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

136

BICIRCULAR QUARTICS.

also by (10)
82= 2r. OT-r2,
whence r2 + 2fx + 2gy + 2 = 2rp...............(11).
Now p2 = a2 cos'2 4> + b2 sin2 b,
whence (11) becomes
(r2 + 2fx + 2gy + 2)2 = 4 (a2 + by2).........(12),
which by (5) is the equation of a bicircular quartic.
When the conic is a circle, a = b, and (12) may be put into
the form
(r2 + 2fx + 2gy + 82 - 2a2)2 = 4a2 (a2- 2fx - 2gy - 82)..(13),
which is the equation of a cartesian.
When the conic is a parabola whose focus is E and vertex X,
p = a sec b, and (11) becomes
(r2 + 2fx + 2gy + 82) = 2a...............(14),
which is the equation of a circular cubic.
The fixed conic is called the focal conic because, as will
be shown hereafter, it passes through four of the foci of the
quartic.
If the quartic (12) be inverted from 0 with respect to a
circle of radius 8, it is inverted into itself. Hence 0 is called
a centre of inversion, and the circle whose centre is 0 and radius
8 is called a circle of inversion. We shall hereafter prove that,
in general, a bicircular quartic has four centres and four circles
of inversion.
Equation (12) contains five independent constants; and if the
origin be transferred to any arbitrary point and the axes be
turned through any arbitrary angle, three more constants will
be introduced. Hence the general equation of a bicircular quartic
contains eight constants, and that of a cartesian seven.
201. The inverse of a bicircular quartic is another bicircular
quartic unless the centre of inversion lies on the curve, in which case
it is a circular cubic.
The general equation is of the form
r4Vo + r2v 4-U2 + u1 + u + = 0...............(15),