FORMAL REALIZAHILITY THEORY — I 235 



7.3* (livcMi a linoar corrcspoiuhMico />, \v(^ make sovcnil definitions: 



y ,.(])) is tlie sot of all reY such that there is a /,• with [v, ^eL(p). 



KiX'p) is the set of all I.eK such that there is a v with |/', L-\eL('p). 



V;,(i(/j) is the set of veViXp) such that 



[v, 0]eL(p). 



Klo(p) is tlie set of keKiXp) «uch that 



[0, k]eL(p). 



7.31* The postulate P2 implies that for each p^Tl , V/.(p), K/ip), 

 '^Loip) and K/.o(p) are all linear manifolds. 



7.32 Vl(p), for example, is the set of veY such that Nz, admits v at 

 frequency p. 



7.4* We now postulate 



P3. There exist fixed linear manifolds Vl C V, K,, C K such that 



(A) For every peT, , W,(p) = V, = (KUp))' 



(I) For every peT, , K,{p) = K. = (Y,oip)Y. 



7.41* We may henceforth write Vlo , K^o , for Ym{p), Klo(p), knowing 

 that, under P3 



V.0 = (K^)°, 



7.42 Linear correspondences satisfying P3 abstract the properties men- 

 tioned in 5.3. The equalities Vl(p) = Vl , Kl(p) = K/, guarantee the 

 frequency-independence of the workless constraints. The equalities 

 Vl(p) = (Klo(p))", K/.(p) = (Vlo(p))° in a sense guarantee that the 

 only constraints imposed upon admissible currents and voltages (as 

 opposed to constraints relating currents and voltages) are those which 

 arise from open or short circuits, i.e., are workless. 



7.43 An illustrative conseriuence of P3, for example, is that if L satisfies 

 P3 and if N^ is such that all of the current amplitudes can be specified 

 arbitrarily, then indeed the voltages are determined by the currents. 

 This will appear as a consequence of 8.1. It is a very general theorem 

 about networks of a kind that this author, at least, has not heretofore 

 encountered. 



* Technical paragraph as e.xplained in Section 2.91. 



