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

140

BICIRCULAR QUARTICS.

But 2p - -r2 = 2,
dp PT
whence dp PT
dr r
Accordingly tan P = PT/QT = tan PQT,
whence PQ and P'Q are the normals at P and P'.
207. If a chord be drawn from a centre of inversion to meet
the quartic in P and P', the locus of the point of intersection of the
tangents at P and P' is a trinodal quartic, having three biflecnodes
at the angular points of the triangle which is self-conjugate to the
circle of inversion and the corresponding focal conic.
From the last proposition, it follows that since QP, QP' are
the normals to the quartic at P and P',,-^P the tangents at these points are also the
tangents to the generating circle whose
lT centre is Q, and will therefore intersect
t Q\ 71^ at a point Q' which lies on the tangent
X MA /.at Q to the focal conic. Draw Q'M
P perpendicular to OM1; then since the
points Q'PQMP' lie on a circle,
~0 ~OM.OQ= OP. OP'= 8,
and therefore Q'M is the polar of Q with respect to the circle of
inversion s.
Let the circle of inversion and the focal conic be referred to
their common self-conjugate triangle; and let (:, q7, ') be the
coordinates at Q. The equation of the circle of inversion is
aa2 cos A + b/32 cos B + cy2 cos C = 0,
and that of the focal conic is
Xa2 + 'p'2 + vr2 = 0.
Since Q'M is the polar of Q with respect to the circle of
inversion, its equation is
aan cos A + b/3'q cos B + cy' cos C = 0,
and the equation of QQ' is

Xac + r + vr-= 0.