POTENTIAL ANALOGUE METHOD OF NETWORK SYNTHESIS 343 



Hence we must find a more general solution of the problem for the circle 

 before we can use the potential analogue method to its fullest extent. This 

 we shall do in the next section. For the moment let us assume that we have 

 solved the problem for the unit circle in the Te;-plane, and thus determined a 

 charge distribution on Ci for which the potential is continuous across Ci . 

 Now, by our definition of T, points on Ci correspond to points on C. Hence 

 we find the distribution on C by an inverse transformation in which the 

 charge at any point on Ci becomes the same charge at the corresponding 

 point on C. This charge distribution on C has the required potential inside 

 C It may be simpler in practice to determine a convenient lumped charge 

 distribution on Ci and then transfer these lumped charges to the corre- 

 sponding points on C. 



It remains to determine T(w)j satisfying the conditions (63). One method 

 is based on the remark above that if C is an equipotential in the />-plane 

 then Ci is an equipotential in the w-plane. Hence T might be defined as the 

 transformation that maps equipotential distributions on C as equipotential 

 distributions on Ci . This transformation has been determined for many con- 

 tour shapes in the classical theory of equipotential distributions. 



At the same time the precise shape of the contour is not usually critical 

 for network purposes, so that it may be simpler to choose a T{w) directly and 

 determine the corresponding shape of the contour. A simple functional form 

 involving two or three parameters might be assumed, for example, 



riw) = aw -^+ -^ (66) 



W TiT 



where the parameters a, b, c will be sufl&cient to give C any length and 

 breadth and a considerable further variation in shape. Illustrative shapes 

 for transformations of the type (66) are shown in Fig. 15. In practice the 

 special case of the ellipse, for which c = 0, is often adequate. 



13. Poisson's Integrals 



We turn now to a general solution of the exterior potential problem for 

 the unit circle in the w-plane, which may be used when the simple inversion 

 theorem is not apphcable. For this purpose we start from Cauchy's integral, 



where C is a simple closed curve in the w-plane and the integration is taken 

 clockwise round C It is assumed that Fe{w) vanishes at infinity at least 

 as 1/w, and then the integral expresses the value of an analytic function Fg 

 at any point outside C in terms of its values on C. 



