CHAP. VII.] OF ELIMINATION. 103 



/(I) = - = - (1 - a?), on development, 

 x u 



The direct interpretation of these equations is 



1st. Whatever individuals are included in the class repre- 



sented by /(I), are not a?'s. 



2nd. Whatever individuals are included in the class repre- 



sented by/(0), are x'a. 



Whence by common logic, there are no individuals at once 



in the class /(I) and in the class /(O), i.e. there are no indivi- 



duals in the class / (1) /(O). Hence, 



/(1)/(0) = 0. (5) 



Or it would suffice to multiply together the developed equa- 

 tions, whence the result would immediately follow. 



8. The theorem (5) furnishes us with the following Rule : 



TO ELIMINATE ANY SYMBOL FROM A PROPOSED EQUATION. 



RULE. The terms of the equation having been brought, by trans- 

 position if necessary, to the first side, give to the symbol successively 

 the values 1 and 0, and multiply the resulting equations together. 



The first part of the Proposition is now proved. 



9. Consider in the next place the general equation 



/(*)- ! 



the first member of which represents any function of a?, y, and 

 other symbols. 



By what has been shown, the result of the elimination of y 

 from this equation will be 



/O, !)/(>, 0) = 0; 



for such is the form to which we are conducted by successively 

 changing in the given equation y into 1, and y into 0, and multi- 

 plying the results together. 



Again, if in the result obtained we change successively x into 

 1, and x into 0, and multiply the results together, we have 



/(1,1)/(1,0)/(0,1)/(0,0)-0; (6) 



as the final result of elimination. 



