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

118

QUARTIC CURVES.

equivalent to one double tangent; but we have shown in ~ 178
that it is also equivalent to two stationary tangents, whence the
reciprocal singularity is a triple point composed of a node and a
pair of cusps.
181. Let (1) be written in the form
a/37yu + S2 =........................(2),
where
S = 12a2 + qm2/2 + n272 - 2mn/y - 2nlya - 21mac......(3),
u=Xa + /43 + vy........................(4),
1/X + m/a + n/v = O........................(5),
then the conic S touches the quadrilateral a, /, y, u at four real
points which are real points of undulation on the quartic.
H
F /
G
B C D K
In the figure, ABC is the triangle of reference; HK is the
line u = 0, and D, E, F, G are the four points of undulation. The
coordinates of G are obtained by solving the equations S = 0, u = 0,
and are determined by
X2a/l = /2/3/lm = v2y/n.....................(6),
whilst the equation of DG is
X ( - -) +m/3- = o..................(7).
We notice that the three straight lines AD, BE, CF meet at
the point la = m/3 = ny. Also four triangles can be formed by
taking any three of the four straight lines AB, BC, CA, HK, and
any of these triangles may be taken as the triangle of reference.
We thus obtain the following theorem:If a triangle be formed by the tangents at anly three real points
of undulation, the lines joining the vertices of the triangle with the
points of contact of the opposite sides meet at a point.