POTENTIAL OF A CIRCULAR LAYER. 351 



We find thus that the coefficients A and B satisfy the general con- 

 ditions 



If we develop the potential on the axis 



as a function of increasing powers of - or - , we obtain the two 

 series 



- TT f _* I / :r \ 2 _lli/^\ 4 i i 3 /-A 6 

 2\a) 2.4\aj 2.4.6\aJ 



(14) 



In order to have the expression for the potential outside the axis 



as a function of the ratio - or of - , we need only remark that if the 

 a r 



density of a spherical layer is symmetrical in reference to a diameter 

 taken about the axis of x, the potential of this layer at a point M 

 only depends on the distance r of this point from the centre O of 

 the sphere, and of the angle which the right line OM makes with 

 the axis. 



From a well-known theorem of Legendre, this potential may be 

 expressed by the general formulas 



/>\ 2 

 - \ 



