This paper is a continuation of four others under the same title†. The paragraphs are numbered following on to those of the fourth paper of the series. In § XXIV we show that, if λ(A) is a linear functional, then there exists a resolution Eμ such that λ(A) = ∫μdr(AEμ), and if B = ∫μdEμ is bounded, then λ(A) = τ(AB)‡ for all A, where τ is the trace. This implies that τ(A) is a linear functional, and that the conjugate space ℒ, i.e. the space of the linear functionals, has a subset ℒ′ which is in (1, 1) correspondence with the original set of operators, and that in this correspondence the linear functional τ(A) is associated with the unit operator.

The following correction has to be made to my recent paper on the adsorption of dipoles. Equation (10) and the sentence immediately preceding it are to be replaced by the following paragraph:

In the polynomial expressions for f1(x) and f2(x) the coefficients of the powers of x depend on η2 and η3 which are functions of T.

The statistical theory of isotropic turbulence, initiated by Taylor (3) and extended by de Kármán and Howarth (2), has proved of value in attacking problems associated with the decay of turbulence. In its application to such hydro-dynamical problems, the theory falls into two parts, a kinematical part and a dynamical part. The kinematical aspect consists in setting up correlations between velocity components, or their derivatives, at two arbitrary points in the fluid, and reducing the form of the tensor thus obtained in accordance with the severely restrictive assumption of isotropic turbulence; the success of de Kármán and Howarth's investigations is largely attributable to their improved treatment of this purely kinematical problem. The dynamical part then consists in applying the implications of the equations of continuity and motion to the functions defining the correlation tensors, in order to obtain information concerning their functional dependence on time and on the displacement between the two points for which the correlations are computed.

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.

First-order calculations are given concerning the possibility of designing a β-particle spectrograph in which the correlation between the photoelectrons of the stronger “natural β-ray lines” and the disintegration particles of the “continuous spectrum” might be investigated. It has been shown that, with a coincidence counting circuit of resolving time 10−6 sec., and with a source of suitable strength, this investigation should be possible in the case of any natural β-ray line the absolute intensity of which is sensibly greater than 0.01. A successful investigation of this kind would provide important data for the analysis of a complex continuous spectrum into the “partial spectra” corresponding to different modes of transformation (and might furnish direct information regarding the internal conversion coefficients appropriate to the stronger γ-radiations involved).

1. The determination of an explicit formula for the postulation of a multiple variety of given characters for primals of sufficiently large order is one which seems to have received attention only in special cases. The postulation of a multiple curve for surfaces in ordinary space is known*, and Roth† has obtained the postulation of a multiple surface in [4], free from singularities, for primals in that space. Apart from these results, and the trivial case of isolated multiple points, the only general results seem to be those of Torelli‡ and Giambelli§ for the case in which the multiple variety is a Vh which is the complete intersection of r − h primals in [r].

The elastic scattering of Yukawa's particles, or mesons, by a static electric field is recalculated by considering the scattering as a process in which the incident meson is absorbed and another meson, i.e. the scattered meson, is created, and Laporte's result is confirmed. This method promises an easier handling of the second approximation.

A similar dependence of the angular distribution of the emitted quanta upon the spin of the incident mesons in a process like Y− + P → N + hν is found. Thus, the angular distribution contains a dependence upon the azimuthal angle in general. For the reverse process, the angular distribution of the emitted mesons, both longitudinal and transverse, contains also a dependence upon the azimuthal angle, which disappears only when the incident light is circularly polarized.

The writer wishes to thank Prof. W. Pauli for calling his attention to the azimuthal dependence in the problem of elastic scattering and Dr C. Møller for help given during the preparation of the paper.

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.

When a diode thermionic tube, having cylindrical symmetry, is placed in a magnetic field parallel to its axis it is commonly called a magnetron. If there is a given potential difference between the anode and cathode of the tube, and if the magnetic field is steadily increased, a sharp and pronounced decrease of anode current occurs when the field reaches a certain value. It is easy to show that, if electrons leave without velocity from a cathode of radius b, they will just graze a concentric anode of radius a at potential V when the magnetic field H has the value given by

It is known † that, if G is any p-group (group whose order is a power of the prime p) the class of which is less than p, then the order of a product of elements of G cannot exceed the orders of all the factors. This result in general ceases to hold when the class of G exceeds p − 1. In the present paper, we prove the following general results.

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.