302 ELECTROSTATICS. [PT. II. CH. VII. 



and if we put at the center of inversion a charge e = Va, 



ea 



is the density of the charge induced on the plane, agreeing with 

 the result of 152. Inverting the distribution induced by a point- 

 charge on the two parallel planes gives us the equilibrium 

 distribution on two spheres tangent to each other, and inverting 

 the distribution on two intersecting planes gives the equilibrium 

 distribution on two spheres intersecting at an angle which is a 

 sub-multiple of two right angles. For the full treatment of these 

 and other examples the reader may be referred to Lord Kelvin's 

 Reprint of Papers on Electrostatics and Magnetism, XIV, XV, 

 and to Maxwell, Treatise, Vol. I, Chapter xi. 



157. Distribution on Spherical Bowl. As a final example 

 we shall work out the solution of the most remarkable problem 

 that has been treated by this method, namely the distribution of 

 electricity on an open spherical bowl, or segment of a sphere. 

 This is the only case in which the distribution on a portion of a 

 geometrical surface has been solved, except in the case of the 

 distribution on a circular plate, the inversion of which gives the 

 circular bowl. We shall not follow the method of Lord Kelvin, but 

 that given by Lipschitz*, who solved the problem independently, 

 being unacquainted with the existence of a previous solution. 



Let R be the radius of the sphere of which the bowl is a segment, 

 Fig. 64, and let the radius of the 

 opening be a. Let the surface of the 

 bowl be denoted by S, and let the plane 

 surface which closes the bowl, of radius 

 a and distance c from the center of the 

 T ' sphere, be denoted by 2. Inverting 

 the figure with respect to the center 

 of the bowl and radius R, let the 

 spherical segment into which the 

 plane inverts be denoted by 2'. 

 Let the space enclosed between S and 

 FlG - 64 ' 2 be denoted by T, that between 2 



and 2' by T", and the remaining portion of space by T'. Let 



* Lipschitz, Borchardt's Journal, Bd. LVIII., p. 162, 1861. 



