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

TRILINEAR EQUATIONS.

109

which is the general equation of a quartic having a tacnode at A
and the line 7 = 0 as the tacnodal tangent.
The condition that A should be a rhamphoid cusp is obtained
from (15) by making A a cusp. This requires that = X2;
whence putting 1 = 0, the required equation is
(m/n2 + Xay)2 + 2mt,3y + y2 (av + v2) = 0..... (17).
By proceeding in the same way as in ~ 163, it can be shown
that the equation of a quartic having an oscnode at A is
(m/32 + + Xa + /ry)2 + 72 (qXy + v2) = 0.........(18),
whilst the equation of a quartic having a tacnode cusp at A is
(m132 + + X7a yy + + ky2)2 + E73 + Fy4 = 0.........(19).
166. A fiecnode is a node, one of the tangents at which is a
stationary tangent.
Since the flecnodal tangent has a contact of the second order
with the branch which it touches, and cuts the other branch which
passes through the node, every flecnodal tangent has a contact of
the third order with the curve.
A biflecnode is a node at which both the tangents are stationary
ones.
The lemniscate (x2 + y2)2 = a2 (X2 - y2) has a real biflecnode at
the origin; and we shall prove hereafter that it has two imaginary
biflecnodes at the circular points at infinity.
Flecnodes and biflecnodes may be real, imaginary or complex;
but the only complex singularity of this kind is formed by a
conjugate point and one or two imaginary stationary tangents.
The reciprocal polar of a flecnode is a double tangent which
has a contact of the first order at one point of the reciprocal curve
and touches it at a cusp at the other; and the reciprocal polar of
a biflecnode is a pair of cusps having a common cuspidal tangent.
Curves of a higher degree than the fourth may have multiple
flecnodes, consisting of multiple points, the tangents at which have
contacts of higher orders than the second with their respective
branches. Thus if a curve of the nth degree has a multiple point
of order k, each tangent may have a contact of order n - k or of
any lower order with its respective branch.