Ball — On Laplace\s Perturbations of Jupiter's Satellites. 659 

 Jupiter's contribution to (i). 



2 



If-- 



''-S 



'■;«=-''- 



„ rd(R) dR 1 



Let r = a + ha + Sr, 



and the expression becomes, retaining only periodic terms, 



1 / 



,8r. 



Sun's contribution to (i^ 



"With Laplace's units we have S -^ IP = M^; and, neglecting the 

 first term, which cannot contribute, 



R = - i Jf2 r^ T- f if- r^ cos (2 ?7- 2t)), 



^r^^ ^^^Jf^«^cos(2Z7-2«;), 



J dt In -2M ^ ' 



r f ^ j = - i if V^ - f if ' a" cos (2 fT- 2^)). 

 Uniting, we have 



- Jf 2^3 _ ^^! — ^^if 2 (?2 cos (2U- 2'v) ; 

 2n-2M ^ ' 



wlience, retaining the periodic terms only, we have 



- 2«if=8r - ^^-^^ j^2 «3 cos {2U- 2v). 

 2n-2M ^ ^ 



3C2 



