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.

In this and the succeeding paper I solve two problems suggested by Prof. Hardy, namely (1) that of proving that Ramanujan's function

has no zeros on the line and (2) that of finding an asymptotic formula

where A is a constant. I also prove similar results concerning the coefficients of general modular forms. I am indebted to Prof. Hardy and Mr Ingham for various suggestions, and in particular to Mr Ingham's paper, “A note on Riemann's ζ-function and Dirichlet's L-functions”.

Suppose that

is an integral modular form of dimensions −κ, where κ > 0, and Stufe N, which vanishes at all the rational cusps of the fundamental region, and which is absolutely convergent for Then

where a, b, c, d are integers such that ad − bc = 1.

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.

The following theorem was proved by Milloux:

Theorem A. Suppose that f (z) is regular and that | f(z) | < 1 in the unit circle. Suppose also that the set of points in the circle | z | ≤ r′ < 1 at which

cannot be enclosed in circles, the sum of whose radii is equal to 2eh−1. Then

where K is a numerical constant.

The object of this note is to extend the proof, for the general canonical varieties of any dimension k < d on a Vd whose existence was established by Dr Todd in a recent paper, of the property of adjunction given by the relation

where S is any non-singular variety of dimension d−1 lying on Vd, and Xk[S] denotes the canonical system of dimension k on S. This formula was proved by Todd in the case when Vd is a general non-singular variety, and S is non-singular and belongs to a sufficiently general linear system. Using this result, I show how to remove the latter restriction. (1·1) cannot be taken, without further proof, as defining canonical systems on virtual or isolated varieties, as is done by B. Segre, since the canonical systems on an effective isolated variety are already well defined.

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.

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.

In this paper the usual theory of quantum electrodynamics is outlined and an alternative method of quantization is introduced. Some of the difficulties of this theory are discussed and a modification is introduced to meet them. The theory is developed so as to satisfy the principle of relativity, but is not quantized. In the following paper an extension of the theory is given which removes this restriction.

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 cross-sections for the elastic scattering of mesons by protons and electrons are calculated by analogy with Møller's theory of collisions between electrons, and the results are compared with those of Bhabha for the collisions with protons and electrons of Fermi-Dirac particles with the same mass as the meson. For collisions with electrons, the difference is inappreciable for energies of the meson less than 100 times its rest energy, but for collisions with protons the difference is important at much lower energies. The ratio of the corresponding cross-sections for scattering by a Coulomb field is found to tend to infinity as the square of the energy of the incident particle, giving a finite cross-section for the scattering of a meson of infinite energy by such a field. It is pointed out that, if an exact solution of this problem is possible for the meson case, the question of the validity of meson theory for high energies would be considerably clarified.

In the present paper application of the work of paper I is made to a specific problem, namely the excitation of an adsorbed atom by the surface electrons of the underlying metal. It is shown that the corresponding mean life-time in a typical excited level is 3·7 × 10−11 sec., which is comparable with the value 1·7 × 10−12 sec. for deactivation by the conduction electrons when the Brillouin zone is only partly filled. Thus, though this is an essentially surface phenomenon, the effect of the surface electrons is begligible. This suggests that their effect can, in general, only become appreciable in semi-conductors or insulators.

Many problems in physics, chemistry and other subjects give rise to differential equations which are difficult or impossible to treat mathematically. It has long been possible to obtain numerical solutions by arithmetic methods, and the development of adding and multiplying machines to their present state of perfection has made these much simpler to carry out. The amount of work involved is, however, still very great.

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 the preceding paper, Blackett and Williams described an automatic curve follower for use with the differential analyser. The device would only operate if the x-coordinate was either continuously increasing or continuously decreasing; it could not handle input functions where the x value pulsated, as may often happen when the x-coordinate is not the independent variable.

By “sensitive” time of a cloud chamber is meant the length of time for which the supersaturation in the chamber remains sufficient to cause condensation along the tracks of ionizing particles. If the cooling expansion is made rapidly this sensitive time depends on the rate at which the temperature of the gas in the chamber rises after the expansion. This problem of the warming of a gas in a vessel, after an initial cooling, has been previously considered by Rayleigh (1). To follow the whole history of the warming until the gas reaches the temperature of the vessel requires an elaborate mathematical treatment which can only be carried out for specially shaped vessels. Rayleigh considers two special cases, viz. a gas contained between two parallel infinite planes, and a gas contained in a spherical vessel. The results obtained by Rayleigh for these cases are, however, not very helpful in dealing with the cloud-chamber problem under consideration. This is so, not so much because specially shaped vessels are assumed, but because the treatment does not lend itself to an evaluation of the rate of warming in the early stages of the process. It is this initial warming that we are concerned with here, since a cloud chamber ceases to be sensitive when the average temperature of the gas has risen by only 5% or less of the cooling produced by the expansion. Rayleigh's result for the average rise in temperature of the gas at any instant is represented by an infinite series which converges very slowly when the rise in temperature is small.