78 PRINCIPLES OF SYMBOLICAL REASONING. [CHAP. V. 



gical interpretations are assigned to the symbols x and y. And 

 it may be remarked, that though functions do not necessarily be- 

 come interpretable upon development, yet equations are always 

 reducible by this process to interpretable forms. 



14. The following Proposition establishes some important 

 properties of constituents. In its enunciation the symbol t is 

 employed to represent indifferently any constituent of an expan- 

 sion. Thus if the expansion is that of a function of two symbols 

 x and ?/, t represents any of the four forms xy, x (1 - y), (1 - x)y, 

 and (1 - x) (1 - y}. Where it is necessary to represent the con- 

 stituents of an expansion by single symbols, and yet to distinguish 

 them from each other, the distinction will be marked by suffixes. 

 Thus ti might be employed to represent xy, t z to represent x(l -y) 9 

 and so on. 



PROPOSITION III. 



Any single constituent t of an expansion satisfies the law of dua- 

 lity whose expression is 



<(i-0 = o. 



The product of any two distinct constituents of an expansion is equal 

 to 0, and the sum of all the constituents is equal to 1. 



1st. Consider the particular constituent xy. We have 



xy x xy 



But x 2 = x, y z = y, by the fundamental law of class symbols ; 

 hence 



xy x xy = xy. 



Or representing xy by t, 



t x t = t, 

 or 2(1 -t) = 0. 



Similarly the constituent x ( 1 - y) satisfies the same law. For we 

 have 



tf = x 9 (l-2/) 2 =l-y, 



.-. [x(l-y)}* = x(l-y), or *(!-*) = 0. 



Now every factor of every constituent is either of the form x or 

 of the form I - x. Hence the square of each factor is equal to that 



