ARE EMPLOYED IN THE THEORY OF TERRESTRIAL MAGNETISM. 263 



24. To abridge, put 



--i d"'P 



1 i 2\U " * y 



and ( i - n 2 Y ' j m _r = r. 



Then we have 



s7 i V n 



t-(n+l)(n+m)pY, 



_ _ - 



- (n + m)(n-m+l) Y. 



Hence (n-m + l) n -( n +l)(n + m) 



- {n (n + 1 ) ( 1 - /r) - wi"} (2ft + 1 ) JC, 



and similarly {n(l-^)-m}^ + {(n+l)(l^^)+m}^ 



= -{n(n+l)(l- n 2 ) - m-} (2n + 1) p Y. 



Substitute for X and Y in the equation for y 2 -, then 



+ {n (n + m) (n-m+l) (1- p?) - m (n + m) (n - m +1)} 



Or, 



