230 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



5.33 They are frequency independent, that is, holonomic in the sense 

 of dynamics. 



5.4 The workless and the dynamical constraints in a physical network 

 are all defined by relations with real coefficients. The space of admissible 

 pairs is then a real linear space. 



5.5 The positivities specified in 4.01 are characteristic of passive sys- 

 tems. They correspond to the fact that the power dissipation and the 

 stored energies are all positive. 



5.6 By definition, finite passive networks contain finitely many lumped 

 elements. Correspondingly, their resonances and anti-resonances are 

 finite in number. 



5.7 We are accumstomed to dealing with networks which have, in addi- 

 tion to the properties listed above, a kind of non-degeneracy, in that 

 the list of admissible pairs [v, k] satisfies: 



5.71 At least one of v or k can be specified arbitrarily — any real function 

 is admitted; 



5.72 When the free number of [v, k] is specified, the other is uniquely 

 determined. 



For these non-degenerate networks, the property 5.2 above is easily 

 formalized: if, say, k is determined by v, then 



v\t) = v\t) for t < k 

 implies 



k\t) = k'it) for t <h, 



where [v\ k'] are admissible pairs, i = 1,2. The general statement of 5.2 

 involves this condition and some discussion of the v'b for which N ad- 

 mits [v, 0], and the dual notions. 



5.8 The reason for speaking in terms of pairs [v, k], instead of in terms 

 of "cause" and "effect," or "impulse" and "response," is hinted at by 

 5.7 above. For the tacit implications of the cause and effect language 

 completely obscure the fact that 5.71 and 5.72 are properties which are 

 not automatically possessed by electrical networks. In fact, the simple 

 four-pole of 2.41 — a pair of unconnected terminals Ti , Ti , and a pair 

 of shorted terminals T2 , T'2 — has neither property, yet it is a perfectly 

 good linear time invariant four pole. Its admissible pairs are 



[(.1 , 0), (0, h)l 



where Vi and /c2 are arbitrary real functions of the time 



