•371] ON A FORMULA FOR THE INTERSECTIONS OF A LINE AND CONIC, &C. 501 
equivalent of course to two equations, and giving by the elimination of (x, y, z), the 
equation 
0[-(A,...$f, v> t) 2 -0 2 ] = 0, 
that is, giving for 6 the foregoing value. And the linear equations then give 
= ?% + 9V-h(+e : S% + ^-9i 
= r >% + M- b ! : + 9f t +b-K, 
= ~ fv : l + 99-°( ■■ Sjg+fz-gt+0, 
where obviously 
-e^z-Ag+Hri + Gg, -e'~ = Hi+B v +Fi, -e^ = Gg+F v + oi. 
By changing the sign of 0, we have of course the coordinates of the other point 
of intersection. The formulae which, singularly enough, have since been given incidentally 
by M. Aronhold( 1 ), may be easily obtained as follows. 
Writing for shortness 
P = ax + liy + gz, 
Q =hx + by + fz, 
R = gx+fy + cz, 
then the equation of the conic gives 
Px + Qy + Rz = 0, 
and combining with this the equation 
&+yy + & =0, 
we have 
x : y : z=Q£-Rrj : R%-P% : Py — Qf, 
or what is the same thing, taking an indeterminate multiplier 6, 
— 0x + Rrj — = 0, 
-0y + PS-R£ = 0, 
— Oz + Qg — Pii = 0, 
1 In his interesting Memoir “Ueber eine neue algebraische Behancllungsweise der Integrale irrationaler 
Differentiale von der Form II (x, y) dx, in welcher n (x, y) eine beliebige rationale Function ist, und zwischen 
x und y eine allgemeine Gleichung zweiter Ordnung besteht.” Grelle, t. lxi. (1862).