404 
A MEMOIR ON CURVES OF THE THIRD ORDER. 
[146 
Article Nos. 29 to 31.—Formula} for the intersection of a cubic curve and a line. 
29. If the line %x + yy + & — 0 meet the cubic 
x s + y 3 z 3 + Qlxyz = 0 
in the points 
(®i, Vi, *i), Oa, V-2, Zi), (x 3> y 3 , z 3 ), 
then we have 
: 2/iMs : = V s ~ : £ 3 - I s : f* - ?? 3 - 
It will be convenient to represent the equation of the cubic by the abbreviated 
notation (1, 1, 1, t§x, y, z) 3 = 0; we have the two equations 
(1, 1, 1, IJx, y, zf = 0, 
gx+yy + £z = 0 ; 
and if to these we join a linear equation with arbitrary coefficients, 
ax + /3y + yz = 0, 
then the second and third equations give 
x : y : z = №-yr) : y: ay-fig-, 
and substituting these values in the first equation, we obtain the resultant of the 
system. But this resultant will also be obtained by substituting, in the third equation, 
a system of simultaneous roots of the first and second equations, and equating to 
zero the product of the functions so obtained 1 . We must have therefore 
(1, 1, 1, l\№~ yv, yf-a£ «V ~ = ( ax i + /%i + 7*i) ( ax 2 + /% 2 + yz 2 ) (««3 + fiy* + yz ? ); 
and equating the coefficients of a 3 , /3 3 , y 3 , we obtain the above-mentioned relations. 
30. If a tangent to the cubic 
x 3 + y 3 + z 3 + Qlxyz = 0 
at a point (x lt y u z x ) of the cubic meet the cubic in the point (x 3 , y 3) z s ), then 
x s : y 3 ■ * 3 = x 1 (y 1 3 -z 1 3 ) : y 1 (z 3 - x 3 ) : z x (x 3 - y 3 ). 
For if the equation of the tangent is %x + yy + gz = 0, then 
x 2 x 3 : y 2 y 3 : z x % = v 3 -? : £ 3 - f 3 : f - y\ 
and 
I : V : Z = xi i + 2ly 1 z 1 : y 1 2 + 2lz 1 x 1 : z x 2 + 2lx x y x . 
1 This is in fact the general process of elimination given in Schläfli’s Memoir, “ lieber die Resultante 
einer Systemes mehrerer algebraischer Gleichungen,” Vienna Trans. 1852. [But the process was employed much 
earlier, by Poisson.]