Let be any transitive permutation group on the n symbols 1, …, n. Let be the subgroup of whose elements leave i fixed. Let ′ be the normalizer of , i.e., the subgroup of the symmetric group on 1, …, n transforming into itself. Let G′, G′1, G′2, etc., denote elements of ′. Finally, let ″ be the centralizer of , i.e., the subgroup in transforming every element of into itself.

According to the Frenkel theory of adsorption, the first molecules striking a surface move about on it like a two-dimensional gas before finally being adsorbed. This surface motion has been demonstrated by Estermann and by Cockcroft in the case of vaporised cadmium.

In the spectrum formed by a grating the red is more deviated than the blue, while in that formed by a prism, which does not exhibit anomalous dispersion in the visible spectrum, the reverse is the case. If, then, white light is passed first through a grating and then through a prism, the nature of the spectrum produced will depend on the relative dispersions of the grating and prism. If throughout the spectrum the dispersion of the prism is less than that of the grating, the sequence of colours will be the same as that in the spectrum due to the grating alone, red being more deviated than blue. If, on the other hand, the dispersion of the prism is throughout the spectrum greater than that of the grating, the sequence of colours will be reversed. On account of the rapid increase in the dispersion of glass near the blue end of the spectrum, it may, however, be possible that the dispersion of the prism exceeds that of the grating in the blue while in the red the reverse is the case. Under these conditions both red and blue will be more deviated than some intermediate colour for which the dispersions of the grating and prism are equal. The spectrum will thus have a sharp edge of this colour at the end of minimum deviation, while in the direction of greater deviation it will at each point consist of two superimposed colours. It is as if an ordinary white light spectrum was folded back on itself about the sharp edge.

In the Wöhler rotating bar fatigue test, in order to examine the behaviour of a specimen under alternating stress, a load is suspended from the end of a cantilever bar, and the bar is then rotated, the load remaining stationary. It is evident that the rotation will set up additional stresses in the bar, so that the limits between which the stress at any point will alternate are not the same as the stress at the corresponding point of a statically deflected bar.

The same perturbation method is found to be valid for a gentle mutual potential between free atom and vibrator as for an infinitely steep mutual potential, provided that the reduced mass of the system is sufficiently small.

A simplified method is discussed for the calculation of transition probabilities. This method is used to calculate the effect of a van der Waals' attractive field.

1. In a previous paper (1) the authors used the theory of the moment generating function to deduce the known random sampling distributions of the estimated variance and co-variance in a system of variables following the normal law of frequency. In this paper we shall go much further. By means of the same general method the simultaneous distribution of the ½p (p + 1) second order moment statistics in a normal system of p mutually correlated variables will be deduced. It will be shown to be completely independent of that of the p sample means, and incidentally the method of proof to be developed will be found, in the special cases p = 1 and 2, to be an improvement on that given in the earlier paper, being independent of previous results reached by other authors.

1. In this note we give a direct evaluation of the integral

whose value has been inferred from the theory of statistics. Here A = Ap = (αμν) and C = Cp = (Cμν) are real symmetrical matrices, of which A is positive definite; there are ½ p (p + 1) independent variables of integration tμν (1 ≤ μ ≤ ν ≤ p), and tμν is written also as tνμ for symmetry of notation; in the summation ∑ the variables μ, ν run independently from 1 to p; k is a real number. A word of explanation is necessary with regard to the determination of the power |A − iT|−k. Since A is positive definite and T real and symmetric, the roots of the equation

The partition function F(θ) is of fundamental importance in the theory of the specific heat of gases. Once it is known, the rotational specific heat of a perfect gas is given by

where R is the gram-molecular gas constant, and θ bears the relation

to the absolute temperature T, k being Boltzmann's constant.

Solutions of the bi-harmonic equation valid in the region bounded externally by parallel lines and internally by a circle midway between the lines have been given by one of the Authors in a recent paper [2]. These solutions were adapted to the requirements of certain problems in the theory of elasticity, but modified solutions satisfying the boundary conditions characteristic of viscous fluid motion are easily derived. These modified solutions will here be given and will be used to find the stream function corresponding to the slow rotation of a cylinder placed symmetrically between parallel walls.

The phenomena which limit saturation in a high pressure ionisation chamber are examined with particular reference to the rôle played by preferential recombination. The complexity of the phenomena is discussed in detail in order to bring out the scope and limitations of the simple theory of preferential recombination and the conditions which a more complete theory would have to satisfy. It is pointed out that in a number of recent papers on the subject fundamental aspects of the problem have been overlooked, and the significance of these omissions is discussed.

The effect of changes in magnetic moment and in the earth's field on the period of invar pendulums used for relative gravity determinations is discussed, and methods of eliminating them by the use of Mumetal screens are described.

Using a thin-walled electroscope, absorption curves in paper and aluminium were obtained for the β-rays of actinium (B + C) and actinium C″. From these curves the range of the β-rays of actinium B was found to be 0·08 ± ·02 gm./cm.2 The energy required for this range is 300,000 ± 50,000 volts, which must be the end-point of the continuous spectrum.

On account of the overlapping of the β-ray spectra of actinium B and actinium C″ the range method can never yield an accurate value of the end-point of the former. Magnetic analysis appears to be the best means of finding this accurately, but the weakness and rapid decay of sources are formidable difficulties.

The theory of the evaporation of adsorbed atoms and ions of the alkali metals from a hot metal surface has been developed from somewhat different viewpoints by Langmuir and Kingdon and by Becker. These two treatments, which yield essentially the same results, are applicable only when equilibrium conditions obtain on the surface, so that the surface concentration is not changing. It is the purpose of the present note to extend Becker's treatment to take account of the rate of attainment of equilibrium on the surface, in order to lead to an interpretation of some experiments on the rate of evaporation of the alkali metals adsorbed on hot tungsten.

It is well known that a beam of light falling on a reflecting mirror forms standing waves. This effect has been very beautifully made use of in Lippmann's colour photography process. The standing light waves, in this case, produce a periodic effect in the emulsion of the photographic plate which, when developed, scatters light and produces a similar colour effect. Instead of using a beam of light, it would seem possible to scatter electrons from the emulsion and obtain a reflection of electrons similar to that of a space grating. But it seemed to us that it would be of much greater interest to consider an experiment in which electrons are reflected from the standing waves of light. The direct scattering of free electronic waves by light has strictly never been observed, and it was thought possible that by this method, owing to the interference of the electrons and to the fact that the scattered electrons are focussed to one spot, the magnification of the phenomenon would be sufficient to make it observable. From the theory developed below, it will be seen that the experiment is just on the verge of possibility, and would be very difficult to carry out. The main interest of the experiment would come from the possibility of observing stimulated scattered radiation which up to the present has never been verified experimentally.