GENERAL INTEGRALS OF PLANETARY MOTION. 9 



We see that, supposing Xq to represent the co-ordinates of the sun or other cen- 

 tral body, ^1 is equal to the co-ordinate of the first planet, which may be any one at 

 pleasure, relatively to the sun, multiplied by a function of the masses, while ti is 

 equal to the co-ordinate of the second planet relatively to the centre of gravity of 

 the sun and first planet multiplied by another function of the masses, and so on. 

 These functions ^j, when divided by the functions of the masses just alluded to, will 

 difi^er from the co-ordinates of the several planets relatively to the sun only by 

 quantities of the order of magnitude of the masses of the planets divided by that 

 of the sun. 



In what precedes we have considered only the co-ordinates x-^. Of course the 

 other co-ordinates are to be subjected to the same transformation. If we represent 

 by )7 and ^ the corresponding functions of y and z, and if in the expressions for ^, y;, 

 and ^ we substitute for x, y, and z, the expressions (3), those quantities will them- 

 selves reduce to expressions of this same form. 



§ 3. Approximation to the Required Solutions by the Variations of the Arhitrary 

 Constants in a First Approximate Solution. 



By the transformation in question we have for the determination of the relative 

 motion of the n-\-l bodies, 3m differential equations, of the canonical form 



d?c,i d£l d^Yii dn dli^i dQ. , , , . 



'df~~d^' 'df~^i' 'W~~di; ^ ^ 



Let us now suppose that we have found approximate solutions of these equa- 

 tions in the form (3), the quantities cc, y, z being there replaced by ^j, 57^, and ^j, 

 that is, solutions which possess the property that, if, on the one hand, each expres- 

 sion is twice diff"erentiated, and if, on the other hand, the values (3) are substi- 

 tuted in the second members of (11), the two expressions shall difi'er only by terms 

 multiplied by small numerical coefiicients. We have to show that when we make 

 a further approximation to quantities ot the first order relative to these coefficients, 

 the solution will still admit of being expressed in the form (3). To do this we 

 shall make the further approximation by the method of the variation of arbitrary 

 constants, remarking, however, that the usual formulae of this method cannot be 

 applied, because they presuppose that the first approximation is a rigorous solution 

 of an approximate dynamical problem, while, in the present case, we are not enti- 

 tled to assume that our approximate solution (3) possesses this quality ; in other 

 words, we are not entitled to assume that any function Hq of the quantities t,-, yj, and 

 ^, can be formed, such that we shall find the Sn equations of the form 



^ _ dOo 



rigorously and identically satisfied by the approximate expressions, both with 

 respect to the time, and the 6ra constants which the solution contains. Conse- 

 quently, we cannot assume the existence of a perturbative function, and must 

 employ other expressions in place of the derivatives of that function. 



We set out, then, with the three sets of equations, having n in each set 



2 November, 1874. 



