236 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



7.5* Continuing toward realizability, we introduce 



P4. If peTL , then peTz.. If [v, k]eL{p), then [v, k]eL(p). 



This postulate embodies most of the reahty properties of networks. 

 It has as an immediate consequence the 



7.51* Lemma: If L satisfies PI, P2, P3, and P4, then all of 



are real. 



Proof: By P4, veY l{v) = V^ implies veW tip) = Vl . Hence Vl is 

 real. Then K^o = (Vl)° is real, and dually. 



7.6* The three remaining postulates on L refer to scalar products. 

 They are concerned with the energy questions related to passivity, 

 rather than with the workless constraint questions. 

 P5. If [w, j]eL{p) and [v, k]eL{p), and if 



(A) u and v are real, or if 



(I) j and k are real, 



then 



{u, k) = (v,j). 



7.61 This is the property which provides the reciprocity law. In its 

 presence, the relations in P3 may be weakened to 



V.(p) = Vz, 3 (K,o(p))", 



KM = K, 3 (V.o(p))'. 



This fact will appear as a consequence of the lemma of Section 12. 



7.7* Lemma: A consequence of P2 and P3(A) is that if 



[v, kr]eL{p), r = 1,2, 



then for any ueY^ , 



(u, ki) = {u, ko). 



For by P2 w^e have that 



[v - V, A-i - A-.,] = [0, A-i - k.2\eL(p), 



hence A;i — k^eKLo . Then however, by P3(A), ueYL imphes ue(K.ij)) , so 

 * Technical [laragraph as explained in Section 2.91. 



