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

CENTRES OF INVERSION.

147

Centres of Inversion.
216. If the centre and radius 8 of the fixed circle whose
centre is 0 be taken as the centre and radius of inversion, it
follows from (12) of ~ 200 that a bicircular quartic is inverted
into itself. We shall now show that the vertices A, B, C of the
triangle, formed by the intersection of the diagonals of the quadrilateral whose angles are the points where the focal conic intersects
the fixed circle, possess the same property; and that the radii of
inversion in the three respective cases are the tangents from A, B
and C to the fixed circle, that is to say the radii rl, r2, r3 of the
circles U, V, W.
Let A be the origin, and AB the axis of x of a Cartesian
system of coordinates; then
U = 2 X + y2 - r12,
V= x2 + y2 - 2cx + c2 - r22,
W = x2 y2 - 2bx cos A - 2by sin A + b2 - r2.
But from ~ 21:.
c2 - r22 = 4R2 sin C (sin C - sin A cos B)
= 4R2 sin C sin B cos A = r12.
Similarly b2 - r32 = r12,
whence if U', V', W' denote the inverses of' these circles when the
radius of inversion is r1, we have
r2U' =-r2U; r2V' = 2V; 2 W' =rW;
which shows that the inverse of the quartic is the same curve as
the original quartic.
Focal Conics and Foci.
217. We have shown that four of the foci of a bicircular
quartic are the intersections of a circle of inversion with its
corresponding focal conic. We shall now prove that the quartic
can be generated by taking any one of the three circles U, V or
W as the fixed circle, and a conic confocal with the original conic
as the focal conic. The intersections of these three circles with
their respective focal conics furnish twelve more foci, making
altogether sixteen.
10-2