SUMMARY Two approaches are given to the problem of approximating the consequences of the types of artificial selection to which domestic animals are subjected. Type and intensity of selection are taken to differ by sex, and it is assumed that selection is imposed over a number of generations. The first approach makes use of a basic recursion formnula and requires that the loci affecting the trait of interest can be partitioned into a few slbgroups, on the basis of similarity in additive gene effects and initial gene frequencies. The second approach is only applicable to the study of selectioni on those traits affected by mainy loci and makes use of mtultivariate-normal theory. The use of the results to explain recently noted differences in estimates of the heritability of dairy-cattle milk production is briefly discussed. The objective in this study is to present mathematical techniques for approximating the consequences of the kinds of artificial selection prevalent among populations of domestic animals, particularly dairy cattle. In these populations it is characteristic that the type and intensity of selection differ for males and females. These differences are a consequence of at least two factors: (i) a male is physically capable of becoming a parent a greater number of times than a female (the importance of this factor has been accentuated by the advent of artificial insemination); and (ii) the trait of ecolomical interest may be expressed in only one sex (e.g. milk production in dairy cattle) and consequently most of the members of the other sex are killed shortly after birth or are otherwise eliminated from the population of potential parents. The need for techniques to determine the effects of these kinds of selection is clear in light of the recent studies of dairy-cattle milk production records conducted by Bradford and Van Vleck [1964] and by Van Vleck [1968]. Using large amounts of data, they found the intraherd heritability of milk production to be approximately 0.4 when estimated by daughter-dam regression, but only 0.25 when estimated by paternal half-sib correlation. Since it is ordinarily held that these two estimates have similar expectations, the difference between them is of considerable interest. Furthermore, the above workers have eliminated many of the possible non-genetic explanations of the discrepancy. A yet largely unexplored explanation of the seeming paradox would be that the difference between the two estimates is due to difTerences between males and females in the intensity and type of selection applied. If culling differs by sex, then it follows from elementary considerations that the two estimates would not have the same expectation. Roughly speaking.