VIII.—Reciprocity. Part II : Scalar Wave Functions

This paper contains an attempt to formulate more rigorously the postulate of reciprocity (Born, 1939) and to solve the problem of finding reciprocal wave functions. We have not yet completely succeeded in this task, though considerable progress has been made; but it turns out that the original discussion has to be modified in several important respects.

XIX.—On the Reciprocation of Certain Matrices

In a recent paper (Frazer, Jones, and Skan, 1937) some methods are discussed for the approximate representation of functions by means of polynomials. The coefficients in the polynomials are determined by the equationwhere c is the column of coefficients, h is a column of known constants, and M is a matrix which depends on the method of representation adopted. The present paper shows how the reciprocal matrix M-1 can be computed rapidly and simply in the two cases where M is a moment matrix or an alternant matrix.

XXI.—The Swimming and Burrowing Habits of the Amphipod Urothoë marina (Bate)

The intertidal and shallow-water sands of the coasts of the British Isles contain, in many cases as the dominant amphipod fauna, species of the three genera Haustorius, Urothoë, and Bathyporeia of the family Haustoriidæ, often coexistent in the same habitat and obtainable in the same samples. The swimming and burrowing mechanisms of Haustorius arenarius have been described by Dennell (1933), and I have described (1939) the mechanisms in some species of the genus Bathyporeia. This paper deals mainly with Urothoë marina; no reference is made to the other species of the genus, although U. brevicornis has been examined and found to be similar to U. marina. Since all species of the genus Urothoë are very similar morphologically and live in similar habitats, it may be assumed that the following description of U. marina will apply to them also, but I have had no opportunity of examining the more southern species U. grimaldii or U. pulchella. Crawford (1937) states that the burrowing habits of U. brevicornis and U. grimaldii var. poseidonis are very similar to those of Haustorius.

XXIII.—Genetic Algebras

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.