Given a succession of nonsingular points which are on a nonhyperelliptic curve of Genus , but are not a group of the canonical series, the number of groups of the first which cannot constitute the group of simple Poles of a Rational Function is . If points next to each other are taken, then the theorem becomes: Given a nonsingular point of a nonhyperelliptic curve of Genus , then the orders which it cannot possess as the single pole of a Rational Function are in number.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 290, 1959.