228 THE BELL SYSTEM TECHNICAL JOURNAL, VL^RCH 1952 



example, when the 2X2 pole is dri\en by separate generators in the two 

 external meshes — these trans^formers can be eUminated. The definition 

 of 2r<-pole requires however that in even,' context the ideal branch con- 

 cept is vahd. 



4.2* The second aspect of the concept of general 2r<-pole is that it im- 

 poses some kind of constraint — other than that imphed by 4.11 and 

 KirchofF's laws — upon the voltages across and currents in its ideal 

 branches. Define the s%Tnbols 



and 



£ = ijt^ = [I'lit), i'-.\t), " , P-(0] 



A- = A-(0 = [hit), hit), • • • , Ut)] 



as the n-tuples. respectively, of voltages across (Tr , Tr) and currents 

 into Tr , I < r < n. These are added and multiphed by scalars bj' the 

 usual rules of vector algebra. If v and k represent simidianeous values 

 of voltage and current in the 27j-pole N — i.e., values satisfying all the 

 constraints — then we say that N admits the pair [v, k]. 



We say that N admits r if there is a /: such that N admits the pair 

 [r, k]. This k is said to correspond to v. Dually, N admits k if there is a 

 V (corresponding to k) such that N admits [r, /:]. 



The constraints imposed by a general 2n-pole N on voltages and cur- 

 rents are completeh' described by the totality of pairs [r, k] which N 

 admits. We shall define a general 2n-pole, therefore, as 



(i) a collection of n oriented ideal branches, as in 4.11, and 



(ii) a list of pairs [r, k] of voltages and currents admitted in these 

 branches. 



Hereafter we shall usuallj' drop the adjective "'general." 



4.21 The definition we have jiLSt given is, in a way, a postulational 

 form of an r< -dimensional Thevenin's theorem. It postulates that a 

 2n-i>ole is a thingt which, as far as the outside world is concerned, can 

 be represented b\' a collection of two-poles ,3, , 1 < r < n, among which 

 there e.xists a comphcated agreement as to what currents and voltages 

 will be admitted. 



4.22 The passive networks (b ) and (c ) of Fig. 1 define 2X2 poles, be- 

 cause they satisfy 2.01 and clearly permit a complete specification of 

 the admissible pairs [r, k]. Any equivalent network would also specify 



* Technical paragraph as explained in Section 2.91. 

 t In fact, at this level of generality, any thing. 



