812 BELL SYSTEM TECHNICAL JOURNAL 



whence 



Ai = a^ — 2aia2, 

 A, = a^Ka^ - 1), 



1 ± Vl +^1 + ^2 



V(i ± Vi + ^1 + A^y - A2 



ai = v(i ± vrr^i + Aif -"3Z 



The upper of the alternative signs usually gives the better reactance 

 characteristic. 



The susceptance of this network is 



B = — a^x 



(1 _!)_(! _,,,),. 



1 + Axx" + A2X^ 

 Case III — n = 3 

 The general conductance expression is 



g^± Vl - x^ 



where 



Zo 1 + Aix"" -\- A2X' -\-AzC<^' 



Ai = ai^ + laias + a^"^ — la^a^ — 1, 

 Ai= a^ai + 2^203 — 201^02^3 — laiChfi^, 

 Az = a^a^ai — a-^a^. 



These equations can be reduced to 



fll + «3 — dxChfl-Z = ± Vl + ^1 + ^2 + Az, 



fli + cfs = Vl + ^1 + 20203, 



010203 = V/I3 + a^cii. 



from which 



Vl + ^1 + 20203 - V^3 + aiai = ± Vl + ^1 + ^2 + Az. 



Upon examining the form of the radicals on the left we see that O2O3 

 is determined by the intersection of a parabola and a hyperbola. 



