CHAP. IX.] METHODS OF ABBREVIATION. 133 



xy + xQ + PQ 

 = xy + (x + P)Q. 



But this expresses the rejection of the term xy from the second 

 factor, and its transference to the final product. Wherefore the 

 Proposition is manifest. 



PROPOSITION III. 



4. If t be any symbol which is retained in the final result of the 

 elimination of any other symbols from any system of equations, the re- 

 sult of such elimination may be expressed in the form 



Et + E(l-t) = 0, 



iu which E is formed by makiny in the proposed system t = 1, and eli- 

 minating the same other symbols ; andE' by making in the proposed 

 system t = 0, and eliminating the same other symbols. 



For let (t) = represent the final result of elimination. 

 Expanding this equation, we have 



#(!)* + * (0) (1 - = 0. 



Now by whatever process we deduce the function $ (t) from the 

 proposed system of equations, by the same process should we de- 

 duce (1), if in those equations t were changed into 1; and by 

 the same process should we deduce (0), if in the same equations 

 t were changed into 0. Whence the truth of the proposition is 

 manifest. 



5. Of the three propositions last proved, it may be remarked, 

 that though quite unessential to the strict development or appli- 

 cation of the general theory, they yet accomplish important ends 

 of a practical nature. By Prop. 1 we can simplify the results 

 of addition ; by Prop. 2 we can simplify those of multiplication ; 

 and by Prop. 3 we can break up any tedious process of elimi- 

 nation into two distinct processes, which will in general be of a 

 much less complex character. This method will be very fre- 

 quently adopted, when the final object of inquiry is the determi- 

 nation of the value of , in terms of the other symbols which remain 

 after the elimination is performed. 



6. Ex. 1. Aristotle, in the Nicomachean Ethics, Book n. 

 Cap. 3, having determined that actions are virtuous, not as pos- 

 sessing in themselves a certain character, but as implying a cer- 



