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

CHAPTER IV.
PLUCKER'S EQUATIONS.
84. WE have already seen that a cubic curve cannot have
more than one double point or a quartic more than three. We
shall now give a series of propositions, due to Plticker, by means
of which the number and species of the different singularities of a
curve of given degree can be determined.
A curve of the nth degree cannot have more than (n1-l)(nz-2)
double points.
Let there be s double points. We have proved in ~ 16 that
when a curve passes through a double point on another curve, it
intersects the latter in two coincident points; hence every double
point counts for two amongst the points of intersection of two
curves. We have also proved in ~ 35 that the first polar passes
through every double point; hence if the first polar intersect the
curve in r ordinary points
(n- l)= 2s +.......................(1).
But a curve of the (n - l)th degree can be made to satisfy
(n - 1) (n + 2) conditions; if therefore the curve has its maximum
number of double points
2(,n-1) (n + 2) = s + r..................... (2),
whence by subtraction
= 1 - )( - 2)....................(3).
Equation (3) gives the maximum number of double points for
a curve of the nth degree; but we shall hereafter show that if the
curve has other singularities, the value of s may be less than the
maximum. When n = 3, s = 1; and when n = 4, s = 3, as we have
proved in Chapter II.