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. 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.

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.

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

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. 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.

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).

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.

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.

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.

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 trivial that the fundamental equations of electromagnetism are invariant under orthogonal transformations of space. This invariance can be brought into evidence by using the calculus adapted to the orthogonal group, viz. the vector-calculus. The equations can be written either in their integral form

or in their differential form

The cross-sections for the emission of radiation by very fast electrons (energy great compared with mc2) and for the production of pairs of positive and negative electrons by very hard γ-rays is calculated, considering the screening of the atomic field in which the processes happen. The main part of the paper deals with the integration of the differential cross-section over the angles. It is found that the integral cross-section increases steadily with increasing energy of the primary particle, reaching an asymptotic value for extremely high energies.

In connection with his work on singularities of surfaces, Du Val asked me to enumerate certain subgroups in the symmetry groups of the “pure Archimedean” polytopes n21 (n < 5), namely those subgroups which are generated by reflections. For the sake of completeness, I have enumerated such subgroups of all the discrete groups generated by reflections (including the symmetry groups of the regular polytopes). The work involved being somewhat intricate, several slips would have been overlooked but for the information that Du Val was able to supply from the (apparently remote) theory of surfaces.

1. In two previous papers the regularity of non-singular surfaces in higher space was investigated by geometrical methods based on theorems which are themselves deducible by algebraic-geometrical methods. Among the results obtained, one of the most important was a consequence of the following theorem, due to Severi:

The problem of the torsion of beams of ⊥- and L-cross-sections has received attention from very few authors despite its important technical applications. The first mathematical solution in this connection was obtained by F. Kötter in 1908 for an L-section both of whose arms are infinite. He attacked the problem by the use of the known solution of the rectangle and by application of the scheme of conformal transformation. Kötter's method, however, does not lend itself readily to the solution of the problem involving more than one re-entrant angle. The first solution for the torsion of a beam whose cross-section is a rectilinear polygon of n sides was published in 1921 by E. Trefftz who also applied his method to an infinite L-section. Recently I. S. Sokolnikoff has suggested a more general method depending upon the fundamental theorem of potential theory that a harmonic function is uniquely determined by the values assigned along the boundary of the region within which the harmonic function is sought, the boundary condition and the region being subject to certain well-known assumptions of continuity, connectivity, etc. As an illustration of his method he has given an approximate solution for a ⊥-section whose flange and web are both infinite.