where the sum of the three probabilities in each row is unity. Such an array of probabilities is known as the matrix of" transition probabilities", from uncle to nephew in this case. Once such a matrix is obtained, the absolute frequencies of the various genotypic combinations of uncle and nephew in the general random mating population may be obtained immediately by multiplying the uncle conditions by their respective" initial" probabilities; that is, multiplying the first row by p2, the second row by 2pq and the third row by q2, where p is the frequency of gene A in the population (q=-p). Therefore, we shall largely deal with the transition matrices of relatives in the following. 2. In order to facilitate the derivation of transition matrices for various types of relatives it is convenient to introduce another con-sideration besides the genotypes. Let X be a gene of the locus under consideration. It may be A or a. We say that it exists in two states. If we select a gene from each of two unrelated individuals and they happen to be both A or both a, we say that these two genes are alike in state. On the other hand, if a parent has an X-gene which is transmitted to his child, we say that these two X-genes (one drived from the other) are identical by descent. Genes of the latter kind are necessarily alike in state, barring mutation, but the reverse is not true. The distinction between the two ways in which two alleles may be alike has been pointed out by a number of geneticists (for example, Cotterman 1941; Malecot 1948; Crow 1954). To trace the origin of a gene, we may label each independent gene by a subscript (a serial number for identification). Thus, in a random mating population, the two parents may be represented by X1X2 X XX4, whatever their genotypes.