240 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



formers, which is described by the Hnear correspondence Lw whose pairs 

 at each p are the pairs 



[W'v, W'k], (2) 



where [v, k]eL{p). 



We refer to Cauer for the proof. It is straightforward. 



9.21 We shall use the second form (9.2) of Cauer's theorem in our 

 reahzation process. Notice that it is in a sense a "physical" theorem, 

 about the way one physical network is related to another. It is used in 

 this way: we shall always solve a realizability problem by finding some 

 network N which is easily realized, and then a W such that N „ , which 

 is now realizable, provides a solution to the given problem. 



9.22* We shall call the 2n-pole N w a Cauer equivalent of N. 



9.3 Although Cauer's theorem will be applied, in a sense, only a posteriori, 

 its effect is fundamental. For it implies that formal physical realizability 

 is a property of matrices which is invariant under the operation (1) 

 or a property of correspondences which is in^'ariant under (2). There is 

 an extensive classical literature on the properties of matrices invariant 

 under operations like that of (1), and the effect of Cauer's theorem is to 

 make these results all available to formal realizability theory. 



9.31* It is worth observing here that we are already well set up to use 

 Cauer's theorem: 



Lemma: If L is a linear correspondence satisfying Pi, • • • , P7, then 

 the correspondence Lw of 9.2 also satisfies PI, • • • , P7. 



Proof: Let M = L^ . Pi and P2 for M are obvious, with Tm = Tz. . 

 By definition of M, 



V.u(p) = W-'Y,{p) = W-'Y, 



KAp) = W'KM = W'K, 



Yuoip) = W-'YUp) = W-'Y,o 



Km(p) = W'KUp) = TF'K^o 



where W~^S for a manifold S consists of all n-tuples W~ v, where veS. 

 Hence in P3, 



yap) = v., = ir^^v. 



K.,(p) =K,, = W'K, 



for fixed manifolds Y m , Kn as defined. 



* Technical paragraph as explained in Section 2.91. 



