If we assign to the relativistic energy of a free particle H the meaning of an energy operator, it can be shown that the phenomenon of ‘Zitter-Bewegung’ (and hence Dirac's equation) is obtainable from quantum mechanics alone.

The success of Stoner's extensions of the theory of collective electron ferromagnetism(1) opens up new possibilities of considerable importance. The background of the theory can be found in standard works on the theory of metals(2) and statistical mechanics(3) and will not be given here.

In Part I of this series it was shown that in a metal with overlapping energy bands, one of which is nearly full and the other nearly empty, there exists a critical temperature below which spontaneous magnetization will be present, Tc being given approximately by the equations

where kθ is the exchange energy term postulated by Stoner, b is the ratio of excess vacancies in the d-band to the maximum possible number of parallel spins, and ζd0 is the equivalent energy depth of the vacant levels in the d-band.

The equations of motion in a surge-tank fitted to a hydraulic pipe-line are non-linear. Various methods of solving them were described by Durand: (1) simplification of the equations and the use of empirical coefficients, (2) simplification by taking the friction forces as proportional to the first power of the velocity, (3) a step-by-step method of solution. As a revolt against the serious expense of time involved in (3) (which was later developed by Jakobsen and by Cole), he was led to devise other methods including the use of models, which he dealt with subsequently in more detail. Nevertheless, there remains a need for a quick approximate method of calculating the maximum surges which may be used with confidence at least in the early stages of design. A promising line of attack was suggested by the success which has attended the approximate analysis of mechanical vibrating systems having non-linear characteristics. This method has been set out by Den Hartog, who dismisses step-by-step solutions as ‘too laborious’.

1. A set of functions {øν (t)} (ν = 1, 2, …) is said to be closed L2 in (a, b), if

implies f (t) = 0 p.p. in (a, b). It is well known that a set of functions is closed L2, if and only if every function of L2 (a, b) can be approximated in the mean square sense as closely as desired by finite linear combinations of the øν (t).

The combustion of wood presents an interesting problem in non-steady heat flow. When wood is heated, the temperature distribution at a given time may be calculated by means of the well-known conduction equation together with the relevant boundary conditions, provided that the temperature is nowhere sufficiently high to cause appreciable thermal decomposition. When this condition does not apply, the calculation becomes much more complicated, since, as has been recognized for a considerable time, the decomposition is exothermic. The general problem, therefore, is to calculate temperatures and rates of decomposition inside a mass of material, the thermal breakdown of which is accompanied by a heat change, given an initial set of conditions, and a known rate of supply of heat to the surface. The theoretical part of this paper aims at solving this problem for the case of sheets of wood heated in a comparatively simple manner. The treatment is, however, general and may be applied to other materials which undergo thermal changes without melting, if the relevant thermal and other constants are inserted.

1. This note originates from a question put to us by Mr W. R. Dean concerning series of the type

where θ is irrational. Series of the type

are familiar: it is well known, for example, that

may have any radius of convergence from 0 to 1 inclusive, according to the arithmetic nature of θ. It is natural to ask how this radius is connected with that of

and, more generally, how those of (1·1) and (1·2) are connected.

In this paper formulae are developed for the first and second focal lengths, and the positions of the first and second principal planes of a type of electrostatic lens which has been the subject of study (mostly experimental) in several previous papers. The lens, which is commonly used in electron optical devices, lends itself to a theoretical study, although this does not appear to have been attempted before. It consists of two equal semi-infinite cylinders placed end to end so that their axes coincide and the ends are separated by a small gap. If the cylinders are at potentials V1 and V2 and we write σ = V2/V1, the system behaves as an electron lens when σ > 0 and as an electron mirror when σ < 0. In the latter case some experimental results have been given by Nicoll(1) who also studied the focusing action in the case σ > 0 and, in particular, the formation of intermediate images when σ ≪ 1 and when σ ≫ 1. But for the precise formulation of the relationship between σ and the number of cross-overs a theoretical study, based on the paraxial equation, would be necessary. The problem will be indicated below. An experimental determination of the lens characteristics for values of σ from about 2 to 15 and for several gap widths has been made by Spangenberg(2), whose results will be compared with those obtained in the present paper. The two-cylinder lens has also been studied by Klemperer and Wright(3) using an experimental and a numerical (trigonometrical) method, and some crude analytical results have been given by Gray(4).

The Eulerian equations obtained by varying a Lagrangian containing certain field quantities qα and their derivatives of any order are cast into canonical forms. A modified canonical form for the equation of motion is given when the conjugate variables are not independent. Finally a special case is discussed.

1. The Petzval field curvature produced in a compound glass optical system of axial symmetry is given by a well-known formula. If the system consists of a number of media, of refractive indices n0, n1, n2, …, having spherical faces whose radii of curvature are r0, r1, r2, …, the formula is [see for example, (1) or (2)]

and this is in common use amongst those concerned with the design of optical equipment. The analogous integral expression in the electron optical case has not, however, received the attention it deserves, in spite of the developments in electron optics during the past decade. In 1935 Glaser(3) presented the third-order error theory of an axially symmetric electron optical system.

Few papers presenting a theory of operation of thermostats can be found. This may be explained by mathematical difficulties which are met when considering the rather complicated construction, from the thermal point of view, of most thermostats. There is, however, a relatively great number of papers describing various mechanical constructions of thermostats. In many cases designers obtain very good practical results by making use of some approximate theories only.

1. Introduction. Recent experiments by Amaldi and his collaborators on the scattering of high-energy neutrons (of 10–15 MeV.) by protons(2) have disclosed a considerable anisotropy in the angular distribution of the scattered particles. Theoretical discussions of this problem show an interesting feature in that the results depend sensitively on the basic assumptions involved with regard to the charge dependence of the neutron-proton interaction. This can be seen in particular from calculations by Rarita and Schwinger(3) and by Ferretti(4). The former authors started from the assumption of a distance dependence of this interaction represented by a square well, while the angular and spin dependence included terms of the axial dipole type. If the charge dependence was further assumed to be of the ‘symmetrical’ type, they found a value for the anisotropy in strong disagreement with experiment, whereas the total cross-section agreed with the measured value; a ‘neutral’ theory, on the other hand, yielded agreement as regards anisotropy, but a total cross-section too large by a factor of the order of 1·5. Ferretti investigated the scattering on Bethe's neutral meson theory(5) and found satisfactory agreement with regard to both angular distribution and total cross-section. It should be stressed that all calculations mentioned were performed in the approximation in which only the contributions of the S- and P-waves are considered.

The paper considers the solution of the general variational equation of relativistic dynamics using the methods of Mathisson with a generalization at one stage. The solution gives us two equations describing the translational motion and rotational motion. The rotational equation is also investigated by setting up a variational equation using the conservation of angular momentum, and this is found to be of the same form as that obtained from Mathisson's variational equation.

In my paper under the same title, to which I shall refer in future as I‡, I generalized the Vitali covering principle from the case of Lebesgue measure to the case of any non-negative additive function. This allowed me to establish the relative differentiation of additive functions. The convergent sequences of sets in this generalized form of the covering principle were restricted to sequences of concentric circles, and therefore the differentiation arrived at was that in the symmetrical sense. In the present paper, I extend the principle to the case of any regular convergent sequences of covering sets; and then establish the relative differentiation of additive functions in the general sense, and in particular the differentiation of indefinite integrals with respect to any measure function. This problem has a complete solution. It is established that indefinite integrals are differentiable at almost all points. In the case of the general measure function, it is not true that the derivative is equal to the integrand at almost all points, but necessary and sufficient conditions are given under which this is true.

A curve Γ has been defined as a normal multiple of a curve C if it contains an involution In, of order n, which has the properties: (1) the sets of In are in (1–1) correspondence with the points of C; (2) each set In consists of n points which are distinct in the birational sense; (3) In is generated by an Abelian group G of order n of birational transformations of Γ into itself. We shall denote the field of complex numbers by k, the function field of C by k(C), and the function field of Γ by k(Γ). Conditions (1) and (3) imply that k(Γ) is a commutative normal (i.e. Galois) extension of k(C). The object of this note is to show how, given the curve C and the Abelian group G of order n, we can construct a curve Γ with the properties (1), (2), (3).