Blichfeldt (2) gave a method of proving the existence of lattice points in certain regions. His method depends on the following lemma (which he stated in a somewhat different form):

Lemma 1. Let K′ be a closed region of volume V. Let K be another region so related to K′ that the difference of any two points in K′ is a point in K. Let Λ be any lattice whose determinant is positive and does not exceed V.

Let x = ℘′(u) and y = ½℘(u), where the Weierstrass ℘-function has invariants g2 = 4A, g3 = 4B and differentiation with respect to the parameter u is denoted by a dash, so that

1·1. There is a familiar method of summation of series usually called the method (R, 1), or sometimes Lebesgue's method. If sn = u0 + u1 + … + un and, as it will be convenient to suppose throughout, u0 = 0, then

when h → + 0§ the convergence of the series for small positive h is presupposed. The method is not 'regular'; there are convergent series not summable (R, 1).

1. Let {μn} be a sequence of positive numbers increasing to infinity. If the series

converges for s > 0, and if f(s) → l as s → 0, then ∑an is said to be summable (A, μn) to l; this generalization of Abel summability, first introduced by Hardy, has long been familiar. Suppose that, for x ≥ μ0, ø(x)is a positive increasing function of x, which tends to infinity as x → ∞. We write throughout λn = ø(μn), and

if this exists.

A function f(x1, x2) of two real variables x1, x2 which are restricted to rational integers will be called discrete harmonic (d.h.) if it satisfies the difference equation

This equation can be considered as the direct analogue either of the differential equation

or of the integral equation

in the notation normally employed to harmonic functions.

1. Let z = reiθ, and let h(z) denote a (regular) positive harmonic function in the unit circle r < 1. Then h(r) (1−r) and h(r)/(1 − r) tend to limits as r → 1. The first limit is finite; the second may be infinite. Such properties of h can be obtained in a straightforward way by using the fact that we can write

where α(phgr) is non-decreasing in the closed interval (− π, π). Another method is to write

where h* is a harmonic function conjugate to h. Then the function

has the property | f | < 1 in the unit circle. Such functions have been studied by Julia, Wolff, Carathéodory and others.

Let the probability density of observations be denoted by φ(x | θ), where x stands for the variables and θ for the parameters. A function t of the observations is called an unbiased estimate of the function ψ(θ) of the parameters if

where dx stands for the product of differentials.

1. The statement of the problem. A problem which arises in certain physical (5) and astronomical (1, 3) contexts relates to the probability distribution of the infinite sum

where the rj's are the position vectors from a fixed origin of an infinite uniform Poisson distribution of points with a given constant mean space density N. While certain special cases of the sum (1) have been considered in the literature (cf. (2)) the study of the general case discloses properties of a class of probability distributions which appear to have some interest. For example, we shall show that the probability distribution of F exists only for that the moments of F (= |F|) exist only for orders 0 ≤ p < 3/n, and finally that an explicit formula can be given for all the finite moments.

1. It is an interesting fact that in many problems of statistical estimation the results given by the theory of inverse probability (as modified by Jeffreys) are indistinguishable from those given by the methods of ‘fiducial probability’ or ‘confidence intervals’. The derivation of some of the important inverse distributions by Jeffreys(3) arouses one's curiosity. It seems that when this agreement is noticed there are usually sufficient statistics for parameters in the distribution. The object of this note is to throw some light, in general terms, on the similarity in form between the posterior probability-density function of the parameters and the probability-density function of the distribution when it admits sufficient statistics. For convenience the following notation in Jeffreys's probability logic is used below:

P(q | p) is the probability of a proposition q on data p.

If ηn is the error in the result of repeating an iterative process n times, and ηn+1 is the iterative process is called Kth order. It is shown that if for a given equation there is an iterative process of the (K + 1)th order, the iterative process of the Kth order is not unique, and conversely if an iterative process of the Kth order is not unique, it is generally possible to construct from two of the Kth order processes a process of the (K + 1)th order. As an example, three second-order processes for a square root are exhibited, and a third-order process is derived from two of them.

Iterative processes for positive and negative integer roots are given, of kinds suitable for use on machines in which division is a relatively slow process and one to be used sparingly. It is shown how a second-order process can be derived from the results of two repetitions of a first-order process. The extension of this iterative process for the solution of differential and integral equations is a development which is urgently required.

1. In a recent paper (1) Hartree has given a systematic classification of iterative processes, and has shown how to construct processes of any order for the solution of equations of the form

where p is a positive or negative integer. It is the purpose of the present note to describe general methods of generating iterative processes, and in particular to show how to construct processes of any order for the solution of algebraic equations.

It is well known to computers that the approximate formula

yields a surprising degree of accuracy even for quite large values of the interval h; for example, if h = 1 the error is one unit in the fourth decimal.

1. Compressible and incompressible flow. Small perturbations in an otherwise uniform stream of compressible fluid moving at supersonic speed are described by the approximate linearized equation for the velocity potential. When the stream flows in the z-direction, the equation assumes the form

where M is the Mach number of the flow and α2 is positive. In cylindrical polar coordinates (r, z), the equation may be written as

which is Laplace's equation in coordinates (iαr, z). We may therefore relate potentials of incompressible flow which are solutions of (1·2) to potentials of compressible flow which are solutions of (1·1).

Harding and Sneddon(3) and Sneddon(6) have shown that the systematic application of Hankel transforms to the problem of the indentation of the plane surface of a semi-infinite elastic solid by a rigid punch, reduces the problem essentially to one of solving a pair of integral equations belonging to a class which has been studied by Titchmarsh(8) and Busbridge(1). The same method has also been applied by Sneddon(7) to the problem of the distribution of stress in the neighbourhood of a circular crack in an elastic solid. In both types of problem the results are expressed in quite simple forms independent of Bessel functions and this suggests that there should be an alternative method of solution. It is now possible to supply such a method with the help of a recent paper by Copson(2) on the problem of the electrified disk. In addition, some generalization of the stress systems considered by Harding and Sneddon can be made since they confined their analysis to stresses which were symmetrical about an axis.

A solution is given of a problem already solved by Inglis, a rectangular plate stressed by the uniform longitudinal relative displacement of two opposite edges, the other pair of sides remaining free. The method used here yields a more accurate solution with greater rapidity than did the previous paper. The general form of the solution is given for a plate of any dimensions, and a plate of sides in the ratio 2:1 is considered in more detail. An approximate solution is obtained by taking four terms of an infinite series, and the consequent errors discussed and compared with those of Inglis.

The search for a theory of the elementary particles which is founded on the well-established principles of quantum mechanics and conforms at the same time with the requirements of the principle of relativity has, in recent years, taken several divergent directions. On the one hand, the second quantization of wave fields derived from a Lagrangian by a variational procedure(1) has succeeded in accounting for the existence and most of the properties of the electron, the photon, and the meson. On the other hand, many generalizations of the Dirac wave equation of the electron(2) have been attempted, with applications to the meson(3) and the proton(4). Heisenberg(5) has considered the much more difficult problem of the interaction between different particles, and has found that the key to the situation is the so-called ‘scattering matrix’, which is nothing other than a limiting form of the relativistic density matrix, as defined in § 2 of this paper. It seems probable that the relativistic density matrix ρ; or statistical operator, as it may be called without reference to representation, will play an important part in relativistic quantum mechanics in the future. It satisfies the same equation as the wave function, but differs from it in being a real linear operator, or a dynamical variable, in the terminology of Dirac.

In this paper, it has been shown that the usual demonstrations of the phenomenon of Bose-Einstein condensation are invalid. Exact formal solutions have been derived for the partition functions and the mean occupation numbers in classical and Bose statistics, and the theory of fluctuations in the case of Bose statistics has been given.

In a second paper, convenient approximations will be derived for the mean occupation numbers in Bose statistics, and it will be shown that for temperatures well above the critical temperature of the supposed Bose-Einstein condensation the well-known result for the mean occupation numbers is valid, whilst for temperatures rather lower than this critical value it will be shown that particles will condense into the lowest available state. The critical temperature T0 is given by where the summation is over all energy levels except the lowest.

A quantitative study of the law of force between bubbles floating on a fluid has been made and the results exhibited in the form of a potential energy-distance curve. The main approximations made are that the inclination of the surface of the fluid is small (so that the equations for the surface become linear), and that the submerged portion of the bubble remains spherical. The latter assumption restricts the validity of the results to small bubbles of about 3·0 mm. diameter and less.

One object of the work was to determine the size of bubble which simulated most nearly a typical atomic interaction curve. This application arises in connexion with Prof. Bragg's ‘bubble’ model of crystal lattices. Exact simulation cannot be expected, for the bubble model is two-dimensional, and also does not exert forces analogous either to Coulomb repulsion or to the metallic bond. It can be shown, however, that the law of force between bubbles of a particular size is a remarkably close fit to the law of force between inert gas atoms, that is, to an interaction consisting of a van der Waals attraction and a short-range or ‘overlap’ repulsion. Fig. 4 illustrates this point. The two full curves are the energy-distance curves for krypton and argon atoms (plotted from Fowler, Statistical Mechanics, 2nd ed., Chapter x, Table 29). The circles mark the corresponding curve for bubbles with α = 1·0 (i.e. of diameter 3 mm. in soap solution). The units of energy and distance are chosen so as to bring the minimum of each curve to the point (1, − 50). It will be seen that the curves are remarkably similar over the range shown.

It has recently been shown (4, 5) that in order to give a true account of the magnetization processes in a ferromagnetic it is necessary to consider the influence of the demagnetizing field of a finite specimen not merely in a quantitative way, affecting the magnitude of the field acting on the specimen, but also in a qualitative way, affecting the actual arrangement of the domains of the ferromagnetic. It is the purpose of this paper to give an account of the process of magnetization in a specimen which has a tension applied in the same direction as the magnetic field. It will be shown that by considering fully the effects of demagnetizing fields results are obtained which agree well with experiment and remove some discrepancies of the older domain theory.

Relations are derived between the dose in r./min. delivered by the γ-ray beam from a betasynchrotron and the mean electron current which is incident on the internal target. Comparison with experiment shows that multiple passage of electrons through the target probably plays an important part in production of the γ-ray beam. Some possible effects connected with the multiple passage of electrons are discussed.