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

PLUCKER'S EQUATIONS.

55

89. We are now in a position to establish Plicker's equations.
We shall denote
the degree of a curve by m,
its class*,
the number of its nodes,, 8,,,',, cusps,, K,,,, double tangents,,,... stationary tangents,,
the deficiency of the curve,, D.
By ~~ 88 and 86, it follows that
Mr = n (n - 1) - 2 - 3c..................(4),
= 3n (n - 2)-68-8/ 8..................(5).
We have also shown that a node corresponds to a double
tangent on the reciprocal polar, and a cusp to a stationary tangent
or tangent at a point of inflexion; also the class of the reciprocal
polar is equal to the degree of the original curve and vice versd.
Whence reciprocating (4) and (5) we obtain
n = m (m - 1) -- 2r-3..................(6),
K = 3m (n - 2) - 6 - 8..................(7),
also by ~ 84
D = - 1) -.................. (8).
Equations (4) to (8) are Plicker's equations, but only four of
them are independent; for if we eliminate 8 from (4) and (5) and
z from (6) and (7) the result in both cases is
3 (n -nm)= c -........................(9).
* Dr Salmon denotes the degree of a curve by im and its class by n; but since n
is usually employed to denote the degree of a curve or of an algebraical expression
the notation in the text is preferable.