** 



102 OF ELIMINATION. [CHAP. VII. 



than one premiss or equation, in order to render possible the eli- 

 mination of a term, the necessary law of thought virtually sup- 

 plying the other premiss or equation. And though the demon- 

 stration of this conclusion may be exhibited in other forms, yet 

 the same element furnished by the mind itself will still be vir- 

 tually present. Thus we might proceed as follows : 

 Multiply (1) by a?, and we have 



* /(1)*-0, . (3) 



and let us seek by the forms of ordinary algebra to eliminate x 

 from this equation and (1). 



Now if we have two algebraic equations of the form 



ax + b = 0, 

 ax + b' = ; 



it is well known that the result of the elimination of x is 



ab'-a'b = Q. (4) 



But comparing the above pair of equations with (1) and (3) 

 respectively, we find 



-/(o), 



'=/(!) '=<); 



which, substituted in (4), give 



/(i)/(o) = o, 



as before. In this form of the demonstration, the fundamental 

 equation x (1 - x) = 0, makes its appearance in the derivation of 

 (3) from (1). 



7. I shall add yet another form of the demonstration, par- 

 taking of a half logical character, and which may set the demon- 

 stration of this important theorem in a clearer light. 



We have as before 



Multiply this equation first by #, and secondly by 1 - #, we get 



/(1)*-0, /(0)(l-)-0. 

 From these we have by solution and development, 



