Extension of covariance selection mathematics.

485 This paper gives some extensions of the selection mathematics based on the covariance function published in Price (1970). Application of the mathematics to 'group s.election' is briefly illustrated. More about applications will be shown in a later paper concerning 'Selection in populations with overlapping generations', which will be submitted to this journal. To facilitate reference in that paper, the equations in this paper are labelled with the letter 'A'. The mathematics given here applies not only to genetical selection but to selection in general. It is intended mainly for use in deriving general relations and constructing theories, and to clarify understanding of selection phenomena, rather than for numerical calculation. In this paper we will be concerned with population functions and make no use of sample functions, hence we will not observe notational conventions Jqr distinguishing population and sample variables and functions. We begin by defining notation for weighted statistical functions. Here we generalize and extend notation defined in Price (1971): ave",x = (~ wixi)/~Wi' i i .-covw(x, y) = [~wi(xi-avewx) (Yi-avewy)]/~wi' i i (A 1) Here w, x and y can be any variables, and summations are over all population members. We call W the 'weight' or 'weighting variable', and we can speak of these functions as 'weighted arithmetic means', 'weighted covariances " 'weighted variances', and so on. If all W i are equal, the weighted functions become ordinary unweighted functions, hence what is said about weighted functions will apply to unweighted functions as particular cases. It should be noted that the value of a weighted function is unchanged if every weight, Wi' is multiplied by the same constant. It should also be noted that the right sides of equations (A 1) to (A 5) are identical with standard expressions for grouped variables, where W i = the number of individuals in the ith group. The sole difference is that we are not limiting W to integral values; but of course with grouped variables we can approximate weighting by fractional amounts as closely as desired by using groups of size CW i , where C is a large constant. Accordingly, any identity that holds 32-3