306 BELL SYSTEM TECHNICAL JOURNAL 



sidering for the moment only the driving-point and open-circuit 

 transfer impedances, the numbers Di, „ and r°x, n are the products of 

 two factors: (i) the number of such impedances measurable for x 

 terminals, which is independent of n and (ii) the number of ways the 

 n terminals may be merged so as to result in x terminals, which is 

 independent of the impedance classes. By equation (1) this result 

 applies also to T\, „ and, as Ux, n is related to T\, „ by a factor 

 independent of n, as will be shown, it applies generally. 

 This leads to the following equation : 



\TA^Y.\t.\ S., n. (2) 



L Un J ^=2 [ «x J 



The small letters are the several factors of the first kind and Sj,, „ is 

 the common second factor. 



The small letters are determined as follows: A driving point im- 

 pedance may be measured between every pair of terminals; hence dx 

 is the number of combinations of x things taken two at a time, that is: 



dx = 



(^l) = hx{x- 1) = i(x)2. (3) 



where (x),- is the factorial symbol x{x — 1) • • • (x — i -\- \). 



For a given pair of driving terminals, there are ( j — 1 



measurable open-circuit transfer impedances since a voltmeter can be 

 connected to every pair of the x terminals except the driving pair; 

 hence, multiplying by the number of driving terminals and by the 

 factor one-half to eliminate reciprocity theorem duplicates: 



t - h"" 



2»-^ 



(4) 



= |[4Cr)3 + (x)4]. 



The second, factorial, form is given for convenience of later develop- 

 ment. 



By equation (1) this serves for enumeration of both open-circuit 

 and short-circuit transfer impedances; the direct enumeration of the 

 latter appears more difficult. 



Considering, for the generalized transfer impedances, a fixed source 

 and an ammeter in a fixed (non-source) position, the voltmeter may 



be connected across ( . 1 pairs of terminals when x terminals are 



