An elementary treatise on cubic and quartic curves, by A. B. Basset.

THE CANONICAL FORM.

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(iii) that the cubic passes through the point of intersection E of
(1, m, n) and (p, l).
109. If a chord BDCF, drawn from a point B on a cubic,
cut the cubic again in D and C, and the polar conic of B in F;
the tangents to the cubic at D and C, and the tangent to the polar
conic at F, all pass through the same point.
Equation (24) shows that (1, m, n) is the tangent at D to the
cubic; accordingly if it intersects AC (which is the tangent at C)
in G, the equation of BG is la + ny= 0.
The equation of the polar conic of B is
y (la + 2nz/3 + ny) + /a2 = 0,
which shows that the line (1, 2m, n) is the tangent to the polar
conic at F. This line obviously intersects AC in G.
110. If any conic be described through four fixed points on a
cubic, the chord joining the two remaining points of intersection of
the cubic and the conic will pass through a fixed point on the cubic.
Let A, B, C, D be the four fixed points on the cubic; let the
equations of A D, CD be /3 + vry = 0 and Xa +,/3 = 0; also let u, v
be any linear functions of (a, /3, 7). Then the equations of the
cubic and the conic may be written
a (/O3 + vr) u + -y (Xa + p,3) v = 0,
a (fOL + vy) + kiy (Xa + Pf) = 0,
where k is a variable parameter.
The first equation shows that the cubic passes through the
point of intersection 0 of the lines u and v; and dividing the first
equation by the second, it follows that the two remaining points
of intersection of the cubic and conic lie on the straight line
v = ku, which obviously passes through 0.
The Canonical Form.
111. It is proved in treatises on Algebra* that every ternary
cubic whose discriminant does not vanish may be reduced to the
canonical form
x3 3+y3 3+z + 61yz = 0...............(25),
where (x, y, z) are linear functions of (a, /3, y). We may therefore
regard (x, y, z) as the trilinear coordinates of a point referred to
* Elliott's Algebra of Quantics, p. 300.
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