Weierstraß's Gap Theorem

Given a succession of nonsingular points which are on a nonhyperelliptic curve of Genus $p$, but are not a group of the canonical series, the number of groups of the first $k$ which cannot constitute the group of simple Poles of a Rational Function is $p$. If points next to each other are taken, then the theorem becomes: Given a nonsingular point of a nonhyperelliptic curve of Genus $p$, then the orders which it cannot possess as the single pole of a Rational Function are $p$ in number.

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 290, 1959.