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

248

THEORY OF PROJECTION.

lines; whilst the sum of the fourth and fifth sets is the equation
of a quartic having three double points.
379. The equation
la3 + m337 + n?3a = 0....................(1)
has been considered by Klein in connection with a septimic transformation in the Theory of Functions*. Putting a = 0, we obtain
/33y = 0, which shows that C is a point of inflexion, and that BC is
the tangent at C; hence A, B and C are points of inflexion, and
CA, AB and BC are the tangents at these points. The peculiarity
of this quartic is that there are three other real points of inflexion
A', B', C' such that A'B', B'C', C'A' are the tangents at these
points, and that the equation of the quartic referred to A'B'C' is
of the form Xa'3y' + py'3/' + vp/3a' = 0; also a conic can be described
through the six points A, B, C, A', B', C'. It will, however, be
unnecessary to consider the general case, since the above theorems
can be proved by investigating the special case of a quartic which
is symmetrically situated with respect to an equilateral triangle,
and then generalizing by projection.
380. The equation of the quartic may now be written
a3 + + + 73v = 0........................(2),
and the equation of the circle circumscribing the triangle of
reference is
/3y + ya+a/3 = 0....................... (3).
To find where (3) cuts (2), eliminate y and put 8//a = k, and we
shall obtain
a3 (1 + k)3 -(1 + )2 k3- k2} = 0........... (4).
The first factor shows that the quartic passes through B and
A; and the second factor gives the remaining five points of intersection of the circle and the quartic. This may be written in the
form
(1 + k + C2) (1 + 2k-k2 -k3)-O.............. (5).
The factor 1 + k + k2, when equated to zero, gives the lines
joining C to the circular points at infinity; whilst the equation
k3 + = 2 + 1..........................(6)

* Math. Ainalen, vol. xiv. p. 428.