AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 197 
and then 
y = + x a /(-|.) • 
and the remaining two values from 
«X 2 — 2nxyXZ — 2 nxtJZ? = 0 
and then 
(91) ; 
(92) , 
(93) . 
Take the factors of the left-hand side of (88) separately, and form first of all the 
part of the discriminant depending on 
CZ + xX- yY, 
Let the roots of (90) be X 1 /Z i and X 2 /Z 3 ; then the corresponding values of Y by 
(91) are 
W(-!) “ d 
Hence 
az x + xX, - yY,) az, + -yY,) 
= «Z, + £X,) (£Z 2 + £X 2 ) 
= ZA(f+{f^ + |) + f|g 
= ZA { 1 % 
In like manner, taking the roots corresponding to (92) and (93), 
aZ 3 + a>X 3 - yY,) (£Z 4 + *X 4 - yY,) = Z 3 Z 4 <* 
Next, taking the other factor of the left-hand side of (88), the part of the dis¬ 
criminant corresponding to the roots of (90) and (91) is 
(3% + xX, - yY,) (3£Z 2 + ^X 2 - yY,) 
= (3CZ 1 + fX 1 )(3CZ a +W 
= Z lZi (^ + 3ff(| + |)+f-||) 
= Z lZi (g{* + ^^) , 
In like manner the roots of (92) and (93) give rise to the following part of the 
discriminant:—■ 
