THEORY OF JUPITER'S SATELLITES. 155 



Substituting the above values in the equation for e 3 we have 



- f 



48 ' 4 < 16 c --c-c -. - 



65 , 

 54* 



65 . 

 23,, 4 9 f , 4 54 



= 48- /C ~8 / " C ' 



l ~l* 



1-^c- 



l'=-2_/v 9 f? 54 



e~384 /C ~6l /C 5~~' 



and from the equation for .e t , 



31 f 4 

 = 48 / ' C ' 



4 31 -, J 



e = ri52^ C - 



We have already found the terms in Sz/a which depend on the term 

 ecospt in rSr/a 2 ; similarly tlie terms which depend on the term 



e l cos (2q p} t 



may be found by writing 2qp instead of p and c l instead of e. If we 

 neglect terms multiplied by f in the coefficients involving e l , we have 



= 2 ce i sn ? -P) t + 2 CCl sn 3 ? ~-^ '' 



and similarly the terms depending upon e 2 will be found by writing 

 for ^J, 



Si i i 



- = - - ce 2 sin (7 +p) < + -- ce, sin (3g +p) t, 



CC Zi 1 \j 



and by substituting for e 1 and e 2 in terms of e, and adding the new terms 

 to the terms in Sz/a previously found, we find the complete value of Sz/a 

 to the first order in e and the first order in f. 



202 



