ON ELIMINATION IN THE CASE OF EQUALITY OF FRACTIONS. 



a x x + b x y + c^z d x a x x + 8 x iv ft, y l 



a b x + b b y + c-z d 5 a 5 x + S b w /3 5 y 6 



a x x + b x y + d x w c x a x x + y x z /3 X o\ 



a b x + b 5 y + d 5 w c 5 a b x + yr> z /3 5 8 5 



a x x + CjZ + djW b x a x x + f$ x y y x 8 X 



= o, 



where 3 of the terms of the numerator of which one is always a m x are kept together, 

 and "2 terms of the denominator of which one is always a m x, and the other the term 

 corresponding to that rejected from the numerator ; the third is 



CtyX + b x y Cj d x a x x + y-jZ + S^v (3 X 



a 5 x + b 5 y c 5 d 5 a b x + y 5 z + S 5 w /3 6 



a x x + c x z b x d x a x x + fi x y + 8 x w y x 



a 5 x + c 5 z b & d b a b x + f3 5 y + 8 5 w y 6 



a x x + d x w b x Cj a x x + fi x y + y x w 8 X 



a h x + d 5 w b b c 5 a 5 .r + fay + y 5 w d b 



= 0, 



where 2 terms of the numerator of which one is a m x are kept together, and 3 terms of 

 the denominator of which one is a m x and the two others the terms corresponding to 

 those rejected from the numerator ; and the fourth is 



a x b x <■■] d x a x x + f} x y + y x z + 8 x iv 

 «5 h H <h a b x + Ps!/ + 7s* + h w 



= 0. 



where the terms of the numerator are all kept separate and those of the denominator 

 are kept all together. 



Instead of always including a m x and a m x when making our selections, we might take 

 any other corresponding pair ; the only difference would be that the factor struck out 

 from the quadric in order to reach the linear equation would not be x, but y or z or w* 



* Had § 2 stood by itself it would of course have been more direct and natural to change its equations into 

 '•("!•'-' + hi) + Ci~) = a- x X + &fl + YiS , 



= {a x -ra x \v + (^ - rl x )y + (7! - rc x )z , 



and eliminate x , y , ~. . 



