348 PARTICULAR CASES. 



from which follows 



The distances r and being constant during the integration, we 

 may write 



(9) 



365. POTENTIAL OF A CIRCULAR LAYER. The potential of a 

 uniform shell with a circular edge may be calculated from the 

 potential of a plane circular layer, or of any given layer spherical, 

 for instance bounded by the same edge. 



Consider, in the first case generally, a layer of revolution about 

 the axis of x. For a point M the abscissa of which is x 9 and which 

 is at a distance /> from the axis, the potential P is a function of x and 

 of p. If this potential be developed in increasing powers of p or 



of - , the series, from symmetry, will only contain even powers of 



P 

 the variable. We may therefore write 



- I + 



the coefficients A , A 2 ..... , B , B 2 ..... being functions of x. 



When x and p are taken as independent variables, Laplace's 

 equation AV = becomes 



V~2 -^~ T = - 

 ox 2 p op Op z 



This condition gives for the first series, a new series developed 

 in increasing powers of p, in which the coefficients of all the terms 

 must be separately null, from which follows the general condition 



