GROUP INVARIANCE OR TOTAL SYMMETRY 1453 



numbers of zeros and ones as in Table VI (a). For such a matrix it is not 

 clear which variables should be primed. It is possible to avoid considering 

 all possible primings by "expanding" the transmission about one of the 

 variables by means of the theorem 



T{xi ,x.2, ■■■ Xn) = XiT{l, X,, ■■■ x„) + x,'T(0, x., , ■ ■ ■ XnY-' 



and then making use of the relation: 



*^ai 1 a-> 1 ' ' ' amv^'l ) •^''' ) " * ' "^n) 



= Xik!)ai_i , a-z—l ) ao— 1 > ' ' ' a„— IV-^'s j ' ' ' X^J 



4- XiSa^ , a-, , •••<!„ ^'2 , • ■ • Xn)^ 



This technique is illustrated in Table \1. The standard matrix for 

 ^(l, Xo , Xz , .T4) has three rows each containing a single one so that 



7X1, X2 , Xi , X4) = Siix2, X3, Xi) 



The transmission 7'(0, X2 , Xs , Xi) has an identical standard matrix so 

 that 



i (0, X2 , X3 , Xa) = 01 (.T2 , Xz , X\) 



This can be written in terms of Xt\ x/, and Xi : 



Sl{X2 , X3 , Xi) = S2iX2, Xs, Xi^. 



Finally 



T{Xi , X2 , X3 , Xi) = XiT{\, X2 , X; , Xi) + XiT{0, X2 , Xz , Xi) 



= XiSi{X2, Xi, Xi) + XiS2{X2, Xz, Xi) = S2{Xi , X2, Xs, Xi).* 



The method just presented for detecting total symmetry is more sys- 

 tematic than the only other available method'' and applies for transmis- 

 sions of any number of variables. 



BIBLIOGRAPHY 



1. Birkhoff, G., and MacLane, S., A Survej' of Modern Algebra, The MacMillan 



Company, New York. 



2. Shannon, C. E., The Synthesis of Two-Terminal Switching Circuits, B. S.T.J. , 



28, pp. 59-98, 1949. 



3. Shannon, C. E., A Symbolic Analysis of Relay and Switching Circuits, Trans. 



A.I.E.E., 57, pp. 713-723, 1938. 



4. Keister, W., Ritchie, A. E., Washburn, S., The Design of Switching Circuits, 



New York, L). Van Nostrand Company, Inc., 1951. 



5. Caldwell, S. H., The Recognition and Identification of Symmetric Switching 



Circuits, Trans. A.I.E.E., 73, Part I, pp. 142-146, 1954. 



* This technique for transmission matricies having an equal number of zeros 

 and ones in all columns was brought to the author's attention bj' Wayne Kellner, 

 a student at the Massachusetts Institute of Technologj'. 



