The purpose of the study of Diophantine equations is to find out as much as one can about the rational solutions of a given indeterminate set of equations. In geometrical language, this is the investigation of the rational points on a given variety. The simplest type of problem for which no satisfactory theory is known is that of the cubic surface—the homogeneous cubic equation in four variables. Throughout this note we shall exclude the case of a cubic cone or cylinder, whose theory follows trivially from that of a cubic curve; however, we do not assume that the cubic surface is non-singular.