333-335] Multiple-valued Potentials 279 



We suppose that we have a single point-charge in the presence of an 

 uninsulated conducting semi-infinite plane bounded by a straight edge. Let 

 us take cylindrical coordinates r, 0, z, taking the edge of the plane to be 

 r = 0, the plane itself to be = 0, and the plane through the charge at right 

 angles to the edge of the conductor to be z 0. Let the coordinates of the 

 point-charge be a, a, 0. 



The Riemann's space is to be the exact analogue of the Riernann's 

 surface described in 332. That is to say, it is to be such that one revolu- 

 tion round the line r = takes us from one " sheet " to the other of the 

 space, while two revolutions bring us back to the starting-point. Thus, for 

 a function to be a single-valued function of position in this space, it must be 

 a periodic function of 6 of period 4-Tr. 



Let us denote by f(r, 0, z, a, a, 0) a function of r, 0, and z which is to 

 satisfy the following conditions : 



(i) it must be a solution of Laplace's equation ; 



(ii) it must be a continuous and single-valued function of position in 



the Riemann's space ; 



(iii) it must have one and only one infinity, this being at the point 

 a, a, on the first "sheet" of the space, and the function 



approximating near the point to the function -p , where R is 

 the distance from this point; 

 (iv) it must vanish when r = oo . 



It can be shewn, by a method exactly similar to that used in 186, that 

 there can be only one function satisfying these conditions. Hence the func- 

 tion /(r, 0, z, a, a, 0) can be uniquely determined, and when found it will be 

 the potential in the Riemann's space of a point-charge of unit strength at the 

 point a, a, 0. 



Consider now the function 



f(r, 6, z, a, a, 0)-/(r, 0, z, a, -a, 0) (260), 



which is of course the potential of equal and opposite point-charges at the 

 point a, a, 0, and at its image in the plane 6 = 0, namely, the point 

 a, -a, 0. 



This function, by conditions (i) and (iv), satisfies Laplace's equation and 

 vanishes at infinity. On the first sheet of the surface, on which a varies 

 from to 2?r (or from 4?r to 6?r, etc.), it has only one infinity, namely, at 



a, a, 0, at which it assumes the value -~. 



i 



From the conditions which it satisfies, the function /(r, 6, z, a, a, 0) must 

 clearly involve 6 and a only through 6 a, and must moreover be an even 

 function of 6 a. It follows that, when = 0, expression (260) vanishes. 



