162 THEORY OF JUPITER'S SATELLITES. [7 



, db { 2s f 2T (cos </> a) cos i<f> , , 



11611(56 da = n jo V <**' 



cos t 



2a cosi<f> 



or ^ = [c { _ , + c i+1 - 2ac J . 



Substitute for the quantities c in terms of b ; 



(1 - a 2 ) ^ = (i + s - 1) &,_, - (i - s + 1) 6 i+1 + 2as& 



i + 2 



a 



+ -) 4 b t -2(i-s+l)b i+1 . 



a / J 



These equations are sufficient for determining -^ *, and thence derived 



functions of higher order; but we can find others which it will generally 

 be preferable to employ. We have 



therefore 



da 



Substitute for -^ the expression in terms of the quantities c found 



above ; then 



and in exactly the same way, 



Differentiate this, and we find 



