NUMBER OF IMPEDANCES OF AN n TERMINAL NETWORK 307 



available; one of these pairs is the source pair measuring a short- 

 circuit transfer impedance which must be excluded; hence, remember- 

 ing that reciprocity theorem duplicates are eliminated in the latter: 



L' X, n — •^ 



= 2 



- 1 



- 1 



■>■ X, n 



-t ° x-\-\, n ^^ 2 



1 



^x+l-Ji+l, n- 



Degrading x by unity to obtain the form of equations (2), the third of 

 the lower case factors is reached as follows: 



'-a)[a)-][Cr'-' 



= i[20(x)4 -f 10(x)5 + We]. 



(5) 



The common factor 5^, n remains for determination. 



Returning to the connection conditions illustrated for three terminals, 

 this number is the number of ways separators may be placed between 

 letters of the collection /i, k- • -tn symbolizing the terminals so as to 

 produce x compartments, symbolizing merged terminals. The ter- 

 minal symbols ti- • -tn may be thought of as the prime distinct factors 

 (excluding unity) of some number and the number Sx, n is then 

 identically the number of ways a number having n distinct prime 

 factors may be expressed as a product of x factors. The enumeration 

 for this latter problem is given by Netto,^ who gives the recurrence 

 relation 



•Ji, n+l ^^ X^x, n \ »Jx— 1, n 



with 



Sn. 



1, 5i, n = 0, X > n, 5o. n = 0, « 4^ 0. 



This is the recurrence relation for the Stirling numbers of the second 

 kind,^ the notation for which has been adopted in anticipation of the 

 result. These numbers are perhaps better known as the "divided 

 differences of nothing," that is, as defined by the equation: 



5x. n = lim-,A^2» = -jA^O^ 

 z=o x! x! 



where A^ denotes x iterations of the difference operator with unit 



3 "Lehrbuch der Combinatorik," Leipzig, 1901, pp. 169-170; Whitworth, "Choice 

 and Chance," Cambridge, 1901, Prop. XXIII, p. 88, gives a generating function for 

 the solution of this problem which, it is not difficult to show, leads to the same answer. 



* Ch. Jordan, "Statistique Mathematique," Paris, 1927, p. 14. 



