Part 1. It is shown that in a large collection of Lepidoptera captured in Malaya the frequency of the number of species represented by different numbers of individuals fitted somewhat closely to a hyperbola type of curve, so long as only the rarer species were considered. The data for the commoner species was not so strictly `randomized', but the whole series could be closely fitted by a series of the logarithmic type as described by Fisher in Part 3. Other data for random collections of insects in the field were also shown to fit fairly well to this series. Part 2. Extensive data on the capture of about 1500 Macrolepidoptera of about 240 species in a light-trap at Harpenden is analysed in relation to Fisher's mathematical theory and is shown to fit extremely closely to the calculations. The calculations are applied first to the frequency of occurrence of species represented by different numbers of individuals--and secondly to the number of species in samples of different sizes from the same population. The parameter ` alpha ', which it is suggested should be called the `index of diversity', is shown to have a regular seasonal change in the case of the Macrolepidoptera in the trap. In addition, samples from two traps which overlooked somewhat different vegetation are shown to have ` alpha ' values which are significantly different. It is shown that, provided the samples are not small, ` alpha ' is the increase in the number of species obtained by increasing the size of a sample by e (2.718). A diagram is given (Fig. 8) from which any one of the values, total number of species, total number of individuals and index of diversity (alpha), can be obtained approximately if the other two are known. The standard error of alpha is also indicated on the same diagram. Part 3. A theoretical distribution is developed which appears to be suitable for the frequencies with which different species occur in a random collection, in the common case in which many species are so rare that their chance of inclusion is small. The relationships of the new distribution with the negative binomial and the Poisson series are established. Numerical processes are exhibited for fitting the series to observations containing given numbers of species and individuals, and for estimating the parameter alpha representing the richness in species of the material sampled; secondly, for calculating the standard error of alpha, and thirdly, for testing whether the series exhibits a significant deviation from the limiting form used. Special tables are presented for facilitating these calculations.