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

54

PLUCKER'S EQUATIONS.

two curves will intersect one another in six coincident points.
Also if any branch of the one curve touches any branch of the
other curve, the two curves will intersect at a seventh point. But
in the present case two of the branches at the triple point on the
Hessian touch one another and also the two branches of the cusp
on the original curve; accordingly at a cusp the curve and its
Hessian intersect one another in eight coincident points, and
therefore the number of ordinary points of intersection cannot
exceed 3n (n - 2) - 8c.
By combining the last two theorems it follows that:If a curve has 8 nodes and K cusps, the number of points of
inflexion is 3n (n - 2) - 68 - 8K.
87. If a curve has 8 nodes, the degree of the reciprocal polar
cannot exceed n (n - 1) - 2.
We have shown in ~ 24 that the first polar of a curve with
respect to any point 0 intersects the curve in n (n - 1) points,
which are the points of contact of the (rn - 1) tangents which
can be drawn from 0 to the curve. Hence the class of a curve,
and therefore the degree of the reciprocal polar, cannot exceed
this number. We have also shown that the first polar passes
through every double point; whence if the curve has 8 nodes the
first polar intersects the curve in n(n- 1)- 2 ordinary points.
Hence not more than n (n - 1) - 2 tangents can be drawn from 0
to the curve, which is therefore the degree of the reciprocal polar.
88. If a curve has K cusps, the degree of the reciprocal polar
cannot exceed n (n - 1) - 3c.
We have shown in ~ 44 that the first polar touches the curve
at a cusp, and consequently at a cusp the curve and its first polar
intersect at three coincident points. If therefore a curve has K
cusps, the curve and its first polar cannot intersect at more than
n (n - 1) - 3 ordinary points, which is therefore the degree of the
reciprocal polar.
By combining the last two theorems, it follows that:If a curve has 8 nodes and K cusps, the degree of the reciprocal
polar and consequently the class of the 5crve is n (n - 1) - 28 -:3K.