It is trivial that a group all of whose elements except the identity have order two is Abelian; and F. Levi and B. L. van der Waerden(1) have shown that a group all of whose elements except the identity have order three has class less than or equal to three. On the other hand, R. Baer(3) has shown that if the fact that all the elements of a group have orders dividing n implies a limitation on the class of the group, then n is a prime. The object of the present note is to extend this result by showing that if M is a fixed integer there are at most a finite number of prime powers n other than primes, such that the fact that all the elements of a group have orders dividing n implies a limitation on the class of its Mth derived group.

1. The particular set of symmetric polynomials known as S-functions has recently been shown to be of importance in a variety of algebraic problems. Many of the applications of these functions depend upon the properties of the operation which Littlewood terms ‘new multiplication’, by which, from two given S-functions {μ}, {ν} of respective degrees m and n, is constructed a symmetric function {μ} ⊗ {ν} of degree mn. Littlewood has devoted a considerable part of his paper (2) to explaining various methods by which this function can be expressed in terms of S-functions of degree mn. None of these methods is really simple (in the sense that the rule for expressing the ordinary product {μ} {ν} in terms of S-functions of degree m + n is simple); and, indeed, it would be unreasonable to expect any simple rule of general validity for writing the resulting expression down since, for instance, a knowledge of the explicit expression for {μ} ⊗ {n} would yield immediately a knowledge of all the linearly independent concomitants, of degree n, of an algebraic form of type {μ} in an arbitrary number of variables. Nevertheless, the evaluation of {μ} ⊗ {ν} in particular cases is often necessary. Of the methods suggested by Littlewood for performing this evaluation the one which seems to be normally the simplest (his ‘third method’) involves a process which is not shown to be free from ambiguity, and which can actually be shown by examples to give, in certain cases, alternative solutions, the choice between which must be made by other considerations. It is therefore perhaps worth while putting on record a quite different method of procedure, which, apart from any intrinsic interest which it may possess, seems to be quite practicable for most of the actual evaluations performed by Littlewood.

The name ‘cardinal function’ was given to

by E. T. Whittaker (1), who considered it as a ‘smooth’ approximation to a function f(x), having the same values as f(x) at the points a + rw (r = 0, ± 1, ± 2, …). It has since been extensively studied (2), mainly from the point of view of interpolation theory. Hardy (3), however, observed that the functions νr(t) defined by

form a normal orthogonal set on the interval (−∞, ∞), for r = 0, ± 1, ± 2, …. This fact suggests a discussion of the cardinal series from the point of view of mean-square approximation.

In this paper we consider the theory of point-curve correspondences on a single surface F, i.e. correspondences in which to each point of F corresponds an algebraic curve of F. The results previously obtained for induced and extended correspondences between two surfaces require some modification here, as we can consider the self-correspondences induced on a curve C of F, and the related theory of extended correspondences, which is now complicated by the existence of the identical correspondence on C. We also develop a theory of correspondences with (non-zero) valency, and show that for a surface whose Riemann matrix is pure and without complex multiplication (and hence for surfaces of ‘general moduli’), all correspondences are valency correspondences, this result being exactly analogous to the well-known theorem for curves. We also consider generalized valency correspondences which are an extension to surfaces of the concept of correspondences of multiple valency on a curve. The analogy between our theory and the known results for curves does break down in one important respect. It is not true that every surface possesses valency correspondences even of the generalized kind, the existence of such correspondences involving restrictions on the intersection group of the surface.

The essential elements in any application of statistical theory are

(1) the set of hypothetical physical states θ which may give rise to the observed phenomena,

(2) the set of all possible observations or samples x,

(3) the set of actions α which may be taken when x is known.

The whole procedure in the common applications can be divided into two parts:

(A) the estimation of θ from the known observation x,

(B) action on the basis of this estimation.

A number of important Markoff processes, with a continuous time parameter, can be represented approximately by a discrete process, interesting in its own right, of the following type. A class of individuals gives rise seasonally (in January say) to a number of new individuals (children), the probabilities of an individual having 0, 1, 2, … children being p0, p1, p2, …. These probabilities are the same for all individuals and are independent. The individuals formed each January are regarded as a new generation, and only this generation is capable of reproducing in the next January. Let

so that F(x) is the probability generating function (p.g.f.) of the number of children of an individual. Clearly the series for F(x) is absolutely convergent when |x| < |1.

In order to determine the mathematical form of the distribution of gene frequencies in a population, R. A. Fisher (1, 2) applied two different but complementary methods. First, differential equations were set up for certain cases and solved. An extension of these results arose from the expression by Wright (10,11,12) of the conditions as an integral equation. But strictly, neither integral nor differential equations can represent the conditions exactly, since integration is substituted for summation, and differentials for minimum steps (1/2n) in gene frequency. The second method employed by Fisher was exact, even though the solutions involved new functions. The properties of generating functions were utilized to yield functional equations, and the special case of rare mutations was dealt with in detail. In the present paper this powerful method is extended to the general case where mutation rates are not necessarily small.

A considerable volume of knowledge is now available on random fluctuations (noise) as regards the behaviour in amplitude. Familiar names in this field are those of Uhlenbeck and Ornstein(7), Fürth(1) and Rice (4), although very many others have made valuable contributions. A particular class of problem, of considerable practical importance, exists when the frequency spectrum is limited to a relatively narrow range. The resulting noise has then the character of a more or less regular oscillation modulated randomly in amplitude and phase. In this case, if we write the fluctuation in the form

(where R(t) and θ(t) are variables changing slowly in comparison with sinω0t), it is clear that the magnitude of the envelope R(t) and the phase θ(t) are now the significant quantities. Rice (4), among others, has made a study of the statistical properties of R, deriving in particular the correlation function R(t) R(t + τ) in terms of the characteristics of the (power) spectrum ω(f). Fürth and the writer (2) have extended this work and carried out a collateral experimental investigation.

The choice of a numerical method for the solution of ordinary differential equations depends on the associated boundary conditions. When all the boundary conditions are specified at one end of the range of integration, one of the well-known step-by-step methods will generally be used, while the method of relaxation is reserved for the case in which boundary conditions are specified at more than one point (1). In the latter, simple but inaccurate finite-difference formulae are used to provide a first approximation to the required solution; this approximation is then used to give an estimate of the errors involved in the use of the inaccurate formulae, and successive corrections are obtained until the full, accurate finite-difference equations are satisfied (1). The same principle is followed in this paper with regard to step-by-step methods, the main difference being the way in which approximate solutions are obtained. In relaxation methods simultaneous equations are solved, while in the use of the step-by-step methods suggested here successive pivotal values are built up by the use of recurrence relations. All the methods of this paper follow this principle, differing only in the method of obtaining a recurrence relation, and consequently in the form of the correction terms.

1. In a steady two-dimensional motion of viscous liquid in the sharp corner formed by the rigid straight boundaries θ = 0, α, where r, θ are plane polar coordinates, it is found that, near enough to the corner, the most important term in the stream-function is of the form rmf(θ). The index m is evaluated in §§ 2–4 for values of α between 360 and 90°, and is found to be complex if α is less than about 146°; the limiting form of the stream-function when α is small is considered in § 5.

It is well known that a solution of the wave equation

for which u = ∂u/∂t = 0 initially outside a surface S0, vanishes at time t in the exterior of a surface St parallel to, and at normal distance ct from S0, so that the wave fronts of disturbances represented by the solutions of the wave equation obey the laws of geometrical optics. Analogous results hold for the solutions of any linear hyperbolic second-order partial differential equation with boundary value conditions of the ‘Cauchy’ type. But the wave fronts of solutions of problems in which some of the boundary conditions are of the type representing reflexion do not seem to have been treated, and in particular the case of diffraction, when there is a ‘shadow’, does not seem to have been considered at all.

In this paper an investigation is made of the properties of material which is under isotropic stress p when in a strained state, and is such that the relation between the stress and the dilatation Δ is

where α is the density, assumed constant in the unstrained state, and c is the velocity of light. It is shown that in this material the waves of dilatation travel with the velocity of light and that a disk or cylinder composed of this kind of matter suffers no change in radius when it is made to rotate.

It is suggested that it is not unreasonable to attach the label incompressible to matter having these properties.

The simplest exact classical theory, of a particle possessing charge and dipole moment proportional to the spin leads to a wave equation with an explicit spin interaction with the field. This wave equation is used in this paper to calculate the scattering probability coefficient in the Compton effect by two methods. The first method uses the unquantized radiation field; the second uses Dirac's scheme of field quantization. The scattering probability coefficient is given in the general case in terms of the charge e, the constant of proportionality in the dipole moment C, and the quantities k, l specifying the spin. Some special results are considered for particles of spin ½ and 1. Formulae are given for the scattered intensity as a function of the angle of scattering, and the total scattering cross-section as a function of the energy of the unpolarized incident radiation. Graphs are given for these formulae when the magnetic part of the dipole moment takes the empirical values in the case of the electron, proton and neutron. In the case of the meson, normal and abnormal values of the dipole moment are used.

1. The method introduced by Dirac (1) in his classical theory of the interaction of an electron with an electromagnetic field has since been extended to the interaction of point particles with more general wave fields, such as occur in the neutral meson theory. Bhabha (2, 3) considered the scattering of neutral mesons by nucleons and gave arguments to show that typically quantum-mechanical effects should be unimportant if the energy of the incident meson is less than the rest energy of a nucleon, so that a classical theory with damping is appropriate to such problems. Now only the charged mesons are observable, and therefore a discussion of the classical theory of the interaction of a nucleon with a charge-bearing field is desirable. Apart from a paper by Fierz (4) which omits the dipole-like coupling terms, there is little published work on this subject.

Three formalisms have been considered for the ‘extraordinary’ field introduced by Podolsky in his generalized electrodynamics.

The c-representation (tildons) for negative energy particles (§ 3), leads to an unstable vacuum in electron hole theory and must be discarded (§ 7).

The d-representation suggested by Podolsky is a hole theory for bosons. This leads at once to inconsistencies and it is impossible to introduce a Fock representation (§ 6).

The least unsatisfactory is the b-representation (tilde photons) used by PKMG. Tilde photons can be created at high energies but decay immediately by pair creation. The introduction of tilde photons does not interfere with any agreement already established between experiment and ordinary quantum electrodynamics. Evidence might be obtained for the value of the constant a from consideration of the 2S level shift of the hydrogen atom and from the theory of showers where the deviation from the usual theory is greatest (§ 7).

However, the formalism involves the introduction of an indefinite metric for the tilde photon field and consequently ‘negative probabilities’ for states in which an odd number of tilde photons are present. Approximate results can be obtained by assuming that ‘negative probabilities’ in the theory are positive probabilities in the real world. This leads to a small but fundamental error and it does not seem possible that a theory in this form can have any deep seated validity (§ 5).

The same analysis applies to the extension of the Podolsky theory to the Proca field by Green (5). In order that the energy should be positive definite it is necessary to introduce an indefinite metric for one of the two superposed meson fields. Fundamentally the difficulties of interpretation are the same but in practice they will be more obvious because the coupling constants are greater.

In Ewald's theory of crystal optics, based on classical principles, an optical wave through a crystal lattice of polarizable atoms (idealized as isotropic oscillators) is a self-sustaining system of vibrations so constituted that, on the one hand, the electric moment of each atom is caused to oscillate by the electromagnetic field and, on the other hand, the electromagnetic field is itself the resultant field due to the superposition of the dipole waves produced by the lattice atoms. Born extended the theory to the case of movable lattice ions and showed that in an optical wave the electromagnetic field is so coupled to the lattice vibrations that each lattice ion vibrates in phase with the local field which, as in Ewald's case, is itself produced by the vibrating ions. In Born's original theory, the motion of the lattice particles had to be treated by classical mechanics. It is shown that the results of the quantum-mechanical treatment of the lattice motion agrees exactly with the classical theory. Not only is the induced current at each lattice point in phase with the local field, but the magnitude of the current is also identical with the classical value, completely independent of whichever vibrational state the crystal might be in.

This paper deals with some details of the application of the Lorentz-Dirac equations of motion of an electron to two simple cases, (a) with no incident field, and (b) with an incident pulse of radiation. In case (a), the field-energy distribution in the self-accelerating motion of the electron when the electron has built up a velocity close to the velocity of light is considered. Numerical and graphical methods are used to form a picture showing how the field energy tends to be concentrated in a disk for this state of motion. A discontinuity of the advanced field variables of the self-accelerating electron is also examined. In case (b), the general solution of the equations of motion for the velocity of the electron is worked out accurately. In the physical solution, this brings out the effects of radiation damping when the pulse has great intensity. Some aspects of the non-physical solutions are pointed out.

A paper under the above title (1) by G. H. Livens was published recently in this journal. The main object of this paper appears to be to show that the treatment of the same subject (2) by myself is unsound. I shall not attempt to refute Professor Livens's opinions. His views concerning magnetic energy have already been criticized by others (3) more expert than myself. I merely wish to draw attention to two completely untrue statements of fact in Professor Livens's paper, namely the following:

(a) On p. 538 “Guggenheim writes therefore

In applying the general theory of intermolecular forces to the complicated problem of ionic solvation, the minimum requirements are an exact knowledge of (1) the force laws governing the interaction of two isolated solvent molecules as a function of their separation and the various angles of mutual inclination, (2) the force laws governing the interaction of an ion with one solvent molecule, also as a function of their distance apart and of their mutual inclinations, (3) the average disposition of solvent molecules in the pure liquid and (4) the disposition of the solvent molecules in the force field of the ion, particularly in its immediate vicinity. None of these theoretical requirements is available for systems of practical interest. The established laws of intermolecular force are those relating to spherically symmetrical fields, applied to gases (J. E. Lennard-Jones; see (1)), in which the molecular arrangements are random, and to crystals (M. Born (2)), in which the arrangements are orderly. In seeking to extend these laws to liquids and solutions, we are hampered at the start by our ignorance of directional forces in general and of the degree of disorder prevailing in condensed fluid systems. What is attempted here is a simplification of the problem of applying intermolecular force theory to ionic hydration by the introduction of assumptions which can claim some experimental warrant. How far the introduction of these assumptions may swamp other factors must remain an open question. At any rate, the attempt removes many anomalies, and allows of the absolute calculation of ionic dimensions in aqueous solution from intermolecular force constants determined without reference to the condensed states of matter.