278 Methods for the Solution of Special Problems [CH. vin 



The appropriate potential-function for a single charge can easily be 

 found. 



As in the problem just discussed, it is clear that the potential due to the 

 single line-charge at (a, a) on the upper sheet is the value of U given by 



U+iV=C + A 

 = C + A 



= C+A log |(Vr eos^r Va cos ~) +i ( Vr sin- \/a sin ) , 

 (\ 2 Z/ \ L 2/) 



so that 



U = C + \A log j( Vr cos - Va cos ~ ) + ( Vr sin - Vasin -r 



(V ^ J \ A A 



= (7 + \A log {r - 2 Var cos \ (6 - a) + a], 



and if this is to be the potential due to a line-charge e, it is clear, on 

 examining the value of U near the point (a, a), that the value of A must be 

 2e. Thus the potential function must be 



C e log {r 2 Var cos ^ (6 a) + a) (258), 



instead of that given by expression (257), namely, 



C - e log {r 2 - 2ar cos (0 - a) + a?} (259). 



It will be noticed that both expressions are single-valued for given values 

 of (r, 6), but that for a given value of z, expression (258) has two values, 

 corresponding to two values of 6 differing by 2-7T, while expression (259) has 

 only one value. Or, to state the same thing in other words, the expression 

 (259) is periodic in 6 with a period Zir, while expression (258) is periodic 

 with a period 4>7r. 



Potential in a Riemanns Space. 



334. Sommerfeld* has extended these ideas so as to provide the solution 

 of problems in three-dimensional space. 



His method rests on the determination of a multiple-valued potential 

 function, the function being capable of representation as a single-valued 

 function of position in a " Riemann's space," this space being an imaginary 

 space which bears the same relation to real three-dimensional space as a 

 Riemann's surface bears to a plane. 



335. The best introduction to this method will be found in a study of 

 the simplest possible example, and this will be obtained by considering the 

 three-dimensional problem analogous to the two-dimensional problem already 

 discussed in 333. 



* " Ueber verzweigte Potentiale im Kaum," Proc. Lond. Math. Soc. 28, p. 395, and 30, 

 p. 161. 



