FORMAL REALIZABILITY THEORY — I 225 



sive networks. Second, wq seek to make mathematically precise those 

 ideas which appear to form the basis of general realizability theory. 

 Sections 4 through 7 introduce the immediate mathematical machinery 

 for this. Section 8 states the fundamental realizability theorem and 

 outlines its proof. At this point the reader has had an introduction to 

 the results of the paper. The remainder of the paper is then devoted to 

 the technical details of proof. Beginning with Section 12, the de\'ice of 

 starring the technical passages will be dropped. 



3.1 Cauer distinguished precisely the class of networks called 2ri-poles 

 from the class of all multi-terminal n(>t works. He also showed that, by 

 the use of ideal transformers, any multi-terminal network is eciuivalent 

 to a network which is a 2/(-pole (for some n) in his sense. We shall in 

 Section 4 define a class of objects to be called general 2n-poles. This 

 class includes all electrical networks which are 2n-poles in Cauer's sense. 

 Its definition abstracts the significant properties isolated by Cauer. 



For the study, alone, of finite passive networks, this definition is 

 uimecessary, since one can in fact so put the arguments as to deal only 

 with 2N-poles which are finite passive networks, and therefore to deal 

 only with concepts already defined in Cauer . The somewhat physical 

 notion of a general 2/i-pole is a convenient backdrop against which to 

 display the important physical properties of finite passive networks, 

 and, indeed, of networks in general. Having it available, we use it 

 throughout the realizability arguments. 



IV. DEFINITION OF GENERAL 2n-P0LE 



4.0* Network theory establishes a correspondence between oriented 

 linear graphs and systems of differential equations. With each node of 

 the graph is associated a potential En = EniO and with each oriented 

 branch a current h = hit). These potentials and currents are constrained, 

 first by Krichoff's laws, and second by differential eciuations which de- 

 pend upon the nature of the branches but not upon the topology of the 

 graph. 



4.01* A finite passive network is one whose graph has the following 

 properties : 



(i) There are finitely many nodes, 1,2, • • • , A''. 



(ii) There are finitely many branches, \, 2, ■ • • , B. 



Technical paragraph as explaiiKMl in Section 2.91, 



