When the differential analyser (1) is being used it is often necessary to supply to the machine information in the form of a functional relation between variables occurring in the equation. This is usually done by keeping a pointer on a graph on an input table. The x-coordinate lead screw of the input table is driven by the machine, and the y-coordinate lead screw is rotated by hand so as to keep the pointer on the given curve. The disadvantages of this method, namely the necessity for the continuous attention of an operator and the possibility of personal errors, would be obviated by the use of an automatic follower instead of a hand control.

In mathematical theories the question of notation, while not of primary importance, is yet worthy of careful consideration, since a good notation can be of great value in helping the development of a theory, by making it easy to write down those quantities or combinations of quantities that are important, and difficult or impossible to write down those that are unimportant. The summation convention in tensor analysis is an example, illustrating how specially appropriate a notation can be.

Let f(x) be a real function of the real variable x, let P be any point lying on the graph of f(x) and let l be a ray from P making an angle θ (− π < θ ≤ π) with the positive direction of the x-axis. We say that θ is a derivate direction of f(x) at the point P if the ray l meets the graph of f(x) in a set of points having a limit point at P.

Es sei f(ξ) eine L-integrierbare und nach 2π periodische Funktion der reellen Variabeln ξ und es sei {sn} die Folge der Partialsummen ihrer Fourierschen Reihe an einer Stelle ξ = x, wo die Funktion stetig ist. Dann ist nach einem klassischen Satze von Fejér

und nach Hardy und Littlewood

Vor kurzem hat L. Fejér gefunden, dass der Hardy-Littlewoodsche Satz (2) Spezialfall eines “gewöhnlichen” Summabilitätssatzes wie (1) ist, der sich aber auf die Fouriersche Reihe einer Funktion zweier Variabeln bezieht. Dieser Satz ist der folgende.

It is shown in this paper and the preceding one that two separate forms of theory can be developed in which a “finite size” is attributed to a charged particle by means of its interaction with the radiation field. The region attributed in this way to the particle is four dimensional and is determined in such a manner that the usual difficulties with relativistic invariance do not arise.

The advantage of such a theory becomes clear when the theory is applied to those problems in which the usual calculations give infinite results. The problem of the method of successive approximations is considered and satisfactory results are obtained provided that the space dimensions of the finite region are of the order of the classical radius of the electron, when the electron is at rest.

It may be noted explicitly that the difficulty that has been associated with the emission of low energy quanta by “Bremsstrahlung” will not arise in the present formulation of the electromagnetic interaction between field and particles. This case is interesting since an infinity arises here which is not analogous to the self energy infinities, but occurs in the direct calculation of a physical process and not in a virtual transition.

The theory seems satisfactory so far as low energy processes (< 137 mc2) are concerned and the real test of its applicability may be expected to arise in discussing processes of high energy. It is hoped to treat these in a later paper.

The problem of the conduction of heat in a solid sphere with a concentric core of a different material, the surface kept at a constant temperature, and the initial temperature of the whole zero, has already been solved in these Proceedings.

The effect of interstellar matter on the sun's radiation is considered with a view to explaining changes in terrestrial climate. It appears that a star in passing through a nebulous cloud will capture an amount of material which by the energy of its fall to the solar surface can bring about considerable changes in the quantity of radiation emitted. The quantity of matter gathered in by the star depends directly on the density of the cloud and inversely on the cube of its velocity relative to the cloud. Thus vastly different effects on the solar radiation can be brought about under fairly narrow ranges of density and relative velocity (ranges that are in accordance with astronomical evidence). In this way the process is able to explain the small changes in the solar radiation that are necessary to produce an ice age and, under conditions less likely to have taken place frequently, the high increase in radiation required for the Carboniferous Epoch. Despite the large effects that the mechanism can bring about, it is shown that the mass of the sun does not undergo appreciable change and hence reverts to its former luminosity once the cloud has been traversed.

The decay period of the α-activity produced when boron is bombarded with neutrons from a (Li + D) source has been measured, and found to agree with the known decay period of 8Li. It has been shown that no activity is produced when (Be + D) neutrons (maximum energy 4·5 × 106 e.V.) are used. The results are explained by postulating the formation of radioactive 8Li according to the process

A short investigation of the number-range distribution for the α-particles emitted by 8Li indicated that a maximum occurred in the distribution at a range of about 6 mm.

The absorption of the harder γ-rays from ionium has been studied, the first, of energy 68 ± 1 k.e.V., by the method of bracketing, making use of the K absorption discontinuities of tantalum and tungsten; the second, of energy 190 ± 20k.e.V., by analysis of the absorption curves. The intensity of these γ-rays is estimated to be 1 quantum in 10−3 disintegration for each γ-ray.

By means of purely qualitative arguments which do not depend on any particular model, the general scheme of stable nuclei and the isotopic breadth of nuclei with odd charge number are explained.

The breadth of the isobaric region can be obtained if the numerical values of certain energies are known. Though these can be estimated only very roughly, the values for the breadth of the isobaric region obtained in this way are in good agreement with the experimental values. The increase in the breadth of the isobaric region from light nuclei to heavier nuclei can be explained, but no plausible explanation has been found for the fact that the breadth decreases again for the heaviest nuclei.

The irreversible heating of iron ammonium alum by an alternating field is shown to be proportional to the frequency of the field. The heating is therefore similar to that arising from hysteresis in ferromagnetics, but no remanence could be found by a ballistic method at the lowest temperature obtained, about 0·075° K. The rate of heating was much increased by the superposition of a steady magnetic field of a few hundred gauss.

If F is a free linear system of surfaces in an algebraic threefold V which is either non-singular or possesses only normal singularities, then F has Jacobian and adjoint surfaces, J2(F) and A2(F), and Jacobian and adjoint curve systems, J1(F) and A1(F), such that

where X2, X2 are the canonical systems of surfaces and curves on V, and X1(F) is the canonical system of curves of F. The imposition of base elements (points or curves) Ei, of assigned multiplicities λi, on F defines a system F1 which we may represent formally by the equation

and it is natural to enquire how the Jacobian systems of F1 differ from those of F, and how we may define adjoint systems A2(F1) and A1(F1) which cut on F1 its canonical curves and sets respectively.

In a previous paper I proved that the density of the positive integers of the form where the letters p, q, and later P, Q, r, denote primes, is positive. As indicated in the Introduction of I, I now give proofs of the following results:

The density of each of the sets of integers

is positive.