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A MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION. 
283 
D 
and 
dd = dd' + cos cf>' (tZ© — da) — cos cf)(dZ — da) + sinS sin cj> d<5>, 
dcf> = — cosec G sin cf,' (¿Z© — da) + cot G sin cf> (dZ — da) + cosec G sin S' cos <// cZ<E>, 
dcf) = — cot G sin cf>' (cZ© — da') 4- cosec G sin cf) (dZ — da) + cosec G sin S cos cf) (Zd?, 
and it is proper to remark, that in obtaining these equations no use has been made 
of the equations da = cos cf> dO, da' = cos cf)' dd'. 
The term in eZ® which contains da', &c. may be written 
(da' — cos 4>' dd') + cos <// dO' + cot d> sin S' dcf)' — cosec sin <£ cos S dO', 
which is equal to 
(da' — cos 0' dd') + cot <i> (sin S' dcf)' — cos S' sin <f>' dd') ; 
and the term in dZ which contains da, &c. may be written 
(da — cos cf> dd) + cos cf) dd — cot sin Sdcf> + cosec <f> sin </>' dd, 
which is equal to 
(da — cos cf) dd) — cot (sin S dcf) — cos S sin cf> dd); 
reductions which depend on 
cos cf>' — cosec <£> sin cf> cos S = — cot <3> sin cf)' cos S', 
cos cf) + cosec <i> sin cf>' cos S' = — cot sin cf> cos S, 
or, what is the same thing, 
cos cf>' sin <f> — sin cf)' cos <i> cos S' = sin cf) cos S, 
cos (f) sin d> — sin (f) COS COS S = — sin c})' cos S', 
which are relations between the sides and angles of the spherical triangle. And we 
then have 
(Zd> = (cos Sdcf> + sin S sin cf) dd) — (cos S' dcf)' + sin S' sin cf>' dd'), 
d%=(da' — cos cf>'dd') — cosec d> (sin Sdcp — cos S sin cf> dd) + cot <3> (sin S' dcf>'— cos S' sin cf>' dd'), 
dZ =(da — cos cf) dd ) — cot (sin Sdcf>- cos S sin <6 dd) + cosec <f> (sin S' dcf)'— cos S' sin cf)' dd'), 
expressions which may be simplified by omitting the terms (da — cos cf>' dd') and 
(da — cos cf> dd). 
Next substituting for da, dcf>, dd, their values, we obtain, 
dr 
d}> 
1 nae sin f 
Vi ^ 
36—2