The method of coincidence counting has been applied to an investigation of the possible production of positron-electron pairs by the high energy β-particles from a source of uranium X in absorbers of aluminium, brass and lead. The results are not inconsistent with the high values recently reported for atomic cross-sections for the effect, nor with the suggestion that the atomic cross-section is proportional to the first power of the atomic number rather than the second. Suggestions are made for the use of the β-particles from artificially radioactive substances in an attempt to increase the sensitivity of the method.

It is shown that slight departures from independence and equal weight of the sample observations introduce a bias into Fisher's z, but that the distributions of the variances “within and between groups”, and of Fisher's z would appear to maintain their form to the first order, although the extent to which the approximations hold in practice has not been considered.

The wave equation for the deuteron in its ground state is solved on the assumption that the mutual potential energy of a neutron and a proton is of the form r−1e−λr. The binding energy of the hydrogen isotope H3 is calculated approximately by the variation method.

The present author recently published what he believed to be a proof and extension of Campbell's theorem on fluctuations. More recently a paper has been published containing a criticism of this proof. It therefore seems desirable that some parts of the proof be amplified. Further explanation of the foundations of the theory seems required, in order that it shall not appear that the theory ignores the randomness of the events, and a more detailed account of the derivation of the mean value of functions of pairs of events (i.e. sums of the type ) is necessary.

In the last chapter of his Introduction to statistical mechanics Gibbs introduces the idea of the grand canonical ensemble. He had previously determined the properties of an assembly or a phase containing a given number of systems by averaging the properties of the assembly over an ensemble of examples canonically distributed in phase, keeping the number of systems in the assembly fixed. This means of course constructing what we now call the partition function for the assembly by summing or integrating e−E/kT over the whole of the accessible phase space.

The recent experiments of Mohr and Pringle on the anomalous scattering of α-particles in hydrogen, deuterium and helium have revealed the existence of appreciable deviations from Coulomb's law at quite large distances (of the order 10−12 cm.). One explanation of this result which immediately suggests itself is that these deviations arise from ordinary electrostatic polarization of the colliding nuclei due to their electric charges, an effect considered originally by Debye and Hardmeier by classical methods to explain anomalous scattering. It is fortunately easy to examine whether this is the correct explanation. If we assume polarization to be the main cause of long-range anomalies in nuclear interaction, it is clear that these anomalies must fall off inversely as the fifth power of the distance. From the observed scattering of α-particles in helium which have √20 units of angular momentum relative to the struck nuclei and so pass at distances where the long-range anomalies are alone important, it is possible to determine the constant in the inverse fifth power law. From this constant the polarizability of the α-particle necessary to give rise to the observed effects can be determined. Now although it is not possible to calculate the polarizability of an α-particle with any degree of accuracy, our knowledge of its structure not being sufficiently definite, we can perform the calculation for the deuteron with some accuracy. Since the a-particle is certainly a more compact structure than the deuteron there is no doubt that it has a much smaller polarizability. Comparison of polarizabilities derived from the scattering and calculated from the known structure of the deuteron can therefore decide for or against this explanation of the anomalies.

The purpose of this note is to draw attention to a certain correspondence between the melting-points of normal paraffins and of fatty acids and to indicate a simple interpretation of this phenomenon. If the number of carbon atoms in a normal paraffin is plotted against the corresponding melting temperature, all the points in the diagram lie very close to a smooth curve drawn among them. If a similar diagram is made for the fatty acids, the points corresponding to an even number of carbon atoms lie on a curve of the same character as that found for paraffins. The points for an odd number of carbon atoms lie on a separate but similar curve, exemplifying the well-known alternation property.

A scheme is presented to show the limiting behaviour of the electronic energy levels in the diatomic one-electron problem for unequal centres, as the ions are separated. This generalizes the well known order of levels in the hydrogen molecule ion. Certain implications of the result for the properties of asymmetric diatomic molecules are briefly discussed.

The methods used to measure separately the electronic and lattice heat conductivities κe and κg in a metal are reviewed, and it is pointed out that care is necessary in interpreting the results in view of the underlying assumptions. The equations given by Wilson for κe and for the electrical conductivity σ are used to plot the theoretical values of the electronic Lorentz ratio Le = κe/σT as a function of T, both for the monovalent metals and for a model metal with 1·8 × 10−2 conduction electrons per atom, which is taken to represent bismuth sufficiently accurately for this purpose. Curves for κe and κg as functions of T are given in both cases, and these, together with a comparison of the observed Lorentz ratio and Le, show that in the monovalent metals κg is unimportant at any temperature, but in bismuth it plays a major part at low temperatures, in agreement with experimental conclusions. Quantitatively the agreement is good for copper and, as far as the calculations go, reasonable for bismuth.

Scattering of lattice waves at the boundaries of single crystals (including insulators) at temperatures of a few degrees absolute is shown to be consistent with the experiments of de Haas and Biermasz on KCl and to be responsible for the rise in thermal resistance in this region as suggested by Peierls.

The assumption in the theory of electronic heat conduction that the lattice energy distribution function has its thermal equilibrium value is examined in an appendix. The conclusion is that it should be satisfactory, though the proof of this given by Bethe is seen to be inadequate.

Considerable progress has been made recently in the study of the adsorption of gaseous molecules in a monolayer on a solid surface, when the molecules are attached to fixed locations on that surface and allowance is made for interaction between the adsorbed molecules. It has been shown in particular that, if there is an attractive field between two adsorbed molecules so that two adsorbed molecules in neighbouring locations are more tightly bound than when they are adsorbed separately at a distance, the adsorption isotherm shows critical phenomena. This feature of the isotherm may be used to give a generally successful interpretation of the well-known critical condensation phenomena discussed first in this manner by Langmuir and Frenkel. A recent example of experiments on this phenomenon is to be found in the work of Cockcroft on the deposition of cadmium on copper. The work of Peierls, who uses Bethe's method for approximating to the partition function for the adsorbed layer, shows in the most convincing way how interactions between neighbours in the single layer lead to critical phenomena for the degree of completion of the layer.

In a previous publication (1) the author described experiments in which the “overshoot phenomenon” reported by Silsbee, Scott and Brickwedde (2) might have been expected but was not observed. The specimens used were straight wires mounted so as to be completely surrounded by the liquid helium bath. Silsbee and his co-workers used wires coiled on paper or glass cylinders. It was concluded that the “overshoot phenomenon” was due to this method of mounting the specimen and was therefore a secondary effect.

The rates of condensation and evaporation of adsorbed particles when they interact with long-range forces are deduced from kinetic considerations, including also the case of adsorption of diatomic molecules with dissociation. The rate of evaporation is compared with experiment, and the variation of the dipole moment with θ required to make the theory agree with experiment is discussed.

Finally the formulae for the rates of condensation and evaporation are applied to the problem of the diffusion of gases through metals, and it is found that for the processes considered the effect of interaction does not alter the conclusion arrived at in an earlier paper concerning the diffusion equation at small p and at large p.

This note is a sequel to two in earlier volumes of the Proceedings, the first by myself and the second by Wilton†.

Suppose that

for |z| <1; that x > 0; that

forr ≥ 0; and that

where τ(x) is to mean 0 if x is not an integer. Thus

where the dash shows that the last term is to be halved when x is an integer;

forr > 1; and

In many radio investigations of the ionosphere it is desirable to use senders working on different wave-lengths and situated in different positions with respect to the receiver. For this purpose it is very convenient to make use of commercial senders which work more or less continuously at high power. It so happens, however, that these senders usually emit Morse signals consisting of dots and dashes of variable lengths and with variable intervals between them, so that an instrument which records the mean signal will give an indication which depends on the speed of sending and on the spacing and length of the signs. In order to obtain a reading which depends only on the signal amplitude, it is possible to employ a quick moving recorder such as a string galvanometer or a cathode ray oscillograph, which will follow the individual Morse signs. With both these types of recorder the record must be made photographically so that changes in the signal strength are not noticed until the record is developed. This is a serious disadvantage in investigations of short-lived disturbances of the “catastrophic” type.

When the approximate methods of Bragg and Williams or Bethe are used in theoretical work on cooperative phenomena it is customary to suppose that only the configurational energy of the assembly depends on the molecular arrangement; the contribution which a molecule makes to the partition function, apart from its share in this configurational energy, is supposed to be independent of the positions of the other molecules.

The theory of heavy electrons recently developed by several authors may be considered to give a satisfactory account of the empirically known neutron-proton interaction. However, it now seems well established that there exists a proton-proton interaction of comparable magnitude which is not accounted for equally well. Owing to the fact that the emission of a heavy electron involves the change of a neutron into a proton or vice versa, the first approximation of this theory gives only an exchange force between unlike particles, whereas a force between like particles must be due to double transitions and thus only appears in the second approximation. It is true that the expansion in terms of the number of particles emitted is actually so badly convergent that the second order proton-proton force at distances of about 10−13 cm. is found to be not essentially smaller than the first order neutron-proton force (see FHK), but nevertheless this does not explain experimental facts, since the calculated second order force is always repulsive.

Note on a Theorem of Myrberg*

On p. 421, line 30, of that note I remark “δ(x) is a lower semi-continuous function of x”. That statement is, in general, untrue. The proof should rather run as follows:

The subset of E1 where δ(x) ≥ a > 0 is clearly closed and, as α tends to 0, tends to h−mE1. Hence, by taking α sufficiently small, we find a closed subset of E1, of positive h-measure, every point x of which has the property that

whenever 0 < d ≤ α. We may call this subset again E1 and the proof, from the top of p. 422, goes as before.

The essential difference between mobile and immobile adsorbed films of gases on solids when there is repulsive interaction between adsorbed particles is discussed. Ordinary statistical methods can be applied to the equilibrium configurations of the particles in a mobile film but not to immobile films. Films formed by the adsorption of diatomic molecules with dissociation are then considered. It is pointed out that for a mobile (equilibrium) film there are differences between the true heat of adsorption and the apparent heat obtained from the temperature variation of the evaporation rate. A detailed comparison is made by obtaining a formula for the apparent heat of evaporation and comparing it with that for the true heat obtained earlier by Wang, and the physical reasons for the differences are discussed. A model suitable for studying the pseudo-equilibrium states in immobile films is described and formulae and diagrams in connexion with the kinetics of the formation of such films and the variation of heat of adsorption with fraction of surface covered are given.