18 GENERAL INTEGRALS OF PLANETARY MOTION. 



according as i and fi represent the same or different indices. But we have akeady 

 found that the expression (?; Cj) vanishes whenever i is different from j, and reduces 

 to unity when those indices are equal. The equations we are considering thus 

 become 



[h c,] = 1, (28) 



while all other combinations [l^, Cj], [?,-, Ij] and [c;, Cj] vanish. 



Let us now return to the integral equations (26), and first form the combination 



The conditions (28) therefore give 



[$„^,]=0 

 and (29) 



the first' equation applying whenever j is different from i, the second when they are 

 the same. 



Let us next consider the combination [l^, Ijl which we know must vanish for all 

 values of i andj. Forming the general expression (27) from the integrals (26), we 

 find:— 



[?, IJ] = [>p„ IV.-] - 1 I [6„ q;,] - p.,, q^.] I + f [6„ />,] = 0. 



This equation being identically zero, the coefficient of each power of t must 

 vanish identically. This gives, in the case of the middle term, 



[&„i!g = [6„qi,]. (30) 



Forming these expressions by the general formula (27), and putting 



we find 



3« I- -\fih. 



ft,.,.J = S.[*.,.I,]g. 



By (29) all the terms of these expressions vanish except that one in the first 

 equation in which Jv=^j\ and that one in the second in which Jc=^i, in both of 

 which the first coefficient reduces to — 1. Hence 



[6«'I'i]: 



= — 



6Cj 



ih;%^-- 



= — 



do- 



eh, 



dhj 





dc, " 



'do' 





and (30) now gives 



(31) 



