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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

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

and where $\alpha_1$ points have one multiplicity, $\alpha_2$ another, etc., and
\Pi=k_1k_2\cdots k_\rho
\end{displaymath} (3)


{\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].
\end{displaymath} (4)


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

© 1996-9 Eric W. Weisstein