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

116

QUARTIC CTURVES.

cannot be done, for if a trinodal quartic has two real biflecnodes
the third node must be a complex biflecnode composed of a conjugate point and two imaginary stationary tangents. Hence a
quartic cannot have more than eight real points of inflexion.
Points of Undulation.
178. We shall commence the consideration of points of undulation by proving the following two theorems.
If the tangent at any point of a curve has a contact of order r,
the tangent is equivalent to r - 1 stationary tangents.
Let the axis of x have a contact of order r with the curve at
the origin; then the equation of the curve must be of the form
y (1+B+x +Cy +... u,_) + xr+l (a + bx + cy +... Vn-r-)= O,
where un, vn are binary quantics in x and y.
A first approximation shows that the form of the curve in the
neighbourhood of the origin is y + ax+l = 0; whilst a second
approximation gives
y + x +1 {a + (b - B) x} = 0,
whence
dX2
If there is a point of inflexion at a point Q in the neighbourhood of the origin, the abscissa of Q will be given by the equation
d2y/dx = 0, and is therefore
ar
(r + 2) (b - B)'
When Q moves up to coincidence with the origin, a = 0, and
consequently when y = 0, the equation of the curve reduces to
xf+2 (Po + plX +... pn-r-2 Xn-r-2) = 0,
which shows that the axis of x has a contact of order r +1 at the
origin.
The preceding theorem shows that every point of undulation
is formed by the union of two points of inflexion; and also, in
combination with ~ 177, shows that a quartic cannot have more
than twelve points of undulation, and that not more than four of
these points can be real.