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

POINTS OF INFLEXION.

59

called the tangentials of D, E, F; and the line D'E'F' is called
the satellite of DEF.
Since the tangents at the points where the harmonic polar
cuts a cubic intersect at a point of inflexion, the tangent at a
point of inflexion is the satellite of the corresponding harmonic
polar.
94. The three points in which a cubic intersects its asymptotes
lie on a straight line.
We have shown in ~ 90 that a cubic has three asymptotes;
hence putting u' =, in (7), where I = 0 is the line at infinity, the
equation
uvw + k w'=0........................ (8)
is the equation of a cubic of which u, v, w are the asymptotes.
The form of this equation shows that the asymptotes intersect
the cubic in three points which lie on the straight line w'= 0.
The straight line which passes through the points of intersection of a cubic and its asymptotes is called the satellite of the
line at infinity.
95. The product of the perpendiculars from any point on a
cubic on to the asymptotes, is proportional to the perpendicular
from the same point on to the satellite of the line at infinity.
It follows from (8) that the equation of a cubic referred to a
triangle whose sides are the asymptotes is
a/gLy + I (X + U + y) = 0................ (9),
where (X, u, v) is the satellite of the line at infinity. But if p
be the perpendicular from any point of the cubic on to the
satellite, p is proportional to Xa + 3 + vy; also I is constant,
whence (9) becomes
aC3y = kp.
Points of Inflexion.
96. If a cubic has three real points of inflexion, they lie on a
straight line.
If in (6) we put u'=v'= w', the equation
uvw + kuz' = 0........................(10)