In this short note my chief object is to prove the following theorem:

If

(1) f(z) is a regular function of z(= ρeiψ) in the angle |ψ| ≤ α, where ;

(2) f(z)= 0(eαρ), where K is a positive constant, throughout this angle;

(3) f(z) is not identically zero;

then

(4)

is a continuous function of ψ for |ψ| < α.

Let σs(n) denote the sum of the sth powers of the divisors of n,

and let , where ζ(s) is the Riemann ζfunction. Ramanujan, in his paper “On certain arithmetical functions*”, proves that, if

and

then , whenever r and s are odd positive integers. He conjectures that the error term on the right of (1·1) is of the form

for every ε > 0, and that it is not of the form . He further conjectures that

for all positive values of r and s; and this conjecture has recently been proved to be correct.

It is shown that a quartic equation may be found which expresses the grid characteristic of a valve accurately over a considerable range, and that for modern dull filament valves the grid current curve is exponential over only a very restricted length.

Expressions are given for calculating rectified current and the amplitude of harmonics produced by a simple grid rectifier when voltages of the following wave-forms are applied: E sin pt, (A + B sin nt), and (A + B cos mt + C cos nt)sin pt. Further, a formula is given from which we may calculate the rectified current produced by a cumulative grid rectifier when a voltage E sin pt is applied.

All these formulae are applicable if the grid characteristic, is parabolic, cubic, quartic or quintic in form.

It is also shown that when an acoustically modulated radio frequency voltage is applied to a cumulative grid rectifier, even if the valve has a parabolic grid characteristic, an infinite number of harmonics of the modulation frequency are produced. These harmonics are due to the. action of the grid leak and condenser. It appears, however, that in practice the amplitude of these harmonics will be very small compared with those produced, primarily, by the curvature of the grid characteristic.

1. The power possessed by water and air in motion of lifting and carrying solids considerably denser than themselves is familiar to all; but it does not seem to have been pointed out that classical hydrodynamics provides a simple explanation. If a solid rests on the bottom of a stream, the points of contact are points of zero velocity; and the velocity just above the solid, by the equation of continuity, must be greater than the general velocity. Hence the velocities produce high pressures under the solid and low ones above it, and the difference tends to lift the solid up. If the resulting thrust exceeds the weight of the solid in the liquid, the solid will be raised, and will be unable to rest in equilibrium on the floor of the stream.

A method of obtaining approximately the high-velocity limits of continuous β-ray spectra by electroscope measurements is established and used for the β-ray bodies of thorium active deposit.

The absence of thorium C γ-rays is confirmed and leads to some interesting results.

The work was carried out in the Cavendish Laboratory, Cambridge. Thanks and gratitude are due to Professor Sir Ernest Rutherford, P.R.S., for his interest in the problem; to Dr C. D. Ellis, F.R.S., who suggested the problem, for continual help and advice; to Mr G. A. R. Crowe for help in preparation of the sources; and to the Department of Scientific and Industrial Research for a Grant during the time the work was being carried out.

The investigations of von Karman dealing with the unsymmetrical double row of vortices in an infinite sea of liquid are well known. He found that the unsymmetrical double row is stable when, and only when, cosh2πa/b = 2, where 2a is the distance between the two rows and 2b is the distance between consecutive vortices on the same row. A detailed account of the stability of the Karman street and of the symmetrical double row has been given by Lamb, and it has been shown that the symmetrical double row is unstable for all values of the ratio a/b. The object of this paper is to investigate the stability of a double row of vortices of arbitrary stagger. We define a double row of stagger 2l to be the system formed by positive vortices at the points (2nb + l, a) and negative vortices at (2mb − l, − a), where m and n assume all integral values from − ∞ to + ∞. The vortices are thus neither exactly “in step” nor exactly “out of step.” When l = 0 the system reduces to the symmetrical double row and when the system is the unsymmetrical double row.

A considerable amount of experimental work has been done in recent years on the properties of the electrons which leave a metal surface when it is bombarded by an incident beam of electrons. Farnsworth, who first studied the energy distribution of the secondary beam systematically, showed that it is made up of two groups. The first has the same energy as the primary beam and may be regarded as consisting of reflected primary electrons; and the second has for the most part a very small energy, of the order of a few volts, and contains the true secondaries derived from the metal. The present writer in a preliminary note has described some measurements of the differential energy distribution by means of the magnetic spectrum method. Owing to an inadequate technique those experiments are considered to be valid only in the upper half of the energy range, where the form of the distribution has several features of particular interest when the mechanism of secondary emission and of electron reflection are considered. In this note, the author called attention to the peculiar depression in the curve immediately preceding the “reflected” peak. It was impossible then to insist on the reality of such an effect, owing to the probable presence of an instrumental cause (which was given there as the explanation). That it is real, however, has been demonstrated in a striking manner by Whiddington and Brown, who have succeeded in recording electrons as slow as 100 volts photographically on a specially sensitised film.

Two alternative views have been expressed in regard to the configuration of quadrivalent atoms. On the one hand le Bel and van't Hoff assigned to quadrivalent carbon a tetrahedral configuration, which has since been confirmed by the X-ray analysis of the diamond. On the other hand, Werner in 1893 adopted an octahedral configuration for radicals of the type MA6, e.g. in

and then suggested that “the molecules [MA4]X2 are incomplete molecules [MA6]X2. The radicals [MA4] result from the octahedrally-conceived radicals [MA6] by loss of two groups A, but with no function-change of the acid residue…. They behave as if the bivalent metallic atom in the centre of the octahedron could no longer bind all six of the groups A and lost two of them leaving behind the fragment [MA4]” (p. 303).

The effect of surface layers in modifying the thermionic emission has been explained by Nordheim on the assumption that the atoms in the surface contamination create an electrical double layer on the surface. If the layer is electro-positive with respect to the metal the work of final extraction of a given electron is reduced. This reduces the value of the work function χ, but at the same time requires the escaping electrons to pass through a region in which the potential energy is greater than their total energy. It follows that the emission coefficient D (W) is decreased at the same time. The details of the connection between χ and D (W) have been examined further by Fowler.

A modification of the alternating field method of measuring ionic mobility in a gas gives an experimental curve showing upper and lower limits to k. From this a distribution curve is derived, which has a calculable resolving power.

The mobility of negative ions in dry air shows a continuous distribution between the limits 2·15 and 1·45, with a peak value about 1·8.

At low pressures the current is resolved into ions and free electrons. From the relative numbers reaching the electrometer it is found that the electron makes an average number 9·4. 104 collisions before capture, independent of field strength and pressure, and therefore independent of the electron speed over a range W = 2×105 to W = 7×105.

The approximation is made that in a many-electron atom each electron is in a stationary state in the field of the nucleus and the remaining electrons, and further that this field is central.

To this approximation it is known that, if the electrons obey Schrödinger's equation, then a complete n, l group of electrons [i. e. 2 (2l + 1) electrons with the same n, l] can be divided into two half-groups for each of which the distribution of charge is spherically symmetrical. To the same approximation it is shown that, if the electrons obey Dirac's equation, a complete Stoner sub-group [i. e. 2j + 1 electrons with the same n, l, j (j = l ± £)] can be divided into two halves, for each of which the distribution of charge is spherically symmetrical.

The distribution of current and the resultant magnetic moment for a solution of Dirac's equation in a central field are obtained, and it is shown that for a complete Stoner sub-group the net current and magnetic moment are zero. The Landé g-formula for one electron is derived from the formula for the magnetic moment by neglecting ‘relativity’ terms.

The formula for the magnetic moment suggests the question whether it is ever possible to specify the direction of the spin axis of the electron (as distinct from the magnetic moment of the whole atom). This is investigated, and it seems justifiable to specify the direction of the spin axis for states for which the magnetic quantum number m has its extreme values ±j, but not otherwise.

In a paper* of some time back I gave the following theorem:

If f(z) is a regular function of z for z inside and on a circle Τ, and C is any convex curve inside Τ, and λ>0, then

Let a finite group Τ be represented as an irreducible group of order N of linear substitutions on n variables,

The variables may be chosen so that the substitutions of the group leave invariant the Hermitian form

The function τ (n), defined by the equation

is considered by Ramanujan* in his memoir “On certain arithmetical functions”

The associated Diriċhlet series

converges when σ = > σ0, for a sufficiently large positive σ0.

When certain elements are bombarded by swift α particles, protons or hydrogen nuclei are liberated which are attributed to the disintegration of the nuclei of these elements. It is now generally assumed that in such an artificial disintegration of an atomic nucleus an α particle is captured by the nucleus. Experimental evidence for this capture of an α particle has been obtained by Blackett in the case of the disintegration of the nitrogen nucleus. Blackett photographed eight disintegration collisions of an α particle with a nitrogen nucleus; in each of these the track of the α particle divided into two branches, one of which was due to the residual nucleus set in motion in the collision, and the other was due to the ejected proton. No trace of a third track to correspond to the track of the α particle after the collision was observed. Blackett therefore concluded that the α particle was captured by the nitrogen nucleus.

If we have one birational representation of a threefold locus V on an ordinary space S3, and if we can project V birationally on to another space S3′, then clearly there will be a Cremona correspondence between the two spaces S3 and S3′. This paper deals, in the first place, with threefold loci in higher space which can be represented birationally on ordinary space by means of cubic surfaces; and Cremona transformations for ordinary space are then obtained, as indicated above, by projecting such loci. In particular it will be shown that most of the more important cubic transformations and their reverses can be obtained very simply by this method.

Recently Yovanovitch and d'Espine using a magnetic spectrum method of low resolving power have found a band of very fast β-rays in the spectrum of radium B + C, having energies up to 7·6 × 106 volts. As early as 1911 Danysz found some evidence of the existence of such high velocity β-rays but they have not been observed by other investigators. A moderate number of such rays, if spread over a considerable range of velocities, might escape detection in most experimental arrangements for the study of β-rays, on account of their low absorption and low ionizing power. As it is important to know how many of these fast β-rays there are on the average per disintegration, the following simple experiment was performed by the writer in an attempt to determine this.

There is a theorem in geometrical optics due to Helmholtz which states that if a ray of light (1) after any number of refractions and reflections at plane or nearly plane surfaces gives rise (among others) to a ray (2) whose intensity is a certain fraction f12 of the intensity of the ray (1), then on reversing the path of the light an incident ray (2)′ will give rise, among others, to a ray (1)′ whose intensity is a fraction f21 of the ray (2)′, such that

The surfaces if not plane must be such that their radius of curvature is large compared with the lateral extent of the ray and this lateral extent must itself be large compared with the wave-lengths in the ray. In view of the many recent studies of reflection and transmission (refraction) of beams of electrons at potential walls, it seems worth while to ask whether there exists an analogy of the optical theorem for beams of electrons (or other particles). It is easy to see that the analogy must exist by using the principle of detailed balancing required by an assembly in statistical equilibrium. But a direct “dynamical” proof is also easy and may prove of sufficient practical interest to put on record here.