MR. LUBBOCK ON CERTAIN TERMS IN THE DEVELOPMENT OF R. 81 



— -^—^ I 3 COS (X - ^3r) + COS (3 X — 3 tsr) 1 



-f ^ 1 3 + COS (2 X - 2 w) 4- COS (4 X — 4 tir) j 

 = log.«- — -32^-^ V + 4'/^^^(^~"'^) + TV^ +-2JC0S(2X-2t!r) 



— — COS (3 X — 3 tjj-) + Q^ COS (4 X — Am) 



d_r Se 



r d e 2 



- y e3 - e (l + -| ^2^ COS (X _ Z57) + -^ (1 + e2) cos (2 X - 2 tjr) 



J cos (3 X ■— 3 bt) + -g- COS (4 X ~ 4 z^). 



It follows from the analysis of M. Poisson, in his Memoire sur le Mouvement de 



la Lune autour de laTerre, that the coefficient of cos (2^—2 nr) in the development 



of the quantity 



r^ rf R 



according to the true longitudes, is the same as that of cos {2m — 2 m) in the develop- 

 ment of R according to the mean longitudes. 



dQ _ ad^Q rd_r dQ _ a,dQ r, d r, 



de da de d Cj da, de, ' 



By means of these equations, and after reductions similar to those of which so 

 many examples have been given in the course of this paper, I find the coefficient of 



cos (2 X — 2 \) in Q = — a"^ a, h^^ 



e^cos(2X - 2X^ + \ — m^) ="2^,^3,1+2^3,2 



e e, cos (2 X — 2 X^ — X + t!T + \ — OT ) ~ "" T ^,2 



15 a* 3 «^ % a^ 



ee2cos(2X-2\-X + r!7 + 2\-2T!r,) . . . =- 32^*5.0 +-i6" ^5,1+32^*5,2 



9 a' 



e^efQ,o^{2'k—2\—2\-\-2w\-2\—2m}Qxe^ef(io%{2m-2xs) = — 54^ *5,2- 



MDCCCXXXV. M 



