159, 160] CONFORM AL REPRESENTATION. 315 



If the two cylinders are external to each other, we must insert 

 the minus sign on the right of (18), so that the equations are the 

 same as before, with S^ replaced by its negative. Accordingly 

 we obtain 



- Rf - d% 



2 log 



The formulae (15), (27) and (29) are important in calculating the 

 capacities of telegraph wires. 



160. Elliptic and Hyperbolic Cylinders. In the pre- 

 ceding examples we were given a function of a complex variable, 

 and from that obtained a conformal representation. We will now 

 consider a case in which we are given a set of orthogonal curves, 

 and we shall seek a function of a complex variable which will 

 make them the conformal representation of orthogonal straight 

 lines. The functions X and ^ defined by the equations 



* * 



' 



a 2 + X b 2 + \ 



' 



are a pair of orthogonal coordinates. Solving for x and y we obtain 



X = A / , 



(2) 



y = V'~b-cf + ^' 



and differentiating logarithmically 



cfo? 1 / d\ 



x 



dy _ 



