166 THEORY OF JUPITER'S SATELLITES. [7 



where /* is the arithmetico-geometric mean between m and n ; and in all 

 cases and 2vr are corresponding, limits for the quantities T, T', ... . 

 Hence integrating between these limits, we get the first theorem. Again 

 by the same transformation we find 



m'(ws*T-s\tfT)dT_ 1 (m - n) sin 2 T'dT' 

 (m 2 cos 3 T+ ri> sin 2 2 1 ) 4 ~ 2 ( m " cos ' T' + n' 2 sin 2 T')* 



substituting sin 2 T' = - - (cos 2 T' sin 2 T'}, and integrating from to 2ir, we 



' 



have 



2r Jo (m 2 cos 2 7 7 +n 2 sin 2 r) 4 /* 



(m /2 - w") ( 2 - (cos 2 T' - sin 2 2") dT' 



~i ^ I ' " 



whence 



v 2 (m /2 - n' 2 ) + 4 (m" 2 - n" 2 ) + . . . 

 p.~ (m 2 n 2 ) ju, 



which is the second theorem. 



To apply these results to the calculation of the quantities b a and b lt 

 for the case when s = -, we notice that if 



a 



then 



_2 (" d<f> _ !_ f 2 * dT^ 



* Jo (l-2acos( + a 2 ) 4 ^Jo {(l+a) 2 cos 2 T+ (l -a) 2 sin 2 T} 4 ' 



& = 2 r- cos <f>d<f> = _2 f 2 - (cos 2 T- sin 2 T 7 ) cZT 7 



^Jo (l-2acos^) + a 2 ) 4 ^Jo {(1 + a ) 2 cos 2 T+(l -a) 2 sin a T} 4> 

 or 



, _2 _2v 



where 



m=l+a, n = 1 a. 



Let us illustrate these formulae by computing a few of the quantities 

 b for the first and second Satellites of Jupiter, for the case s = - . 



