DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
351 
(1 — cos® cos®jH-2 cos i cosj cos 0— cos® 
cos i+f cosi+Ji^<^'>}=<+’- 
.... 
_(C- A)i 
dV 
dcos 
dY 
d cosj 
(i-) 
2ACh 
+ ^{^ + 
f 
cos 2 COS 0 — COS^ 
Q sin 9 
d6\=^. 
C Q. 
In order to get rid of the troublesome integration involved in the term we 
may (1) eliminate this term between the first and last of these equations, and (2) 
eliminate between the first and seeond. We thus find the two following equations, 
“cos i cos 9 — cosy (3 C — A , 
(ii.) 
cosj|(p+J 
cos t cos 9— cos^ 
Q sin 9 
dd\Jr 
J sin U 
-Q 
2h, , «cos? 
■j{t + T) 
which last, combined with the preceding, gives 
'sin 9</9 , N acosf + jScosy 
=x(^+’') k 
I' 
(iii.) 
and we may take (i.), (ii.) and (iii.) as expressing the solution of the problem. 
Now we have cos j— cos7cos9 ^ from which it is easy to find (observing the 
sin^ sm 9 j \ o 
conditions which determine the sign of Q) 
^ j cos i cos 9 — cosy 
-dJ = 
Gl sin I 
cosy cos 9 - 
cos 1 
Gl sin 9 
d6, and similarly. 
d & ; 
we have moreover 
de= 
sm ( 
Q 
All the integrations may therefore now be performed immediately; and we may 
take for the inferior limit of & any value which satisfies the equation Q=0, or 
(cos S— cos i cosj)®— sin® i sin®J=0 ; 
this is satisfied by which evidently corresponds to 1=0, J=0, 0=0, and it 
is manifest that equations (i.), (ii.), (iii.) will thus become identical with equations 
(R) of art. 44. 
I do not regret however having introduced the rather prolix investigation of arts. 
39 and 43, because it is interesting to know the actual value of V (equation (48.)), 
which the method just given leaves undetermined. 
