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

CIRCULAR CUBICS.

153

and the double point will be a node, a cusp or a conjugate point
according as
g2 > or = or < 4a (a +f),
that is according as the point 0 is without, upon or within the
focal parabola. In this case the cubic will be the inverse of a
conic with respect to a point on the curve.
All circular cubics have only one real asymptote, viz. the line
x + 2a= 0.
From (2) we obtain
rr2 = 2.............................(4),
(r, + r2)=- (fcos + g sin 0 +asec 0)........(5).
Equation (4) shows that if OPQ be any chord, the rectangle
OP. OQ = 82; also that the lengths of the tangents drawn from
0 to the curve are all equal to the radius of the fixed circle.
Equation (5) shows that the locus of the middle point of PQ is
the circular cubic
r2x + (a +f)x2 + gxy + ay2 = 0,
whose node and nodal tangents are identical with those of the
first cubic when 8= 0.
224. We have shown in ~ 121 that one of the forms of the
equation of a circular cubic in trilinear coordinates is,S = -Iu..........................(6),
where u, is a ternary quantic of degree n.
The form of (6) shows (i) that the cubic passes through the
two circular points at infinity; (ii) that it passes through the
point where the line ul intersects the line at infinity, from which it
follows that the line uG = 0 is parallel to the asymptote; (iii) that
the cubic passes through the points of intersection of the conic
u2=0 with the circle S= 0 and the line u, = 0. It also follows
that a circle cannot intersect a circular cubic at more than
four points which are at a finite distance from one another.
225. The following proposition is of fundamental importance
in the theory of circular cubics.
If a circle intersect a circular cubic in four points A, B, C, D,
the three straight lines which respectively join the points, where the
three pairs of straight lines AB, CD; BC, AD; CA, BD again
intersect the cubic, are parallel to the asymptote.