7] THEORY OF JUPITER'S SATELLITES. 139 



This gives Oj to the second order when c, is known to the first order, 

 (160, - 12a/) g 2 = [a, - -Jc*a, -/c'c.J . 



This gives a 2 when a t and Cj are known to the first order. 



Again 



dsz 



-J-, = acg 2 [sin gtf + 9Ci sin 3qt + 25c. 2 sin 5qt], 

 ctt 



^ = nr 1 + 3! cos2qt + 3a 2 cos 4gtf + - aj 2 (l + cos qt) 



x [sin g + c 1 sin 3</f 4- c 2 sin 5qt~\ 

 = ^ sin g + c, sin 3^^ + c 2 sin 5^^ 



3 3 



+ - a a ( - sin 5^ + sin 3qt) + - a^ (sin qt + sin 5(^) 



^j <-> 



S 33 1 



+ - a 2 ( sin 3qt + sin 5g) + - / sin gi + - a/ 2 ( - sin 3qt + sin 5^<) . 



Z, Zi 4 



^ = ~- [1 + 5! cos 2<^] [sin r/ + c l sin 

 ? ot 



sin 5< + f! sin 3g< + - a, ( - sin g^ + sin Bqt) , 

 \_ & J 



) c, sin 3qt\ 



[1 + 7a, cos 2gt -3 sn g - sn 3^^ + 1 - cos 

 + 7a, cos 2g<] - sin qt-- sin 3g + - c t sin 





. 



-- c, sin qt -c^ sin 

 4 ^4 



- - - " sin - - sn 8? ' - c - sn < 



21 7 1 



+ a t ( - sin at + sin 3^) - r a, (sin qt + sin 5^<) 



o o J 



~l 

 . 



J 



3 , 7 fc 



c^-^smbr/ci i. 



/ J 



182 



