CHAP. V.] PRINCIPLES OF SYMBOLICAL REASONING. 79 



factor, and therefore the square of the product of the factors, i. e. 

 of the constituent, is equal to the constituent ; wherefore t repre- 

 senting any constituent, we have 



t z = t, or (1-0 = 0. 



2ndly. The product of any two constituents is 0. This is 

 evident from the general law of the symbols expressed by the 

 equation x (1 - x) = ; for whatever constituents in the same ex- 

 pansion we take, there will be at least one factor x in the one, to 

 which will correspond a factor 1 - x in the other. 



3rdly. The sum of all the constituents of an expansion is 

 unity. This is evident from addition of the two constituents x 

 and 1 - x, or of the four constituents, xy, x (1 - y), (1 - #)y, 

 (1 -x) (1 -y). But it is also, and more generally, proved by 

 expanding 1 in terms of any set of symbols (V. 12). The consti- 

 tuents in this case are formed as usual, and all the coefficients 

 are unity. 



15. With the above Proposition we may connect the fol- 

 lowing. 



PROPOSITION IV. 



If V represent the sum of any series of constituents, the separate 

 coefficients of which are 1, then is the condition satisfied, 



Let 1, t 2 . . . t n be the constituents in question, then 



F= *! + t z . . . + t n . 



Squaring both sides, and observing that t? = t lt ^ 2 = 0, &c., we 



have 



F 3 = *! + t z . . . + t n ; 

 whence 



F= V\ 

 Therefore 



V(\ - V) = 0. 



