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

MULTIPLE CONTACT.

13

The foregoing remarks only apply to equations which represent
proper curves, that is to say equations which are incapable of
being resolved into factors which represent two or more curves of
a lower degree than that of the equation. For example, if a cubic
equation be capable of resolution into a linear and a quadric
factor, the two points of intersection of the straight line and the
conic satisfy the analytical conditions of a double point; and by
parity of reasoning it appears that whenever a curve of the nth
degree has more than the maximum number oP double points, the
equation representing the curve breaks up into factors, each of
which represents a curve of a lower degree than the nth.
20. Before proceeding further it will be desirable to give a
few definitions.
The deficiency D of a curve is the number by which the
number of double points, real or imaginary, falls short of the
maximum.
The class of a curve is the number of tangents, real or
imaginary, which can be drawn from any point to the curve. We
shall denote the class by the letter m.
A point of inflexion is a point, which is not a double point,
where the tangent has a contact of the second order with the
curve. The tangent at a point of inflexion is sometimes called a
stationary tangent.
A point of undulation is a point, which is not a triple point,
where the tangent has a contact of the third order with the curve.
No curve of a lower degree than a quartic can have points of
undulation.
A double tangent is a line which touches a curve at two distinct
points. Since a double tangent intersects a curve in four points,
no curve of a lower degree than a quartic can have a double
tangent; but curves of a higher degree than the fourth may have
multiple tangents of a higher order. Also a multiple tangent
may have a contact of a higher order than the first. Thus a
sextic may have (i) a triple tangent having a contact of the first
order at three distinct points, (ii) a double tangent touching the
curve at a point of undulation and at a point at which the contact
is of the first order, (iii) a double tangent touching the curve at
two points of inflexion.