of the Moon's Motion. 87 



that plane. Then s — sin i sin (v — 3) = x sin t; — A cos i;. 

 We have by known formulae, 



These formulae cannot be investigated here, they are found 

 on the assumption that -j-dx + -z-dx = 0; consequently -j- 



, . . .dv A d?& d& , ... .d 2 v 



— (xcosu + Asm v)-r- 9 and -7-3 -f- -j-zS = (xcosv + Asmo-7-3 



dv dv 1 dv 2 v 'dv 1 



( dx . d x\ dv rn\ - l . P dv* . 



•+■ I cos v —r- + sin -7- I -j-. 1 his becomes, it -r-^ aA + 4, 

 V dv dv / dv dv 2 



where A is constant, <p periodic, 



d 2 s d* v 



srs + As = (x cos v + A sin v) -r-s — s 

 dv 1 v • dv z 



g 4 cos 2 z / dR . dR\ 



+ V^v cos ^~ s,n ^ 



* • • (3.) 



All the terms of the second member of this are of the order of 



the disturbing force. 



For correcting the values of the elements x and A, we have 



ds ds 



dv dv . 



x = 5 sin v + -7— cos v, A = — s cos v + -7- sin v, 

 dv dv 



dv dv 



When we know the parts of s and v depending on the first 



power of the disturbing force, we can find those of x and A 



depending on the same by taking the finite variation of the 



preceding equations ; and so on for the higher powers of that 



force. This method may be employed for correcting the value 



of R when we do not wish to find x and A separately. I have 



employed x and A in preference to the elements i and 3, but 



am not quite sure that it is better to do so. 



To find the reduction to the fixed plane, let © be the long. 



of the node on this plane, v x the long, on it, and make v — 3 



= Vl _ © + A, or A = (v — S) — (», — ©). Then, if $ 



= lat. sin A = sin (v — •&) cos {v x — %) — cos (v — S) sin^ — ©) 



sin 2 -|- sin 2 (v — ■&) 



= tan — tan 4> cos (v — •&) = } and A sb' 



2 v ' cos<$ 



