The present volume is an account of the analytic principles of the theory of curves, of the rational functions belonging thereto and of the integrals of these functions, with a brief account of the methods, by loops and by Riemann surfaces, for dealing with the periods of these integrals. But the theory of correspondence, and some necessary references to involutions in a plane, find themselves in the succeeding volume, which is mainly devoted to the theory of surfaces.

It is perhaps desirable to explain the origin of these volumes. In the last fifty years a remarkable advance has been made in the theory of surfaces, and of algebraic loci in general; the English reader may find a description of the nature of this in a Presidential Address to the London Mathematical Society given in November 1912 (Proceedings, Vol. XII). But attempts, since the War, to expound these new results have continually shewn the necessity for a precise appreciation of the ideas out of which this advance has developed; in mathematics it is not sufficient to know the enunciation of a result; it is necessary to understand the proof. These two volumes have grown up in the attempt to meet this need. The further need of a volume explaining the applications of topological theory, especially to the periods of the integrals belonging to the higher loci, may, I hope, appeal to another.