254 CELESTIAL MECHANICS, I. vii. 27- 



¥ COS "^ — 0^+2/* sin cos •^) -\- (cos H—siu H). {g sin ^ + h 

 cos ^). 



Now since the first members of each of these equations 

 must vanish, the second will vanish also, and we have sin 6 



< (a- — h- sin ^^ cos ^|.+/(cos -aj/— sin 2 4,):= cos (Ji sin ^—g 



^\ ,, sin 6 

 cos ^|/), consequently — 



h sin -^ — q cos -^ ■ _ . . 



:. — ± 1 ; and [smce sm cos 6 



(a^— 6^) sin cos i^ + f{cos"^—&m^^) 



(a2sin-^^4-&2 cos H— c« +2/'£in cos 4.) = sin 2 fi— cos 2 5 



(^ sin 4/ 4- A cos 4^), whence 



a^sin-T^ + S^cos^^.— c2+2/'sin4'Cos4'_sin fi jcosj 



gsm-^-\-hcos-^ cos 5 sin fi' 



subslitutingMfortang-vj/zi , having divided by zz 



cos -^ a2 sin 2 4/ + 52 ^os ^ -^ — c^ + 2/'sin ^|/ cos -4/ __ 

 cos 2vf.' gu+h 



hu—g _\ {a^—h^)u+f(l—u^) ^ cos2^|, 



(a2sin24, + 52cosH— c2+2/sincos^l.)5(a2— i2)M^y(^l_ 



cos H (^M + 70 ; or 0=:(gu + Ji), (Am— g)^— |(a^— 52)^+/ 

 (1— M*)|(^(a2-62)M+y(l_,^2)|cos 2^(gu-\-h) + (a2 



sin 2^1.4-62 cos^^—c^ +2/" sin cos %|.)) Am — g I V in which 



the latter part becomes — c^ (Jiu — g) + a^ (u cos 24, (^^ + A) 

 + sin 24. (Jiu—g)—b- (u cos 24. (^gu-\-7i)~-cos 24. {7iu—g) +J' 



j (1 — m2) cos 24^5rM + A) +2 sin cos 4'(Am— g) | z: -^c^Qiu—g) 



