78 BELL SYSTEM TECHNICAL JOURNAL 



to require less. This will be done by showing that there are not enough 



2" 

 networks with less than (1 — e) — branches to go around, i.e., to represent 



n 



all the 2^" functions of n variables, taking account, of course, of the different 

 assignments of the variables to the branches of each network. This is only- 

 possible due to the extremely rapid increase of the function 2" . We require 

 the following: 



Lemma: The number of two-terminal networks with K or less branches is 

 less than (6K) . 



Any two-terminal network with A^ or less branches can be constructed 

 as follows: First line up the K branches as below with the two terminals 

 and b. 



a. 1—1' 



2—2' 

 3—3' 

 4—4' 



b. K—K' 



We first connect the terminals a,b,l,2, ■ ■ ■ ,K together in the desired way. 

 The number of diferent ways we can do this is certainly limited by the num- 

 ber of partitions oiK-\- 2 which, in turn, is less than 



for this is the number of ways we can put one or more division marks between 

 the symbols a,\, • ■ ■ , K,b. Now, assuming a, \,2, • ■ ■ , K, b, intercon- 

 nected in the desired manner, we can connect 1' either to one of these ter- 

 minals or to an additional junction point, i.e., 1' has a choice of at most 



A+ 3 



terminals, 2' has a choice of at most A -f 4, etc. Hence the number of 

 networks is certainly less than 



2^+i(iC -i- 3) (A + 4) (A + 5) • • • {2K -f 3) 



<{6KY K>2> 



and the theorem is readily verified for iv = 1,2. 



We now return to the proof of Theorem 7. The number of functions of n 



variables that can be realized with elements is certainly less than 



n 



the number of networks we can construct with this many branches multi- 



