A discussion is given of Mayer's theory of the condensation phenomenon. It is shown that the theory is not restricted to classical statistical mechanics, nor that it is necessary to assume the additivity property of the intermolecular forces. The theory is based on Ursell's development of the partition function in powers of the volume ( Β§ 2). Then the reasoning of M a y e r is given ( Β§ 3), which leads to the fundamental equations (I) and (II), which represent the equation of state of a non-ideal gas. In Β§ 4 the analogy is shown between these equations and the equations of Einstein for the ideal Bose gas. For this latter case Einstein had predicted already a condensation phenomenon. Attention is drawn to the analogy between his argument and the reasoning, by which M a y e r explains the condensation of a vapour ( Β§ 5). Assuming certain properties (a--e, p. 19) of an analytic function X(z), which is characteristic for the behaviour of a real gas, a rigorous derivation of the condensation phenomenon is given ( Β§ 6). Β§ 1. Introduction. In a series of papers M a y e r and collaborators 1) have recently tried to explain the phenomenon of condensation on the basis of classical statistical mechanics. The contribution which M a y e r has made to this fundamental problem, although quite important, was far from convincing, especially from the mathematical standpoint. B o r n *) has succeeded in simplifying and improving the arguments of M a y e r. We have been working on the same lines and have partially reached the same results. Our consideraΒ’ions however are somewhat more general than those of Mayer and Born, in so far that they are not restricted to classical statistical mechanics. Furthermore we shall show a) the *) The first paper of Born 2) was presented at the Van der Waals Centenary Congress in Amsterdam. The discussions there with him and other members of the congress have been very clarifying for us. A second paper of B o r n and F u e h s will appear in the Proc. roy. Soc. We are greatly indebted to Prof. B o r n for showing us