238 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



By an application of Kron's method (described by Synge ), it will then 

 be shown that the imposition of Kirchoff's laws preserves the postulates. 

 This work is most efficiently performed after the full machinery of the 

 sufficiency proofs is available, and will be done in Section 19. 



8.04 The sufficiency of PI through P7 can be deduced — and we will do 

 so — from the lemmas to be quoted below. Apart from Section 19 on 

 necessity, the remainder of the paper is devoted to the proofs of these 

 lemmas. 



8.1* Lemma: If L is a linear correspondence satisfying PI, P2, P3, and 

 P4, then there exists a fixed real nonsingular matrix W such that 



8.11 The list L wip) of all pairsf 



[W-\ W'k], 



where [v, k]eL(p), describes a linear correspondence Lw satisfjdng 

 PI, P2, P3, and P4. 



8.12 The 2n-pole Nir (= ^lw) associated with Lw consists of 



(i) Some number r of open-circuited terminal pairs (Ti , Ti), ■ • • , 



(ii) Some number s of short-circuited terminal pairs (Tns+i , Tn-s+i), 

 (iii) A set of w = n — r — s terminal pairs (T'r+i , Tr+i), • - • , 



\J- r+m ) -t r+m) • 



8.13 Either m = 0, or the terminal pairs in (iii) are those of a 2??i-pole Ni 

 which has a nonsingular impedance matrix Zi(p). 



This lemma, and the following, will be proved in 13.2. 



8.2* Lemma: If L satisfies Po, P6, and P7, then Zi(p) is a positive 

 real| matrix, that is, Zi(p) satisfies (i), • • • , (iv) of 1.1. 



8.3* Lemma: If a 2m-pole Ni has a positive real impedance matrix, then 

 Ni is physically realizable. 



This is the sufficiency half of the matrix realizability theorem 1.1. 

 Part II will be devoted to its proof. 



8.4* Lemma: If N n- is physically realizable, then N can be constructed 

 from it by the use of ideal transformers. 



This is Cauer's Transformation Theorem" about which we shall say 

 more in Section 9. 



* Technical paragraph as explained in Section 2.91. 



t W'^ and W are respectively the reciprocal and the transpose of W. 



X Gewertz's terminolog}-*, by now traditional. 



