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

FLECNODES AND BIFLECNODES.

113

172. Every line through a biflecnode is divided harmonically
by the curve and the harmonic polar.
Let BC be the harmonic polar; then v1 = 0 and the equation
of the quartic becomes
a22L + u = 0........................(10).
Let 3 = ky be any line through A; then its points of intersec —
tion with (10) are given by
a2u' + 7u4 = 0,
where u2', u4' are what t2, u4 become when 8 = k, y =1. Hence
al/yl + a2/72 = 0,
from which it follows from ~ 100 that the line is divided harmonically by the curve and the harmonic polar.
173. If two straight lines be drawn from a biflecnode to meet
a quartic in four points, and their extremities be joined directly
and transversely, the points of intersection will lie on the harmonic
polar.
Let the equation of the quartic be
222 - 4 =........................(11),
where u2 = (12, m, n2%f, 7)2,
u4 = (\2, X\\, p, ), 2, 7)4,
so that BC is the harmonic polar of the biflecnode A.
Let AB, A C be any two lines through A cutting the quartic
in P, Q and p, q respectively; then putting 7 =0 in (11), the
coordinates of P and Q are given by
la= + ~ X3........................... (12).
Putting / = 0 in (11), the coordinates of p and q are given by
na= + pry...........................(13).
Let the upper signs refer to the points P, p and the lower to
the points Q, q; then the equations of Pp and Qq are
n (la - x/3) - Ivry = 0,
n (la + X/3) + rvy = 0,
which obviously intersect on the line BC. In the same way the,
equations of Pq and Qp can be shown to be
n (la - /3) + Ivy = 0,
n(la + X/)- lv = O,
which also intersect on BC.

B. C.

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