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

122

QU ARTIC CURVES.

represents a quartic having a pair of real and a pair of imaginary
points of undulation on BC and AC, and four imaginary points on
AB. Also the equation
c(/8y (Xa + /Au3 + vy) = (la + m/3 + ny)4
represents a quartic having four real points of undulation on the
line (1, m, n).
Double Tangents.
186. We have shown in ~ 180 that
ac yuL + S" = 0...........................(1)
is the equation of a quartic, four of whose double tangents are the
lines a,,,,,; and that the points of contact are the intersections of these lines with the conic S. Let
- S +,k/7..........................(2);
then S = 0 is the equation of another conic which passes through
the four points of contact of the double tangents /3 and y with the
quartic. Substituting from (2), (1) may be written
/3y (at- 2kS1- k23) + 2= 0............... (3).
Since the terms in k cancel one another when the quartic is
written out at full length, k may have any value we please; if
therefore k be determined so that the discriminant of the conic in
brackets vanishes, the latter will be the product of two linear
factors vw, and (3) becomes
yvw + = 0...........................(4).
We therefore obtain the theorem:A conic can be drawn through the eight points of contact of any
four double tangents to a quartic.
The discriminant of the conic when equated to zero furnishes
a quintic equation for k which involves k as a factor. The solution k = 0 reproduces the conic S, whilst the four roots of the
quartic factor furnish four conics of the type E. Since each of
these five conies passes through the four points of contact of the
double tangents / and 7, it follows that:Through the four points of contact of any two double tangents
five conics can be described, each of which passes through the four
points of contact of two other double tangents.