A method is described for improving the approximate calculation of energy levels for atomic configurations consisting of a set of closed shells and a single ‘series’ electron. This involves the inclusion of a ‘polarization’ potential in the Hartree-Slater-Fock equation, caused by the series electron. The form of potential found in two cases investigated, namely, Si+3 and Ca+, is described and discussed.

The paper deals with the behaviour of finite non-homogeneous Markov chains having regular transition matrices. Two types of ergodic behaviour are distinguished, and sufficient conditions are given for each type. Chains with commutative transition matrices are specially investigated. By means of expansions in terms of the characteristic roots several types of behaviour of chains which are not ergodic are studied.

The paper contains some remarks on the problems treated by Hammersley(9). With the aid of a new and simple method the exact solutions of Hammersley's problems and the asymptotic behaviour of the solutions are found. The problems in question are: A Poisson process is transformed into a new process by a Type II counter, and this new process is transformed again by a Type I counter. The distribution of the number of events occurring in each transformed sequence is to be determined.

A correction is made to a previous paper(2) on clumps formed by random placing of laminae, a useful property of such placings is deduced and formulae are found for clumps formed when convex bodies are randomly placed in space and for clumps formed by the orthogonal projections of such placings.

The Tamm-Dancoff theory of meson-nucleon scattering (pseudoscalar coupling) leads to a series of singular integral equations. The asymptotic behaviour of the solutions of these equations is obtained here for all scattering states. For attractive states there is a critical coupling constant beyond which no normalizable solutions exist for these equations; for repulsive states another critical coupling constant appears, beyond which the solutions oscillate infinitely often but are still normalizable. It is concluded that the renormalization procedures proposed previously (3) are consistent and successfully define finite vertex-functions, and that the equations obtained for these vertex-functions have satisfactory properties for numerical work.

1. Pólya's walk in the Euclidean plane is the motion of a particle, which takes successive independent random steps of unit length parallel to the coordinate axes, the probabilities of each of the four possible directions at each step being ¼. We imagine the particle starts at the origin, and that, in taking the ith step (i = 1,2,…) from (xi-1, yi-1) to (xi, yi), it leaves a straight track of unit length between these two points. Let Tn denote the track thus marked out in the first n steps; so that Tn is a connected chain of n straight segments each of unit length. Define the interior of Tn to be the set of points, which neither lie on Tn nor can be connected to the point at infinity without crossing Tn at least once; and let An denote the area of the interior of Tn. Associated with each walk ω, there is a sequence of random numbers {An(ω)}; what can be said about the behaviour of this sequence, especially for large n? We shall prove Theorem 1. For all ω,0 ≤ An(ω) ≤ 1/16 n2.

The interaction of conduction electrons with a vibrating metallic lattice has been considered by Ziman in terms of non-adiabatic perturbations to the adiabatic approximation of the Born-Oppenheimer theory. This programme is continued to calculate the self-energy of the electrons. When account is taken of the dependence of the energy levels of the adiabatic Hamiltonian itself upon the electron configuration, the final result is the same as the standard one.

The statistic is applied in several different situations to detect departure from uniform dispersion. The given sample points may lie on a square lattice (without repetition), a rectangular lattice (repetition being allowed), or within rectangular or circular plane boundaries. The problems considered include an analysis of cosmic ray shower structure, and the counting of dust particles and bacterial colonies.

In a previous paper (3) the method of conformai transformations was employed to study the behaviour of a universe assuming the perfect cosmological principle, i.e. that apart from local irregularities the general aspect of the universe is the same at all points independent of place and time. To conform with Einstein's general theory an anisotropic cosmological term had to be postulated.

A disjunctive combination of a finite set of two-person games Γ1, Γ2, …, Γk may be defined thus: The players play alternately, each in turn making a move in one and only one of the individual games. If, in addition, the conditions are imposed that

(i) a player loses if unable to move (in any game),

(ii) the games are impartial, i.e. the allowable moves from any position do not depend on which player is about to play (or on the previous moves, though these can be ‘built in’ to the position if necessary),

(iii) the games are of bounded play, i.e. for each game Γi corresponding to any initial position Pj there is an integer bij such that the game must terminate after at most bij moves, then Grundy (6) has shown that there is a function G(P) (called by him Ω(P)) of the positions P with the following properties:

(1a) G(P) = 0 for a terminal position, from which no move is possible; for other positions G(P) is the smallest non-negative integer different from all values of G(Qi), where there is a permissible move from P to Qi,

(1b) for a disjunctive combination of games, G for the combined position is the nim-sum of the G's the individual positions. By the nim-sum, we mean that each separate G is to be written in the scale of 2, as Σar 2r, and then in forming the sum, the ar's are to be added mod 2 for each value of r, as in the theory of Nim((1), (7), (8), pp. 36–8).

Preliminary self-consistent field calculations without exchange have been performed for the ground states of the structures In3+ and Sb3+. The results are presented.

The exact solution for a trapezoidal potential barrier is given in terms of Airy functions and trigonometrical functions. Numerical values are found for the ratio of the amplitudes of the transmitted and reflected waves. These are compared with those given by the WKBJ approximation.

The modified Thomas-Fermi method reported in (2) has been applied numerically to calculate the distribution of valence electrons in CC14. Slater wave functions have been used to describe the Cl inner-shell electrons, whilst a description of the C ls electrons based on the Thomas-Fermi model for atoms has been adopted. The electron distribution thus found is compared with the two approximations discussed in part I of this series (3). The first of these, in which valence and inner-shell electrons are treated in the same way, is shown to give a distribution of valence electrons in good agreement with the present results. The second approximation, that of effective nuclear charges, is less satisfactory and could only lead to a distribution of valence electrons in reasonable agreement with the present work if the effective nuclear charges were considerably reduced. This seems to throw some light on the results of Bowers for the force constants of tetrachlorides and tetrabromides.

The overall distribution of electrons obtained here seems to be the inevitable result of assuming the valence electrons to move in any plausible central field. It is, of course, by no means certain that such an approximation is near to the truth. In fact, the results of (3) and the present paper leave little doubt that the electron cloud obtained by such an approximate method will be too diffuse. This seems at least partly responsible for our finding that rather more than 20 of the 28 Cl valence electrons lie outside the sphere drawn through the Cl nuclei.

An attempt is made to calculate the first few angular terms in an expansion of the electron density for the phosphine molecule in Legendre polynomials. Such an expansion is appropriate for a model in which the three hydrogen nuclei are smeared to form a circular line charge. The Thomas–Fermi approximation has been used in conjunction with the variational method. The variational density employed includes p and f angular terms. An approximate charge density map is constructed for a plane containing the molecular axis in order to demonstrate the effect of the angular terms.

Various authors have discussed the ‘disjunctive compound’ of games with the last player winning (Grundy(2); Guy and Smith (3); Smith (4)). We attempt here to analyse the disjunctive compound with the last player losing, though unfortunately with less complete success. The problem was explored tentatively by the first named author, but the rigorous development is due to the second.

The properties of centre-of-mass systems at high energies are discussed, and the single time form of field theory is used to set up state vectors that describe the simpler centre-of-mass states of fields. These states, which are labelled by the eigenvalues of total angular momentum, are expanded in terms of the spherical harmonic resolution of the fields about a fixed point. A discussion of the application of the method to pion-nucleon scattering is given.

A very simple model, consisting of N particles moving in a one-dimensional assembly divided by potential ‘humps’ into M cells, is studied. The process of passing from a quantum-mechanical description of such an assembly to the equation of diffusion type that governs it in practice is shown to consist of at least three separate steps: ‘averaging over phases’, and letting N and M become large. The effects of these steps are considered separately. Strict irreversibility in time appears after the first step, but the assembly remains ergodic until after the second step and fluctuations persist until after the third step.

Classical dynamics is generally considered as the prototype of a deterministic theory; the equations of motion determine the coordinates q and the momenta p at any time provided they are given at an initial instant. It is pointed out that this is an unrealistic assumption, for it is inevitable that there should be small uncertainties Δq, Δp; a point of the mathematical continuum has no physical significance. The uncertainties can be taken into account without violating the deterministic equations by introducing a probability density P in phase space p, q. The function P satisfies the partial differential equation which expresses Liouville's theorem. Thus it can be shown that an initial uncertainty spreads out in phase space, so that finally all states of the system consistent with the mean initial values of the constants of the motion are equally probable. This property holds for one degree of freedom just as well as for many, and is not a consequence of our ignorance concerning large numbers of particles. The essential difference between classical mechanics and quantum mechanics consists not in the physical necessity of a statistical interpretation, but in the further introduction of a probability amplitude, the square of which is the probability density; thisimplies the restriction of the initial uncertainties as expressed by the uncertainty laws, and the phenomenon of interference of probabilities which makes necessary a revision of our concepts of physical reality. It is remarkable that the product of the uncertainties of a properly chosen pair of variables is constant also in classical mechanics, although of course its value is not a universal constant as in the quantum case.

By a generalization of methods recently applied in electrodynamics, it is proved that the ‘bare-particle’ coupling constant in renormalizable meson theories becomes imaginary at infinite cut-off. Analytical assumptions made are discussed.

The calculation of the strengths of forbidden lines in the 1s22s22p2 configuration is extended to include the effect of interaction with the 1s22p4 configuration. For this case it is shown that if observed energy levels are used instead of theoretical energies in line-strength formulae which neglect configuration interaction the major part of the effect of configuration interaction is nevertheless taken into account. The results are illustrated by detailed computations on the Ca xv spectrum. The 1s22s22p4 configuration is also discussed.