7] THEORY OF JUPITER'S SATELLITES. 153 



which is nearly equal to jfc 2 , and this is to be substituted in the first 

 equation in order to obtain the final equation for p to the second order in f. 



We may see that with the values thus found for eje, e 2 /e terms of 

 the second order in f will be left outstanding involving cosines of the 

 angles (4g p)t and (&<].+ p)t. In order to get rid of these terms suppose 

 e. 3 cos (4^ p) t and e t cos(4:q+p)t to be two additional terms in r8r/a"; 

 in forming the left-hand side of the final equation due to these terms we 

 may omit all quantities involving /, so that the equations for determining 

 e 3 , e t will be 



-.f * _ 

 e 4S- 7 ' 



^ 



, e 



23 \ 21 g" ,63 



^ ~48- 7 ' T" 4 - 1 S 22+^(42+^) + 16- 7 



and the determination of these quantities will complete the solution of our 

 problem. 



At first neglect the terms in f- in the first equation (divided throughout 

 by e), and put p = q in the terms containing / in the first power, and we 

 have approximately 



Next take into account terms in / 2 , putting p = <l in tnese terms, and in 

 the terms which involve / in the first power, putting 



(f < _ ^ _ 



4^P 2 ~ (f (3 + 4/- 5/c 2 ) ~ 3 ~ 9 

 A. II. 20 



