The following theorem, first enunciated by Dr Zeeman in 1899, has attracted the attention of several writers:

If five points A, B, C, D, E in three dimensions are such that the feet of the perpendiculars from E on the faces of the tetrahedron ABCD are coplanar the relation between the five points is symmetrical.

The locus of a point the feet of the perpendiculars from which on the faces of a tetrahedron are coplanar is known to be a cubic surface having nodes at the corners: the pedal planes, by which I mean planes containing the feet of four perpendiculars, do not seem to have been much discussed. I propose to prove two theorems concerning them and make some remarks on their envelope.

The circles in a plane can be represented by the points of space of three dimensions in such a way that the points (point circles) of the plane correspond to the points of a quadric Q in space and two orthogonal circles in the plane correspond to two points conjugate with respect to Q. An ∞2 system σ of circles in the plane corresponds to a surface S in space and the reciprocal Sσ′ of circles in the plane closely and reciprocally related to the system σ. So far as I know only trivial examples of such systems σ and σ′ have hitherto been noticed and my object is to direct attention to an illustration drawn from the modern geometry of the triangle which is less simple and perhaps more interesting.

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 theorem that if two triangles be inscribed in a conic their six sides touch another conic is, of course, to be found in all the text-books; it is apparently due in the first place to Brianchon. The further remark, that if three triangles be inscribed in a conic the three conics obtained from them in pairs have a common tangent, is to be found in Taylor's Ancient and Modern Geometry of Conics; it was made independently by Wakeford.

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 paper deals with the eddy currents in thin circular cylinders of uniform conducting material, due to periodic currents in conductors lying parallel to the axis of the cylinder, or to the rotation of the cylinder in a two-dimensional field of force. The first of these problems was discussed by Mr M. B. Field in a paper entitled “Eddy current losses in three-phase cable sheaths,” read before the British Association at their Cambridge meeting in 1904. The solution proposed, however, although probably sufficient for the object, is mathematically defective, in that the field due to the current carried by the cable is assumed as the total field, the effect of the eddy-current field on the eddy currents themselves being left out of account.

Varying ideas of the mechanism of the propagation of light and of the constitution of matter have led at different times to different formulae for the dependence of the refractive index n of a medium on its density ρ, notably those of Laplace, , and of Lorenz and Lorentz, . Recently Macdonald has given a theory which seems more in accord with the modern concept of atomic structure and finds a formula for a non-magnetic medium. It has therefore seemed worth while to examine a number of experimental results to see whether this new formula gives as satisfactory agreement as that of Lorenz and Lorentz; it is found that it does in most cases, and is occasionally the better.

Before the introduction of interpolation formulae, beyond linear interpolation by proportional parts, the presentation of the numerical values of mathematical functions was much restricted, for the labour of computation and the cost of printing, to say nothing of the inconvenience of handling a bulky volume, had to be increased quite disproportionately with every increase in accuracy. A four-figure logarithm table occupies two small pages, Chambers's seven-figure table takes 150 pages, while Vega's ten figure table requires 300 pages twelve inches long.

The method described gives the differential energy distribution curve directly from readings of a galvanometer and a voltmeter. When adjusted, the arrangement is convenient in operation and very good “resolution” can be obtained. The existence of a maximum frequency of secondary electrons at a finite small energy has been established, and its position is shown to be independent of the energy of the primary beam over a fairly wide range. The effect of deposition of tungsten on a clean copper surface is studied with respect to the secondary emission properties of the surface. From these experiments some interesting conclusions concerning the mechanism of the process may be drawn.

I am very grateful to Prof. Sir Ernest Rutherford for the encouragement he has given and the interest he has taken in this work.

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.

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.

The following is an attempt to account theoretically for the observed specific heat of water vapour. The usual theory of the vibrational specific heat together with the theory of the dissociation of the H2O molecule at high temperatures is found to account satisfactorily for the observed facts. It is shown, also, that at temperatures near 100° C. there is an effect due to polymerisation.

We recently had occasion to have a right-angled quartz prism made with the optic axis of the quartz perpendicular to one of the short sides of the right-angled triangle (Fig. 1). The prism has the property that it gives two images when light enters perpendicular to one of the faces and, after internal reflexion at the hypotenuse, passes out at right angles to the other face. It was at first sight rather difficult to see why this doubling of the image occurs, since the incidence on both faces of the prism is normal and quartz is a uniaxial crystal, there can be no double refraction occurring.

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.

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.

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.

The early experiments of Lenard(1), Leithauser(2), and Becker(3) on cathode rays led to the exponential law of absorption; those of Rutherford(4), Schmidt(5) and Hahn and Meitner(6) on β-rays indicated that the rays from some radioactive bodies were absorbed exponentially and that those from others were not. It was considered by Hahn and Meitner that the exponential law of absorption was an indication of the homogeneity of the rays. It was found also that the exponential law was not disturbed by the thickness of radiating material and this led to the conclusion that the rays are not appreciably reduced in velocity on their passage through matter. The theories of McClelland(7) and Schmidt(8) attributed the absorption to the stoppage of particles in mid-career.

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.