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

THE ELLIPTIC LIMACON.

187

the circular points, it belongs to the ninth species of quartics for
which Pliicker's numbers are n = 4, 8 = 1, c = 2, n = 4, r = 1, t = 2,
D = 0, and therefore has a triple focus, which is the point of intersection of the cuspidal tangents at the circular points.

280. To find the position of the triple focus.
Transform (1) into trilinear coordinates by writing
/ =x+ty, 7=X —y, I=1,

and it becomes

{/3 + b (/3 + y) I2 = a12-3.

The cuspidal tangents at the circular points (I, I) and (y, I)
are 27+ bI= 0 and 23 + b= 0, or in Cartesian coordinates
x- y + fb =0, x+y + - =0, which intersect at the point
x=- b, y= 0, which is therefore the triple focus.
From (1) we obtain
FA = a- b, FB = a + b, AB = 2a,
from which it follows that if a point A' be taken on the opposite
side of F such that FA'=FA, and S be the triple focus,
FS= A'B. Also the distance FF1 can easily be shown to be
equal to 2 (a2 - b)/b, from which it follows that dAF1 = (a - b)2/2b,
and is therefore positive; whence F1 lies outside the curve. The
vectorial equation of the curve is
ar - br1 = (a2- b2)....................(2),
which shows that F and F1 possess the properties of foci.