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

In a recent paper, Poisson's summation formula

was proved very simply by integration by parts, subject to the conditions:

(α) for all real values of x, f(x) and f′(x) are continuous and f (x)→0, f′ (x)→0 as |x| → ∞.

(β) f(x) and f″(x) are such that the integrals, converge.

Hobson has given a proof of this theorem in its fullest generality. The present note gives an alternative for part of Hobson's argument. The theorem may be stated in two forms. If f(x) is a function of x, monotone when a ≤ x ≤ b, and φ(x) is integrable over the same range, then

where a ≤ X ≤ b,

(ii) the same holds with a < X < b except in some trivial cases where f(x) is constant in the open interval a < x < b. The form (ii) is not mentioned by Hobson.

The problem here considered is a particular case of the more general problem of finding the number of [k]s, lying in a space of dimension n(n > k), which satisfy a certain number of conditions of the following type: viz. to meet a given [a0] in a point, a given, [a1] containing the [a0], in a line, etc., finally to lie in a given [ak] containing [ak−1]. The number of such conditions is such that there are just a finite number of [k]s which satisfy them all. The method which we employ is the one introduced by Schubert and commonly known as the “degeneration method” it is explained very briefly below.

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.

It is a familiar fact in Solid Geometry that the problem of finding a twisted cubic curve with four given tangents is poristic. In fuller statement four prescribed tangents are apparently sufficient to determine such a curve yet

(A) In general there is no twisted cubic touching four given lines,

(B) If there is one there is an infinite number.

1. The following proof of Bateman's expansion may be of interest since it avoids the use of Appell's hypergeometric functions of two variables, though it suffers from the defect that it presupposes a knowledge of the expansion.

Miquel's Theorem, as generalised by Clifford, states that the four circles through the intersections of four straight lines, taken in triads, meet in a point; that the five such points derived from five lines, taken in sets of four, lie on a circle; that the six such circles, determined by six lines, meet in a point……. The process continues indefinitely.

The solution of problems in diffraction by an elementary application of Huyghens's principle is discussed. The obliquity function is investigated, using the criterion that the formula used must give the right result when integrated for the case of an undiffracted plane wave. It is shown that this is satisfied for distant points by any function which makes the integrals converge, but that to satisfy it completely a constant obliquity function is necessary. This makes a consideration of the distant boundary essential, as the integrals do not converge in this case. It is shown that a boundary distributed in a Gaussian way is completely satisfactory. The integral for the case of diffraction by a straight edge is solved exactly, leading to the usual result. Finally, the results are discussed in relation to the teaching of the subject.

I am indebted to Dr H. Jeffreys and to Mr F. P. White for their interest in this paper; also to Mr N. F. Mott, who has contributed much to the course of its development.

It is well known, having been pointed out in the first instance by Plücker, that a plane quartic curve consisting of four unifolia, in Zeuthen's sense, i.e. four even circuits each containing one concave portion or bay, has all its twenty-eight bitangents, and all their fifty-six points of contact, real; since each bay has a bitangent whose points of contact bound it, and each pair of circuits have four common tangents. It has not, so far as I know, been remarked that the algorithm of Hesse and Cayley, in which each of the bitangents is indicated by a pair out of the symbols 1 2 3 4 5 6 7 8, can be applied in a very simple way to such a curve.

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.

1. The form of the absorption curves of homogeneous β-rays in aluminium has been investigated by means of a Geiger counter. It is found, in agreement with Schonland, that the main portion of the curve is linear, with pronounced initial and final flattenings. Similar initial flattenings were found for copper and silver, but not for gold.

2. The range of β-rays emergent at different angles has been determined, and it appears that the loss of range is due to the increased path consequent upon scattering.

3. Some evidence has been obtained of the angular distribution of plurally scattered particles, and of the existence of a most probable angle of scattering, similar to that found for α-rays by Geiger.

The tetrahedral complex of lines in space of three dimensions may be defined as the aggregate of lines which meet four given planes in a range of points which is related to some fixed range. In the present paper we consider the more general system of lines, in space of r dimensions ([r]), which meet a certain number k of given primes (or [r−1]) in a range of points which is related to some given range. The paper is divided into three parts. In the first part, after considering successively the cases r = 2, 3, 4, we proceed by an induction argument to the general case. The section following treats the matter from an algebraical point of view, and in it we obtain in a concise form expressions for the locus (when such exists) generated by the lines. In the final section the systems of lines are shown to arise naturally as projections of known configurations in higher space.

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.

In classical mechanics the state of a dynamical system at any particular time can be described by the values of a set of coordinates and their conjugate momenta, thus, if the system has n degrees of freedom, by 2n numbers. In quantum mechanics, on the other hand, we have to describe a state of the system by a wave function involving a set of coordinates, thus by a function of n variables. The quantum description is, therefore, much more complicated than the classical one. Let us consider, however, an ensemble of systems in Gibbs' sense, i.e. not a large number of actual systems which could, perhaps, interact with one another, but a large number of hypothetical systems which are introduced to describe one actual system of which our knowledge is only of a statistical nature. The basis of the quantum treatment of such an ensemble has been given by Neumann. The description obtained by Neumann of an ensemble on the quantum theory is no more complicated than the corresponding classical description. Thus the quantum theory, which appears to such a disadvantage on the score of complication when applied to individual systems, recovers its own when applied to an ensemble. It is the object of the present note to examine this question more closely and to show how complete the analogy is between the quantum and classical treatments of an ensemble.

The problem of the collision between an electron and an atom was first considered on the Quantum theory by Born, who has worked out in detail the case of atomic hydrogen, and has obtained formulae giving the variation of scattering with angle both for elastic and inelastic collisions. Born's solution is only approximate. The purpose of this note is to discuss the physical nature of the approximations used by him, and also to extend his results to Helium, for which experimental evidence is now available. It is found that the theoretical curve agrees with the experimental as well as the approximations used would lead one to expect. We shall confine ourselves to elastic collisions.

The work of Heitler and London has shown that the first order perturbation energies of two atoms taken as a function of the separation between the nuclei gives a fairly good description of the behaviour of the atoms with regard to possible molecular formation. For example, the dissociation energy of the hydrogen molecule calculated in this way is of the same order of magnitude as the accepted observed value, being in fact about 25% too small. The theory has been generalised by Heitler, and also by London and others, to the case where each atom contains many electrons, but detailed calculations do not appear to have been made for systems more complex than that of two hydrogen atoms.

From measurement of tracks formed in an expansion chamber range distribution curves are obtained relative to 2134 α particles from thorium C′ and 729 α particles from radium C′. These curves exhibit marked differences in the proportion of short range particles which each contains. The origin of these differences is discussed. Estimates are made of the linear straggling coefficients relative to the absorption of the two groups of particles in air (in the experimental arrangement other media entered as well). These results very definitely confirm the conclusions reached by Briggs from measurements of the straggling of velocity of the particles. The sources of error which enter when the expansion chamber is used for the measurement of straggling coefficients are briefly reviewed.

The purpose of the present paper is to illustrate the use of the Compton Quadrant Electrometer for current measurement by the rate of deflection method. There are several points in the theory and technique for this instrument which do not appear to have been mentioned in any previous papers. With the Compton Electrometer the rate of deflection method gives accurate measurements of currents of 10−14 amperes in less than a minute. The instrument is of course more difficult to use than the Dolezalek type, and it is for this reason that an attempt is made here to discuss and illustrate its use. The theory is given in Compton's original paper and there is also an important note on the subject by Cox and Grindley.

The purpose of this note is to extend in two respects some previous work of the writer on the wave equation of a particle in a central field of force which is a Coulomb field for sufficiently large r.