250 CELESTIAL MECHANICS. I. Vli. 26. 



^e — ^£-=1 — dd. (cos ■4' cos ^(p + cos sin t{/ sin cos (p 

 — d^. (sin cos 5 sin -^ cos "<p — sin Q cos %|/ sin 



cos ^) 

 4- d^. sin d sin ij' 

 7^= — d5. sin ~& cos %^ 

 y'Z— d5. cos ^d cos -^ — d%|/. sin cos 5 sin 4^ 

 y^ — yii— — d^. cos ij/ -|- d^. sin cos d sin xf^ =: C 



We have here B' + C — A'z=. — d5 (cos -^ (cos ^(p — sin ^(p 

 + l) + 2 cos Q sin -^ sin cos (p) — d^ (sin cos sin -^ (cos ^(p — 

 sin ^(p — ] ) — 2 sin Q cos -^ sin cos <p) — — dd (2 cos 4^ cos ^(p + 

 2 cos fl sin ^ sin cos (p) + d-4'. (2 sin cos Q sin \[/ sin ^cp 2 sin 

 6 cos x|/ sin cos 9) twice the coefficient of A ; whence we 

 obtain, as before, for the other coefficients, — d5. (cos %|/ 

 sin "(p — cos Q sin -4. sin cos (p) + d>^. (sin cos Q sin a|/ cos "(p — 

 sin B cos -^ sin cos ^), and — d^/. sin cos 5 sin ^^ + d^. sin Q 

 sin 4' ; which we must compare with — q cos (p cos -^ — q cos 

 6 sin (p sin t^, with r sin ^ cos -^ — r cos 5 cos <p sin 4^, and 

 with — p sin d sin ^ respectively : of these the first becomes 

 + d5, (cos ^(p cos 4' + cos 5 sin cos (p sin i) — d^^. (sin sin cos 

 <p cos 4/ + sin cos sin "(p sin 4^), the second d5. (sin ^.p cos 4^ 

 — cos & sin cos <p sin 4^) + d4^. (sin sin cos <p cos -^ — sin 

 cos & cos 2 <p sin 4^) and the third d4'. sin cos 5 sin 4^ — ^<P' 

 sin sin 4^; agreeing in .each instance with the reduction 

 here detailed, which is inserted more for the sake of pre- 

 serving the uniformity of system, and of leaving nothing 

 undemonstrated, than for its immediate importance to a 

 student.] 



346. Theorem. Retaining the notation 

 of the last propositions, and making 4, infi- 

 nitely small, or oc infinitely near to the plane 

 of <r" and y\ putting also Cp—p'^ Aqzzq\ and 

 Br=zr\ we obtain the equations 



