In three earlier papers the author has developed the theory of the operational wave equation, which was originally suggested by Professor Eddington as the most extensive generalisation possible of the linear wave equation devised by Dirac. The wave function,ψ, plays a very minor rô1e in the development of this theory and, in reality, it is introduced simply to provide an operand which shall be patient of the action of the wave operators, A1, A2, A3, A4. The object of this paper is to show that the wave function may be entirely eliminated from the theory, which then takes the form of a “ matrix mechanics,” i.e. a set of relations between matrices representing the coordinates, the momenta and the spin operators.

The intensity of stifling is estimated. Under ideal conditions the absorption should vary as the secant of the angle of incidence of the sound.

It is well known that the rotational specific heat of a diatomic gas is given by

where R is the gas constant, σ = h2/8π2AKT, h is Planck's constant, T is the absolute temperature, K is Boltzmann's constant, and A is the moment of inertia of the molecule.

The equation of state of an ionised gas has been discussed from time to time by various writers*. It is of importance chiefly in astrophysical applications. The simplest form of the problem, and the only form considered in this note, is to find the pressure in a gas of given temperature in which the numbers of atoms per unit volume at the different possible stages of ionisation are supposed known. In describing the ionisation an electron is counted as “free” when its total energy is positive. If this problem can be solved we should have a good estimate of the relative importance of the various physical factors at work, without tackling the more difficult, though more natural, problem of finding the ionisation for given temperature and pressure.

In previous communications to the Society the author has discussed the eddy current losses in the sheaths of cables supplying a polyphase alternating current system; and has given equations from which the loss could be computed in any case. The present communication deals with the particular case of the cables leading from the supply-transformers to the electrodes of a mercury vapour rectifier, and derives interest from the fact that the loss in the sheaths is found to be substantially independent of their nature.

A type of automatically interrupted triode oscillations is described which depends on the interaction between an oscillating triode circuit and. a circuit containing a non-linear resistance and a time-constant device. The theory of the circuit is developed and tested experimentally by means of oscillograms. The theory of the rise and decay of currents in a circuit containing an inductance and a non-linear resistance is dealt with in an appendix.

The characteristic relationship subsisting between anode current ia, anode potential ea, and grid potential eg in a well-evacuated thermionic triode, viz.

leads to very simple analysis of the circuit conditions when μ, is a constant and φ is a linear function. In all the usual applications of triodes, μ is very nearly a constant; in some applications, e.g. in good acoustic amplifiers, over the working range φ is very nearly linear; in others, e.g. in rectifiers, non-linearity is essential to the performance. In oscillators, and in power amplifiers used with them, both conditions are met. Where a sinoidal oscillation is desired, i.e. an alternating current as devoid of harmonics of the fundamental frequency as can be contrived, the working range must be sensibly linear. A feature of such a régime is that the major portion of the power supplied to the triode must be dissipated in heating the anode. Where higher efficiency is required, the cycle must be made to extend beyond the linear range, and harmonics are necessarily introduced.

According to the views which we expressed recently (Proc. Roy. Soc. 1929, 124 A, 322) the solid condensed, liquid condensed, expanded and vaporous types of unimolecular films on the surface of aqueous solutions are the two-dimensional analogues of the solid, liquid crystal or smectic, liquid and vapour states of bulk matter. The previously-unsuspected observation that the films show triple point phenomena supports these conclusions.

I know only one case in mathematics of a doctrine which has been accepted and developed by the most eminent men of their time, and is now perhaps accepted by men now living, which at the same time has appeared to a succession of sound writers to be fundamentally false and devoid of foundation. Yet that is quite exactly the position in respect of inverse probability. Bayes, who seems to have first attempted to apply the notion of probability, not only to effects in relation to their causes but also to causes in relation to their effects, invented a theory, and evidently doubted its soundness, for he did not publish it during his life. It was posthumously published by Price, who seems to have felt no doubt of its soundness. It and its applications must have made great headway during the next 20 years, for Laplace takes for granted in a highly generalised form what Bayes tentatively wished to postulate in a special case.

Many measurements have been made of the complete absorption curves for β-rays in various substances, and it has been found that, for homogeneous β-rays, the curve is not exponential, and, in particular, when aluminium is the absorber, the curve is linear over a considerable portion of its range (e.g. Schonland, etc.).

If a light quantum has an electric moment, we should expect its axis to coincide with the direction of the electric vector, and therefore to be perpendicular to the plane of polarisation. Now an electric doublet of moment μ on entering a uniform electric field E with its doubletaxis parallel or antiparallel to the field will have its energy changed by an amount ∓ μE. For a light quantum this is equivalent to a change in wave-length

when in the field. The method adopted to detect this change was to set up a grating with a strong electric field normal to its surface, and then have a plane polarised parallel beam of light fall on the ruled surface with the electric vector parallel to the direction of the field. The spectra formed near the normal to the grating were then examined both in the presence and absence of the field.

Experiments on the deformation of soap bubbles in an electric field show that until the bubble begins rapid vibrations there is no difference between positive and negative bubbles.

The positive bubble always bursts very shortly after beginning to vibrate even when resting upon a large supply of soap solution.

The negative bubble, however, when resting on ample soap solution, continues vibrating until a much higher field is reached and usually does not burst until some protuberance on the positive plate allows a spark to pass, which may then strike and destroy the bubble.

The values of the field for which a particular bubble begins rapid vibrations, so that in continuous light it appears to be conical, are found to be the same within experimental error for both positive and negative bubbles. The fields for bubbles of different sizes are found to vary inversely as the square root of the radius.

Photographs are given illustrating the shape of bubbles in various fields.

These experiments were carried out at the suggestion of Professor C. T. R. Wilson, and I am greatly indebted to him for his continued advice and interest throughout their progress.

The exact distribution of the simple correlation coefficient was established by Dr R. A. Fisher in 1915 (Biometrika, X, p. 507). If ρ is the true correlation in the population, and r the sample value, then the frequency element is given by

where θ = cos− (−ρr).

When a floating body performs free simple harmonic vibrations in a vertical plane it will create a field of pulsating current flow in the liquid, and the total kinetic energy of the whole moving system will correspond to that of the floating body with an enhanced mass. The total apparent mass of the body is often called the “virtual mass” and the apparent increment the “added mass,” and these terms will be used subsequently. We are concerned in this paper with finding the “added mass” of rectangular and triangular prisms floating in water. The authors' concern with this problem is related to their investigation of the natural frequency of lateral vibration of a ship's hull, and they have contributed already two papers to these Proceedings which should be associated with the present one. In their ultimate problem the body is not moving as a whole in a vertical plane but is flexing about two or more nodal points, but in the present problem the floating body is supposed rigid and all parts of it are moving vertically with the same velocity. It is believed the present investigations are applicable to the lateral vibrations of ships' hulls, but whether or not this is so the investigation is complete in itself: for example it may be used to calculate the natural frequency of a punt or pontoon which has, say, been relieved suddenly of a dead load.

For the quantitative examination of the behaviour of two-dimensional insoluble films on water surfaces the trough apparatus devised originally by Langmuir has been hitherto employed. With the aid of this apparatus it has been possible to demonstrate the existence of two-dimensional films in the solid, smectic, liquid and vaporous states, to observe the phenomenon of two-dimensional dimorphism, to study the conditions of phase equilibria and to measure the entropy changes associated with the process of two-dimensional fusion, vaporisation and sublimation. No very definite concept of molecular orientation has been obtained from the force area diagrams. Apart from the fact that the polar groups adhere to the surface and that the hydrocarbon chains in long chain compounds adhere to one another, our picture of all forms of films except possibly the solid and the highly attenuated or vaporous states is by no means clear and the various suggestions made as to the molecular architecture are somewhat problematical. Two other methods suggest themselves for gaining a further insight into molecular structure: optical methods on the lines outlined by Rayleigh and by Raman and measurements of the interphasic potential differences. While it may be possible in the future to develop the former into a method of precision, we have succeeded in rendering the latter as accurate as the method of the Langmuir trough. That interphasic potentials between air and liquid exist and are of not inconsiderable magnitude has long been known, although the first systematic investigations on changes in these potentials on the addition of capillary active materials to the liquid phase were made by Kenrick (Z.P.C. xix, 625,1896). The technique was considerably improved by Guyot (C.R. clix, 307, 1914, Ann. d. Phys. x, 2501, 1924) and by Frumkin (Z.P.C. cix, 34, 1924, cxi, 196, 1924, cxv, 485, 1925).

Several suggestions have recently been made that the course of radioactive transformation is not, as is usually believed, independent of such changes of the physical environment as may be effected on earth. To the particular locality, to the concentration of the source, and to the action of intense γ radiation have been attributed effects which their investigators have been unable to ascribe to other contingencies. In particular, Pokrowski has developed a tentative hypothesis to explain the complicated anomalies which he had already observed.