Let X be a metric space and τ a non-negative function on the subsets of X. By the well-known Carathéodory process, we generate outer measures μδ(τ), for δ > 0, and (see §3). When, for every A ⊂ X, τA = (diamA)s for s ≥ 0, μ(τ) is the Hausdorff s-dimensional measure, and, if τA = h(diam A) for a monotone continuous function h with h(0) = 0, μ(τ) is the Hausdorff h-measure. In both of these cases, μ(τ) has been extensively studied.

The Local Uniformisation Theorem was proved by O. Zariski [5] in 1940, and, for the general case, it is so far the only existing proof of the theorem.

Let V be an irreducible manifold defined over a ground field of characteristic zero, and let Σ be its function field. Suppose V lies in an affine space An, and D is any subvariety of V not at infinity. Let J be the integral domain of V and ρ be the prime ideal in J defining D. Then we denote the quotient ring of D by Q(D|V), and by this we shall mean the quotient ring Jρ [1; p. 99]. Thus when we deal with subvarieties of two birationally equivalent manifolds V and V', then the quotient rings will always be subrings of the same representation of the function field of V and V'. Let B be any valuation of Σ whose centre on V is C. The Local Uniformisation Theorem states that there exists a birational transform V' of V such that the centre C' of B on V' is simple and Q(C/V) ⊆Q(C'/V').

The results given here represent an extension of previous work [1, 2] in which the author considered the oscillations of a plane current-vortex sheet in an ideal perfectly conducting fluid. In this paper we consider the effects of curvature of the sheet in a direction transverse to the velocity and magnetic field direction. This problem may be regarded as that of finding longitudinal small oscillations on a jet of fluid which moves along the lines of force of an impressed magnetic field. For oscillations, whose wavelength is small by comparison with the radius of curvature of the section of the jet, it is to be expected that the criterion for stable or unstable oscillations will be the same as for the plane case examined previously, and this is verified. When one considers the other extreme, in which the wavelength of the oscillations is large, the analysis shows that the magnetic field aligned to the jet has the effect of stabilising the jet, irrespective of the magnetic field strength. The magnetic field thus behaves for large wavelengths in the same way as a surface tension does for small wavelengths. For values of the applied magnetic field which would make the current-vortex sheet without curvature unstable, it is seen that there is a single transition from instability to stability as the wavelength increases. It is shown also that when small wavelengths are stable, in addition to large wavelengths, it does not necessarily mean that the jet will be stable for all wavelengths. Criteria are deduced to distinguish this case from another in which the jet remains unstable for a simple bounded range of intermediate wavelengths.

Let k be a finite field of q elements. The equation f(x, y) = 0, where f(x, y) is a polynomial with coefficients in k, may be construed to represent a curve, C, in a plane in which x, y are affine coordinates. On the other hand, this equation can be thought of as denning y as an algebraic function of x, where x is transcendental over k. The purpose of this paper is to show that, for a certain class of curves, corresponding in the classical case to curves having n distinct branches at x = ∞, if the degree, n (in y), of the polynomial f is large compared with q, then the genus† of C cannot be too small. We infer this result from a theorem about the genus of a function field; for we can think of C as being a model of such a field.

In this note we prove:

Theorem. Let Λ be a normal number field of absolute degree n. Then there exist infinitely many normal number fields K with the following properties.

(i) The Galois group of K over the rational field Q is the symmetric group Sn on n symbols.

Siegel defined an E-function to be a function which admits a power series expansion

with complex coefficients βn, belonging to some algebraic number field K (of finite degree over the rationals Q), satisfying the following conditions:

(i) Given ε > 0, the maximum of the absolute values of the conjugates of βn, satisfies

Let I be a homogeneous ideal in the polynomial ring R = Λ[X1, X2, …, XN], where Λ is a field or, more generally, an Artin ring†. Then R|I has an induced structure as a graded R-module and its homogeneous elements of degree n form a Λ-module of finite length. If this length is denoted by H(n, R|I), then H(n, R|I), considered as a function of n, is often known as the Hilbert function of the ideal I although, in other contexts, it is called the Hilbert function of the graded module R|I. We shall adopt the latter terminology.

Suppose we have a set of n points P1, …, Pn in a unit square. Let

and let Sn be the upper bound of d1+…dn taken over all positions of P1, …, Pn in the square. We prove that

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.

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.

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.

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.

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.

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.

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.

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.

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 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 .