7] THEORY OP JUPITER'S SATELLITES. 161 



Consider next the relations between the coefficients that arise by giving 

 different related values to the quantity s. Let us write 



1 



S a ~ 1 = - C + G! COS <j) + C 2 COS 2<f> + . . . , 

 a 



then it is required to investigate the relations between the quantities b and c. 



-ntT i f 2/lr COS 



We have 



hence from the equation 



d /sin i<f>\ _ i cos i<f> sa sin i<j> sin 



i cos i<j) sa cos (il)(f> sa cos (i + 1) <j) 



we deduce i6 i = sa[c i _ 1 c i+1 ]. 



For the case ^ = 0, we must replace this by 



In virtue of the sequence equation connecting the quantities c, 



( 1 + a 2 ) ic t -a(i + s) c^ -a(i-s)c i+l , 

 we may write this result in the two forms 



(i + s )b t = s[(l + a 2 ) G( - 2oc i+ J, 

 (i s) b i = s\_2ac i _ 1 (1 +a 2 ) cj. 

 Change i into (i+1) in the latter expression; then 



Thus from two consecutive members of either of the series of quantities 

 6, c, we can obtain the values of all the members of the other series. 



Let us now investigate the relations between the functions b, and their 

 derived functions with respect to a. 



We have b t = - 



Jo o- 



A. Q. 21 



