Electric Field in the neighbourhood of a Spherical Bowl. 45 

 not a constant potential, but for the point C on the bowl it is 



v . po 



However, by concentrating at P a quantity of electricity 

 — Y b, the potential on the bowl becomes null ; and at B we 

 have, writing E for — V b, 



v- A -EfJL 4. a6 ' \ h\ 



V ~PB 7T IPB^PB'.OBJ* " ' ■" K) 



This accordingly is the expression for the potential at any 

 point B in the electric field, determined by a charge E at P, 

 the bowl being connected by a thin conducting wire with the 

 earth. What still remains to be done is to express 9 and 6 r 

 as functions of any parameters, which are capable of defining 

 the positions of P and B with reference to the bowl and to 

 each other. 



This purely geometrical problem requires for its solution 

 rather lengthy calculations. At least I have not been able to 

 compress them within narrower limits than is done in the 

 following paragraph. 



§ 6. The problem before us may be stated thus : — 



Being given in space a circle and two arbitrary points ; 

 draw all possible circles through the two points and a point 

 of the given circumference : it is required to find the greatest 

 angle enclosed between two of these circles. 



We shall solve this problem by the method of the reciprocal 

 radii, of which the theory of electrical images is the extension. 

 The figure defined in the statement of the problem we shall 

 transform by the said method into fig. 1. 



To this effect we assume a coordinate system with B for 

 origin, and the line drawn from B to P for positive z axis. 

 We suppose the circle to be given as the section of a sphere 

 through B, and a plane. 



Let the equation of the plane be 



az + j3y + yz — p=0, . . . . . (4) 

 with the condition 



«* + /3 2 + 7 2 = l; 

 that of the sphere be 



{x-u) 2 +(tj-v) 2 +(z-iv) 2 = V 2 , ... (5) 



with the condition 



u 2 + v 2 + w 2 =Y 2 . 



To invert this figure with B for centre and PB = R for 



