274 Methods for the Solution of Special Problems [OH. vm 



The value of z p is of course x p + iy p , or .Z?P. Thus from the equation 

 just obtained, equation (247) may be thrown into the form 



"* = ;: (log fc- log 6) 



If the lines of force were not disturbed by the bend, we should have 



O-'fe 



f p 



Equation (252) shews that I ads is greater than this, by an amount 



J E 



Let us denote the distances between the plates, namely TrA A/- and T 



\r CL 



by h and k respectively, so that A /- =T. Expression (253) now becomes 



V a K 



so that the charge on the plate EP is the same as it would be in a parallel 

 plate condenser in which the breadth of the strip was greater than EP by 



I 



^ 



When h = k, this becomes 



g-log.2) or -279A. 



MULTIPLE-VALUED POTENTIALS. 



330. There are many problems to which mathematical analysis yields 

 more than one solution, although it may be found that only one of these 

 solutions will ultimately satisfy the actual data of the problem. In such a 

 case it will often be of interest to examine what interpretation has to be 

 given to the rejected solutions. 



The problem of determining the potential when the boundary conditions 

 are given is not of this class, for it has already been shewn ( 186 188) 

 that, subject to specified boundary conditions, the determination of the 

 potential is absolutely unique. But it may happen that in searching for the 

 required solution, we come upon a multiple-valued solution of Laplace's 

 equation. Only one value can satisfy the boundary conditions, but the 

 interpretation of the other values is of interest, and in this way we arrive 

 at the study of multiple-valued potentials. 



