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

126

QUARTIC CURVES.

Let the lines i and y be chosen so that they are two of the
tangents to the quartic from the nodes B and C; then
N=2L c, M=2Lv,
and (1) becomes
La3 (La + 2v/3 + 2wy) + X2/32y2 + /2722 + 2a2/32 + aLryu = 0...(2).
The equation of the line joining the points of contact of / and
qy with the quartic is
La + v/3 + ly = 0,
and (2) may be written in the form
{X/3y + (Lc + /a ( + u y)}2
+ a/7Y {(I - 2LX - 2gv) a + (m - 2VX) 3 + (n - 2X/) ry} = 0...(3).
The form of this equation shows that
(I - 2LX - 2/,v) a + (m- 2vX)/3 + (n - 2X/) y = 0......(4)
is one of the double tangents; also since (2) is unaltered when
the sign of X is changed, another double tangent is
(I + 2LX - 2v+) a + (m + 2vX) / + (n + 2X/c) y = 0......(5).
Equation (2) also remains unaltered when the signs of L, F,, v
are changed; but this would merely reproduce equations (4) and
(5).
Since (3) is of the form S2 + uvwt = 0, the remaining six double
tangents can be found by the method explained in ~ 180. If
however the quartic has a cusp at B, m = + 2Xv; taking the upper
sign, it follows that (4) is not a double tangent, but one of the
tangents drawn from the cusp; and the double tangents consist of
(5) and three others. If C is also a cusp, n = 2X/j, and the only
double tangent is given by (5).
Trinodal Quartics.
191. Every trinodal quartic has four double tangents, which
will however be reduced in number if any of the nodes become
cusps; also since the curve is of the sixth class, only two tangents
can be drawn from a node to the curve. The bitangential curve
is obviously a conic.
To find the equations of the four double tangents and of the
bitangential conic*.
* H. M. Taylor, Proc. Lond. Math. Soc. Vol. xxvIII. p. 316.