It is well known that if, in a continued fraction

there is a subsequence of the an which increases very rapidly, then ξ is a transcendental number. A result of this kind follows from Liouville's Theorem on rational approximations to algebraic numbers, but the most precise result so far established is that which was deduced from Roth's Theorem by Davenport and Roth [1]. They proved (Theorem 3) that if ξ is algebraic, then

where qn is the denominator of the nth convergent to (1). Thus if

ξ is transcendental.

Let G be a graph with b+v vertices, each of the b vertices P1, …, Pb having valency k and each of the remaining v vertices Q1 …, Qv having valency r, and each edge joining a vertex Pm to a vertex Qn. Suppose also that b≥v; then r≥k since bk = vr.

This paper is concerned with infinitesimal transverse displacements of homogeneous isotropic elastic plates. The method uses moments of the fundamental equations of orders 0, 1, 2, 3. Assuming a form for the shear stresses tα3, these equations enable one to determine the mean values of the transverse displacements instead of the weighted mean values associated with plate theories of all but the classical type. The relevant moments of the stresses and displacements are expressed in terms of three functions satisfying three differential equations of the fourth order, the solutions of which may be expressed in terms of six independent functions. Thus six boundary conditions may be satisfied. Equating two, three and four of the above functions to zero in turn gives plate theories involving four, three and two boundary conditions respectively. The method is illustrated by assuming that the shear stresses are quadratic functions of the distance from the mid-plane of the material.

This paper is a sequel to a previous paper [1] on axially symmetric torsion-free stress distributions in isotropic elastic solids and applies the methods used to investigate these distributions to distributions in solids under torsion. The basis of these methods is that in a solid of revolution containing a symmetrically located crack the stresses set up in the neighbourhood of the crack by forces applied over the crack can be found by perturbing on their values in an infinite solid containing a crack of the same radius and under the same applied forces, provided the radius of the crack is small compared with a typical length of the solid of revolution. The problem of determining the stresses in the solid of revolution is shown to be governed by a Fredholm integral equation of the second kind, which holds whatever the ratio of the crack radius to the typical length, but which, when this ratio is small, is readily solved by iteration to give stresses perturbing on those in an infinite solid. A similar method can be applied to an infinite solid containing two or more cracks when the crack radii are small compared with a typical length of the crack array.

A formal solution of the coupled linear thermoelastic equations representing oscillatory motion of an elastic body is obtained, and the decay of the normal modes of vibration examined in detail. The main results of Zener's theory of the thermal damping of a vibrating body are then derived, and somewhat amplified, and consistency with the foregoing normal mode analysis established.

If f(s) is the analytic function defined by the Dirichlet series and if where 0 ≤ b < 1, then the series converges for Re s > 1 and f(s) is regular in the half plane Re s > b except for a simple pole with residue C ≠ 0 at a s = 1. Thus f(s) has a Laurent expansion at s = 1 and it has been shown [1] that under these conditions

where

The purpose of the study of Diophantine equations is to find out as much as one can about the rational solutions of a given indeterminate set of equations. In geometrical language, this is the investigation of the rational points on a given variety. The simplest type of problem for which no satisfactory theory is known is that of the cubic surface—the homogeneous cubic equation in four variables. Throughout this note we shall exclude the case of a cubic cone or cylinder, whose theory follows trivially from that of a cubic curve; however, we do not assume that the cubic surface is non-singular.

In this paper we shall consider some axisymmetrical problems involving the determination of a harmonic function which satisfies prescribed conditions on two given spherical surfaces. The latter may be exterior to one another or one inside the other. The classical problems which come under this heading are the electrostatic two sphere condenser and the motion of two spheres along the line of centres in an unbounded inviscid fluid. The early investigators of these problems used the well known method of images (Kelvin [1], Hicks [2]). In the electrostatic case (Dirichlet boundary conditions) the images are sets of point monopoles, and in the hydrodynamic case (Neumann boundary conditions) sets of dipole sources. Later Neumann [3] and Jeffery [4] gave solutions using a bipolar coordinate system.

Archbold [1] has shown how a “distance” can be denned in an affine plane over the field GF(2n) of 2n elements. In terms of this distance, he has shown how to define a group, R(2, 2n), of 2×2 “rotational” matrices which have certain properties of ordinary orthogonal matrices. In the present note we find a standard form for such matrices. Using this standard form, we show that the order of R(2, 2n) is 2n+1+2 and that it has a “proper rotational” subgroup, R+(2, 2n), of index 2. The multiples of R+(2, 2n) by elements of GF(2n) are shown to form a field, which is necessarily isomorphic to GF(22n). The groups R+(2, 2n) and R(2, 2n) are then shown to be cyclic and dihedral groups respectively.

Several recent papers † have been devoted to the problem of the solubility in integers of a homogeneous equation (or of simultaneous equations), or the representation of an integer by a form (homogeneous polynomial). Such problems can be regarded as extensions of Waring's Problem: that of representing an integer as x1k+…+x8k with positive integral x1 …, x8. The methods used are developments of the Hardy-Littlewood method, or if not they employ auxiliary results proved by that method. The number of variables needed to ensure success is usually very large when the degree k of the homogeneous form is large. In this note we draw attention to a somewhat special problem for which a comparatively small number of variables, namely 2k+1, suffices.

A slow steady motion of incompressible viscous liquid in the space between two fixed concentric spherical boundaries is considered. The motion arises from two point-sources of strengths ±2m at arbitrary points A, B on the outer sphere r = a. The velocity is calculated as the vector sum of the velocities in two simpler motions in each of which there is an axis of symmetry so that a stream function can be used. The force exerted by the liquid on the inner boundary r = b is similarly the resultant of two forces, each passing through the common centre of the spheres; it can be simply expressed in terms of a, b, m and the vector .

If f(z) is a function of the complex variable z, regular in a neighbourhood of z = 0, with f (0) = 0 and f' (0) ≠ 0, then the equation w = f(z) admits a unique solution, regular in some neighbourhood of w = 0, given by

where C is an appropriate contour encircling z = 0. These formulae are well-known, being stages in the proof of the classical reversion † formula

due to Lagrange.