A quartic with one node and two cusps is a curve of the fourth class,
having one double tangent and two points of inflection (see Salmon).
Hence its reciprocal is also a nodal bicuspidal quartic, a fact of
which frequent note will be made in this section.
The inverse of a conic with respect to a focus is a curve called
Pascal’s Limaçon. From the polar equation of a conic, the focus being
the pole, it is evident that the polar equation of the limaçon may be
written in the form:
e 1
r = —— cos_x_ + —— ;
p p
where e and p are constants, being respectively the eccentricity and
semi-latus rectum of the conic.
From the above equation it is readily seen that the curve may be traced
by drawing from a fixed point O on a circle any number of chords
and laying off a constant length on each of these lines, measured
from the circumference of the circle. The point O is the node of the
limaçon; and the fixed circle, which I shall call the base circle, is
the inverse of the directrix of the conic. This is readily shown as
follows:—the polar equation of the directrix is r = p/(e cos_x_). Hence
the equation of its inverse is r = (e cos_x_)/p, which is the equation
of the base circle of the limaçon.
The envelope of circles on the focal radii of a conic as diameters
is the auxiliary circle. Inverting:—the envelope of perpendiculars
at the extremities of the nodal radii of a limaçon is a circle with
its centre on the axis and having double contact with the limaçon.
Projecting:—from any point on a nodal bicuspidal quartic draw lines to
the three nodes and a fourth line forming with them a harmonic pencil;
the envelope of all such lines is a conic through the two cusps and
having double contact with the quartic; the chord of contact passes
through the node and cuts the line joining the cusps so that this
point of intersection, the two cusps, and intersection of the double
tangent with the cuspidal line form a harmonic range. Reciprocating:—on
any tangent to a nodal bicircular quartic take the three points where
it cuts the two inflectional tangents and the double tangent, and a
fourth point forming with these a harmonic range; the locus of all such
points is a conic touching the two inflectional tangents and having
double contact with the quartic; the pole of the chord of contact is
on the double tangent; join this last point to the intersection of the
inflectional tangents and join the node with the same intersection;
these four lines form a harmonious pencil.
If the tangent at any point P of a conic meet the directrix in Q, the
line P Q will subtend a right angle at the focus O; the circle P O Q
has P Q for a diameter and hence cuts the conic at P at right angles.
Inverting:—from any point P on the limaçon draw O P to the node O;
draw O Q perpendicular to O P meeting the base circle in Q; P Q is
normal to the limaçon at P. Projecting:—from any point P on a nodal
bicuspidal quartic draw lines to the three nodes and a fourth harmonic
to these three; from O draw lines to the two cusps and a fourth
harmonic to these two and the line O P; the locus of the intersection
of the fourth line of each pencil is a conic through the three nodes.
Call this the basal conic of the quartic. Reciprocating:—on any given
tangent to a nodal bicuspidal quartic take its points of intersection
with the double tangent and the inflectional tangents, and a fourth
point harmonic with these; on the double tangent take its points of
intersection with the given tangent and the inflectional tangents, and
a fourth point harmonic with these; the envelope of the lines joining
the fourth point of these two ranges is a conic touching the double and
inflectional tangents.
The locus of the foot of the perpendicular from the focus on the
tangent to a conic is the auxiliary circle. Inverting:—draw a circle
through the node tangent to a limaçon; draw the diameter O P of this
circle; the locus of P is a circle having double contact with the
limaçon, the axis being the chord of contact. Cor.; the locus of the
centre of the tangent circle is also a circle. Projecting:—through the
three nodes of a nodal bicuspidal quartic draw any conic touching the
quartic; the locus of the pole with respect to this conic of the line
joining the two cusps is a conic; draw the chord O P of the first conic
through the node O and the pole of the line joining the two cusps; the
locus of P is a conic through the cusps, having double contact with the
quartic.
If chords of a conic subtend a constant angle at the focus, the
tangents at the ends of the chords will meet on a fixed conic, and
the chords will envelope another fixed conic; both these conics will
have the same focus and directrix as the given conic. Inverting:—draw
two nodal radii of a limaçon O P and O Q, making a given angle at O;
the envelope of the circle P O Q is another limaçon; the locus of the
intersection of circles through O tangent to the limaçon at P and Q is
another limaçon. These two limaçons have the same node and base circle
as the given one. Projecting:—through the node O of a nodal bicuspidal
quartic draw a pencil of radii in involution; let O P and O Q be a
conjugate pair of these nodal radii; the envelope of the conic through
P, Q, and the three nodes, is another quartic of the same kind: also
draw conics through the three nodes tangent to the quartic at P and
Q; the locus of their point of intersection is another quartic of the
same kind. These three quartics all have the same node, cusps, and base
conic.
Every focal chord of a conic is cut harmonically by the curve, the
focus, and directrix. Inverting:—every nodal chord of a limaçon is
bisected by the base circle. Projecting:—every nodal chord of a nodal
bicuspidal quartic is cut harmonically by the quartic, the base conic,
and the line joining the two cusps. Reciprocating:—from any point
on the double tangent of a nodal bicuspidal quartic draw the other
two tangents to the quartic and a line to the intersection of the
inflectional tangents; the fourth harmonic to these lines envelopes a
conic.
Since the limaçon is symmetrical with respect to the axis, it
follows that the two points of inflection are situated symmetrically
with respect to the axis. Hence the line joining the two points of
inflection is parallel to the double tangent. Therefore by projection
we infer the following general theorem for the nodal bicuspidal
quartic: the line joining the two cusps, the line joining the two
points of inflection, and the double tangent meet in a point. Also the
fourth harmonic points on each of these lines lie on a line through
the node. Reciprocating:—the point of intersection of the cuspidal
tangents, the point of intersection of inflectional tangents, and the
node all lie on a right line. From the node draw a fourth harmonic
to this right line and the tangents at the node; draw a fourth line
harmonic to this right line and the inflectional tangents; draw a
fourth harmonic to the cuspidal tangents and this right line; these
three lines all meet in a point on the double tangent.