NUMBER OF IMPEDANCES OF AN n TERMINAL NETWORK 305 



measurement, the total of six transfer impedances being halved to 

 eliminate reciprocity theorem duplicates. With two of the terminals 

 connected by an ammeter, there are again three driving-point and 

 three transfer impedances, the latter being short-circuit transfer 

 impedances, for there are three ways of connecting pairs of terminals 

 and one driving-point and two transfer impedances for each, the 

 total of transfer impedances again being halved to eliminate duplicates. 



There are no generalized transfer impedances because with an 

 ammeter connected, there is only one measurable voltage, the driving- 

 circuit voltage. 



With terminals designated by /i, t^ and /a, the conditions arising 

 from connection of terminals may be exhibited as follows: 



Terminals distinct ti\ti\ tz 

 Pairs connected txt^ I tz 



tMh 



t\\t%tz 



the lines of separation dividing the terminals into groups such that 



the terminals in any group are merged into a single terminal. Paying 



attention only to the number of terminals in each group, the groups 



illustrated may be designated by the partition notation (111) or (P) 



and (21), the numbers in the designation being partitions^ of the 



number 3. 



The enumeration for three terminals may then be exhibited as 



follows : 



Measurable Impedances 



It will be noted that the open-circuit and short-circuit transfer 

 impedances satisfy equation (1), that is, T°z, 3 = T\, 3. 



This table and its correspondents for larger values of n show that 

 the impedances may be expressed as sums with respect to x, where x 

 is the number of terminals defined as in equation (1), from 2 to w; 

 thus e.g., Z)„ = Y^Dx, n where Dx, „ is the number of driving-point 

 impedances measurable for all conditions of merging of n terminals 

 such that the resulting number of terminals is x. Moreover, con- 



^ A partition of a number n is any collection of positive integers whose sum is 

 equal to n. It may be noted that the number of parts of a partition is the number 

 X of equation (1) ; the partition (1^) has three parts corresponding to the three distinct 

 terminals; (21) has two parts corresponding to two terminals, each merged pair of 

 terminals counting singly. 



