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

CUBIC CURVES.

57

91. In ~ 41 we have discussed several forms of the general
equation of a curve of the nth degree in trilinear coordinates,
and we shall now consider these special forms when the curve is
a cubic.
The general equation may be expressed in the form
u0oC3 + u la2 + uza + U3 = 0...................(1),
where Un is a binary quantic in 3 and y, or in two other forms in
which a, /3, 7 are interchanged.
The equation of a cubic circumscribing the triangle of
reference is
a2 + 32 + y27v + c +,8y = 0................... (2),
where u, v, w are the tangents at A, B and C, and are consequently
linear functions of /3, y; 7, a: a, f3 respectively.
The equation of a cubic having a double point at A is
Cal +,3 = 0...........................(3),
also if the cubic pass through the points B and C, its cannot
contain /3 and 73; hence the equation of a cubic circumscribing
the triangle of reference and having a double point at A is
au2 +,8/ (i3 + vy) = 0..................(4).
The equation uz= 0 is the equation of the tangents at the
double point; hence the latter will be a node, a cusp or a
conjugate point according as the roots of ',t, regarded as a
quadratic in /3/y, are real, equal or complex. The line //3 + vy = 0
is the line drawn from A to the third point where BC cuts the
cubic.
If A is a point of inflexion, the tangent at A must meet the
cubic in three coincident points. Hence ou = 0, and u, must be
a factor of 2,; whence the equation of a cubic having a point
of inflexion at A is
'ucai + uiva + '13 = 0......................(5).
92. If three tangents be drawn to a cubic from a point of
infiexion, their points of contact lie oni a straight line.
By (5) the polar conic of A is
dF/da = u, (2a + v,) = 0,