de Jonquières Theorem

The total number of groups of a $g^r_N$ consisting in a point of multiplicity $k_1$, one of multiplicity $k_2$, ..., one of multiplicity $k_\rho$, where

$$\sum k_i = N$$ (1)

$$\sum (k_i-1) = r,$$ (2)

and where $\alpha_1$ points have one multiplicity, $\alpha_2$ another, etc., and

$$\Pi=k_1k_2\cdots k_\rho$$ (3)

is

$${\Pi p(p-1)\cdots(p-\rho)\over\alpha_1!\alpha_2!\cdots} \lef... ...i\over\partial k_i\partial k_j}\over p-\rho+2}+\ldots}\right].$$ (4)

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 288, 1959.