FORMAL REVLIZABILITY THEORY— I 229 



the same 2X2 pole, because — by its very equivalence — it would admit 

 the same pairs. The closest association we can make between a 2n-pole 

 and a network, then, is to identify the 2n-pole with an equivalence 

 class of networks. 



4.23 The completely symmetric role played by voltages and currents 

 in this definition of general 2/<-pole will make it possible to take early 

 advantage of the well-known duality principle of network theorj'. We 

 shall do so freely. 



■4.3* We shall call a 2n-pole physically realizable if its admissible pairs 

 [v, k] are the solutions of a .system of differential equations obtained 

 from a finite passive network, admitting the limiting elements: ideal 

 transformers, open circuits, and short circuits. 



V. PHYSICAL PROPERTIES OF XETWORK.S 



5.0 There are clearly a great many properties of finite passive networks 

 which are not \'et possessed by the general 2n-poles now introduced. It 

 is instructive to examine these properties physicallj'. 



5.1 In the fin«t place, the d>'namical coiLStraints (a), (h), and (c) of 4.01 

 are expressed by linear, time invariant, differential equations. Accord- 

 ingly, the 2n-poles of network theory' are: 



5.11 Linear, in that the class of admissible pairs [v, k] is a linear space; 



5.12 Time invariant, admitting \N-ith each [v(t), k(t)] also all 

 [v(t -\- t), kit -{- t)] for aribtrary r. 



5.2 In the second place, a physical network N cannot predict the future, 

 i.e., it cannot respond before it is excited. This can be formalized in 

 terms of the pairs [v, k] admitted by N, but to do so would require some 

 digression. The rea.sons \\ill be .seen under 5.7 below. 



5.3 We ha\e already mentioned the constraints imposed on voltages 

 and currents in a network by the topologj' of the network, through the 

 medium of Kirchoff's laws. The.se constraints have three important 

 properties : 



5.31 They are workless, since they are impased by resistanceless 

 connections, leaklass nodes, and, in the formal theor>', by ideal 

 transformers. 



5.32 Though it .seems scarceh' necessar\' to .say it, they are the onh' 

 workless constraints. All other constraints are djmamical and have 

 powers or energies associated with them. 



* Technical paragraph as e.xplained in Section 2.91. 



