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

110

QUA RTIC CURVES.

167. In ~ 20 a point of undulation was defined as a point
where the tangent has a contact of the third order with the curve.
This definition must be understood to mean that the tangent has
a contact of the above order at a point which is not one of the
preceding singularities. It will be shown in ~ 180 that the
reciprocal singularity is a triple point composed of a node and
a pair of cusps.
On curves of the nth degree points exist where the tangent
has a contact of any order which is not higher than the (n - l)th.
Also multiple tangents may exist, which have contacts of orders
r, s, t, &c., at different points, where these quantities may have any
integral values subject to the condition that
r + 1 + I + t +1 1 &c.
is not greater than the degree of the curve.
Flecnodes and Biflecnodes.
168. We shall now proceed to discuss the properties of
flecnodes and bifleenodes of a quartic, but the following preliminary proposition will be useful.
The curve which is the locus of points, whose (n - r)th polars
break up into a straight line and a curve of degree r - 1, passes
through every point on a curve where the tangent has a contact of
the rth order.
The equation of a curve which passes through the vertex A of
the triangle of reference is
u, OLn-1 + uan-2. -= +............... (1).
Now ul is the tangent at A, and if this tangent has a contact of
the rth order with the curve, ut must be a factor of all the u's up
to ur; whence (1) becomes
u1 (v0o-1 + v1a'-2 + - +......,._1n-r) +....... n = 0.
The (n - r)th polar of A is d'-'F/dcan-, which breaks up into the
tangent at A and a curve of degree r- 1, which proves the proposition. In the case of a quartic, the proposition becomes: The
locus of points, whose polar cubics break up into a conic and a
straight line, passes through every point where the tangent has a
contact of the third order with the quartic.