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

66

CUBIC CURVES.

then if D, E, F be the points in which (1, m, n) cuts BC, CA, AB,
then D, E, F are points of inflexion, and BC, CA, AB are the
tangents at these points. The coordinates of E are /= 0,
la -+ ny = 0; whence the polar conic of E is
/3 (n - la) = O.
The second factor equated to zero is the harmonic polar of E,
which obviously passes through B the point of intersection of the
tangents at D and F.
106. The harmonic polars of three collinear points of inflexion
pass through a point.
By the last article the harmonic polars of the three points
D, E, and F are
m/3 = ny, nt = la, la = m/3,
which obviously meet in a point.
107. If a cubic has a double point, each harmonic polar passes
through it.
If A be a point of inflexion, the cubic is given by (5); also
if B be a double point, the terms involving /3 and /32 must be
absent. Whence vl = nry, U = 72 (/3 + vy); and the harmonic
polar of A is 2a + n7 = 0, which obviously passes through B.
Since only one tangent can be drawn from a point of inflexion
to a nodal cubic, it follows that the harmonic polar is the line
joining the node and the point of contact. When the cubic is
cuspidal, the harmonic polar is the cuspidal tangent.
108. If two tangents be drawn to a cubic from a point A on
the curve, the tangent at the third point where the chord of contact
intersects the curve cuts the tangent at A at a point on the curve.
Let B and C be the points of contact of the tangents from A;
let AE be the tangent at A, and DE the tangent at the point D
where the chord of contact cuts the curve. Then the equation of
the cubic must be of the form
/3y (la + m/3 + nry) + a2 ('ua + py) = 0...........(24).
The form of (24) shows (i) that the line (1, m, n) is the
tangent DE at the third point D, where the chord of contact
cuts the cubic; (ii) that the line (/a, v) is the tangent AE'at A;