332, 333] Multiple-valued Potentials 277 



may be represented by a region on one of the other sheets of the Riemann's 

 surface. 



To take the simplest possible illustration, suppose that in the f-plane we 

 have a line-charge e along the line represented by the point P, in front of 



f- plane z surface 



P +e P (upper sheet) 



B O 



P'- e ?' (lower sheet) 



FIG. 94. 



the uninsulated conducting plane represented by the real axis AB. The 

 solution, as we know, is obtained by placing a charge e at the point P', 

 which is the image of P in AOB. The value of the potential (U) is given, 

 as in 318, by 



Let us now transform this solution by means of the transformation 



?=** ................................. (256). 



The conducting plane AOB transforms into a semi-infinite plane OB, which 

 may be taken to coincide with the branch-line of the Riemann's surface. 

 The charge e at P becomes a charge at a point P on the upper sheet of the 

 surface, while the image at P' becomes a charge at a point P' on the lower 

 sheet. Thus we can replace the semi-infinite conductor OB in the -plane 

 by an image at a point P' on the lower sheet of a Riemann's surface, and we 

 obtain the field due to a line-charge and a semi-infinite conductor in an 

 ordinary two-dimensional space. 



From the transformation used, the potential is found to be given by 



77 -rr V^-Va 



U + iV = A log-^ -=, 



V z V a 



in which V is the potential, z a is the point (a, a) on the upper sheet, and 

 z = a is the image on the lower sheet. 



In calculating a potential on a Riemann's surface, we must not assume 

 the potential of a line-charge e at the point (a, a) to be 



C-2elogE .............................. (257), 



where R is the distance from the point (a, a). In fact, this potential would 

 obviously have an infinity both at the point (a, a) on the upper sheet, and 

 also at the point (a, a) on the lower sheet, and would be the potential of 

 two line-charges, one at the point (a, a) on each sheet. 



