The methods of solution of the wave equation for a central field given in the previous paper are applied to various atoms. For the core electrons, the details of the interaction of the electrons in a single nk group are neglected, but an approximate correction is made for the fact that the distributed charge of an electron does not contribute to the field acting on itself (§2).

For a given atom the object of the work is to find a field such that the solutions of the wave equation for the core electrons in this field (corrected as just mentioned for each core electron) give a distribution of charge which reproduces the field. This is called the self-consistent field, and the process of finding it is one of successive approximation (§ 3).

Approximations to the self-consistent field have been found for He (§ 4), Rb+ (§ 5), Na+, Cl− (§ 9). For He the energy parameter for the solution of the wave equation for one electron in the self-consistent field of the nucleus and the other corresponds to an ionisation potential of 24·85 volts (observed 24·6 volts); this agreement suggests that for other atoms the values of the energy parameter in the self-consistent field (corrected for each core electron) will probably give good approximations to the X-ray terms (§4).

The most extensive work has been carried out for Rb+. The distribution of charge given by the wave functions in the self-consistent field is compared with the distribution calculated by other methods (§ 6). The values of X-ray and optical terms calculated from the self-consistent field show satisfactory agreement with those observed (§ 7).

The wave mechanical analogue of the case in which on the orbital model an internal and an external orbit of the same energy are possible is discussed (§ 8).

This paper studies the steady motion of an electrically conducting, viscous fluid along channels in the presence of an imposed transverse magnetic field when the walls do not conduct currents. The equations which determine the velocity profile, induced currents and field are derived and solved exactly in the case of a rectangular channel. When the imposed field is sufficiently strong the velocity profile is found to degenerate into a core of uniform flow surrounded by boundary layers on each wall. The layers on the walls parallel to the imposed field are of a novel character. An analogous degenerate solution for channels of any symmetrical shape is developed. The predicted pressure gradients for given volumes of flow at various field strengths are finally compared with experimental results for square and circular pipes.

The steady motion of a viscous fluid confined between two coaxial rotating disks is discussed both experimentally and theoretically. It is found experimentally that when the disks rotate in the same sense the main body of the fluid rotates as well, but if they rotate in opposite senses the main body of the fluid is almost at rest. An adequate theory is found to explain this.

The exponential distribution is very important in the theory of stochastic processes with discrete states in continuous time. A. K. Erlang suggested a method of extending to other distributions methods that apply in the first instance only to exponential distributions. His idea is generalized to cover all distributions with rational Laplace transforms; this involves the formal use of complex transition probabilities. Properties of the method are considered.

It is well known that a beam of light falling on a reflecting mirror forms standing waves. This effect has been very beautifully made use of in Lippmann's colour photography process. The standing light waves, in this case, produce a periodic effect in the emulsion of the photographic plate which, when developed, scatters light and produces a similar colour effect. Instead of using a beam of light, it would seem possible to scatter electrons from the emulsion and obtain a reflection of electrons similar to that of a space grating. But it seemed to us that it would be of much greater interest to consider an experiment in which electrons are reflected from the standing waves of light. The direct scattering of free electronic waves by light has strictly never been observed, and it was thought possible that by this method, owing to the interference of the electrons and to the fact that the scattered electrons are focussed to one spot, the magnification of the phenomenon would be sufficient to make it observable. From the theory developed below, it will be seen that the experiment is just on the verge of possibility, and would be very difficult to carry out. The main interest of the experiment would come from the possibility of observing stimulated scattered radiation which up to the present has never been verified experimentally.

I know only one case in mathematics of a doctrine which has been accepted and developed by the most eminent men of their time, and is now perhaps accepted by men now living, which at the same time has appeared to a succession of sound writers to be fundamentally false and devoid of foundation. Yet that is quite exactly the position in respect of inverse probability. Bayes, who seems to have first attempted to apply the notion of probability, not only to effects in relation to their causes but also to causes in relation to their effects, invented a theory, and evidently doubted its soundness, for he did not publish it during his life. It was posthumously published by Price, who seems to have felt no doubt of its soundness. It and its applications must have made great headway during the next 20 years, for Laplace takes for granted in a highly generalised form what Bayes tentatively wished to postulate in a special case.

The paper is concerned with the solution of the algebraic problems arising in the partial wave treatment of electron-hydrogen atom collisions. Explicitly antisymmetrized wave functions are used throughout. The boundary conditions are written in S-matrix notation and expressions for total and differential cross-sections obtained. The algebraic coefficients fλ and gλ occurring in the continuous state Hartree-Fock equations are expressed in terms of Racah coefficients, and tabulated as functions of the total angular momentum for atomic s, p and d electrons and all angular momenta of the scattered electron. Expressions are given for the calculation of first-order corrections to the results obtained using approximate wave functions.

The differential analyser has been used to evaluate solutions of the equation

with boundary conditions y = y′ = 0 at x = 0, y′ → 1 as x → ∞, which occurs in Falkner and Skan's approximate treatment of the laminar boundary layer. A numerical iterative method has been used to improve the accuracy of the solutions, and the results show that the accuracy of the machine solutions is about 1 in 1000, or rather better.

It is shown that the conditions are insufficient to specify a unique solution for negative values of β a discussion of this situation is given, and it is shown that for the application to be made of the solution the appropriate condition is that y′ → 1 from below, and as rapidly as possible, as x → ∞. The condition that y′ → 1 from below can be satisfied only for values of β0, greater than a limiting value β0, whose value is approximately − 0·199, and which is related to the point at which the laminar boundary layer breaks away from the boundary.

If the average spacing of the resonance levels is very small as compared with the range of energy in which the spacing or width of the levels changes appreciably on the average, one can speak of a statistical distribution of the level spacings and widths. The question then comes up naturally, whether the ‘distribution law’ for width and spacing is different for protons, neutrons, etc., i.e. whether it depends on the long-range interaction of the scattered particle. It is pointed out that while the average width must depend on the long-range interaction (on account of the penetration factor) the distribution of the widths, if these are measured in terms of their average, can be expected to be independent therefrom. In the case of the level spacings, not only the distribution about the average, but the average itself is also independent of the long-range interaction. It is pointed out, incidentally, that the logarithmic derivative of the wave function at the surface of the nucleus will have, on the whole, a positive value between resonances because of the greater effect of the high-energy resonances than that of the bound states.

This paper considers the correlation P(r) between the fluctuating pressures at two different points distance r apart in a field of homogeneous isotropic turbulence. P(r) can be expressed in terms of the fourth moment of the velocity fluctuation, which is evaluated with the aid of the hypothesis that fourth moments are related to second moments in the same way as for a normal joint distribution of the velocities at any two points. The experimental evidence relevant to this hypothesis (which cannot be exactly true since it gives zero odd-order moments) is examined. The alternative hypothesis made by Heisenberg, that the Fourier coefficients of the velocity distribution are statistically independent, has identical consequences for the fourth moments of the velocity, although it does not lead to such convenient results.

The pressure correlation is worked out in detail for the important special case of very large Reynolds numbers of turbulence; the mean-square pressure fluctuation is found to be . The mean-square pressure gradient is evaluated, from the available data concerning the doublevelocity correlation, for the cases of very small and very large Reynolds numbers, and a simple interpolation between these results is suggested for the general case. Finally, the relation between the mean-square pressure gradient and rate of diffusion of marked fluid particles from a fixed source is established without the neglect of the viscosity effect, and the available observations of diffusion are used to obtain estimates of which are compared with the theoretical values.

In Part I of this paper (6) we proved various index theorems for manifolds with boundary including an extension of the Hirzebruch signature theorem. We now propose to investigate the geometric and topological implications of these theorems in a variety of contexts.

Let us consider a discontinuous bivariate distribution. That is, let us consider N × Q non-negative values pνq (ν = 1, 2, …, N; q = 1, 2, …, Q), being the theoretical probabilities of the νth value of a variate Xμ (μ = 1, 2, …, N) concurring with the qth value of a second variate Ys (s = 1, 2, …, Q).

It is advantageous in automatic computers to employ methods of integration which do not require preceding function values to be known. From a general theory given by Kutta, one such process is chosen giving fourth-order accuracy and requiring the minimum number of storage registers. It is developed into a form which gives the highest attainable accuracy and can be carried out by comparatively few instructions. The errors are studied and a simple example is given.

This paper contains some contributions to the analytic theory of ideals. The central concept is that of a reduction which is defined as follows: if and are ideals and ⊆ , then is called a reduction of if n = n+1 for all large values of n. The usefulness of the concept depends mainly on two facts. First, it defines a relationship between two ideals which is preserved under homomorphisms and ring extensions; secondly, what we may term the reduction process gets rid of superfluous elements of an ideal without disturbing the algebraic multiplicities associated with it. For example, the process when applied to a primary ideal belonging to the maximal ideal of a local ring gives rise to a system of parameters having the same multiplicity; but the methods work almost equally well for an arbitrary ideal and bring to light some interesting facts which are rather obscured in the special case. The concept seems to be suitable for a variety of applications. The present paper contains one instance which is a generalized form of the associative law for multiplicities (see § 8), and the authors hope to give other illustrations in a separate paper.

If the probability differential of a set of stochastic variates contains k unknown parameters, the statistical hypotheses concerning them may be simple or composite. The hypothesis leading to a complete specification of the values of the k parameters is called a simple hypothesis, and the one leading to a collection of admissible sets a composite hypothesis. In this paper we shall be concerned with the testing of these two types of hypotheses on the basis of a large number of observations from any probability distribution satisfying some mild restrictions and their use in problems of estimation.

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.

A vector identity associated with the Dirichlet tessellation is proved as a corollary of a more general result. The identity has applications in interpolation and smoothing problems in data analysis, and may be of interest in other areas.

The previous paper (1) omitted the proof of the theorem on the connective constant of a crystal. This paper supplies the proof. For ease of reference, we first recall the relevant nomenclature and enunciate the theorem; to this extent the present paper is self-contained.

We begin with some definitions.

A cubical graph is a simplicial 1-complex in which each 0-simplex is incident with just three 1-simplexes.

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.