## Abstract

The problem of finding the theoretical values of covariances in diploid populations under selfing is solved first for the case of one locus with an arbitrary number of alleles and then for the case of an arbitrary number of loci, each with an arbitrary number of alleles and with arbitrary epistacy, in the absence of linkage and selection. It is found in the one-locus case that the covariance of the $r\text{th}$ generation and $s\text{th}$ generation progeny means of individuals in the original population can be expressed as $\sigma _{GG}-(K_{r}+K_{s})$ $\sigma _{GH}+K_{r}K_{s}$ $\sigma _{HH}$ where $\sigma _{GG}$, $\sigma _{GH}$ $\text{and}$ $\sigma _{HH}$ are defined and $K_{4}=2(1-1/2^{R})$. It is also found in the one-locus case that the covariance within $(k-1)$th generation families of $k\text{th}$ generation individual progeny means in the rth and sth generations is expressible in the simple form $\frac{1}{2^{k-1}}\sigma _{1}^{2}+\frac{1}{2^{r+s-k-1}}(\sigma _{\epsilon}^{2}+\mu _{\epsilon}^{2})$. For the general case the convariance of $r\text{th}$ generation and $s\text{th}$ generation progeny means of individuals in the original population is expressible as $\sigma _{yy}$+$\underset m=1\to{\overset n\to{\Sigma}}(K_{r}^{m}+K_{s}^{m})(-1)^{m}\sigma _{ym}+\underset l,m=1\to{\overset n\to{\Sigma}}K_{r}^{l}K_{s}^{m}(-1)^{l+m}\sigma _{lm}$. The relationship of the simple results to previous work is discussed.