140 THEORY OF JUPITER'S SATELLITES. [7 



Now equate coefficients of corresponding terms, dividing throughout 

 by ac 



3 3 ,/ 5 \ ,,/! 3 21 



2 a = ~ 4 a '" + * ( l + 2 "/ + ^ C " (4 



3 21 \~| 

 ~ 2 ' ~ ~8 "') J ' 



to O Q /O !? \ ~1 



c 2 + 2 A + - a. 2 + - a, 3 + 5/c 2 ( - c a + - a, j J . 



The first of these gives the relation between q~ and when a, and 



a 



c, are known ; 



the second gives c, in terms of c^ and a., ; 



the third gives c, in terms of < ,, 2 . 



Now substitute the value of <f given by the first of these latter 

 equations in all the rest and divide out by the common factor /i/a 3 , and 

 we have the following equations for the determination of the coefficients 

 ctj, a,, c,, c. 2 , taking into account terms of the second order, 



1 5 



40i - 60,' + 12/a, - 15/c 2 a, = a, -/a, - -/c 2 + -/crj, +fc\, 



or 



(1). 



or 



(2), 



or 



3 3 3 27 



Sc^-a.--^--^^ a^ 



