7] THEORY OF JUPITER'S SATELLITES. 149 



+ T G ^ W^W--P) G -j 





Now substitute this value of 8z/a in the first equation and also 

 e cos pt for rSr/a 2 and we shall thus include all the terms which are 

 of the second order in f which arise from the assumed term in rSr/a~. 



Hence if we transpose all the terms of the equation 



d 2 /rSr\ .. A'3r\ r , ., Sz r . . , > -, 



_-..-= -0-- [l + &c.l + <r--[sm<7n2/c + &c.) ...] 

 dt* \ a" / ^ \ a 2 / - a L 



to the left-hand side, the terms arising from the assumed term in r8r/a 2 

 will be 



> 2 ~ 6/V + /^ [cos (2(7 -^) * + cos (2q +p) t] 



r 23 ~i 



- /V J [cos (4? -;?) t + cos 





j osp - cos 2 - 



- 3/+ /c 2 [cos p* - cos (2q +p) t] 



16 



4 



cos 2 +> * - cos 



T <f [7 ~ / O g2 J [ cos ( 2 2 -^) * - cos ( 4 ? ~ 



-p) L4 i> (2g- -p)J 

 /0 . , .. v 2 2 Ff + nr^< \\ [ 



4 J (2q+p)(4q+p)* \_4 p (2q +p) J L 



