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

THE HARMONIC POLAR.

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whence
71 + y2= 2,y3
al a2 a3
from which it can be proved as in ~ 100 that
1 1 2
DP + DQ DE '
104. If two straight lines be drawn through a point of
inflexion to meet a cubic in four points and their extremities be
joined directly and transversely, the two points of intersection lie
on the harmonic polar.
Let A be the point of inflexion, and let AB and AC be the
two straight lines which meet the cubic in B, D and C, E
respectively. Then the equation of the cubic is
a (m/f + ny) (Xa + / f + vy) + 3ly (MON + ry) = 0...(20),
and the harmonic polar of A is
2Xa + 8 + vy = 0......................(21).
Let BE, CD intersect in G and BC, DE in H. Putting, = 0 and y = 0 in (20), the equations of BE and CD are
Xa + vy=0 and Xa +, ==0...............(22),
which show that the equation of DE is
X2 + / +vy = O......................(23).
Equations (21) and (22) show that BE and CD intersect at
the point Xa = - ft, = - Pry, which by (21) lies on the harmonic
polar; whilst (21) and (23) show that DE intersects the harmonic
polar at the point where it cuts the line BC.
If AB coincides with AC, the lines BC and DE are the
tangents at B and D, whence:-Tangents at the extremities of any
chord drawn through a point of inflexion intersect on the harmonic
polar.
105. The tangents at any two points of inflexion intersect on
the harmonic polar of the point of infiexion which lies on the line
joining the other two.
Let the equation of the cubic be
a/Iy + (la + m/ + nry)3 = 0,

B. C.

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