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

CIRCULAR CUBICS.

155

Let A, B, C, D be the points where any circle intersects the
cubic; let AB, CD intersect the cubic A
in a and c; and AC, BD in R and R'.
Let B move up to coincidence with a,
and D with c. Then Aa, Cc are the Bs -- C
tangents at a and c, and the four
points AacC lie on a circle. a
When B and D coincide with a and c, BD coincides with ac;
but since a line which is parallel to the asymptote cannot cut the
curve in more than two points at a finite distance from one
another, the point R' must move off to infinity. Hence the line
RR', which by ~ 225 is parallel to the asymptote, cuts the cubic
in only one finite point R, and therefore it must be the asymptote.
230. Let the points A and C coincide; then:If a tangent be drawn to the cubic from the point where it is
cut by its asymptote, and if from the point of contactA two tangents
be drawn to the cubic touching it in a and c, the circle circumscribing Aac will touch the cubic in A, and the line ac will be parallel
to the asymptote.
231. If a circular cubic 2 be inverted from any point 0 on
itself into a circular cubic Z', the osculating circle of E at 0 will
invert into the asymptote of Z', and vice versa.
The osculating circle intersects the cubic in three coincident
points at 0, and one finite point P; whence the circle inverts
into a line cutting the inverse cubic in one finite point P' and
touching it at two coincident points at infinity; whence the
inverse of the osculating circle is the asymptote of 2'.
232. If the cubic be inverted from the point 0 where the
asymptote cuts the curve, the point 0 will be a point of inflexion on
the inverse curve.
It follows from (6) that the equation
(ax + by)S + X (ax + by)2 + ex +fy = 0
represents a circular cubic whose asymptote is the line
ax + by = 0.
The inverse cubic is

(ax + by) S' + Xk2 (ax + by)2 + r2 (ex +Jy) = 0,