This single volume contains the two original volumes, which were written ten years apart from each other. The first volume is a preliminary discussion of multiple integrals and the associated geometry. The authors treat the connections with algebraic surfaces and integrals of total differentials. The last two chapters of Volume I are devoted to the study of numerical invariants introduced by Clebsch and Noether, and to the double integrals attached to them.The second volume is a more developed discussion of both the geometry and the analysis of algebraic functions, including results obtained by Picard, Castelnuovo, and Enriques in the time after the publication of Volume I. Most of the geometry revolves around linear systems of curves, either in the plane or on a surface. There is also a special chapter on the geometry of hyperelliptic surfaces. The major part of the analysis in Volume II is centered on integrals of the second kind, including their periods and the number of distinct such integrals in specific instances. The book concludes with a long section of notes on various interesting topics.