282 
A MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION. [212 
Calling the sides A, B, G, and the opposite angles a, b, c, the general formulae for a 
spherical triangle give without difficulty, 
sin A sin B sin c 
dC = — cos b dA — 
cos a dB + 
and conversely 
7 sm b 7 . cos Osina 7 „ , 
da = -r—~ dA + 7—- rT - dB + 
sm 6 sm G 
77 cos G sin b 7 , sin a 7 0 , 
db = . yy dA + -r—- n dB + 
sm G sm 0 
de = cos B da + cos A db 4- 
dA = 
sin B 7 cos c sin A 77 
-7—da— 7 db — 
dB = - 
sm c 
cos c sin B 
sin c 
da + 
sm c 
sin A 
sin c 
db 
sin G 
sin A cos b 
sin G 
sin B cos a 
sin G 
sin a sin b sin G 
sin G 
sin a cos B 
sin c 
sin b cos A 
sm c 
do, 
dc, 
dc, 
dG, 
dC, 
dC, 
which, in the present case, become 
dA> = cos S deb — wo \aj\ij \ . . 
“ T sm <P 
lci . sin 5 7 , COS 6 sin S' 7 sin 6 cos S in 
dS d * + —£r^~ dG > 
n, 7 ,. sin 6 sin cb' sin G 
cos S' dcf) + — r dG, 
sin <P 
sin <P 
cos<6sm$ 7 . sm S 7J/ , sm0cos$ jn 
dS= . — dcb + — 7 deb'+ V——— dG, 
sm cf> sm (p sm <p 
and 
dG = cos <f>' dS' — cos <£ dS + 
7 , sin cb' 7 cos G sin <6 7 ^ 
dcb = - dS + . n dS + 
T sm G sm G 
sin S sin S' sin ( t> 
sin G 
d&, 
sin S cos cb' 7 * 
—i r ~-d<P, 
sm 6r 
= _ cost? S i n f rfg dS + sm IS cost 
^ sm G sm G sm G 
and we have also 
sin G 
dS = d% — da, 
dS' = dS-da', 
dG = dd -dd'. 
dcf), 
Hence, observing that sin S sin </> = sin S' sin cf)', the preceding equations may be 
written 
= cos S d(f> — cos S' dcf)' + sin S sin cf>dd— sin S' sin <// dd' 
d© = da'— cosec <i> sin S dcf) + cot cf> sin S' dcf)' + cosec <E> sin cf)' cos S (dd — dd'), 
d% = da — cot <P sin S dcf) + cosec cf) sin S' dcf)' + cosec <3> sin cf) cos S' (dd — dd'),