10 Prof. J. Larmor on Electromagnetic Induction 



And by expanding the potential of the field in a series of 

 harmonics of different orders, we obtain the general solution 

 as the sum of the corresponding special solutions for each 

 term. 



Now \r is usually very small, so that we may write 



i= L ^r{ 1+ <&)}' 



when we have 



wher 



B 1 = 



P=[-A ,» + B 1 (X,)»{l + ^ T) }]Y^ 



(2?i + 1)«"- 1 A 



{ ( « +1) .-> + !}{ M . + ig|}' 



or 



(2n + l)A 



Thus, finally, for a sphere in a varying field of force given 



\ttic 

 by P 0? for which t\ 2 = '- is a small quantity, the value of P 



inside is given by 



i | 2(2>i + 8) »- , 



1 + (XaY A °' lnS ' 



L ■ + a(2n+l) J 



or 



Xfjr M A r»Y^ 



2V2« + 3 2» + l/ ° " ' 



or, taking real parts only, if the field is given by 



P =A r"Y,*cos/tf, 

 we have 



inside the sphere; and 



-P- 4tt«A a 2n+3 v . 



F -""(2»+l)(2n+3)cr t^+» ** sm ** ... (7) 



outside the sphere. 



