syMnE:iis or s\\ itci/i.xg cimclits 87 



Theorem 9: Any fwiclion f{Xi , X2 , • ■ ■ , Xn) may be realized with a 

 series- parallel circuit with the following distribution: 



(1,2,4, ••■ ,2-2), 2-2 



ill terms of transfer elements. 



This we prove by induction. It is true for ;/ = 3, since any function of 

 three variables can be realized as follows: 



f{X, Y, Z) = [X + /i (F, Z)][X' + /2 (F, Z)] 



and/i(F, Z) and/2(F, Z) can each be realized with one transfer on V and 

 one on Z. Thus /(A', I', Z) can be reahzed with the distribution 1, 2, 2. 

 Now assuming the theorem true for {n — 1) we have 



/(Xi ,X2,--- ,Xn) = [Xn + /i (Xi , X2 , • • • , Xn-l)] 



[X„ -{- f2{Xi , X2 , • ' - , Xn-l)] 



and 



2, 4, 8, • • • , 2-3 

 2, 4, 8, • • • , 2-3 



4, 8, 16, • • • , 2-2 



A simple appHcation of the Lemma thus gives the desired result, Many 

 distributions beside those given by Theorem 9 are possible but no simple 

 criterion has yet been found for describing them. We cannot say any 

 distribution 



(1, 2, 4, 8, • • • , 2-2, 2-2) 



(at least from our analysis) since for example 



3, 6, 6, 7 = (2, 4, 8, 8) 



cannot be decomposed into two sets 



ai , 02 , 03 , 04 = (1, 2, 4, 4) 



and 



bi,b2,b,,b,= (1,2,4,4) 



It appears, however, that the almost uniform case is admissible. 



As a final example in load distribution we will consider the case of a net- 

 work in which a number of trees in the same variables are to be realized. 

 A large number of such cases will be found later. The following is fairly 

 obvious from what we have already proved. 



