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].

S. T. Tsou and A. G. Walker have defined the I-extension of a given Lie algebra as a certain Lie algebra on the Cartesian product of the given algebra and one of its ideals (Tsou 1955). I-extensions have been studied also in connection with metrisable Lie groups and metrisable Lie algebras. The definition can be applied immediately to any anti-commutative algebra, and in this paper properties of such I-extensions are established. A list of all proper I-extensions of dimension not greater than four over a field of characteristic zero is also given together with a set of characters.

The synthesis of 7-bromo- and 7-chloro-fluoranthene from the readily accessible 1, 2, 3, 1ob-tetrahydro-3-oxofluoranthene is described. Oxidation of the chloro-compound yields 1-chlorofluorene-9-one-8-carboxylic acid and thus opens up a method for the preparation of 1, 8-disubstituted fluorenes and fluorenones.

Thanks are expressed to the Department of Scientific and Industrial Research for the award of a Maintenance Grant to one of us (D. A. C).

In order to estimate the number of partitions of a multi-partite number, the components of which are all large and of approximately the same order of magnitude, it is necessary to evaluate for ℛ(z1) > o(l=1, 2, …,j — 1) the integral

where

for o < u < 2π min (1, |z1|−1, …, |zj−1|−1) and Asymptotic expansions are obtained for I when the z1 are small. Simple expressions give an approximate value of I when every zl is real and exact formulæ are derived when every Zl is real and rational.

An asymptotic formula is given for the number r(s, P; N) of representations of an integer N as the sum of s non-negative squares, where each square does not exceed P2. The numbers s, P and N are large and are subject to certain conditions, one of which is that N is approximately ⅓sP2.

Sets of integers are constructed having the property that n members are in arithmetical progression only if they are all equal; here n is any integer greater than or equal to 3. Previous results have been obtained only for n=3. The problem is generalized in various ways. The analysis can also be applied to construct sets for the analogous problem of geometrical progressions. These sets are of positive density, unlike those of the first kind, which have zero density.

The roots of these equations are of importance in several theories and various authors have studied certain of their properties. Here we solve the equations in the sense that we define two numbers z(1), z(0), and a sequence {zn} which include the roots of both equations. Except for a small, finite number of values of n, we find a rapidly convergent series for zn whose terms are alternately real and purely imaginary. We give a number of expansions and a variety of practical methods which enable us to calculate the small number of remaining roots to any required degree of accuracy.

The main results of this article have been announced without proof or details in Wright 1960.

The preparation of 1-benzhydrylnaphthalene, 1, 2, 3, 4-tetrahydro-1-benzoylnaphthalene, and 1-diphenylmethylenetetralin is described.

Thanks are expressed to the Department of Scientific and Industrial Research for a maintenance grant (to D. K.); to the Royal Society of London for a grant (to A. J. P.); to Miss E. W. Robertson for help with the experimental work; and to the Education Committee of Stirlingshire for permission to use laboratory facilities.

Algebras which are nilpotent and anti-commutative are studied. Canonical forms are found for all such algebras of dimension n whose centres have dimension n−r (r < 3), and characters are given which enable any two non-isomorphic algebras to be distinguished.

A metrisable Lie algebra is a Lie algebra for which there is a non-singular, symmetric, adjoint-invariant bilinear form a(λ, μ), and such an algebra is reduced if its centre is contained in its derived algebra. The importance of the reduced algebras follows from the fact that every metrisable Lie algebra is the direct sum of a reduced metrisable Lie algebra and an abelian Lie algebra. Tsou (Thesis 1955) introduced metrisable Lie algebras, and obtained canonical forms for all real reduced metrisable Lie algebras whose derived algebras have dimension 3. We conclude this paper by providing an alternative derivation, two of the algebras being nilpotent.