In the first part of this paper we discuss the radiation from a single charged particle moving in an arbitrary central field of force and obeying Dirac's equation. We consider the electric quadripole and magnetic dipole radiation as well as the electric dipole. We derive the selection rules for the magnetic dipole radiation and collect together for reference the corresponding selection rules for the electric dipole and quadripole radiations. In the second part we discuss the relative intensities of the various types of radiation, treating in detail the cases where the selection rules for magnetic dipole and electric quadripole are simultaneously satisfied. Finally we show that these results have an important bearing on the theory of internal conversion of γ-rays. The internal conversion of soft γ-rays occurs with such high probability that the theory is unable to account for the experimental results unless it is assumed that the radiation is largely magnetic dipole in character. On the other hand, Fisk and Taylor (loc. cit.) were unable to account for the presence of magnetic dipole radiation in appreciable amounts. We show that this is due to the fact that, of the two possible transitions (a and e of § 2) in which both magnetic dipole and electric quadripole radiation can be emitted, Fisk and Taylor considered only the second. In the case of the second, corresponding to a transition between two distinct terms, we show that Fisk and Taylor were correct in predicting a negligible amount of magnetic dipole radiation, but in the case of the first, corresponding to a transition between two levels of one multiplet term, we find that there is indeed a high percentage of magnetic dipole radiation.

1. This note contains a second proof of a remarkable theorem, announced by W. Killing and first proved by Eugenio-Elia Levi, according to which any infinitesimal group G, in the sense of Lie, contains a subgroup which is simply isomorphic to the factor group G/Γ, where Γ is the greatest integrable invariant subgroup in G§. Also the theorem is sharpened in that it is proved for groups depending on real as well as on complex parameters. In the final section it is seen that the methods used in proving this theorem can be applied to the problem of solving certain equations which are stated in terms of any semi-simple infinitesimal group of linear operators.

Some apparent anomalies in oscillograms obtained in the course of electrical impulse tests of a material used in the construction of lightning arresters have led to a fuller investigation of the behaviour of the test circuit as applied to the study of a material with a non-linear voltage-current characteristic.

This investigation has been carried out by evaluation of solutions of the equations of a simplified form of the circuit numerically and by means of the differential analyser, and by further oscillograms taken under conditions suggested by the behaviour of these solutions.

The apparent anomalies which originally gave rise to the investigation have been shown to be a real feature of the response of the circuit, and not a spurious effect due to the recording apparatus, and the study of the initial response of the circuit to the impulse leads to the interpretation of another feature of the oscillographic records which had been previously regarded as an instrumental error.

In this paper I prove some new Pflastersätze for r−dimensional sets in the n−dimensional Euclidean space Rn belonging to one and the same family, in which Lebesgue's fundamental lemma represents a “0-dimensional” case out of r + 1 possible cases of dimensions 0, 1, 2, …, r.

1. Let f(x) be a complex function belonging to LP (−∞, ∞); i.e. let f(x) be measurable, and |f(x)|p integrable, over (−∞, ∞). The function

is called the Fourier transform of f(x), if the integral on the right exists, in some sense, for almost every value of y. It is well known that, if 1 ≤ p ≤ 2, the integral (1) converges in mean, with index p′ = p/(p – l)† i.e. that

where

1. Ellis(1) has proposed energy level systems for the bodies of the Thorium series, but the values of the γ-ray intensities forming the basis of this system fail to account satisfactorily for the total energy of the γ-rays emitted by Th C and its products. This is a serious discrepancy which needs examination. Since the publication of these level systems, we have obtained new values for the intensities of the γ-rays from Th C and Th Pb. The details of these measurements are given in the second part of this paper. These γ-ray intensities are, of course, deduced from the intensities of the discrete β-groups by dividing by the appropriate internal conversion coefficient. The changes which we have made in the γ-ray intensities are due first to revised values of the β-ray intensities from Th Pb and secondly to a revised set of values of the internal conversion coefficient (2). We have not however found it necessary to make any change in Ellis and Mott's (3) allocation of lines to dipole or quadripole type. The contributions of the two factors are made clear in Table I.

It is generally agreed that instants are mathematical constructions, not physical entities. If, therefore, there are instants, they must be classes of events having certain properties. For reasons explained in Our Knowledge of the External World, pp. 116–20, an instant is most naturally defined as a group of events having the following two properties:

(1) Any two members of the group overlap in time, i.e. neither is wholly before the other.

1. The problem of calculating the polarizability of molecular hydrogen has recently been considered by a number of investigators. Steensholt and Hirschfelder use the variational method developed by Hylleras and Hassé. For ψ0, the wave function of the unperturbed molecule when no external field is present, they take either the Rosent or the Wang wave function, while the wave functions of the perturbed molecule were considered in both the one-parameter form, ψ0 [1+A(q1 + q2)] and the two-parameter form, ψ0 [1+A(q1 + q2) + B(r1q1 + r2q2)], where A and B are parameters to be varied so as to give the system a minimum energy, q1 and q2 are the coordinates of the electrons 1 and 2 in the direction of the applied field as measured from the centre of the molecule, and r1 and r2 are their respective distances from the same point. Mrowka, on the other hand, employs a method based on the usual perturbation theory. Their numerical results are given in the following table.

1. Hasse's second proof of the truth of the analogue of Riemann's hypothesis for the congruence zeta-function of an elliptic function-field over a finite field is based on the consideration of the normalized meromorphisms of such a field. The meromorphisms form a ring of characteristic 0 with a unit element and no zero divisors, and have as a subring the natural multiplications n (n = 0, ± 1, …). Two questions concerning the nature of meromorphisms were left open, first whether they are commutative, and secondly whether every meromorphism μ satisfies an algebraic equation

with rational integers n0, … not all zero. I have proved that except in the case (which is equivalent to |N−q|=2 √q, where N is the number of solutions of the Weierstrassian equation in the given finite field of q elements), both these results are true. This proof, of which I give an account in this paper, suggested to Hasse a simpler treatment of the subject, which throws still more light on the nature of meromorphisms. Consequently I only give my proof in full in the case in which the given finite field is the mod p field, and indicate briefly in § 4 how it generalizes to the more complicated case.

If we consider, by the method of small oscillations, the stability of a viscous fluid flow in which the undisturbed velocity is parallel to the axis of x and its magnitude U is a function of y only (x, y, z being rectangular Cartesian co-ordinates), and if we assume that any possible disturbance may be analysed into a number (usually infinite) of principal disturbances, each of which involves the time only through a single exponential factor, then it has been proved by Squire, by supposing the disturbance analysed also into constituents which are simple harmonic functions of x and z, and considering only a single constituent, that if instability occurs at all, it will occur for the lowest Reynolds number for a disturbance which is two-dimensional, in the x, y plane. Hence only two-dimensional disturbances need be considered. The velocity components in the disturbed motion will be denoted by (U + u, v). Since only infinitesimal disturbances are considered, all terms in the equations of motion which are quadratic in u and v are neglected. When u and v are taken to be functions of y multiplied by ei(αx−βi), the equation of continuity becomes

and the result of eliminating the pressure in the equations of motion then gives the following equation for v, where ν is the kinematic viscosity of the fluid:

The purpose of this note is to calculate the specific heat and paramagnetic susceptibility of an electron gas obeying the Fermi-Dirac statistics for all temperatures, including those temperatures for which the gas is partially degenerate. The results are applicable to the electrons in a metal, whether free or moving in a periodic field, provided only that the number of electronic states per gram atom with energy between E and E + dE can be expressed in the form

as for free electrons.

The paper deals with the problem of the effect of two-dimensional first order disturbances on the linear and parabolic flows of a viscous fluid. If the laminar flow is in the direction x and is bounded by the planes y = ±h, and if the stream function of the disturbance is assumed to be of the form , it is found that, for small Reynolds numbers R (= Uh/v), C can be developed in the power series

The first term in (1) corresponds to a disturbance in still water and is known from Rayleigh's investigation of that problem; the succeeding C's are real and can be obtained by an application of Schrödinger's perturbation theory of wave mechanics. In the case of the parabolic flow, expressions for C0 and C1 were thus derived for both the symmetrical and anti-symmetrical types of disturbance.

The imaginary part of C, as represented by in this approximation, is found to be positive. When α is of the order of unity, the value of R which makes the second term of the first is about 50 for the symmetrical disturbance and 100 for the antisymmetrical disturbance. C0, the first term in the expression for the phase velocity, is less than 1 and increases with α, which implies a group velocity greater than the phase velocity.

In the case of the linear flow it is found that the C's with even subscripts vanish, implying a vanishing phase velocity. This verifies the assumption to that effect made by Southwell and Chitty, and it also suggests that the region of convergence of (A) is limited to the range of values of R covered by these authors. Expressions are derived for C1 and C3 and are evaluated for α=1. The formulae thus obtained for the damping constant give an initial decrease with R, which is not shown in the corresponding curve of Southwell and Chitty.

An alternative method of determining C is outlined in the last section. Use is made of the orthogonal system of the characteristic functions for the vorticity of a disturbance in still water. The equation determining C is reduced to an infinite determinant in which C occurs only in the diagonal elements. When R→0, the determinant can be evaluated and the roots C are found to coincide with the first two terms in (A).

Finally it is shown that the two-dimensional laminar flow is stable for values of R less than the smaller of the quantities 2α and α3.

1. By expanding the radiation field in a series of spherical waves the complete expressions for the field emitted by an electric and magnetic 2l-pole are obtained (§§ 1, 2).

2. It is shown that the wave emitted by a (electric or magnetic) 2l-pole has an angular momentum about the z-axis Mz = mU/ν (U = total energy), where m, according to the orientation of the 2l-pole in space, can assume the values −l, −l + 1, …, + l (§ 3). The angular momentum is contained in that region of the field in which the product EH decreases as r−3.

3. By quantizing the waves it is shown that the angular momentum of a light quantum emitted by a 2l-pole behaves like the angular momentum of an electron in a central field of force without spin (commutation relations, etc.) (§ 4). The angular momentum of a single light quantum is an integral multiple of ħ.

The errors occurring in a large number of Jeffreys's phases for neutral chlorine have been determined. A discussion of the source of these errors is given, from which it appears that errors in the phases of zero order and of order unity are due to a cumulative error over the range of integration. A method of correcting for this error is obtained. This method is of easy practical application, and it reduces the error almost to zero for phases of these orders.

The error occurring in higher order phases is found to be mainly due to errors in the two π/4 terms, which take into account the exponential tails of the waves in the totally reflecting region. For phases of high order these errors in the two π/4 terms cancel each other, and consequently the total error in phases of high order is practically zero.

A correction graph for the zero-order phase is given from which the error in this phase for atoms of atomic number up to 36 can be read off. The graph is such that an extrapolation to higher atomic number would not involve much error.

1. In a recent note I showed that Langmuir's adsorption isotherm

where θ is the fraction of the surface covered by adsorbed gas, p the gas pressure in equilibrium with it, and A (T) a specified function of the temperature, can be derived as a theorem in statistical mechanics without any appeal to the mechanism of deposition and re-evaporation. Necessary and sufficient assumptions for the truth of (1) are that the atoms (or molecules) of the gas are adsorbed as wholes on to definite points of attachment on the surface of the adsorber, that each point of attachment can accommodate one and only one adsorbed atom, and that the energies of the states of any adsorbed atom are independent of the presence or absence of other adsorbed atoms on neighbouring points of attachment. Under these assumptions the explicit form of (1) is

where m is the mass of the adsorbed atom or molecule, bg(T) the partition function for its internal states in the gas phase, and vs(T) the partition function for its set of adsorbed states. These sets of states are to be so specified that the energy zero is assigned tot the lowest state of each set in constructing bg(T) and vs(T), and then X is the energy required to transfer a molecule from the lowest adsorbed state tot the lowest gas state. Quite another adsorption isotherm was shown to hold when adsorption of a molecule takes place as atoms and requires two or more points of attachment.

These experiments were performed at the Cavendish Laboratory with the high voltage apparatus of Cockcroft and Walton and may be considered as a continuation of the experiments of Cockcroft, Gilbert and Walton on the production of induced radioactivity by high velocity protons and deuterons.

A new form of the variation principle is given using the sum T of the Lagrangian L and the Hamiltonian as an action function. This new form of the variational principle enables us to find a new special action function, which conserves the chief features of Born's theory while changing some of its former results. To a given charge correspond two static solutions with central symmetry, one giving a finite, the other an infinite energy. The potential of the one (light) particle is analogous to that in Born's theory while the potential of the other resembles a potential barrier. Also, by using the new action function, the symmetry between electric and magnetic fields ceases to exist.

The various processes occurring at the surface when hydrogen is adsorbed on tungsten are considered together with the dissociation equilibrium of hydrogen in the gas phase. The form of the adsorption isotherm is deduced from the principle of detailed balancing and is in agreement with that obtained by Fowler using a statistical method. A detailed interpretation of the experimental results now available shows that either (a) measurements of the rate of removal of the adsorbed film of oxygen on tungsten do not measure the rate of evaporation of oxygen atoms or (b) it is not possible to obtain a general first approximation formula giving the rate of evaporation of adsorbed atoms in terms of the heat of desorption. The desorption of hydrogen from tungsten is discussed and it is shown that the agreement between the temperature at which the film evaporates at an appreciable rate and that deduced from a desorption formula of the type mentioned in (b) assuming that the hydrogen evaporates as atoms must at present be regarded as a coincidence.