BOOLEAN ALGEBRA AND CIKCUIT DESIGN 295 



111 ('(M'taiu cases the use of theorem (14b), iioi-mally employed to reduee 

 I he contacts of a particular relay to a siiifi;le make and break, can pro- 

 duce simplifications difficult to accomplish ()tiiei\vis(>. This is shown 

 below, with the theorem applicnl with respect to i-elay /.' since E tended 

 otherwise to be hea^•ily loaded. 



(li + (" + D'){ir + r + K')iB + (" + K)((" + ly + K) 



[E + (B + C + I)'){B' + (' + \)(B + (" + 0)((" + jy + ())] 



[A" + (7^ + C + iy){B' + (' + 0)(/i + (_" + 1)(C" + // + 1)] 



(14b) 



(K + (" + Bfy)[E' + (B' + C)(B + C + 7^')] (9a, 10a, 3b, 9b, 5b) 



By modifying the first factor of the final expression in accordance with 

 theorem 1 la, this equation can be put in bridge form as shown on Fig. 4c. 



[E + (" + B(B' + iy)W + (/^' + (')(B + (" + D')] 



The above equation uses the same contacts as the previous expression, 

 and although the right hand member is in a slightly different form, the 

 expression is equivalent to the one obtained earlier. 



When it is known that output conditions are inconsequential for some 

 relay combinations, these inconseciuential relay combinations may be 

 combined with valid combinations to eliminate contacts in the network. 

 Inconsequential means that the output during these particular combina- 

 tions does not affect the proper functioning of the circuit. Four such 

 combinations are listed in Table III. Only those inconsequential combi- 

 nations which will combine readily with the actual coml)inations, thereby 

 resulting in a reduction in the nvmiber of contacts, are to be used. Al- 

 though the use of all the all-relays-released condition may be helpful in 

 certain cases, it will not be used in the circuit under consideration since 

 its use makes the recjuirement that no tie shall exist between output leads 

 until the second pulse is received hard to meet. 



With this in mind the "0" lead network is again examined. Note the 

 use of another "shorthand" rule which states that if a part of the 2" 

 possible combinations is used in closing a path, the negative of the unused 

 part of the 2" possible combinations is e(iuivalent to the original com- 

 binations. Thus if in the case at hand three of the possible four combina- 

 tions of the B and C relays occur in series with the same combination of 

 the D and E relays, the expression used is that for the series })ath of the 

 D and E relays plus the negative of the missing combination of the B 



