In the second paper under this general title, it was shown how a theorem about the torus could be deduced by a limiting process from a theorem on finite abelian groups. The object of this paper is to prove a similar continuous analogue of H. B. Mann's (α+β)-theorem. It was found that the limiting process used in the second paper could not easily be modified to apply to the present problem, and an alternative method had to be found. The method is, roughly, to prove the result first for open sets satisfying certain conditions, then for closed sets by taking intersections of open sets, and finally for arbitrary measurable sets, since every measurable set contains a closed set of almost equal measure.

Ever since Mendel promulgated his famous laws, probability theory and statistics have played an important role in the study of heredity (9). Etherington introduced some concepts of modern algebra when he showed how a nonassociative algebra can be made to correspond to a given genetic system (1, 4). The fact that many of these algebras have common properties has led to their study from a purely abstract point of view (2, 3, 5, 6, 11, 12). Furthermore, the techniques of algebra give new ways of attacking problems in genetics such as that of stability.

Let be a covering of a topological space X and ℱ a sheaf of abelian groups over X. By a well known result of Leray, (3) II theorems 5.2.4. and 5.4.1., if is open, or closed and locally finite, there exists a spectral sequence {Er} satisfying isomorphisms and for some filtration of the graded group H*(X, ℱ). ℋq(ℱ) denotes the system of coefficients over : s→Hq(| s |, ℱ).

A class of dual integral equations involving Bessel functions is solved by formal application of Mellin transforms.