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



while the lines of force are confocal hyperbolic cylinders. On taking V 

 as the potential, we get a field in which the equipotentials are confocal 

 hyperbolic cylinders. 



II. Schwarzs Transformation. 



322. Schwarz has shewn how to obtain a transformation in which one 

 equipotential can be any linear polygon. 



At any angle of a polygon it is clear that the property that small elements 

 remain unchanged in shape can no longer hold. The reason is easily seen to 

 te that the modulus of transformation is either infinite or zero (cf. figs. 24 

 and 25, p. 61). Thus, at the angles of any polygon, 



dW 



-j = or oo . 

 dz 



The same result is evident from electrostatic considerations. At an angle of a 

 conductor, the surface-density a- is either infinite or zero ( 70), while we have the 

 relation ( 313), 



R 



dW 



~fa 



Let us suppose that the polygon in the ^-plane is to correspond to the 

 line V in the TT-plane, and let the angular points correspond to 



U=u l} U=u 2 , etc. 

 Then, when W=u l} W = u 2 , etc., 



must either vanish or become infinite. We must accordingly have 



u^ ..................... (236), 



where \, X 2 , ... are numbers which may be positive or negative, while F 

 denotes a function, at present unknown, of W. 



Suppose that, as we move along the polygon, the values of U at the 

 angular points occur in the order u lt u 2 , .... Then, on passing along the 

 side of the polygon which joins the two angles 7=^, U=u 2 , we pass along 

 a range for which F=0, and u l < U<u 2 . Thus, along this side of the 

 polygon, W u 1} W u 2 , W u 3 , etc. are real quantities, positive or negative, 

 which retain the same sign along the whole of this edge. It follows that, as 



we pass along this edge, the change in the value of arg 



by equation (236), is equal to the change in argjP, the arguments of the 

 factors 



undergoing no change. 



