The sequence of orthogonal functions derived from Laguerre-polynomials is known to be complete, and hence closed, in L2 (0, ∞) if (a) > − 1. In a recent paper Dr Kober introduced a generalisation of this sequence, which enables him to extend the known results also for (a) < − 1. Kober's guiding principle seems to be the following one: The Laguerre orthogonal functions form, for (a) > − 1, a complete system of self and skew reciprocal functions of the Hankel transformation of order a. Now, if (a) < − 1, the ordinary Hankel transform has to be replaced by so-called cut Hankel transform. Hence the system of functions which has to replace-Laguerre orthogonal functions when (a) < − 1, should be a complete system of self and skew reciprocal functions of the cut Hankel transformation of order a, such that it reduces for (a) > − 1 (when the cut Hankel transform reduces to the ordinary one) to the sequence of Laguerre orthogonal functions. This, of course, is by no means a unique definition; nevertheless, together with what one would call the permanence of the Mellin transform, it enabled Kober to find a sequence of functions which (i) reduces to the sequence of Laguerre orthogonal functions when (a) > − 1, m = 0, (ii) is a complete set of self and skew reciprocal functions of the cut Hankel transformation with kernel Ja, m and (iii) has the required qualities of completeness and closedness.

The process of duplication of a linear algebra was defined in an earlier paper, where its occurrence in the symbolism of genetics was pointed out. The definition will now be repeated with an amplification. Although for purpose of illustration it is applied to the algebra of complex numbers, duplication will seem of no special significance if attention is fixed on algebras with associative multiplication and unique division; for duplication generally destroys these properties. The results to be proved, however, show that it is significant in connection with various other conceptions which appeared in the discussion of genetic algebras; namely baric algebras and train algebras (defined in G.A.), also nilpotent algebras, linear transformation and direct multiplication of algebras.

We suppose throughout that f(t) is periodic with period 2π, and Lebesgue-integrable in (− π, π).

We write

and suppose that the Fourier series of φ(t) and ψ(t) are respectively cos nt and sin nt. Then the Fourier series and allied series of f(t) at the point t = x are respectively and , where A0 = ½a0, An = ancos nx + bnsin nx, Bn = bncos nx − ansin nx and an, bn are the Fourier coefficients of f(t).

It has been pointed out that, in the paper by H. W. Richmond (Proceedings of the Edinburgh Mathematical Society (2), 6 (1940), 190-191), lines 3 and 4 of page 191 should read as follows:—Now the pencil includes two ruled cubic surfaces, one of them having 8′ as its double line and 7′ for the second line which the generators meet.