THE THEORY OF TERRESTRIAL MAGNETISM. 427 



The factor in large brackets in this expression 



= 2w 2 (n + I) 2 + n (n + 1) [m (TO - 2) - m (m - 1) + m (m + 2) - m (m + 1) + 2m 2 ] 



+ m 2 [2 (m 2 - 1) - (m - 2) (m - 1) - (m + 2) (m + 1)] 

 : 2n~ (n + 1 )" + 2n (n + 1 ) m 2 6m 2 

 = 2 [n 2 (n + 1 )'-' + (nn+i - 3) m 2 ]. 

 Hence 



-3) 



(2n-l)(2n 



n 

 J ' 



Also (l-^Y{dp. + m t (H^dp. + (n + lY l - p?) (H?)> 



-3) 



(n+l) (nnTV + m 2 - 1)] 



We also readily find that 



-i ri 



(18). 



18. We have seen above (p. 413) that, when n^ is less than n, 

 1 ^ m - l - (n-m-1)! fo 



T H:+ 



and 



J -i 



542 



