7] THEORY OF JUPITER'S SATELLITES. 165 



An alternative method of reaching these results is given by Gauss 

 (Determinatio Attractionis etc., 16, 17). 



Write 



II P 



P 2rrjo 



dT 



f- JL I*" (cos>T-sm*T)dT 



1 cos 2 T+ n 2 sin 2 Tf M ^ Jo (m, 2 cos 2 T+ n* sin 2 

 then if 



1 



m = - (m + w) , w = 



m" = - (m' + n'), w" = N/m'w', 



^2 



and we proceed till we find a common limit to the quantities m and w, 

 we shall see that this limit is p. \L is called the arithmetico-geometric 

 mean of m and n. Again if 



\(m*-n^ = \, jK-n*) = V, .... 



so that 



X 2 X' 2 



. 



then it will appear further that 



/2 + 4\" 2 +8X'" 2 +.. 





The first of these may be proved by making the substitution 



. ,_ _ 2m sin T' _ 

 ~ (m + n) cos 2 T + 2m sin 2 T' ' 



when we find 



dT dT' 



(m 2 cos 2 T+ n" sin 2 T) 4 (m /2 cos 2 T' + w' a sin 2 T')* ' 

 making a second similar substitution 



dT' 



_ = _ _ _ 

 (m /2 cos 2 T + ri* sin 2 T') 4 ~ (^" 2 cos 2 T" + n m sin 2 T") 4 



d _ d 



"* in.^* /* : 



