1. There is a lemma given by Severi which is of importance because it is used by him in his proof that the number of finite Picard integrals belonging to an algebraic surface is equal to the irregularity of the surface; it is also used by Castelnuovo † in his proof of the same result. The lemma is: If upon an algebraic curve there is an irreducible algebraic series, ∞1, of sets of s points, of index r; and if the sets of sr points which consist of all the r sets which contain a given point (this point taken r times) move in a linear series as this point varies, then any one of the r sets of s points separately moves in a linear series. The proof given by Severi was held satisfactory by Castelnuovot, but I find difficulty in stating it with precision; and Castelnuovo gives an entirely different proof, founded on an enumerative formula due to Schubert (loc. cit. p. 341); this is also the course adopted by Enriques-Chisini.