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.

In a previous paper (2) by one of the authors some types of periodic potential functions were obtained and were applied to problems in which the boundaries were cylinders or spheres arranged at regular intervals in a row or ring. It is now our object to extend the methods of that paper to a number of other cases.

In a recent paper Dirac has shown that by passing from the ordinary Euclidean space to a four-dimensional conformal space, some of the equations of physics can be written in a tensor form, the indices of which take on six values. Those equations which can be written in this form are then invariant under conformal transformations of the Euclidean space. Among the equations of physics which have this more general invariance are the Maxwell equations, as was proved by a direct transformation a long time ago by Cunningham, and Bateman, so that Dirac's paper provides an alternative and more general proof of this result. Certain errors ∥ in Dirac's paper, however, necessitate a reformulation of the proof. Before we do this in § 2, we briefly recapitulate in § 1 some of the general results derived there. In § 3 we investigate further the conformal invariance of the wave equation for an electron in the presence of a general electromagnetic field.

The modified field equations proposed by the author can be derived from the energy density

(where is the electric displacement, the magnetic induction, the Poynting vector) with the help of the commutation laws for these vectors.

The disintegration of nitrogen by slow neutrons has been studied in photographic emulsions of different sensitivity, which enable an unambiguous distinction to be made between the emission of α-particles and protons. Evidence has been obtained that the disintegration takes place according to the reaction

with a cross-section of about 10−24 cm.2

The steady state theory of sound propagation in exponential loud-speaker horns is well known. The problem of transients, however, has not been treated hitherto, so in this paper it is proposed to consider what happens when the air particles at the throat of the horn are suddenly set in motion. To avoid difficulties which arise in treating sound propagation in the horn when the linear dimensions of the cross-section are comparable with the wave-length of the sound, the following

artifice is postulated. The axis of the horn is linear and the air column is divided up into a large number of frictionless conduits by very thin rigid partitions as indicated in Fig. 1. The cross-section of each conduit expands exponentially from the throat towards the mouth of the horn. The major linear dimension of the section of a conduit at the mouth is small in comparison with the wave-length of the highest frequency to be reproduced.

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.

In a previous paper (afterwards referred to as Paper I) tests have been given for the significance of some quantities found statistically. The results are given in the form P(q | θh)/P (˜ q|θh); here h denotes the previous knowledge and θ the experimental evidence used, while q is the hypothesis that all the variations outstanding can be attributed to accidental error or random variation, and ˜q the hypothesis that at least part of them is systematic. It has been supposed in the analysis that q and ˜q are equally probable on the information h; but if they are not, the only alteration is that the ratios evaluated now represent

If successive batches of relevant information are available the total effect on the probability of q can therefore be got by multiplying the values of

given by the investigations separately. In each case the assumption that q has prior probability ½ is really a practical working rule rather than a statement of fact.

A repetition has been made, with the introduction of electrical counting methods, of the experiment of Bothe and Geiger on the simultaneous production of a scattered quantum and of a recoil electron in the scattering of X-rays. Evidence has been obtained that such coincidences do actually occur, in agreement with the results of the original experiment, and with the Compton theory of scattering.

The paper first scrutinizes thoroughly the variety of compositions which lead to the same quantum-mechanical mixture (as opposed to state or pure state). With respect to a given mixture every state has a definite probability (or mixing fraction) between 0 and 1 (including the limits), which is calculated from the mixtures Statistical Operator and the wave function of the state in question.

A well-known example of mixtures occurs when a system consists of two separated parts. If the wave function of the whole system is known, either part is in the situation of a mixture, which is decomposed into definite constituents by a definite measuring programme to be carried out on the other part. All the conceivable decompositions (into linearly independent constituents) of the first system are just realized by all the possible measuring programmes that can be carried out on the second one. In general every state of the first system can be given a finite chance by a suitable choice of the programme.

It is suggested that these conclusions, unavoidable within the present theory but repugnant to some physicists including the author, are caused by applying non-relativistic quantum mechanics beyond its legitimate range. An alternative possibility is indicated.

It is well known that the eigen-functions corresponding to the free electrons in a one-dimensional lattice are of the form

where C (x) is a periodic function with period c (the lattice constant), and Kis a continuous variable.

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.

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.

A source of ions consisting of a low voltage arc, yielding homogeneous ion currents of the order of ½ milliampere at low potentials is described. With such a source, atomic disintegrations have been observed with ion beam energies of less than 8kV. Approximate values of the absolute yield for protons from the deuteron on deuterium disintegration, and α-particles from the protons on lithium disintegration are obtained.

Second order focusing can be obtained with an apparatus, which contains a circular magnetic field and differs little in its dimensions from the apparatus at present in use. The lay-out is shown to scale in Fig. 4: the essential data are collected at the end of Section V, and indications of the accuracy to be expected are given at the end of Section III.

1. As is well known, the electrical resistance of a metal is very greatly in-creased by the addition of a second metal with which it forms a solid solution. The increase Δρ in the resistivity due to the addition of a small percentage of the second metal is in general independent of the temperature (Matthiessen's rule), though there are oertain exceptions (e.g. Cr in Au). The quaritum-mechanical explanation of both these facts was first given by Nordheim, and may be expressed as follows: the electrical conductivity of any metal may be written in the form

where τ is the “time of relaxation”, equal to half the time between collisions, and N is the effective number of free electrons per unit volume: hence, for the resistivity, we have

The method of determining the focal length f of a thin converging lens by finding the minimum distance Δ0 between image and object is well known. When the thickness of the lens is negligible, f = ¼Δ0.

In the general coaxial system, the media at the two ends have refractive indices μ1, μ2, and there is a finite distance t between the principal or unite plances. The two focal lengths are unequal, since, f1/f2=μ1/μ2, and the nodal points do not coincide with the principal points.

The dissociation of hydrogen by a hot tungsten filament has been studied under conditions such that all the atomic hydrogen produced is effectively removed by reaction with molybdenum or tungsten oxide. The rate of production of atomic hydrogen is many times greater than was inferred from earlier work. With the tungsten at constant temperature the rate of dissociation is proportional to the square root of the pressure. A formula is given for the rate of production of hydrogen atoms per sq. cm. of the tungsten per second.

The intensities of X-ray spectral lines resulting from transitions between the K, L and M shells have been calculated using screened relativistic wave functions. The relativistic modifications are shown to decrease the Lβ3β4 doublet ratio but do not alter the Kα1α2 and Kβ1β3 ratios, in agreement with experiment. The calculated intensities of the forbidden lines arising from quadrupole transitions are in good agreement with experiment. Lines due to transitions arising from the magnetic dipole moment of the electron are found to be very weak, but the LI → K transition might just be observable for heavy elements.

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.