294 Methods for the Solution of Special Problems [CH. vni 



where p , , < are the coordinates of P, being positive, the point P' is the optical image 

 of P in the disc, a is given by the equation 



cos a = cosh p cosh p - sinh p sinh p cos (< - < ), 

 and the smallest values of the inverse functions are to be taken. 

 Prove that the total charge on the disc is qdo/rr. 



Explain how to adapt the formula for the potential to the case in which the circular 

 disc is replaced by a spherical bowl with the same rim. 



93. Shew that the potential at any point P of a circular bowl, electrified to potential 

 C is 



AB OA . , (OP AB \} 

 + OB Sm (OA ' AP+BP)} ' 



where is the centre of the bowl, and A, B are the points in which a plane through P 

 and the axis of the bowl cuts the circular rim. 



Find the density of electricity at a point on either side of the bowl and shew that the 

 capacity is 



-(a+sina), 



7T 



where a is the radius of the sphere, and 2a is the angle subtended at the centre. 



94. Two spheres are charged to potentials F and V l . The ratio of the distances of 

 any point from the two limiting points of the spheres being denoted by <P and the angle 

 between them by , prove that the potential at the point , rj is 



where rj = a, 77 = - /3 are the equations of the spheres. Hence find the charge on either 

 sphere. 



