┌ ┐
│ (x² + 2xy + y²) (x² - 2xy + y²) │ 4aexyz(x + y) b²x²y²
z² │ ———————————————— - ———————————————— │ - ————————————— - —————————
└ a² b² ┘ a² a²
4aexyz(x + y) b²x²y²
- ————————————— - ————————— = 0,
a² a²
which is the equation of the quartic referred to the triangle formed by
the three nodes. We are now able to determine the nature of the node at
the vertex (y, z). Factor x² out of all the terms which contain it; and
arrange thus:
┌ ┐ ┌ ┐
│ z² z² 4aeyz b²y² │ │yz² yz² 2aey²z │
x² │——— - ——— - ————— - ——————│ + 2x │———— + ———— - ———————│
│ a² b² a² a² │ │ a² b² a² │
└ ┘ └ ┘
y²z² y²z²
+ —————— - —————— = 0.
a² b²
The quantity which multiplies x² represents the two tangents at the
double point (y, z); but this quantity is a perfect square and hence
we have a cusp. In this way the point (x, z) may be shown to be a
cusp. Lastly, when a parabola is inverted from the focus, we obtain a
tricuspidal quartic.
The trinodal quartic can be generated in a manner analogous to that
shown for the nodal cubic. Let two projective pencils of rays have
their vertices at A and B, the locus of intersection of corresponding
rays is a conic through A and B. Invert from any point O in the plane,
and we obtain two systems of co-axial circles, O A being the axis of
one and O B of the other. The locus of intersection of corresponding
circles is a bicircular quartic having a node at O. Projecting the
whole figure we have the following theorem:—two projective systems of
conics through O P Q A and O P Q B generate by their corresponding
intersections a trinodal quartic having its nodes at O, P, and Q, and
passing through A and B.
It is evident that the quartic generated in this way may have three
nodes, one node and two cusps, two nodes and one cusp, or three cusps,
depending upon the nature of the conic inverted and the centre of
inversion. Making this the basis of classification we thus distinguish
four varieties of unicursal quartics. To these must be added a fifth
variety, viz: the quartic with a triple point. Each of these varieties
will be considered separately.
The method of treating unicursal quartics given in this and the next
four sections is in some respects similar to that suggested by Cayley
in Salmon’s Higher Plane Curves. But the method here sketched out is
very different in its point of view and much wider in its application,
yielding a multitude of new theorems not suggested by Cayley’s method.