360 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



tion of charge Qi{s) as it stands gives rise to non-physical networks with 

 poles in the right half -plane. However it is possible to add to Qi(s) sl dis- 

 tribution of charge producing a high enough uniform cross-axis field (flat 

 delay) so that the total charge distribution Q{s) yields physical networks. 

 For the time being consider just the equalization of one. singularity. If 

 we solve this simple problem the Qi(s) for the general case of any number of 

 singularities can be obtained by adding up the charge distributions for the 

 individual singularities. For the sake of concreteness imagine the singularity 

 to be a unit positive charge at ^o = —a-\r ih in the left hand />-plane. What 

 is needed is a distribution q\{s) of charge on C which produces inside the 

 contour the complex potential 



I'^'^p + pt 



corresponding to a charge — | at />o and a charge + J at — /)o . By the phase 

 invariant transformation, these two charges give the same field across the 

 oj-axis as a unit negative charge at pQ , while along the axis their fields cancel. 

 Note that we have reversed the sign of the charge at pQ . This is because the 

 shielding distribution on C due to any set of exterior charges must be such 

 that its potential inside C exactly cancels the potential of the charges, that 

 is, it matches the potential that would be obtained if the signs of all charges 

 were reversed. 



Now the complex potential of a point charge Q at — />o , outside C, isF(p) = 



— Q log {p — po). When this is mapped on the w-plane by a transformation 

 p = T{w) which maps C on the unit circle Ci the transformed function may 

 be separated into two parts, analytic respectively inside and outside Ci , 



F.W = -Q log {w - w,), F,(w) = -Q log rW - rfae) ^ 



W — Wq 



where Wo is the w-p\sine mapping of po , defined by po = r(wo), and Wo is 

 outside Ci . We have seen that the mapping of the charge distribution q(p) 

 on C into the charge distribution q'{w) on Ci is determined by Fa(w), and in 

 the present case Faiw) represents the complex potential of a point charge Q 

 located at Wo . It follows that the required shielding distribution on C in the 

 presence of exterior charges maps into the shielding distribution on Ci in 

 the presence of the mappings exterior to C\ of the exterior j>-plane charges. 

 Thus in the w-plane our equalization problem is to determine the shielding 

 distribution on C\ due to a charge +^ at Wq and a charge — J at i^o , where 



— pt = T(wo). Since we are considering only one singularity in the /)-plane, 

 and ignoring the physical requirement of an equal singularity at the con- 

 jugate complex point, we cannot apply the simple form of the inversion 



