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

138

BICIRCULAR QUARTICS.

trinodal quartic which possesses a pair of biflecnodes at the
circular points, in which case the conic is a rectangular hyperbola and the centre of inversion is the centre of the hyperbola.
It only remains therefore to consider the case in which the quartic
has a pair of flecnodes at the circular points.
Equation (5) of ~ 169 is the equation of a trinodal quartic
having a pair of fleenodes at B and C. Putting a = I, / = x + ty,
y = x- y, it will be found that in order that the resulting
curve should be real, we must have 1 (p + q) = 1; whence, putting
I/nq = A, lq = B, the equation of the curve becomes
(+2 + y2)2 + 2Ax (2 y + A2B {(3 - 2B) x2 - (1 - 2B) y2} = 0,
which is the equation of a bicircular quartic having a pair of
fleenodes at the circular points. The origin will be a cusp
when 2B = 3; but if 2B= 1, the curve degenerates into the
square of a circle.
Comparing the last equation with (16), we find that the
centre of inversion is given by the equations
(a2_- b2)= (a2+b2)2 g =0,
hence this point is determined by the following construction.
From either focus draw an ordinate cutting the director circle
in P, and let the tangent at P intersect the transverse axis of the
conic in T, then T is the required point.
When the conic is a parabola, the equation of the quartic is
(X2 + y2)2 +t 2Ax (c2 + y2) + 3A2y2 = 0,
and the point T lies on the opposite side of the directrix at a
distance equal to that of the focus.
204. The pedal of a central conic with respect to any point
in its plane is a bicircular quartic having a third double point
at the origin, which is a node, a cusp or a conjugate point according as the origin lies without, upon or within the conic; but
the pedal of a parabola is a circular cubic.
The pedal of a central conic with respect to any origin, whose
coordinates with respect to the centre are (f, g), is
(r2 +fX + gy)2 = a2x2 + b2y2.
The origin will accordingly be a node, a cusp or a conjugate
point according as
f2/a2+gl/b > or = or < 1.