The paper deals with weak ergodicity, i.e. the tendency for a chain to ‘forget’ the distant past. This may occur in non-homogeneous chains even if the probabilities of being in a given state do not tend to a limit as the number of trials increases.

Investigation of the general conditions of weak ergodicity leads to the definition of a special subclass of regular matrices. Conditions for chains involving regular matrices are also given (§§2,3).

Section 4 discusses similarities in asymptotic behaviour between chains whose transition matrices differ only by small amounts.

In an earlier paper (Helliwell and Mackie(3)) it was shown that steady two-dimensional flow patterns of a compressible inviscid fluid at high subsonic speed past a finite wedge could be determined quite simply when sonic velocity is attained at the shoulder of the wedge and thereafter the flow breaks away from the shoulder with a free streamline. In a subsequent paper (Helliwell (4)) a similar method of analysis has been applied to determine a flow pattern of the same general type past a finite wedge symmetrically placed in a channel, from which the case of the wedge in the free stream may be deduced as a special case. However, in a general investigation into transonic flow past wedges (Mackie and Pack (5)) it was argued that when the wedge angle or the free stream (subsonic) velocity is too small no supersonic region would develop on the wedge side, and the flow would break away from the wedge shoulder with some higher subsonic velocity, giving a free stream line. The present note examines the flow pattern which develops under these conditions for a wedge symmetrically placed in a channel with parallel walls.

A pair of variational principles are formulated for two-dimensional scattering by obstacles. The first of these is in terms of the obstacle boundary values, and it is shown that a simple ‘optical’ trial function leads to an incorrect frequency dependence for the scattering cross-section. In the second, the obstacle is viewed as the analogue of an aperture coupling two half spaces. The geometric optics part of the cross-section can then be made explicit and is split off to leave a stationary form for the frequency correction. The zero-order calculation for the cross-section of a circle, using corresponding ‘optical’ trial functions, is found to have the correct (Ka)−2/3 frequency dependence.

The behaviour of two disk dynamos coupled to one another is examined in relation to the earth's magnetic field. It is found that reversals of electric current and magnetic field occur, unlike the case of a single disk dynamo.

On assuming a simple engine that drives a disk dynamo, the coupling between electro-magnetism, motion and heat is discussed. A possible range of temperature fluctuation associated with a dynamo action is also estimated very roughly.

The scattering of acoustic and electromagnetic radiation by random refractive index fluctuations of the scattering medium (e.g. the troposphere, the ocean) is a problem of considerable contemporary interest. For brevity, we shall refer to these refractive index fluctuations as dielectric noise. The purpose of this paper is to compare two approaches to the problem of the scattering of incident plane waves by a confined region of dielectric noise. In the first approach, which is the conventional one, the scattering dielectric noise is represented as a ‘section’ of a homogeneous (i.e. spatially stationary) random process. In the second approach, which is based on a recent generalization of the notion of stationarity (1), the scattering dielectric noise is represented as a locally homogeneous random process. The advantage of representing the dielectric noise as a locally homogeneous process, rather than as a homogeneous process, is that one can thereby take into account in a natural way the spatial variation of the average noise power, without departing from the physically reasonable requirement that the process be approximately homogeneous in suitably restricted domains.

Dirac's wave equation of an electron enables one to solve for the wave function of an electron moving in an electromagnetic field. The wave function ψ has 4 components ψ1, ψ2, ψ3, ψ4, and the electromagnetic field is described by a four-vector Aμ, consisting of a scalar potential φ and a vector potential A.

In the usual problem, we solve for the wave function ψ when the potential Aμ is given. By doing so, we obtain the wave function of an electron which moves in a given electromagnetic field. It is of some interest to consider the reverse question: given the wave function ψ, what can we say about the electromagnetic potential Aμ, which is connected with ψ by Dirac's equation? Is Aμ uniquely determined, and if not, what is the extent to which it is arbitrary ?

A train of long waves travelling along a canal of depth h2 encounters a vertical step at which the depth changes to h1. The wave-length is much greater than h1 or h2. The classical Long-Wave theory where vertical acceleration is neglected is clearly inapplicable, although a treatment has been given by Lamb. In the present paper the problem is treated rigorously by the linearized theory of surface waves. A singular Fredholm integral equation of the first kind is obtained for the horizontal velocity above the step and is converted into a regular equation of the second kind with a kernel which tends to zero as the wave-length tends to infinity. To achieve this we first take the formal limit λ = ∞ in the integral equation of the first kind. This corresponds to a fluid motion between rigid boundaries which can be solved explicitly by a conformal transformation. In this way we obtain the inverse operator corresponding to λ = ∞, which is then applied to the original equation where λ < ∞. An equation of the second kind results which has a kernel tending to zero as λ tends to infinity and which is soluble by iteration for large λ. Although the Long-Wave method of Lamb fails to describe the details of the motion correctly, his predicted reflexion coefficient appears as a first approximation.

The paper studies the problem of elliptical whirl of perfectly alined, finite, flooded, journal bearings, with incompressible isoviscous lubricant; it bases itself on Reynolds's equation. Explicit solutions are obtained, using the long (L/D≫ 1) and short (D/L ≫ 1) bearing approximations.

The physical situations corresponding to the solutions obtained are those of unloaded and constant-loaded bearings, with elastic, massive, frictional suspensions. The agreement with empirical findings is generally good, and the paper is published in the hope that further experimental research will be stimulated by it.

The general equation for the adiabatic second-order derivative of the energy En of an eigenstate with respect to parameters λ and λ′ occurring in the Hamiltonian ℋ is

The applications of this equation to molecules (λ, λ′ = nuclear position coordinates) and to enclosed assemblies of interacting particles (λ = λ′ = volume) are discussed, and the classical analogue of the equation for a micro-canonical ensemble is derived.

In this paper the solution is found of certain boundary value problems appropriate to the flow of an invisoid, compressible fluid past a wedge-shaped profile. The most general hodograph equation in the form

is used. Expressions are found for the position coordinates and the drag. The resulting solutions are then discussed for functions k(σ) appropriate to Chaplygin's equation, Tricomi's equation and the equation of Tomotika and Tamada. The case of Helmholtz flow past a wedge with sonic velocity upstream is treated in detail, and the resulting expressions for the drag enable an estimate to be made of the error which results in using approximations to Chaplygin's equation.

In a previous paper by the author (3) it was shown how the theory of distributions of L. Schwartz enables a mathematically consistent formalism to be given for a system composed of point charges interacting through their classical electromagnetic field. In the present paper a definition of the energy and momentum of the field lying in a space-like surface is given, and it is shown that from this four-vector it is possible to derive the usual equation of conservation of total energy and momentum.

Within the limitations of the small-deflexion plate theory, complex variable methods are used in this paper to obtain an exact solution for the problem of a thin circular plate supported at several interior or boundary points, and subjected to a certain normal loading spread over the area of an eccentric circle, the boundary of the plate being free. The load considered includes as a special case a linearly varying load over the circle and, as the radius of the loaded circle tends to zero, this load can be made to tend to a couple nucleus at its centre. As limiting cases the procedure adopted provides us with solutions appropriate to a circular plate, an infinitely large plate and a half-plane having free boundaries and acted upon by any normal system of concentrated forces and concentrated couples in equilibrium. Formulae for the moments, shears and deflexions relating to special examples are worked out in detail.