Introduction. Attempts have been made recently by Albanese' and Severi2 to extend to surfaces some of the classical results of the theory of correspondence between algebraic curves, and in particular, the theory of correspondences with valency. Albanese's investigations are concerned primarily with the behaviour of continuous systems of curves on the surfaces under the transformation induced by the correspondence, and he makes extensive use of the Picard varieties associated with the surfaces. He is thus enabled to define correspondences of valency zero between two surfaces, and correspondences with valency on a single surface, and to prove that on surfaces with general period-systems for the integrals of total differentials of the first kind these are the only correspondences which exist. Severi, on the other hand makes use of his series of equivalence in defining valency. His concept of valency is essentially less general than that of Albanese, which relates to the wider class of regular series. Both authors then consider a formula for the number of coincidences in a correspondence with valency on a surface, which was first obtained, in some special cases, by Zeuthen. In this paper we consider the general question of algebraic correspondences between two Vd, distinct or coincident. Our method is to study the effects of the correspondence on the linear cycles of the Vd, and upon the associated Riemann matrices (period-matrices of the simple integrals of the first kind). For d = 2 this is the analytical equivalent of Albanese's geometrical reasoning, but it seems rather more concise. We obtain generalisations of many of Albanese's theorems as consequences of a single matrix equation. A simple topological interpretation of valency is also obtained. Finally, we show that the Zeuthen coincidence formula is a very special case of the general topological coincidence formula due to Lefschetz, which is valid for correspondences on arbitrary topological manifolds.