+ 2m' 



(n - ti'f - N^ 



Ball — On Laplace^ Perturbations of Jupiter^ s Satellites. 563 



a^—r- + ; aA"-^^ cos («;-«;), 



\ da 71 -n j 



7Z ^T^i — ^. « -IT- + / (^^^^ cos (3y - Zv'). 



It only remains to introduce the value of 8a into N". 



dF_ 1 



da »* 



1 ^, 1^ ,<?^t'') 



whence, 



1 / (^\ 1 i/^- »' ^ , dAC^ 



ha = — p a -— - + - 2,m — ; — ; 



3«V 2 Q 71" 6 (?« ' 



but 



{a + 6a/ rt' ' " ' 



3 dA'"^ 1 (^^^f 



+ ^ 



2« (/rt 2 6?a^ /' 



a a^\ 2 \2a da 2 <^a^ ' 





This process appears to be free from the difficulties which beset 

 Laplace's analysis. 



On the Pertiirhations in the Longitudes of Jupiter's Satellites which are 

 Independent of the JSecentricities and the Inclinations. 



The following approximate method seems to have something to 

 recommend it as regards directness and simplicity, when contrasted 

 with the elaborate formulae of Laplace : — 



We have 



div di)~~'dv' 



