This paper deals with the branch of statistical ecology which studies the pattern of variation in the abundances of the different species present in a habitat, as distinct from the actual abundances of a fixed set of species. Much of the work in this field has been done by ecologists, the leading contributor being C. B. Williams, who has published a monograph on the subject (Williams, 1964). A consequence is that while the practical phenomena which stimulated the development of the theory are well documented, the mathematical reasoning behind the results is sometimes left implicit or unclear. For instance the empirical observation that in a random collection of insects the numbers of species represented by 1, ..., s specimens are often approximately of the form