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

102

QUARTIC CURVES.

159. When the equation of a quartic is of the form u4 + u3 = 0,
the origin is a triple point, the three tangents at which are given
by the equation u3 = 0. Since this is a cubic in y/x, the tangents
are (i) all real and distinct, (ii) one real and distinct and two real
and coincident, (iii) all real and coincident, (iv) one real and two
imaginary. Hence there are four species of triple points; and we
shall now show that every triple point is formed by the sirnultaneous union of three double points.
Let A, B, C be three crunodes. When the nodes coincide, the
tangents at A and B to the branch AB coalesce into a single
tangent. Similarly the tangents at A and C to the branch A C,
and those at B and C to the branch BC respectively coalesce into
two single tangents. Hence the three pairs of tangents at A, B
and C coalesce into three single tangents at the point at which
A
A A
I 2 3 4
B B
AS A4'A C A
C
5 6 7 8
the three nodes ultimately coincide, and therefore this point is a
triple point. The forms of the curve before and after union are
shown in figures 1 and 2.
The second kind of triple point is composed of two crunodes
and a cusp; and the forms of the curve before and after union
are shown in figures 3 and 4. The triple point consists of a cusp
which lies on the curve.
The third kind of triple point is composed of two cusps and a
crunode; and the forms of the curve are shown in figures 5 and 6.
The point scarcely differs in appearance from an ordinary point on
the curve.