POTENTIAL ANALOGUE METHOD OF NETWORK SYNTHESIS 347 



does not contribute to the discontinuity in the stream function. In fact the 

 charge distribution on Ci is now determined by 



q'W = - i^aW - ^a(^o)] + Qo , (78) 



TT 



where Qq represents a constant charge density in the zc-plane, and ^a is 

 that part of the stream function on the circle contributed by Fa{w). 



Certain functions Fi{p) lead to very simple separation formulas for any 

 contour shape, provided T{w) has been expressed in analytic form. A simple 

 example is the linear phase function, 



Fd.p) = -Kp, (79) 



for which 



F'i{w) = -KT{w). (80) 



By the definition of T{w) this function has a pole at infinity, but is otherwise 

 analytic outside C^ . Inside Ci it will have poles at any poles of T(w). We 

 can separate out that part of Fi{w) which is analytic inside Ci by considering 

 the value of the derivative dV/dw at w = co . This will have a finite value 

 Too , and we write 



F'i{w) = {-KtL)w + [-KT(w) + {KtL)w]. ' (81) 



The first factor is analytic inside Ci and the second outside Ci , hence the 

 exterior function is 



F^iw) = -^ + [-KTiw) + {KtDwI (82) 



w 



while at the charge point w = e'^ the integrated charge is 



q'(^) = -^sin^ + Qi (83) 



TT 



A more general example of the use of the separation theorem will be 

 found in the next section. 



15. Elliptic Contours 

 The unit circle Ci in the w-plane is mapped on the />-plane ellipse C of 

 Fig. 17 by the transformation 



p = T{w) = i^J''^ - -) , (84) 



where the major axis of the ellipse is along the real frequency axis with foci 

 at ±/coo , the intercepts on the co-axis are at zhiioioii + k), and the inter- 

 cepts on the ^-axis are at ±io;o(, — k). This transformation will map the 

 outside of Ci on the outside of C if coo and k are real positive constants with 



