7] THEORY OF JUPITER'S SATELLITES. 173 



Now the functions with which we shall have to deal are 



r dr ' i* dO ' 



that is to say 



and 



- ,- -- r- - 



a da a da 



d /I dQ\ . fd?Ai I dA\ 



-- - -- = --'-- 



. i 



-j- - - 7 - )= S -,-,'-- ,-- \co8i (I -I'), 

 da \a da/ \aa a da J 



, d /I (/^\ a' d-A { . n . 



a j-/(- j 5 = S- j ; , J -^ cos i (I -I'), 

 da a da a dada 



d /I dQ\ ., 1 dA, . . 

 31 (- T - = -2- -T-'tai 



at \a / a aa 



Again 



.,- 

 - dl J \a- a- da 



, d /I t^\ v a' 

 ' - 



7-7 TT 7~T l sm l ' - 

 da' \a- dl / a 2 rta' 



^-o /7 7/\ 



JT I -; -jT I =2 :, A-,1- COS I (/ /'). 



dl \a- dl / a- 



In these formulae, under the sign 2, ^ A stands in place of A a . 



We have thus shewn completely how to express numerically the 

 coefficient of any periodic term in the disturbing function. 



