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

172

SPECIAL QUARTICS.

to any diameter CA, and PN perpendicular to the tangent at A.
Join CN and let it intersect PM at Q. Then the locus of Q is
the curve in question, and its equation is
_2)........................(1).
The curve has a biflecnode at the origin, and a tacnode at
infinity, and therefore belongs to species VII. To prove the
latter statement, transform to trilinear coordinates so that the
axes of x and y are the sides BC, BA, whilst the line at infinity is
the third side of the triangle of reference; then (1) becomes
4 = a2 ( 2 _ 2) 2........................(2).
Now if in (16) of ~ 165 we interchange 8/ and 7, the resulting
equation represents a quartic having a tacnode at A and the line
3= 0 or AC as the tacnodal tangent; and if in the resulting
equation we put X =, = 0, vv = - v0y2, it reduces to (2).
The Oval of Descartes.
259. The oval of Descartes is the locus of a point P which
moves so that its distances from two fixed points F, F2 are
connected by the relation
FP + mF1P = a,
where m and a are constants.
The two points F, F1, as well as a third point F2 (see ~ 262)
will be provisionally called the foci; and we shall prove in ~ 273
that these three points satisfy Plicker's definition of foci.
Let FP = r, F1P = ri, FF1 = c, PFF1 = 0, then the polar equation of the curve is
r2 (1 - m2) - 2r (a - 2c cos ) + a2 -2c2 =......(1).
If this equation is written in the form
{r2 (1 - m2) + 2m2Cx + a2 _ m2c2}2 = 4a2r2
it is identical with what (12) of ~ 200 becomes when a = b; and is
therefore a cartesian.
If the curve be defined by the equation
r + mnr = a...........................(2),