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

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.

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.

1. It is known, from the theory of the Riemann theta-functions, that the canonical series of a general curve of genus p has 2P−1 (2P − 1) sets which consist of p − 1. points each counted twice. Taking as projective model of the curve the canonical curve of order 2p − 2 in space of p− 1 dimensions, whose canonical series is given by the intersection of primes, we have the number of contact primes of the curve. The 28 bitangents of a plane quartic curve, the canonical curve of genus 3, have been studied in detail since the days of Plücker. The number, 120, of tritangent planes of the sextic curve of intersection of a quadric and a cubic surface, the canonical curve of genus 4, has been obtained directly by correspondence arguments by Enriques. Enriques also remarks that the general formula 2P−1 (2p −1) is a special case of the formula of de Jonquières, which was proved, by correspondence methods, by Torelli§.

We now consider some properties of “almost all” series

where the numbers vary independently in the interval (0, 1). What we need is a definition of measure in the space of sequences or, what is the same thing, in the space

of infinitely many dimensions. Such a definition has been given by Steinhaus. He defines a correspondence between the cube (7.2) and the interval (0,1) in the following way:

The integral

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

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.

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.

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

.

1. A description of a simple method of demonstrating electric lines of force around a charged body is given.

2. The instability of liquid surfaces in an electric field is used to obtain an idea of the order of magnitude of the charge developed on tourmaline on cooling through about 140° C.

3. A new method of measuring the electric moment of tourmaline is described.

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 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.

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.

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).

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 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.

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.