SYNTHESIS OF SWITCHING CIRCUITS 81 



PART II: CONTACT LOAD DISTRIBUTION 



4. Fundamental Principles 

 We now consider the question of distributing the spring load on the relays 

 as evenly as possible or, more generally, according to some preassigned 

 scheme. It might be thought that an attempt to do this would usually 

 result in an increase in the total number of elements over the most economi- 

 cal circuit. This is by no means true; we will show that in many cases (in 

 fact, for almost all functions) a great many load distributions may be ob- 

 tained (including a nearly uniform distribution) while keeping the total 

 number of elements at the same minimum value. Incidentally this result 

 has a bearing on the behavior of m(«), for we may combine this result with 



Y' 



X' 



Fig. 19 — Disjunctive tree with the contact distribution 1, 3, 3. 



preceding theorems to show that /i(«) is of the order of magnitude of — - as 



w— > 00 and also to get a good evaluation of m(«) for small n. 



The problem is rather interesting mathematically, for it involves additive 

 number theory, a subject with few if any previous applications. Let us 

 first consider a few simple cases. Suppose we are realizing a function with 

 the tree of 'Fig. 9. The three variables appear as follows: 



W, X, Y appear 



2, 4, 8 times, respectively 



or, in terms of transfer elements* 



1,2,4. 

 Now, W, X, and Y may be interchanged in any way without altering the 

 operation of the tree. Also we can interchange X and Y in the lower branch 

 of the tree only without altering its operation. This would give the dis- 

 tribution (Fig. 19) 



1,3,3 

 * In this section we shall always speak in terms of transfer elements. 



