DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
339 
observing that -^=EQ, &c. (see equation (99.)), we have evidently 
?=-(! + 
d£l 
,|))EC.-(|-)E,a 
and it only remains to find the values of namely, the values of ^5 &c., 
which correspond to the assumed conditions y=y^= 0 . 
Now in any spherical triangle of which a, h, c are the sides, and A, B, C the oppo- 
site angles, if a be considered as a function of c, A, B, by virtue of the equation 
cot a sin c— cot A sin B-j- cos c cos B, 
we have by differentiation 
(sin a)^(cot a cot c-\- cos B) 
da 
dk 
= cos a sin a cot c-l-(l — (cos aY) cos B= cosB-|- cos a cot C sin B 
^ sin B sin a sin B 
/sm«\"‘ sin B 5 
\sinA/ sine 
sin C 
^ _ _ (sin a) _ ^ g _J_ sin a cot C ; 
sin c 
and if the sides of the triangle be indefinitely diminished, these expressions become, 
in the limit, 
da sin(B + C) da ^ da ^ 
Tc sin^ ’ d%~~ ’ 
provided the angle C do not vanish. 
If these results be applied to the triangle of which the sides are y, y^, v—v, and the 
opposite angles 5 r—/p /, I, it is easily seen that the values of are as follows : 
( du\ sin '/ / ‘ 
dv) sinl’ \dvj sin F \dv ) sini’ \dvj sin I 
(f)=(^)=(l')=(^')=«’ 
provided 1 be not = 0 . We have therefore 
^ = - EQ + ^ (sin . EQ + sin 1 . E,Q) 
> 
dSl dGi _ 
exactly in the same way we find 
— E.Q.— if(sin /.E,Q,+ sin 1 .EQ.) 
dvf ' ' sm 1 '' ' ' ' ‘ 
dSli dD^i 
dtf d\ 
2 z 
(100.) 
( 101 .) 
MDCCCLV. 
