An Archimedean solid (in three dimensions) may be defined as a polyhedron whose faces are regular polygons of two or more kinds and whose vertices are all surrounded in the same way. For example, the “great rhombicosidodecahedron” is bounded by squares, hexagons and decagons, one of each occurring at each vertex. Thus any Archimedean solid is determined by the faces which meet at one vertex, and therefore by the shape and size of the “vertex figure,” which may be defined as follows. Suppose, for simplicity, that the length of each edge of the solid is unity. The further extremities of all the edges which meet at a particular vertex lie on a sphere of unit radius, and also on the circumscribing sphere of the solid, and therefore on a circle. These points form a polygon, called the “vertex figure,” whose sides correspond to the faces at a vertex and are of length 2 cos π/n for an n-gonal face. Thus the vertex figure of the great rhombicosi-dodecahedron is a scalene triangle of sides .

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.

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.

In the “special” or “restricted” theory of relativity, for which the line-element ds of the “world” of space-time is given by , the geodesics of the world are straight lines, and the null geodesics (i.e. the geodesics for which ds vanishes) are the tracks of rays of light. When Einstein discovered the “general“ theory of relativity, in which the effects of gravitation are taken into account, he carried over this principle by analogy, and asserted its truth for gravitational fields; it was, in fact, the basis of his famous calculation of the deviation of light at the sun. The law was, however, not proved at the time: and indeed there is the obvious difficulty in proving it, that strictly speaking there are no “rays” of light—that is, electromagnetic disturbances which are filiform, or drawn out like a thread—except in the limit when the frequency of the light is infinitely great: in all other cases, diffraction causes the “ray” to spread out over a three-dimensional region.

When the inner integrals in a pair of repeated integrals are absolutely convergent, inversion of the order of integration is a not too difficult problem and there are a number of general theorems which, under comparatively unrestricted appropriate conditions, can be applied to give a solution. When, on the other hand, the inner integrals are only conditionally convergent, the problem is of a very different nature and the solution, if it exists at all, is usually only to be obtained by special devices.

Let f(z) be a single-valued function of z, defined in the domain D given by ; suppose that f(z) is integrable in the general Denjoy sense round all circles with centre ζ lying in D, and along all portions of radial lines intercepted between two such circles.

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.

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.

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.

The use of Pitot tubes in exploring fields of fluid flow is well understood but the limitations of the method have not been very systematically explored.

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.

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.

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).

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 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.

Photographs have been taken under controlled illumination of the tracks of α particles in a cloud expansion chamber and a calibration of the photographic plates employed has been carried out. Systematic photometry of the track images has made possible the calculation of the variation of the light scattering power of an α particle track over the last two centimetres of its length in standard air, and the variation of this quantity has been identified with the variation of ionisation along the track.

Photographs of tracks in air, helium and hydrogen have been examined. In these three gases the maximum ionising efficiency of the α particle occurs when it possesses the velocity respectively appropriate to the distances 3·0, 2·55 and 2·25 mms. from the end of its path in standard air. This common velocity parameter has been employed throughout the discussions which are appended to the experimental results.

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 interferometer designed by Michelson has been adapted in recent years to the testing of various types of optical systems, one form being used for the testing of prisms and another form—the “lens interferometer”—for the examination of the symmetrical optical system. The details of the instrument may be found in a paper by Mr F. Twyman, in which are given some photographs of the results of its application to certain lenses of known aberration. Essentially the interferometer ensures that a plane and undistorted wave of light falls upon the lens or system to be examined, after refraction through which it falls upon a convex mirror whose centre is at the focus of the converging wave; by which, therefore, the wave is returned through the optical system and emerges as a plane wave if the system under examination be ‘perfect,’ or free from aberration. Otherwise the final wave is not plane, but is distorted by the aberrations impressed upon it by the system under test—through which, it will be noticed, it passes twice; and this emergent wave is superposed upon a portion of the incident (plane) wave, obtained by previous passage through a partly silvered mirror; with which, therefore, interference effects are observed. And these effects depend only upon the aberrations produced by the system under examination, for they measure, in fact, the departure of the finally emergent wave from the ideal plane form. Various interference contours are obtained in this way, and their form and size enable the observer to deduce the aberrations of the system and to measure their magnitude.