In recent years several papers [Grad(9), Condiff and Dahler(5), Baronowski and Romatowski(2), Eringen(8), Allen and de Silva(i)] have been written dealing with the possibility of antisymmetric stress in a fluid and its relationship with the internal micro-structure of the fluid. In the classical fluid dynamics of the Navier–Stokes equations, the molecules of the fluid element are treated as material points devoid of any internal spin motion and the only type of angular motion that the macroscopic elements of the fluid possess is the usual vorticity ½ curl v, the velocity of the fluid being v. In the case of fluids whose molecules are not regarded as material points, but are treated as micro-structures having internal spin, the total angular velocity of a macroscopic volume element will be equal to the vector sum of spins of the micro-structures and the part due to the vorticity.

1. Introduction. In a paper on the many nucleon problem (l), which we shall henceforth call I, the determination of the irreducible representations of the four-dimensional unitary group were found from a decomposition of its infinitesimal ring U04 The method of decomposition made use of the four primitive four-component idempotents (projection operators) of the Dirac ring each of which, as was recognized long ago by Eddington (2), can be identified with a possible charge-spin state of a Dirac particle. Some experimental justification for the representations was also provided, and it is the purpose of this paper to apply the same tools to the many electron problem. In particular, matrices will be derived for the spin multiplets of a system of r electrons, and it will be shown how the model can account for the atomic shell structure and orbital angular momentum.

A method is given for evaluating matrix elements of three particle operators in many nucleon states of [44…4] symmetry.

The existence of a steady vortex ring close to Hill's spherical vortex is established, and an approximate description of its boundary is given. The vorticity in the ring is proportional to the distance from the axis of symmetry. The core propagates steadily in an unbounded fluid at rest at infinity. The boundary of the vortex ring is close to an interior stream surface of Hill's vortex.

.This paper deals with small amplitude waves in inhomogeneous warm electron plasmas. The waves are coupled electromagnetic and electron-acoustic waves, and are described by Maxwell's equations together with single-fluid hydrodynamical equations. Here, previous work is generalized by including the effect of a static pressure gradient. Coupled wave equations are obtained and specialized to the case of a planar stratified plasma. Then, as a preliminary to a treatment of wave coupling, the behaviour of the solutions of the uncoupled wave equations in a coupling region is investigated. The static pressure gradient complicates the behaviour of the uncoupled field components; singularities occur at two points which coalesce as the static pressure gradient is allowed to tend to zero.

An identity of fundamental interest in General Relativity is the breakup or ‘decomposition’ of the scalar curvature density into an ordinary divergence in addition to a term constructed solely from the metric tensor and its first derivative. It is proven that this identity is the most elementary example of an identity satisfied by all members of a particularly important class of Riemannian invariants. For a subset of these invariants, the decomposition identity undergoes a simplification that connects this material with some problems of current research interest.

It is shown that the n-dimensional simple wave flow of a perfect gas, that is, flow in which the velocity components, pressure and density depend on only one generating function, must be isentropic. This result is then employed to explain why two independent investigations of apparently different types of simple waves have led to the same results.

This paper is concerned with the linear theory of the transmission of sound through a vortex sheet separating two fluids in relative motion, but with the same density and sound speed. For harmonic excitation asolution is determined, with particular attention to transition regions where large effects might be expected, and it is found that Helmholtz instability plays no role in this solution. However, this harmonic field does not lead to a solution of the initial value problem which satisfies causality. When causality is complied with an additional field must be superimposed which gives waves growing exponentially in space in the harmonic case and a singularity, which is more severe than has been previously encountered, in the time-dependent problem. The consequent effects of this instability are discussed.

Extremum principles are obtained for the laminar flow of a conducting liquid in a straight pipe of arbitrary cross-section with conducting walls of arbitrary thickness, which is subject to a uniform applied magnetic field. These extrema provide upper and lower bounds for the mass-flow rate in the pipe. These bounds are particularly useful in the construction of asymptotic estimates for the flow rate at high Hartmann number. The method is illustrated by its application to the pipe of circular cross-section and to pipes with thin conducting walls.

An existence theorem is proved for Robin's integral equation for the density of electric charge on a closed surface, under the assumptions that the surface is convex, smooth and twice continuously differentiable. The technique is essentially Neumann's method of the arithmetic mean, used by Robin himself to show that the solution, assumed to exist, can be successively approximated by a sequence. In order to facilitate the main argument of the proof, it is assumed initially that the Gaussian curvature is everywhere positive, but this restriction is subsequently removed.

A minimum principle associated with a class of magneto-elastic boundary-value problems is presented. The principle depends on a result on integral inequalities which appears to be new. An accurate variational solution is obtained for an illustrative one-dimensional problem.

The compression of fairly short solid cylinders under axial load is considered. Radial expansion is prevented over a central region of the outer surface by a rigid constraint concentric with the cylinder. Physically the situation arises when a fairly soft rivet is expanded into a relatively rigid plate. Two types of elastic material are considered; firstly, rubber-like materials governed by a strain energy function of the Mooney form, and secondly, metals which have a quadratic strain energy function. In the former case a finite axial compression is permitted prior to contact between the cylinder and the constraint. In both cases the irregularities introduced by the constraint are sufficiently small that they can be described by infinitesimal elasticity theory. The analysis utilizes displacement potential functions and is reduced to solving a set of dual cosine series. The particular case in which the cylinder height and diameter are equal and the contact height is equal to the radius is examined in detail and the contact stresses are given graphically.

1. Introduction Born and Infeld originally proposed their non-linear theory of electromagnetism (1) as one derivable from a Lorentz invariant Lagrangian and, in contradistinction to the usual Maxwell theory, possessing solutions of the ‘point charge’ type without singularities at the position of the charge. Subsequently, examination of the theory by means of iterative methods (2) has demonstrated the self-scattering of outgoing radiation and the existence of wave tails. In the present paper it is shown that the theory, under suitable asymptotic conditions, gives rise to peeling of the type exhibited by Maxwell's theory as well as by the asymptotically flat space-times of general relativity (3, 4, 5). It is also shown that certain conserved quantities exist paralleling those exhibited in the above theories. The Maxwell theory exhibits a sequence of such conserved quantities, the truncation of which is examined in detail for the Born-Infeld theory.