CHAP. V.] PRINCIPLES OF SYMBOLICAL REASONING. 73 



tion adequately represents the function /(#), whatever the form 

 of that function may be. For x regarded as a quantitative sym- 

 bol admits only of the values and 1, and for each of these 

 values the development 



assumes the same value as the function /(#). 



As an illustration, let it be required to develop the function 



- . Here, when x = 1, we find/(l) = ^ , and when x = 0, 

 we find/(0) = =- , or 1. Hence the expression required is 



- - = - x + 1 - x\ 

 1 + 20 3 



and this equation is satisfied for each of the values of which the 

 symbol x is susceptible. 



PROPOSITION II. 



To expand or develop a function involving any number of logical 

 symbols. 



Let us begin with the case in which there are two symbols, 

 cc and y, and let us represent the function to be developed by 



/<*,?) 



First, considering / (x, y) as a function of x alone, and ex- 

 panding it by the general theorem (1), we have 



0(1-*); (2) 



/GO =/() + (/' (0) + 77^ + i^i + &c ' } * (2) 



But making in (1), x = 1, we get 



/(I) =/(0) +/' (0) +^ +{^ + &c. ; 

 whence 



/' (0) +4^ + &c - ^/c 1 ) -/()' 



and (2) becomes, on substitution, 



/G0=/(0) +{/0)-/(0)}*, 



the form in question. This demonstration in supposing/(ar) to be developable in 

 a series of ascending powers of x is less general than the one in the text. 



