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.

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.

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.

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 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.

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.

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.

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.

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.

The source of light used as a background is an important factor in determining the convenience and accuracy of ultra-violet absorption work, etc. If a source of light of constant intensity is available, a direct comparison method can be used and it is only necessary to calibrate the plates. If the source of light is not constant in intensity, it is necessary to divide the light into two beams and use one to check the variations of intensity while the other goes through the absorbing substance or (during calibration) the reducing sector or wedge. This latter method requires much more complicated apparatus and if the variations in the source are at all large it becomes inaccurate. In addition to being constant in intensity a good background for ultra-violet absorption spectra should possess the following qualities:

(1) Most of the energy should be emitted in the form of a continuous spectrum.

(2) It is desirable to be able to use one photograph of the whole region to be investigated. For this purpose it is necessary that the variations of intensity in different parts of the spectrum should be small enough for it to be possible to arrange the exposure so that all parts of the spectrum are within the correct exposure range, i.e. it must not be necessary to over-expose any part in order to get a strong enough intensity at another wave-length.

The hydrogen continuous spectrum possesses both these qualities and is an excellent background for the region on the short wave-length side of 3200 A.U. It may be used for longer wavelengths, but the hydrogen secondary lines are apt to prove trouble-some unless a fairly large dispersion is used.

This still is a modification of Hulett's pattern in which mercury is distilled in a current of air at greatly reduced pressure. The current of air has two uses; first it prevents bumping and so is led in beneath the surface of the boiling mercury, and secondly, it oxidizes most of the zinc, cadmium and lead which go over. The oxides float on the distillate and are easily removed by straining it. A still made according to Hulett's directions and having a boiler of half a litre capacity will distil satisfactorily about 750 grams per hour. If the rate of distillation is increased much, some of the other metals go over, and if more air is admitted to prevent this, the boiling mercury is thrown about very violently in the boiler, which breaks sooner or later.

Two alternative forms of the CO2 molecule have been suggested by various authors who have discussed the band spectrum data. The specific heat curves based on these models are considered here. It is found that neither is quite satisfactory over the whole range of temperature and we discuss the difficulties for the low temperature and high temperature portions separately. In order to get agreement for low temperatures we find it necessary to introduce a further hypothesis about the molecular model which also seems to explain one or two outstanding difficulties in interpreting the fine structure of the bands. This assumption does not make any difference at higher temperatures where we show the error in one of the curves to be of the order we should expect to be accounted for by a centrifugal stretching of the molecule.

The alkali metals, sodium, potassium and rubidium can be distilled easily in a good vacuum and obtained reasonably free from occluded gas in the following way. As potassium is now used for the absorption of mercury vapour and many experiments are being done on the others, an account of a convenient way of preparing pure specimens may be of some service to experimenters.

In the course of an investigation of the β-ray spectrum of Ra (B + C) by the ionisation method, Chadwick(1) concluded that the line spectrum is superimposed on a strong continuous background. The ionisation method, however, is not one which admits of high resolving power, and doubts have been entertained whether the continuous spectrum has in fact any real existence. In this connection the case of Ra E is one of special importance, since no line spectrum from this body has been detected by the more sensitive photographic method. Experiments have therefore been undertaken with the object of determining the distribution with velocity of the numbers of particles in the β-ray spectrum of Ra E.

The usual method of calculating the probability of a switch between stationary states by the Quantum Mechanics is really equivalent to finding the intensity of the dipole radiation. Other switches of the system not involved in this radiation may be possible but are not taken account of by the calculation. A method of calculating the probability of switches due to radiation by the quadripole moment seems to be supplied by Dirac's recent theory of the interaction between matter and radiation in his paper “The Quantum Theory of Dispersion.” Equations from this paper will be denoted by a D. The results are probably of a purely theoretical interest and the intensities found probably too faint to be observed; but the present considerations do appear to emphasise the need for including the radiation in the exact theoretical treatment of a Quantum Mechanical system.

Any two real numbers p and q will be called conjugate if

1. The complete Denjoy-Fourier integral is of the form

where

the generating function ƒ(t) being integrable in the general Denjoy sense in every finite interval, and the integrals (1·2) being interpreted as principal values or in some other suitable way.

The β-ray type of disintegration is accompanied in general by two forms of radiation, the β-rays and the γ-rays. Radium E is noteworthy in that it gives a large amount of γ-radiation that it can only be detected by careful measurements. Yet, since it differs in this way from β-ray elements such as radium B and radium C, it is important for theoretical discussion that we should obtain as much knowledge as possible about this small amount of γ-radiation.