svxTfiEsis or swirciiixc. cikci its 97 



a much hotter limit than the corresponding 



2"-' + 18 



for the general function. 



Table I 

 CiRori' Invarianck Involving Two Variables (Superscripts Refer to Fig. 22) 



-V„„ {x y) (v xy* 



Nox {x y'T'* (y x')^* 



Xio (x'y)^ iy'x)^* 



Xn {x'y'r (y'x'y 



Table II 

 Group Invariance Involving Three Variables (Superscripts Refer to Fig. 23) 



9. Partially Symmetric Functions 



We will say that a function is "partially symmetric" or "symmetric in a 

 certain set of variables" if these variables may be interchanged at will 

 without altering the function. Thus 



ATzir + (AT' + A'Dir + irz' 



is symmetric in A' and Y. Partial symmetry is evidently a special case of 

 the general group invariance we have been considering. It is known that 

 any function symmetric in all variables can be realized with not more than 

 n~ elements, where // is the number of variables.'' In this section we will 

 improve and generalize this result. 



Theorem 15: Any function f(Xi , X-2 , ■ ■ ■ , X„, , I'l , I'o , • • • I'„) sy)n- 

 metric in A'l , A'o , • • • , X„, can be written 



/(A 1 , A^2 , ■ ■ ■ , X,n , I 1 , I 2 , • • • 



= [5„(Xi,A%, ••■ ,A„,)+/,.(I'i, Y,, 



{s,{x,,x,, ■■■ ,x„.) +/i(ri, Y,, 



[S^X, ,X,,--- , A„,) + /„,(I'i , 1'2 , ■ ■ • , Y,,)] (6) 



.'here 



A-(Fi , r^ , • • • , F„) 

 = /(o, 0, ••• ,0, 1, 1, ■•• , 1, r, , r.,--- , r„) 



y^O's (m - k) Vs 



