The discussion of the distribution of the gene ratio of the author's paper of 1922 is amended by the use of a more exact form of the differential equation to be satisfied. It appears that the time needed to halve the variance by random extinction of genes in the total absence of mutations should be 1·4 instead of 2·8 times the number of potential parents in each generation. Either value shows that the loss of variance due to this cause is too trifling to be appreciable in the balance of causes which maintain the actual genetic variability of species.

The same correction alters the distribution appropriate for the maintenance of variability at a fixed level by mutations in the absence of selection. The new solution closely resembles the form previously obtained and now confirmed for the practical case in which selection is present. The method of differential equations, however, fails to deal satisfactorily with these cases, owing to the failure of the integrals to converge at the termini representing cases in which one or other allelomorph is extremely rare.

A method of functional equations is developed for dealing with the termini, and is shown to lead to the same solutions as the amended differential equations in the central portion of the range for which the latter are valid, and further to give the terminal distribution of rare allelomorphs. The method requires the investigation of a continuous function uv of argument v satisfying the recurrence formula

From the asymptotic form of this function its expansion in the neighbourhood of u = 0 is derived, giving the frequencies of the required distributions.

Exceedingly minute values for the selective advantage or disadvantage make a great difference to (i) the chance of success of a mutation and (ii) the contributions of such mutations to the specific variance. The order of magnitude to be considered is the inverse of the population of the species. The neutral zone of selective advantage in the neighbourhood of zero is thus so narrow that changes in the environment, and in the genetic constitution of species, must cause this zone to be crossed and perhaps recrossed relatively rapidly in the course of evolutionary change, so that many possible gene substitutions may have a fluctuating history of advance and regression before the final balance of selective advantage is determined.