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.

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 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.

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.

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.

1. The problem of determining the distribution of stress in the neighbourhood of a penny-shaped crack defined in terms of cylindrical coordinates (ρ, φ, z) by 0 ≤ ρ ≤ α, z = 0, has been considered by Sneddon ((2)) and Sack ((1)). In the latter paper the solution is derived only in the case in which the stress field is due to the application of constant pressure to the faces of the crack. In the former paper the analysis given applies to an axisymmetric distribution of pressure p(ρ) applied to both the upper and lower face of the penny-shaped cavity, but the calculation of the stress intensity factor

and of the energy W required to open up the crack is a complicated matter even in the case in which p(ρ) is a constant.