Since a triple point is analytically equivalent to three double points,
a quartic with a triple point is unicursal. Such a quartic is obtained
by inverting a unicursal cubic from its node. The equation of such
a cubic may be written u₂ + u₃ = 0, where u₂ and u₃ are homogeneous
functions of the second and third degree respectively in x and y. Hence
the equation of the inverse curve is u₃ + u₂ (x² + y²), which shows
that the origin is a triple point and the quartic circular. By
projecting this all other forms may be obtained.
The nature of the triple point depends upon the relation of the line at
infinity to the cubic before inversion. Thus the line at infinity may
cut the cubic in three distinct points all real, or one real and two
imaginary, in one real and two coincident points (an ordinary tangent),
or in three coincident points (an inflectional tangent). Hence the
quartic may have at the triple point three distinct tangents all real,
or one real and two imaginary, one real and two coincident, or all
coincident.
This quartic may be generated in a manner similar to that used for the
curves already discussed. We showed in the section on nodal cubics that
a system of conics through A, B, C, D, and a projective pencil of rays
with its vertex at A generate by the intersection of corresponding
elements a cubic with a node at A. Invert the whole figure from A
and then project:—the pencil of rays remains a pencil; the system of
conics becomes a system of unicursal cubics having a common node at A
and passing through five other common points; the cubic inverts and
projects into a quartic with a triple point at A, passing through the
five other common points of the system of cubics.
The three points of inflection of a nodal cubic lie on a right line.
Inverting:—there are three points on a circular quartic with a triple
point whose osculating circles pass through the triple point, and
these three points lie on a circle through the triple point. Let these
three points be designated by A, B, and C. The lines from the triple
point O to the points A, B, C, and the common chord of the osculating
circles at two of them form a harmonic pencil. Through one of these
points, A, and the triple point draw a circle touching the quartic; the
point of contact is on the common chord of the osculating circles at B
and C.
From theorems which we have already proved for a system of cubics
having a common node and passing through five others fixed points,
we can infer other theorems for a system of quartics having a common
triple point and passing through seven other fixed points. For example,
any conic through the common double point and two of the fixed points
is cut by the cubics in pairs of points which determine at the node a
pencil in involution. Hence any cubic having its node at the common
triple point and passing through any four of the fixed points is cut
by the quartics in pairs of points which determine at the common
triple point a pencil in involution. Again, the pairs of tangents to
the cubics at the common double point form a pencil in involution, the
two cuspidal tangents being the foci of the pencil. Inverting:—the
line at infinity (which passes through two of the fixed points, i. e.
the circular points) cuts the system of circular quartics in pairs
of points in involution. Projecting:—a line through any two of the
seven fixed points cuts the system of quartics in pairs of points
in involution. Since the line at infinity touches the inverse of a
cuspidal cubic, it follows that any line through two of the fixed
points will touch two of the quartics of the system; these points of
contact are therefore the foci of the involution.
Other theorems on such a system of quartics will be given in the next
section.