Given a succession of nonsingular points which are on a nonhyperelliptic curve of curve 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 curve genus p, then the orders which it cannot possess as the single pole of a rational function are p in number.