Sewall Wrightʹs Equation Δq=(q(1−q) ∂w/∂q)/2w

An equation of Sewall Wrightʹs expresses the change in the frequency of an allele under selection at a multiallelic locus as a function of the gradient of the mean fitness “surface” in the direction in which the relative proportions of the other alleles do not change. An attempt to derive this equation using conventional vector calculus shows that this description leads to a different equation and that the purported gradient in Wrightʹs equation is not a gradient of the mean fitness surface except in the diallelic case, where the two equations are the same. It is further shown that if Fisherʹs angular transformation is applied to the diallelic case the genic variance is exactly equal to one-eighth of the square of the gradient of the mean fitness with respect to the transformed gene frequency