POTENTIAL ANALOGUE METHOD OF NETWORK SYNTHESIS 349 



When n is odd each series ends in the first power of its argument; when n is 

 even Fa ends in a constant (which may be ignored in determining the charge 

 distribution) while Ft ends in a term in w~ . 



We have seen that the charge distribution on Ci is determined by Faiw), 

 and from equation (78) we find 



fro\ ^ \^ /wo\Tsin M sin {n — 2)^ . ~| 



(88) 



Corresponding to each power ^" in Fi{p) we have a finite Fourier sine series 



for q\^). Conversely, the powers of p from to n, for each value of it, may 



be summed in such proportions that the resulting w*^ degree polynomials, 



Fi(p), correspond to charge distributions sin nd^ on Ci . The actual form of 



these polynomials may be determined by considering the formulas we have 



just derived. 



sin n^ 

 If the charge distribution is Cn — -. — , the corresponding term in Fa{w) 



is Cn{w/ky, and this is matched by the term Cn{—k/wY in Fb(w). Hence the 

 interior function for this charge is 



*,=e.[©- +(-!)■ 



(89) 



Now on the real frequency axis, p = iw, the solution of equation (84) for w 

 in terms of co is 



w = ke^, 6 = sin"'-. (90) 



coo 



This means that the real frequency axis in the />-plane in the region j u) | < ojo 

 corresponds to a semicircle of radius k in the w-plane. Substituting from (90) 

 in (89) we have 



F,M = CnW"' + (-)V-'»'|. (91) 



Hence, corresponding to a charge distribution 



//„\ ^ ^ sin w^ , ^/ 



in the w-plane, we have, on the real frequency axis in the /i-plane. 

 Fi(to) = I Z CAe'"' + (-)" e-""] + Co 



00 *o 



= 2 C2m cos 2md -i- iJ2 C2m+i sin (2w + 1) 5 



m=0 rn=0 



