In this section we extend the definition of an E-set, so that it includes sets of the type

where the only restriction on the Ei is that they be non-singular. We now consider matrices of the type

where each ei takes independently the values 0, 1, …, n − 1, while the a(ei) are either complex numbers or else matrices of order r, the product a(ei) E(ei), in the latter case, being interpreted as the direct product of the two matrices a(ei) and E(ei).

A period of a function f(z) is defined to be a number w(≠0) such that

is identically zero; and it is not a difficult matter to show that an integral function may either have no periods or else a single sequence kλ, k = ±1, ±2, ….

Much has been written, from the algebraical as well as from the geometrical standpoint, on the subject of pencils of quadrics: algebraically the problem consists of the canonical reduction of a pencil of quadratic forms, and the classical paper on the subject is by Weierstrass. But among the different kinds of pencils of quadratic forms there is the “singular pencil,” in which the discriminant of every form belonging to the pencil is zero; interpreted geometrically this means that every quadric belonging to the pencil is a cone. This case was expressly excluded from consideration by Weierstrass, and the canonical reduction was only accomplished later by Kronecker. But, although Weierstrass and Kronecker together solved completely the problem of the canonical reduction of a pencil of quadratic forms, a much clearer insight into the nature of the problem was gained when Segre gave the geometrical solution. He published two papers, one dealing with the non-singular pencils and the other with the singular pencil.

In a recent series of papers, Straneo has introduced into Riemannian space-time a teleparallelism defined by an asymmetric connection, and in this manner attempted to develop a unified theory of gravitation and electricity. The essential idea underlying his work is, therefore, similar to that of the Einstein-Mayer theory of 1929–31.

In a very remarkable work on the operational Calculus, Dr Balth. van der Pol has introduced a new function, playing with respect to Bessel function of order zero the same part as the cosine- or sine-integral with respect to the ordinary cosine or sine. He showed that this function—which he called Bessel-integral junction—can be used to express the differential coefficient of any Bessel function with respect to its index. But he did not investigate the further properties of his new function. I propose to give here some of them, which appear to be interesting, and to introduce and study the functions connected, in the same way, with Bessel functions of any order.

In connection with their proof of the fundamental relation

Whittaker and Watson state that “several proofs of this important proposition have been given, but none are simple.”

In the Mathematische Annalen, 11 (1877), 440–444, E. Toeplitz has proved the following theorem: If three quadrics are polar quadrics of three points with respect to a cubic surface, then their (2, 2, 2) invariant vanishes; the invariant being of the second degree in the coefficients of each of the three quadrics.

The alternant with powers of variables for its elements

has often been investigated and its properties are well known. In the present note an “alternant” with factorial elements,

will be discussed. Here a, b, c, …, z, α, β, γ, …, ω are integers such that

and

with the convention that a(0) = 1, O(0) = 1, and that, where negative indices occur (in the reductions of §2), the factorial is to be regarded as zero.