1. A “Fourier kernel” means here a function K(x) which gives rise to a formula

of the Fourier type. Thus

are Fourier kernels. If K(x) is a Fourier kernel, λ is real, and a positive, then

are Fourier kernels.

The best of the theorems on continued fractions, to be found in Ramanujan's manuscript note-book may be stated as follows:

where there are eight gamma-functions in each product and the ambiguous signs are so chosen that the argument of each gamma-function contains one of the specified numbers of minus signs. Then

provided that one of the numbers β, γ δ, ε is equal to ± n, where n is a positive integer; and the products and sums on the right range over the numbers α β, γ δ, ε.

1. In some recent papers I have developed the theory of what I have called harmonic integrals associated with an algebraic variety. My object has been to provide an apparatus with which to investigate the properties of the classical integrals

of total differentials which are finite everywhere on an algebraic variety Vm of m dimensions

These integrals are referred to in what follows as “Abelian integrals”, as it is desirable to have a single name to denote all integrals of this type. The usual qualification “of the first kind” is omitted, and is always to be understood. In the present paper it is shown how the theory of harmonic integrals can be applied to deduce certain results concerning Abelian integrals, and a number of interesting problems are suggested by these results.

1. There are two well-known theorems on the limit of a bounded function at a point.

Montel's Theorem. Suppose that f (z) is regular for | arg z | ≤ α, | z | ≤ 1, except perhaps at z = 0, and that f(z) is bounded in that region. Suppose also that f(z) → l as z → 0 along arg z = β, where | β | < α. Then f(z) → l as z → 0 uniformly for | arg z | ≤ α − δ for every δ > 0.

In the first section of this paper we illustrate the use that can be made of higher space in dealing with the problem of resolving a given Cremona transformation into the product of simpler Cremona transformations. In the second section we restrict ourselves to a particular large but finite class of Cremona transformations of [3], those of genus one, and show that these can all be built up from the four following simple types:

(1) The bilinear transformation T3,3, determined by three equations bilinear in the coordinates of the two corresponding spaces; in the most general case of this both the direct and the reverse homaloidal systems consist of cubic surfaces passing through a non-degenerate sextic of genus three;

(2) Three transformations Tn, n (n = 2, 3, 4) in which the homaloidal surfaces may in each case be obtained by taking in [4] a primal V of order n which has two (n− l)ple points, and projecting on to a given [3] from one of these points the sections of V by primes through the other; for n = 2 we have the familiar quadroquadric transformation determined by quadrics through a conic and a point.

1. Let P = 0, Q = 0, U = 0, V = 0 represent four independent quadric surfaces, and consider the equation

The general solution of Born's new field equations is found for the two-dimensional electrostatic case, by which the coordinates are expressed as functions of the field vectors, Conditions for inversion are discussed. Special cases are worked out, namely: singnle charge, two charges, charge in a constant field. Expressions are given for forces acting on the charges. A singular solution is also discussed, with reference to the neutron. The implication of the solutions on the general theory and the equations of motion is discussed in the conclusion.

1. This paper is supplementary to an article by M. H. A. Newman on the superposition of manifolds. Guided by his main idea, I extend Newman's theorem that two equivalent manifolds have a common subdivision from equivalent manifolds to equivalent complexes of an arbitrary nature. The theorem is also sharpened by the possibility of leaving a certain subcomplex unaltered, and by substituting a partition for a general subdivision.

The object of this paper is to extend the ordinary theory of integral equations of the second kind to certain classes of more general kernels and functions, with special reference to equations of the Volterra type.

Owing to a chance observation some interesting phenomena were observed during the splashing of water in an electric field. If a water drop falls into water, there is a characteristic splash of which the details were investigated photographically by Professor A. M. Worthington in the latter years of the last century. The final account of this work was given in two papers (1) and also in a book entitled A Study of Splashes (2). The splash cannot be examined with the unaided eye because the various stages of the splash follow one another too quickly. For the moment it may be said that an important stage of the splash consists of the rising up of a cylindrical column of water to a considerable height. The radius of the column is comparable with the radius of the drop.

It is well known that an electron possesses a magnetic moment. Frenkel has further deduced by relativistic arguments that the electron must also possess a real electric moment. Dirac has deduced, from his linear wave equation, a relation which indicates formally the existence of an imaginary electric moment as well as the real magnetic moment. It is shown that Frenkel's argument is not sound, since we now know that a relativistic state of affairs may not admit of tensorial representation, and that Dirac's reasoning is not sufficiently precise. When it is replaced by precise reasoning the magnetic moment remains as before, but the terms representing the electric moment disappear from the result. The electric moment therefore does not exist.

It has been shown by Chadwick and Lee(1), using a high-pressure ionization chamber, that if one neutrino is emitted by each disintegrating Ra E nucleus, then the neutrinos do not produce more than one pair of ions in 150 km. of air at N.T.P. Calculations based on the wave mechanics show that the ionization due to a neutrino having a magnetic moment of one Bohr magneton would be very easily detectable(2), whereas it has peen estimated that if the neutrino has no magnetic moment at all its encounters with nuclei will be as scarce as one in 1016 km. of water(3). I have investigated the matter again, using two Geiger-Müller counters, instead of an ionization chamber. The counters have the advantage of giving a discharge for a single pair of ions(4). The two counters of 15×5cm. were connected in parallel, and filled with air at 76mm. pressure. They were shielded on all sides by 45 mm. of lead. A source of 7 mg. of Ra (D, E, F) was placed at a distance of 11 cm. from the centres of the two counters, so that the total solid angle subtended by the counters at the source was 0·10, expressed as a fraction of the complete sphere. Assuming that both the disintegrating atoms of Ra D and Ra E give one neutrino, then the number of neutrinos crossing the counters per minute is 3·1 × 109. The average effective length of the path in the counter, reduced to N.T.P., is 0·3 cm., so in one min. the total range in air of all the neutrinos crossing the counters is 9·3 × 108 cm.

The number of ions produced by a neutral particle of small mass with a magnetic moment equal to n Bohr magnetons has been calculated (I–III) and found to be 103n2 per km. path in air at N.T.P., being practically independent of the mass and energy of the particle (IV). A comparatively large fraction of the secondary electrons produced have high energy and may therefore be counted if they are produced in the wall of the Geiger counter instead of in the counter itself. This fact should facilitate the experimental detection of the ionization considerably (V).

Bell and Wolfenden have published a theory of diplogen concentration by electrolysis, which, to a considerable extent, explained the empirical result, that the efficiency of diplogen separation is constant. This theory depends on the assumption that a quantity, γ, introduced by Gurney is constant. In a later paper Bell writes that this assumption would “probably not be justified in a strict examination of the problem”, but he gives no quantitative estimate.

Experiments on the β-ray spectrum of thorium C + C″ by the Wilson cloud chamber method were made. From a certain number of photographs taken, the upper limit of the continuous spectrum was found at 9250 Hρ and no β-particles of higher energy were observed. This value for the upper limit agrees fairly well with the result obtained by Henderson's recent experiments by the method of Geiger-Müller counter and focusing magnet.

Experiments are described for measuring the lateral deviation of wireless waves after reflection from the E and F regions of the ionosphere. It was found that the greatest lateral deviation observed, 20° or more, was that due to the e region, and the least, about 0·5°, was due to the normal E region in the case of a distant transmitter.

The time variation of amplitude of a reflected wave was found to be consistent with a random scattering at the ionosphere.

In the theoretical discussion it is shown that changing horizontal irregularities, ion clouds, are a very important cause of fading. Values are calculated for the average fading periods which would result from the horizontal winds in the neighbourhood of the E region known to exist from other evidence. These calculated periods agree with the observed and it is inferred that horizontal winds are a very important cause of fading.

The discovery of Fermi and his collaborators, that neutrons are much more readily captured by various atoms, e.g. silver, when their kinetic energies, originally of the order of 106 e.v., have been reduced by collisions with hydrogen nuclei, has been confirmed.

The process of slowing down of the neutrons has been studied in some detail, and the nuclear cross-section for collisions with hydrogen nuclei has been determined for the primary neutrons and for the slow neutrons which are readily captured by silver atoms.

The nature of the “gas” of slow neutrons was also discussed, but it was not found possible to reach a definite conclusion as to the energy of the neutrons concerned.