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

CHAPTER VI.
SPECIAL CUBICS.
120. IN the present chapter we shall consider various special
cubics, and shall commence with the discussion of a certain class
of circular cubics. It will be shown hereafter that every circular
cubic is a degenerate form of a bicircular quartic; hence the
theory of circular cubics is best studied as a particular case of
these curves. This will be done in Chapter IX.; but the discussion of the circular cubics which are the inverses of conic
sections with respect to their vertices deserves separate treatment.
Circular Cubics.
121. A circular cubic is a cubic which passes through the
circular points at infinity.
From this definition it follows that the trilinear equation of
every circular cubic is of the form
vS + Iv =........................(1),
where S is a circle, I the line at infinity, and v, is a ternary
quantic in a, /, y. Also since the line vI intersects the cubic in
two points at a finite distance from the origin and one point at
infinity, this line is parallel to an asymptote.
122. To find the equation of a circular cubic in Cartesian
coordinates.
Since I is a constant, (1) may be written in the form
(v1 + v0) (r2 + w, + wo) + V= 0,