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

76

SPECIAL CUBICS.

Let the equation of the conic be X2/A2 + y2/B2 = 2x/A; then
inverting with respect to a circle of radius k and putting a = 1k2/A,
b = lk2A/B2, the equation of the curve becomes
x(x + y2) = ax2 + by2.....................(3).
When a and b are both positive, the curve is the inverse of an
ellipse; when a = 0 the curve is the inverse of a parabola and is
called a cissoid; when b is negative the curve is the inverse
of a hyperbola; and when a=-b, the curve is the inverse of a
rectangular hyperbola and is called the logocyclic curve. The
latter curve has been discussed by Dr Booth in connection with
the geometrical origin of logarithms.
The cubic obviously cuts the axis of x at the origin 0, which
is a double point, and also at the point A, where OA = a, which is
called the vertex; and the line x = b is the only real asymptote.
The lines y = (a/3b) x cut the curve in two points of inflexion,
which are real or imaginary according as the curve is the inverse
of an ellipse or a hyperbola. The remaining point of inflexion,
which is necessarily real, is at infinity.
The different forms of the curve, according as the conic is an
ellipse, a hyperbola or a parabola, are shown in the accompanying
figures.
0 A B BA 0 B
126. Iffrom the vertex A a straight line is drawn cutting the
curve in P, P', then AP. AP' = A 02; and the locus of Q the middle
point of PP' is the circular cubic
2x (x2 + y2) = (b - a) y2 - 2ax2.
Transfer the origin to the vertex and then change to polar
coordinates and we shall obtain

r2 _r {(b- a) sin2 0- 2a os2 0} sec 0 + a = 0......(4),