104 OF ELIMINATION. [CHAP. VII. 



But the four factors of the first member of this equation are 

 the four coefficients of the complete expansion of /(#, ?/), the 

 first member of the original equation ; whence the second part of 

 the Proposition is manifest. 



EXAMPLES. 



10. Ex. 1. Having given the Proposition, "All men are 

 mortal," and its symbolical expression, in the equation, 



y = vx 9 



in which y represents "men," and a: "mortals," it is required to 

 eliminate the indefinite class symbol v, and to interpret the 

 result. 



Here bringing the terms to the first side, we have 



y - vx = 0. 

 When v = 1 this becomes 



and when v = it becomes 



y=0; 



and these two equations multiplied together, give 



y - yx = 0, 

 or y(l-#)=0, 



it being observed that y z = y. 



The above equation is the required result of elimination, and 

 its interpretation is, Men who are not mortal do not exist, an 

 obvious conclusion. 



If from the equation last obtained we seek a description of 

 beings who are not mortal, we have 



' 



y 



Whence, by expansion, 1 -x = -(1-y), which interpreted gives, 

 They who are not mortal are not men. This is an example of 



