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

112

QUARTIC CURVES.

which shows that A must be a biflecnode. In order that the
quartic may be real, it is necessary that the sign of one of the
constants should be different from those of the other two; whence
writing -v for v, it follows that if X, /L, v are all positive, the
nodal tangents at A and B are real, whilst those at C are imaginary.
To prove the second part, let u, v, w be any real or imaginary
straight lines forming a triangle; and consider the quartic
Xv2w2 + /'WV2t2 +,t2V2 = 0..................(8).
Let u=a, v =,/3+ ky, w =/3 -L^y,
2,' = /l, + V, 2v' =, -- tv,
then (8) becomes
X ( + k2y2) + (2 + (722 + 2 ( 1h - _LC272) =0......(9).
Equation (9) represents a quartic having a real biflecnode at
A and two imaginary ones at the points where a intersects
v and w.
To find what (9) becomes when the imaginary biflecnodes are
the circular points at infinity, let A be the origin of a pair of
rectangular axes; then since the lines joining A to the circular
points are x +~ y = 0, we must put
/ =x, 7=y, k=1, a=I
in (9), which becomes
X (X2 + y2)2 + 2 {/I (x2 - y2) - 2vxy} = 0,
or r2 = a2 cos 20,
which is the lemniscate of Bernoulli.
171. We shall now prove that a biflecnode possesses a variety
of harmonic properties analogous to those possessed by a point of
inflexion on a cubic.
From (3) it follows that the polar cubic of A is
u2 (2avo + v,) = 0,
and therefore consists of the biflecnodal tangents and the line
2av0 + v, = 0. This line, for reasons which will appear in the next
section, is called the harmonic polar of the biflecnode.