558 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



and positive definite matrix aS and find the a described in 4.31. Then the 

 matrix 



Zi(p) = Z(p) - aS 



is PR and has a zero of its real part at p = t'coo . 



Proof: Clearly Zi(p) is symmetric. By 2.09, then, Zi(p) is PR if the 

 function 



^i(p) = (Zi(p)A-, /c) = (Z(p)k, k) - a(Sk, k) (3) 



is PR for each A'. Clearly this function is rational and has no singularities 

 in r^ . It suffices then to show that its real part is non-negative on 

 p = to:. By (1) of 4.3 



Re ^i(fco) = (R{io:)k, k) - a(Sk, k) 



and this is non-negative by (i) of 4.31. 



That Zi(p) has a zero of its real part at p = iwo is (ii) of 4.31. 



4.33 Let Ni be the 2n-pole whose impedance matrix is the Zi(p) of 4.32. 

 We define the operation Res as that which produces Ni from N. It is 

 evident from (3) above that the poles of Zi{p) are exactly those of Z(p), 

 hence 7 = 2 for Ni . Nothing can be said of the admittance matrix for 

 Ni . 5(Zi) = 8{Z) by 2.14 and 2.15, and rii = n by construction. The 

 claims in 4.04 are now established for Res, and dually for Con. 



The relation 



Zip) = Z,(p) + aS 



shows that N is a series combination of Ni and a device with the im- 

 pedance matrix aS. Since a > 0, this latter is a realizable resistance 

 network (3.1). Hence N is realizable if Ni is. 



4.34 We observe that no reactive elements are used in the network 

 between Ni and N (2.12, 3.12). This verifies 4.07 for Res and Con. 



4.4 We now turn to the piece de resistance of the generalized Brune 

 process, the operation IB and its dual. Consider a 2n-pole N in the 

 category (2, 1 -f- 2) — i.e., its impedance matrix Z(p) exists, is not 

 constant, is non-singular on p = iu, and has a zero of its real part at 

 some p = iiioo . We have for some /c e K such that k 7^ 0, 



R(io}o)k = 0. (1) 



Here, R(p) is as defined in 4.3. 



4.41 We now assert that we may assume that < ojo , and icoo 5^ 0° in 



