ARE EMPLOYED IN THE THEORY OF TERRESTRIAL MAGNETISM. 257 

 Also from equation (17) we get 



rm _ (n-m-^ 1 ) (n - m) rm _, 

 (2n-3)(2n-l) 



or putting (?i+l) for n and (m+l) for m, we get 



l ^- m 



We have 



Equation (10) gives us 



< +1 - 2w/t ( 1 - /x 2 )-* e' + (n - m + 1 ) (?i + m) QT 1 = 0. 



Hence since 



OM _ l_^Ji^2n-l) 



" 1.2.3 ... (n-m) '" 



we get 



(n - m) H: +I - 2mn ( I - ^ 2 )~* H% + (n + m) H'^ 1 - ............ (20), 



Again since H? = G ( 1 - ^) T , 



we get (n - m) (1 - /A 2 ) G^ 1 - Zm^ + (n + m) G~ 1 = ............... (21). 



Putting TO = 0, we get 



(i-X)fli + fl?-o. 



We have also (see Art. 14) 



rl(~l m /-7"' +1 A* M+l Jm P m-1 



<' -"''* ' -<)'/. (i -rt ; 



therefore (l - ^ =- = (w - m) H? l -mp (1 - 

 hence from equation (16) 



A. n. 33 



