FORMAL RKALIZAHILITY THEORY — II oGS 



may take the first m current vectors, /v'l , h , • ■ • , k,^ , specifying this 

 frame, to span J. It follows from the matrix form of G then that the 

 corresponding dual vectors Vi , ■ ■ ■ , v,,, span the range of G — i.e., the null 

 space of Z^"\iwo)- We shall adopt such a frame for the further discussion. 

 Let Ki be the space spaiuied by k^+i , • • • , kn , and Vi that spaimed 

 by i;„!+i , • • • , Vn , ill this frame. Then 



K = J © Ki 



(1) 

 V = U © Vi , 



say, where U = J*, Vi = Kf [Cf. (I, 10.0)]. 



If M is the name of any given 2n-pole discussed in the paragraphs 

 4.4 to date, we let M denote the Cauer equivalent of M in this new 

 frame. 



4.51 Let N,; be the 2//?-pole whose matrix in the new frame is the in X w 

 non-singular admittance matrix which, when bordered, gives the matrix 

 of the operator 



G^V) = -rf-2 G. 

 V + Old 



The 2n-pole whose matrix is (j(p) then obtains by adjoining n-m open 

 circuits to No . The matrix of No operates from U to J and has an inverse. 



4.52 Fig. 5 shows a diagram, which n = 5, m = 3, of the manner in 

 which we now have N represented. The terminals on the extreme left 

 are those of N. N is obtained from N by a transformer. The horizontal 

 current paths cut the dotted section A-A at points which may be inter- 

 preted as the terminals of N. Ideal transformers, as in Fig. 1 of I, can be 

 introduced here as needed. Putting them in the diagram merely com- 

 plicates the picture 



)nnection of Nx 



are on B-B. N^^\ again, is the parallel connection of a 2n-pole obtained 

 from No by the adjunction of open circuits, and N^^\ The latter has its 

 terminals on C-C. Again, N^^^ is the series connection of Ni, and N^^\ 



4.53 Let M.^ be the device between A-A and D-D of Fig. 5. This device 

 has n terminal pairs on A-A and n more on D-D. We may suppose that 

 ideal transformers are attached at each terminal pair as in Fig. 1 of I, 

 since including them in the construction of N Avould not alter its be- 

 havior. Then M.^ is a 2(2w)-pole. 



M AD is constructed from certain 2r poles (with various r) as indicated 

 in the diagram of Fig. 5. The ideal graph* of this diagram (rather, of 



*Cf. (I, 4.1). 



N is the series connection of Nx and N'^\ The terminals of the latter 



