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.

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.

Consider the pencil of [4]'s through the [3] which contains l15, l25, l34, any [4] of the pencil meets the V25 again in a conic, q.

Most of the results of this paper have been given before by Professor T. H. Havelock, who obtained them by the solution of a difficult integral equation. The method used here is, however, much easier to handle than that given by Professor Havelock.

1. The subject-matter of this communication I believe to be new, but after Lemma 1 the method is classical; Lemma 1 is itself a particular case of a theorem which I have given elsewhere, and is a straightforward extension of a well-known result.

Experiments have been conducted by Gutton, and later by Kirchner, and by Gill and Donaldson upon electrical discharges through gases under the influence of high-frequency oscillations of the order of 107 cycles per second. It was found that the peak voltages required to maintain bright luminous discharges were of the order of 100 volts even when the pressure was as low as that in a soft X-ray tube. The present paper deals with some further studies of these phenomena.

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.

The Mathieu functions of period π and 2π have recently been constructed by the help of analysis similar to that developed by Laplace, Kelvin, Darwin and Hough to find the free tides symmetrical about the axis of a rotating globe. The purpose of this note is to show that a similar construction can be carried out for the second solution of the Mathieu equation, when one solution is periodic in π or 2π, by the help of analysis similar to that used for forced tides. The construction is effected in a form suitable for numerical computation.

Suppose that ƒ1, ƒ2, ƒ3 are three quadratics in the complex variable z with complex coefficients, say

and consider the equation

the α's being real quantities. It determines two values ζ, ζ′ of z and thus two points P, P′ in the complex plane. Clearly when P is given the ratios α1: α2: α3 are in general determinate and accordingly P′ also. This is the involutory transformation I propose to discuss.

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).

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.

The success of the kinetic theory in accounting for the properties of gases, and that of the ionic lattice theory in accounting for those of crystals, have been such as to warrant the belief that the main features of the constitution of gases and crystals are now understood, and that future development will proceed mainly on the lines already laid down. For liquids, on the other hand, we have hardly the beginnings of a theory; amorphous solids are in a slightly better state. The present article does not claim to have gone far towards the construction of such a theory, but it is hoped that it will at any rate indicate what are some of the chief problems to be faced when one is constructed.

Surfaces of order n in space of n dimensions, for 3 ≤ n ≤ 9, were discussed by Del Pezzo, who showed that the prime sections of such a surface are represented on the plane by cubic curves through (9 − n) base points.

In a recent communication to the Society, the author referred to cable-sheath losses, and gave formulae for computing them in certain cases. These appertained to power cables in which were comprised a group of conductors, arranged symmetrically and encased in a single conducting sheath. In some distribution systems, however, the conductors for the several phases are encased in separate lead sheaths, which are either laid in proximity as separate cables, or grouped and comprehended in an outer sheath. The analysis previously given does not include such cases directly. Moreover, it is common practice either to lay the elementary cables with sheaths in contact, or to bond the sheaths together at the ends of suitable sections, in order to prevent differences of potential between them; and, when this is done, a circulating current flows in the circuit of the sheaths and bonds, sufficient to maintain equality of potential between the several sheaths. This current, to which reference was made in the former paper, is additional to the eddy current discussed therein, the integral of which over the cross section of the sheath is zero. It is for convenience here referred to as the “circulating current,” to distinguish it from the “eddy current,” although there is no such distinction between them as the names imply.

Briefly summing up we see that when a point source is employed the slit of the spectroscope is illuminated by light coming from one point only in the wedge, and consequently clear distinct bands will be obtained in the spectrum irrespective of which end of the wedge is towards the source, and irrespective of the angle of incidence or of the distance between the wedge and spectroscope.

With a large source, however, the slit of the spectroscope receives rays coming from many different points in the wedge. The path difference between the interfering components of each of these rays is in general different. Therefore each ray will produce a complete set of interference bands in the spectrum. These sets of bands are relatively displaced, the superposition of all causing uniform illumination.

It has been shown, however, that there is one special case in which the path difference for the interfering components of all rays is the same. In this case, each ray still produces, a complete set of bands in the spectrum but these sets are no longer relatively displaced; bright bands are superimposed on bright bands and dark bands coincide with dark bands. There is thus both an increase of intensity and definition.

It has been shown that the conditions necessary for this special case are:

(1) The thick end of the wedge is towards the source.

(2) The slit of the spectroscope is placed at a distance of

from the wedge, the distance being measured along the central ray entering the slit of the spectroscope.

The case of an air wedge has been discussed, and a possible advantage to be gained in using such for spectroscopic purposes in place of parallel plates has been mentioned.

The velocity of bimolecular reaction at a catalyst surface is often proportional to the concentration of either reactant when this is small, but at greater concentrations the rate of reaction falls off considerably, and finally the increase of concentration may actually retard the reaction. The maximum reaction velocity does not occur in general when the partial pressures of the reactants are in the ratios suggested by the chemical equation of reaction.

The fascinating chain of theorems due to Clifford and recently extended by Mr F. P. White originates in the elementary fact that the circumscribing circles of the triangles formed by four lines meet in a point. From a like simple germ, namely the fact that the centres of the above-mentioned circles lie on a circle, an infinite chain of theorems was first evolved by Pesci and may be enunciated as follows:

(i) The centres of the circumcircles of the triangles formed by four lines lie on a circle; thus from four lines we derive a point, namely the centre of the latter circle.

(ii) Five lines give five sets of four and the five derived points lie on a circle; thus from five lines we derive a point, namely the centre of the latter circle.

(iii) Six lines give six sets of five and the six derived points lie on a circle.