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

150

BICIRCULAR QUARTICS.

220. The foci of the focal conic are the double foci of the
quartic.
In (12) of ~ 200 write / = x + ty, y = x- y, and make the
resulting equation homogeneous by multiplying each term by the
proper power of a; then the equation
{/37 fy (/3 + y) - tgo (3 - y) + 822Itj2 = a2 ta2 (3 + 7)2 - b2 (3 _ 7)2}
represents a quartic having a pair of imaginary nodes at the
points B and C of the triangle of reference.
The nodal tangents at B and C are
(7 +fa -- ta)2 = (a2 b2) a2
and (/3 + fa + ga)2 = (a2 - b2) a2.
Retransforming to Cartesian coordinates and putting a = 1, it
follows that the nodal tangents at the two circular points are
x +f f ae - (y +g) = 0,
x +f + ae + L (y + ) =0,
which intersect at the two real points
= ae- f, y=-g,
x='-ae-f, y=-g.
These are the coordinates referred to 0 of the foci of the focal
conic, and therefore, by ~ 79, these points are the two double foci
of the quartic.
Putting e = 0, it follows from ~ 200 and the preceding paragraph that when the focal conic reduces to a circle, the centre of
the latter is the triple focus of a cartesian.
221. If r1, r2, r3 be the distances of any point on a bicircular
quartic from any three real foci, then
lr, + mr2 + nr3 = 0.
Let the point r, be taken as the origin, and let the axis of x
pass through r2; then
r12 = r2,
r22 = r2 - 2ax + a2,
r32 = r 2 (ex +fy) + e2 +f2,