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.

Generalities. Suppose that

is real and L2(0, ∞), that

is its Mellin transform, and that

Then K(1)(x) generates a “Watson transformation”, with a “Fourier theorem”

valid for every F(x) of L2(0, ∞).

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.

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.

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.

When γ-radiation passes through a gas it ionizes by means of fast electrons which produce clusters of secondary ionization at intervals along their paths. At high pressures a considerable amount of recombination of ions takes place in these clusters; in the present paper a theory is described which enables the proportion of ions escaping recombination to be calculated as a function of the gas pressure and collecting field. A review of the available experimental data concerning the variation of ionization current with pressure and collecting field is given, and it is shown that the predictions of the cluster recombination theory are in satisfactory agreement with these experimental data.

Jaffé has given a theory of initial recombination valid for a columnar distribution of ionization, and has shown that this theory is in agreement with experiments in which α-particles produce the ionization. A number of authors have applied Jaffé's equations to ionization produced by fast electrons regardless of the initial localization of the ions in clusters. It is shown in the present paper that such measure of agreement with experiment as is obtained by this procedure is only obtained at the expense of assigning incorrect values to certain known constants. Also in the case of X- and γ-rays the columnar theory predicts a variation of the proportion of recombination with the wave-length of the radiation which is much too rapid. It is further shown that the method based on Jaffé's equations which is used by Clay and his colleagues to extrapolate experimental ionization currents at finite collecting fields to saturation currents at infinite fields is liable to systematic error in a direction likely to lead to the deduction of a spurious wall effect or the exaggeration of an existing wall effect.

1. The description “approximate”, as applied to properties of a measurable function at a point, has come to mean, roughly speaking, “with the neglect of sets of measure zero”. If a function has, at a point x0, a certain property, such as differentiability or continuity, then it has the same property in the approximate sense, but not conversely.

It is well known that a single neutron may cause a nuclear reaction chain of considerable magnitude, if it moves in a medium in which the number of secondary neutrons which are produced by neutron impact is, on the average, greater than the number of absorbed neutrons. From recent experiments it would appear that this condition might be satisfied in the case of uranium.

Difficulties associated with the evolution of stars by radiation alone are briefly discussed. It is clear that some other process is also affecting the stars and it is shown that the stars are capable of adding to their mass by the process of accretion of the cosmical cloud. The gravitation of a moving star causes additional collisions of the atoms of the interstellar matter and the motions become randomized to such an extent that the star probably captures all material passing within the distance at which the velocity of the star relative to the cloud is the parabolic velocity. This rate of accretion of mass of a star is 4πγ2ρM2/ν3, and is accordingly of great importance for stars of low velocity. Stars of high velocity are least affected by accretion and therefore in general remain of low mass, while stars of low velocity must attain great mass. The periods of time involved in bringing about appreciable changes in the mass of a star are of the order of 5 × 1010 years and are in agreement with independent estimates of the time scale, as deduced, for example, from the companion of Sirius. The evolution of the components and orbits of binary stars are consequences of the accretion process. The more massive component increases in mass more rapidly than the less massive component in the case of wide pairs, and may therefore in general continue to emit more ergs per gram. The orbit evolves in such a way that the total angular momentum remains constant. For equal masses the separation is proportional to the inverse cube of the mass, and the period to the inverse fifth power, so that great changes of separation and period occur. The evolution of the stars is governed almost entirely by their velocities relative to the cosmical cloud. In the case of double stars the evolution takes the form of decreasing period and decreasing separation. Such features as galactic concentration and the correlation between spectral type and velocity are direct results of accretion.

Nunn May (1) has developed the theory of the mode of action of the Geiger-Müller counter in terms of the photoemission produced in the electron avalanches constituting the discharge of the tube. An avalanche of electrons in the counter produces excitation and ionization of atoms and molecules in the neighbourhood of the positive central wire: the photons liberated by these in their turn release from the walls of the counter the photoelectrons which initate the following avalanche. Direct experimental evidence that such a sequence of events may occur is to be found in the work of Greiner (2).

The exchange forces between two heavy particles (neutrons, protons) have been treated on the basis of the meson theory by Yukawa, Sakata and Taketani (5), and by Fröhilch, Heitler and Kemmer (2), assuming the existence of positive and negative mesons only. The calculations have been extended by Kemmer (3) for a meson field which includes in addition neutral particles (neutretto's). The first order approximation of the exchange potential, i.e. the potential due to the emission and subsequent absorption of one meson only, gives qualitatively the correct interaction between the unclear particles.

This paper is a continuation of three others under the same title. The paragraphs are numbered following on to those of the third paper.

It has been observed that the Behrens and Fisher test of the difference of the means of two samples gives a smaller percentage of significant results than might be expected on the analogy of the ordinary t test with a pooled estimate of variance. The cause of this apparent anomaly is explained, and it is shown that the criticisms of the test to which the anomaly has given rise have their origin in (a) neglect of the relevant information provided by the estimated values of the variances, and (b) an insufficient appreciation of the fiducial basis of all tests of significance (including the ordinary t test) on small samples.

It is pointed out that Sukhatme's table (constructed for the Behrens and Fisher test) also provides a test for the weighted mean of the means of two sets of observations, concerning whose relative accuracy no prior knowledge is available.

Although the existence of an electromotive force in apparently homogeneous circuits under asymmetrical temperature distributions has been questioned by only a minority of workers in this field, all attempts at explaining the effect have been little more than speculations. These speculations may be divided roughly into two types, first(2), those which agree with the contention of Benedicks that the E.M.F. is an essentially new effect not explicable in terms of the classical theory of Kelvin, and secondly(3), those which contend that the effect is essentially spurious, to be explained away as due either to hidden sources of error, or to unsuspected heterogeneities in the circuit. The somewhat fantastic claim of Benedicks' school that the effect was an inverse of the Thomson effect, which proved entirely false upon examination in the light of Sommerfeld's work(4), did much to increase scepticism. Benedicks' later claim to have found yet another inverse of his original effect(5), the “electro-thermal” effect, has also been refuted(6), and this does not inspire confidence in the genuine nature of the original effect.

1. Let .

When r is a positive integer, various writers have considered sums of the form

where ω1 and ω2 are two positive numbers whose ratio θ = ω1/ω2 is irrational and ξ is a real number satisfying 0 ≤ ξ < ω1. In particular, Hardy and Littlewood (2,3,4), Ostrowski(9), Hecke(6), Behnke(1), and Khintchine(7) have given best possible approximations for sums of this type for various classes of irrational numbers. Most writers have confined themselves to the case r = 1, in which