152 THEORY OF JUPITER'S SATELLITES. [7 



Now if p and the ratios of e, t\, e. 2 be so chosen as to make the 

 coefficients of cosjtf, cos (2q p) t, coa(2q+p)t vanish, we have 





= e \p* ( 



p 



(f 



p 2 (2q -pY 4 ' " p 2 (4r/ - p 2 ) 1 6 ' 



- - 



(2q-p)(4:q-p) 



and 



r. ^r- 

 - e q- -/c - 6/V + 



p(2q-p) 



These two last equations give the ratios of e l and e 2 respectively to 

 e in terms of p, and the substitution of the results in the first equation 

 gives the final equation for determining p. 



It is readily seen from a remark made before that the term in e t in the 

 first equation contributes nothing of the order of f' ; and the approximate 

 value of e^e given by the third equation is 



2 J p(2q+p)(Sq+p)' 



