Two classes of linear algebras, generally non-associative, are defined in § 3 (baric algebras) and § 4 (train algebras), and the process of duplication of a linear algebra in § 5. These concepts, which will be discussed more fully elsewhere, arise naturally in the symbolism of genetics, as shown in §§ 6–15. Many of their properties express facts well known in genetics; and the processes of calculation which are fundamental in many problems of population genetics can be expressed as manipulations in the genetic algebras. In cases where inheritance is of a simple type (e.g. §§ 10–13, 15) this constitutes a new point of view, but perhaps amounts to little more than a change of notation as compared with existing methods. §14, however, indicates the possibility of generalisations which would seem to be impossible by ordinary methods.