Let σ1(n) denote the sum of the tth powers of the divisors of n, σ(n) = σ1(n). Also place

where γ is Euler's constant, ζ(s) is the Riemann ζ-function and x ≧ 2. The function Δ(x) is the remainder term arising in the divisor problem for σ((m, n)). Cesàro proved originally [1], [6, p. 328] that Δ(x) = o(x2 log x). More recently in I [2, (3.14)] it was shown by elementary methods that . This estimate was later improved to in II [3, (3.7)]. In the present paper (§ 3) we obtain a much more substantial reduction in the order of Δ(x), by showing that Δ(x) can be expressed in terms of the remainder term in the classical Dirichlet divisor problem. On the basis of well known results for this problem, it follows easily that . The precise statement of the result for σ((m, n)) is contained in (3.2).

The purpose of this paper is to call attention to a very simple example of intersections on a singular variety, and to its effect on intersection theories relating to ambient varieties with singularities; in particular it will be shown how the example—to which we refer throughout as Example A—invalidates an aspect of the theory put forward by P. Samuel ([1], Ch. VI).

Let m, n, q denote positive integers, p a prime, and a, b, h, r, s, t, u, v integers. If (r, q) = 1, let [r, q] be the integer s for which 0 < s ≤ q and rs ≡ 1 (modq). Let

It is shown that the greatest value of the resultant shear in the Saint-Venant torsion problem for an aeolotropic material possessing digonal elastic symmetry occurs on the boundary of the cross-section.

In a recent paper (Shail [1]) the present author considered the problem of finding a two-centred expansion of the retarded Helmholtz Green's function This work formed an extension to that of Carlson and Rushbrooke [2] (also Buehler and Hirschfelder [3]) on the Coulomb Green's function and arose out of considerations of the interaction energy of two charge distributions taking account of electromagnetic retardation. The two-centred expansion obtained in [1] took the form of a double Taylor series, each term being interpreted as a Maxwell multipole—multipole interaction energy between two “charge” distributions coupled through a retarded scalar field. The first few terms in the expansion were also given in spherical polar coordinates.

A solution is obtained by real variable methods for a Cauchy problem for the generalised radially symmetric wave equation. A solution of this problem has been given by Mackie [1] employing the contour integral methods developed by Copson [2] and Mackie [3] for a class of problems occurring in gas dynamics. The present approach employs a simple definite integral representation for the solution and reduces the problem to solving an Abel integral equation. The real variable approach avoids the unnecessary restriction of the initial data to be analytic and also avoids the difficulty encountered in the complex variable approach in continuing the solution across a characteristic. The solution is in fact obtained in a form valid everywhere in the region of interest.

Let D(f) denote the discriminant of a binary cubic form

having integral coefficients. We restrict ourselves to forms with D(f) ≠ 0, and further (as a matter of convenience) to forms which are irreducible in the rational field. It is a problem of some interest, in connection with the approximation properties of transcendental numbers, to estimate the sum

as H→∞, where ‖f‖ = max(|a|, |b|, |c|, |d|). Since |D(f)| ≪ H4 when ‖f‖ < H, there is the trivial lower bound

In an earlier paper [2] by the present writer, a solution of the equations of elasticity in complete aeolotropy was found under the assumption that the stresses and therefore the strains are linear in the third cartesian coordinate z. In the present paper, the solution is extended to the case where the stresses and therefore the strains are polynomials in z. This provides a wider scope of applications to the problem of elastic equilibrium of a completely aeolotropic cylinder under resultant end forces and couples and a general distribution of tractions on the lateral surface of the cylinder. These tractions may take any form consistent with the elastic equilibrium of the cylinder, provided they are polynomials of any degree in z. Of the many applications of the present theory, Luxenberg [1] considered the very particular case of torsion of a cylinder made up of a material having a plane of elastic symmetry, under a constant lateral loading.

For any (real or complex) transcendental number ξ and any integer n > 0 let ϑn(ξ) be the least upper bound of the set of all positive numbers σ for which there exist infinitely many polynomials p1(x), p2(x), … of degree n, with integer coefficients, satisfying

where ‖pi‖ denotes the “height” of pi(x), i.e. the maximum modulus of the coefficients. Plainly ϑn(ξ) serves as a measure of how well (or how badly) the number zero can be approximated by values of nth degree integral polynomials at the point ξ. It can be shown by means of the “Schubfachprinzip” that, at worst,

if the transcendental number ξ is real, and

if it is complex, i.e.ϑn(ξ) is never smaller than these bounds. Furthermore, a conjecture of K. Mahler may be interpreted as stating that for almost all real and for almost all complex numbers the equations (2) and (3), respectively, are actually true; in other words, almost all transcendental numbers have the worst possible approximation property for any degree n.

The main object of this paper is to give a self-contained elementary proof of a result (Theorem 1, below), which could be deduced from a theorem of Siegel ([1], Satz 2). It seems worth while to do so, because Siegel's proof is long and difficult, though his result is deeper and more precise than mine.

Let be a plane bounded convex set whose width in the direction φ is . It has been shown by L. A. Santaló that

is invariant under unimodular affine transformations. Santaló [1] established a number of properties of this invariant and conjectured that if the area of is A() then and that equality characterizes triangles. In this note Santaló's conjecture is shown to be true.

1. Introduction. Let I0 be a closed rectangle in Euclidean n-space, and let ℬ be the field of Borel subsets of I0. Let ℱ be the space of completely additive set functions F, having a finite real value F(E) for each E of ℬ, and left undefined for sets E not in ℬ. In recent work, we used Hausdorff measures in an attempt to analyze the set functions F of ℱ. If h(t) is a monotonic increasing continuous function of t with h(0) = 0, a measure h-m(E) is generated by the method first defined by Hausdorff [2].

The Siegel conjecture on the rational approximation to algebraic numbers was proved a few years ago by K. F. Roth [1] with the following theorem: Let α be any algebraic number, not rational. If has an infinity of solutions in integers h and q (q > 0) tehn k ≤ 2.

This paper is chiefly concerned with inequalities for the numbers of subgroups of a finite p-group. The following are typical results. Let G be a p-group of order pa, and let nG(pk) denote the number of subgroups of G of order pk.

Papers I, II of this projected series lay the algebraic foundations of the theory of the Clifford groups; I deals with the case p > 2, II with the case p = 2. The present introduction refers to both papers. Our theory has applications in group theory, geometry and number theory.

If A and B are two complex matrices and if U is a complex unitary matrix such that UAUCT = B (where UCT denotes the conjugate transpose of U), then A and B are said to be unitarily similar. Necessary and sufficient conditions that two matrices be unitarily similar have been dealt with in [5] (from the point of view of group representation theory) and in [2] (from the point of view of developing a canonical form under unitary similarity).

In the controversy which has been continuing over the “clock paradox”, it has been claimed by supporters ([1]) of the orthodox interpretation that the difference in behaviour of clocks carried by observers in motion relative to each other can be traced to differencesin the motion relative to the universe. Whilst A remains unaccelerated relative to the universe, B for a portion of his journey does not. Professor Dingle, in reply ([2]), remarked that he would “not dispute a possible slight effect” but could not understand why, if the acceleration of B were so important, it did not appear explicitly in the answer.

If f(x) is a real indefinite quadratic form in n variables with determinant d ≠ 0, we set for any real α.

The present paper deals with the Clifford groups in the case p = 2. For the most part, it runs parallel to the previous paper I ([1]) on the case p > 2, and a number of proofs are therefore either given in outline or omitted. A general introduction to both papers is given in I, § 1.