GENERAL INTEGRALS OF PLANETARY MOTION. 13 



da 



ttj { ottj ^ da-j j 



dy/j_^ \ d{bJc), 



dUi 



S. 



cosiV, — (hk\ ^ t sin N, \ 



daj da-j j 



^, cos N\ — (bh'), ^ - tsinN, > 



da, ^ da, ) 



} (18) 



By changing a^into Oj in the thi'ee equations (17), and making the reverse change 

 in (18), we have the complete expressions necessary to form any term of the ex- 

 pression 



We see at once that this expression will be of the form 



'i" S\, I A^, sin (iV„ — N,) + At + A't^ \ 



Since the expression is known to be independent of t, we must have, to quaziti- 

 ties of the first degree of approximation, A = Q and A' = by the condition that 

 ^, >7, and ^ satisfy the original differential equations, and the coefficient A^i,v must 

 vanish, unless we have 



N,j, — iV,= constant. 



The coefficients 6i, &2 ^'am being supposed incommensurable, this can only 



happen when we have in (3)' 



%^ = ly, j ^2/^ — - ''2n GtC, 



and hence 



when sin {N^ — iV,,) will itself vanish. Hence, (a,,, Uj) containing no constant 

 term whatever, we must have 



(«.,«,) = 0. (19) 



Again, differentiating the equations (16), the first three with respect to I,, and 

 the last three with respect to Ij, we find 



^h 

 61, 



' — — S^ (^y^)/^ sin iV^ 

 ~ S,, {iji:)f, cos N^ 

 = S,. (i,/.-)^ cos N'f^ 



^^ = — S, (ijbkl cos iV; 





— — S, {ijhlc\ sin N, 

 ^ — S, {jjb'h\ sin W,. 



