Ball — On Laplace's Perturbations of Jupiter's Satellites. 561 

 We now substitute in the general formula 



1 d-r"^ a 



+ 2 



2 dt^ r 



UdR\ IdR 



\dtl^'\l^^'''''^" 



retaining only periodic terms. 



A delicate point has here to be carefully attended to. The left- 

 hand of the equation will assume the form 



dt 



iYmust be very accurately determined to the second order. It is 



therefore necessary that the values of O in these two terms be 



1 d'r"^ 

 absolutely equal. We have the term - ^j^y where r = a -{■ S« + 8r ; 



therefore 



2~dF'' 'dt' ' 



O is therefore {a + S«) Sr. If we had here omitted S» we should have 

 altered JV^, which depends iipon ha. This is the most critical point 

 in the whole of this part of the analysis. The shortest method is as 

 follows : — 



dt' [ci + bay 



/ 3 dA'"^ 1 d'-S"^ 

 \2a da 2 da~ 



-2aM-hr 



6n - 3M „, , . ^ ^ . 

 M-a~ cos (217- 2v) 



2n~2M 



^ , I 2n , dAv > ,, , ,, 



+ 2^m A,. + a —— cos (hv - nv ). 



\7i- w ' da ' 



This may be written in the form 



d'{a + 8a) 8r 



