688 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



/i2, /23, and /i3 . We determine them, as shown in the last three columns of 

 Table I, by setting /»y = whenever terminals i snxd j are to be connected 

 and fij = 1 otherwise. Our methods of two-terminal network synthesis will 

 produce the fij networks. 



Our criterion (P > H — log H) for deciding which of the two synthesis 

 methods to use (case 1 or case 2 of theorem II) is not the best rule when H 

 is as small as it is in this example. By actually trying the different ways of 

 apportioning switches with 2, 2, and 4 positions between a tree and a net- 

 work to produce all functions, one finds that the most economical way is to 



Fig. 8 — Network with the 3-terminal switching function f{x, y, z) of Table I. 



put a two-position switch, say x^ into a network which provides all functions 

 of X (0, 1, ac, and ic'), and the other switches into a tree. When this pro- 

 cedure is adopted we next express /,j in the form of identity (2). For example 



fu = ly + e2iz)][y -h ei(z) + x][y + e^iz) + x][y + ^2(2) + x][y + ^1(2)]. 



The synthesis method described in the text then leads directly to a net- 

 work for fu which is shown joining terminals 1 and 3 in Fig. 8. The net- 

 work for /12 which is shown in Fig. 8 was obtained by the same process. 

 For the sake of illustration the f2z network was found using a tree only. 



