Electric Field in the neighbourhood of a Spherical Bowl. 51 



(1) A 2 V = 0, A 2 V = 0. 



(2) for r=a, 6<a, V' = V = V . 



(3) for r=a, 6>u J V' = V. 



(4) for r=a, 0>«, (©=(£) 



Since every term is a spherical harmonic, the first-named 

 property is evident. To prove the exactness of the other 

 statements, we take as a starting-point the function 



(l+z 2 -2zcos0)-> 



of the complex argument z. This function has two points of 

 discontinuity, namely, 



z = e± lfi , 



whose moduli are equal to unity. We may write thus 



(l + MV ta — 2Me»cos0)~*=2Mv ua n for M<1, 

 and 



(l + MV--2Me-cos^)~ l =^X('g)V ta ^ for M>1. 



Since the function is continuous for M = l (unless # = «), and 

 since both expressions become identical in that case, we 

 have generally 



00 



(1 +e 2ta — 2e ia cos Q)~~ 2 = te ma <f> n , 

 o 

 with the only condition 6^.a. Or 



a . a 



cos-x- — tsm — 



V 



9 a 9 6 o 



cos 4 -r — cos 4 



2 ~"~ 2 

 and, multiplying by e ia , 



V 



a .a 



cos -~ + 1 sin -^ 



9 a 9 6 o 



S " ~a ~ C0S o" 



Separating the real and imaginary parts of these formulae, 

 we must make a distinction between the cases 6 < a and 6 > a. 

 We find, 



E 2 



