72 PRINCIPLES OF SYMBOLICAL REASONING. [CHAP. V. 



function ^-^j- , /(I) will represent ^-^ , and / (0) will repre- 

 sent - . 

 a 



9. Definition. Any function f(x) , in which x is a logical 

 symbol, or a symbol of quantity susceptible only of the values 

 and 1, is said to be developed, when it is reduced to the form 

 ax + b (1 - #), a and b being so determined as to make the result 

 equivalent to the function from which it was derived. 



This definition assumes, that it is possible to represent any 

 function /(#) in the form supposed. The assumption is vindi- 

 cated in the following Proposition. 



PROPOSITION I. 



10. To develop any function f (x) in which x is a logical symbol. 



By the principle which has been asserted in this chapter, it 

 is lawful to treat a? as a quantitative symbol, susceptible only of 

 the values and 1. 



Assume then, 



f(x\ = cue + b (1 #), 



and making x = 1, we have 



/(!) = . 

 Again, in the same equation making x - 0, we have 



Hence the values of a and b are determined, and substituting 

 them in the first equation, we have 



/(*)=/(!) a+/(0)(l-*); (1) 



as the development sought.* The second member of the equa- 



* To some it may be interesting to remark, that the development of /(JT) 

 obtained in this chapter, strictly holds, in the logical system, the place of the 

 expansion of /(#) in ascending powers of a: in the system of ordinary algebra. 

 Thus it may be obtained by introducing into the expression of Taylor's well- 

 known theorem, viz. : 



/GO =/(0) +/ (0)* +/" (0) ~ 4/'" (0) y-l^, &C. (1) 



the condition * (1 - ar) = 0, whence we find .r 2 = x, x 3 = x, &c., and 



