The distributions of velocity and temperature in the wake behind a heated body of revolution have been determined in a low-turbulence wind tunnel. Difficulty was experienced in obtaining a truly symmetrical wake, and observations have been reduced to mean values, curves of which are given.

The probability of an electron having an energy greater than a certain fraction of its initial energy after traversing a certain thickness of matter is calculated for different values of the initial energy. Whereas in previous calculations the energy losses by radiation and by ionization have been treated separately, and the results added, the two types of losses are here combined and their effect on each other included. The results may be compared with those of experiment for values of the initial energy up to a few times the initial energy up to a few times the critical energy of the substance traversed. The theoretical values obtained are found to give better agreement with the results of experiment than do those obtained by the less accurate method.

1. Particular series of conal and toroidal functions have been studied by various authors*, but so far no method has been given of expanding an arbitrary function in such a series. Ordinary methods fail, since conal functions (and also toroidal functions) are not orthogonal, although they form a complete set. In this paper I give an expansion theorem.

13. Formulae have been obtained for the moment of the fluid forces which act on a plane plate which is gliding on the surface of a stream of finite or infinite depth. It has been shown that the formulae for the lift and moment forces acting on a plate which is gliding on a stream of infinite depth may be obtained by a limiting process from the formulae for the general case of gliding on a stream of finite depth. Also, by another limiting process, the lift and moment forces for the Rayleigh flow past a flat plate in an infinite stream have been obtained from the lift and moment forces for gliding on a stream of infinite depth.

Numerical work has been carried out for the case when the angle of incidence of the plate to the stream is 5° and this has shown the variation of the lift and moment forces and the position of the centre of pressure with a range of values of the length of the plate, the depth of the stream and the height of the trailing edge of the plate above the bed of the stream.

The measurements of Hall and Hislop (see the preceding paper) were made in a wind tunnel considerably less turbulent than has usually been used in such measurements.

This paper may be regarded as a sequel to a previous papers(1) in these Proceedings. The vector and matrix notation of that paper used for a statistical sample is systematized somewhat further, so that while a sample S refers as before to the matrix of nm values (a sample of m observations in one variate only being a row vector), we write

for the linear regression formula between the dependent and independent variates into which a sample is supposed partitioned (in place of equation (12) of (1)). More generally, a third submatrix S0 is partitioned off, and its effect eliminated (corresponding to equation (13) of (1)), but without loss of generality we assume that S2 in equation (1) above can always stand for S2.0 if necessary.

Solutions of Fock's equation for the self-consistent field with exchange have been carried out for the negative ions Na−, K−. The results indicate that these negative ions can certainly exist; and calculations of the energy values, and estimates of the errors of these values, indicate that these ions are probably stable and not liable to autoionization.

The wave functions for these negative ions are required in calculating atomic cross-sections for the capture of electrons by the neutral atoms.

The calculations of the velocity and temperature distributions for the turbulent flow in axially symmetrical and plane jets are described.

For axially symmetrical jets Tollmien's results derived from calculations based on the momentum theory are stated; the calculations on the basis of the modified vorticity transfer theory and on the vorticity transfer theory with symmetrical turbulence are carried out. Unfortunately it seems to be true that none of these theories provides results in good agreement with experiment for both the velocity and temperature distributions.

For plane jets the results obtained by Tollmien and based on the momentum transfer theory are stated; the calculations on the basis of the vorticity transfer theory are carried out. The velocity distributions on both theories are identical and in fair agreement with experiment. In order to test the validity of the theories it is necessary to compare the theoretical results for the temperature with experiment. No results for the latter seem to have been published.

By solving the fundamental integral equation an expression is obtained for the electrical resistance which takes into account, to the first approximation, the mutual influence of the impurities and the lattice vibrations. It is found that deviations from Matthiessen's rule are to be expected and that these deviations are surprisingly large. The formulae derived only indicate the trend of the resistance curve, but this trend is not confirmed by experiment. Whereas the theory indicates that the mutual influence of the impurities and the lattice vibrations should decrease the electrical resistance, the experimental results of Grüneisen on copper show that, when deviations from Matthiesen's rule occur, the resistance is increased.

The method of the self-consistent field is applied to a discussion of the energy and wave-functions of the ground state of the hydrogen molecule; an analytical expansion of the wave-function is given in terms of spheroidal coordinates, and the distribution of charge is determined. Two simpler, though less accurate, wave-functions for the molecule are also included, and the possibility of using the method for more complex molecules is discussed.

The notion of fiducial probability was introduced by R. A. Fisher and is now widely used in statistical work involving a single unknown parameter. Fisher has also considered the joint fiducial distribution of several parameters for which there exists a sufficient set of statistics, and has derived the fiducial distribution of the two parameters of the one-variate normal law*. Since Fisher's account is brief, we give a more extended description of the fiducial distribution of several independent parameters possessing a sufficient set of statistics.

The theory of heavy electrons recently developed by several authors may be considered to give a satisfactory account of the empirically known neutron-proton interaction. However, it now seems well established that there exists a proton-proton interaction of comparable magnitude which is not accounted for equally well. Owing to the fact that the emission of a heavy electron involves the change of a neutron into a proton or vice versa, the first approximation of this theory gives only an exchange force between unlike particles, whereas a force between like particles must be due to double transitions and thus only appears in the second approximation. It is true that the expansion in terms of the number of particles emitted is actually so badly convergent that the second order proton-proton force at distances of about 10−13 cm. is found to be not essentially smaller than the first order neutron-proton force (see FHK), but nevertheless this does not explain experimental facts, since the calculated second order force is always repulsive.

An elementary theory of the diffraction of light by supersonic waves was put forward by Sir C. V. Raman and myself (4) to explain many of the important features of the phenomenon observed by Bär and others. Recently Mueller (2) pointed out the essential idea that the optical diffraction effects in solids may be interpreted on the basis of this theory if due account is taken of the photoelastic effects arising from the periodic strains caused by the sound waves. He was thus able to give an explanation of the results of Schaefer and Bergmann (5, 6). Mueller's idea, however, required a still further refinement, since it did not consider the phase relationships between the differently polarized components of the diffraction orders. This refinement has, however, been recently incorporated by Mueller and myself (3). This led us to offer an explanation of the diffraction effects observed by Hiedemann and Hoesch (1) in glass blocks.

According to quantum mechanics there exists in any non-metallic crystal a band of allowed electronic energy levels which are unoccupied when the crystal is in its state of lowest energy. We call this band the conduction band; the crystal can conduct electricity if electrons are raised into the conduction level from lower levels. According to the theory of semi-conductors given by Wilson, there exist in these substances lattice imperfections at which an electron can exist in a bound stationary state below the conduction band, electrons being raised from these levels into the conduction band by the thermal agitation of the surrounding atoms.

The wave equation for the deuteron in its ground state is solved on the assumption that the mutual potential energy of a neutron and a proton is of the form r−1e−λr. The binding energy of the hydrogen isotope H3 is calculated approximately by the variation method.

The velocity distribution in the wake behind a body of revolution calculated by Miss Swain on the momentum transfer theory with l constant over a cross-section of the wake is not in accordance with experiment.

On the generalized vorticity transfer theory the equation for the first approximation to the velocity defect u far downstream is

There are two methods of simplifying this equation on the analogy of the successful simplification of the corresponding equation for the two-dimensional case. One method gives the same equation as if the turbulence were symmetrical; the other gives the equation of the modified vorticity transfer theory.

For the former there is no real solution if l is constant over a cross-section; for a real solution l must be infinite on the axis of symmetry. The equation was integrated with l ∝ r−1 over a cross-section; this variation of l was suggested by putting the change in the area enclosed by a circular vortex-line proportional to the area of the section of the wake. The agreement with experiment is fairly good over the middle portion of the wake, but not near the edge, where the theoretical value of ∂u/∂r is not zero. If l2 ∝ rp, ∂u/∂r vanishes at the edge of the wake only if p = − 3. The solution is carried through for this case; it shows only fair agreement with experiment over the middle of the wake, and none at the edge.

The calculation on the modified vorticity transfer theory is carried out with l constant over a section; in this case there is no definite edge to the wake, nor does u fall off sufficiently rapidly to make the momentum integral for the drag converge. (u ∝ r−2 for large r.) Even over the middle of the wake there is a discrepancy with experimental results, rather like that shown by the momentum transfer theory.

The temperature distribution behind a heated body was calculated according to each of the foregoing theories.

The temperature distribution was also calculated from the experimental velocity distribution, without any assumption being made concerning the kinematic coefficient of turbulent diffusion, which was eliminated between the equations for the velocity and temperature. The vorticity transfer theory with symmetrical turbulence gives an impossible result. The modified vorticity transfer theory gives θ/θmax < u/umax when the experimental velocity distribution is used.

A scheme is presented to show the limiting behaviour of the electronic energy levels in the diatomic one-electron problem for unequal centres, as the ions are separated. This generalizes the well known order of levels in the hydrogen molecule ion. Certain implications of the result for the properties of asymmetric diatomic molecules are briefly discussed.

By using Bragg and Williams' approximations and Bethe's interaction energies between nearest neighbours, it has been shown that, for alloys of the AB3 type, (1) the maximum value of the critical temperature Tc must occur when c = ½ (2) for all temperatures within the range 0 < T < (Tc)max there is always a concentration range within which the alloy exists in a two-phase state, the one phase being partially ordered and the other disordered. This two-phase region is small at all temperatures and vanishes both at T = 0 and at T = (Tc)max. In particular for T/(Tc)max = 0·82, which is the value of Tc for c = ¼, the limits of the two-phase region are c = 0·245 and c = 0·256. The theoretical phase diagram has therefore qualitatively the shape, given in Fig. 5. For a fixed concentration there is thus a certain temperature range in which a disordered phase and a partially ordered phase exist in equilibrium with one another. The temperature at which the disordered phase completely disappears is that usually given by experimental determinations of the critical temperature. It is somewhat higher than Tc, but its maximum value also occurs when c = ½, so that the discrepancy between theory and experiment for this type of superlattice still remains.

Calculations are made to investigate the behaviour of ordering near the critical temperature for alloys with a superlattice of the type AB for various compositions. No fluctuation among the sites of the same kind, which may lead to the possible existence of some other types of superlattice, say, AB3, is considered. Results showing the change of ordering with T near Tc are plotted.

The theory is also applied to the adsorption of gases upon a solid surface, where the sites of accommodation are regularly arranged. It is shown that under certain conditions, a superlattice formation in the adsorbed layer is possible.

In a recent letter to Nature F. London(3) has called attention to a discontinuity in the derivative dCV/dT of the specific heat with respect to temperature which arises in a perfect Einstein-Bose gas at low temperatures. When discussing the properties of a perfect gas satisfying the Bose statistics, Einstein(1) remarked that at low temperatures something resembling a condensed phase should appear. The temperature at which this condensation should begin is given by the equation

where n is the number of particles of mass m per unit volume and the other symbols have their usual significance.