GENERAL INTEGRALS OP PLANETARY MOTION. 15 



By giving j all values in succession from 1 to 3?i, we shall have 3n equations to 



determine the variations of Cj, Co, Cg,,, from which the variations of a^, ag, 



«.„ are to he obtained by the Zn equations (21). But, for our present pur- 

 poses, it will be more convenient to consider the c's as the fundamental elements, 



and to consider a^, a^, a^^ to be replaced by Cj, c^, c^n ^i-^ the original 



equations. 



The second class of differential equations (16) will, by (19), be represented by 



K, ^0 -i^r K' ^^) -i, + etc. = ^-^^^ i ^^ ^, + ^ ^a- + m^ Sa, \ 



Substituting for the coefficients in the first member their values (23), we shall, 

 have 3)1 equations represented by 



Putting k successively equal to 1, 2 Sn, we shall have 3re equations of this 



form. Let us multiply the first of these equations by —^ the second by ^ , the itJi. 



by^, and so on to the Snth, and add all the products, noticing that the theory of 

 functional determinants gives 



2 ^^ = + 1 orO 



i = 1 Otti dCj. 



according as li is or is not equal to j. Then, by putting 



^_*r Sd^^kiQJ^ ,Q\iQni ,Q%QSi\ ^Q' 



cU ---^'^ 



we shall have 



dk 



dt 



dh _ 



di - — ^2 (24) 



dhn ^/ 



These Zn equations, combined with the 3h equations (23), will give, by simple 

 integration by quadratures, the perturbation of the 6w constants, which, being 

 substituted in the original equations (12), will give values of the variables which 

 satisfy the original differential equations to terms one order higher than they were 

 satisfied by (12) originally. 



It will be observed that if our functions of the time and 6n arbitrary constants, 

 which we have represented by ^j, ni, and ^,-, possessed the property that a function 

 ^0 of ^, r„ and ^ could be found such that for all values of i 



