CHAP. IX.] METHODS OF ABBREVIATION. 139 



Or, by Prop. 3, the result of the elimination of t and p from the 

 above equation will be of the form 



wherein E is the result obtained by changing in the given equa- 

 tion y into 1, and then eliminating t and p ; and E r the result 

 obtained by changing in the same equation y into 0, and then 

 eliminating t and p. And the mode in each case of eliminating t 

 and p is to multiply together the coefficients of the four con- 

 stituents tp, t(l- p), &c. 



If we make y = 1, the coefficients become 



1st. w (1 - s) + s (1 - w). 



2nd. 1 + w (1 - sr) + s (1 - w) r 9 equivalent to 1 by Prop. I. 



3rd and 4th. 1 + w, equivalent to 1 by Prop. I. 



Hence the value of E will be 



w(l - s) + s(l -w). 



Again, in (5) making y = 0, we have for the coefficients 



1st. 1 + w (1 - s) + s (1 - w), equivalent to 1. 



2nd. w (1 - sr) + sr (1 - w). 



3rd and 4th. w. 



The product of these coefficients gives 



E' = w(l-sr). 



The equation from which y is to be determined, therefore, is 

 {w (1 - s) + s (1 - w)} y + w (1 - sr) (1 - y) = 0, 

 w (1 - sr) 



w ( 1 - 



and expanding the second member, 



y = - wsr + ws (1 - r) + - w (1 - s) r + - w (1 - s) (1 - r) 



+ (1 - w) sr + (1 -w) s (1 - r) + - (1 - w) (1 - s) r 



"O v * 

 whence reducing 



