The random pure radiation field postulated in an earlier paper is set up in a relativistically invariant manner. The requirement that this field be isotropic in three dimensions makes much of the formalism identical with the theory of isotropic turbulence, as has been noted by a previous author. It is found, however, that the more stringent requirement of invariance under the Lorentz group, together with the fact that the field components satisfy Maxwell's equations, mean that, to within a multiplicative constant, the entire random process is uniquely specified.

In accordance with results obtained previously, the constant is effectively identified as Planck's constant. There then results a striking resemblance between the formalism of the present theory, and that of the quantized pure radiation field, with random variables taking the place of operators, and correlations between these random variables taking the place of commutator brackets.

The infinite field energy, which arises also in quantum field theory, where, however, it is usually considered to be purely formal, remains a central difficulty of the random theory. It can be shown to be a natural consequence of assuming Maxwell's equations, and it therefore becomes necessary to consider non-linear modifications of these.

The connexion between the present results and the inconsistencies of classical and quantum electrodynamics is discussed. Also some new information is obtained concerning the behaviour of thermodynamic quantities under a Lorentz transformation.

Introduction. Infinite systems

are discussed in which k(θ) and b(n) are differentiable. The aim is to determine the behaviour of a(n) for large n and to reduce (1) to a finite system. This involves solving

for the high order a(n), n ≥ N, in terms of the preceding a(n), with a large enough N to facilitate the iteration procedure employed. The desired N − 1, N − 1 system follows on substitution for the high order a(n) in the first N − 1 equations (1).

In a previous paper (3), to which this is a sequel, a central limit theorem was presented for the homogeneous additive processes defined on a finite Markov chain, a class of processes treated extensively by Miller (4). A typical homogeneous process {R(t), X(t)} takes its values in the space

and is described by a vector distribution F(x, t) with components

and an increment matrix distribution B(x) governing the transitions. The present paper treats the motion of the process in the same space when its homogeneity is modified by the presence of a set of boundary states in x. Such bounded processes have many applications to the theory of queues, dams, and inventories. Indeed, this paper and its predecessor were motivated initially by a desire to discuss queuing systems with many servers and many service phases. We will treat both absorbing boundaries and associated passage time densities, and reflecting boundaries. For the latter our main objective is an asymptotic discussion of the tails of the ergodic distribution.

The theory of unsteady two-dimensional compressible boundary layers is studied in two parts. Part I deals with the solutions when the main stream is steady and the body is in an arbitrary motion while Part II deals with the solutions when the main stream is steady and the body is at rest with an arbitrary unsteady suction or injection perturbing about a zero mean. General methods of obtaining solutions are given when the coefficient of viscosity is proportional to temperature and the Prandtl number is unity. The main feature of the work in this paper lies in subjecting the equations to a transformation similar to that of Howarth and in linearization similar to that of Lighthill. Following the work of Sarma in incompressible flows two types of solutions are developed, one for large times and the other for small times. General expressions for the skin friction, the pressure ratio, the heat transfer and the difference between the normal coordinates of compressible and incompressible flows are obtained.

If P is a point in space-time, and P′ is a point whose space coordinates are continuous functions of time and whose speed in space is always less than the speed of light, then the time, T, at which the square of the interval between P and P′ is R, is a two-valued real function of R and the space-time coordinates of P for a range of R that includes R = 0. For fixed R in this range, each branch of T defines a scalar field over the whole of space-time, and, as R → 0, these fields reduce to the retarded and advanced times of P′ at P. Any function of T is also a field that reduces to a retarded or advanced field as R → 0. An interesting property of these fields is that there are identical relations between their derivatives with respect to R and the space-time variables, which enable the space-time derivatives to be expressed in terms of the R-derivatives; moreover, these relations are linear in the derivatives and take the same form for every such field.

Recent papers by J. R. Ellis and the author contain formulae for the fields of moving dipoles which are expressed in rather different forms in the two papers, and it appears that the author's results are more general than Ellis's. Here, the results are re-derived from a consideration of the Hertzian six-vector potential for a distribution of moving dipoles, and a comparison is made with the previous forms. It is found that Ellis's results are essentially as general as the author's. The expressions for the potentials and the field tensors are treated as weak functions in Temple's sense, and very few restrictions have to be imposed on the dipole moments and velocities, which may be expressed either as functions of coordinate time or as functions of proper-time.

The quantum mechanical kernel K(q″,t″; q′,t′) in Feynman's form of a sum over paths is evaluated by means of the canonical transformation to the system Q, P for which the Hamiltonian vanishes identically. The kernel is found to be completely determined by the classical paths, and is given by

The sum is over all classical paths leading from q′ at t′ to q″ at t′. S(q″, P, t′; q′, P, t′) is the classical action along the classical path characterized by the constant momentum P.

The unsteady two-dimensional thermal boundary-layer equation linearized as by Lighthill is studied. Two different problems are considered mainly, one in Part I and the other in Part II. Part I deals with the solution when the temperature of the main stream is constant and that of the wall is unsteady and Part II when the temperature of the main stream is constant and the heat transfer from the wall is unsteady. Unified methods are developed from which the results for the stagnation flow and the flow along a flat plate, etc., can be derived as special cases. The results of the unsteady velocity boundary-layer equations analysed by Sarma are used and solutions are obtained in two cases, first, when the main stream is in steady motion and the wall is in an arbitrary motion and secondly when the main stream is in unsteady motion and the wall is at rest. The flat plate problem is considered in detail; the results agree with those given by Lighthill and Moore.

The flow following a sudden change in the wall temperature in a laminar boundary layer, which was originally incompressible and steady, is studied. The temperature to which the wall is raised is supposed large enough to bring about density changes in the boundary layer. The details of the flow immediately after the change in wall temperature are analysed for a general boundary layer. Two specific examples are considered and for these the manner in which the new steady state is achieved is also investigated.

In a recent paper ((2)) Saffman gave a representation of the flow of a dusty gas in which the dust was described in terms of a number density of small particles with very small volume concentration but appreciable mass concentration. In the same paper the problem of the stability of plane parallel flows to small disturbances was formulated, and subsequently the author ((1)) gave some approximate results for the stability of plane Poiseuille flow.

Cohen and Coulson (4) and Cohen (5) have obtained wave-functions and energy levels for the hydrogen molecular ion in a spherical polar representation. They expanded the wave-function in terms of spherical harmonics, and their expansion may be written as

where S = 2k for even states, and S = 2k + 1 for odd states.

An infinitely long canal with uniform cross-section is filled with inviscid fluid. It is required first to show that any small two-dimensional motion of the fluid can be represented as the superposition of normal mode disturbances. A suitable generalized Green's function G(x, y; ξ) is constructed and is used to set up an integral equation (2·9) for the velocity potential on the free surface. It is shown that the eigenfunctions are complete and so are their (possibly time-dependent) extensions to the whole canal, in the sense that an arbitrary disturbance possesses a unique representation. In section 5, it is required to find asymptotic approximations to the large eigenvalues of (2·9). For this purpose a different integral equation (5·5) is set up on the canal, the kernel of which is the sum of a degenerate kernel and a small kernel. The solutions of this equation can therefore be obtained by iteration. The form of the mth eigenvalue is shown to be

for sufficiently large m.

The transonic, two-dimensional flow past a thin airfoil is treated by approximating the transonic small perturbation equation by a linear equation valid in the neighbourhood of the airfoil. The solution is derived by representing the body by a line of suitable sources. Pressure distributions are given for airfoils with maximum thickness both forward and aft, and agreement with experiment improves as the position of maximum thickness moves aft.

It is shown, for a classical dynamical system with a Lagrangian, that the existence of an ignorable coordinate is equivalent to the vanishing of a certain Lie derivative. On this covariant description is based a new definition of steady motion. A definition given earlier by Synge is criticized.

It is shown how to convert the integral equation for the pressure distribution on a circular disc in an axisymmetric sound wave into a singular integral equation of the first kind. This singular integral equation has a simple kernel and can be transformed easily into an integral equation of the second kind which is especially useful at high frequencies. Although a direct iteration on this last integral equation fails, an indirect method is devised which ensures that each iterate is of lower order than the preceding at high frequencies. The form of the general term in the iteration is given together with the first terms of its asymptotic behaviour. It is relatively simple to estimate the error caused by stopping at any particular iterate.

Detailed calculations are made for a plane wave striking the disc at normal incidence.

The dispersion equation describing the character of the equilibrium of two viscous incompressible superposed fluids under the action of gravity is considered in detail. Approximations for extreme wave-numbers are obtained and exact numerical results for particular cases are given. The existence of certain secondary modes is also considered.