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THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



erations. They are, however, expressible in terms of the basic operations 

 of juxtaposition (I, 17) and restriction (I, 18). 



For example, the series connection of Ni and N2 is formed by first 

 juxtaposing Ni and No , to get a 2 X 2n-poIe N. Let J be the 2n dimen- 

 sional space of 2?i-tuples 



j = [jl , • • ' , jn , ^1 , • • ■ , 4]. 



We interpret this j as a 2n-tuple of currents in the 2 X 2/(-pole N, where 

 jr is the current in the r*^ terminal paii* of Ni and 4 that in the r*^ pair 

 of N2 , 1 < r < n. Let K be an n-dimensional space. Given an n-tuple 

 k e K, say 



k = [m , • • • , Kn\, 



we define the operator C from K to J by 



J ^ Ck = [ki , • • ' , Kn , ki , • ' • , kn\. 



Restricting N by C gives the series combination N of Ni and N2 . The 

 details may easily be supplied by the interested reader. 



2.41 Representing the series and parallel connections in terms of juxta- 

 position and restriction makes the lemma, 2.2, an inmaediate conse- 

 quence of the lemma of (I, 17.2) and the theorems of (I, 17.3, 18.3). 



2.5 We report here for record a curious property of PR operators which 

 has so far found no application: 



Lemma: If Z{p) is a PR impedance operator from K to V = K*, 

 and Y{p) any PR admittance operator from V to K, then the operator 



1 + Y(p)Z(p) 





N, N2 N, N2 



SERIES CONNECTION OF N, AND N^ PARALLEL CONNECTION OF N, AND N2 



Fig. 2 — Series and parallel connection of 2N poles: Series, left, and parallel, 

 right. 



