FORMAL REALIZABILITY THEORY — II 595 



Finally, we note the determinant relation 



1 Wr(q) I = I A(ci) II Z,(c,) II B(q) \ = (constant) X I Z,(q) \ , 



since the determinants of .1 antl B are constant. Now Zi(q) lias no pole 

 at 9 = 00 , hence its determinant is finite there. The same is true of 

 Z7 (q), so indeed 



Now by direct calculation 



I "^'W I = I'll ' ' ' 1" n - 



Since this is finite and not zero at q = oo, the numerator and denom- 

 inator are of the same degree. Hence 



8(Z) = 5(Zi) = degree W,) = degree (He,) = d{ZT') = 5(Z~'). 



VI. THE EXACT COUNT OF REACTIVE ELEMENTS 



6.0 We showed in the inductive argument of 4.07 that the Brune proc- 

 ess constructs a realization for a given Z(p) which uses exactly 8{Z) 

 reactive elements. To establish 2.18, we must still show that no net- 

 work with fewer than 8(Z) reactive elements can do this. To prove this, 

 we shall show that if Z(p) is the impedance matrix of a network con- 

 taining X reactive elements, then 



8{Z) < X. (1) 



We shall, in fact, in this Section show somewhat more than (1). The 

 demonstration of (1) requires enough calculation that is as easy to prove 

 the following extension of 2.18. 



6.01 Theorem: Given any linear correspondence L, (I, 6.2), which 

 PR, (I, 16.71), there exists a number 8{L) such that 



(i) The realization process outlined in (I, 8) and 4.07 of this Part 

 constructs with 8{L) reactive elements a network realizing a 

 member of the Cauer class of L. 



(ii) If L is the correspondence established by the Cauer class of a 

 physical network which contains x reactive elements, then 



8{L) < X. 



The proof is divided among the remaining paragraphs of this Section. 

 We maintain here a strict distinction between geometric objects and 

 their concrete coordinate representations. 



