Some factors which influence the choice of population size in selection aimed at improving composite populations of maize as sources of inbred lines are considered. The consequences of different effective population sizes on progress from selection within a population with a specific genetic model are examined, using a new mathematical approach described by Curnow and Baker (1968) [but see correction and Curnow and Baker (1969) and later errata notice]. The results obtained suggest that reasonably rapid progress from selection can be expected with small effective population sizes. Substantial added progress may be obtained if selection can be practiced within each of a number of replicate lines developed out of the same original population, followed by selection of the best replicate lines.The results obtained also suggest that the selected lines should be tested in all possible combinations to identify outstanding lines which can then serve as sources of new inbred lines or as foundation material for new crosses in which selection can then be practiced.The effect of departures from the assumed genetic model are discussed. However, the work reported reflects results expected with a specific genetic model. The most important finding is that conditions can exist where recommendations on population size, at present basic in the planning of a selection program, should be reconsidered.

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Kojima's (1961) approximate formulae for the mean and variance of the change in gene frequency from a single cycle of selection applied to a finite population are used to develop an iterative method for studying the effects of repeated cycles of selection and random mating. This is done by assuming a particular, but flexible and probably realistic, approximate form for the distribution of gene frequencies at each generation.The method gives for each generation the first two moments of the gene frequency distribution, the expected gain from selection, the probabilities of fixation and also the variability of gain. The variability of gain is of considerable importance in evolution, selection experiments and in plant and animal breeding programmes.Kojima's (1961) formulae have been extended to allow for differentiation between males and females. Hence different selection intensities and population sizes for the two sexes can be studied. Selfing with selection is considered separately. Extensions to cover simple examples of multiple alleles, linkage and epistasis are possible. Reference is made to previous work using transition matrices.