164 The Effect of a Spherical Conducting Shell on Induction. 

 We thus find 



ke- k "J (kp)J l (kf)dk, 



j 



as before. 



Appendix. 



It may be useful to add the proof of the transformation 

 used in Case II. 



If yLt=COS 0, 



P« satisfies 



(l-^-2^ +n (, i + l)P,, = 0. . . (1) 



Let s = sind. 



(1) transforms into 



ao> d r n , 1 — 4S dZn , i\t» /-. 



- s ^ + —r-di +t < n+l ^ n =°- 



Assuming 



~P n =a + a l s + a 2 s 2 + . . . 



we find in the usual manner 



V-„fl "("+ 1 ) J , (n-2)n(n + l)(n + 3) , 4 V. 



r * — "0 | X M 6 "• 02 Z2 " * * J 



But V n = 1 when 5 = ; 



.'. a =l. 



Let n-=ka. s= - , 



a 



and let n and a become infinite, k and p remaining finite. 

 n(n + l) 2 _kY 

 2 2 ~ 2 2 ' 

 ultimately 



(n-2)n(n + l){n + 3) 4 _ *y 

 2 2 . 4 2 * — 2 2 . 4 2 ' 



&c. = &c. 



.*. ultimately 



P_i_^!4. *V__ *V 



JTn—l 22 f 2 2.4 2 2 2 .4 2 .6 2 



=Jo(*/>). 



