When a foreign gene is being bred into a pure line, or when a population with a lethal gene, incomplete sex linkage, heterostylism, or self-sterility is being inbred, the population is kept heterozygous for a particular gene or chromosome segment, but otherwise inbred. In these cases expressions are found for the probability of introducing a gene linked with the given gene, or for finding such a linked gene still heterozygous after inbreeding. Expressions are also found for the mean length of foreign chromosome introduced, or the length of chromosome remaining heterozygous (Table VI). The difficulty of introducing a single gene without any linked genes is emphasised.