CHAP. IX.] METHODS OF ABBREVIATION. 141 



Wherefore if the full development of (x, y) be represented 

 in the form 



axy + bx(\ - y) + c(\ -x)y + d(l-x) (1 - y), 



the coefficients #, 5, c, c? must each be 1 or 0. 



Now reducing the equation t = (#, y) by transposition and 

 squaring, to the form 



and expanding with reference to a? and y> we get 



b) + b(l -t)} x(\-y) 



whence, adding this to (1), we have 



(A + t(l-a) + a(l-t)}xy 



+ {B + t(l-b) + b(l-)} x(l-y) + &c. = 0. 



Let the result of the elimination of a? and y be of the form 

 Et + E'(l-t) = 0, 



then E will, by what has been said, be the reduced product of 

 what the coefficients of the above expansion become when t = 1 , 

 and E the product of the same factors similarly reduced by the 

 condition t = 0. 



Hence E will be the reduced product 



(A + 1 - a) (B + 1 - b) (C+ 1 - c) (D + 1 - d). 



Considering any factor of this expression, as A + 1 - , we see 

 that when a = I it becomes A, and when a = it becomes 1+^4, 

 which reduces by Prop. I. to 1. Hence we may infer that E will 

 be the product of the coefficients of those constituents in the de- 

 velopment of V whose coefficients in the development of (#, y) 

 are 1. 



Moreover E' will be the reduced product 



(A + a)(B + b) (C+c)(D + d). 



Considering any one of these factors, as A + , we see that this 

 becomes A when a = 0, and reduces to 1 when a = 1 ; and so on 

 for the others. Hence E' will be the product of the coefficients 



