INTRODUCTIONA problem of primary importance in population genetics is that of determining the genotypic frequencies in a large population of sexually reproducing organisms mating at random in nonoverlapping generations with negligible mutation and selection. In populations of disomic organisms where the sexes enjoy equal conditions (i.e. sex-linked inheritance is neglected and both sexes are evenly distributed amongst the various genotypes and possess the same frequencies for the modes of gamete formation), Hardy (1908) showed that gene frequency remains constant and that the genotypic distribution for any Mendelian factor reaches a stationary state in the first generation of random mating. Some years after the discovery of genetic linkage, Jennings (1917) andRobbins (1918) showedthat withtwo linked factors the population approaches an equilibrium state in which the factors are associated at random. They determined the rate of approach to equilibrium and gave the genotypic distribution after any finite number of generations. Geiringer (1945) has given a similar analysis for three factors. With more than three linked factors the problem is much more complex, the basic recurrence relations for the gametic frequencies then forming a system of quadratic instead of linear difference equations. A general but involved method of procedure has been outlined for these cases by Geiringer (1944). In the present paper we develop a general representation of random mating in disomic species as a linear transformation of a set of rational integral functions of the gametic frequencies.The principal components provide an elegant form of solution to the problem. Two other questions are considered from the same standpoint: (i) a modification of the above problem in which allowance is made for possible frequency differences between the sexes, and (ii) random-in-time mating in a population of virus (or haploid) particles.

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IN most plant species in which reproduction is effected both by crossand seif-fertilisation, the relative contributions from these two kinds of matings depend to some extent on climatic and other environmental conditions and so vary from one generation to the next. In the present article, however, we shall confine our attention to the simplified but fundamental situation in which these contributions bear a constant ratio to one another throughout all generations. Consider a large population of plants reproducing in non-overlapping generations in such a way that there is a constant probability s that any plant will be self-fertilised and a probability is that it will cross with some plant chosen at random from the population. We shall suppose that all crosses are equally fertile and all genotypes equally viable. With this system of mating, gene frequency clearly remains constant. Suppose A and a are the genes present at some REFERENCE