FORMAL REALIZAHILITY THIOOKY 1 275 



we consider the individual two-polos and the system of coupled coils 

 as separate unconnected networks. Then [Z,i{p)\ is composed of a 

 (c — n) X (c — n) diagonal mat rix in 1 he upper left field and a (d — c) X 

 {d — c) matrix in the lower right, with zeros elsewhere. 



19.11 The diagonal part of [Z,i(p)] has elements drawn from the follow- 

 ing list : 



(i) /(?>) = P 



(ii) f{p) = 8p 



(iii) f(p) = \p 



where p, 8, \ are non-negative constants, possibly different for each 

 branch. 



19.12 It is shown in texts on electromagnetic theory that the matrix 

 representing a system of coupled coils is of the form 



p[G], 



where [G] is a real, constant, symmetric, and semi-definite matrix. 

 The lower right field of [Zd{p)] is then such a matrix. 



19.13 It is obvious from this description that [Zd{p)] is PR. It therefore 

 describes a PR correspondence between (d — n)-tuples of current and 

 voltage. 



19.2 We must at last consider ideal transformers in detail. Let Vi and 

 Ki be m-dimensional spaces represented as aggregates of m-tuples. 

 Let pi , P2 , • • • , p,n be m real numbers. Let Vt consist of all m-tuples 



[a] = [oi , • ■ • , am]eYi such that 



fll (22 (I'm 



Pi P2 Pm 



We interpret these relations as follows: 



(a) If any pr = 0, then a^ = 



(b) If any two pr , Ps are not zero, then 



ttr _ tts 

 Pr p. 



(c) If only one pr ^ 0, then Ur is arbitrary. 



Let Kr consist of all 7M-tuples [b] = [bi , • • • , 6,„]eKi such that 



pJh + P262 + • • • + Prnbm = 0. 



Vr and Kr are linear manifolds. 



