Professor A. Landé has kindly sent me a manuscript of a paper in which he develops some most important ideas on the meaning and the value of the number 137. His work is founded on the principle of reciprocity which I recently suggested (Born, 1938). Landé adds to this principle a new assumption about the wave functions, representing an electron, and he finds a linear integral equation for these containing a parameter the value of which may be expected to be 137. It is Landé's great merit to have shown that the reciprocity principle leads to a condition for the Planck constant ћ (the value of which in natural units, c = 1, e = 1, is 137), and to have discovered the root of the peculiar order of magnitude of the number. But some difficulties which I encountered led me to reconsider the problem.

1. The integration of systems of linear partial differential equations of hypergeometric type has been the subject of a great number of recent investigations (e.g. Appell, 1925; Appell-Kampé de Fériet, 1926). Especially Professor Horn (see references) and his pupils, for instance Borngässer (1933), have done much valuable work in this field.

The first method that suggests itself for integrating systems of this type is to try power-series expansions of the solutions in the neighbourhood of the singular points of the system. This method, however, does not yield in general a complete set of solutions. Those singular points of the system in which more than n singular manifolds intersect, n being the number of independent variables, give rise to considerable difficulties.

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.