z=E' 



248 CELESTIAL MECHANICS. I. vii. 26. 



^£zz — d5. (cos -5 sin 4^ cos -^— cos fi cos ^ sin cos <p) 



+ d^. (sin cos sin -^ sin cos ?>~sin cos \|/ sin ^(p) 

 /3'£= d5. sin -5 sin 4^ cos ^(p— dvf. . (sin cos 5 cos tjx cos ^(p + 

 sin 6 sin 4' sin cos (p) 

 + d^. (sin cos Q sin 4' sin cos ^ + sin 5 cos -^ cos 2^) 

 ^e' — 0Bz=. — dd. (sin 4/ cos^ <p — cos cos 4' sin cos 9) 

 + d4' . (sin cos S cos 4' cos ^^ + sin 6 sin 4^1 



sin cos (p) 

 — d(p. sin d cos 4' 

 y^ =: — dd . sin ^5 sin 4. 

 y'C= dfl. cos^d sin4' +d4'. sin cos 5 cos 4' 

 y^ — 7'^— — dS , sin 4^ *-d 4^ sin cos Q cos 4^ = C 



Hence B' + C — -4' = -- d 5. (sin 4. (cos ^(p — sin ^cp) — 

 2 cos B cos 4^ sin cos cp + sin 4^) + d4' . (sin cos Q cos 4^ (cos^ 

 q> — sin ^<p) + 2 sin 5 sin 4' sin cos (p — sin cos 6 cos 4') = — 

 d5 . (2 sin 4' cos ~(p — 2 cos cos 4^ sin cos i^)- — d4' (2 sin cos 6 

 cos 4^ sin^ <p — -2 sin fi sin 4^ sin cos <p), half of which is in the 

 coefficient of J. in the value of — N', 



For that of B, we subtract this half from C, and ob- 

 tain — dfi (sin 4^ (1 — cos ^cp) + cos 5 cos 4^ sin cos (p) + d 4^ (sin 

 cos B cos 4^ (sin ^(p — 1) — sin B sin 4^ sin cos (p)z:z — d5 (sin 

 4/ sin ^<p + cos cos 4^ sin cos ^) — d4' (sin cos 6 cos 4^ cos ^^ 

 4- sin B sin 4^ sin cos (p) : and subtracting it from B\ we 

 have d4'. sin cos B cos 4' — d^. sin 5 cos 4^. 



It is easy to perceive that these are the coefficients al- 

 ready assigned for the value of N\ divided by d^: for 

 qdt being = — dB. cos <p + d4'. sin 6 sin <p, we have for q 

 cos B sin (p cos 4' — q cos ^ sin 4^, — dB. (cos 5 cos if' sin cos 

 ^ — sin 4^ cos ^(p) + d4'. (sin cos B cos 4^ sin ^^ — sin 5 sin 4^ 

 sin cos ^) ; and, since rdtzidB . sin ^ + d4'. sin B cos ^, for r 

 cos B cos <p cos 4^ + ^' sin (p sin 4'> d5 . (cos B cos 4' sin cos p 

 rj- sin 4^ sin ^(p) + d4' . (sin cos B cos 4^ cos ^^ -|- sin B sin 4' sin 



