74 PRINCIPLES OF SYMBOLICAL REASONING. [CHAP. V. 



wherein /(I, y) represents what the proposed function becomes, 

 when in it for x we write 1, and/ (0, y) what the said function 

 becomes, when in it for x we write 0. 



Now, taking the coefficient / (1, ?/), and regarding it as a func- 

 tion of?/, and expanding it accordingly, we have 



o)(i-y), (3) 



wherein /(1, 1) represents what /(I, y) becomes when y is made 

 equal to 1, and /(I, 0) what /(I, y) becomes when y is made 

 equal to 0. 



In like manner, the coefficient /(O, y) gives by expansion, 



/(0,y)=/(0,l)y+/(0,0)-(l-y). (4) 



Substitute in (2) for /(I, y), /(O, y\ their values given in (3) 

 and (4), and we have 



/(*, y) =/(!, 1) xy +/(!, 0) (1 - y) + / (0, 1) (1 - ) y 



+/(0, 0) (1 - x) (1 - y), (5) 



for the expansion required. Here /(I, 1) represents what/^, y) 

 becomes when we make therein # = 1, y = !;/(!, 0) represents 

 what f(x, y) becomes when we make therein x = 1, y = 0, and 

 so on for the rest. 



1 x 

 Thus, if/(#, y) represent the function - - , we find 



whence the expansion of the given function is \" 



Jay + 0*(1 - y) + J(l - ) y + (1 -*) (1 -y). 



It will in the next chapter be seen that the forms - and -, the 



former of which is known to mathematicians as the symbol of in- 

 determinate quantity, admit, in such expressions as the above, of 

 a very important logical interpretation. 



Suppose, in the next place, that we have three symbols in 

 the function to be expanded, which we may represent under the 

 general form/(.r, y, z). Proceeding as before, we get 



