The following is a preliminary account of experiments made in an attempt to obtain information about the relation between the energy of a positive ion and the effect produced on a surface which it bombards. The energy relations which hold in collisions between rapidly moving positive ions and the molecules of a gas or of a solid are of extreme importance in the theory of gaseous discharges at low pressures, in particular of the initiation of the self-sustained discharge.

In the light of the hypothesis that the helium atom can exist in a hydrogen-like form in which one electron is relatively far removed from the nucleus with respect to the other, a series of experiments have been carried out with a view to determine whether or not helium could be obtained in the form of a compound similar to certain known hydrides. The mass of evidence points to a positive result. A gaseous compound was shown to form when excited helium was passed over a strong source of the active deposit of radium. Compound formation was proved by counting scintillations caused by the transportation of the radioactive material from the discharge tube into a bulb containing a zinc sulphide screen. That the effect observed in the bulb was due to the presence of a gaseous radioactive helide and not to radium emanation or solid particles in suspension was definitely proved; that it was due to hydrogen impurities in the helium was shown to be very unlikely. The compound in hydrogen was more efficiently condensed by a carbon dioxide-ether cooling mixture than that in helium. The catalytic effects of tap grease and mercury vapour on the reaction were noted.

In conclusion the writer wishes to thank Professor Sir Ernest Rutherford, O.M., P.R.S., for his interest in the work; Dr J. Chadwick, F.R.S., for advice and helpful suggestions; Mr J. A. Ratcliffe, for the wireless set used in producing the electrodeless discharge; Mr Crowe, for the preparation of the radioactive sources; and Mr Brown, for help in counting scintillations.

The Statistical Mechanics which has been developed in accordance with the requirements of the new Quantum Theory is concerned with distribution laws over energy values only—over, that is, the characteristics of Schrödinger's equation. To obtain a space distribution law, even for the Classical limit, some use must be made of the characteristic functions. A formula has been suggested by Fowler, but it has not been shown that this formula gives the Classical law for gases at ordinary temperatures and pressures. In this paper we shall show that this is so, but before doing so we shall sketch the analogous method of obtaining the law, on the Classical theory.

Experiments have been performed to measure the scattering of alpha particles by surfaces on which a source of radioactive material was deposited and on surfaces in the neighbourhood of the source. It was hoped that, where a very narrow but not necessarily homogeneous beam of alpha particles was required, the effective strength of a source could be increased by including particles which are scattered out of the activated surface at glancing angles. The effect was found to be too small to be of any great practical importance.

Experiments were also performed on the scattering of alpha particles by the walls of a long glass tube down which a beam of particles was projected. This is essentially a repetition of some experiments by Lawson and Hess in 1918, whose results indicated that the particles in being scattered by the walls of the tube lost definite amounts of energy. The present experiments indicate that they can lose any fraction of their total energy.

The experiments indicate that the most probable angle of scattering in such experiments is large, and that the average particle spends at least half and probably a greater fraction of its range in the scattering material.

Particular cases of the solution of the equation of tidal-wave propagation in canals of varying section have been obtained by various writers. It is believed that the cases which we are going to consider now have not been investigated before.

There are two general methods of measuring the elastic constants of bodies; one involves a study of the static deformation produced by the appropriate kind of stress, and the other a measurement of the period of oscillation of a system of known inertia under the elastic forces.

T. Hori has recently carried out a careful study of the ultra-violet bands of the hydrogen molecule and deduced numerical values for the energy levels involved in their production. Mr R. H. Fowler pointed out to the writer the desirability of re-calculating the specific heat of H2 by using Hori's empirical values of the energy. The present note gives the result of the calculation, which is of course similar to the investigation of Kemble and van Vleck.

If A0, A1, A2, A3 are four arbitrary circles, it is well known that there are eight circles which cut them all at equal angles, and that these eight circles fall into two tetrads B0, B1, B2, B3, C0, C1, C2, C3 which are in desmic position to each other and to the tetrad of circles orthogonal to threes of A0, A1, A2, A3.

In the theory of the specific heats of gases of diatomic molecules the function

plays a well-known and important part. The rotational specific heat Crot of a diatomic molecule, which is susceptible of representation as a rigid body with two equal principal moments of inertia, without spin about the other principal axis, is given by

where R is the gram-molecular gas-constant and

the pair of equal moments of inertia being equal to A. Whether such a model is or is not an adequate representation is a matter for determination by a detailed study of the structure of the band spectrum, particularly of the nature of the normal electronic and vibrational state. It is known to be applicable to normal molecules of the halogen hydrides, to CO and other molecules which have a normal state of type 1S including (but for a certain special feature) H2.

An attempt is made in this paper to find the nature of terms that occur in the arc spectrum of antimony. The spectrum is analogous to those of the singly ionized elements of the sixth group. Prof. A. Fowler investigated the spectrum of ionized oxygen (O II). The classification was shown to be in exact accord with the “Hund-Heisenberg theory of complicated spectra” by R. H. Fowler and D. R. Hartree. Recently G. R. Toshniwal has classified the arc spectrum of bismuth.

The paper is concerned with the practical determination of the characteristic values and functions of the wave equation of Schrodinger for a non-Coulomb central field, for which the potential is given as a function of the distance r from the nucleus.

The method used is to integrate a modification of the equation outwards from initial conditions corresponding to a solution finite at r = 0, and inwards from initial conditions corresponding to a solution zero at r = ∞, with a trial value of the parameter (the energy) whose characteristic values are to be determined; the values of this parameter for which the two solutions fit at some convenient intermediate radius are the characteristic values required, and the solutions which so fit are the characteristic functions (§§ 2, 10).

Modifications of the wave equation suitable for numerical work in different parts of the range of r are given (§§ 2, 3, 5), also exact equations for the variation of a solution with a variation in the potential or of the trial value of the energy (§ 4); the use of these variation equations in preference to a complete new integration of the equation for every trial change of field or of the energy parameter avoids a great deal of numerical work.

For the range of r where the deviation from a Coulomb field is inappreciable, recurrence relations between different solutions of the wave equations which are zero at r = ∞, and correspond to terms with different values of the effective and subsidiary quantum numbers, are given and can be used to avoid carrying out the integration in each particular case (§§ 6, 7).

Formulae for the calculation of first order perturbations due to the relativity variation of mass and to the spinning electron are given (§ 8).

The method used for integrating the equations numerically is outlined (§ 9).

It is a familiar fact that an important part is played in the Analytic Theory of Numbers by Fourier series. There are, for example, applications to Gauss' sums, to the zeta functions, to lattice point problems, and to formulae for the class number of quadratic fields.

In a recent paper in these Proceedings the writer suggested the possibility of a transition from one molecular form to another in CO2. The suggestion is embodied in the equation (10) and the resulting specific heats for low temperatures given. He greatly regrets that it was not till after those results were published that he found they gave a high and altogether impossible maximum in the specific heat curve for higher temperatures before it returns to the neighbourhood of the unmodified curve Cv′.

The methods of solution of the wave equation for a central field given in the previous paper are applied to various atoms. For the core electrons, the details of the interaction of the electrons in a single nk group are neglected, but an approximate correction is made for the fact that the distributed charge of an electron does not contribute to the field acting on itself (§2).

For a given atom the object of the work is to find a field such that the solutions of the wave equation for the core electrons in this field (corrected as just mentioned for each core electron) give a distribution of charge which reproduces the field. This is called the self-consistent field, and the process of finding it is one of successive approximation (§ 3).

Approximations to the self-consistent field have been found for He (§ 4), Rb+ (§ 5), Na+, Cl− (§ 9). For He the energy parameter for the solution of the wave equation for one electron in the self-consistent field of the nucleus and the other corresponds to an ionisation potential of 24·85 volts (observed 24·6 volts); this agreement suggests that for other atoms the values of the energy parameter in the self-consistent field (corrected for each core electron) will probably give good approximations to the X-ray terms (§4).

The most extensive work has been carried out for Rb+. The distribution of charge given by the wave functions in the self-consistent field is compared with the distribution calculated by other methods (§ 6). The values of X-ray and optical terms calculated from the self-consistent field show satisfactory agreement with those observed (§ 7).

The wave mechanical analogue of the case in which on the orbital model an internal and an external orbit of the same energy are possible is discussed (§ 8).

The great increase in the speed of chemical change at the surfaces of specific substances has always been one of the most mysterious of all chemical phenomena, and the collection of evidence which will throw light on the nature of the process is research which lies at the foundation of chemical science, because it concerns the ultimate nature of chemical combination and dissociation.

To the conditions of reaction in homogeneous systems is added yet another condition, the adsorption of the reactants on the centres of activity of the surface. The surface is treated as if it were homogeneous, reaction taking place as if those centres on which the heat of activation is smallest were alone responsible for chemical change.

The general equation is worked out in terms of the rate of bombardment, and the mean lives of the molecules on the surface. It is incidentally shown that the same areas associated with the forward reaction must inevitablycatalyse the backward reaction.

The general solution is impracticable, so the special cases of irreversible synthesis and decomposition are considered and results are obtained.

The number of α-particles emitted per second by thorium (C + C′) has been determined by an ionisation method. The number obtained is 4·26 ± ·08 × 1010 α-particles per second per curie equivalent γ-ray activity when in equilibrium with radio-thorium and when measured by the γ-rays of thorium C″ through 18 mm of lead. The result agrees within 1 per cent, with that extrapolated from Shenstone and Schlundt's values.

The same apparatus was used to determine the slope of the Bragg curve over the first three centimetres of the range. The data fit the curve as given by I. Curie and Behounek within the accuracy of experiment. The curve given by G. H. Henderson falls too rapidly as it approaches the axis of ordinates.