iV-TERMINAL SWITCHING CIRCUITS 



675 



tions / can be obtained by connecting to the functions f{yi , • • • , jj-i ,1), 

 • • • Jiyi , • • • , Ji-i , py) through eiiyi), • • • , e^^iyi) as shown in Fig. 5. In 

 this way a new network is found which produces all functions of the j vari- 

 ables and uses ypj contacts where 



(4) y^i - ^y_i < p(2^^---^' - 2^'-"^-'). 



If we now assume that formula (3) holds for ^y_i we obtain 



Thus the theorem will follow by induction when we prove (3) for the case 

 L = 1. Since ^o = (no contacts are needed to synthesize the two functions 

 and 1) the inequality (4) reduces, when L = 1, to 



^1 < P(2^^ - 2) 



and the theorem is proved. 



Fig. 5 — Network to produce all functions of (yi , . . . , yy) . 



The induction process we have just described will use up the smallest 

 number of contacts when the large switches are used up first and the small 

 switches last. If, in the process, pj > pj+i , then the number of contacts 

 which would have been saved by making switch yy+i precede switch yj is 

 found to be 



(pj - Pj+i)2 



P1---PJ. 



X2''' - 1)(2 



P/+1 



1). 



By adding switches in order of decreasing size in the induction process, the 

 factor P in (3) can be reduced to nearly pi. , the smallest of the L ranges. 

 This refinement is unnecessary for the theory which follows. 



