iV-TERMINAL SWITCHING CIRCXriTS 677 



we have F < 1. Also, F 9^ since any pi satisfies 



Pi<P<H-2log H. 



Since F 5^ it follows that F > 1/P. For if < F < 1/P, adding another 

 switch to the collection S will increase XI^ pi without making it exceed 

 F - 2 log H. 



Using (3) and (5), the number of contacts in the network is less than 



^ {H - 2 log H)F -\H'^H-2 log H/ ' 

 Since P < H -2\ogH, 



W H H - 2logH 



and the theorem is proved. 



Only a small fraction of the functions will use up this many contacts. In 

 any particular case, the number of contacts used will be about 



(hi) 



H - 2 log H 



and, if many different sizes of switches are used in the network, one should 

 be able to make 1/F much closer to 1 than P. Even when all the switches 

 are the same size, one expects 



in about half the cases. 



Part II: TV-Terminal Networks 

 Synthesis 



Let the accessible terminals be labelled 1, 2, • • • , N.To each pair i, j of 

 terminals of an iV- terminal network there corresponds a hindrance function 

 Bij{xi , • • • , Xm) which tells whether or not there is a closed path between 

 i and j. The Bij satisfies a consistency requirement 



(7) "Bia + Bab+ '-' +Bde+Bej=0 impUcs Bij = 0". 



The number of consistency requirements (7) is 



^ N\ ^eNl 

 h2{N - r)\^ 2 ' 



