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

ON A SPECIAL QUARTIC.

251

Also A = C= BC'A
= BC'A' + AC'A'.
Whence C' = BC'A' + CA C',
accordingly BCB' = CA C'
= CB'C' = CBC',
whence BC' is parallel to B'C and AA'. Accordingly
BC'A' = BAA' =ABC',
therefore ~ r = B = CBC' + ABC'
= BC'B' + BC'A'= C',
whence the triangle A'B'C' is equilateral.
382. The quartic (1) is anautotomic; for its discriminant is
equal to 261mn, which cannot vanish unless one of the constants
1, m, n is zero, in which case the quartic splits up into a cubic and
a straight line.
By tracing the symmetrical quartic, it can easily be seen that
the points A, A', &c. are the only six real points of inflexion; and
also that the quartic has only three real double tangents which
touch the curve at real points, and that the six real points of
inflexion and the six points of contact of these double tangents lie
on two concentric circles. If in addition we eliminate y between
(2) and the line at infinity, we obtain (a2 + a/3 + /2)2= 0, which
shows that this line is a double tangent whose points of contact
are the circular points. If therefore we generalize by a real
projection, we obtain the theorem:The six real points of inflexion, and the six real points of
contact of the three double tangents lie on two conics which touch one
another at the two imaginary points where a fourth real double
tangent touches the quartic.
The projection may be accomplished as follows. Let the
equilateral triangle ABC be projected into the triangle A'B'C';
and let (a, /3, y) and (a', 3', y') be the coordinates of two corresponding points P and P' in the two planes referred to the triangles
ABC and A'B'C' respectively; also let the projection be such
that
a=a/X, 3=fl'/,A, = Y'/v,