NUMBER OF IMPEDANCES OF AN n TERMINAL NETWORK 301 



impedances which are equal by the reciprocity theorem; the doubling 

 of Tn in forming the total is due to the equality in number of open- 

 circuit and short-circuit transfer impedances. The numbers increase 

 rapidly with n, reaching almost 126,000,000 for ten terminals. The 

 number of representations, which is the number of combinations of 

 the measurable impedances |w(« — 1) at a time less the number of 

 non-independent sets, at a guess increases even more rapidly, indicating 

 a variety of equivalents, few of which seem to have been investigated. 



TABLE I 



Measurable Impedances of an m-Terminal Network 



Because the field of the work is somewhat unusual, considerable 

 space is given to details in the formulation of the problem before 

 proceeding to the enumeration proper. The enumerating expressions 

 obtained are found susceptible of some mathematical development 

 which, though subsidiary to the main object of the paper, seems of 

 sufficient interest to justify the relatively brief exposition given. The 

 arrangement is such that readers not interested in this mathematical 

 half may obtain the substance of the paper without it. 



Formulation of the Problem 



The enumerating problem is essentially one of combinations, as 

 indicated schematically in Fig. 1, which shows the n terminals of a 

 linear passive network together with the apparatus required for 

 impedance measurement, that is, a source, a voltmeter and an ammeter, 

 each supplied with two terminals (shown solid to distinguish them 

 from the network terminals). Each of these latter may be connected 

 across any pair of the n terminals except that the ammeter, which 

 constitutes a short circuit, may not be connected to terminals to which 

 either the source or voltmeter is connected; in the former case no 

 current will be supplied to the network and in the latter the voltmeter 

 will read zero. The ammeter may be connected in series with the 

 source to read the source current, of course. 



