The state of a real fluid is completely specified by its velocity, density, pressure and temperature fields. When the fluid is in turbulent flow, all these quantities fluctuate in a disordered manner. The method of space Fourier spectra is used to show that these field variables separate into two physically distinct groups, one corresponding to fluctuating acoustical waves, or random noise, and the other to fluctuating vorticity, or eddy turbulence. The corresponding decomposition of the spectral and correlation tensors in a homogeneous field of turbulence is given. The noise Fourier components are shown to be coupled to the eddy Fourier components only through the non-linear inertia terms in the dynamical equations of the fluid; whereas the former propagate as acoustical waves, the wave character of the latter is due entirely to the mean motion of the fluid. The measurement of the noise component, its attenuation through absorption by walls and its effects on the eddy component are discussed. Finally, the dynamical equations for the eddy component of the velocity spectral tensor in a homogeneous field of turbulence are compared with the corresponding equations for an incompressible fluid.

It is proved that the matrix algebra for any relativistic wave equation of half-odd integral spin can be factorized as the direct product of a Dirac algebra and another, called a ξ-algebra. The structure and representation of ξ-algebras are studied in detail. The factorization simplifies calculations with particles of spin > ½, because the ξ-algebra contains only one-sixteenth as many elements as the original matrix algebra.

In a contribution (3) to these Proceedings the author considered stochastic problems in physics where one had to deal with a stochastic variable representing the number of particles distributed in a continuous infinity of states characterized by a parameter E, and where the distribution varied with another parameter t which might be continuous or discrete (if t represents time or thickness, it is of course continuous). The author introduced the concept of product densities and derived some general results relating to the functions representing these densities. Recently, Janossy(2), using a certain mathematical model for a nuclear cascade, introduced certain functions which bear a close relation to the product-density functions. The object of this note is to establish a complete correspondence between these two sets of functions and apply them to the specific problem of the development of a nucleon cascade. The diffusion equations involving product densities can be derived from the diffusion equations involving Janossy's functions.

In a previous large-sample treatment of sequential estimation (1), it was shown that in certain circumstances, when there was only one unknown parameter in the distribution of the observations, an estimation formula valid for fixed sample sizes remained valid when the sample size was determined by a sequential stopping rule. The proof was heuristic, in that it depended on an application of the central limit theorem of which the justification was not obvious. Another proof has recently been given by Cox (2) (in the course of deriving a correction term to my result). Dr Cox pointed out to me that this work suggested that fixed-sample-size formulae might be valid generally for sequential sampling, provided the sample size was large. In establishing a proposition to that effect, I have now been able to by-pass some of the complexity of my previous approach by concentrating attention on a condition of ‘uniform continuity in probability’ to be satisfied by the statistic used. Theorem 1 is the basic result, which is applied in Theorem 2 to determine a sequential stopping rule giving required accuracy of estimation of an unknown parameter. Theorems 3–6 indicate some situations in which the uniform continuity condition postulated in Theorems 1 and 2 is satisfied. A few examples are discussed.

When the electric intensities on two parallel planes, of which the two perfectly conducting sides of a wave-guide of finite length and infinite width are portions, are taken as unknowns, the problem of the diffraction of a plane harmonic electromagnetic wave polarized parallel to the edges of the guide leads to two integral equations. By means of the Laplace transform these equations are converted into others suitable for solution by successive substitutions. The series thus obtained is too complex for practical purposes, and so an approximate solution is found for the case when the length of the guide is large compared with the wavelength. Finally, there is a brief discussion of the difference between the distant fields when l is large and when l is infinite.

In an earlier paper the author considered the problem of the turbulent diffusion, relative to a fixed origin, of a cloud of marked fluid whose initial position is given. This was found to be determined by the initial shape of the cloud and the statistical properties of the displacement of a single fluid particle. The present paper is concerned with the relative diffusion of the cloud, i.e. with the tendency to change its shape, or, more precisely, with that part of the relative diffusion which is described by the probability that a given vector y can lie with both its ends in marked fluid at time t. This aspect of the relative diffusion is found to be determined by the initial shape of the cloud and the statistical properties of the separation, at time t, of two fluid particles of given initial separation. The statistical functions introduced to describe the relative diffusion are found to be related to Richardson's distance-neighbour function.

The relative diffusion of two particles is a more complex problem than diffusion of a single particle about a fixed origin because the relative diffusion depends on the initial separation. The closer the particles are together, the smaller is the range of eddy sizes that contributes to their relative velocity; for the same reason, relative diffusion is an accelerating process, until the particles are very far apart and wander independently. The hypothesis is made that if the initial separation is small enough, the probability distribution of the separation will tend asymptotically to a form independent of the initial separation, before the particles move independently. This hypothesis permits various simple deductions, some of which make use of Kolmogoroff's similarity theory. The important question of the description of the relative diffusion by a differential equation is examined; Richardson has put forward one suggestion, and another, based on a normal distribution of the separation, is made herein.

The conditions under which bound states associated with atoms in the surface of a metal may exist are investigated, using the tight-binding approximation. These states arise as a result of modifications in the parameters of certain atoms. The modifications required to produce (a) bound states associated with all the atoms in the surface (surface states) and (b) bound states associated with particular small groups of atoms are found for the simple cubic lattice. It is also shown that most of the simpler crystal structures do not exhibit surface states without such modifications; in the graphite and diamond lattices, however, surface states exist solely by virtue of the lattice geometry.

The so-called ‘clock paradox’ is concerned with the difference in the time-intervals reckoned by two observers in relative motion for the lapse of time between two encounters. In this paper the problem is treated purely by general relativity by considering a particular example in which the two observers are attached to two test-particles moving freely in the field of a gravitating mass; one of these makes complete revolutions in a circular orbit while the other moves radially outwards and inwards. The time-interval between two successive encounters is shorter in the reckoning of the former than in that of the latter. The difference is found to agree qualitatively with a naïve application of special relativity.

A relativistically rigid rod is defined as one along which any disturbance travels with the fundamental speed c without dissipation of energy. A necessary consequence of this definition is that the rod should satisfy a certain ‘equation of elasticity’. If this equation is satisfied, it is shown in accordance with relativistic mechanics that a disturbance of any amplitude is propagated along the (unstressed or stressed) rod with speed c and without dissipation. Consequently, the equation provides also a sufficient condition for rigidity in the sense defined.

A detailed account is given of the classification of the electromagnetic fields arising from a vibrating charge-current distribution. The work owes much to that of Berestetzky and Fierz, but differs from their treatments in that the exact expansions are taken as a starting point. Hence the general forms of the potentials and fields and their approximate forms are obtained directly. Formulae appropriate to large distances are given in §§2·1 and 3·1. The relation to the approach of Condon and Shortley (4) is discussed.

The problem discussed in this paper is the enlargement of a circular hole in a flat sheet of which the material is in plastic flow under constant shear stress; inertia terms are taken into account in the equilibrium equations so as to permit the occurrence of high velocities.

The treatment is in some ways analogous to that of problems in one-dimensional compressible gas flow. The characteristic equations are seen to represent straight lines under certain initial conditions; this facilitates consideration of any mode of enlargement; in particular, uniform acceleration is illustrated by numerical calculations. There appears a circular shock-front, i.e. a discontinuity in thickness and velocity moving ahead of the hole. The initial conditions for the existence of this ‘simple wave' solution involve a singularity at the centre of the plate.

In general, the characteristic curves will not be straight lines and the equations are best solved step by step. Their characteristic form is well adapted to such a process; this is outlined for the interesting case of initial enlargement with uniform velocity followed by expansion under constant acceleration.

Curves illustrate some of the general results.

1. The problem of this paper was first attempted some years ago at the suggestion of Sir G. I. Taylor, to whom the writer makes grateful acknowledgement, in connexion with experimental work in which he and others (5,6) had used half-pitot tubes in order to measure fluid velocities as near as possible to a fixed boundary.

The convergence of the perturbation expansion for the S-matrix in interaction representation of a three-boson contact interaction is investigated. A lower bound is obtained for the integrals corresponding to irreducible graphs when the total rest mass of the system is insufficient for bare particles to be created. It is shown that in this case the perturbation expansion cannot converge no matter what value the coupling constant has. A discussion is given of the bearing of this result on the general problem of the convergence of perturbation expansions for the S-matrix in renormalized field theories.

Problems involving the addition of heat to a flowing gas have received considerable attention in recent years. The practical interest lies in the application to ram jets and similar engines, and the presence of entropy gradients makes the problem of theoretical interest.

The S-matrix formulation of quantum electrodynamics, as developed by Feynman (3) and Dyson (1), expresses the matrix element for any process as a power series in the coupling constant, the coefficients of the series being, in general, rather complicated multiple integrals. These integrals contain singularities in their integrands, and in certain circumstances the coincidence of such singularities gives the S-matrix a non-analytic behaviour as a function of, for example, the total energy. The threshold behaviour of the S-matrix in the neighbourhood of energy values at which this phenomenon occurs has been investigated by Eden (2), who shows that the non-analytic behaviour is connected with the possible commencement of new physical processes, such as the creation of a particle.

The object of this paper is to investigate the influence of the initial values on the behaviour of the one-dimensional unsteady isentropic motion of a perfect gas, and in particular the occurrence of singularities. For this purpose it is expedient to develop an unfolding procedure in the theory of partial differential equations of hyperbolic type. This not only resolves a difficulty in the classical treatment of the initial value problem associated with this type of equation, but also simplifies the presentation.

The geometrical properties of the singularities (branch and limit lines, edges of regression, etc.) occurring in two-dimensional steady flow have been well discussed already, and their counterparts in one-dimensional unsteady motion have been indicated by Stocker and Meyer(5). However, here a method for finding the explicit analytic connexion between the occurrence of such singularities and the prescribed initial conditions is developed.

Assuming quite general initial conditions, two examples are considered. In the first, a conjecture of Riemann is verified; in the second a special type of periodic solution is discussed, and it is shown that this solution cannot be continued to all values of t > 0. Finally, this latter result is shown to be true for arbitrary periodic initial conditions.

We consider the perturbation of the velocity of an inviscid incompressible fluid rotating about an axis with uniform angular velocity Ω, due to the slow uniform motion, after an impulsive start, of a sphere of radius a along the axis with velocity − V. This problem was first considered by Taylor (4), who obtained a family of solutions for the case of steady motion of a sphere along the axis of rotation of the fluid. All the solutions satisfied the boundary condition at the surface of the sphere and also the condition that the relative velocity should vanish at infinity. It was thought possible that the indeterminacy lay in the manner in which the motion was started. However, an experiment carried out in connexion with the problem showed that if V/aΩ was less than about 0·16 a column of liquid of the same diameter as the sphere was apparently pushed along in front of it. This agreed with Proudman's observation (3) that if the perturbation was small and steady it would be two-dimensional, while in Taylor's solutions the relative motion of the fluid was not small. Taylor (5) suggested that three possibilities presented themselves. Either the motion never becomes steady, or it becomes steady but not small, or it becomes steady and two-dimensional. Of these he preferred the latter, which seems to occur in practice.