412 THE THEORY OF TERRESTRIAL MAGNETISM. 



Also from equation (3) above (p. 246) by differentiating m times we get 





, n n + +1 



Hence 



'P = n+ - {(2n + 

 2n+ 1 |/ 



p )>-] p^ w h en n i s e ven, 

 '"'"'P, when n is odd. 



D m 'P when n is even, 

 i is odd.JJ 



i. e. f,/)"' F H = (i - m + 1) D'"- 1 /^ + (2n - 3) D m ^P n _ 



rZ>"'"'P when ?t is even, 

 ^ LsZ) 1 " 'P, when ?i is odd. 



Also we have seen above (see p. 249) that 



( 1 - ^) D m+l P n = 2mnIPP n - (n + m) (n -m+l) D m ~ l P n . 

 .-.(I- f) Z)" l+1 P n =-(n- m} (n-m+l) D m ~' P n + 2m (2n - 3) Z)"' 1 P n _, + &c. 



+ 2m (2 - 4r + 1) D"'" P B _ sr + &c. 



m-\ 



Multiplying by (I-/* 2 ) 2 and putting 



<?: for (1-^f D m P n , 

 we get Q: +1 = - (n - m) (n - m + 1) Q:~ l + 2m (2n - 3) Q'^ + &c. 



2m (2n - 4r + I ) <^ + &c. 



and since -' - l} ^T, we get 



Hence ^= -^.2^^ 



----...-- .. . / g v 



(27i-l)(2/i-3)...(2n-4r+3) " 



