<-, 



144 THEORY OF JUPITER'S SATELLITES. [7 



^ = 4 [1 + 5! cos 2qt + 5a 2 cos 4g< + 5a, 2 (1 + cos 

 x [sin qt + c l sin 3qt + c., sin 5g<] 



= 4 (1 + 5<*i 3 ) s i n <7^ + c i s i n 3<^ + c 2 sin 

 a \_ 



5 5 



+ - ! ( sin qt + sin 3qt) + - a,Cj (sin qt + sin 



(5 5 \ ~| 



- a, + - a,- 1 ( - sin 3<jt + sin 5qt) 

 4 z, I J 



ucf/5 

 = ^ I ( 1 - 2 a i 



1 



^ 



J 



5 / 5 5 5 



' + *& sin V' + c> + a> ~ a -~ 



5 5 5 \ 



O tt l C l + Q ^ + O 



Z Z Z 



2 65 r , 35 5 



/. . o 

 " - 



^ v -/ v++ sin 



65 , 2 655 ,. 

 36 S~ C ~ + 288 / 



is found by multiplying this by fa\ 



j- = r [1 + 7ttj cos 2qt~] [sin g< + c^ sin 

 * 



15z/c r 7 ~| 



= j- sin 9? + Cj sin 3(/^ + - Oj ( sin g^ + sin 3qt) 



