346 LE VEURIER ON THE PERTURBATIONS OF PLANETS. 



or, by putting generally for all the values of i, 



K« = 



22'-' sin (2i— 1) — sin(2i— 2)— ... sin — 



>■ 



(24.) 



22''-'sin(2i — 1) —sin (2/— 2) 

 we shall have 



(25.) 



«. ^'-(f+i) _ J. a;-i(i+4) = 2 [?:] 2. J 



These equations, on replacing c, and bi by their values in terms 

 of A; and B,-, and on having again regard to the relations (6.), 

 uill become 



Ai cos i {2i— ] ) ^ — B; sin i (2?— 1) - = [z] ,, 

 Ai cos i (2z -h 1 ) ^ - B; sin i{2i + l)^ = [/] 2, 



and we shall thence deduce 



Ai=H, 



sin i(2z + l)- sini(2i— 1)- 



B,-=Hr 



sinz« 



cose(2i + l)- 



-m, 



-H. 



sin la. 



cosi{2i—l)~ 



(26.) 



Thus then, (i)j and (i)2 being determined by the formula (22.), 

 we shall deduce from it [i]i and [t]^ by the formulae (24,), then 

 Aj and Bj by the formulae (26.). 



12. The calculation relative to the detennination of A;_, and 

 of Bi_ 1, will be simpler by having recourse to the second and third 

 formula of the system (19.), rather than to the first and second. 

 These equations will become, by transformations wholly similar 

 to those which we have just developed, 



^VT^f/ sm(2?-3)-...sm- 

 / 2 \-'-^ « . c 



■ =2(z-l)2v/-l. 



