458 THE THEORY OF TERRESTRIAL MAGNETISM. 



On integrating this expression from p-= 1 to /* = 1, we get 



(n^}}\ l _^-^H n H n d, 



= i e- (n + , + 2) (n + n, + 4) (n + ,-!)( ^H n H n da. 



4 J-i 



This vanishes, except when ?i, = n 2, 



or when n 1 = n + 2, 



or when n = n l . 



When n 1 = n 2, the value of the coefficient of a n _, is 



t 2n(n+l)(2n-3) (n + m}\ (n - m}\ 

 ~~2n+ 1 [1 ."3TyT".".~(2w" -T) J ' 



and when n 1 = n + 2, the value of the coefficient of a n+ , is 



) ! (n + 2 - m) ! 





_ 

 "~2n + 5 [1,3.5 ~~ 



Similarly the coefficient of /3 ni in the final equation for a n is 

 _ 77 _i_ i 



" 



-WX+ 1) (w-r^ 

 ~ 



. 



hence, when nj = n 2, the coefficient of /8 n _ 2 vanishes. 

 When n 1 = n + 2, the coefficient of /8 n+2 is 



Similarly the coefficient of a ni in the final equation for is 



i -i 

 i l n + 3)(n 1 n 2) 



