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

GENERAL EQUATION IN TRILINEAR COORDINATES. 23

On the General Equation in Trilinear Coordinates.
41. The general equation of a curve of the nth degree may
be written in the form
F (a, /, /3) = u0an U^ n- + u u2n-2 +... un = 0..... (27),
where un is a binary quantic in / and y. The equation may also
be written in two similar forms by interchanging the letters a, /3
and ry.
If the curve pass through the vertex A of the triangle of
reference, (27) must be satisfied by / = y = 0, which requires that
u = 0. Hence if a curve pass through the angular points of the
triangle of reference, the terms involving the nth powers of a, /3, y
are absent.
If, in addition, we seek the points where the line u, = 0 cuts
the curve, we find by eliminating y that the resulting equation
contains /32 as a factor, which shows that the line 8 = 0 or CA
cuts the curve at a point where ai has a contact of the first order
with it. From this it follows that if a curve pass through the
angular points of the triangle of reference the coeflicients of the
(n - l)th powers of a, 3 and y equated to zero are the tangents at
these points.
If the point A be a double point, ul as well as u0 must be zero;
and u2= 0 is the equation of the tangents at A.
If therefore the angular points of the triangle of reference are
double points, the coefficients of the (n - 2)th powers of a, /3, ry are
the tangents at the double points.
If A be a point of inflexion, the tangent at A must meet the
curve in three coincident points. If therefore in (27) we put
u = 0 and eliminate 7y, the resulting equation must contain /33 as
a factor. This requires that u2 = uv, and (27) becomes
UCl-1 + Ululan-2 + u3alt-3 +... un = 0......... (28).
The last result enables us to prove the following important
proposition.
42. The points of inflexion are the points of intersection of a
curve and its Hessian, and their number cannot exceed 3n (n - 2).
By ~ 29, the polar conic of A is
dn-2F
da n —2