The integral

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

The vertical gradient of temperature needed to produce convection currents in a layer of incompressible liquid has already been investigated in several instances. For a compressible fluid instability cannot arise until the gradient exceeds the adiabatic one; it has been assumed usually that what matters is the excess of the actual gradient over the adiabatic. Thus the excess needed in a compressible fluid is to be found by the same formula as gives the gradient needed for instability in an incompressible fluid. It is desirable, however, to investigate the validity of this assumption.

The expansion of a general function in a series of squares of Bessel functions has been considered by Neumann. As, however, there appear to be only a fairly small number of known expansions of this type, it is perhaps worth while adding to their number. I give here some new expansions of this type, and deduce some simple inequalities satisfied by Bessel coefficients of low order.

Let c0, c1,…, cn,…be a sequence of real constants, and ƒ0(x), ƒ1(x),…, ƒn(x),… a sequence of functions defined, for example, in the interval (0, 1). In this paper we shall investigate some of the properties of the series

which may be obtained from the standard series

by interchanging the signs of the terms in a quite arbitrary way.

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.

For r>-1, let denote

If

we say that the series a0 + a1 + a2 +…+an+… is summable by Cesàro mean of order r, or more shortly summable (C, r) to sum s. If r >−1, and

we say that the series is summable by Rieszian mean of order r to the sum s. It has been shown that these two methods of summation are equivalent. Throughout this paper I shall deal with the Rieszian mean, but I shall retain the symbol (C, r). It is known† that if a series is summable (C, r), it is also summable (C, r′) to the same sum for all numbers r′ greater than r.

A function of sets γ(E) absolutely additive is termed by Radonée Poussin continuous if γ(E) vanishes whenever E consists of a single point only. It is known that if a function of sets is not continuous there exists a countable set ε1 and a continuous function of sets such that

We consider in space [3] a curve C of order n and genus p without multiple points. If we represent the lines of [3] by the points of a quadric Ω in [5], the chords of C will be represented by the points of a surface F of order (n−1)2−p lying on Ω. This surface has a triple curve M (with multiple points) corresponding to the ruled surface of trisecants of C (and the quadrisecants) of order ⅓(n−1)(n−2)(n−3)−p(n−2). It is the object of this note to find the genera of M and of a prime section ϑ of F; these being also the genera of the ruled surface of trisecants of C and of the ruled surface of chords of C which belong to a linear complex.

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.

When a vessel of liquid has been emptied and put aside, a thin film of liquid clings to the inside and gradually drains down to the bottom under the action of gravity. The layer being thin, the motion is very nearly laminar flow, and the curvature of the surface in a horizontal direction may be ignored. Thus the problem for a cylindrical vessel is reducible to that of a wet plate standing vertically.

The present paper is a companion to one which appeared in these Proceedings some months ago. Since that time the characteristics of the Hyper-Octohedral group of order 2n.n! have been calculated by Dr Alfred Young, who has kindly allowed me to make use of his results.

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.

In the geometry of the cubic curve Γ in space, we have to study the quadrics Q which stand in certain special relations to the curve. We shall find it convenient to refer to these relations as A, B, B′, C, C′; these are defined below.

Mr Ursell has given a proof in these Proceedings (vol. 24, p. 445, (1928)), that there exist non-combining groups of wave functions of an assembly of similar systems. His proof was not sufficient; but it can be amended so as to be rigorous. It will be necessary to suppose, however, that the wave function ψt is analytic for all real values of the time, for each set of space coordinates: q1, q2, q3, q4….

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

.

For dealing with atoms involving many electrons the accurate quantum theory, involving a solution of the wave equation in many-dimensional space, is far too complicated to be practicable. One must therefore resort to approximate methods. The best of these is Hartree's method of the self-consistent field. Even this, however, is hardly practicable when one has to deal with very many electrons, so that one then requires a still simpler and rougher method. Such a method is provided by Thomas' atomic model, in which the electrons are regarded as forming a perfect gas satisfying the Fermi statistics and occupying the region of phase space of lowest energy. This region of phase space is assumed to be saturated, with two electrons with opposite spins in each volume (2πh)3, and the remainder is assumed to be empty. Although this model hitherto has not been justified theoretically, it seems to be a plausible approximation for the interior of a heavy atom and one may expect it to give with some accuracy the distribution of electric charge there.

It is generally believed that isolation has played an important part in evolution. If an organism is to evolve so as to adapt itself to a special type of environment, e.g. a cave or a desert, it must not be swamped in each generation by migrants from the original habitat.

It has long been known that positive ions are capable of ejecting electrons from a metal surface, and it has recently been shown by Oliphant that metastable atoms of helium are capable of producing the same effects. In view of the importance of these phenomena in the theory of discharge processes and of their intrinsic interest, it is desirable to consider the theory of the effect in the light of the Sommerfeld theory of metals and the analytic methods of wave mechanics.

1. We consider a small cube of the solid, of edge l, in the actual strained position. If the stresses were removed and the cube cooled to the absolute zero, the displacements of its particles would be (−u, −v, −w). Then the particles may be considered to have displacements (u, v, w) from a standard position; and the cube has strain components of the usual forms

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.