FORMAL REALIZABILITV THEORY — II 551 



IB: This is the analog of the step in the Brune process for scalars 

 in which the reactance of a minimum resistance structure is 

 tuned out to create a zero. The details are intricate in the 

 generalization to 2/i-poles. 

 AB: This is the dual operation to IB. 



F: A 2wr-pole Nr which has a constant PR matrix (admittance or 

 impedance) is realizable at once, by 3.1. The operation F de- 

 notes this realization. 

 To each Nr , one of these nine operations is to be applied. The effect 

 of the last (F) is clearly salutary. That of each of the others is to split 

 off a realizable piece of N^ and leave a 2nr+i-pole N^+i to which again 

 some one of the operations is to be applicable. 



Exactly which of these operations to apply at any stage depends upon 

 the properties of the Nr in cjuestion. We shall first devise a notation for 

 describing the relevant properties of Nr , and then in 4.04 present a table 

 which summarizes what is to be proved in the paragraphs 4.1 et seq. 



4.02 Definition: We say that Z(p) has a zero of its real part at p = v'coo 

 if for some A" e K, k 9^ 0, w^e have 



[Z(t«o) + Z(-ia)o)]A- = 0. 



4.03 Let / be an integer describing a 2n-pole N as follows: 

 7 = if N has no impedance matrix. 



/ = 1 if N has a non-constant impedance matrix which has no poles 



on p = ico, and no zeros of its real part on p = t'co. 

 7 = 2 if N has a non-constant impedance matrix with a zero of its 



real part on p = z'w, but no poles on p = iw. 

 7 = 3 if N has an impedance matrix with a pole or poles on p = ico. 

 Let A be an integer describing the admittance category of N in a 

 dual way (e.g., A = if N has no admittance matrix, etc.). 



Let (7, A) denote the category of 2w-poles N for which the indicated 

 values of both 7 and A are true. Let 



(7i + U , A, + .42) (1) 



denote the category of 2n-poles N for which either 7i or I2 is true and, 

 simultaneously, either Ai or Ao is true, with a similar definition for 

 more summands. Then for example the category (1) above consists of 

 the logical union of the following: 



(h , A,), ih , A,), (I, , .40, (72 , A,). 



Let C denote the category of 2n-poles N which have a constant ma- 

 trix, impedance or admittance. 



