I n tr o d u c t io n J. E. M ayer (1937) has published, together w ith some collaborators, several papers under the same title as the present one.* We consider these papers as a m ost im p o rtan t contribution to statistical mechanics, and this opinion was shared by the In tern atio n al Conference held in A m sterdam , 26 N ovem ber 1937, in com m em oration of Van der W aals' birth. One of the present authors gave to this m eeting a rep o rt on M ayer's work (published in Physica, 1937)

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... h was followed by a vigorous dis cussion on the question as to w hether M ayer's explanation of th e pheno mena of condensation is correct. D oubts abou t this point were raised by the referee, because it is difficult to com prehend how a m ethod of approxi m ation such as th a t of M ayer, startin g from the gaseous state, can lead to the discontinuity of th e density on an isotherm al curve which corresponds to condensation. The usual m ethods for treatin g the equilibrium of two phases introduce the equation of state of both phases and derive the con dition for their co-existence. M ayer's theory does nothing of this kind, b u t treats all possible molecular arrangem ents w ith th eir proper weight, as if there were only one phase. How can the gas molecules " k n o w " when they have to coagulate to form a liquid or solid? M ayer's m athem atical m ethod is too involved to m ake this point quite clear. We have devoted a considerable effort to control and clarify these calcu lations, m aking am ple use of the theory of complex functions, and we believe th a t we have succeeded in showing rigorously, and in a somewhat simpler way th a n M ayer himself, th a t his statem ents are completely correct. And we have succeeded by our m ethod in going a little farther, in so far as we can derive the conditions for co-existence of several phases (triple point). We have had the privilege of corresponding about these questions w ith Professor G. E. Uhlenbeck (U trecht, Holland), to whom we * We have to thank Dr Mayer for sending us the manuscript of another paper (written in collaboration with S. F. Harrison) before publication; this article which has meanwhile appeared (1938) contains the most complete presentation of his theory. T he statistical m echanics of condensing system s * This theorem has been found and proved by J. E. Mayer and Maria Goeppert-Mayer; but no complete proof has been published. We think that our proo£ is somewhat simpler than that which we tried to reconstruct from Mayer's indications.