The geodetic problem presented by a quadratic form is identical with the dynamical problem in which the same quadratic appears in the kinetic energy, and potential energy is not represented. If the quadratic in space-differentials is equated to kdt2, the problem is essentially the same whatever constant value is given to k; the corresponding feature is the arbitrary constant value of kinetic energy.

When a triangle ABC is in perspective with A′B′C′ and with B′C′A′, so is it also in perspective with C′A′B′, and the two triangles may then be said to be in cyclic perspective. The centres of perspective then form a third triangle XYZ, such that, of the three triangles, every two are in cyclic perspective with the vertices of the third for centres. The figure thus arising is the figure of Pappus' Theorem with nine points and nine lines incident by threes.

The figure in three dimensions consisting of two sets of three non-intersecting lines, such that each line of one set intersects each line of the other set, will here be called a “net.” All nets are projectively equivalent to each other: and there are just 72 projective transformations that change one net into another.

The problem of the scattering of radiation by a free electron has been treated by the author on the basis of Heisenberg's matrix mechanics, which was first modified to be in agreement with the principle of relativity. The main point of this modification is that, whereas in the non-relativity theory one deals with matrices whose elements vary with the time according to the law eiwt, in the relativity theory the elements of the matrices must vary according to the law eiwt′ where t′ = t − (l1x1 + l2x2 + l3x3)/c if they are to determine correctly the radiation emitted in the direction specified by the direction cosines (l1, l2, l3), x1x2 and x3 being the coordinates of the electron at the time t. These matrices were obtained by writing the Hamiltonian equation of the system in the form

where W′ is a variable canonically conjugate to t′ and H′ commutes with t′, and then using H′ as an ordinary Hamiltonian function of a dynamical, system that has W′ for its energy and t′ for its time variable.

1. Polarisation measurements are described on the Hg lines λ 4358, 5461, 5770, 5791 emitted from a low-pressure electron-maintained arc to which a longitudinal field of up to 3000 gauss may be applied.

2. Two types of polarisation effect have been found. The first is typified by the line λ 5461. It is a polarisation of the light with the electric vector perpendicular to the lines of force. Its maximum value is 9% but it decreases to zero (a) with decreasing field strength and (b) with decreasing current in the tube. It is suggested that this effect is due to a self-reversal of λ 5461 owing to the metastability of the 2 3p2 state and that the polarisation may be caused through a slight non-uniformity of the magnetic field.

The second type of effect is found for λ5770 and 5791. The polarisation is in a plane parallel to the discharge, and is found in a zero magnetic field. This effect is further discussed elsewhere. The line λ 4358 is unpolarised under all conditions tried.

3. Neither of these effects implies a polarisation of the total light emitted directly from atoms in the Transverse Zeeman Effect when the atoms are excited under isotropic conditions and the experiments give definite evidence against the existence of such a polarisation effect.

1. It has been shown that the action of alpha particles on paraffin is independent of the state of aggregation of the substance.

2. Defects in the customary procedure for the decomposition liquids and solids by the emanation tube method are discussed.

3. A lecture demonstration of the chemical effect of alpha particles is described.

There are two ways of investigating the velocity distribution of an electronic emission—the retarding potential and the magnetic deflection methods. These have been applied to the case of the photo-electrons by Millikan and Ramsauer respectively. On one important point the results disagreed; Millikan found a definite maximum velocity, while Ramsauer obtained an asymptotic falling off of the number of electrons with increasing velocities. In a critical discussion of the two methods Klemperer has shown that they agree when photoelectric activity of the collecting electrode in the first, and when electronic reflection at the walls and slits in the second, are eliminated.

The glow discharge between cold aluminium electrodes in air, oxygen, nitrogen and hydrogen has been analyzed by Langmuir's method, for pressures between 0·1 and 0·4 mm. Hg, current densities of from 0·02 to 0·2 mA./sq.cm., and applied potentials between 300 and 700 volts. An annular exploring electrode has been used. It has been found that whilst practically the whole fall of potential is localized across the cathode dark space at the lower pressures, a fall of as much as 40 volts can exist across the remainder of the discharge at the higher pressures. Reversal of the electric field has been found in the negative glow, and in certain cases in the Faraday dark space, when conditions are favourable for passage of an electron current by diffusion against the field. In several instances the negative glow was at a higher potential than the anode. Two groups of electrons occur in the negative glow, together with a single fast group at the anode boundary of the cathode dark space, and a single slow group in the Faraday dark space.

The theoretical calculation of observable atomic constants is often only possible if the effective electric field inside the atom is known. Some fields have been calculated to fit observed data but for many elements no such fields are available. In the following paper a method is given by which approximate fields can easily be determined for heavy atoms from theoretical considerations alone.

The tensile deformation of single crystals of aluminium was examined by Taylor and Elam, using test-pieces composed of one crystal only. Miss Elam kindly supplied the author with a test-piece containing three large crystals, and an investigation of the effect of the constraint at the crystal boundary was carried out upon it as described below.

A method has been devised by Dr Alex Müller for determining the orientation of a single-crystal of metal by photographic measurement of the reflection of characteristic X-rays from surface layers. The incident beam passes perpendicularly through an axis of rotation around which the crystal is turned until a reflection is obtained with one of the component wave-lengths of the X-rays.

There are two effects involved in the sucking action of the diffusion pump. One is the penetration of gas molecules into the mercury vapour jet, and the second is the existence of a gradient of the partial pressure of the gas in the direction of the jet, so that the pressure at the origin of the jet is much lower than at the end of it.

In the application of the Characteristic Function of Hamilton, or of any allied function, to the computation of a symmetrical optical system three steps are necessary. The performance of the system as a whole must be considered, and from this it appears that the aberrations may be derived from certain ‘aberration coefficients’ which occur in the expansion of an ‘aberration function’. In the second place, relations must be obtained giving the properties of the complete system in terms of the properties of the component systems, which, in general, will be single refracting surfaces; and finally, an evaluation of the coefficients must be made for the simple system—a single surface. For the first and third of these steps reference may be made elsewhere, and also for a general investigation of the second; a simple derivation is given, in the present note, of the necessary relations between a composite system and its components in the case where first order aberrations only are considered.

This investigation is to discuss some of the conditions which influence the absorption spectra of various “saturated” and “unsaturated” organic compounds. A condensed cadmium spark was used as the source of radiant energy; alcoholic solutions of the substances were employed, except for stearic acid, which was dissolved in ether. The vapours were examined in a quartz tube 200 mm. long, placed in a bath which could be heated to various temperatures.

The relation between the reaction velocity of catalytic dehydrogenation and the rate of passage of alcohol vapour over the catalyst has been found by experiment. At low rates of flow the concentration of the aldehyde and hydrogen in the effluent gas, and therefore over the catalyst, is comparable with that of the reactant. The whole of the surface is thus not covered with the reactant, and the adsorbed reaction products cause marked decrease in the reaction velocity.

When the concentration of the reaction products becomes sufficiently small, that is, when the rate of passage of the alcohol vapour exceeds some limit, the velocity is approximately independent of the rate of flow of reactant. With very high speeds the reaction velocity tends to fall again, but this is probably due to the difficulty of keeping the temperature of the catalyst constant.

The formula previously deduced connecting the partial pressure of the reactant in a mixture with the fractional reaction velocity has been verified for mixtures of alcohol vapour with water, acetone and benzene; and the composition of the adsorbed gas film with the fractional partial pressure of the gases for mixtures of carbon monoxide with oxygen on platinum.

The effect of mixture of vapours on the temperature coefficient of reaction is considered, and equations deduced to show the alteration produced. If the heat of desorption of the reactant is greater than that of the diluent, then the temperature coefficient is diminished, if they are equal there is no alteration, and if less the temperature coefficient is increased. Experiments show that water present in alcohol has little effect on the temperature coefficient of the reaction.

Expressions are found for the progress of slow selection in a Mendelian population where generations overlap. The changes are very similar to those which occur when generations are separate.