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

POINTS OF INFLEXION.

115

Points of Infiexion.
176. It appears from ~ 158 that an anautotomic quartic
cannot have more than twenty-four points of inflexion. We shall
now prove that the maximum number of real points of inflexion is
eight.
Let 0 be a node on a curve; then it follows from ~~ 46 and 85
(i) that 0 is a node on the Hessian, (ii) that the nodal tangents
at 0 are common to the curve and its Hessian, (iii) that the curve
and its Hessian intersect in six coincident points at 0. Hence
each nodal tangent is equivalent to three stationary tangents.
If 0 is a conjugate point, all six tangents are imaginary;
hence a conjugate point reduces the number of imaginary points
of inflexion by six.
If 0 is a real cusp, the curve and its Hessian intersect in
eight coincident points at 0; hence the cuspidal tangent is
equivalent to six imaginary and two real stationary tangents. It
therefore follows that a cusp reduces the number of imaginary
points of inflexion by six and the number of real ones by two.
If the cusp becomes a crunode, two of the imaginary stationary
tangents move away to some other points on the curve, and each
nodal tangent is equivalent to one real and two imaginary
stationary tangents. Hence a crunode reduces the number of
imaginary points of inflexion by four and the number of real ones
by two.
If a node or a cusp is imaginary, all the tangents are imaginary; but since imaginary singularities occur in pairs, it follows
that a pair of imaginary nodes or cusps reduces the number of
imaginary points of inflexion by twelve and sixteen respectively.
177. To prove that a quartic cannot have more than eight real
points of inflexion.
We have already shown that a crunode reduces the number of
real points of inflexion by two; hence a real biflecnode reduces
the number by four. Now if it were possible for a quartic to have
ten real points of inflexion, the fourteen constants could be determined so that the points A and B should be real biflecnodes, and
the point C a real crunode; but we have shown in ~ 170 that this
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