232 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



P2. If [v\ k']eL{p) and [v', k%L(p), then 



[aiv + avv", ttik + a2k']eL(p) 

 for any complex numbers ai , a-i . 



6.21* Given such a linear correspondence L, we can always describe a 

 2n-pole Nl by: 



Nl admits [v, k] at frequency p if and only if [v, k]eL{p). 



That is, we can always interpret the pairs [v, k] generated by (3) and 

 (4) from the [v, k]eL(p), for each peTL , as the voltages across and 

 currents in a set of n ideal branches. We call N/, the 2n-pole associated 

 with L. 



6.22* We call Tl the frequency domain of L (or of N^). 



6.23 From here on, the words "2n-pole" can with some strain be re- 

 garded as suggestive but unnecessary. We in fact deal with linear corre- 

 spondences — having properties as yet unspecified — and shall show how 

 physical networks can be constructed which admit the pairs [v, k]eL(p). 

 Actually we use freely the concept of general 2n-pole and thereby avoid 

 some elaborate circumlocutions. 



6.24* We identify two correspondences Li and L-z as being the same if 

 (i) their frequency domains differ only by a finite set, and (ii) for each 

 p where both are defined the lists Li(p) and L2{p) are the same. 



6.3 The simplest linear correspondences are those generated by ma- 

 trices. For example, let Z(p) be an n X n matrix with, say, elements 

 Zrs(p) which are rational functions of p, 1 < r, s < n. Let Tl consist of 

 all the values of p at which Z(p) is defined. For peTt , define L{p) as 

 the class of all pairs 



[v, k] (5) 



obtained by letting k range over K, where for each k, v is defined by the 

 matrix equation 



V = Z{p)k. (6) 



This kind of matrix equation will be used throughout to symbolize the 

 n component equations 



n 



Vr = T. Zrs(p)ks, I < r <n. (7) 



The list of pairs L{p) defined by (5) clearly satisfies PI and P2. It 

 can therefore be used to define a 2n-pole Nl . It is easy to see that N/, 



* Technical paragraph as explained in Section 2.91. 



