POTENTIAL OF A SPHERICAL BOWL. 



141 



The value of the potential at P of an element d$ of the layer 

 situate at the point M, on a radius which makes the angle with the 

 axis, is 



MP (x* + u*-2ux cos 6)* 



According as we have x^u, we may express this potential by one 

 or the other of the two converging series, 



(40) 



MP u L I + ^ Xl + ^ X2 + '"J 



^sr u U* -i 



= I + - A, + A 2 + . . . . 



X I X X* 



If <j> is the angle of the two planes OCH and OCM, and putting 

 = cos 0, we may write 



= i sn 



The potential U of the spherical layer will be obtained by 



S 



integrating the expression of in respect of <f> from to 2ir v and 



in respect of p from /x = i to ^ = cos p. We have thus the two values 



(41) 



U =27TK 



27T 

 X 



r i ^ n ^2 ri 

 4t + - x//x + - x^fi + ... 



uh u *h 



i u r 1 ^ r 1 



V/X+- x^+- 



* ^JM *Vii 



