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

TACNODES.

103

The fourth kind of triple point consists of two conjugate points
and a crunode. The forms of the curve are shown in figures 7
and 8, and the point does not differ in appearance from an ordinary point*.
No quartic can have a triple point composed of three cusps;
for if such a point existed, the quartic would belong to species X.,
and therefore its reciprocal polar would be a nodal cubic having
three coincident points of inflexion; but on referring to ~ 98 it
will be seen that the equation for k cannot have three roots equal
to zero unless n vanishes, in which case the cubic breaks up into
three straight lines.
160. Since imaginary singularities occur in pairs, no cubic
can have an imaginary node or cusp; but such singularities may
occur in all curves of a higher degree than the third. We shall
also see that, in addition to the triple point, certain other singularities exist which are formed by the union of two or more
simple singularities. We shall therefore require the following
additional definitions:
(i) The simple singularities are four in number, viz. the node,
the cusp, the double tangent and the stationary tangent.
(ii) A compound singularity is one which is formed by the
union of two or more simple singularities. Compound singularities
are real, imaginary or complex, according as the simple singularities of which they are composed are all real, all imaginary, or partly
one and partly the other.
In the case of an ordinary triple point, the three double points
are supposed to move up simultaneously to coincidence; but if
two double points first move up to coincidence and the third one
afterwards moves up to coincidence with the first two, we obtain
certain singularities which are not triple points. These will now
be considered.
Tacnodes.
161. A tacnode is formed by the union of two nodes.
In the figure let the two nodes A and B coincide, whilst C
remains stationary. The portion ADB, which lies on the side of
* In the case of a quartic, the two conjugate points must lie outside the
portion ABC, and must be so situated that no line can be drawn through either of
them so as to cut the curve in more than two points.