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

POINTS OF UNDULATION.

117

179. Every quartic may be expressed in the form S2 = uvwt,
where S is a conic, and u, v, w, t are straight lines.
The general equation of a quartic may be written in the form
of a ternary quadric in U, V, W, where these quantities equated
to zero represent three conies. The simplest way of proving this
is to recollect that every ternary quadric can be expressed as the
sum of three squares by means of a linear transformation. The
quartic can accordingly be expressed in the form
IU2 + mV2 + nW2= 0,
and if the terms be multiplied out it will be found that the
equation contains fourteen independent constants. It also follows
that any form of the equation of a conic in trilinear coordinates
will represent a quartic if U, V, W be substituted for a, /3, y.
Every quartic may be regarded as the envelope of the conic
X2U+ 2XV+ W= 0, where X is a variable parameter; for the
envelope is the quartic V2 = UW, which by the last paragraph is
one of the forms to which every quartic may be reduced.
The equation V2 = UW is equivalent to the equation
{X/k U+ (X + /) V+ W12 =(X2U 2X+ Wv+ ) (,2U+ 2V + W),
where X and, are arbitrary constants, as can at once be seen by
multiplying out. The left-hand side is the square of a conic, and
by determining X and,I so that the discriminants of the two
factors on the right-hand side vanish, the latter may be reduced
to four linear factors. Hence any quartic may be reduced to the
form S2 = uvwt, where S is a conic and u, v, w, t are four straight
lines. This form is due to Plicker, and furnishes a means of
determining the double tangents to a quartic.
180. We have shown in the last article that every quartic
may be written in the form
S2 + vwt = 0........................(1).
The four straight lines u, v, w, t obviously touch the quartic at
the eight points where the conic cuts it, and are therefore double
tangents to the quartic. If, however, u, v, w, t touch S, the
points of contact will be points of undulation on the quartic, and
they may be all real, all imaginary, or two real and two imaginary.
From this it follows that every tangent at a point of undulation is