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

FOCAL CONICS.

141

Eliminating (|, V, ') by means of the equation of the focal
conic, the locus of Q' is of the form
P/a2 + Q/32 + R/y2 = 0,
which is the equation of a quartic having three biflecnodes.
208. Every bicircular quartic can be expressed in the form of a
ternary quadric of U, V, W, where these quantities are the equations
of three circles.
By means of a linear transformation any ternary quadric can
be reduced to the sum of three squares; hence the equation in
question may be written in the form
IU2 + mV2 nW2 = 0..................(18).
Now U=r2 + u + uo &c.; whence substituting in (18) it will be
found that the equation reduces to (4).
209. We shall now examine the relations of the fixed circle
to the focal conic.
The equation
U+ V+ vW = 0..................... (19)
obviously represents a circle; and it can be shown by the usual
methods that (18) is the envelope of (19), where (X,,p, v) are
subject to the condition
X2/ + 2/ + /n =0................... (20).
Let the vertices A, B, C of the triangle of reference be the
centres of U, V, W; then the distances of their centres from BC
are bsin C, 0, 0; whence the distance of the centre of (19) from
BC is Xb sin C/(X +,~ + v). Accordingly if a, f, y be the trilinear
coordinates of the centre of (19)
aa/X = b/p1L = cy/v....................(21).
Substituting in (20) it follows that the centre of (19) lies on
the curve
a2a2/l + b22/m + c2y2/n = 0................. (22),
which is a conic to which the triangle whose vertices are the
centres of U, V, W is self-conjugate.
210. We shall now prove that the circle which cuts U, V and
W orthogonally, cuts (19) orthogonally.