The derivation given by Hoyle and Lyttleton for an accretion formula proposed by them is examined. A number of arguments against its validity are put forward, especially that on the one hand their capture radius depends on the theorem that if the velocity of certain masses of gas after collision is less than the velocity of escape at the point, they will not in fact escape, while on the other hand it is clear (and is now admitted) that the gas cannot in fact move with this velocity at all. It is also shown that since, ex hypothesi, the individual molecules will all, on the average, retain their hyperbolic velocities, there is not the compelling reason for their capture that there appeared to be in Hoyle and Lyttleton's argument, where only the mean radial velocity of the centre of gravity of the mass was considered. Further, it seems improbable that the temperature of the interstellar matter can be low enough for the initial assumptions of their theory to hold.

In a recent paper, Fröhlich, Heitler and Kahn(1) discussed the deviation from the Coulomb law for the proton on the basis of the meson theory. Instead of being considered as a point charge, the proton was assumed to be surrounded by a meson field, which interacts with an external electrostatic field. They deduced the result that the Coulomb attraction between the proton and a negative point charge goes over into a strong repulsion for distances less than one-sixth of the electronic radius. This has been found to agree with the experiment by R. C. Williams on the fine structure in the spectrum of the hydrogen atom and the interpretation of this experiment by Pasternack. Later experiments(2), however, seem to indicate that the effect is much smaller than the theory predicts.

Calculations have been made of the cross-section for ionization of the inner shells of atoms by electron impact in the cases of the K-shells of nickel, silver, mercury and of the three L-shells of silver and mercury.

The agreement with experiment is reasonably good for the K-shell ionization, but only fair in the case of the rather meagre experimental data available for the L-shell. The values obtained for the relative ionization in the K- and L-shells are in good agreement with those to be expected from experiment.

On the assumption that the potential energy of the three cubic lattices of the Bravais type consists of two terms, an attractive one proportional to r−m and a repulsive one proportional to r−n, n > m, stability conditions are expressed in the form that two functions of the number n should be monotonically increasing. These functions have been calculated numerically for n = 4 to 15, and are represented as curves with the abscissa n. The result is that the face-centred lattice is completely stable, that the body-centred lattice is unstable for large exponents in the law of force, and that the simple lattice is always unstable,—in complete agreement with the results of Part I.

The penetrating component of cosmic radiation is now believed to consist of mesons—particles with a mass about one-tenth of that of a proton—while the soft component is attributed to electrons. By collisions with atomic electrons a fast meson occasionally produces fast secondary electrons, “knock-on” electrons, thus giving rise to a certain amount of the soft component. It was first suggested by Bhabha(1) that fast electrons produced in this way may be responsible for the electron showers known to be associated with the penetrating particles. In the present note we calculate the ratio, which we denote by R, of the average number of electrons which accompany a meson due to the knock-on process, making use of the simplification which results from the fact that the rate of loss of energy by fast electrons in ionization and excitation and also the emission of soft quanta are nearly independent of their energy. In virtue of this, the total length of electron track, and therefore the magnitude of R, is only slightly affected by the cascade process of shower production, a process which causes the energy of a fast electron to be shared among a number of electrons. The difficulties associated with a treatment of the cascade process may therefore be avoided in calculating R.

The theory is given of the structure and behaviour of immobile adsorbed films on a simple quadratic lattice in which each particle is so large that it occupies more than one site and makes it impossible for adsorption to occur on sites which are neighbours to that on which it is adsorbed. The results are obtained partly by statistical methods and partly by the use of a model of the film.

The final or complete film of this type is not homogeneous and has gaps in it. The sites on the surface can be divided into two groups, the gaps occurring where regions of the surface occupied by particles adsorbed on one group and those occupied by particles on the other group meet. The number of these gaps or “boundary sites” is determined, and the possible configurations of particles in the boundary regions are discussed in detail.

The results are considered in connexion with some experiments on the adsorption of oxygen on tungsten. In this case a second layer is formed over the boundary sites and the amount that can be so adsorbed or the number of “second layer sites” to which the boundary sites give rise is obtained. It is shown that numerically the experimental results for oxygen are in better agreement with the view that the first or very stable layer is a molecular layer of the above type than with the view that it is atomic.

Expressions are obtained for the scattering of neutral mesons by protons according to the quantized theory. It is found that the scattering due to the g1 (point charge) interaction is much less than for charged mesons, but that the anomaly in the scattering due to the g2 (dipole) interaction remains. The bearing of this upon recent attempts to modify the theory by introducing nuclear particles with charge 2e or − e is discussed.

The foregoing paper by Dr Atkinson relates to the new formula for the rate of accretion of mass by a star that has recently been proposed by us. The derivation of this formula was made by means of a dynamical argument, certain physical aspects having to be neglected.

The energy density of a cubic lattice, homogeneously deformed by a force acting in the direction of one axis, is calculated, and the equilibrium conditions and the stability conditions for any arbitrary small additional deformations are derived. A special assumption is made as to the law of force between the atoms, and the numerical calculations are performed for the face-centred lattice. In this way the strain as a function of the deformation is calculated and, from the stability conditions, the tensile strength is determined. The results are not in agreement with the experimental facts, and the possible reasons for this disagreement are discussed.

In the present paper Bethe's approximation is applied to the statistical treatment of the adsorption of dipoles considering (a) the electrostatic forces alone, and (b) the electrostatic and van der Waals forces jointly. In each case formulae are obtained for the adsorption isotherm and for the variation of the heat of adsorption with the fraction of surface covered. In case (a) the heat curves are compared with those obtained by Wang who used a different approximation to take into account the electrostatic interactions due to particles adsorbed on outer sites. The curves are of the same general shape; sources of differences in them are discussed. In case (b) the results are compared with those obtained by Roberts using a different method. This treatment confirms his result that the electrostatic and van der Waals forces give contributions to the variation of the heat of adsorption which are of opposite sign and almost counterbalance one another, so that the resultant variation in the heat of adsorption is very much less than would be expected from a consideration of forces of one type only. This comparison shows further that the distribution of particles on the surface, which is taken into account in the statistical method used in this paper, has a considerable effect on the variation of the heat of adsorption.

The rate of accretion of interstellar matter by stars as proposed in a previous paper is further discussed. It is shown that this amount, while sufficient for the evolution of the majority of stars, is insufficient by a factor of the order of 10 or more to give a satisfactory description of the general evolution of massive stars and close binary systems of small mass. Consideration of the possibility of increasing the rate of accretion for such exceptional stars leads to the conclusion that this can be carried out satisfactorily only by a corresponding increase in the density of the cloud. Although we were led to this view by considering all the factors involved in accretion and showing that only a change in the density could possibly produce the required increase, it is at once clear from the accretion formula, without detailed discussion of the other quantities involved, that the density is the only factor through which effects could be introduced that do not apply to all stars quite generally. By investigating the various factors in the galaxy affecting the density, it is shown that within 100 parsecs of the galactic plane, and also in local regions, the density may rise above 10−21 g. per c.c., which gives an increase of order 100 times the normal rate for stars lying in these regions. These suggestions receive independent corroboration from investigations by Jeans relating to extra-galactic nebulae which led to average densities also of order 10−21 g. per c.c., while a further argument from geological evidence shows that the average density of material along the sun's track must be higher than 10−21 g. per c.c. It remains to be seen whether future observations will succeed in confirming these suggestions indicated by the requirements of this theory of stellar evolution.

It is shown that the usual molecular orbital treatment for the mobile electrons in unsaturated hydrocarbon molecules is not always satisfactory, since it does not yield a self-consistent field. The conditions under which the field is self-consistent are analysed, and are shown to be satisfied in many cases.

If Tαβ is the energy-tensor, then, for any vector field ξα, the equation

follows from the energy equation

Suppose first that the physical system considered is complete, i.e. that the energytensor vanishes beyond some world tube Z. Let L be a time-like world line running inside the tube. We integrate both sides of (a) over a portion of Z, and we transform an integral over a four-dimensional region into a linear integral over L. We obtain the variational equation

in which the m's are tensors characteristic of the physical system. An essential feature of the m's is that they are symmetrical in their two last superscripts. Equation (c) has to be satisfied by every field ξα, provided that the ξ's vanish, with all their derivatives, at the ends of the integration path.

In order to explain the phenomena of melting, tensile strength, etc., we have to investigate the stability of crystals for finite deformations, for which deviations from Hooke's law occur. Although these deviations are in most cases of an irreversible character, it is necessary, for a systematic study, to develop mathematical methods for treating the mechanical (reversible) case of a highly strained crystal lattice, where terms of higher order than the second in the deformation energy must be taken into account.

Ewing's original theory of hysteresis, though not admissible for metals, seems to apply to paramagnetic salts. The peculiar form of the interaction between dipoles implies that a certain fraction of them may be incapable of following small changes in the external field. The agreement with experiment is satisfactory, considering the rough nature of the theory.

In conclusion, I should like to thank Messrs E. S. Shire and H. M. Barkla for kindly discussing their results with me before publication, and also Prof. R. H. Fowler, Dr J. D. Cockcroft and Mr A. H. Wilson for helpful discussions. Also I should like to thank the Provost and Fellows of King's College for the award of a studentship, during the tenure of which this work was carried out.

It frequently happens that we wish to evaluate the total energy of the mobile, or π, electrons in an unsaturated hydrocarbon molecule. According to the method of molecular orbitals(1) this energy is the sum of individual electronic energies ∊r, and the summation is over all the occupied orbitals. The energies ∊r are the roots of the secular determinant, and, if we write

and suppose that there are n unsaturated carbon atoms, then the equation that gives the energies zr is

The critical-field curves of pure aluminium, gallium, thorium and zinc were measured by a magnetic induction method. An unusually marked supercooling effect was found in aluminium and to a much smaller extent also in gallium; some experiments on this effect are discussed. The first three of these metals were found to have “ideally pure” superconducting properties, and thorium, although belonging to the “hard” group of superconductors, had a value of dHc/dT of only 190 gauss/degree K. Only a very minute fraction of the volume of a titanium specimen was found to become superconducting at temperatures down to 1° K., although Meissner had found a titanium wire to have zero resistance below 1·8° K.; an explanation of this discrepancy is put forward. Molybdenum and uranium did not become superconducting down to 0·3 and 0·98° K. respectively, though, in the case of one uranium sample, a small fraction of the volume showed superconducting effects as in titanium.

The paper contains some ionization-height curves for an isothermal ionized atmosphere in which the mechanism of electron disappearance is attachment to oxygen atoms at a rate proportional to the product of the number of oxygen atoms and of electrons present. Chapman's formula for the rate of production of ions is used. The curves were obtained by means of a differential analyser.

The paper describes a method of integrating differential equations by means of the Mallock linear equation machine. The method, which is based on one described by Hartree, is applicable to pairs of second order simultaneous linear ordinary differential equations in two unknowns. Any required degree of accuracy in the solution can be obtained.

The interaction between an external electromagnetic field and a nuclear system can be expressed in terms of the multipole moments. The electric quadripole and the magnetic dipole moments of the deuteron have been calculated, taking into account the exchange forces as given by the meson theory. The cross-section of the photomagnetic effect of the deuteron has been calculated.

This work was carried out under the guidance of Dr Heitler and Dr Fröhlich. The writer wishes to express his sincerest thanks to them for suggesting the problem and many valuable comments. The writer is also indebted to Dr Kahn for discussions during the early stages of this work.