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.

This note is a sequel to a former one, a knowledge of which will be assumed. We first prove a theorem which is analogous to O Theorem I, and is indeed a simple consequence of it. As an easy inference from this theorem we obtain a necessary condition that the series allied with a Fourier series may be summable (C, k) for all k >p, where p is any number greater than 1. We then show that the condition thus obtained is sufficient. We finally show that the same method can be applied to Fourier series. This note thus contains a solution of the Cesàro summability problem for Fourier series and for the allied series, which is the most precise one yet obtained.

This investigation owes its origin to a remark by Professor Lennard-Jones that it should be possible to calculate the approximate force between two atomic systems at large distances apart by a fairly direct method. The inter-atomic force is known to be of the form kR−7 at a large distance R, and an attempt to obtain the value of the constant k for two hydrogen atoms was made by Wang. The method suggested to the writer of this paper by Lennard-Jones was used by London and Eisenschitz for this case, and they obtained a value ofk differing from that given by Wang. The value given by London and Eisenschitz has been confirmed by following the method given in this paper, but further applications to the calculation of the inter-atomic force at large distances were postponed as the problem of calculating the polarizability of helium provided, on Lennard-Jones' suggestion, a simpler case on which the method could be tested.

The integral

where n is a positive integer, was obtained recently by Van der Pol by operational methods.

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.

This paper is devoted chiefly to the consideration of the surface oscillations of water contained in a vessel in the shape of a circular cylinder with its axis vertical, when the motion is slightly disturbed from a uniform rotation about the axis of the vessel. The work was undertaken with the hope of finding some indication of the effect of the depth of the water in the vessel on the period of the surface waves, and for the purpose a vessel of circular cross-section was naturally chosen. It is shown that a slight change of shape does not affect the periods of the oscillations. The solution of the corresponding problem when the surface oscillations take the form of “long waves” or “tidal waves” is well known, and the present paper deals only with “short waves,” for which the horizontal velocity is not the same at all depths.

I think that some light may be thrown on a recent paper of Professor Whittaker's by giving concrete examples. The first example considered belongs to a transformation, which I gave some years ago, whereby a constant acceleration is incorporated in Einstein's form. Opportunity is taken to develop some features of the kinetic form of solution, in which all space coordinates are relative to a source, and the time of the latter is the only time appearing. This is carried as far as the treatment of energy.

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).

Let A, B, C, D, P, Q be any real constants such that

.

A well-known theorem by Minkowski states that integer values of x, y exist such that

.

This note is a sequel to a former one, a knowledge of which will be assumed. We here develop the methods of that note to give a proof of Jordan's Theorem. We write ind C (P) for 1/2π times the absolute value of the change in log (z − P) as z describes the continuous arc C. If C is a Jordan curve, ind C (P) is either 0 or 1. Further, if C is a polygonal line, the index is a continuous function of P. If C is a closed continuous curve, interior to a circle to which P is exterior, then ind C (P) = 0.

It is shown that, by use of the Thomas-Fermi atom, effective scattering cross-sections may be obtained from a single curve for all atoms (provided that the atomic number is not too small). This curve is calculated. The range of validity of such values has been obtained and Born's formula shown to be inaccurate for representing the scattering of electrons with velocities less than 400 volts except by atoms of very low atomic number. For the case of high velocities an approximate expression is obtained for the angular distribution allowing for screening by the extra-nuclear electrons.

Let f(t) be a function periodic in 2π, and absolutely integrable in the interval (0, 2π). We write

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.

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.

An electron, according to relativity quantum theory, has two different kinds of states of motion, those for which the kinetic energy is positive and those for which it is negative. Only the former, of course, can correspond to actual electrons as observed in the laboratory. The latter, however, must also have a physical meaning, since the theory predicts that transitions will take place from one kind to the other. It has recently been proposed that one should assume that nearly all the possible states of negative energy are occupied, with just one electron in each state in accordance with Pauli's exclusion principle, and that the unoccupied states or ‘holes’ in the negative-energy distribution should be regarded as protons. According to these ideas, when an electron of positive energy makes a transition into one of the unoccupied negative-energy states, we have an electron and proton disappearing simultaneously, their energy being emitted in the form of electromagnetic radiation. The object of the present paper is to calculate the frequency of occurrence of these processes of annihilation of electrons and protons.