CHAP. IX.] METHODS OF ABBREVIATION. 131 



which it involves, and not at all upon the actual values of the 

 coefficients (VI. 5). Now let x be any term of the series, and 

 xy any other term having a; as a factor. The expansion of x with 

 reference to the symbols x and y will be 



and the expansion of the sum of the terms x and xy will be 



But by what has been said, these expressions occurring in the 

 first member of an equation, of which the second member is 0, 

 and of which all the coefficients of the first member are positive, 

 are equivalent ; since there must exist simply the two constituents 

 xy and x (1 - y) in the final expansion, whence will simply arise 

 the resulting equations 



xy = 0, x (1 - y) = 0. 



And, therefore, the aggregate of terms x + xy may be replaced by 

 the single term x. 



The same reasoning applies to all the cases contemplated in 

 the Proposition. Thus, if the term x is repeated, the aggregate 

 2x may be replaced by x, because under the circumstances the 

 equation x = must appear in the final reduction. 



PROPOSITION II. 



3. Whenever in the process of elimination we have to multiply 

 together two factors, each consisting solely of positive terms, satisfying 

 the fundamental law of logical symbols, it is permitted to reject from 

 both factors any common term, or from either factor any term which 

 is divisible by a term in the other factor ; provided always, that the 

 rejected term be added to the product of the resulting factors. 



In the enunciation of this Proposition, the word "divisible" 

 is a term of convenience, used in the algebraic sense, in which xy 

 and x (1 - y) are said to be divisible by x. 



To render more clear the import of this Proposition, let it be 

 supposed that the factors to be multiplied together are x + y + z 

 and x + yw + t. It is then asserted, that from these two factors 

 we may reject the term x, and that from the second factor we 

 may reject the term yw, provided that these terms be transferred 



K2 



