288 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



taken from The Design of Switching Circuits by Keister, Ritchie and 

 Washburn*. The development of the algebraic expressions from the 

 sequence of operations table will be in exact parallel to the methods 

 suggested in the aforementioned text. 



. The symbolism adopted in the following development is l)asically that 

 oi using the notation A for all the make contacts on the A relay, and A' 

 for all the break contacts on the .4 relay. Contacts or groups of contacts 

 in series are related by the symbol of addition (+) and contacts or groups 

 of contacts in parallel are related by the symbol for multiplication (•) 

 which may or may not be explicitly written, as in ordinary algebra. 

 Therefore (.4 + B') symbolizes a series contact path that is closed 

 when the A relay is operated and the B relay is released, while {AB') 

 symbolizes the parallel contact path that is closed when either A is 

 operated or B is released. S\^dtching algebra includes only two numerical 

 values, and 1, with the quantity assigned to represent a closed path 

 and 1 to represent an open path. For the tabular notation of Table I, 

 is used to indicate that the relay listed at the head of the column is oper- 

 ated and 1 is used to indicate that the relay is released. 



As stated earlier, the present application of switching algebra utilizes 

 the sequence of operation chart of Table I. The operate and release 

 combinations for controlling the A, B, C, D and E relays can be selected 

 from this table by observing where each relay to be controlled changes 

 state. For example, the operate combination for relay D is relay combi- 

 nation 8 and the release combination for relay D is relay combination 24. 

 It is not necessary to include the contacts of a relay in its own operate 

 and release combinations. Note that the A and B relays which serve as 

 a pulse divider can be controlled solely by the L relay and contacts on 

 A and B without reference to C, D, E. However the C, D and E relays 

 are internally controlled by all five counting relays. The development 

 of all these control paths uses the following abbreviations: 



g{X) = operating combinations for the X relay 



r(X) = releasing combinations for the X relay 



h(X) = holding combinations for the X relay 



X = make contact on the A' relay 



Furthermore as expressed by theorem (6a and 6b) the negative of a 

 contact network X is defined as a network which is a closed path under 

 all conditions for which X is open, and is open under those conditions 



* D. Van Nostrand, 1951. The Bell Telephone Laboratories Series. 



