CONSTANT RESISTANCE NETWORKS 189 



shunt arm {Pmn'\^ + l)/amX, then for m odd, that is, for a series 



element, 



1 

 X = ^^^ 



(m41)/2 



m/2 



VVheti m = n, or for the last series arm, a special relation is necessary-, 

 readily obtained by the limiting value of reactance as x approaches 

 zero. This gives 



(fll + 03 + • • • + an)x = (1 + ^pm)x 



or 



0„ = 1 + l.pm — («! + «3 + • • • + an-2). 



To use these relations it is necessary to know the expression for Xm, 

 the reactance to the left from the successive points in the network. 

 To determine this in terms of the elements already known use may be 

 made of the following theorem : 



If the impedance looking to the left into a network is Z, the impedance 

 to the left from A, any point ivithin the network is the negative of the 

 impedance to the right from A when the network is terminated on the 

 right by an impedance — Z. 



For example, referring to Fig. 5, to determine az it is necessary to 

 know the reactance to the left starting with a^x.- By the theorem 

 this is 



1 



X3 



a^x 1 



1 — PiV Oix — tan |3i 

 and when x — I/P2 we have for as 



1 _ az _ 1 



^ ~ Pi" - Pi2 ~ ai - P2(tan ^1)2 ' 



The impedance at that end of the filter terminated by the resistance 

 is of interest. Its value of course depends upon the terminating 

 impedance at the junction of the two filters, but assuming that this 

 impedance and the separate terminating resistances are all Ro, the 

 impedance from the load of the first filter is Ro tanh (02 + ijSi) if 

 terminated in a series arm and Ro coth (a2 + i^i) if terminated in a 

 shunt arm. Note that the impedance of the first filter depends upon 



