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

10

THEORY OF CURVES.

Contact.
14. When two curves intersect one another in r + 1 coincident
points, they are said to have a contact of the rth order with one
another.
When two curves have a contact of the first order with one
another at two distinct points, they are said to have a double
contact with one another.
15. Every straight line through a double point has a contact of
the first order with the curve at that point.
Let any two points be taken on the curve in the neighbourhood
of a double point; then the straight line through these points
intersects the curve in at least two points. Accordingly by making
the two points move up to coincidence with the double point, it
follows that every straight line through the latter intersects the
curve in two coincident points.
16. Every tangent at a double point has a contact of the second
order with the curve.
Take any point P on the curve near the double point; then
the line through P and the double point ultimately becomes a
tangent at the latter. But since every line through a double
point intersects the curve in two coincident points, the tangent at
the double point intersects the curve in three coincident points.
In the same way it can be shown that if a curve A passes
through a double point on a curve B, the former has a contact of
the first order with the latter at the double point, but the curves
will not touch one another unless the curve A intersects one of the
branches of B, which passes through the double point, in two
coincident points. In this case the curve A will have a contact of
the first order with the particular branch, and a contact of the
second order with the curve B at the double point.
17. A tangent to a curve is usually defined as a line which
intersects the curve in two coincident points, or as a line which
has a contact of the first order with the curve; but this definition
is only applicable to curves of the second degree. For we have
just shown that every line through a double point satisfies the
preceding definition of tangency, whereas there are only two