1. The problem in question is to find a necessary and sufficient condition which numbers co, …, cm must satisfy in order that there shall be a non-decreasing function σ(t) such that

where (a, b) is an unbounded interval. (When (a, b) is a bounded interval, the problem has been solved. See Achyèser and Krein, Communications de la Société Mathématique de Kharkoff, Série 4, 12 (1935), 13–35, Satz C.)

We shall first give some definitions concerning parametric surfaces. Denote by H a closed circle (disk) and by M a variable point on it. Let P = Ф(M) be a continuous function on H whose value P is a point in three-dimensional space. The symbols Ф(E), Ф−1(P), where E is a set of points on H and P a point in the three-dimensional space, will have their usual meaning. Ф−1(P) is a closed set. Any saturated continuum in Ф−1(P) or any point of Ф−1(P) that does not belong to such continua is called a Ф-element of H. Thus to any continuous function Ф(M) corresponds a representation of H in the form of the sum σQ of Ф-elements. The set of the pairs (P, Q), where Q runs through all Ф-elements of H and, for any Q, P = Ф(Q), is called a parametric surface, and any pair (P, Q) is called a point of the parametric surface. We shall often speak of a point Ф(M) of the parametric surface, by which we shall mean either the point (P, Q), where P = Ф(M) and Q is the Ф-element containing M, or the point P = Ф(M) of the three-dimensional space. The exact meaning will always be clear from the context. If there are exactly k points of the parametric surface whose first member is P0 we say that P0 is a point of multiplicity k. If k = 1, P0 is a simple point.

In 1914 Carathéodory defined m–dimensional measure in n–dimensional space. He considered one-dimensional measure as a generalization of length and he proved that the length of a rectifiable curve coincides with its one-dimensional measure.

A sequence of papers by P. Hall ((1)–(4)) on some fundamental properties of soluble groups was followed in 1940 by his account of a new construction theory for such groups (5). We shall have frequent occasion to refer to these papers, and we begin with a brief account of their contents.

The determinantal quartic primal in [4], represented on [3] by quartic surfaces passing through a decimic curve of genus eleven, has already been the subject of a number of papers. The primal for which the base curve is a general curve C10 has twenty nodes, mapped by the twenty quadrisecant lines of the curve. If the C10 breaks up into a number of curves of lower order, the primal has, in addition, other nodes, mapped by the neighbourhoods of the intersections of two simple branches of the base curve. When the C10 consists of ten lines, having twenty mutual intersections to preserve the virtual genus eleven, the primal has forty nodes; the two primals of this type which contain no planes are the subject of a paper by Todd (3). Further specialization of the ten lines, giving more additional nodes, is possible; it has been shown in another paper (Todd (4)) that there is a quartic primal with forty-five nodes (the maximum number) which is determinantal. The figure in [3] formed by the base curve is described there; for a description of the configuration formed by the nodes in [4], and many other interesting properties of this primal, the reader is referred to two recent works (Baker (1) and Todd (5)). Further details about the general quartic primal, and references to a number of special cases, may be found in Room (2), chapters XV and XVI.

1. Introduction. Modern computational machines and methods are conveniently divided into two classes:

(A) Those depending on physical analogues, and therefore restricted in accuracy.

(B) Those depending on digital processes, and so capable of giving any desired accuracy.

A method is given for the solution of transient differential equations of the type

where D is a differential operator in one or more dimensions. These equations arise in vibrational problems and problems on the conduction of heat. Some examples on the application of the method are given.

1. The boundary value problem of Laplace's equation for two spheres is a classical one, and has been the subject of discussion by many mathematicians (1). The earliest attempt to solve a boundary value problem of this type is due to Poisson (2), but his analysis is applicable only to the electrostatic problem. The first of the methods which can be successfully applied to both electrostatic arid hydrodynamical problems was developed later by Lord Kelvin (3); this procedure, which is known as the ‘method of images’, was first applied to the problem of the motion of two spheres in a perfect fluid by Hicks (4). Another method of great generality, that of transforming Laplace's equation to bipolar coordinates and studying the solutions in these coordinates, was developed about the same time by Neumann (5) and much later by Jeffery (6). More recently a new method has been developed by Mitra (7) for the solution of the problem of two spheres in a potential field. It makes use of two sets of spherical polar coordinate systems; the solution is expressed in terms of infinite series whose coefficients satisfy an infinite set of linear algebraic equations. The chief interest of Mitra's method lies in the fact that he has found it possible to derive exact solutions of this infinite set of equations. All of these methods suffer from the disadvantage that the potential function is obtained in the form of an infinite series so that any numerical calculations are rendered cumbersome.

In this paper, the general hydrodynamic theory of visco-inelastic, incompressible, non-Newtonian fluids, developed in a previous paper (l), is applied to the problem of the flow of such a fluid through a tube of circular cross-section. It is found that the absolute values of the normal stress components are no longer uniform over a cross-section of the tube normal to its axis, as in the case of Newtonian fluids obeying the laws of classical hydrodynamics.

However, the pressure difference, between points at equal radii on two planes normal to the axis, is independent of the position of these points on the planes.

1. Introduction. A considerable amount of attention has been paid to the problem of determining the conditions which decide whether a liquid heated from below is stable or unstable. The motion consequent upon the disturbance of an unstable ideal gas does not, however, seem to have been treated so far, and this problem forms the subject of the present paper. Heat conduction and viscosity are at first neglected, and we are therefore dealing with the small motions of a gas slightly disturbed from a position of equilibrium under the influence of gravity. The condition for the stability of such a gas is well known, namely, the temperature gradient must be less than the adiabatic gradient. Furthermore, it is known that there is a sharp distinction between slow large-scale (meteorological) and rapidly varying small-scale (acoustical) phenomena. The present paper confirms these points and derives the time scale of meteorological phenomena. Heat conduction and viscosity are then shown to set a lower limit to the dimensions of such disturbances, while the effect of the earth's rotation is shown to be negligible.

An attempt is made to interpret quantum mechanics as a statistical theory, or more exactly as a form of non-deterministic statistical dynamics. The paper falls into three parts. In the first, the distribution functions of the complete set of dynamical variables specifying a mechanical system (phase-space distributions), which are fundamental in any form of statistical dynamics, are expressed in terms of the wave vectors of quantum theory. This is shown to be equivalent to specifying a theory of functions of non-commuting operators, and may hence be considered as an interpretation of quantum kinematics. In the second part, the laws governing the transformation with time of these phase-space distributions are derived from the equations of motion of quantum dynamics and found to be of the required form for a dynamical stochastic process. It is shown that these phase-space transformation equations can be used as an alternative to the Schrödinger equation in the solution of quantum mechanical problems, such as the evolution with time of wave packets, collision problems and the calculation of transition probabilities in perturbed systems; an approximation method is derived for this purpose. The third part, quantum statistics, deals with the phase-space distribution of members of large assemblies, with a view to applications of quantum mechanics to kinetic theories of matter. Finally, the limitations of the theory, its uniqueness and the possibilities of experimental verification are discussed.

1. When crystals of a metal are highly perfect, their elastic limit is low, plastic flow taking place when the applied shear stress is very small. The elastic limit reaches higher values as the perfect structure is progressively broken up by cold-work. A steady state is finally reached where further distortion of the metal does not increase the elastic limit. The stress at which a metal passes beyond the elastic limit and begins to yield is ill-defined, as in general the crystal begins to ‘creep’ appreciably near this point and the time element enters into the definition of the stress-strain curve. Nevertheless, there is a fairly well-defined stress beyond which a metal, which has been brought into a steady state by being saturated with cold-work, ceases to behave elastically and undergoes permanent deformation. An attempt is made in this paper to derive an expression for the ultimate elastic limit or yield stress of a cold-worked metal, in terms of its structure and elastic constants.

1. This paper and an earlier paper (l) were written at the suggestion of W. L. Bragg in connexion with his theory of yield in metal crystals (2).

Investigations of reactions of the type

are of interest for their own sake, and also for the fact that, if is a radioactive nucleus, as it usually is, being a stable isobar, activation of a few different A nuclei in the same fast neutron flux will yield some information about the neutron spectrum. This follows from the fact that these (np) reactions (and the similar (nα) reactions) are nearly all endothermic, showing a threshold of neutron energy below which no activation takes place. Thus knowledge of the excitation functions of a few such reactions would enable the above-mentioned crude neutron spectroscopy to be realized. The detectors (A nuclei) would have a standard shape and size, and the induced β-activity would be measured with a standard counter in fixed geometrical conditions. Though exact spectroscopy would not be possible, any change in the relative activations of the detectors would be a quite sensitive indicator of a change in the incident neutron spectrum.

Recent papers by Néel (8) and Lawton and Stewart (7) have provided an explanation of the process of magnetization in a single crystal, using the well-known ideas of the domain theory but bringing out the important role played by the demagnetizing field. These authors applied their ideas to two shapes of single crystals of iron, long rods and oblate spheroids, but in the latter case the only crystal orientation worked out in detail was that in which the equatorial plane was (100). The case of oblate spheroids of iron and iron-silicon where the equatorial plane is a (110) plane of the crystal, and the magnetic field is applied in this plane, are considered in detail in this paper. Calculations are made for In, the component of magnetization perpendicular to the applied field He, and comparison made with experimental measurements. This particular problem was considered by Bozorth and Williams (4) who calculated the torque per unit volume (i.e. the product InHe) experienced by the crystal in a magnetic field. Their calculations are, however, incomplete, because their picture of the magnetization process is not a valid one in moderate fields. Moreover, the demagnetizing field was taken into account only in assessing the size of the field H acting within the crystal, but not its direction.