svxTiiEsis or swnnnxc; circuits 



63 



The following are some further identities for general functions: 



A'+/(.v, y,z,---) = A-+/(0, V,Z, •••) 

 A"+/(.v, i',z, ■••) = x'+/(i, r,z, •••) 



Xf(X, Y, /,-••) = A7(l, I',Z, •••) 

 X'fiX, Y, Z, ■••) = A7(0, Y,Z,---) 



X + f(X,Y, Z,W) 



X + f(0,Y, Z,W) 



'o— o o— o o— 

 f (X,Y, Z,W) 



l— o o— o o— o 



Xf(l,Y,2,W) + X'f(0,Y,Z,W) 



= XYf(l, 1,Y, Z) + X Y'f (1,0,Y, Z) + X'Y f (0, 1,Y,Z) + X' Y' f (0, 0, Y,Z) 



Fig. 4 — Examples of some functional identities. 



The network interpretations of some of these identities are shown in Fig. 

 4. A little thought will show that they are true, in general, for switching 

 circuits. 



The hindrance function associated with a two-terminal network describes 

 the network completely from the external point of view. We can determine 

 •from it whether the circuit will be open or closed for any particular position 

 of the relays. This is done by giving the variables corresponding to operated 

 relays the value (since the make contacts of these are then closed and the 

 break contacts open) and unoperated relays the value 1. For example, with 

 the function / = W[X + Y{Z + A^')l suppose relays A' and Y operated and 

 Z and W not operated. Then/ = 1[0 + 0(1 + 1)] = and in this condition 

 the circuit is closed. 



