FORMAL REALIZAlilLITV THKOltV — II 567 



We can now compute tlie voltage across Mad eorrespoiulinp; to the 

 ciirreut (10). Let w be tli(^ /(-tuple of voltages appeai'iuf;- at the section 

 B-B or C-C' of Fig. (i, with its coiupoiieiits listed in the a|)pro|)riate 

 order. Then we may iiitei|)i'et ir as a xcctor in V, and write it 



IV = ;/„ © To (12) 



where »o c U, ro e Vi . Now by KirchofV's current law applied to the 

 slumt arms in the upper channels of Fig. (>, the current into No is 



ii + h , 

 and therefore 



«o = G-\p)(j,+j-;). (13) 



B}^ Kirchoff's voltage law applied to a typical mesh which begins on 

 A-A, goes through N.v to B-B, and then through a shunt arm and i-eturns 

 to A-A, the voltage /(-tuple appearing at A-A is 



X(p)(ii + k) -f w. 

 Referring to (12), let us use Vo also to denote the vector 



^/o © e V, 

 and I'o to denote 



© Vo € V. 



Interpreting (13) in this way we get 



X{p)(j, + k) + G-'(p)in + J2) + Vo (14) 



as the voltage n-tuple on A-A. 

 A similar calculation gives 



L{pKJ2 - k) + G-\p)(jr -f i,) + Vo (15) 



as the voltage n-tuple on D-D. The ordered pair (14), (15) then gives 

 the voltage 2M-tuple corresponding to (10), in the notation analogous 

 to (5). 



4.57 X(p), L(p), and G~\p), respectively, are defined in (6) of 4.44, 

 (11) of 4.491, and (11) of 4.5G. Each one is finite except at p = and 

 p = CO , Let Fm be the complex plane from which these two points are 

 deleted. It is now possible to show that the linear correspondence whose 

 pairs, for each p e Fm , are the voltages (14), (15) e V and the ciu'rents 

 (10) € K', satisfies PI through P7 of (/, 6, 7)— that is, is PR (I, 1G.71). 



