m, the order,
n " class,
[delta] " number of double points,
[kappa] " " cusps,
[tau] " " double tangents,
[iota] " " inflections;
and this being so, Plucker's "six equations" are
(1) n = m(m - 1) - 2[delta] - 3[kappa],
(2) [iota] = 3m(m - 2) - 6[delta] - 8[kappa],
(3) [tau] = 1/2m(m - 2)(m^2 - 9) - (m^2 - m - 6)(2[delta] + 3[kappa])
+ 2[delta]([delta] - 1)
+ 6[delta][kappa] + (9/2)[kappa]([kappa] - 1),
(4) m = n(n - 1) - 2[tau] - 3[iota],
(5) [kappa] = 3n(n - 2) - 6[tau] - 8[iota],
(6) [delta] = 1/2n(n - 2)(n^2 - 9) - (n^2 - n - 6)(2[tau] + 3[iota])
+ 2[tau]([tau] - 1) + 6[tau][iota]
+ (9/2)[iota]([iota] - i).