160 THEORY OF JUPITER'S SATELLITES. [7 



We can exhibit b f as the solution of a differential equation ; thus it 

 may be verified that 



a- -y- 



f<S->\.L. /g-\ , - , e _, 



-y- - ( A s ) + -jj, (/b ) + - ^ - a -j- (o 8 ) - 0> 8 = ; 



da 2 ^ d(j>- x 1 a' da v 1 a 2 



substitute for $~ s its development, and equate to zero the coefficient of 

 cos i(f> ; then 



ji 

 " 



I a'" tZa 1 a" 



An expression for b t in the form of a definite integral is of frequent 

 use ; we have 



S~ s = - 1> + 6, cos </> + b., cos '2<j)+ ... +b { cos i$+ ____ 



Multiply both members of this equality by cos i(f> and integrate with 

 respect to (f> between the limits and 2ir ; then 



This leads to a Sequence Equation connecting the quantities b { for 

 consecutive values of i. We have 



d /sin i<f)\ _ i cos i<f> (s 1 ) a sin ij> sin < 

 d<l>\S'- 1 ) = ^S^ "3 >s " 



i (l + a 2 ) cos i< ia cos <j> cos i^> (s 1) a sin <ft sin z'^ 

 ^^ 5* ~S^~ 



i ( 1 + a 2 ) cos i<f> a (i + s 1) cos(i 1) <f> a(i s+l)cos(i+l)<f) 

 ^ 8 ~ ~~&~ ~&~ 



Integrate with respect to < between the limits and 2ir ; observing 

 that the right-hand member vanishes, 



= (l+a?)ib i -a(i + s-l)b i _ l -a(i-s+l)b i+l . 



This equation enables us to deduce the values of all the quantities 

 b { when the values are known for any two consecutive values of i. 



