THEORY OF JUPITER'S SATELLITES. 177 



Making use of the relation 



we may write the coefficient of a it 



f(n-n'Y-N\ 



where 



_.__. 

 2 J a 3 2 a da 



Where the highest accuracy is not required we may put 



2/n 2 in place of -2ffj./a\ 



m'n" 



. 



a -= in place of - -j- ; 

 da a da 



but if we wish to be as exact as possible, we must include a further 

 small correction to the value of N* for the following reason. The quantity 



m' dQ i m' dA 



-j contains the constant term - - -j ; hence a small correction xa 



a da 2 a da 



,i i c .,, . . , ,. d fl m' dA,\ m! dQ 



to the value ot r will introduce a correction xa -=- 1 - - -p- m 



da \2 a da / a da 



which contains the term 



n ;/\ r l //*4> 1 dA t \~\ 

 a { cos i (I - V)\ - - m' -~ -._-.-!), 

 2 \ofa a aa/J 



to be added to the right-hand member of the first equation. Terms 

 multiplied by a i _ 1 , a i+J , etc., are also introduced, but we shall ignore 

 them ; thus we get the more exact value of N", 



--- ) 



2 J a 3 2 rfa 2 a da J 



and then a i( ^ are found from 



a 2 



, 2W A 1 



+ - ; r-4< > 

 w n * J 



2n 1 m' 



" 



~ t(n-n') * i(- 

 The coefficients a ^ gain in importance if the divisor i 1 (n n') 2 n 2 be 



small. Now i 2 (-n / ) 2 - w2 = {*( n - ri/ )- n }{*( w ~ n/ )+ w }' an( ^ in tne case ^ 

 A. n. 23' 



