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.

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).

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).

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.

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.

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’.

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.

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.