POTENTIAL ANALOGUE METHOD OF NETWORK SYNTHESIS 333 



outside C the integral vanishes. In potential theory it is shown that the 

 , potential F on C may be interpreted as a double layer of charge of strength 

 ^ V, while the normal derivative of the potential on C may be interpreted 

 as a single layer of charge of density dV/dn. Thus Green's formula ex- 

 presses the potential inside C as due to a single and double layer of charge 

 on C, the charges being determined by the known values of V inside C. 



Green's formula represents a very simple and general solution of the 

 charge distribution problem. The simplicity is due primarily to the con- 

 stancy of the potential outside C; and this in turn is made possible by the 

 double layer of charge, which suppUes the discontinuity between the vari- 

 able interior and constant exterior potentials. Unfortunately, from • the 

 network synthesis point of view, it is not a practical solution, for double 

 layers of charge lead to zero-pole combinations which are not easily realiza- 

 ble. A double layer might be approximated by two closely-spaced strings 

 of positive and negative charges, but the resulting zeros and poles would 

 be in addition to the zeros and poles for the simple layer of charge. Hence 

 the associated network would be difficult to design, and would also be 

 unnecessarily compUcated and wasteful of network elements. 



It is well-known, however, that V and its normal derivative cannot both 

 be assigned independently on C, and that the potential inside C is deter- 

 mined when we know the values of V alone on C. This would make it pos- 

 sible to eliminate the double layer of charge, if we could obtain the analytic 

 continuation of V on both sides of C. Then we should have a potential 

 which is continuous across C, and this would be consistent with the exist- 

 ence of a simple layer of charge on C whose density is determined by the 

 discontinuity in the associated stream function, as we saw in Section 3. 



We might remark that if V(P) is a given function of P outside C, the 

 integral (33) will again express V{P) at points outside C in terms of the 

 values of V and dV/dn on C. In this case V{P) must vanish at infinity at 

 least as 1/p, and the value of the integral will be zero at all points inside C. 

 Hence if we retain both single and double layers of charge it is possible to 

 obtain a charge distribution on C for which the gain characteristics are 

 assigned over the entire frequency axis. With simple layers the gain may 

 be assigned only over that part of the frequency axis which Ues inside C. 

 Then we must accept its values on the remainder of the axis, though it 

 may be possible to control these values to some extent by varying the 

 contour C. 



19. The Exterior Transmission Function 

 We have just seen that for a simple layer of charge on C we have to 

 determine the analytic continuation of the transmission function on both 



