II.—A Matrix Notation for Mendelian Populations

Abstract

Those who have had experience of problems involving inbreeding or the correlation of relatives in a population with specified proportions of Mendelian genotypes will have realised what difficulties reside in the absence of a convenient notation. In the absence of such a notation difficulties arise more from the unwieldiness of the expressions obtained than the abstruseness of the problems encountered. The familiar chessboard diagrams of Mendelian hybridisation and the form of a contingency table for relatives alike suggest that a matrix notation might be generalised to take into account both sex-linked inheritance and the more usual type of transmission. A determinant form would meet some of the requirements of autosomal transmission; but is not adapted to the asymmetrical case, when the male sex cannot be heterozygous. In extending to more remote relationships a previous inquiry (3) into fraternal and filial correlations involving sex linkage, the writer has found it useful to employ a system of operations applicable to other types of inquiry. The former investigation is not yet complete. In the meantime it seemed possible that the method used might prove suggestive to other workers. A preliminary account of the fundamental operations is here given in so far as they are relevant to single gene substitutions.