A tricuspidal quartic is a curve of the third class with one double
tangent and no inflection. Its reciprocal is therefore a nodal cubic.
We shall begin by reciprocating some of the simpler properties of
nodal cubics. Since the three points of inflection of a nodal cubic
lie on a right line, it follows that the three cuspidal tangents of a
tricuspidal quartic meet in a point. The reciprocal of the harmonic
polar of a point of inflection is a point on the double tangent, found
by drawing through the point of intersection of the three cuspidal
tangents a line forming with them a harmonic pencil. Three such lines
can be drawn and it is not difficult to distinguish them. All six lines
form a pencil in involution, the lines to the points of contact of
the double tangent being the foci. I shall call such a point on the
double tangent the harmonic point of the cuspidal tangent. Since any
two inflectional tangents of a nodal cubic meet on the harmonic polar
of the third point of inflection, it follows that any two cusps of a
trinodal quartic and the harmonic point of the third cuspidal tangent
lie on a right line. Since the point of contact of the tangents from a
point of inflection of a nodal cubic is on the harmonic polar of the
point, it follows that the tangent to the tricuspidal quartic at the
point where it is cut by a cuspidal tangent passes through the harmonic
point of that cuspidal tangent.
The inverse of the parabola from a focus is the cardioid; and the
inverse of the corresponding directrix is the base circle of the
cardioid. The cardioid projects into a tricuspidal quartic and its base
circle projects into a conic through the three cusps which has the same
general properties as the base conic of the nodal bicuspidal quartic.
The circle circumscribing the triangle formed by the three tangents
to a parabola passes through the focus. Inverting:—three circles
through the cusp, and tangent to a cardioid, intersect in three
collinear points. Projecting:—three conics through the three cusps
of a tricuspidal quartic and touching the quartic intersect in three
collinear points. Reciprocating:—if three conics touch the three
inflectional tangents of a nodal cubic and the cubic itself, their
three other common tangents intersect in a point.
Circles described on the focal radii of a parabola as diameters touch
the tangent through the vertex. Inverting and projecting:—from a point
on a tricuspidal quartic lines are drawn to the three cusps and a
fourth line forming a harmonic pencil; the envelope of this fourth line
is a conic through the three cusps and touching the quartic at the
point where the latter is cut by one of the cuspidal tangents. There
are three such conics, one corresponding to each cusp. At any cusp the
tangent to its corresponding base conic, the cuspidal tangent, and the
lines to the other two cusps form a harmonic pencil. Reciprocating:—on
any tangent to a nodal cubic take the three points of intersection
with the inflectional tangents and a fourth point forming with these a
harmonic range; the locus of this fourth point is a conic touching the
three inflectional tangents and the cubic. The tangent to the cubic
where it is touched by the conic goes through a point of inflection.
On any inflectional tangent the point of contact of this conic, the
point of inflection, and the points of intersection of the other two
inflectional tangents form a harmonic range.
The circle described on any focal chord of a parabola as diameter will
touch the directrix. Inverting:—the circle described on any cuspidal
chord of a cardioid will touch the base circle. Projecting:—through a
cusp C draw any chord of a tricuspidal quartic meeting the quartic in
P and O; draw a conic through P, O, and the other two cusps so that
the pencil at P formed by the tangent to the conic and the lines to
the cusps is harmonic; all such conics will touch the base conic of the
cusp C. Reciprocating:—from O, on any inflectional tangent of a nodal
cubic, draw two tangents P and Q to the cubic; draw a conic touching
the tangents P and Q and the other two inflectional tangents so that
the range on one of these tangents formed by the point of contact of
the conic and the intersection of the three inflectional tangent is
harmonic; the envelope of all such conics is a conic touching the three
inflectional tangents.
The directrix of a parabola is the locus of the intersection of tangents
at right angles to one another. Inverting and projecting:—through any
point P on the base conic of a cusp C of the tricuspidal quartic, two
conics can be drawn through the three cusps and touching the quartic;
their two tangents at P and the lines to the other two cusps form a
harmonic pencil; their two points of contact lie on a line through
C. Reciprocating:—from any point on one of the inflectional tangents
to a nodal cubic draw the two tangents P and Q; draw two conics each
touching the cubic and the three inflectional tangents, one touching P
and the other Q; the envelope of their other common tangent is a conic
touching the three inflectional tangents; the two points of contact of
any one of these common tangents and the points where it cuts the other
two inflectional tangents form a harmonic range.
Any two parabolas which have a common focus and their axes in opposite
directions cut at right angles. Inverting:—any two cardioids having a
common cusp and their axes in opposite directions cut at right angles.
Projecting:—two tricuspidal quartics having common cusps and at one
of the cusps the same cuspidal tangent, but the cusps pointed in
opposite directions, cut at such an angle that the tangents at a point
of intersection and the lines to the other two cusps form a harmonic
pencil. Reciprocating:—two nodal cubics have common inflectional
tangents and on one of them the points of inflection common, but the
branches of the curve on opposite sides of the line; any common tangent
to the two curves is cut harmonically by the points of contact and the
other two inflectional tangents.
Circles are described on any two focal chords of a parabola as
diameters; their common chord goes through the vertex of the parabola.
Inverting:—circles are described on any two cuspidal chords of a
cardioid; the circle through their points of intersection and the cusp
goes also through the vertex of the cardioid. Projecting:—through one
of the cusps of a tricuspidal quartic draw two chords; draw conics
through the other two cusps and the extremities of each of these
chords so that the pole of the line joining the other two cusps with
respect to each of these conics is on the corresponding chord; the
conic through the points of intersection of these two conics and the
cusps passes also through the point where the cuspidal tangent of
the first mentioned cusp cuts the quartic. Reciprocating:—on one of
the inflectional tangents, of a nodal cubic take two points P and Q;
draw a pair of tangents from each of these points to the cubic; draw
two conics each touching a pair of these tangents and the other two
inflectional tangents, so that the polars of the point of intersection
of the other two inflectional tangents with respect to each of those
conics pass respectively through P and Q; the conic touching the common
tangents to these two conics and the three inflectional tangents
touches also the tangent from the first mentioned point of inflection
to the cubic.