym- metrical determinant, then the product formed by combining its columns with the rows of the conjugate to the second, is also symmetrical determinant. PROPERTIES OF THE INVOLUTION SYSTEMS. 113 This proposition has an important application to our differ- ential equations. If we consider q as only a parameter in (14), then D a)p is a functional determinant. If (14) are now solved for the p's so that they are expressed in terms of and q, we get (21) p< = pi(<*i, 2 , . , gi,g a ,... 3) (*' = 1,2, ...n), then F p is the functional determinant of (21), and The derivatives are enclosed in parentheses to distinguish them from (9) and (10), from which they are evidently entirely different. The substitution of (14) in (21) gives identically Pf=p f . If (21) is differentiated in this sense with reference to 3?, ^/s, ^, C a , Cj8, C, these being so taken that with six other corresponding equations giving ^ and in terms of 2/ and 2?. 118 THEORIES OF PLANETARY MOTIONS. The constants and /? must satisfy the conditions r , 1/5, from which it appears that the expressions (13) i/? 2 ft 2 , 2 / 5 3 /V 3 , 3/ 5 i /Vi have a common value, which we shall take = 1. We shall here denote the P/s corresponding to , >y, C by A, //, The equations (9) then give fuj = V /M/s wM, W 2 = - 2 ^a - & ^ - m^ , w 3 = 3 A a fi a Ap m^ . If we put, (15) M = mj + ?/i 2 + ?n 3 , it follows, by reversing and using (12) and (13), that (16) Mx } = + Ca(/ 5 2 w 3 /5 3 m 2 ) s^(a 2 m 3 3 ra 2 ), with eight other analagous equations. From these {17) 0i #2 ) = ?a/? 3 + ^ s, etc. Also (18) r x 2 2 = (^ x 2 ) 2 + ( yi 2/ 2 ) 2 + (^i * 2 ) 2 In the same way we may form r 2 2 3 and r& and, hence, the six distances are entirely independent, of , ^, C. If the six constants satisfy the new equation (19) A + _A + ^a = m l m 2 m 3 and if, for brevity, we put JL- =a ;^.j i V:_i-5L m a m, m 2 ?7i 3 ' kinetic energy becomes, by (14), _ = , _ , / m^ r^! m 2 ra 3 ' CANONICAL SYSTEM OF DIFFERENTIAL EQUATIONS.' 119 ( 2i) X^ ^ + /i m, ra a V + /V + The function H falls therefore into two parts, rr'- W i w 2 m 2 m 3 m 3 m 1 V-* v a , V -f /^ 2 -f v^ " r5 " " ri " ' r 3 2 2m a 2m"~ ~ Hence, for , 77, C, A, /a, v, we get the system of the sixth order (23) -.jl, |=0, etc., which can at once be integrated. For the twelve other varia- bles, we can limit H to the part H' and obtain a system of the twelfth order, The system (23) can be at once integrated arid gives proposi- tions concerning the center of gravity. The system (24) fur- nishes the relative motions of the points, and is a complete canon- ical system of the twelfth order. It is worthy of notice that there are only four equations between the six constants and /? and that some discretion is also possible in their selection, which under some circumstances proves serviceable. Four integrals of the system (24) are known, those giving the kinetic energy and the propositions concerning areal veloc- ity. By introducing these in (24), it is- possible to reduce these equations to other ones, also having the canonical form, but with only eight variables and t. Finally by eliminating t and aoother variable in this new system, a canonical system of the sixth order can be obtained, but the symmetry disappears. This system has been formed in many ways, but with difficulty in retaining the canonical form, and the function H on which everything depends, losing its original simplicity. This process has not given us any further reduction, for Lagrange had reduced the problem to the same order. And Lie, by the more complete representations of the the theory of groups, has shown that further reduction of the known integrals is impossible. f 120 THEORIES OP PLANETARY MOTIONS. 16. THE PARTIAL DIFFERENTIAL EQUATION OF HAMILTON AND JACOBI. In the integration of the systems d Pi dH dq t dH W d*- = 8* ^t~~ -&*' (*=--!>*. ) 2n constants, a i? a 2> . . . , 19 /3 2 , . . . ft n are introduced. In 13, it was shown that these constants can always ba selected such functions of Pi and q i9 which may also contain t, 9 , as to form an involution system. From 14, it follows that, if a and p be regarded as the independent variables of (2), and if /? and q be expressed in terms of them, the resulting equations take the from where W denotes some function of p, a and t, (4) W=W(p l ,...p n ,a 1 ,...a H ,t). We shall now examine the function W more closely. Th& first system of equations (3) furnish n integrations of the sys- tem (1) with n arbitrary constants . If we substitute the values of q f in (5) H=H( Pl ,...p n ,q,,...q n ,t), it becomes (6) H H( f D,, . . , ) a pi op n If we also substitute the values of q f in the first of the sys- tems (1), we get a system of n differential equations in which Pi and t are the only unknown quantities. If the same substi- tution is made in the second of systems (1), a similar set of- differential equations, between p { and t, is obtained, which is not independent of the first system. Now, the total differen- tiation of (3) with respect to t, by the aid of (4), gives HAMILTON AND JACOBl's EQUATION. 121 dq { _ v rr , dt ~~ dpidt ^dpidp^^tt' or, by (1), dH _ d 2 W t ^ d 2 W dH dpi ~~ dp, dt therefore d 2 W dH dp^t ~^"dpi where, again, the values of q\ are to be substituted in { -^ I . dH In the formation of ^ , the expression (5) is fundamental. But we may differentiate (6) with respect to p it which here appears doubly inasmuch as it is also contained implicitly in W, and if we denote the partial derivatives of (6) with respect to p, by ( ^ J, in order to distinguish them from the preceding, it follows that and by the introduction of the former equation, this takes the simple form where H has the form given by (6). The equation (7) shows that the expression ,8) ' *jf + H,A is entirely indedendent of p { . We can further show that it does not contain t . Differentiating (8) with respect to ,-, we get 8 a, ~ J 8g A 8 , \ S 122 THEORIES OF PLANETARY MOTIONS. But the second member of this is 0, since it is the total deriv- O -nr ative of -^ , that is of a constant with respect to the time. Hence () o= The equatious (7) and (9) show that (8) is a pure function of the time with no arbitrary constants. But, since the derivative of this function f(t) vanishes in the differential equations (1), it may be subtracted from H, and then dW . This equation, in which p and t are the independent variables and W the function to be determined, is the partial differential equation of Hamilton and Jacobi. Jacobi has shown that its complete integral immediately furnishes the integrals of the system (1). This will now be proven. By a complete integral, or a complete solution of (10), is meant, according to Lagrange, a solution which has as many independent arbitrary constants as there are independent vari- ables. The number in this case is w + 1. One of these is a constant belonging to W and this may be neglected. Let the remaining n constants be a , a 2 , . . . , so that W has the form (11) W=W(p 1 ,...p n ,a l ,...a n ,t). Introducing /io\ w *=%? we can at once give the integrals of the system (13) dpi_dH dt ~ dq, > in which the value of q it obtained by differentiating (12), is to be used. (In the second member of (13), p and t represent the variables and the 's the parameters). If (10) is differentiated HAMILTON AND JACOBl'S EQUATION. 123 with respect to a parameter a i} it follows, since it is contained only in W, that 8 W 8 = dt and hence, by (13), = or d 'dt Hence, the expressions, (14) dw "W^l represent the n integrals of (13), and by solution for p { will give these as functions of the time t, the arbitrary parameters , and the constants of integration /?. By the total differentiation of (10) with respect top,-, we get 0= * . r dH or, by (12), (15) dH Equations (13) and (15) reproduce the complete system (1) ; and (12) and (14) are its integrals with the 2n arbitrary constants a t and &. 124 THEORIES OF PLANETARY MOTIONS. The fundamental idea in these investigations consists in the disclosure of the intimate connection between the theory of partial differential equations of the first order, and the theory of simultaneous differential equations. This connection, for linear equations and also for equations containing two inde- pendent variables, has been known since the time of Lagrange. By a slight generalization, it easily appears, that the one problem can be transformed into the other. The reader is referred to Jacobi's masterly investigations. 17. H NOT CONTAINING THE TIME. Up to this time, H has been a function of p t , q { and t. If it does not contain t, the partial differential equation (10) of the preceding paragraph becomes , 8 W 9 W Since (1) does not explicitly contain t, a peculiar transfor- mation may be used, in which the transformation formulas depend upon the yet undetermined W, instead of given funct- ions. Substitute, in (1), a new variable for t, such that <> i?-.-* (3) t = t( Pl ,...p n ,h), and for the function W, a new function V, such that (4) V =W-(t-) where is a constant selected arbitrarily. Kemembering that (5) W=W( Pl ,...p n ,t) t and making the substitution (3) in (4), we have, by differentia- tion 8V_8W8t 0W0t _ ~ 0t dh 0t dh ( > dh H NOT CONTAINING THE TIME. 125 dv _dw dwdt dt _dw d Pi ~d Pi "* dt dp^ d Pi ~Wi* Accordingly, equation (1) becomes In this case, Pi , . . . p n and h are the independent variables. The derivative of F with respect to h does not occur; hence, this equation can be integrated as if h were constant. In this sense, the complete solution of (8) has in its expression (9) V=V( Pl) ... Pn) h,a 1) ...a n _ 1 ), n 1 arbitrary constants a,, . . . a n _ l besides the additive one depending on V. The latter, we shall not consider. With equal propriety a 1} . . .a n _ l may be regarded as constants or as arbi- trary functions of h. We shall consider them as constants. Returning to the primitive function TF, we have If the substitutions (9) and (7) are made in this, W takes the form (11) W= W( Pl , . . .p., a,,... _,, *_.), so that the earlier constants !,..., now have the values (12) i, 2 ,...a w _ 1 ,e, Therefore, the quantities (12) are half of a system of canoni- cal constants of integration. The others are determined by the equations (14), 16, which here, have the form and dw , d w : ** --ST " + ^r : ~ h - Eut the derivatives of V in the form (9) can also be at once sub- stituted in (13). For, according to (10), dh dV _ _ , ~ ' 126 THEORIES OF PLANETARY MOTIONS. Hence the substitution, which changes W into V, need not be taken into consideration, and we get*the following final result: If there is a system of total differential equations ( 16) *&=!* *' = -|* (, = !,. ..n), dt Qq t dt dp^ in which H does not contain the time t } the complete integral V of the differential equation (8) can be formed, in which i, 2 > a n-\ are the arbitrary constants. The solution of (16) will be given by the formulas dV &= g^7> (* = !,. '..n), dv ( . - -, A = 7=r-, (* = 1, ...n 1), and the n pairs of canonical constants of integration are a l,ftl', a 2,fa -!, ftli > - fe - For the special system (8), 10, the partial differential equa- tion (8) is and the equations (13) represent the final equations between the coordinates, while the velocities are at once determined by the formulas dxt dV dy t dV dz t d V (18) m f -rr = 7r , m i T = -^ , ra; -=-^. dt dxS dt dy { dt dz f The partial differential equation (17) was first obtained by Hamilton, but he did not see the deeper conclusions to be drawn from it, and the way in which he proceeded was quite another. He defined V as the "action," and we shall see how easily his definition may be drawn from (17). If V be determined from (17) as a function of the coordinates, and its total derivative with respect to / be taken, we get H NOT CONTAINING THE TIME. 127 t 6 dt 80?, dt dy t dt . ' df and * F = 2 Tdt+C, and, if we suppose, that F vanishes for a definite epoch f , (20) The action F is, accordingly, equal to twice the double inte- gral of the kinetic energy with respect to the time. Moreover, this integral can be at once obtained from (21), 6. In this case V = ^ IL . If we consider only the relative motion of the " center of gravity, C'=h, and therefore, since T= V'+h, * y (21) V = 2(V'+h)dt= -. d * mT *\ -Zh(t-t'). Finally we will give Hamilton's derivation of the partial dif- ferential equation (17), because it is the best known and depends upon a well known principle the so-called Hamiltonian Princi- ple. It has, however, been shown by Jacobi that this derivation is made with an unnecessary limitation which tends to cover the true relations. Hamilton started with the equations /OON d * x i U d 2 y f 6U d 2 z f dU m <=> m *= m >-= (23) If we multiply the equations (22) in order by the virtual vari- ations 8x {J dy it dz tf and, for brevity, put 128 THEORIES OF PLANETARY MOTIONS. we get Now d*x ^ d ( dx ^ \ dx ^dx d ( dx ^ ~\ ^f dx~\ z _ _ o 'y* < I (j / y* I o I o'Y* I -"t o I I df ~dt\.dt J dt dt~dt\~dt J \.dt J ' The preceding equation, therefore, becomes and hence, on integration with respect to t, between the limits t' and t, (25) t t = 8(U+T)dt = d (U+T)dt. This equation, in which dx it ... denote any selected infini- tesimal variations, was Hamilton's starting point. The coordinates and velocities are functions of the time and of 6n constants. For the initial values of these, corresponding to the time f, we shall take a?/, j//, /, u-, v-, w- so that the final equations take the form (26) *, = *,(*/, 2//, zS, . . . <, vj, w/, . . . (tf) ). By the differentation of (26) with respect to t, we at once obtain (27) u f = u^xi, 2/j', zS, . . . w/, <, w^, ...(tt')). H NOT CONTAINING THE TIME. 129 The action t '=2 I = 2 / Tdt t> then takes the form (28) V = V(x l f . . . HI ... (t t r ) ). We will transform (28), by obtaining u', v', w r from (26) and sub- stituting them in it. This gives (29) F = V[x lt y lt z l} ... x,', y/, */, (tf) ]. The equation for the kinetic energy is, in this case, (30) If we substitute for u^v^Wf their values given by (27) and then for u- t v-, w/ their values given above, h becomes a func- tion of the old and new coordinates and of the time (t t'). If from this we find (t t') and substitute it in (29), we get, finally, (31) F = V(x lt y lt z lt ... x,', yi ', z/, . . . h). This is the form of the equation, which Hamilton took as the basis of his investigations. We shall now assume, that, in (25), the variations flee, fly, dz are not entirely arbitrary, but determ- ined in the following manner. Consider the motion of a system from the time t' to the time t. It is first determined by the coordinates and component velocities. If we suppose that these are changed infinitesimally, we have another motion which is similarly determined. From one instant to the next, the con- figurations and velocities will change infinitesimally. The changes or variations will be represented by 8x it dy it dz it $u it tot, 8w it and the initial variations by flo?/, fly/, flz/, du i9 flu/, flw/. Then from (26) and (27), we have fla? v , etc., in the form $xj duj ^ By the aid of (30), the integral in the second member of (25) becomes 130 THEOBIES OF PLANETARY MOTIONS. t t- = (ZTh)dt = 2Tdt h(t t')=V h(t f), *' . *' With the limitations now made, t (5 (U+T)dt=dVd[h(tt')~\ It can now be shown, that, (32) W = <*-*> By the substitution of the expression (33) (t -t') = (t-t') (x 1 ,...<,.../ i ), the element h in (31) is brought back into the form (29) Therefore, 3V _ dV d(t-t') ~dh~d(tt') dh ' where V in the second member is taken from (29). Now t t V=2 Tdt= t' tr consequently, where 0(T+U)dt H NOT CONTAINING THE TIME. 131 Further, it follows from (25), that, if the upper limit t changes, so that the final variations 3x t , dy i} dz { become u { dt, Vjdt, 2T=P, and hence -(t f>\ ' } d(tt r y and, consequently, dV__ dV d(t t' dh ~Q(tt') dh~ The last two terms of the equation preceding (32), therefore and by substitution in (25), we get (34) 2 m, (u< ^ + v< dy t + w { 3 Zi ) Sm, (uj dx? + 1?/ 9y! + w/ 3 Now, the initial variations of the coordinates and component velocities were taken at will, consequently the number of initial and final coordinates may be taken the same. Therefore, for every value of Sx t , . . . 8x1 ., the equation (34) must consist of two parts dv (35) 8V If V, the action, as assumed in (31), is a function of the initial and final coordinates and of h, the second system of (35) gives at once by differentiation, 3n integrals, and the first sys- tem gives the components of the velocities, while (32) deter- mines the time employed. With this limitation the action V, cannot be used for the solution of the problem, formation of 132 THEORIES OF PLANETARY MOTIONS. (31) requires that the problem be already solved. The second system of equations (35) can, of course, furnish only (3n 1) equations between the current coordinates x it y i} z it and the relation between them may be at once obtained, by noticing that the equation (30), which, by means of the first of equa- tions (35), becomes , , must also subsist for the initial values, or Qvv, f 8FY ' (37) h = ** [ i t' *V -^ m i This last equation may be regarded as a second partial differen- tial equation in V. 18. THE PARTIAL DIFFERENTIAL EQUATION OF HAMILTON AND JACOBI FOR THE MOTION OF THE PLANETS AROUND THE SUN. In this case, the function H has the value and the differential equations of the motion are /o\ dx_dH dy_dH dz __ dH dt~du' ~di~lh' di~dw' -du_ dH dv_ dH dw __ dH dt~ ~'fa 9 dt~ ~1^> ~dt~ ~l3z' and Jacobi's partial differential equation is It is now necessary to find a solution of (4), which contains two arbitrary constants besides h aad a constant depending on V. In his Vorlesunger ilber analytische Mechanik, Jacobi has MOTION OF THE PLANETS AROUND THE SUN. 133 shown how this can easily be done by the introduction of elliptic coordinates. We shall, however, here use V in Hamilton's form, that is expressed by h, x, y, z and the initial values of the coordinates, x', y f , z', (5) V=V(x t y,z t x',y',z',h). Now, it is clear, that the action V, depends only on 1. Semi-axis major of the conic section, therefore on h - , 2. The distance of the initial place from the sun, 3. The distance of the final place from the sun, r = V ^ 4. The distance of the final place to the initial place, For, it is easy to show that these four magnitudes are sufficient to determine the form of the conic section described by the planet and also the velocity at each point in the orbit, as well as t t = 2 Tdt= ( u * iv 2 ) dt. We may, therefore, write V in the simple form (6) , V=V(r,r f , f>,h). Consequently, dV_dVdr dVdp _dVx dV(x-x') dx ~ drdx + dp dx ~ dr ~r + $T p ' etc< If we use the equation x(x x') + y(y y') +z(z z') =r 2 ( xx' -j- yy' -j- zz' ) then 134 THEORIES OF PLANETARY MOTIONS. p r Therefore the equation (4) takes the form <') -- The corresponding differential equation for the initial coor dinates, therefore, becomes <> --^ In these differential equations r, r' and p are the independent variables and V the function to be determined. It is convenient first to integrate the equation (7), and then to specialize the solu- tion in such a way that the condition (8) is satisfied. Since derivatives with respect to r and /> only occur in (7), r' may be regarded as a constant parameter and the integration effected by the method given by Lagrange for this case. The solution may also be obtained in other ways, for example, by the intro- duction of new variables. If we put and, i|ierefore, <9a) p=pq, we get dr ~ ~ and, therefore, MOTION OF THE PLANETS ABOUND THE SUN. 135 likewise, If, for brevity, we put the equation (7) becomes, '--"= A special solution can be obtained at once, by putting These two equations can plainly exist together. If we take the first root positive and the second negative, we can put for V r + r'+p P 2 - f I1T~ ~ C la (13) v= V2 / ~/-.4-Acte'= l V-2 / <\ - J \ x J V g r-\-r f p Since this expression for V is symmetrical with respect to T and /, it also satisfies the partial differential equation (8). 136 THEOBIES OF PLANETARY MOTIONS. Since it vanishes for x = x', y = y', z = z', it appears that (13) is- the action of the system from the initial to the final configura- tions, and it is in the desired form. The time required by the planet to describe the arc of the orbit between the two points, is found by (32), 17, r + r' + p 2 1 r 1 dx- 2 - l r ' ^ 2 / /*j_ J ^x + h r+r' p V2 V Vi- r-f r x p , . /M- 8F 1 r__J_ ,,- 1 r c?x ~~~Wh~ */% nr A/ai / /i r 2a' This is Lambert's Theorem. It was obtained in an entirely different manner on page 23. From it, it follows, that the time required by a planet to pass from one place in its orbit to another, depends only on 1. The semi-axis major, a, 2. The sum of the distances from the sun, r -f- r' t 3. The chord connecting the two places, p. Moreover this time is the same as that required by a planet in falling in a straight line towards the sun, from the distance -J(r -f / + />) to the distance %(r-\-r r p). The expression for the time is especially simple when h = 0,. that is when the orbit is a parabola. Equation (5), 3, then gives In this limited case, the proposition was known to Euler,, (Miscell. Ber< A. T., S. 20); in it's most general form, it was dis- covered by Lambert. For a long time, the proposition was regarded as a curiosity. Its true source was shown by the inves- tigations of Hamilton and Jacobi. The investigations also give noteworthy equations for the orbit and for the component veloc- ities. If for brevity we put MOTION OF THE PLANETS ABOUND THE SUN. the equations (35), 17, give 137 dv dx ~ / /* v- , o I + 1 dx r-\-r' p 8x and, therefore, ox dx (17) and likewise (18) = A ( y \,r A C z = A( - V. r ^C z z z'~\ ^1 --- -I, Vr t> J If the initial coordinates and component velocities a?', y', z r , u f , v f , w', and, consequently, h are given, the equations (18) give three final equations between x, y, z of. a very noteworthy form. They are equivalent to two independent equations, since the equation must be identically satisfied if we substitute for u', v' y w', their values from (18). Further if we eliminate A and B from (18), we obtain (19) = x(y' w'z'v') + y(z' u' x' w') + z(x' v' y' u'\ *The coordinate x must, in this case, not be confounded with the x under the sign of integration. 9 138 THEORIES OF PLANETARY MOTIONS. which is the equation of a plane passing through the sun. It is not so easy to see from (18), that the orbit is a conic section. It can be shown in the following manner: If we find A and B from the first two of equations (18), and substitute the values in the expression A 2 B 2 , we get W-x) -u'(y'-y)-] (-yf x' + u' y') '~ and in the same manner two other analogous expressions for A z J5 2 can be formed. If, for brevity, the successive numera- tors be represented by ^ , A 2 > ^3 > and their sum by A , it appears that The denominator can be easily transformed into (a 8 + y 2 + * 2 ) O*' 2 + y" + *' 2 ) (xx 1 + yy' + We also have, from (16), ,2 jf _ - Substituting this value in the above equation, and multiplying by "~ jQ or Now, finally and since A is linear with reference to x, y, z, we obtain the final equation in the form r-ax-\- by -\-cz-\-d which is nothing else than the equation (25), 1. It is also not difficult to verify the other equations of the same section, although no one would have succeeded a priori in MOTION OF THE PLANETS ABOUND THE SUN. 139 getting the notable equations (17) and (18) from those 1. This example shows how a general theory may throw a new and sur- prising light on an old and special problem. Jacobi has shown that the partial differential equations (4), can be integrated in other ways, thus giving the solution of the problem a great many forms, which could hardly have been dis^_ covered in any other way. We will finally consider a simple geometrical meaning of the action in the case of an elliptic orbit, as it was given in a note in the Quarterly Journal of Mathematics, 1866. (Note on the Action in an Elliptic Orbit). If we denote the velocity of the planet P, by V, (see figure on p. 13), then dV = v 2 dt = ^ds. dt The increment, dS, of the area described by the radius vector to the sun, is e a ), or, otherwise, 2dS =fds, where / denotes the perpendicular from the focus F upon the tangent, and therefore, dt ~V / If /' is the perpendicular from the other focus upon the tan- gent, by a known property of the ellipse, Hence dV=- ds= =if' ds J VI y, ZH, win denote the coordinates and masses of n bodies forming a system. Let , ^, C denote the coordinates of the center of gravity of the system. Besides the n bodies forming the system, let there be an exterior body whose coordinates and mass are X, Y, Z, M, and whose distance L from the center of gravity of the system is much greater than the greatest dimen- sion Z of the system. And, further, let the coordinates, when referred to the center of gravity of the system, be denoted by primed letters. Then The equations of motion of the exterior body are 144 THEOBIES OF PLANETARY MOTIONS. The quantity under the radical sign is equal to I?2(X'x' + Y'y' + ZV) + (^ 2 + 2/' 2 + A in which the orders of the second and third terms with refer l C I V ence to the first term are and I I . Remembering that Sma/ = Sra?/' = Sraz' = 0, the use of the binomial theorem gives d*X (l~\* J) Now or and the above equation becomes C l\ 2 By neglecting terms of the I I th and higher orders with respect to the leading term, the following result is obtained: The motion of the center of gravity of a system of n bodies, due to the attraction of a very distant exterior body, is the same as if the masses of the n bodies were united at their center of grav- ity. We shall next investigate the relative motions of the n bodies with respect to their center of gravity. We have C . Mmd + THE SOLAR SYSTEM AS A. SYSTEM OF n BODIES. 145 In the first term V again denotes the potential of the system of n bodies. The order of the second term with the reference to the first is M 2m (1)'. the third term is of a still higher order. Hence, if M is not" very great compared with 2m, we have, very approximately, A very distant body has no influence upon the relative motions of a system of n bodies about their center of gravity. Instead of a single exterior body, consider a system of bodies or many systems, whose dimensions are very small compared with their mutual distances. From what precedes, it is easy to see that any such system will attract as if the bodies composing it were united at their center of gravity, and that the relative motions of the bodies composing it will not be disturbed by the other systems. The solar system, including the sun, the planets, and their satellites, the comets and the smaller bodies revolving around the sun, is such a system. Some of the comets move away from the sun to indefinite distances, but aside from this, the greatest dimension of the solar system is probably less than one four- thousandth of the distance' to the nearest fixed star. At such a distance, the influence of a star on the relative motions of the bodies forming the solar system is of the order ' m 1 2m* 64,000,000,000' which is probably not appreciable, It is possible, and indeed probable, that in the coarse of centuries, the fixed stars may -change the direction and velocity of the otherwise uniform motion of the center of gravity of the solar system. This motion is now directed towards the constellation Hercules. While we may safely neglect the fixed stars as perturbers of the relative motions in the solar system, we can not neglect the fact that the system itself is composed of bodies instead of points. The dimensions of these bodies are, however, very small compared with their mutual distances, and excepting the motions 146 THEOKIES OF PLANETARY MOTIONS. about their own centers of gravity, they may be regarded as material points. Moreover, the form of the heavenly bodies enhance the approximation. They are nearly spherical or are composed of nearly homogeneous spherical shells. Now, it is rigorously true that a homogeneous sphere or a body composed of homogeneous spherical shells, attracts an exterior material point as if its mass were concentrated at its center of gravity.. The departure of the planets from the spherical form is so slight that it sensibly influences the motions of the satellites only. It appears, therefore, that with close approximation, the sun, the planets and their satellites may be treated as material points which have no further influence upon each other than their mutual al traction. At the same time, there are in the system certain secondary systems, formed of the satellites revolving around a planet as a primary. In these cases it will be neces- sary to test the degree of approximation. We will take at first the sun, earth and moon. The earth is about four hundred times as far from the sun as from the moon. The influence of the sun, on the relative motions of the earth and moon, is, therefore, m UOO ' in which M represents the mass of the sun, and m that of the earth and moon .together. This is approximately 1,000,000 J_ 3. 64,000,000" 192' so that the sun's disturbing effect is small, although its mass is more than 300,000 times the combined masses of the earth and moon. Yet it is entirely too large to be neglected. Indeed, the numerous inequalities in the moon's motions are nearly all due to the sun. The influence of the sun on the other secondary systems is much less, partly because their dimensions are relatively less,, and partly because the primaries have greater masses and are at greater distances from the sun. The moving force of the sun on the common center of grav- ity of the earth and moon is THE SOLAR SYSTTM AS A SYSTEM OF H BODIES. 147 Mm -w +d > R being the distance of the center of gravity from the sun.. The order of 5 with reference to the first term is (400) 2 " 160,000 ' In this case the order is really less than this because the mass of the earth is more than eighty times that of the moon. In order to determine the resulting decrease of the error <5, we will again assume n bodies, with the same notation as before. Then in which d' can be shown to have the value +"2 m -~ -^JT- ~) + ' in which contains the terms of order higher than those devel- oped. Now, let one of the bodies, for example the one whose mass and coordinates are m^ x ly y^ z lt have a mass far exceed- ing the sum of the masses 2m of the others. Then the part of o f which depends upon m l will be very small compared with ^V^ the others. For its coordinates will be of the order - , and -^i "S, 2 their squares and products of the order ( J . If terms of this order be neglected, d r will be of the order M2m 1} In comparison with the principal term, viz., - ^ -, this- term is of the order m l - or of the order 2m 148 THEORIES OF PLANETARY MOTIONS. Since the mass of the earth is somewhat more than eighty times that of the moon, the approximation on neglecting d rises C 1 Y from the order (^TQA J t tne order f Y -- 1 U-OO/ '80 "12,800,000" In what follows, by the position of a planet, will be under- stood the position of the center of gravity of the secondary sys- tem of which the planet is the principal member, and by its mass will be meant the combined masses of the bodies forming the system. The masses of the asteriods and comets, as well as the other bodies circling about the sun, are so small compared with the masses of the sun and major planets, that they may be neglec- ted. These smaller bodies do not sensibly influence the motions of the larger ones, while the latter may, and in many cases do, influence the motions of the former very much. From the entire solar system, we will select nine bodies, the Sun, Mercury, Yenus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune, and as in the problem of nine bodies, treat them as mathematical points. 21. THE ORBITS OF THE PLANETS. THEORY OF ABSOLUTE PERTURBATIONS. The general integrals and principles of the preceding sec- tions are, notwithstanding their great significance, not sufficient to solve the problem of n bodies in its most general form. On the contrary, on the supposition that the coordinates and veloci- ties are given for a certain instant, the developments are made in ascending powers of t, as proposed by Euler. Let x be any coordinate and x its value for t = t , then the development takes the form . x = * + V- The coefficients are found by differentiating the differential equations, substituting lower derivatives for higher, and elimi- nating the latter down to the first order. THEORY OF ABSOLUTE PERTURBATIONS. 149 Series (1) converges so long as (t 1 ) does not exceed a cer- tain limiting value. This follows from Cauchy's investigation on the convergence of those series which represent functions defined by differential equations. The theory is beautifully given in Briot and Bouquet's Thgorie des fonctions doublement p&riodiques. The series represented by (1) can be used as long as conver- gent, and a new series must be begun before it ceases to be con- vergent. A long interval of time must be divided into suitable parts and then, advancing step by step, the coordinates compu- ted for each epoch. It is evident that this method suffers from many imperfec- tions. The development of the higher derivatives is very labo- rious; Gasparis has carried them out to n = 4. Besides it is not certain but that the errors made in every computation of the series may finally, in the aggregate, amount to such a quan- tity as to make the results, after long intervals of time, entirely illusory. Finally this progressive development would mask any simpler laws which might exist in the nature of the motion. Hence, this method is to be resorted to only when other meth- ods are not available. Fortunately, in the case of the solar system, two other methods are known, developed by Euler and Clairaut and so perfected by Lagrange and Laplace as to leave nothing for later mathematicians to do so far as the ground-work of the theory is concerned. The unexpectedly rich results afforded by the use of these methods are due to certain impoitant circumstan- ces which gave rise to their development. The first of these is found in the controlling position of the sun. It is due to its predominating mass, which is upwards of seven hundred times as great as that of all the planets. Astronomers, therefore, except Lagrange in part, have referred the relative motions not to the center of gravity of the system, but to that of the sun. If we follow thenij we must employ equations (28), 6. The differential equations to be used here, are of the form * __A, *~ 150 THEOEIES OF PLANETARY MOTIONS. in which (3) &= ^ V K *) 2 + (2/A 2/a) 2 + (Z^ -2 a =2i in which may have any value, not A , from 1 to n. In the differential equations (2), the first terms in the second member contains the sun's mass, and is very much larger than the second terms containing R. The first approximation may, therefore, be obtained by neglecting JB n JB 2 , . . . R M . The equa- tions (2) then become the same as those treated in 1, etc., and as the quantities /* differ from each other, and in consequence Kepler's three laws are very approximately true. In fact these laws so perfectly represent the motions of the planets, that Kepler was able to deduce them from observations. Yet the influence of the quantities B 19 R 2 , . . . R n is appreciable in the -course of time, and for this reason Kepler's laws are not rigor- ously true. These quantities are called the perturbing func- tions and the following investigations are devoted to them. If in R, we write m 2 , m 2 , . . . am n for m 19 m 2 , . . . m n respect- ively where is any selected number (Cauchy's E^gulateur), then It will be multiplied by and equations (2) become /K . d 2 x } u l x l . dR l ' - + a ' etc - If in these equations a be taken as an analytical magnitude the coordinates which satisfy these differential equations, will contain, besides , 6n constants of integration. If 0, these equations reduce to the forms given in 1, etc. The essential principle upon which the first method that of absolute perturbations developed by mathematicians depends, -consists in the assumption that the coordinates in (5), which are THEOEY OF ABSOLUTE PERTURBATIONS. 151 functions of a and of the Qn constants of integration, can be devel- oped in ascending powers of a. Any coordinate x may, therefore, be represented in the form 2 (6) X = X + adx + ^, d*x+... A\ If a = Q, this gives, naturally, the coordinates of Kepler's ellipse. It remains, therefore, to fix the remaining terms 3x Q , 8*x, ... of the series (6). If, finally, we put =1, we obtain the actual coordinates in terms of the time in the form ... The determination of &#, d*x, . . . , which are called the abso- lute perturbations of the first, second, etc., degrees, can be found by a process of induction, for the perterbations of the first degree are deduced from the original expressions X Q , etc. ; then those of the second degree from the first and the original expressions; and, in general, those of any degree from all the lower degrees with the original expressions. If we substitute the coordinates given by (6) in (5), the latter become completely identical. Therefore, if we put (8) x, = a for = 0, we again obtain, as must be the case, ( 9 > * ^ = -wf x i9 y\> z \ are determined as in 1. If now, we take so small that the developments of both members of (5) in ascending powers of converge, the coeffi- cients of the equal higher powers in the two members, must be equal. In the first of equations (5), the coefficient of ^ in the first member (10) df The term ^~ = - ^_. X3 gives as the coffi. * l +Vi--r&) 152 THEORIES OF PLANETARY MOTIONS. a t> cient of j an expression whose form is somewhat complicated, but whose formation is easy by the help of the polynomial the- orem. But it can be seen at once that that part of the coeffi- cient which depends on the perturbations of the p ih order, is at once given by ( 11 ) - ft + (*,***, + yfS'y, + zfS'z). VI ) Vi ) As to the term ^- J -, we can, because of the factor a reach th a the coefficient of by taking the series (8) up to the (p l) degree. If we denote the sum of these coefficients and that part of the preceding which depends only upon the perturba- tions to the (p l) th degree, by Xf, Yf, Zf, and if we write simply x lt y lt z l for xf, yf, z,\ we get rl^ftP'Y (12) * ? = - If the original coordinates and the perturbations are known up to and including the (p l) th degree, then Xf t Yf, Zf are- known functions of the time and it is a question of the integra- tion of the differential equations (12) in which 8 p x^ o^y^ d p z l are the unknown quantities. It is to be noted that the successive differential equations to be integrated have always the same general form, and when the general form of the solution has been obtained on the single supposition that Xf t Yf, Zf are given functions of the time, we get, by the uniform repetition of the same process all the per- turbations and also the coordinates as functions of the time. Of course the resulting series must always be converging. Since appeared in the equations (5) as a factor of the masses m u w 2 , . . . , the quantities 8 p x, . . . become homogene- ous functions of the p th degree, (at least, we can so represent them), and the degree of the perturbations is that in which the disturbing masses enter their expressions. SOLUTION FOE THE ABSOLUTE PERTURBATIONS. 153 22. SOLUTION OF THE DIFFERENTIAL EQUATIONS FOR THE ABSOLUTE PERTURBATIONS. The question now relates to the integration of Jinear differ- ential equations of the general form (1) * = - in which a?, y, z are such given functions of t as satisfy the dif- ferential equations * . =i*r-ra + d*-ar + --- and by means of them, equations (4), by differentiation, take the form 'dx dx . as if A, B, . . . were constant. The differentiation of (6) gives dx dx dA Jfa.dB ~de ' ~ ~ dt and by substituting (7) and (4) and using (3), the equations (1) take the forms dA du dB du /o\ _dA dv dBdv z __ dA dw dB dw dt da dt de in which u, v, w are, as before, the component velocities. Equa- tions (5) and (8) are the six equations which determine the vari- ables A, J3, . . . . If they are solved for , , , they Ct> v CLT> immediately give by the use of (60), 10, developments of the form dA ^da , ^da , n da dt (9) SOLUTION FOR THE ABSOLUTE PERTURBATIONS. 155 These equations can be at once integrated and give A, B, . . . , as functions of t, which, when used in (4) furnish the perturba- tions dx, dy, dz. This completes the solution so far as the integration of the equations (1) is concerned. But, on account of the develop- ments which follow, equations (9) will be transformed by the employment of the principles developed in 10. In this case, the equations (58) by the use of (62) and (63) become da du da dx Qx By the substitution of (10) and the corresponding equations in (9), and for brevity putting, 0R dx 0a Qx dy Qa dy Qa^ da' 8z we obtain the system dA (13) dB and, since Poisson's expressions are independent of the time, it at once follows from (13), that (14) 156 THEORIES OF PLANETARY MOTIONS. The perturbations dx, <%, 3z receive, by another use of (10), the forms (15) da fdR,. . de fdR _- I K- , y = r sin v cos

, we get a? = r [cos ( + /) cos 2 J& + cos ( )sin 2 ^i], y = r\_ sin (fl -\- at) cos 2 -J- * -f- sin (fl ai) sin 2 1 i ] , and therefore, -f- cos 2 ij ij sin 2 ^ ^ cos (fij & 2 -f- o^ -|- ^ + sin 2 J*! cos 2 Ji 2 cos (fl 2 ^i .+ w i + W + sin 2 J ^ sin 2 ^ i 2 cos (fl, i2 2 a* 1 -f- ^ 2 -f | sin I, sin * 2 cos (>! cu 2 ) ^ sin ii sin ^ cos (at l -f- o> 2 ) J , or, + 2/i2/ 2 + i 2 ) = n r 2 [ m cos (^ o> 2 ) + n sin ( Wj cw 2 ) in which m = (1 + cos ^ cos i 2 ) cos (^ : i2 2 ) -f sin ^ sin 4 , n = (cos ij -f- cos z' 2 ) sin (^ ^ 2 ), o = (1 cos ^ cos ia ) cos (fij fi 2 ) sin % sin i 2 , p = (cos e\ cos i 2 ) sin (Q 1 i2 2 ). The four quantities m, n, o, p depend upon the three angle i lt ij, &! 2 . A relation must, therefore, exist between them. If we introduce the angle between the planes of the orbits J, we have (5) cos J = cos ij cos i 2 -\- sin ^ sin i 2 cos (& l ^ 2 ), DEVELOPMENT OF THE PERTURBING FUNCTION. 169 and an easy reduction gives m 2 -}- n 2 = (1 -[- cos J") 2 = 4 cos 4 |-J, consequently, Let //! and /7 2 be the angles between the nodes of the planes of the orbits on the xy plane and the intersection of the planes of the orbits, and we have the equations, _-- cos -i co / IT n \ _ n _ ( cos *i ~l~ cos *a) g i n (^i ^2) x "" 2) " ~A/m 2 4-n 2 ~ 1 + cosJ ~' _ (1 cos % cos ia) cos (fi t i2 2 ) sin ^ sin z' 2 1 cos J - P - (cos ir- cos i 2 ) sin (Q l Q - sin ( i ~p Jv 2 > -- 1 COSJ and finally, (7) x 1 x 2 + y 1 y z -\-z 1 z 2 = r 1 r z lGOB(, + 2r 1 r 2 (cosF cosW)z']~^ = p~~ ^ r l r 2 ( cos V cos W ) z p~ % + I [_ r i r 2 (cos V cos W) z Jp~ * I [TI r 2 (cos V cos W) z Jp~ % On account of the smallness of z 9 only a few terms of this development are needed. To complete the form (12), the pow- ers of (cos V cos W) may be arranged in terms of the cosines* of the multiple angles. This gives DEVELOPMENT OF THE PERTURBING FUNCTION. 171 2 (cos V cos wy 2-f cos2 V + cos2 W 2cos( V W) (16) < 4 (cos V cos W) s = 9 cos V 9 cos W+ cos 3 F cos 3 W -3cos(2F+TF)-f 3cos(y-f 2TF) -3cos(2 V W} 4- 3cos(F 2 W). Further, let A= + oo (17) /o-r = (r 1 a + r a a 2nr a cos V) ~f = J ^ ( n *) cos ( A F )' A= oo This formula, in which it is assumed that (18) K) = (n-), is fundamental in the analytical development of the perturbing function, and on this account we shall treat it in detail. If

and from this we get a four-fold infinite number of terms of the form (34) Kz a . l?i . IJ* . cos (h, I, + h 2 1 2 + d l M l + 3 2 M 2 ) , where , ^h* remain unchanged as in (28), ^ and # 2 take all possible positive and negative integral values, r\ and ^ 2 are pos- itive numbers satisfying the conditions that r\ M and 7-2 M are even, positive, or zero. In this way the perturbing function r# l is to be developed in ascending powers of the quantities, z = sin 2 J, e lt e 2 , and in such a way that the angles increase proportionally to the time. By the degree of (34) is meant the number (35) = 2 Therefore, by (29) and (34) (36) g[ must be even, positive or zero. If we undertake to form systematically the term (33), a definite limit for the degree must be decided upon, beyond which the terms may be neglected. The perturbing function (37) rr 2 l = SX^ej* e^ cos (h, 1,+h, 1 2 + d,M, + <5 2 M 2 ) may be expanded into a finite number of simple infinite series. For if n is the limit set for the degree g, then it follows from (35), that a, YD Yi can contain only a finite number of systems 176 THEORIES OF PLANETARY MOTIONS. of values, and the same is true from (29) and (34) for h } -\-h 2) . *i, V If we now put 24, 7i 2 = 2J h l9 and write simply h for h 2 , then for a proper system of six num- bers - (38) a, ft, ft, <*i, &s, ^ the expression 7i = + oo (39) 2^%^*-^- cos[^ (ft 2 JH + M-fi + ^L ; = oo where /i passes through all integers from oo to + oo while the system (38) remains unchanged gives such a simple infinite system as has been referred to; and, since, as we have seen, there are only a limited number of systems (38) for which the degree g^n, we have, if (39) is a term, broken the perturb- ing function into a limited number of terms. At the same time it must be noticed that the expression for k still contains the whole number h left as yet undetermined. Moreover, it is not difficult to determine the number of terms in (39) whose degree is g. It is only necessary to determine the number Sg of the system (38) of the six whole numbers, where (40) <7 = and where (41) 2 [2J], n-M, r 2 - must be even, positive or 0, while J, d l9 Tit 7'2> ha fyj> ^i, where /7 X and // 2 are to be determined from (6). If the incli- nations, longitudes of perihelia, longitudes of the nodes, and the mean longitudes are to be expressed explicitly in the per- turbing function, each term of the perturbing function (42) K&e or (43) where for brevity, (44) must be expanded in a three-fold infinite series. The three angles J, .'A , /7 2 are to be eliminated by the sub- stitution of their values from (5) and (6) and the result reduced to a suitable form, which can be done in the following way. From (6) it follows that (45) 2 , in which _ cos J it cos-J% ~"~ sn cos * 2 __ ' in which i= \f^I and e = 2 .7182818284, and, therefore, DEVELOPMENT OF THE PERTURBING FUNCTION. 181 where A and /* run through all whole numbers which have the same sign as p (0 included), for which Likewise e * _ - A]] & [/AI] e - (A ' i where ^ and /->-! run through all whole numbers which have the same sign as q (0 included), for which From this it follows, finally, that ( 46 ) Kz a e l y^ e 2 ^ cos (A -f px -f g?/) ,. FA] 1>|>] /v [A If we now put a, = fe,H-*i, bj = a lf 1 ^2 = - ^2, so that (47) a. + the angle in the second member of (46) becomes (48) a 1 C 1 +a 2 C 2 + b 1 r 1 + b 2 7T 2 + c 1 i2 1 + c 2 i2 2 = By the help of equation (45) the product becomes (- 1 ) M1 (1 _ ; , )M (cosjyw [MI (cos^g W * [^ x (sin J^M + M( sin Jta)^] + [MI], The product of the last four factors can be developed in ascend- Z - LJ ing powers of ^ and i 2 . The first factor r= - r^i" caD > since [#] is not negative, be developed in positive and ascend- ing powers of z. But z itself and its powers can be at once arranged in cosines of multiples of Q l 2 and the coeffi- cients become functions of ^ and i 2 and can be easily arranged 182 THEORIES OF PLANETAEY MOTIONS.' in ascending powers of these quantities. If these are all intro- duced into the above product and this again in the term (46), the development of the perturbing function finally receives the following form: (49) r^ l S K eji e 2 vi ifi if* cos L. L has here the the form (48) in which the whole numbers <*!,... satisfy (47) but otherwise can take all positive and nega- tive values, except that c x + C 2 must always be even. The expo- nents ft, ft, i, 2 are all positive and satisfy the conditions that ri [&iL ra = [& 2 ], i [cj, 2 [c 2 ] . must all be even, positive or zero. The coefficients K are homogenous functions of the order 1 of c^ and a 2 , in whose general expressions also enter the above whole numbers. The degree g of a term is here 9 = n + 72 + i + 2 i|>i + o 2 ]. The second part (2) of the perturbing function gives terms of the same form as (49), but in this case [cjH-[c 2 ] = or 2. The computation of this part is relatively simple. If we put Q! + d 2 = 4 , there is only a finite number of the systems of numbers ft, r*> a i> "2, &i> ^2, c 1} c 2 , ^, for which g does not surpass a certain limit. If, with restriction to such a limit, the perturbing function is resolved into simple series, of which each consists of an infinite number of terms agreeing with the above system of numbers, the number of series is limited. Their number s ff for a given degree can be easily obtained by the formulas 4.3.2 . 5.4.3 1.2.3 4.3.2 , 5.4.3 3) 1.2.3 ~ DEVELOPMENT OF THE PERTURBING FUNCTION. 183 where, in general, 3 except for k = 0, for which .= ! From these we find If Leverrier had selected the complete expeditious form (49), he would have had no less than 1659 terms to compute. Many mathematicians have occupied themselves with the development of the perturbing function. Cauchy introduced the eccentric anomalies JE lt E 2 and developed in terms of the cosines of their multiples. Bessel's functions then enabled him easily to substitute the mean for the eccentric anomalies. His numerous investigations on this subject appeared in the Comptes rendus and are collected in his Oeuvres completes. While the methods hitherto used by astronomers for the development of " are direct, it does not follow that this development may not be reached indirectly. For brevity, put F = r^ 1 - An indirect method of development would be that of forming certain differ- ential equations which are satisfied by F as a function of the twelve elements of the two planets and which serve to determ- ine F. If we consider F as a function of the coordinates x l , x 2 , . . . and of the component velocities u^ , u 2 , . . . , then F will satisfy the six partial differential equations (50) 0f - ^- dw ~ ^ BF @F which express the fact that F is independent of the component 184 THEORIES OF PLANETARY MOTIONS. velocities. We have also the following six partial differential equations (51) \ 8*3' dydy, ' dF dF dF dF dF dF of which the first three show that F depends on x l x 2 , y l y 2 , z l z 2 alone, and the last three that F is a function of r 12 . The six equations (51) are not independent, an identical relation exists between them. Finally, as a homogenous function of the coordinates of the order 1, F satisfies the partial differen- tial equation (52) dF . d dF dF dF - = Equations (50), (51), (52) determine F with the exception of a constant factor c, and from them it follows that If, now, the elements are substituted for the component veloc- ities, the partial differential equations pass into others having the element a t , . . . C 1? a 2 , . . . C 2 as the unknown quantities. For example, the Aquation dF takes the form 8^801 , 8^8e_i , dFdi, da,, du v 8 Ci du. 8 d in which the coefl&cients 7 , x-^ > - r ^ to be considered f unc- 8t*j tftt, tions of -Oj , 6 t , . . . . If for J' 7 we now take the form (49), the thus modified partial differential equations will determine K up to a single constant factor which remains over but which can then be easily determined. DEVELOPMENT OF THE PERTURBING FUNCTION. 185 Two of the partial differential equations can be at once writ- ten out. One is (52) which now takes the form dF . dF and from which it follows that every coefficient K satisfies this differential equation. The other is a combination of the two last of equations (51) and takes the form dF dF dF dF dF _ d^d^-rQ^e^d^d^- from which the equation of condition (47) follows for each term of (48). The remaining ten equations (50) and (51) are some- what more complicated for we know that the K homogeneous functions of the degree 1 in a^ and 2 serve to determine these functions and to form an infinite number of relations between them and their partial derivatives with respect to a, and a 2 . The very expeditions form (49) is often unnecessary. When it is only necessary to represent the perturbing function as an analytical function of t which is contained in Ci and C 2 only, we can collect all the terms of (49) for which c^ and a 2 have the same values, and the form becomes (53) r^ 1 =: in which a x and a 2 denote all positive and negative numbers and it is assumed that -O-_ QI ,_ a 2 ^^ -A OJ j a 2 j -& % > a 2 -~ & i > ag * The coefficients A and B can be at once expressed in double defi- nite integrals, if we employ the process used by Fourier for the trigonometrical functions of an angle. We have . 27T 2ff zn UTT B i > * = 21? / / T2 ' sin (a, f , + a 2 C a ) dC, dC, . 186 THEORIES OF PLANETARY MOTIONS. The numerical computation of these coefficients can be per- formed in two ways. On the one hand, after Liouville, (Sur le calcul des ingalits priodiques, Jour, de Math. J, 1836), we may expand the double integrals into a very rapidly converging series of simple integrals which we may determine by mechan- ical quadrature. Or, we may, after Leverrier (Recherches astronomique) employ a process of interpolation. Finally, in the articles already referred to in Comptes rendus, for the case in which [QI + d 2 ] is a large number, Cauchy has given a special process which he has applied practically to a definite term in the theory of Pallas and Jupiter. Finally we will also notice that Hansen has given an entirely novel manner of developing the perturbing function. It con- sists in using the mean anomaly of one body and the true anom- aly of the other when the latter, as is the case with the comets and some of the planets, etc., has an eccentricity so large that the series is slowly convergent or entirely fails. This method of development has its chief value in relation to the moon, the asteriods and comets. It will not be given here, since we are considering only the sun and major planets. 25. THE DEVELOPMENT OF (a? 2a 1 a 2 cos , we obtain, by differentiating (1) with respect to d, or, if both members are multiplied by 2/> + 00 +GD S ( a* -f- a 2 2 ) ^y t( s* ) si n ^ 2aj a 2 cos 5 ^> i( s*' ) si n - 00 - GO But and if in the second member the summation index i be ex- changed with (i 1) and correspondingly with (i-f-1) it follows that In the same manner DEVELOPMENT IN A TRIGONOMETEIC SEEIES. 189 the summations always extending Trom GO to + oo. There- fore saa . s 1 - s* sn id The coefficient of sin id in the first member is, according to (2) r changed by replacing i by i and the same is true in the sec- ond member. The coefficients must therefore be equal. Hence (8) i sa ,a 2 [( S '-')-(s'' + 1 )] = (a," + a/)i (a') - a, a, [(i - 1) (s- ') + (i + 1) (+')]. If we put (9) it follows that _ By means of this formula any (s*) may be obtained from the two immediately preceding. For example, from (s) and (s 1 ) we have 4)] (s 1 ) The equation (10) affords the relations between the coeffi- cients of a single series (1). There are also such relations between the coefficients of two such series for which the s's- differ by an even number. These are sufficient, for in astron- omy, the only cases which occur are those in which s is odd. Multiplying (1) by />, we get -l a > = s 2)0 cos id = ( ttl 2 + a 2 2 ) |2 (') cos id 190 THEORIES OF PLANETARY MOTIONS. Since, by (2), the coefficient in braces remains unchanged when i is exchanged with i, we mast have (12) ((-2)') or, by (10), (13) ((g _ 2 )') In order, conversely, to express the quantities (s)*' by ( (s 2)'), exchange i with (i 1) in (12) and use (2). Then (14) (( g -2V-) = ( It follows from (13) and (14), that (-8 + 2)0, o (s From this an important conclusion may be drawn. If s is positive, (s*) is, by (5), positive for every value of i. If s > 2, then ((s 2)*) and ((s 2) t '~ 1 ) are positive. Therefore, if we select i positive and so great that both ( s + 2* + 2) and {s + 2?' 4) are positive the second member of (15) becomes negative and If we put, for example, s = 3, i = 2, it follows ,* _ 3 [(F) + (!')] hence (16) (3 2 )<(3'). This inequality will have an important application in the following paragraphs. If we solve (13) and (14) for (s*) and (s 1 '- 1 ), we get <-* - 2(- g + 2 + 2) ((82)0 + fc(-g- 2i + 4) ((s- ( s + 2) ttj a 2 (fc 2 4) Finally we will show how to express the derivatives of (s f ) with respect to a^ and a 2 in terms of (s*). If we differentiate ( 1 ) with respect to a^ , we get TERMS OF THE DEGREES 0, 1 AND 2. 191 or and, therefore, and a*lso by (12), if we write (s+ 2) for and, finally, by (17), n qx d(0 _ [ V By exchanging the subscripts 1 and 2, we get the derivatives The derivatives of a 2 can also be obtained from those with respect to c^ from the fact that (') is a homogeneous func- tion of the ( s) th degree in a x and a 2 , and hence the partial differential equation must be satisfied. 26. THE TERMS OF THE PERTURBING FUNCTION OF THE DEGREES 0, 1 AND 2. The analytical development of the coefficents of the perturb- ing function requires, if we pass beyond the second degree, days, weeks and* even months of labor in computing them. If the computation is confined to the zero, first and second degrees, the necessary computations are not excessive and the reader can, for himself, easily verify the formulas. The equation (15), 24, then becomes 192 THEOEIES OF PLANETARY MOTIONS. (3) + Jr,r a S(3 A ) cos (^F + TP). The quantities jp and g become The (S A ) are liere functions of r t and r 2 and may be denoted by r2 . On the other hand (S A ) is the (s x ) ai , ,. The equation (1), in which p and q need be substituted only in the first term of the second member, now becomes If this is now developed in ascending powers of p lt p 2t Qi, ft then their values given above are substituted, it follows that A. = cos eg + e 1 e. 2 2 E^cos -f ! 2 ^A cos ( ? 1 +M" 2 ) + 0^x008(^ + where for brevity, and where the coefficients have the following values (3) TERMS OF THE DEGEEES 0, 1 AND 2. 193 when for brevity the following symbols are used: / - (W // - a 8(P) '/ - - )j J * .' -^- By equation (20), 25 and the results obtained by differentiat- ing it with respect to a^ and a 2 , the following relations may be found to exist among the coefficients. ' T ' I f> - by the aid of which the coefficients may be further transformed. We find, for example Further, it follows from (12), 25, that and, therefore, J A ). If we wish to produce the entire expeditious development of the perturbing function, we must proceed as in 24, page 180,. The angle x is small, of second order, and all terms except the first can be neglected. We have here to put cos x = 1,, sin x | z, i' 2 sin (^ fl 2 ), so that we obtain A cos p = where for brevity Further, the term 2;6r A cos^ separates into the three terms V'' + (Si fi a )], while the last term .L A cos(?> + ^ + ^) separates into + ^ + Ca 2fi,) + Ji a a i A 006(0 + :! + ^ 2fi 2 The equation (2) now changes into 13 194 THEORIES OF PLANETARY MOTIONS. (6) r^ A= oo + e 2 C A cos ( + C 2 ?r 2 ) + e^ > A cos a7^X~c S >)- The coefficient (6) becomes zero (8) (1) if a, = Q 2 = 0, and, therefore, for all secular terms. Then by integration of (5) with respect to t, a term of the form (9) is obtained, that is a term proportional to the time. And zero (10) (2) if 1 :n s = a 8 :-a 1 i that is when the two mean daily motions, and therefore when the two periods are in commensurable ratio to each other. We have seen that there is much discretion in the selection of the original elements. We can choose the periodic times in such a way that equation (10) is never fulfilled, or fulfilled for such large entire numbers di and a 2 (exact or approximate), that the degree of the term is very large, and therefore the coefficient K ANALYTICAL EXPRESSIONS FOR THE PERTUBATIONS. 199 very small, so that this term for very long intervals of time may be neglected. We shall see later that there is a very suitable selection of elements and that with this the periodic times of the planets are commensurable only for very large values of QI and 2 . We see, then, that practically, only the secular terms of the perturbing function give proportional terms by integra- tion with respect to the time. We will, therefore, divide the perturbing function into the so- called periodic part and the secular part and designate them by (R) and [12]. In 24, it was showu that R l can be developed into a sum of the form (49). (R^ is then the sum of the terms of R! for which the two integrals a lt a 2 are not both zero at the same time, while [9ftJ is the sum of those terms for which a t and a 2 are zero together. Let (11) Ke^e^i, eii^cos^+o^ + b,r 1 -hb a 7r a -}-c 1 fl 1 + c a fl a ), or briefly (12) HcosL, be a term of (R^ where foi integral numbers, the conditions of 24 hold, that is (13) a 1 + o a + b 1 + b a + c l + c a = 0, and the four differences i [&i], a [VI, A [Ci],/5 2 [c a ] are even, positive, or zero. Further, let (14) kefi' ip' cos (a/ f , + b/ TT, + c/ fl,), or briefly, H' cos L' be a term of x, so that according to 4, a/+b/-fc/=l, and /-[6/], A'-Cc,'] are even, positive or zero. If we now denote the part of dx which depends on (R^ by (fla^), it follows by the use of the equations (15) and (11) of 22, and the table (22), 11, that 200 THEORIES OF PLANETARY MOTIONS. (15) (teO = 2 [M, cos (V + L) -f M 2 cos (,' L)] , in which (15a) M, = L=r n x H H -, - o " a l aa, a, 1 If IT , VI e, 2 , , ,, ' + ~~ ( ~ "' ' + "" - i 1 VI e \ - 2 2t 1 smi 1 v I-T-CI The second term M 2 cos(I/ I/) is formed from the first by exchanging the six integral numbers in L with their opposites. The symbol 2 in (15) refers not only to the integral numbers in (11) and (14) but to those of all the disturbing planets. If (15a) is developed*in ascending powers of the eccentricities and inclinations, the law of formation is at once seen to be (16) (dx,) = 2 (K) ep e^ ifl if* cos (a, C t + a 2 C 2 where the whole numbers t , 2 fulfill exactly the same con- ditions as before, except that the sum (13) is not now = but =1. (tyi) follows at once from (dx^ by putting sine for cosine. Finally (tei) has the same form as (dy^ except that for (dz^ the sum (13) again = 0, while c a + C 2 is odd. If, for brevity, we put tXd C a )=0 and. limit ourselves to the consideration of terms of the first degree, the following terms result: (17) (^) = 2[( J E'/)coB(04-C 1 ) + e^K?) cos (^ + TT,) + l (Kf) cos (0 + 2^ ^ + eJ(Kf) cos (0 + 7r 2 ) + e 2 (^ 5 ) cos (0 + 2: 2 - 2 ), ANALYTICAL EXPRESSIONS FOR THE PERTURBATIONS. 201 (18) (%) = S [(*:/) sin (0 + d) + e,(K?) sin ( + e 2 (*:/j sin (4> + *,,) + e 2 (^, 5 ) sin (t (19) fa) = S[i 1 (^)8in(0+ C! fi,) + W)sin(^+: 2 fi a )], where (.BQ represents coefficients depending only on a t and a 2 and which are to be formed from what precedes. It is to be noted that, in order to take account of the perturbations up to the n th degree, the perturbing function must have been developed to the (n + I) ih degree. We now pass to the secular part [-RJ of the perturbing func- tion and to the corresponding terms pa?, ] , [fly, ] , [<^] . For this part 8: and m From this it follows that (20) [te,] = Ww etc , tJic. } aj Corresponding expressions can be obtained for [<%i] and [&i]. By the development of (20) we now get The same conditions hold here for the integral numbers as in (16) except a 2 is always zero. [<^/i] is formed from [&EJ] by substituting cosine for sine. p#i] has the same form as [%i], differing from it only as (flzj) differed from (^). Finally six arbitrary constants appear in the six integrals 8.R - dt, etc., of equations (15), 22. These constants will be UCtt 202 THEORIES OF PLANETARY MOTIONS. denoted by \a : \, . . . They give another term { From (1) and (2) it follows that (3) n _ dx da dx de ~~ $ y da dy de dadt ^ de dt 'dz d a i @ z ae da dt + de dt The second drivatives # d 2 x _ d' 2 x dx' da ^ dx' de d? ~ W 8o~ dt ' de dt >'*' are obtained by differentiating (2). and on substitution in the equations and by the use of * v" x - fJ.X become dx' da dx' de dy' da dy' de 8a dt ~^~ de dt ~^~ " (W' de __ .di _ QR da de The equations (6) are to be solved for , , ... According to dt dt 10, these solutions are da N dR , QR QR de dR and by the use of (21) and (22), 11, da di de dt a dR / 1 \l o e \f au @ s s V d ! J - OK !g(dR~\ , Vr^e" 2 (l Vl e*)dR - _9 l a (3R 'dt' ~ *\~\3a + | 1 cos?: QE sin i \/ /j--a(l e 2 ) di ' dt ~ e \ afi di _ 1 cos i dR 1 cos i dl\ dt sin i V / J - (1 e 2 ) ^ sin i V /*a(l e 2 ) ff 1 ^^ sin i i dR THE VAEIATION OF ELEMENTS. 207 These equations (8) form the foundation of the theory of the variation of elements. According to Poisson's method they can be obtained easily and directly. Laplace obtained them in a more difficult way and gave them in a supplement to the sec- ond volume of Mgcanique c&leste. They have the advantages that the second members depend only on the perturbing functions and, consequently, are relatively small, that the differentation is with respect to the elements, and that the coefficients do not explicitly contain the time. Still simpler equations could be obtained by using the canon- ical constants introduced in (7), 13. The corresponding equa- tions are dctj dR da 2 dR da 3 dR ~dt ~ ~~8/V ~dt~ ~~8/V ~dt~ ~~0/V dpi _ dR dft 2 _ dR d/?3 dR dt da-i' dt 8 2 ' dt da 3 ' But since R has been developed in ascending powers of e and i, the equations (8) are more convenient for us and we will use them. The derivative of R with respect to a is put in paren- theses because it is to be taken as complete, that is, both iu so far as a is explicitly contained in R and also in the relation (9) C'F?'*+*. If, as was done by Tisserand, (Exposition, d'apr&s les princi- pes de Jacobi, de la mgthode suivi par Delaunay Journal de Math&matiques pure et appliqu6es, 1868), we introduce the mean longitude instead of e, we avoid the complete derivatives with respect to a and thus the explicit appearance of the time. We have dR dR V .dR By the differentiation of (9) we also get dZ 3 , / /j. da . ds = n 1 -+ I r H dt %\a*dt^dt 208 THEORIES OF PLANETAKY MOTIONS. da If we also put (10) K=R we get VI ^2 I . / "I p2 ft T?' & -L f^ J_ ^ L/J-w a;j. e oe 1 cosi ftn' _ _ sin *V ,"(! e 2 ) 0i R' can also be put in the place of R in the other equations (8).. Equation (11) has the advantage that it gives at once the deriv- atives of the mean longitude with respect to the time. Another circumstance to be mentioned is that - - and at at become infinite when e or i=0. It is evident that when the eccentricities or inclinations are very small, small changes in the orbit cause great variations of the perihelion or node. To overcome this difficulty Lagrange substituted for e, i, x t tt,, four new variables by means of the equations . ? = From these dh de h . d dl _ de I dx dt~dt^e~dt ' dp .di . dti da . .di dQ -= qeo ti--- p , and further a = aR z _aR 8J? r dR , dR APPBOXIMATE INTEGRATION. 209 By introducing (12) into (8), we get a less synoptic form, but there is the advantage that with vanishing eccentricities and inclinations no infinite coefficients appear. The coefficients can be developed in powers of h, I, p, q and the form holds for the perturbing function R. The substitutions (12) will be applied later in a case limited to the second degree of the inclinations and eccentricities. 29. APPROXIMATE INTEGRATION or THE DIFFERENTIAL EQUA- TIONS FOR THE VARIATION OF THE ELEMENTS. The differential equations (8) of the preceding section are special forms of the original system (2), 21, and it is evident that in this case also the actual integration is impossible. Yet, for two reasons, equations (8) possess a very great advantage when it is a question of approximate integration. The first is the smallness of R and the second, the analytical form to which it is reduced. For example, from the first of equations (8), it follows that (1) a But we have not yet gained anything, for in order to com- plete the integral, the elements must have been already deter- mined in terms of the time, or the problem must have been already solved. In (1) it is as if with the equation x n = x-\-a, the unknown quantity x were to be obtained by the formula x \f x-\-a. And exactly as this formula can be used to find x by a process of approximation, beginning with a known approx- imate value, can formula (1) be used with the skillful employ- ment of the development of R. It has already been shown that R separates into a periodic part (R) and a secular part [R]. Let (2) be any periodic term of R lt (We will from this on use sub- scripts to distinguish the disturbing from the disturbed planet. ) According to (8), 28, this term produces a term of the same 14 210 THEORIES OF PLANETARY MOTIONS. form (2) in every derivative of the elements, and hence in the elements themselves terms of the form If the elements were not variable this could be integrated at once and would give W + w cos I il Cl + ** C2 i It can' be shown, however, that since the disturbing masses are small, (4) is very approximately the integral of (3). For cos sin The second term of the second member is an integral which, according to (11), 28, depends only on the second powers of the disturbing masses, since k as well as ^ * * . 2 * 2 ^ -- 1 L(i 1 n 1 -\-i 2 n 2 )dt is of the first order with reference to those masses. Neglect- ing, therefore, the second powers of the disturbing masses and integrating by parts, we get /7 cos sin , j *i Ci -f- i 2 C 2 cos ' is also of the second order with reference to dt the disturbing masses. Neglecting the integral in the second member this passes into (4). It appears, therefore, that a peri- odic term of the perturbing function produces perturbations in THE SECULAR VALUES OF THE ELEMENTS. 211 the elements which depend on the same argument (v^i + ^Q. k The coefficients - - are proportional to the disturbing masses and, therefore, do not in general reach any value of con- sequence. Hence A periodic term in R alternately increases and decreases the elements but does not permanently change them. The secular term [JR] has a different action. It produces similar terms in the derivatives of the elements and if the ele- ments were constant, it would produce terms proportional to the term containing the time t. As the elements are variable this does not actually occur, but it is easy to see that the action of the secular term goes far in time and may result in an entire change in the elements. The approximate integration of (8), 28, may be made in the following manner. Limit R to its secular term \_R~\ and substitute this is (8). Integrate the resulting simplified differ- ential equations. The resulting values will not be the true ones, that is, those actually existing at any moment, but the so-called secular values of the elements. The actual elements may be obtained by adding to the secular elements the periodic terms k sin 1 1 I 2 2 COS in which the secular values may be used. The elements are now complete and the coordinates may be computed from them. This process is not rigorous, but the error for long periods is so small that it gives a very close approximation. 30. THE SECULAR VALUES OF THE ELEMENTS. DEVELOPMENT OF THE ElGOROUS EQUATIONS BETWEEN THEM. In 26, it was shown that the secular part of the perturbing functions R lt R 2 , . . . depends only on the one quantity If we denote this by W and replace the elements by their secular values, it follows from (8), 28, since 212 THEORIES OF PLANETARY MOTIONS. (2) that >.^ = 0, dCi la,dW . /L e \(lVl e*)dW ^ l -rr = m l n l 2 A /-^ h -%/ <^t \ /-*i (7^! \ /-/j ttj 6j OCj 1 nna i. r) T?7 (3)]^=-]-^^!^, dlr, 1 COS*, 8TF , 1 Jl e*dW mi dT = sint V an^T^^r e"i V^^T^r d^ 1 cos^ 8TF 1 d^ Gin * -A./ ( a 1 a 1 (l ^j") ^^i sin^/x/^j dw These equations are to be formed for every planet and from them the elements (or rather their secular values) are to be determined. Since is not contained in W, the second of equations (3) is to be dropped and after the elements a, e, -, i, &. have been found, the mean longitude C is to be determined by pure quadrature. The first of the equations (3) is at once integrable, and gives (4) a, = constant, and therefore The major axis of each planet, aside from periodic changes,, remains invariable. This important result was first obtained by Laplace, but only as an approximation, since he neglected the higher powers of the inclinations and eccentricities. Lagrange succeeded in perfect- ing the theory, by the stroke of a pen, says Jacobi, by prov- ing this leading proposition of the whole theory of perturbations, in all its generality. From the last four of equations (3), it follows that THE SECULAK VALUES OF THE ELEMENTS. 213 i i i , y dWdQ l _ d* + 8^ dt ~di^ dt + d^ ~dt ~ In the same manner, the variations which W undergoes by reason of the changes in the elements of the other planets are aero. Hence (5) we get the following final equation f or g 1 , (21) I.*! -"-J t" [1,2], [1,3], L**J -"-J 5 [2,2]- 3 , [2,3], [3,2], . . . . [3,3]-flf, . . . = 0. This equation furnishes n values gv, g 2 , ... g n of g. Each root g\, according to (20), corresponds to a system of values, (22) 1,A,2, A, .,A, for whose more exact determination we also have the equation (23) a, , A 2 + 2 , A 2 + 8 , , 2 + . . . -f- a H , A 2 = 1. Since the determinant (21) is symmetrical alid also all the quantities [A, /*'] are real, the roots g are real. Consequently the coefficients of the substitution (16) are also real, and when the g's are all different, it is easy to show that they satisfy the conditions (18). But when (21) has a multiple root #x, appa- rently the transformotion becomes illusory because in this case each of the equal roots g\ appears by reason of (20) and (23) to correspond to only one system of values (22). It has been shown that this is not so. For if all the [ A , /./.] are real, as is here the case, not only the determinant (21) but all of its minors to the (/ l) th degree vanish when there is a multiple root g^ , so that, from (20) and (23), as many systems (22) appear for these roots as equals the multiplicity of g^. Hence, the substitution (16) is always possible, and it is indifferent whether (21) has equal roots or not. The quantities g are also all positive, because

l gcos/ It is here not easy to recognize the geometrical law of change of r A with the time t. But it can be shown, on the whole, that the perihelia advance; that is that they have a positive motion. By differentiation, it follows from (32), that 222 THEORIES OF PLANETARY MOTIONS. (33) e^m x 1 g 2 )t+d 1 <5 2 ] 4- . . . The constant terms in the second member are positive. The periodic terms are sometimes positive, sometimes negative, so that d~x can be negative, but only at times, and as an interrup- tion of the generally positive sign. This follows from the inte- gration of (33), which gives . . . )t The term proportional to the time increases continously, while the periodic terms never surpass a definite limit. But there still appears a missing step in the argument leading to the conclusion that on the whole - A increases. For if two gf's, say g l and g 2 , are equal, the denominator g l g 2 vanishes. Yet even in this case d- A must, on the whole, increase. For the sec- ond member of (33) can also be written )-f a A , 2J BT 2 V 2 sin so that for g l = g 2 the last term entirely vanishes. The addition of all the equations arising from (33), by ex- changing A gives ^ " / ~^''A -TT o , T 7- 9 l g l + K/g 2 + Kg % -f . . . , A and, since the second member is positive, it follows that all the perihelia cannot retrograde together. The numerical results show that in this, thousand years all the perihelia have positive motion except Venus for which the motion is small and retro- grade. If in (32), the absolute value of any coefficient, say [0^,1-^1] is greater than the sum of the absolute values of the others APPROXIMATE CALCULATION OF SECULAR VALUES. 223 [a A3 oJ^ 2 ] + [ a \ 53-^3] H~ 5 is e &sy to show that ?TA oscillates con- stantly about a mean value git-\-8 l or 0i 4- <^i + ~ , according as is positive or negative. For it follows from (32), that where A = A and _B = A , ^-[-6^,2^2 COS [( + A , 3 ^3 COS [((/ 3 9i)t 4- (^3 *l The denominator, by the assumption made, can never vanish and hence tan [7^ (0i-Mi)] can never become infinite, so that this angle in fact oscillates about or - and the magnitudes of the oscillations are always less than ^-." If g 1 is the greatest root of (21), TTx increases continuously. When the equations (29) with the 2n constants of integra- tion K and 8 have been formed, the next step is to determine these constants, when the eccentricities and longitudes of peri- helia, and therefore the tis and Ts or the H*s and Z/'s, have been given for any instant. This is best done by equations (25), which by the substitutions (28) become { \ By division these give (35) tan (^ + ^ When ^ A has been found from (35), either of equations (34) will give the arbitrary factor K^ . Moreover the determination of <5 X is double, since 8^ can be increased by - whereby the sign of .STx will be changed. This completes the theory of the secular variation of the eccentricities and longitudes of perihelia. There are corres- ponding developments for the inclinations and the longitudes of the nodes. The part of the perturbing function W which is concerned in this, is 224 THEORIES OF PLANETARY MOTIONS. By the introduction of new variables PA and Q\, by the equa- tions (36) PA V WA V^OA = PA, q^ ro x VA^ = $A , the differential equations between p and g become passes into (41) (x, , x 2 , . . . x n ) = /'i 2/i 2 The final equation (21) is, for this case (42) I2,2|- r , |3,2|, |2,3[, |3,3|- r , This gives the n roots ri t r zs n, . . . r- It can at once ba shown that one of these roots 0. The function ! i , v ni z V A* 2 2 , ... Then 4> can be written in the form (x\ Xp. \ 2 71= / ~ 7?= - , - - 1 \ W A V A^a* \ m^ V /V a M ' It follows at once from this form that the determinant of

are negative and we have now to operate with these exactly as with the roots of equation (21). The equations corresponding to (29) are Pi \m l V/*i i = A , i EI sin ( p-2 _ \m. 2 W 2 2 = ^ 2 , i KI sin (TI t -f (5/ (46) l4" ^'+i*'+V + W fe+^1 a , V - H HI i (X , j/jj^axpO^X PA' gA 2 +px^ + gA^) I J 228 THEORIES OF PLANETARY MOTIONS. The first line of the second member of this equation, in which ij. takes all values from 1 to n exclusive of A, is constant. The two following terms are homogenous quadratic functions of h and I If the values of h and I from (29), 31, are substitu- ted, the terms become homogenous quadratic functions of K. If they are arranged in terms of the cosines and sines of the angle (gt-\- #) they give, first, constant terms; second, terms of the form 2acos[(g a gp)t-{-(8 a fy)], where a is a constant coefficient. Constant terms and terms of the form 2 b cos [(/a rp) t -J- (i(l [ e i] 2 ) a 8 PJ cos in ft) 1 cos ft] Oj cos sin ft]V^i[a,](l [ej 2 ) a sin ^ (1 cosft])^^, , ^ it ,, gm [aja-h^sinftj^-cos THE PERIODIC TEEMS IN THE ELEMENTS. 231 The corresponding periodic term of C, remains to be devel- oped. The first term of -^ in (2), 32, namely n lt does not con- Git tain the disturbing masses as factors, and in the integral fn^dt we must take account of the periodic terms (n^. Now and, therefore, The term of the integral corresponding to the term (2) in the perturbing function is, by the first of equations (4), =2 sn and hence the periodic term in C corresponding to (2) is sn sn COS M! Oja COS - ei sn ^ __ 1 cos[t' 1 ] __ dk 1 sin ,., sin [t\] V^[a,](l [e] a )0R] cos By the substitution of these values in (1), we get the elements as completely determined functions of the time and these can, therefore, be computed for any given instant. From the ele- ments the coordinates can be obtained by 5, exactly as if the elements were constant. The elements are thus divided into two parts, the secular and the periodic. The last are constantly proportional to the dis- turbing masses and are therefore small. According to the devel- opments in 30 and 31, the first are also periodic, but their periods are quite different, since the mean longitudes appear in the trigonometrical functions in the periodic terms. The secu- lar periods embrace thousands of years and cause a very slow 232 THEORIES OF PLANETARY MOTIONS. change in the secular values of the elements, except, of course, the terms in [C] which are proportional to the time. We will now see how the periodic terms may be very conveniently sepa- rated from the secular in the final expressions for the coordi- nates. Let x be any coordinate. It is a function of the elements, therefore, x = x(a, e, i,-, fi, C), or, by substituting the values of the elements from (1), (6) * = *([a] + (a), [] + (),...,[:] + (:)). If the second member of this equation be developed in pow- ers of the quantities (a), (e), . . . , which are small and proportional to the disturbing masses, and limit the development to their first powers, we get (7) a = *([al[e],[i],[* 8x . Sx . . dx The terms u r -.()+ . . . are, by 22 and 28, exactly the same as the purely periodic terms in the absolute perturbations of the first degree. Hence (7) becomes (8) a> = a ( [a], [e],... ) + (*<*). From formula (8) flows the following remarkable combina- tion of the two different methods for obtaining the motions of the planets, the theory of variation of constants and that of abso- lute perturbations. Imagine first a fictitious (say a secular) planet, which moves in an ellipse with the secular mean motion of 32, and whose elements are the secular values of the real elements. The real planet will always be a small distance from the fictitious planet the distance depending on the periodic terms. The departures of the real planet from the fictitious place may be regarded as due to the departures of the elements from their secular values, and these may be treated, according to (8), directly as absolute perturbations. In this way the variation of constants is used to THE STABILITY OF THE SOLAE SYSTEM. ' 233 get the effects of the secular terms of the perturbing function, while the theory of absolute perturbations is applied only to its periodic terms. It is remakable that this combination of the two methods was used earlier than that of the pure variation of the elements. Euler and Laplace recognized the fact that the secular terms of the perturbing function exercised a vital influence on this varia- tion and they fixed their attention on this alone. Lagrange first gave the theory of the variation of elements in all its purity. Astronomers, in calculating the positions of the planets, usually employ a combination of the two methods, and they thus get a system of formulas which are very convenient. 34. THE STABILITY or THE SOLAR SYSTEM. The important results, which the theory of the variation of the elements brings to light, are not only a close approximation to the truth, but they are also of the greatest importance in solving a problem of the highest interest the problem of the stability of the solar system. With the limitation that an approximation, although next to mathematically exact, can never take the place of a mathematical certainty, the theory enables us to affirm the stability of the solar system. The planetary system certainly undergoes important changes in the couse of centuries, yet these changes do not affect the two elements which with entire propriety have been called the prin- cipal ones, the mean distance and the periodic time. If the last has been obtained from hundreds or thousands of observed periods, we may be positive that this mean exactly represents the invariable periodic time. This element, with Kepler's third law, corrected as in 32, gives the exact mean distance, which is incapable of direct observation. While the directions of the axes, the eccentricities, nodes, and inclinations may undergo great changes in the course of time, yet the stability of the sys- tem is even here assured in so far as it depends on the eccentric- ities which always remain small, and upon the inclinations which are small when referred to the invariable plane. 234 THEORIES OF PLANETAEY MOTIONS. The recognition of the stability of the system is the finest result reached by the investigations of Lagrange and Laplace. The dread that the mutual attraction might eventually cause a collision is completely relieved. The planets will always circu- late as regularly about the sun as if each alone were the only planetary member of the system. 35. THE EFFECT OF NEGLECTED SECULAR TERMS OF THE PER- TURBING FUNCTION WHOSE DEGREES WITH EESPECT TO THE ECCENTRICITIES AND INCLINATIONS ARE HIGHER THAN THE SECOND. In 31 the secular values of the elements were developed for the terms of lower degree of the perturbing function. By the omission of terms of higher degree it was possible to change the differential equations between h, Z, p, q, and the time t into- linear equations with constant coefficients. If the eccentricities, and inclinations are very small, the neglected terms are also very small, as compared with those not neglected. At the same time, it is evidently possible that a very small additional term in a differential equation may have a notable effect in the integral,, especially if the time be sufficiently extended. So far as the author knows, Leverrier was the first to take into account the terms of higher order in the secular value of the perturbing function. This was done in his work, Integra- tion des Equations diff&rentielles, dont dependent les in&galits s&cu- laires, en tenant compte des termes, qui sont du troisidme ordre par rapport aux excentricit&s et aux inclinaisons. If W is developed to the fourth degree inclusive, then (1) W=W Q +W 2 +W>, where W , W 2) W, are the terms of the th , 2 nd and 4 th orders. If h, i, p, q are introduced, (2), 31, gives at once W Q and W 2 . The formation of W t is more complicated. In every case h, I, p, q occur only in the combinations (2) ftxM-Zx 2 , ftx^+U,, px 2 H-gx 2 , THE EFFECT OF NEGLECTED SECULAR TEEMS. 235 and in such a way that each term is even both in reference to h and Z, and also p and q. W 4 can then be represented as a homo- geneous quadratic function of the combinations (2), but never so that one of the first four is multiplied by one of the last two. When the second members of equations (3), 31, include the terms of the third degree, they become 8 r tf +L dW, 2 ~ l dh 01 dq dq_ dW, , fp* w^- -g- These equations are to be formed for each planet, and the problem is to integrate them, even if only approximately. If the terms in brackets were equal to 0, then h, I, p, q would have the forms given in 31. We will use Lagrange's method and assume that the same form occurs here with the difference that K, d, K', d' are no longer constants but undetermined functions of t. (The a's and gr's and the /S's and A's in the equations (29) and (45), 31 remain invariable since they are not constants of integration but depend upon the invariable axes.) If we put (4) we have, by 31, (5) Wt 236 THEORIES OF PLANETARY MOTIONS. W takes the form (6) TF 4 = fc + Sfccos>l, where the coefficients k are homogeneous functions of the fourth degree and / an angle of the form (7) x = a, 6^ + 0303+... +&iA + & 2 A+... The quantities a and b are here positive or negative whole numbers, not included, which satisfy the conditions (8) ai +a a +... +&! + &,,+ ... =0, (9) c^ -f- 02 +...-{- a,, = an even number, (10) [a,] + [aj + . . . + [6,] -f- [6 2 ] + . . . - 4, or 2, or 0. If for brevity the bracketed quantities in (3) be represented by (11) [H], [L], [P J [01 we obtain by the substitution of (29) and (45), 31, (12) [H] = Sfccos/, where the &'s are homogeneous functions of the 3 rd degree in K and the A's angles of the form (7), except that the second mem- ber of (8) equals 1, the sum (9) is odd and the sum (10) equals 3ori. It likewise follows that (13) [] = 2 A? sin A, where the fc's and /'s have the same values as (12). Finally P^Sfc'cosx', Q = Sfc'sin/', where the fc's denote homogeneous functions of K of the third degree and the angles /' have the form (7) except that the sec- ond member of (8) equals 1, the equation (9) remains unchanged, while the second member of (10) equals 3 or 1. The equations (34), 31, now become (14) Kx sin <3 A = "2 M ? A \ m M V AV a M = 1 fX=M (15) Kx COS (TA = a M A \ W M V /V tt M THE EFFECT OF NEGLECTED SECULAE TEEMS. 237 Differentiating these equations and remembering that K and are variable, it follows that dK, . dt sin (TA + K), cos G\ - cos 6r A = , dhp v PH&P d j Substituting the value of - - from (3) and remembering that the term g\K^cosG^, by 31, must vanish because of the term -^71 u,A QW* . ,. , the preceding equation becomes "= di^ and similarly, dt and, th (16) . COSO-A AASinGrA x^^ / / dt dt <^* \in^ V /-vc^ erefore, ' t H = n d]\-\ ^_ ^ *^u A - "^ / ==([^1^ JSIU (TA -f- L-L/ r *' X| m^ v ! J -u. a n n**i and jU. . n d$\ ^^ ^i f t-u i A c ^J \ / m^ \ r p a M u = l If the values of [fi"J and [I/] are substituted from (12) and (13), the second members of these equations take the forms where again the fc's are homogeneous functions of the J^T's and K"s of the third degree and indeed of an even degree with ref- erence to the K"s, and further the angle A has the form (7). The corresponding equations are 238 THEORIES OF PLANETARY MOTIONS. /? , ^ (17H / M V/v< / V /V a/t where the second members again have the forms (18) 2fcsinA, 2fccosA. In order to integrate these equations approximately, distin- guish the so-called secular-secular in the second member, that is, the terms independent of G and F from the secular-periodic dK terms, that is, the terms containing G and r. Since and ctt at here have the form 2 ft sin ^, and the secular terms fail, it fol- lows, when limited to these, that dK dK>_ dt~'^ ~dt~ Therefore, the former constants of integration K and K' remain constant as before. It is otherwise with the derivatives of the for example r n , is zero. Hence F n = d n '= a constant. The terms of (18) for which / 2/", or 4^ are, therefore, to be 240 THEOBIES OF PLANETARY MOTIONS. regarded as secular-secular terms. Hence the equations = u/t d,K r and - are not fulfilled and this is entirely correct. In at fact, these two equations are not generally correct; the invaria- ble plane should be selected as the plane of xy, because, as in 31, the terms dependent on Yn then vanish in the first approx- imation. With this selection of coordinates it is also evident that no secular terms would appear in -~ , so that the new CLtf Y n ' = Y n -j- SY* would vanish. In the same way still higher terms could be brought into the account and, finally, a formal development of h, I, p, q would be obtained like the following. The 7i's and Z's would take the forms (24) 7i where A is an angle of the form (7) with the condition that and the numbers a and b including all possible integers which satisfy the condition that c^ -f- - 2 + -f ,, is odd. By the selection of the invariable plane as that of xy, b n is discarded. Likewise, the values of p and q have the same forms (25) p = S where A' has the same form as A except that the sum a^ -f a. 2 -f- . . . -\-a lt is even. The angles A and A' are linear functions of the time because G and r are such. The solutions (24) and (25) are, to be sure, only formal. There appear for instance, in (22) arid (23), denominators which eventually vanish, or at least become very small. Leverrier and Lnhinann have specified several such denominators and in combination with numerical computation have come to the conclusion that the influence of terms of higher order is decidedly greater than was supposed by La- grange and Laplace. If, overlooking these difficulties, the periodic terms intro- duced in 33 are added to the elements, we get, when these are TERMS OF LONG PERIOD. 241 introduced into the expressions for the coordinates, the follow- ing general schematic representation of the coordinates: (26) x (27) z The K's and K"a are coefficients independent of the time and the angles L are of the following form : (28) i = a 1 c 1 + o 2 : 2 +...+b 1 G 1 + b 2 G 2 +... + Cl r 1 -f-c 2 r 2 +... The integral numbers a, b, c satisfy the condition The angles L' are of the same form (28), except that for them the sum of the integral numbers 0. In spite of their very different origin C, 6r, and r have the common property that they increase proportionally to the time. We will resume this representation of the coordinates in 40 and the following sections. 36. TERMS OF LONG PERIOD AND THE COMMENSURABILITY OF THE PERIODIC TIMES. In the separation of the actions of the secular and periodic terms on the elements or coordinates, a fundamental difficulty appears which the efforts of mathematicians and astronomers have not succeeded in overcoming. If (i) a,:, mu is a periodic term of the perturbing function, then, by formulas (4), 33, similar terms must appear in the elements. In these (2) a^ + a^ appears in the denominator. In the mean longitudes the term /ON 3 djA; sin , , , -?( fl| n 1 + a.^'-eo8 (a ' C ' + fl ' C ' ) occurs and in its denominator the square of this binomial appears. It is these denominators that cause the difficulty mentioned. Even when w, and n 2 are irrational with respect to each other, a 16 242 THEORIES OF PLANETARY MOTIONS. series of whole numbers can be so determined that the equation (4) 0^ + 08*18 = is almost exactly fulfilled. This is best done by developing in a continued fraction and determining approximately the numer- ators and denominators. The periodic perturbations, which depend on the term (l),then beeome disproportionately enlarged by these denominators, and although the original coefficient k may have been very small, the corresponding coefficients in these perturbations become important and may become infinite. This difficulty is generally met as follows: The mean daily motion n in our system has the property that equation (4) is approximately or exactly fulfilled only when a : and a 2 are very large numbers. In this case the coefficient k of the perturbing function, which is of the degree [c^ + a 2 ] with reference to the eccentricities and inclinations, becomes so extremely small that the term has no sensible influence on the elements for very long periods of time. We will, however, assume that c^ ^ -{- ct 2 n 2 is very small for integral numbers d and a 2 which are not too large, and investi- gate more closely the action of the corresponding term (1) of the perturbing function on the perturbations. The period of this term is (5) ?= d First it is to be noted that such a term appears in R l and also in .R 2 , and we will compare the coefficients k in the two cases. Leaving the other planets out of consideration, we have / m * (ey x * ( x i~ a? 2) 2 +(2/i 2/ 2 ) 2 + (i 2 2 ) 2 p - ( l \ \S(xi- Fpr brevity put' d Cj -f- a 2 C 2 = A. Then let the periodic term with the argument A in n TERMS OF LONG PERIOD. 243 1 P and Q, with respect to the eccentricities and inclinations, are of the degree [a x + aj , while p and q are of the degree [aj -f- [a 2 ] 2, so that, if c^ and Q 2 are large numbers with opposite signs, (which must be the case since a! ^ -f- a 2 n 2 ), is to be very small the coefficients p and q are in general smaller than P and Q. By (16), 26, the corresponding terms become in R l \( c^ = wia IP --- s L-V. f fj. 2 in jR 2 We can here put /^ = //- 2 M. The factors of m 2 and w, then become nearly equal, for their difference is = (.".-^)(a.". + ^) (pcos; . + ^^ art Not only are p and q small in relation to P and $ but Q : n^ + a 2 w 2 must also be relatively small. Consequently, the above terms become in R l (7) = m 2 (P' cos A + $' sin A) = 2 C cos (/ + e), in P 2 (8) = m t (P' cos / + Q r sin A) = Wl C cos (A + e). The terms in the mean longitudes ^ and C 2 corresponding to the periodic perturbation terms (3), are, therefore, Their 244 THEOBIES OF PLANETARY MOTIONS. _g 1 m 2 a 2 2 a 2 Wja 1 2 ' Since c^ w, + a 2 W 2 is to be very small, we have, approximately and the above ratio becomes ra 2 It is therefore constant and negative. If one mean longi- tude is increased by the periodic perturbation depending on the argument A, the other is decreased. These deviations are inversely as the masses and the square roots of the mean dis- tances. As the duration of /the period is long, it can only be recog- nized as a period, by observation, after a long series of revolu- tions about the sun. One revolution would appear accelerated, another retarded, until finally the relation would be reversed and the acceleration would pass into a retardation and vice versa. This phenomenon is known with accuracy for Jupiter and Saturn. Representing Jupiter by the subscript 1 and Saturn by 2, we have in this case a very small fraction of n x and n 2 and about = HK. The angle increases very slowly, and since Saturn's period is about thirty years, it takes about 900 years for A to increase to 2-, and, con- sequently, for the restoration of the previous relation between the corresponding terms of the perturbations of the mean longi- tude. Since [a, 4- a 2 ] = 3, C in (7) and (8) is of the third degree with respect to the eccentricities and inclinations, and must be relatively small. On the other hand, the coefficients in (9) and (10) increase so much, through the square of the small denominator 5n 2 2w n that they are not only appreciable, but in fact the greatest of all the periodic perturbations of the TEEMS OF LONG PERIOD. 245 solar system. The irregularities of the periodic times caused by these perturbations were noticed at an early time by a com- parison of the periods of revolution for different epochs. It was thought that the smallness of the coefficients warranted the omission of these terms of the perturbations, and early astrono- mers, including such men as Euler and Lagrange, were at a loss to account for these irregularities. They suspected the existence of something foreign to Newton's law of gravitation until Laplace succeeded in showing the real cause in the appearance of the square of a small denominator, the smallness of which was due to the approximate commensurability of the two periodic times. We will now go further and study a case, not actually plan- etary, but of the secondary systems of the Earth and Jupiter, where, under changed circumstances, it plays an important part. Assume two periodic times so nearly commensurable that the coefficients of the terms (9) and (10) in the mean longitudes assume so great a magnitude that it puts in jeopardy the process of integration which depends on the smallness of the periodic perturbations and their relatively short periods. Preserving the previous symbols, and limiting the consideration to a seventh or eighth term of the perturbing function in (8), 28, we have .dn^ 3n l da l _3a 1 m 2 . dt'~ -Zadt'- -- dt 2 a 2 dt a 2 2 Further, the total differentation of (11), 28, with respect to t gives d*Ci __ dni d^ 2 __dn 2 dt* - dt H "*" W~ dt 2 ^'* 2 ' where ^ and k 2 are of the second order with respect to the dis- turbing masses. Neglecting these (and always with limitation to periodic terms of the perturbing function depending on A), we get (12) 246 THEORIES OF PLANETARY MOTIONS. and, therefore, since / = d ^ + a 2 C 2 , If we put (14) A (15) and neglecting ^ , which is likewise of the second order with Cut respect to the disturbing masses, equation (13) passes into (16) where p can always be taken positively, since, in the opposite case, only , therefore also V, needs to be increased by * . Equation (16) is not exact; in the second member, it lacks the other periodic terms and the terms of the second degree with respect to the disturbing masses. Only the terms depend- ing on V have been taken into account. If we take p as con- stant (it depends only on the elements and changes very slowly), equation (16) is that for the vibration of a pendulum in a plane. Assume, now, that for a certain instant V = 0. If is CLv then positive, V is positive and, by (16), , is negative. Ctb therefore decreases. If is negative, V is negative and dt dt T5- is positive and the absolute value of still decreases. Equation (16) therefore shows that the tendency of both plan- ets is to lessen the increase, positive or negative, of V. If we take the first case, that in which V = the velocity ~-=r is positive. Then while V increases, - - decreases, and V (tt . dt grows more and more slowly as it becomes larger. If does dt not become 0, or if its value is too large for T = 0, then it TEEMS OF LONG PEEIOD. 247 decreases only so long as V < * . From here V grows again with accelerated velocity from - to 2^, and so on. This case corresponds to the perfect circular pendulum and is found only among the heavenly bodies. If becomes for a definite angle V Q9 then V returns, CtL at first slowly and then more rapidly to 0, then it becomes nega- tive, reaching finally a value V Q . Then it turns back, and so on. This is the analogue of the oscillating pendulum. When ^ is negative, the same treatment gives the same two cases. dV By multiplying (16) by and integrating, we get (17) dV If c is positive and greater than p, V increases without limit. (ft But if c is negative, or at least less than p, then becomes ( for a definite angle F which is determined by the equation P But or, suppressing the terms depending on the disturbing masses, dV _ dt * whence (<*! Wj + Q 2 w 2 ) 2 = 2c + 2pcos V. If (?ij) and (n 2 ) are the values of n^ and n 2 corresponding to V = 0, then 248 THEORIES OF PLANETAEY MOTIONS. and (17) becomes and, if V oscillates, (a^) + a 2 (nz)Y < and, therefore, (18) [o 1 (w 1 )+a 2 (n a )]< When (18) is fulfilled, V oscillates about 0, and hence QJ d + 2 C 2 about e . Aside from these oscillations a x d + Q 2 C 2 remains constant and, if [nj and [n 2 ] are the mean values of the mean daily motions, it follows that (19) a i rn 1 ] + a a [n a ] = 0, [n 1 ]:[n a ] = o a : a 1B From this the following important conclusion results: Let there be two planets with approximately commensurable periodic times, when the sun only affects them. If the approxima- tion surpasses a limit defined by (18), the mutual attraction of the two planets will cause the approximate commensurability to become exact. This limit is not reached for Jupiter and Saturn. Relatively small changes in their major axes would, as Laplace has shown in the M6canique celeste, cause the limit to be passed, and we would then have the remarkable phenomenon, for the two largest planets of the solar system, that when one had revolved five times about the sun, the other would have completed exactly two revolutions. An oscillation through a small angle about a mean value is called a libration in astronomy. The duration is determined by the complete elliptic integral o = 4 f J It is evident that, in this case, the term of the perturbing function which depends oh V is no longer properly periodic, and that its influence is exerted in a totally different manner on EXACTNESS OF FORMULAS FOR VARIATION OF ELEMENTS. 249 the elements, than is that of the other periodic terms. Laplace made some studies of this case in which he showed that, taking the extreme excursion V as small, all the elements take part in the libration. The investigations which have been made in an exact com- mensur ability, are, of course, founded on assumptions which are themselves not exactly fulfilled. The terms of the perturbing function which do not depend on V and those of higher degree for the disturbing masses are neglected. Nevertheless, the results are of permanent value because they show that even a close approximation to commensurability, existing in two periods would bring with it no danger to our planetary system, but would rather bind the planets in still closer bonds, as is shown by the entire coincidence between the times of revolution and axial rotation of the moon. 37. THE EXACTNESS OF THE FORMULAS FOR THE VARIATION OF THE ELEMENTS. In the preceding sections the elements have been put in the form (1) The first terms in the second members are the secular values of the elements and the second the periodic values and are pro- portional to the disturbing masses. Each of the latter is of the frequently employed form (11) 27, in which the secular values are to be substituted for the elements a, e, . . . - In this sense (a), (e), . . . are functions of [a], [e], . . . and hence the ele- ments a, e, . . . are functions of [a] , [e]> ... Formulas (1) are only approximate and equations (8), 28, are not entirely satisfied by substitution of them. In conse- quence, certain differences remain which are now to receive attention. If (1) are differentiated from the above point of view, it follows, when subscripts are introduced, that 250 THEORIES OF PLANETARY MOTIONS. dj _ d[ttj] d(a } ) _7 t -t t -ji (2), , , . . t "1a[a l ] d* -SR] d* h r I 8[oJ ~f" 1 or- ~\~ji 1" There are corresponding equations for the derivatives of the other elements. Taking the second members of (2), we will first bring the expression into a very simple form. First = Q. Then if ^ a, d+ a 2 : 2 ctt sin is a term of - 1 arising from the periodic term of ^ , then by 29, we have for the corresponding term in (a), K sin The corresponding term in s??i[ n i]+ ^fFnC^s] is again cos ( ~\ K . \ QiCi+a 2 C 2 I. The expression (3) is then equal to the sec- sin v. / EXACTNESS OF FORMULAS FOB VARIATION OF ELEMENTS. 251 oncl member of the first equation (8), 28, if the secular values are used for the elements. This amounts to the following: Let the perturbing function R l be expressed in terms of the elements, or (4) R = R(a,,ei, . . . , a 2 , e. 2 , . . . ). If we substitute the values of these elements from (1), we get (5) B=JB(([o 1 ]+(a 1 )),....([a,] + (a 1 )) f ...). Now put, for the elements, only their secular values and indi- cate the result of this substitution by R. Then , (6). S = -B([a 1 ],[e 1 ],...[ a ],[e2],...). Then (3) becomes _ 2 /JXl dR, , V * 8[c, Considering further the remaining terms in the second mem- ber of equation (2), it is seen that they are all of the second order with respect to the disturbing masses, because each one is formed by two factors, each proportional to these masses. Since the first factor is purely periodic and the second purely secular, the product is purely periodic. If the sum of all these purely periodic terms be represented by { { a } \ , that is 6[a,] dt + the first of equations (2), becomes ,ox dt~ where, in the formation of *, it is to be remembered that a t ctt contains the time not only explicitly, but also implicitly, in so far as quantities appear in a x which were constants in (1) but are now functions of t If the partial derivatives of a, with respect to t+ in so far as t is explicit in a, are now denoted by ^- we get by the preceding section 254 THEORIES OF PLANETARY MOTIONS. (3) |p- By subtracting (2) from (8), it follows that Similarly The second members of these equations are functions of t and of the variables given in (1). The first members contain the parts of , , . . . which remain when the explicit deriva- clt d/t tives with respect to t are removed and they therefore, vary only for the quantities (1). From the Qn equations (4) and (5), the derivatives of the quantities (1) can be obtained. Yet, with close approximation the secular values of the elements may be substituted in second members of (4) and (5) for the elements themselves. The equation (4), for instance, may be written, by(l),37, Now ( ^ ) is of the first degree with reference to the disturb- ing masses, and hence - ~i is of the third degree since the derivatives of the former constants (1) are, by (4) and (5), of the second degree. Neglecting terms of the third degree, we get, therefore, and similarly (7) ^_^] =SJeiH _ M , etc. Equation (6) is in exactly the form taken by the derivative of a previous constant, namely [aj. It is L = 0, and hence VAEIATION OF CONSTANTS. 255 (8) ^ = U.f}-[{a.$]. Designating the constants of ( 1 ) in order by p^ , p 2 , . . . p 6H , we have _ = i , 2, ~ ~ ~* h dt dt ~ 8pi dt 8p 2 dt ' 8p 6w dt ' and (7) becomes W^t + ^f+"-+i^HWJ-[W],.* From (9) the derivatives of thep's can be obtained. It is not the purpose here to complete the determination but rather to make some general remarks on it. The coefficients 80J 80i] f the equations (9) contain only secular terms. The .... will, therefore, contain onlv secular terms if such at occur in [ { ^ f] , [ { e l \ ], . , . (We know from the preceding sec- tion that they do not occur in { \a^\ \, etc.). In accordance with the principles of 29, we will here also limit the consideration to the secular terms in the second mem- bers of (S) and (9), that is to those independent of C, and we will pass on later to the periodic members which will be treated by the methods used before. The next section will show that there are no secular terms in [ja^] and that with limitation to these, equation (S) becomes d[a 1 ]_ dt ' so that the previous constants [a] still remain constant. With this determined, the remaining equations (8) can be approxi- mately integrated by the previous methods. I will not enter into the minuter details, but will only note that K and K r also remain constant, if, to the eccentricities and inclinations other secular terms are added, whose arguments are formed by integ- ral combinations of G and F and whose cofficients contain the disturbing masses as factors. Also g' and f are increased by small fractions, which are likewise of the first power of the masses. 256 THEORIES OF PLANETARY MOTIONS. 39. THE INVARIABILITY OF THE MAJOR AXES. Poisson proved that the major axes are invariable even when, as in the preceding sections, account is taken of terms of the second degree with reference to the disturbing masses. (See Poisson's paper Sur les ingalit6s s&culaires des may ens mouvements des plan&tes. Journal de VEcole polytechnique, Tome VIII, 1809). In what follows, Laplace's demonstration is used, though in a modified form. After the preceding section, it remains only to be shown that in (1) HO.}] there are no terms independent of the mean longitudes. Bepre- senting, since there is now no chance for confusion, the [a] , [e], . . . by a, e, ... and also R by R, the quantity [jc^f] con- sists, by (10), 37, of terms of the three following forms: (2) <.)+sSr (,)++ 0C,8a, / T rj \ L. //v \ i L_ (p \ i i V -"- 1 / >r > V a 2J H^ O^ ^ \y2) "f* T* Each term in (2) is a product of two factors. R is here taken as developed in the usual manner, as in form (49), 24. The quantities (a), (e), . . . are of the same form. If the fac- tors are multiplied out term by term and use is made of the formula COS a COS /5 = -|( COS ( + /?) + COS (a ,3 ) ), it becomes clear that there are only two such terms that can give secular terms, both dependent on the same argument (3) ;^a 1 r 1 -fa 2 : 2 in which both integral numbers c^ and Q 2 must not be 0, at the same time, because only periodic terms appear in (a) , . . . Let the term in R, corresponding to this argument, be (4) k sin A -f- 1 cos /. THE INVARIABILITY OF THE MAJOR AXES. 257 Then the corresponding terms in (a), (e), ... are to be formed according to 29, and substituted in (2). Take first the term I of (2). The term in (c^) depending on A is by (4), 33 , . dR and in -^ 9C, ' (^(fccosA Zsin/l). By multiplication the purely periodic term is formed in (I). Take the sum II next. If the first term in the expression for (C) is disregarded, the products of the terms in II, ,which depend on the same argument /, all vanish. On the other hand the pro- 02r> duct of this term with ... ^ is <7- 1 C'- B2 3a 3 Zsin/) oAOiW, + a 2 2 ) 2 and, hence, also a periodic term. The treatment of III is somewhat more difficult. Since .R, is here differentiated with respect to the elements of the plan- et with subscript 2, the part of R^ involved is __ 1 __ Xi^ + yjh+ZizA V(*,-a*)'+^^'+&-*) T r * 3 r and since E! is also differentiated with respect to Cj and only such values of A can enter as are dependent on d, the terms in (a 2 ), (e 2 ), . . . depending on Ci must also be dropped. (a 2 ), (e 2 ), . . . arise from jR 2 and in this we have then to take into account only the part 17 258 THEORIES OF PLANETARY MOTIONS. Let 1 (5) .. X2 , , , 2 . , =^ 2 = 2(P sin A + Q eos A ), V (! acaX + (2/1 - 2/2) + (i a) (6) ^^yi& + ^2 S(.p sin A + g cos A). The factors mj and m 2 in .Rj and R 2 may be omitted as imma- terial in the present treatment. If now, Rf and Rf represent the terms dependent on the argument A, then, by (7) and (8), 36, (7) R ^ (8) B 2 " - (P- ?&p ) sin /. + ('- ^ g ) cos A . If Rf ^ 2 X } no secular terms will arise in .III. The proof of this is made exactly as for II, and in the formation of (a 2 ), (e 2 )j the consideration may be limited to the difference (9) ^ = B/-^ = -(^-^0 (psinx + gcosA). Turning next to the elements as they are contained in (7) and (9), aside from the coefficients P, Q, p, q, it appears that theVonly ones occurring in ^ and n 2 are c^ and a 2 . Under lim- 02j itation to these we have in - the term (10) < i^aa^pcos/ g sin A). The corresponding term in (a 2 ) is by (4), 33, The product of (10) and (11) gives only periodic terms. It is further to be observed that a 2 which occurs in (9) is contained in w 2 2 . And its product with di&zV , -2-2 ( p sm A -f- q cos /), f*2 &R the part of ^ ^ which arises from the term THE INVARIABILITY OF THE MAJOR AXES. 259 a 2 n 2 (p sin A -|- q cos A) of -R 1? likewise gives only periodic terms. Hence, only the following suppositions can be made: First, Rf is limited to (PsinA + $cos/) and F A retains its value (9). Second, R } * and V x are both limited to (p sin A -j- gcosA). . . The second supposition affords only periodic terms which follow exactly as for II. It remains to investigate only the first supposition, namely that in which only the term (12) remains in ^ and only the term ( 13) in J?2- The value of R 2 can be put in another form. For undis- turbed motion it is 'GH p lrp , -Tto \** J V- In this case it is permissible to put ^ = A 2 , since their differ- ence depends only on the disturbing masses. The above equa- tion then becomes ( 1 * vr 2 3 r-j 8 __ J^^ If we put (14) W=J then (15) w=(x 1 x 2 '- f*i 260 THEORIES OF PLANETARY MOTIONS. This integral of the perturbing function must be introduced into the expressions for the periodic terms. Kepresenting, as before, Poisson's expressions by (a, a), (a, e) , ... we have (16) dw (a,) = (a 2; e 2 ) --> + (a, , e 2 ) -^ oe 2 0%| (e 2 ) = (e 2 , a*) + (ea, 2)-F- Finally after an easy reduction, which is due to the differen- tiation of n. 2 (which contains a a and by (15) appears in x 2 , yj, z. 2 '), with respect to a. 2 we get (17) ( ^ = ( Introducing this into III and for brevity putting (18) g=r, C7Vj. the expression III becomes (19) ^ After the coordinates and component velocities have been introduced into this expression, it can, by formula (2), 12, be changed into dVdW dV dW dVdW dV dW a^^ e^raT^8y' %jay, dvdw d It only remains to prove that there is no secular term in (20). This can be done as follows. Omitting the factor p., we have (21) W = x,x 2 f and hence, (20) becomes THE INVARIABILITY OF THE MAJOR AXES. 261 ?? _L^Z -L-^-Z 4- '+ '+3' The quantity F from (18) must now be substituted in (22). We have ''-VC-gy+to-yy-K.-,).-^ and therefore, _ dR l _ dR l dx l dR : dy l dR l dz l Consequently dV dV_ dV dxj ~ dyj ~~~ dz,' and (22) reduces to dv , dv (24) ^ + W. yi ^* 1 ' This expression can now be so changed that the absence of secular terms is at once visible. V is, by (23), a homogeneous function of degree 2 in x ly y lt z l9 x zt y zy z zt and depends only on (a?, x 2 ), (^2/2), (2i z 2 ). Hence and and (24) becomes V = -^- contains no term independent of Ci . Hence *no such term can appear in (25), for after substituting for Fits value (18), we can at once write (25) as follows: (26) 262 THEORIES OF PLANETARY MOTIONS. Therewith the proof is completed that can contain no sec- ular term even when the second powers of the masses are taken into account, and we can now, with greater emphasis than in 30, state the proposition Excepting periodic perturbations the major axes remain con- stant. 40. THE FORM IN WHICH THE ELEMENTS AND COORDINATES APPEAR AS FUNCTIONS OF THE TIME. In the preceding sections, the elements a, h, Z, p, q, Z of the planets have been represented by aid of a definite number of angles, all of which are linear functions of the time. They are {1. The n secular values [C J , [C 2 ] , . . . [C n ] of the mean longitudes. 2. The n angles G lt G 2t ...G n . 3. The (nl) angles r M r 2 , . . . r n _ lt We will now disregard the manner in which these angles have been introduced ( T for instance, in the first approximation to elliptic motion and G and r through secular perturbations). We will also disregard the fact that the coefficients of the time in G and r are very much smaller than in C and we will regard the above angles as, so to speak, of equal value. Designating them in order by /o\ ill i it appears that any angle l\ is of the form / O \ - , . _ where a A and &A are constants. The x coordinates are expressed by an infinite series of the form (4) x-. where the numbers a u a 2 , . . . can take all values which satisfy the condition (5) cii-f a 2 -f . . . o 8M -i = 1. ELEMENTS, ETC., AS FUNCTIONS OF THE TIME. 263 Further (6) y = ^k sin (a^ + a 2 1 2 with the same coefficients k. Finally (7) z = 2A where, however, (8) The invariable plane is taken as the plane xy. The fc's and fc"s are functions of only (3n 1) constants, namely, fl. The n secular values of the mean distances [X ], [a 2 ] . . . , 2. The n constants of integration K^ , . . . K n , 3. The (n 1) constants of integration, KJ, . . . K n _ l f . The (3n 1) coefficients a x in the Z's are, by 31 and 32 likewise functions of these (3n 1) constants (9). The constants can, therefore, be expressed by the quantities a\ and we may even say that the coefficients k are functions of a A . The quan- tities &A. are constants additive to the Ts and do not effect the coefficients k. In the forms (4), (6) and (8) it is especially noteworthy that the time appears only in periodic form and that all the angles are linear functions of the time. It appears from this that each term, in so far as it is not constant, returns to the same value in equal intervals of time, and that its course is shown by a regular wavy line. Our planetary system must turn about its center of gravity with the greatest regularity, but periodic motions must be innumerable and their periods and magnitudes must show the greatest diversity. There is no initial condition in the past from which it has gradually devel- oped, nor is a final condition necessary towards which it tends. It is hardly necessary to warn the reader that these remarks are of force only when all other agents are excluded except gravitation. In fact there are others, but the duration of active observation is not more than enough to afford a slightest trace of them. It is not difficult to analyze their action, but 264 THEORIES OF PLANETARY MOTIONS. they are not included in the purpose of this work which is devo- ted solely to the study of motions resulting from gravitation. Yet, with this limitation, the doctrine of the eternal stability of the solar system is very far from standing on the foundation of certainty due to perfectly rigorous deduction. No pains have been spared in bringing clearly into view the terms omitted in the integration of the differential equations, and no one has succeeded in showing that the general form of the expressions means more than that it is an approximation good for so many thousands of years, and so near the truth that it represents very approximately nearly all observations, in so far as the constants of integration have been suitably determined. On the other hand there is a circumstance which continu- ally warns us against a too early celebration of success. It is the reappearance of that denominator which is summoned m "n by the integration of terms of the form (< + ) The most COS able astronomers and mathematicians have given it their care. Yet, notwithstanding the keenness of their investigations, inclu- ding as they do those of Laplace, given in 36, the case has not been completely analyzed. It is known, to be sure, that this denominator can only be great or infinite in our system when the coefficient of the respective term is very small, yet this does not at all dispose of the doubt which has recently led to the investigations of Professor Gylden. When the terms referred to are neglected in the differential equation this part is eliminated. Yet it is certain that the neg- lected terms will have a sensible influence in the course of thou- sands of years, and in any case, by their neglect the solution is not rigorous, and not good for all time. The way in which the matter stands at present may be stated as follows: The equilibrium of our planetary system has, by the admirable labors of the mathematicians, been shown to be true for periods of time which are very long, viewed from a human standpoint, but it has not been proven to exist forever. The forms (4) to (8) for the coordinates were reached as an approximate result from the theory of variation of elements ELEMENTS, ETC., AS FUNCTIONS OF THE TIME. 265 by means of certain circumstances, the smallness of the per- turbing masses, and the smallness of the eccentricities and inclinations. If this represents the facts, is it because it expresses a mathematical law, and not because it is merely an interpolation form? According to my idea, this question is of fundamental significance, for an affirmative answer to it would give a brilliant justification to the acute researches of Laplace and Lagrange. It has been made a matter of reproach to them that their methods of integration lacked mathematical rigor and it has caused a feeling of uncertainty with a regard to neg- lected terms, a feeling which may be fallacious, and which, in fact, has proven fallacious in one case, as shown in 26. Many mathematicians, led by the facts, have hesitated to admit the value of their results. But they undoubtedly felt, more strongly than .others, all scruples which could be raised, and if any im- provement had been possible, they would have made it. If we succed in proving in any other way the laws which they discov- ered by imperfect deduction, to them belongs the great service of having pursued, undisturbed by doubts, and of having finally solved the great problem of the motions of heavenly bodies submitted to Newton's law of gravitation. This other way so far as the anthor knows, is as yet hardly begun. Certain formal results have been obtained, but the question of convergence has not been taken up. Forms (4) to (8) are apparently quite general for they hold without the limitation to small masses relative to the controlling mass, and to small eccentricities and inclinations. The prepon- derance of the mass of one point does not guarantee the stability when the latter is taken from the general standpoint already defined. If stability means a peculiarity of the system by virtue of which all the distances of the bodies from each other are always finite (never becoming or infinite) this can be fulfilled, however large the masses may be, if the initial condi- tions have been suitably chosen. With this generalized concep- tion of the problem, it is plain that we cannot take refuge either in the method of absolute perturbations nor in that of variation of elements. Hence it seems to me that forms (4) and (8) 266 THEORIES OF PLANETARY MOTIONS. form a guiding line for further investigations, that the above mentioned theories are to be abandoned after they have been exhausted and that the problem of n bodies, in its general form, should be undertaken entirely independently without recourse to Kepler's laws or other intermediary as initial points. That astronomers have to-day reached this point is shown by the fact that the following offer of a prize, published in Acta Mathematica in 1885. The proposal is as follows: "Any system of material points subject to Newton's law being given, with the conditions that there shall be no collisions, to develop the coordinates of each point in an infinite series of terms composed of known functions of the time, which series shall converge uniformly for a period of unlimited duration. "That the solution of this problem, which must afford us in its course an insight into the most important of the phenomena of our system is not only possible, but is attainable with the analytical helps already existing, is shown by the statement of Lejeune-Dirichlet,* who, just before his death, confided to a mathematical friend that he had found a general method for the integration of the differential equations of mechanics, and that by the application of this method, he had succeeded in rigor- ously proving the stability of our system. Unfortunately we know nothing of Dirichlet's method except the indication that the small oscillations afforded him a foothold in the discovery. It may be taken as certain that it does not consist in difficult and complicated analysis, but in the introduction of a simple fundamental idea, the rediscovery of which may follow an earnest and persistent search. " If the problem offers difficulties which cannot be overcome for the time, the prize can be given for a work in which any other important problem in mechanics is completely solved in the manner above indicated." *Kummer, Gedachtnissrede auf Lejeune-Dirichlet. Abliandlungen der Konigl. Akademie der Wissenschaftenzu Berlin 1868, p. 35. GENERAL FORMULAS RELATING TO COEFFICIENTS. 267 41. SEVERAL GENERAL FORMULAS RELATING TO THE COEFFI- CIENTS IN THE DEVELOPMENT OF THE COORDINATES IN TRIGONOMETRICAL SERIES. In 36, 26 and 27, the coordinates are developed with ref- erence to the invariable plane (or rather a plane parallel to it and passing through the sun) in the following form y x = 2.K A (a 1 ,a 2 , . . . a 8n _,) sin (a 1 r A 2fc A (b 1? b 2 ... b 3rt _j S & A sin ft, in which it is to be noted that a. In the first and second formulas the S relates to all positive and negative values of the integral numbers cti, . . 'flsn-i which satisfy the condition (2) !* + ...+a 3w _ 1 =l. 6. In the third the numbers b are such that \ / 1 I Li I * I 3w 1 c. The -ST A (a 15 . . . ^ n _^} and also A? x (b t ,b2, . . . b 8 ,,_i) are constant coefficients. d. Each angle h is of the form (4) h = *J+ ?,. In order that the coefficients & A , may be single-valued we will assume (5) ** = .( b lf b 2 , . . . b 8H _,) = fc^A, b 3-i). Let it be assumed that equations (1) are not simply approx- imate, as was done in the preceding section, but that they are rigorous. Also let it be assumed that (1) are uncondition- ally convergent, and, therefore, that the sums of the absolute values of the coefficients are likewise convergent. The terms in (1) will naturally be arranged in the order of the integral numbers a and b. 268 THEORIES OF PLANETARY MOTIONS. The angles I are all linear functions of the time, and the same must be true of H and h. Let (7) where (8) 2V=a 1 a 1 (9) n = b^ The angles H may, from now on, be briefly designated as those of the first kind, the angles h as of the second kind. If each term in (1) is differentiated, trigonometrical series of the same form appear which we will assume are also conver- gent. Let this be the case however often they are differentiated. Then we get (10) a> A '= SJTxtfsin.H, y A ' = ^K^Ncos H, z*'=2>k^n cosh, or (11) OJ A ' = 20 A sinfl, y;/ = 20Aeos.H, **' = 2 A cos ft, if we put In the same way (12) x"=2, z" 2fc A w 2 sin h, etc. These series are to converge and to be unconditionally con- vergent. That is the sum of their absolute values, are to converge for each integral exponent p. From (1) it follows that (13) r A 2 = xj+yf + z, 2 = 2 2 K^ cos (HH'} where the accent means that -fiT A and J5T A ', etc., belong to any two terms of (1) and the double summation signifies that the summation is to be carried out for the integral numbers in each term. All the angles appearing in ( 13 ) are of the second kind, and it appears that r A may be developed in the form GENERAL FORMULAS RELATING TO COEFFICIENTS. 269 (14) 2k COSh. The same form (14) appears in the development of the squares of the distances of the planets from each other. One further assumption will be made and that is, that all positive and negative powers of the distances can be developed in trig- onometrical series of the form (14). The major axes of the orbits and the expressions (12), 28, representing h, I, p, q can also be developed in trigonometrical series of the forms (1) and (14). In the narrower sense, the expression stability is now to be defined as follows: 1. For the major axes, the constant term is decidedly larger than the sum of the coefficients of the periodic terms. 2. The sum of the coefficients in the trigonometrical series for the expressions (12), 28, is small The constant angles /?* which are contained in Z A replace (3n 1) constants of integration. The coefficients A. in (4) and the coefficients K and k in (1) contain besides only (3n 1) constants of integration. These are the secular values of the major axes of the previous paragraphs and the factors K and K' of 31. Altogether there are (Qn 1) constants of integra- tion or two less thaD is required in the complete solution with arbitrary choice of the system of coordinates. The expressions (1) are obtained on the assumption that the origin falls at the sun's center. If the origin is put at the cen- ter of gravity of the (n -f- 1) bodies, (the sun and n planets) and this is convenient in the investigations that follow, the new coordinates are linear functions of the old and the forms of equations (1) remain unchanged. In equations (1), let the sign S extend to all integral num- bers a or b. If the summation be extended to all the (n -f- 1) bodies, we may, to avoid confusion, use the symbol S. Then (15) Sm^x ' X = 0, flfmxtfA = 0, This gives, by (1), the relations (16) Sm x -fir*(a 1 ,a 2 ,...a 1Nt _ 1 ) = ^ 270 THEORIES OF PLANETARY MOTIONS. which must hold for each system of numbers a and b. Further (17) Sm The first member is constant. The periodic terms in the second member must therefore vanish and thus a new series of relations arises between the coefficients K, while the second member is reduced to a constant term, making {18) Sm A (^2/ A ' yiXi')=S2miKSN=S2miKiGi. Further, we get {19) SmifazS ZM')=$S^miKiki(N+n)Bin(H h), (20) Sm*(zM f x^)= $StttHiKiki(N+n)s(H h). The second members contain only terms of the first kind, hence no constant terms. They must, therefore, vanish. That is, (21) Sm x (2/x x' z x y\ ) = Sm x (z x #/ x x z x ' ) = 0, as must be the case when the invariable plane is the plane of ocy. As shown before the expressions (1) contain (6n 2) con- stants of integration, half of which are formed by the constant angles (3 contained in I and which do not appear in K, k and a. The other half which appear in K, k and , can be represented by {22) c 1? c 2 , . . . c 3M _ 1 . By Lagrange's theorem, (30), 10, the expressions fa a -\-Srn \ dx * dx * [a,,a ; .J- w - - . 0yx0yx ; 0yx0y/ da, daj daj 6a, _ da t daj da, d where a,-, , are any two of the above (6n 2) constants of inte- gration are constant. These may be developed by the method employed by Newcomb (Thgorie des perturbations de la lune qui sont dues a V action des plan&tes. Journal des Math&matiques pures et appliquges, 1871, and Sur un probldme de mgcanique Comptes rendus, 1872. GENERAL FORMULAS RELATING TO COEFFICIENTS. 271 If we introduce the notation dA6B dAdB (24) {Ay B\ = ^^--^--, and substitute (1) and (11) in 23, it follows that (25) \a it aj] = S2^m^{K, G'\ sin (H + kg f {h,h'\sm(hh')~\. If we put (a) a, = &, a,=ft, only the last term in (25) remains, since K, k and a are inde- pendent of p. By (24), we have here \H,H'\ =a i a j f q,a/, {h,h'\ = b.-b/ b/b/, and therefore <26) [A, ft] = SSSm A [(a,o/ a,a/)^(?' sin (HH 1 ) + (b) b/ b, b/) A?^' sin (^ V) ] . This is to be constant. All periodic terms destroy one another. There is no constant term. Hence (27) [fc,ft] = 0. (&) a< = /^, a ; . = c/. We have The remaining combinations { A, B\ in (25) vanish, hence (28) [/5,,cJ 272 THEORIES OF PLANETARY MOTIONS. H') Since [&,0J is constant, the terms multiplied by t must vanish and the above must reduce to the constant term. Hence (29) (c) Here we have (30) [c< ,cJ ,G' \G'\K } N\)cos(HH f ) (k\n,g'\g'\k,n r l)ooB(hh r y] \N,N'} sin(H-H') + kg'{n,n'\ sm(h-h')']. The periodic terms and all terms multiplied by t and t* must destroy one another. A really constant term does not occur in (30). He^ce (31) [c,,c,]=0. The vanishing factor in the term proportional to t is cos *-*' When the values of 2V and n given in (8) and (9) and equa- tion (29) are used, the above equation takes the form (32) GENERAL FORMULAS RELATING TO COEFFICIENTS. 273 Equation (32) can be used for the deduction of a noteworthy theorem. The quantities c are not more closely determined than that they appear in the constants K, k, a. If desired the (3n 1) quantities may be considered as the c's. But any (3n 1) independent functions of K, k, a may be taken for the c's and it is convenient, in order to make (29) as simple as possible, to put (33) c^S^m^a^N +%<]?, C by the transformation formulas The transformation coefficients can be expressed in terms of the three angles /, ?, d t as follows: a : = cos / sin 5, &! = sin / cos

cos #, GI sin / sin

cos ^ + (?/) cos ^ -\-z sin ^ sin 5, = (a?) cos {p cos 5 (T/) sin

, /?,] =j 0, and, finally, ( 44 ) \y , d] = ^m x ( ajy' ya/ ) sin 5 = S 2 m x ^^T' sin (5 . If the motion is referred to a stationary system of coordi- nates, the coordinates of the center of gravity must be added. These are, by (9), 6, linear functions of the time. It appears, therefore, that (45) [o,oj = [>,&,] = [<,,,]= - All of Lagrange's combinations have now been formed. The system of (6w+6) constants of integration can easily be changed into a cannonical one, but we will not occupy ourselves with it. 276 THEOBIES OF PLANETARY MOTIONS. The fundamental idea in the present problem is, therefore, the following: Take the system dp, dH dq t dH f . - ~ in which H does not explicitly contain the time. Change P^ # by a canonical transformation into the new variables Pi , ft and the new system is ,47) f_ _ _ d* " 8ft' dt ~ dP t ' H is to be expressed as a function of the new variables. If we can now succeed in selecting such a canonical trans- formation, that H contains only half of the new variables, OT7 say Pt the problem is solved. For then 7 - = 0, and, therefore, utyi P { = a constant. At the same time the would be constant dt and the ft's would be linear functions 61 the time. In our case the present P t is the former c,- and the ft the l t . These are the results to which we are led by the assumption that form ( 1 ) of the solutions obtained by induction is a math- ematical law. If they do not suffice actually to give a solu- tion, they afford a new view of Celestial Mechanics. A few brief remarks follow. The coordinates have been represented as trigonometrical functions of the (3n 1) angles, Z 1? 1 2 . . . tgn i. These can be replaced by any (3n 1) others as 7 ' 7 ' 7 *1 *S | *ta H which are formed from the preceding by linear substitutions, with integral coefficients. For example provided the substitution determinant = 1. For then the solu- tions of (48) become (49) / X = b, , A Z/ + 6 2 , A ? 3 ' + . . . + &,'_ A ? 3 n-/, GENEKAL FORMULAS RELATING TO COEFFICIENTS. 277 containing also integral coefficients. If H is a linear function of the Z's with integral coefficients, then, by (49), it is changed into a similar function H' of the Z"s and vice versa. If the problem of (n + 1) bodies is limited to the distances only, angles of the second kind, h, appear and for these the sum of the integral numbers vanishes. Hence, these angles can also be written so that they depend only on the (3n 2) angles a 0,03 *i)...(^-i o. Any angle of the first kind can be changed into Therefore: If ( n -f- 1 ) bodies are moving about their center of gravity, their distances ivill be trigonometrical functions of (3n 2) angles, all of which are linear functions of the time. In order to get the coor- dinates, another angle of the same form must be added. For n = 1, this result is completely verified. The distance of the planet from the sun is a trigonometrical function of the mean anomaly. In order to get the coordinates x and y (z here = 0), another angle is needed which is here constant and is equal to the longitude of perihelion. For n = 2, Lindstedt has formally deduced the same result, leaving questions of convergence aside, from Lagrange's differ- ential equations for three bodies, (38) and (45), 7. (See his Sur la determination des distances dans le probldme des trois corps. Annales de V6cole normale, 1884). If the coefficients a of the time t in are incommensurable, the original coordinates never reappear. If they have a greatest common measure C, they are integral multiples of c, and so are N and n. The coordinates can then be developed in trigonometrical functions of a single angle ct, 278 THEORIES OF PLANETARY MOTIONS. and after the time T = , the system returns to its original position, so that the motion is purely periodic, exactly as with two bodies. The special cases of the problem of three bodies have been given in 8. Liouville has shown (Memoire sur un cas particu- lier du probl&me destrois corps, Journal des Math6matiques pure et applique, 1842, page 110 and 1856 page 248 ) that the special case in which the three bodies remain constantly in a straight line and describe similar ellipses in space, is unstable in so far as a very small change in the initial position, such as would fol- low from the smallest thrust or force from without, would, in time cause the three lines joining the bodies to become fully separated. From what precedes this would be expected. For if, in a special case, the 's are commensurable, they cease to be so by a very small departure from the original position. The ratios of the periods are, then, not expressible in whole num- bers and therefore the special condition, that the three points shall lie in a straight line, must finally be lost. If the a's are incommensurable, " the motion is not purely periodic. Every angle I has, to be sure, its period, but these periods do not agree, and consequently the motions presented to the eye are very complicated, such for example, as are exhib- ited by the paths of the components of many of the multiple stars. 42. BRIEF HISTORICAL EEVIEW OF THE THEORIES OF PERTURBATIONS. The history of the theories of perturbations can be carried back to Newton, who, in the third book of his Principia explains the chief of irregularities of the moon's motions. The next advance was in 1747 when Clairaut and d'Alembert presented to the French Academy of Sciences two memoirs on the prob- lem of three bodies, in which they gave methods of integrating the differential equations when the central force is the control- ling one. They developed the fundamental parts of the theory of absolute perturbations and Clairaut soon had an opportunity HISTORICAL REVIEW OF THEORIES OF PERTURBATIONS. 279 to give a brilliant test to his theory. Halley's comet had been seen near perihelion in 1531, 1607 and 1682, and it was settled that its period was about 75 years. Its return was expected, therefore, at the latest, about the end of 1758. Glair aut com- puted its perturbations due to the known planets and informed the French Academy that its return would be delayed until about the middle of April 1759. This announcement was made on November 14, 1758. He added that the small terms neg- lected in his calculation of the retardation, which he put at 690 days, might make a difference of a month. In fact, the re-ap- pearance was on March 12, 1759, within the limits which he had set, and he would have been more exact had he known Saturn's mass more closely. The founder of the theory depending on the method of the variation of constants was Leonhard Euler. In three memoirs, which were crowned by the French Academy in 1748, 1752 and 1756, he developed by regular steps the method of the varia- tions of constants. . His formulas were not entirely rigorous for among the six constants there were some which he did not vary. Yet for the case of two planets, he succeeded in showing the existence of secular variations of eccentricities, inclinations, perihelia, and nodes. Unfortunately, numerous errors crept into the numerical developments, completely vitiating the results and possibly causing Euler to devote himself to other investigations. Besides the foundation of the general theory, he gave, in his papers, the fundamental propositions for the development of the perturbing function in a trigonometrical series whose arguments are the mean anomalies. Euler' s ideas, which laid the foundations, were developed to their limits by Lagrange, whose services to astronomy alone are enough to give him an indestructible memorial. His first work appeared in the Melanges de la Soci6t6 de Turin, Tome III, 1766. He here treated the eccentricity and longitnde of perihelion, in the expressions for the radius vector, as variable, and in this sense differentiated totally with respect to the time. The part of the derivatives which depends on the variation of these two constants, he equated to zero, differentiated again, then intro- 280 THEOEIES OF PLANETAEY MOTIONS. duced the perturbing function, and thus got two equations for determining the derivatives of these two elements. This pro- cess is undoubtedly inexact, because it disregards the variations of the major axis and of the epoch of perihelion. Yet he formed, in the case of two planets, the correct final equation for for determining the secular periods, which Euler had given erroneously. He also obtained correct formulas for the inclina- tions and nodes, so that his numerical results in an application to Jupiter and Saturn were nearly correct. On the other hand in the determination of the mean longitudes of the two planets, two terms proportional to the squares of the times were inclu- ded which are entirely wrong and which were due to the inac- curacies which the great mathematician permitted in this first attempt in this uncultivated field of science. In 1773, Laplace offered to the French Academy of Sciences his first work on the theory of the planetary system, and in 1776 this was printed in the M&moires des savants Strangers. He determined with entire accuracy the formulas for the deriva- tives of the elements, but these did not possess their present elegant form. While he was carrying on the numerical computations for Jupiter and Saturn, he found to his great surprise that the sec- ular terms in the values of the derivatives of the major axes, mutually vanished. He succeeded in proving that this phenom- enon was not a play of chance, but founded in the formulas when their development was carried out to terms of the second degree in the eccentricities and inclinations. The theory of secular variations of the elements originated fundamentally in the circumstance that the usual process of integration by series brought to light terms proportional to the time. Every device was employed by Euler, Lagrange, and Laplace to get rid of these terms. Any one who studies their works, must feel that aside from mathematical considerations, a sort of metaphysical idea directed these earliest explorations which bridged the gap between entire ignorance and complete clearness. It soon appeared that the theory of absolute per- turbations was not the best, and the variation of constants grad- HISTORICAL REVIEW OF THEORIES OF PERTURBATIONS. 281 nally displaced it, though curiously enough, not in its purity, but wonderfully mixed with the first theory. It was seen that the secular terms must eventually cause great changes in the ele- ments, but instead of determining these directly, an indirect course was taken. This appears plainly in Laplace (Mgcanique celeste, first volume), where, in the expressions for the absolute perturbations, the terms proportional to the time are forcibly removed, and then after introducing the complete theory of variation of constants, the natural path of integration is taken, retaining only the secular terms of the perturbing function. Another advance was made by Lagrange in 1774, in a paper which appeared in the Annales of the French Academy, where he introduced the elements h and I in place of the eccentrici- ties and perihelia and p and q in place of the inclinations and nodes. The differential equations became linear by this trans- formation. In 1780, in a paper on the perturbations of comets, which was crowned by the Berlin Academy and published in their memoirs in 1782, he treated the expressions for elliptic coordinates for the first time in their most general form, that is, not as functions of the time alone, but also of the six constants of integration. Here for the first time the theory of the variation of constants was demonstrated with entire rigor and purity, and the formulas for its use simplified. When Lagrange applied his general theory to the expression for the major axes, he recognized at once that in only periodic terms at appear, and thus, with the stroke of a pen, says Jacobi, the proposition was proven, that the influence of the perturbations on the major axes is proportional to the disturbing masses only, and is expressible by periodic terms so long as only the first powers of the disturbing masses are considered. In 1809, Poisson in his paper Sur les in&galites seculaires des moyens mouvements des plandtes (Journal de V dcole polyt. I) showed that this proposition still held if the second powers of the masses were included. He obtained this result by a peculiar use of the theorem of kinetic energy. Laplace gave in Mecanique celeste another and more complete demonstration which has 282 THEORIES OF PLANETABY MOTIONS. been followed here. Lagrange also tried to provide a proof,, but, unfortunately his process suffered from an error in sign, which Serret has pointed out. Endeavors have been made to extend this demonstration to the third powers of the disturbing masses. Mathieu, (Memoires sur les inegalites seculaires des grands axes des orbites des planetes, Crelle's Journal, 1875), carried out investigations from which he concluded that the secular terms fail even when the third pow- ers are included. Later Tisserand, Haretu, Gasparis, GyldSn, and others have gone over the same ground with a result some- times favorable, sometimes unfavorable, so that the question is not finally settled. After the formulas for the derivatives of the elements had been developed, their notable simplicity in the case of the axes led to a search for equally simple expressions for the other elements. Laplace, by skillful changes succeeded in producing the fundamental formulas, as given in (8), 28. The discovery was published in 1808. By a remarkable coincidence, Lagrange published the same formulas at the same session of the Bureau des longitudes. He had reached them in an entirely different way, namely, by the introduction of the expressions (a x , a M ) and he at the same time discovered the peculiar origin of these im- portant formulas. With these the theory won a wonderful sim- plicity and elegance, as compared with the earlier forms, and astronomers were enabled to replace the derivatives of the per- turbing function with respect to coordinates, by those with respect to the elements. The denominator which appears in the periodic terms on integration is a very notable phenomenon, to which reference has already be*en made. Laplace deserved the credit of having first called attention to it, and of having investigated the remarkable increase of an otherwise small periodic term, caused by this denominator when small. He applied the theory to Jupiter and jSaturn and thus explained a very enigmatical phe- nomenon. Halley had compared ancient observations with those of the middle ages, and had drawn the conclusion that the motion of Saturn about the sun had been retarded, that of Jupi- HISTORICAL REVIEW OF THEORIES OF PERTURBATIONS. 283 ter accelerated. On the other hand, Lambert had found that in modern times the opposite is the case. Laplace saw that the explanation of this apparent contradiction of the law of the invariability of motions was to be found in a term of the per- turbing function, neglected up to that time because of its min- uteness, and he fixed the period of this perturbation at 930 years. These purturbations of long periods, as they are called, are the source of many others, and they especially affect the secular elements. If the periods are yet more nearly commensurable, the period continually becomes longer and the coefficient continually in- creases. Finally a phenomenon occurs which actually takes place in another form in our system. The two planets affect each other in such a way that their periods become exactly com- mensurable, and a close bond is formed between them, which continues forever unbroken. It lacked but little, as Laplace showed, of occurring in the case of Jupiter and Saturn. To assure it, it would only be necesssary to decrease Saturn's- mean distance by r iU and to increase Jupiter's by T3 inr . After the theory had been thoroughly grounded, and our knowledge had passed from the darkness of its origin into the stadium of perfect clearness, the effort was to bring the analyt- ical and numerical developments to the highest degree of sharp- ness and exactness. When the perturbing function is devel- oped to higher powers of the eccentricities and inclinations, the labor involved rapidly increases. This induced the most emi- nent mathematicians to occupy themselves in improving the solution. Cauchy published in the Comptes rendtts a series of many articles in which he advanced entirely new ideas, and used his celebrated theory of residuals. He expressed the coor- dinates in terms of the eccentric anomalies and developed the perturbing function in trigonometrical functions of them. It was then easy, by Bessel's functions, to pass from the eccentric to the mean anomalies. His formulas are of especial elegance and simplicity when the term, the coefficients of which are to be determined, is of high order. His labors in this field do not appear to have found much appreciation among other astrono- 284 THEORIES OF PLANETARY MOTIONS. mers although he showed, in a controversy with Leverrier, that his results are of practical value. When it is only the question of the numerical value of a coefficient, a double integration will answer, as has been shown. It it noteworthy that Liouville (Note sur le cacul des inegalites peri- odiques du mouvement des planetes. Journal des Mathematiques pure et appliquees, 1836, page 197), offered a process by which this double integral can be reduced to a simple one with a high degree of exactness. Leverrier, at a later time made a searching investigation with the numerical and also analytical development of the perturbing function, and he reached a remarkable degree of accuracy. In his Recherches astronomiques, he calculated all the coefficients up to and including the seventh degree, and performed a labor which only he can judge who has once undertaken such computations. Besides the astronomers and mathematicians already named, Bessel, Lubbock, Encke, Hansen, Gylden, Newcomb and others have worked at the development of the perturbing function. Hansen, who pre- fers his own method, believes that, when the eccentricity and inclination of the disturbed planet are relatively great, the most convenient process is to develop the perturbing function with respect to the eccentric anomaly of this planet and the mean anomaly of the disturbing planet. Laplace and Lagrange developed the secular values of the elements with limitation to the second degree of the eccentrici- ties and inclinations, but it was left to Leverrier, so far as I know, to undertake the estimate of the influence of the neg- lected terms. His investigations were continued by Lehmann, who died before the computation of the numerical work. Lev- errier and Lehmann concluded that the influence of the neg- lected terms is greater than had been assumed, and that possi- bly great improvements in the secular periods can be made as they were determined by the roots of the secular equations. The theories of perturbations afford much opportunity for changes and transformations, and this is employed to attack the problem in new ways. Hansen, for instance, introduced a mov- ing system of coordinates in order by its use to refer the mo- HISTORICAL BEVIEW OF THEOEIES OF PERTURBATIONS. 285 tions of the planet about the sun to one plane. He imagined each point of the orbit connected with the sun and thus formed a very flat cone with the sun at the apex. On this cone rolls a plane without slipping, so that at each instant a point in the plane coincides with a point in the orbit. In this manner the orbit is described in one plane. Two differential equations of the second order determine this plane, and three of the first order determine its rotation. In addition, he uses two symbols t and T, for the time, the one direct in the elliptic coordinates, the other indirect in the perturbations, and he develops certain views in which he varies the latter term T. Since his investi- gations relate to the moon, minor planets and comets, for which they are of great importance, they will not be noticed further, though they are no't without mathematical interest. The same is true of Delaunay's investigations. His process is to fix the attention on a single term, and especially the most important of the developed perturbing function, and to integrate the differential equations thus obtained. This is possible with mathematical rigor. With the aid of these integrals the differ- ential equations are changed and another prominent term is considered. This is done repeatedly until the remainder is so small that it may be expeditiously treated by the usual methods. More recently Glyde"n has attacked the planetary problem in a different way. In order to get the special perturbations, he divides the orbit into parts and develops the purturbations for each part separately, then introduces elliptic functions into the elliptic elements. The latter he favors especially because, with their help, the integrations converge more rapidly. He has also introduces a peculiar conception, that of the intermediate orbit, by which he means an orbit from which the planet never widely departs. In so far there is nothing new in this conception; it is already introduced in 33. It is to be noted, however, that he does not get his intermediate orbit from Kepler's ellipse but reaches it directly. The reader can find a part of his prin- ciples in Die intermediare Bahn des Mondes (Acta Math. VIII), and in a paper by Andoyer, Contribution a la thorie des orbites intermediare s, (Annales de la faculte des sciences de Toulouse, 286 THEOEIES OF PLANETARY MOTIONS. 1887, I). He has lately undertaken studies on the convergence of the infinite series used in astronomy, which have, however- so far led to no satisfactory conclusion. One of the chief aims of mathematical effort of the present, time consists in the investigation of the properties of functions denned by differential equatipns. A glance at the immense number of investigations, which the theory of linear differen- tial equations alone has called out in very recent times, and is daily calling out, will give a clear idea of the great magnitude of the problems here involved, and will afford a conception of the difficulties which lie in the way of a complete solution of the problem of three bodies in its most general form. The ques- tion here is, certainly, concerning real variables only, and this circumstance simplifies the problem greatly. What interests the astronomers is not the investigation of all the properties of the analytical expressions, in finding their nodes, acnodes, cusps, etc., but the proof that such points do not exist when the time is made the independent variable and extends over a real path uninterruptedly from oo to -j- GO . The question, therefore, with this limitation to one branch, or rather to one line of a function, relates to the development of the coordinates in con- verging series, and these must converge without fail. It is easy to see that this can be done without embracing the great reverse problem of differential equations, which affords us already so many noble results. There must be a direct road, in* the above sense, to the solution of the problem of many bodies, a path which will lead to correct answers to the questions which, in spite of all investigations, still remain unanswered. Purely theoretical discussions are not sufficient for the prac- tical aims of astronomy. The numerical computations, also, which the author of this book has not undertaken, but has forced entirely in the 'background in order to preserve the ana- lytical character of the work, require great skill and greater patience. The calculations take on, in fact, almost unmanagea- ble dimensions when, as Leverrier has done, in the Recherches astronomiques (Annales de V Observatoire de Paris, Memoires), all terms are taken into account which change the geocentric place HISTOEICAL REVIEW OF THEORIES OF PERTURBATIONS. 287 -of the planet by the tenth of a second. The comparison of the computed places with the observed afford the best touchstone for the correctness of the theory, and it has very often led to the discovery of neglected but influential terms. It also affords a means of continuously correcting the masses and elements of the planets. When Leverrier had completed his comparison of the com- puted and observed places of the planets, the result was found to be very satisfactory. The differences fell usually within the limits of error of observation and in only a few small mat- ters was further discussion necessary. For instance, in the case of Mercury's perihelion he found that observation gave a some- what greater movement than computation. He concluded that between Mercury and the Sun there was one planet, or perhaps many, which had so far escaped observation, and which would cause the acceleration of Mercury's perihelion. This intra-mer- curial planet, to which the name Vulcan had already been given, has not yet been observed beyond a possibility of doubt, though it has been actively sought. Small, dark points have been seen, several times, to cross the sun's disk but it appears probable that they were either sun-spots or an error was involved in the observation. The planetary tables constructed on Leverrier's formulas are to-day generally recognized, except that for Uranus and Nep- tune Newcomb's are better. Of course, Newton's law of gravitation must submit to the test of observation, out of which it was originally created. Whether gravitation is a true prima causa or whether it results from other forces, either impulses, for instance, or electric, attraction, cannot now be decided. Possibly Newton's law is only approximate, though remarkably close. Possibly other forces are at work in celestial space, such as the resistance of the ether which, according to physicists at present, not only occupies stellar space but interpenetrates* matter. Encke adopted the idea of such a resistance as a result of his investi- gations of the orbit of the comet named after him. 288 THEORIES OF PLANETARY MOTIONS. Certainly the law of gravitation is, as already said, very nearly fulfilled, so nearly that mathematicians must assume it rigorously true and investigate its consequences. How closely it agrees with observation was shown in the most brilliant way by the discovery of Neptune, forty or fifty years ago. Sir W. Hersch el discovered Uranus in 1781. It was then continuously observed and its elements computed. The neces- sary calculations were made by Bouvard according to Laplace's formulas. It appeared, however, that in the course of time the new planet departed in a regular manner from its computed place, and that the amount of departure increased with the time. It also appeared that the planet had been observed several times as a fixed star and that it was impossible satisfactorily to repre- sent these old observations. Bouvard himself had concluded as early as 1821 ( Tables astronomiques, les mouvements d* Uranus annoncent ^existence d'une plan&te perturbatrice ext&rieure), that a planet exists outside of Uranus. It was beyond Uranus, because otherwise it would appreciably affect Saturn. It grad- ually became a familiar idea that the mass and elements of the new planet could be computed from the observed purturbations, making the reverse of the ordinary problem. If the smallness of the perturbations be considered, they amounted to only a few minutes, and the complicated analytical form in which the ele- ments appear in perturbing function, it will be evident that the problem was one of great refinement. Bessel interested him- self actively in it and his letters show that he was about to take it up when death put an end to his activity. It was solved, almost simultaneously, by two other astronomers, Leverrier and Adams. The first published his preliminary investigations in the Comptes rendus and explained his research in the greatest detail in Recherches sur les mouvements de la plan&te Herschel, ( Connaissance des temps pour 1849) after the new planet had been found in 1846 by Galle within a degree of the place he had predicted. The study of this work affords great pleas- 'ure because in it are combined great discretion, painful accur- acy in analytical and numerical developments, and especially clearness in the development of the line of thought. The first HISTORICAL REVIEW OF THEORIES OF PERTURBATIONS. 289 question was whether any terms of the development had been neglected which could cause an appreciable error. In fact, there were several suspected, but they were shown to be unim- portant. Then the inquiry was, could the errors be eliminated by small changes in the elements selected and a negative result was obtained. These preparatory studies put beyond doubt the perturbing influence of an unknown cause, and, after the rejec- tion of other possibilities, the perturbations were traced to a planet more distant than Uranus from the Sun. The major axis, the epoch of perihelion, eccentricity, longitude of perihelion and the mass of the new planet were introduced as the five unknown quantities, while the inclination with reference to Uranus was neglected because of the small perturbations in latitude. Each observation gave an equation of condition between five unknowns, and since the number of observations was very large, they were divided into groups. The major axis appeared in a very com- plicated form, hence Leverrier assumed, using Bode's law as a single foothold, that the new planet was twice as far from the sun as Uranus. The four other unknowns were then determined. Finally, he found that a decrease in the adopted value of the major axis decreased the unavoidable errors. He then determined the amount of the decrease and deduced the final value of the elements* Undoubtedly Leverrier would have reached much closer results if he had not been misled by Bode's law and adopted at the beginning, a value much too great for the major axis of the unknown planet. It is evident that an increase in distance required an increase in the mass of the planet and as the unknown planet was nearly in conjunction with Uranus the error in distance had its greatest effect. It is due to this that the mass obtained by him was nearly double the real value and hence all the elements were entirely wrong except the mean lon- gitude, which, in this case, was the most important, because the one fixing the place in the sky. The same was true of the elements obtained by Adams, who published his investigations in An Explanation of the Observed Irregularities in the motion of Uranus, (Memoirs of 19 290 THEORIES OF PLANETARY MOTIONS. the Royal Society of London, 1847). As he found a still greater major axis his mass was even larger than Leverrier's. Since his results were communicated to Airy some months before Lever- rier's investigations appeared, the planet might have been dis- covered earlier, had not Airy, who had some doubts as to the correctness of Adams's results, decided on making a catalogue of the stars in that part of the sky. It is one of the best possible proofs of Newton's law of .gravitation that it led to a discovery which might otherwise have been delayed many years. 43. NOTES ON THE TABLES. The unit of distance is half the major axis of the earth's orbit, or the mean distance of the earth from the sun. This is not the semi-axis which we have called the secular value, but the one which follows from the observed periodic time and .Kepler's third law. The secular value is obtained by applying the correction given in 32. Of the masses, those of Mercury and Yenus are the most uncertain. As these planets are not known to have satellites, their masses can be obtained only by ihe perturbations they produce on comets which pass near them and on the earth. It is quite possible that the mass of Mercury, especially, requires a very appreciable correction. The secular variations of the longitude of perihelion, of the ascending node, and of the inclination are not referred to a fixed plane, but to the moving ecliptic on which the vernal equi- nox is selected as the initial point for longitudes. The vernal equinox retreats yearly on account of precession, a distance of 50" . 2113 but is free from nutation. If these variations are referred to the ecliptic of 1850, then during the current thou- sand years all perihelia advance, except that of Venus, while all the nodes retrograde, except those of Jupiter and Uranus. The fifth table gives the roots of the equations (21) and (42) 31; the sidereal year is taken as the unit of time. The amounts by which G and /' annually increase aod decrease is therefore, measured in seconds. The absolute maximum of the NOTES ON THE TABLES. 291 increase is /v It' T is the time taken for the proper angle A = nt~\-