164 THEORY OF JUPITER'S SATELLITES. [7 



so that 



(1 - 2a cos <$> + a 2 )* = ( 1 - 2Aa cos - 2a" sin 2 + a 2 )* 



= (A 2 - 2Aa cos + a" cos 2 0)* 

 = A a cos 0. 



Hence 



(l-2acos<+a 2 ) 4 A ' 



C08<f)d<f> I a asin 2 0\ ,,, 

 ^ = I cos 6 H ) d6. 



Now let 6 vary from to 2ir; then since < can never exceed 

 the angle whose sine is a, it varies from continuously to again, and 

 therefore <f> will vary with 6 from to ITT. Hence, remarking that 



cos = , 



o 



we have 



, i c^de , i 



b =- , &,=- 

 7rJ A ai 



or writing as is usual 



or 



we have 



TT ttTT 



From these expressions 6 , & t may be computed; thus it is known 

 that if a, a', a", ... be a set of moduli derived in succession by the 

 formula 



where 



then 

 and 



1 .1 , ,. 

 -aa' + -oaa 



