Random walk on a d-dimensional lattice is investigated such that, at any stage, the probabilities of the step being in the various possible directions depend upon the direction of the previous step. The motion may be characterized by a generating function which is here derived. The generating function is then used to obtain some general properties of the walk. Certain special cases are considered in greater detail. The existence of recurrent points is investigated in particular, and the probability of returning to the origin after 2n steps. This latter function is evaluated asymptotically for the cases d = 1 and d = an even integer.

Consider a Markoff chain (by which is meant a stochastic process defined for discrete values of both variable and parameter, and whose probability dependence does not extend to more than a unit interval) with completely specified transition probability matrix P = (pij), where pij is the conditional probability that the (r + 1)th observation belongs to the state Ei, given that the rth observation belongs to the state Ei for r≥ I. We assume that there are a states E1, E2, …, Ea, where a is finite. We also assume an ergodic property for the stochastic process, that is, we consider non-periodic chains for which all the possible initial states remain permanently available. This defines what has been called the positively regular case. Also under the above assumption the moduli of all the latent roots λr of the transition probability matrix P = (pij) other than the first simple latent root λ1 = 1 are less than one (see Bartlett(1)).

Before the war, self-consistent field calculations for the Au+ ion had been carried out by W. Hartree but were left still unpublished at his death (see prefatory note in (5)). These results have been used by Brenner and Brown (1) in a relativistic calculation of the K-absorption edge for gold, and they were also used in obtaining initial estimates for the partial self-consistent field calculations for thallium of which results are given in §§3–5 of the present paper. In the meantime an independent calculation for Au+ has been carried out by Henry (6), and his results agree closely with those of W. Hartree. However, it still seems desirable to publish the latter, since they give directly the radial wave function P(nl; r) at exact values of r, whereas Henry used log r as independent variable, as had been done for similar calculations for Hg(4), and has tabulated r½P(nl; r) which is the natural dependent variable to use with log r as independent variable (2); in some applications it is more convenient to have the radial wave functions themselves.

It is shown that the infinity arising in the second Born approximation to the scattering amplitude of a particle scattered by a Coulomb field is due to the distortion at infinity of Coulomb waves. Ways of subtracting out this infinity, which does not lead to any observable effects, are indicated.

Detailed energy calculations for CH4, SiH4 and GeH4 have been carried out using the Thomas-Fermi approximation together with a nuclear field which is averaged over all orientations about the central nucleus. The energy, including exchange, has been calculated using the unmodified Thomas-Fermi electron densities as variation functions, and in this way bond lengths and force constants for the three molecules have been obtained. The effect of exchange is very marked, agreement with experiment being quite reasonable when it is included in this way.

In order to test the accuracy of this procedure, a scale factor has been introduced and used as a variational parameter. In this refined approximation, calculations have been made for SiH4, thus enabling improved charge distributions to be obtained. Energy curves are also presented for this molecule, showing separately the variations of kinetic, potential and exchange energies with internuclear distance. It is shown that whilst the scale factor has an appreciable effect on the separate energy curves, the total energy is virtually unchanged.

Finally, the reason why the exchange energy is of considerable importance in the theory is briefly discussed, and some observations are made concerning the rather abnormal bonding in GeH4.

Eigenvalues and eigenvectors of the total Hamiltonian in field theory are determined by the method of direct diagonalization, using canonical transformations. The difficulties of this method are discussed in detail.

A new formulation of Hamilton's principle for the case of an ideal fluid is proposed which is claimed to be the uniquely proper form for such a system. The resulting derivation of the equations of motion on varying with respect to the position of the fluid particles is free from the difficulties encountered in previous treatments based on incorrect forms of Hamilton's principle. The treatment also differs from previous treatments in transforming to the fixed (a, b, c) space before carrying out the variation, and in allowing for the equations of mass and entropy conservation by means of undetermined multipliers. The necessity for conservation of entropy, if Hamilton's principle is to be applied, is emphasized. Finally, the method, first used by Eckart, of deriving the equations of motion for an ideal fluid by means of a variational principle of the same form as Hamilton's, but varying with respect to the velocities of the fluid particles, is extended to the general case of rotational motion.

Isotropic Gaussian processes, of which we shall give a formal definition presently, arise in various practical problems. The present inquiry arose from the consideration of the variability found in the yields of plots in agricultural field experiments. Samples of such patterns of variability can be obtained from uniformity trials, whereby a piece of land is treated uniformly throughout and the crop is harvested in small units or plots. The results from such trials have been widely used to determine optimum plot sizes for future experiments with the crop concerned, but it has never been clear how valid is the generalization from a uniformity trial to future experiments on other sites. One difficulty arises from the lack of a suitable model to express the variability; another difficulty arises from the formidable analytical problems besetting any attempts to apply deductive reasoning to even simple models. We may mention two approaches that have been made to the uniformity trial problem: Fairfield Smith (3) has supplied an empirical law connecting the variance of contiguous groups of plots with the size of the group, and Quenouille (6) and Whittle (8) have considered the fitting of two-dimensional isotropic Gaussian processes to uniformity trial data. Whittle has pointed out that a satisfactory model may have to incorporate an additional random element at each point, and has outlined the difficulties of estimation which arise when such an element is introduced. The connexion between Fairfield Smith's law and the approach via two-dimensional stochastic processes, if any, seems to be quite unknown.

In Parts I and II of this series (March(6); Ballinger and March(1)), we considered in detail the application of the Thomas-Fermi (T.F.) approximation to molecules with tetrahedral and octahedral symmetry. In these papers, following the work of Buckingham, Massey and Tibbs(2), who obtained results for CH4, we averaged the nuclear field over angles (the ‘smoothing approximation’) and considered the electrons as though they moved in the resulting central field. In this way, it was possible in (6) to carry through self-consistent (T.F.) calculations giving the electron distributions and potential fields in tetrahedral and octahedral molecules.

The work of which this paper is an account began as a study of differential equations for functions whose values are random variables of finite variance. It was intended that all questions of convergence should be treated from the standpoint of strong convergence in Hilbert space—familiar to probabilists from the writings of Karhunen(11) and Loève(13) as mean-square convergence. The more general Banach-space approach now adopted was made possible by the discovery of a theorem (Theorem 1 of this paper) which Mr D. G. Kendall, its apparent author, kindly communicated to us.

The convective effects of isotropic turbulence on arbitrary vector and scalar fields satisfying certain conservation conditions are studied under the assumption that all fourth-order cumulant tensors are zero. In the absence of molecular conduction, the behaviours of such vector and scalar fields are related to those of material lines and surfaces respectively, and the diffusive action of the turbulence results in a stretching of these lines and surfaces. The present analysis leads to a definite prediction for the corresponding strain rates which suggests that an initially spherical material volume element is drawn out into a long thin ribbon of constant width.

The S matrix is expressed in terms of momentum-space operators and its Cayley transform exhibited. A new graphical formalism called O-graphs is developed in which the vertices are temporally ordered and the lines represent real particles. This is used to prove a conjecture of Gupta concerning the relation of real and virtual processes to the Cayley transform. The transformation operator for a semi-infinite time interval is also considered and some of its properties demonstrated. In conclusion, another graphical formalism with temporally ordered vertices is developed and renormalized.

Many models of the universe have been proposed, by de Sitter, Milne, Bondi and Gold, Hoyle and others. The observed data being insufficient, the models are usually based on some simple hypothesis. The simplest is the cosmological principle, namely, that apart from local irregularities the universe presents the same general aspect at every point. Milne (5) has used a restricted form of the principle, namely, that the aspect is independent of spatial position but is dependent on the observed time from some fixed epoch in the past. Bondi and Gold(i) have proposed the ‘perfect cosmological principle’ that the aspect is completely independent of space and time.

The readings Φ(x) of an instrument with the apparatus function g(x) are related to the true values f(x)of the observed function by the equation

A three-point interpolation operator can be chosen to simulate the spreading action of a class of functions g(x). Under certain conditions the interpolation operator has an exact inverse which, when applied to Φ(x), gives a function which is a better approximation to f(x). The accuracy obtained is discussed, and the effect of the method on the standard errors in the measurement is found.

For elastic deformations beyond the range of the classical infinitesimal theory of elasticity, the governing differential equations are non-linear in form, and orthodox methods of solution are not usually applicable. Simplifying features appear, however, when a restriction is imposed either upon the form of the deformation, or upon the form of strain-energy function employed to define the elastic properties of the material. Thus in the problems of torsion and flexure considered by Rivlin (4, 5, 6) it is possible to avoid introducing partial differential equations into the analysis, while in the theory of finite plane strain developed by Adkins, Green and Shield (1) the reduction in the number of dependent and independent variables involved introduces some measure of simplicity. Some further simplification is achieved when the strain-energy function can be considered as a linear function of the strain invariants as postulated by Mooney(2) for incompressible materials. In the present paper the plane-strain equations for a Mooney material are reduced to symmetrical forms which do not involve the stress components, and some special solutions of these equations are derived.

The usual formula for the rate of loss of energy by a fast particle contains a factor z2, the square of the charge, which might suggest that He+ should lose energy at a quarter of the rate that He++ does. It is shown that when Born's approximation is applicable the energy-loss is reduced instead by a factor of about ⅝.

If r̄nl is the mean radius for the radial wave function of a complete (nl) group in an atom of atomic number N, the variation of 1/r̄nl with N is nearly linear. Further the variation of a given (nl) radial wave function with N is such that for a given value of (r/r̄nl), the variation of the quantity (r̄nl)½P(nl; r) with r̄nl is nearly linear. These relations between the radial wave functions for different atoms are examined from the point of view of using them as a means of interpolating, with respect to atomic number, between results for atoms for which solutions of Fock's equations have been carried out.

The maximum in thermal resistivity exhibited by superconductors in the intermediate state is found to be particularly marked in tin at 0-5° K., at which temperature heat transport in the pure superconducting state is almost entirely by phonons. A simple theory of the effect is developed, based on the idea that phonon free paths are considerably shorter in the intermediate than in the superconducting state, owing to the high scattering caused by normal regions. Theory and experiment are in satisfactory agreement.

There has recently been some interest in atomic wave functions for the Mn+2 ion in two different physical contexts, namely neutron diffraction (Shull, Straussen and Wollan (12), and Halpern (3)) and luminescence in solids (Williams (14)). In the neutron diffraction context the particular substance concerned is MnF2, for which the nuclear scattering is small so that the main contribution to the diffraction of slow neutrons is through the magnetic scattering by the incomplete (3d)5 group of the Mn+2 ion.