THE THEORY OF TERRESTRIAL MAGNETISM. 429 



19. To find the expressions for X, Y and Z io terms of powers 

 of .. 



so that the general term of P H is 



(2n-2r)l 



_ 



v ' 



rl(n-r)l (n-2r)l 



If this be divided by the coefficient of the first term in P n so as 

 to reduce the coefficient of p n to unity, we have the general term in 



L 2n ~ 2r ) ! - 



or 



Similarly the general term in D m P n is 



(2n-2r)l 



= !(-!)' (2n-2r)S ^_ m _ 



and the first term in the same quantity will be 



I n\ 2nl n . m= l_ 2n\ M . 

 2 (n-m)l n\nV 2 n (n-m)ln\^ 



therefore dividing by this so as to reduce the coefficient of /u,"~ m to unity, 

 we have the general term in H 



= ( 1 nfl ( - IY 

 Hence 



- & c . 



+ ( - 1 V ( n ~ m+1 ) ! nl (2n-2*-) I tt _ m _ 2r+1 l 

 ' (n m 2r + 1)1 2n\ rl (n r)\ " J 



