FORMAL REALIZARILITY THEORY — I 273 



nit ion of juxtaposition 



I^ut tluMi/ = PJ*j, u = Eu, and In- (k^tinition of M 



Tlioroforo for every peT m , Mip) = L\{p). Therefore there is a fre- 

 (luency domain (F.v/) for Li such that Li is ni. 



XIX THE NECESSITY PROOF 



19.0 Fortunately for this section, those parts of network tlieory which 

 we recfuire have recently been very succinctly stated by J. L. Synge '. 

 We shall paraphrase them here, referring the reader to the source " for 

 details of definition. 



19.01 First, we observe that in Cauer's definition'', which w(> shall 

 repeat in detail below, an ideal transformer with m windings is a 2w-pole 

 whose terminal pairs are the termini of the respective windings. 



A system of m coupled coils is a 2w-pole with similarly defined terminal 

 pairs. 



19.02 Given a 2n-pole N which is a finite passive network, let us adjoin 

 ideal transformers as in Figure 1(b). We then draw the ideal graph of 

 this network. Adjoin to the graph ideal generator branches 71 , • ■ • , 

 7„ , jr between 7V and TI , I < r < n. Let 0r be the ideal branch repre- 

 senting the transformer winding between Tr and Tr , I < r < n. Enu- 

 merate the remaining branches of the graph /3„+i , ■ ■ ■ , (3b . 



19.03 The branch 7^ is in a mesh with (3r and no other branches. Let us 

 call this the r-th external mesh. Any basic set of meshes must include 

 each of these. 



19.04 Let fi , • ■ • , fn be the currents in the generator branches, 

 Ai , • • • , kb the currents in the branches I3i , ■ ■ • , ^b and 



Let Wi , ■ ■ ■ , Wn be the voltages across the generator branches, 

 t'l , • • • , ^6 the currents in the /3i , ■ • • , f5b and 



[w] = [Wi , ■ ■ ■ , Wn , Vi , ■ ■ • , Vb], [v] = [Vi , • • • , Vb]. 



19.05 Let us choose a basic set of meshes, let ji , • • • , js be the respec- 

 tive mesh currents, and 



[J] = [jl, '•• , js]- 



