A sequence of papers by P. Hall ((1)–(4)) on some fundamental properties of soluble groups was followed in 1940 by his account of a new construction theory for such groups (5). We shall have frequent occasion to refer to these papers, and we begin with a brief account of their contents.

The convergence of the perturbation expansion for the S-matrix in interaction representation of a three-boson contact interaction is investigated. A lower bound is obtained for the integrals corresponding to irreducible graphs when the total rest mass of the system is insufficient for bare particles to be created. It is shown that in this case the perturbation expansion cannot converge no matter what value the coupling constant has. A discussion is given of the bearing of this result on the general problem of the convergence of perturbation expansions for the S-matrix in renormalized field theories.

When waves are propagated through a medium whose velocity of propagation varies gradually from place to place, the wave direction and intensity vary according to the laws of refraction. Although the geometry of ray-paths has been well explored, and so also the laws governing the intensity of a coherent train of waves, little attention has apparently been given to the variation in intensity of an incoherent beam having a broad spectrum. The transformation of the energy spectrum is of practical importance in branches of geophysics, for example, in the study of sea waves entering shallow water, or of microseismic waves propagated through inhomogeneous regions of the earth's crust. Accordingly, it seems worth while to state and prove the rule governing the transformation of the two-dimensional spectrum function of a wave disturbance undergoing refraction.

It is shown that a real scalar function in n which is of class Cn and either has zero mean on all spheres of unit radius, or has zero mean in all balls of unit radius admits a unique expansion in terms of eigenfunctions of the Laplacian operator. In a similar manner, a suitably smooth vector-valued function in n which has zero flux through all spheres cf unit radius is shown to be decomposable as the sum of a solenoidal part and a series of conservative parts that are eigenfunctions of the Laplacian. Applications are given, including some in complex analysis.

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.

For an elastic solid the constitutive law can be written in terms of the deformation gradient α and its conjugate nominal stress s ≡ s(α), and also in terms of the right stretch u and its conjugate stress τ ≡ τ(u). It is shown that for isotropic elastic solids s(α) is invertible, in the local sense, for all u in the domain of u-space where τ(u) is locally invertible, with the exception of certain configurations which correspond to planes in τ-space.In the global sense a given s corresponds to four distinct τ's, and s is invertible to give four distinct α's when the corresponding τ's are uniquely invertible. That there are four branches of the inversion α(s) is of fundamental importance in that it clarifiesthe extent to which non-uniqueness of solution of boundary-value problems can be expected.

The implications of these results in respect of the complementary variational principle are discussed, and the controversy surrounding the use of nominal stress in this principle resolved.

Consequences of the required restrictions on τ(u) are examined and discussed in relation to inequalities which may be regarded as entailing physically reasonable response. It is intimated that τ(u) is invertible in the domain of elastic response.

An attempt has been made in the work described in the present paper to use the method initiated by Hartree, for the numerical solution of the Schrödinger wave equation for an atom with a non-Coulomb field of force, to estimate the number of dispersion electrons (hereinafter denoted by “ƒ” for brevity), corresponding to the lines of the principal series of the optical spectrum of lithium, and also to the continuous spectrum at the head of the series. Various attempts have been made to do this for hydrogen and other atoms by an application of the Correspondence Principle, but the first successful attempt at a complete description was made by Sugiura†, who has calculated ƒ for the Lyman, Balmer, and Paschen series and the corresponding continuous spectra, by using the known analytical solutions of the wave equation for an electron in a Coulomb field. The same author has also calculated ƒ for the first two lines of the principal series of sodium, by the utilisation of an empirical field of force in the atom calculated from the observed term-values by a method based on the old quantum theory. He has estimated the contribution to Σƒ (summed for the whole series) due to the continuous spectrum by the theorem that Σƒ = 1 in the one-electron problem§. This property provides a useful check on the work when ƒ for the continuous spectrum is also calculated. In the present paper ƒ for the continuous spectrum is actually calculated and it is found that Σƒ = 1 to a good approximation.

In this paper we shall investigate the occurrence of almost-primes in arithmetic progressions and in short intervals. These problems correspond to two well-known conjectures concerning prime numbers. The first conjecture is that, if (l, k) = 1, there exists a prime p satisfying

In the present article we prove a result characterizing the Jordan analogues of B*-algebras among the complex Banach Jordan algebras in terms of the algebra and norm structures alone.

Let u(x, t), where x ∈ 3, t ∈ , be a C∞ solution of the wave equation on {|x| > a} × for some a > 0, and suppose that u = 0 for |x|> a, t < 0. One then also has u = 0 for |x| > a + t, t > 0, so that u(.,t) can be thought of as a wave expanding into a previously undisturbed medium. One can now ask for a description of the asymptotic behaviour of u as |x| → ∞. It turns out that there is a v0(θ, τ) ∈ C∞ (S2 × ) such that

in the topology of C∞ (S2 × ). This limit may be called the radiation field of the expanding wave u. (See (2) and the earlier papers quoted there.)

Stokes and Wilson(1,2) have shown that the integral breadths of the Debye-Scherrer lines produced by small or imperfect crystals of the cubic system are given by

or, what is the same thing, that the apparent particle sizes are given by

In these equations λ is the X-ray wave-length, θ is the Bragg angle, Vt is the volume common to the crystal and its ‘ghost’ shifted a distance t in the hkl direction(1), and Jt is the mean value of the product FF of the structure amplitudes of two cells separated by a distance t in the hkl direction (2). The purpose of this note is to outline an argument which establishes the same result for crystals of any symmetry.

Burnside ((1), p. 241) has proved the following theorem:

If G is a non-metabelian p-group, then the centre of the derived group of G cannot be cyclic. In particular, a non-Abelian group of order p3 cannot be the derived group of a p-group.

We are concerned with the existence of smooth time functions on connected time-oriented Lorentzian manifolds. The problem is tackled in a more general abstract setting, namely in a manifold M where is just defined a field of tangent convex cones (Cx)x ∈ M enjoying mild continuity properties. Under some conditions on its integral curves, we will construct a time function.

Our approach is based on the definition of an intrinsic length for curves indicating how a curve is far from being an integral trajectory of Cx. We find connections with topics pertaining to Hamilton–Jacobi equations, and make use of tools and results issued from weak KAM theory.

The object of this paper is to determine all the solutions of the wave equation

which are of the simple form

where F denotes an arbitrary function. It will be shown that, in addition to the obvious cases of plane or spherical progressive waves, such solutions exist only when the wave fronts

are certain algebraic surfaces of the fourth order, the cyclides of Dupin. These include, as degenerate cases, the sphere, the plane, the cylinder, the cone, and the torus.

Let A, B be real symmetric n × n matrices having orthogonal reductions

where ΛA = diag (λ1(A), …, λn(A)) and ΛB = diag (λ1(B), …, λn(B)) and the sets of latent roots are each in descending order. (Thus we may say that the latent vectors of A, B are ordered with respect to respectively.) Let the multiplicities of the roots and in order of occurrence be respectively and , so that

Consider a linear elastic solid occupying a bounded regular three-dimensional region B with smooth surface ∂B. The components of displacement ui referred to cartesian axes xi are then well known to satisfy the system of governing equations

in which t denotes the time variable, x = (x1, x2, x3) denotes the position vector, ρ(x) is the non-homogeneous density, assumed positive,. i(x, t) are the Cartesian components of body force per unit mass, and cijkl(x) are the non-homogeneous elasticities, which apart from certain smoothness conditions stated later, are assumed to possess the symmetry

Throughout this paper, all suffixes range over the values 1, 2, 3 and the usual converition of summing over repeated indices is adopted. Except where it is in the interest of clarity we avoid explicit mention of the dependence of functions on their arguments.

It was shown by Baber and Hassé(1) that the Schrödinger equation for the single electron in H+2 with fixed nuclei is separable in spheroidal coordinates. The separated equations were integrated by Bates, Ledsham and Stewart (2) and their solutions may be regarded as exact. However, several attempts have been made to derive approximate solutions for this, which is the simplest of all molecular problems, generally with the hope that it might be possible to extend the analysis to larger diatomic systems. Recently a single-centre expansion has been much favoured, since it avoids the difficult two-centre integrals; but unfortunately this advantage has been offset by the fact that the rate of convergence for the lowest states has been rather slow since the deviations of diatomic H+2 from the united-atom H+e are serious. But successful approximations to some excited states suggest that a single-centre approach may still be useful, provided that the trial functions are suitably chosen.

The equilibrium of a layer of a viscous incompressible fluid of density ρ variable in the vertical (z) direction, the fluid extending indefinitely in the horizontal directions and being confined between rigid horizontal boundaries, is examined by the usual method of studying the initial behaviour of a small disturbance. Diffusion effects are ignored. Chandra-sekhar has shown quite generally that a variational procedure for solving the resulting characteristic value problem is admissible, suggesting a method whereby an approximate solution can be obtained analytically. In this paper, by making use of appropriate approximate methods of solution, three types of density configuration, ρ(z), are studied. The first corresponds to two superposed fluids of great depth, the upper and lower fluids haying densities and coefficients of viscosity ρ2 and μ2, and ρ1 and μ1, respectively. Chandrasekhar has given an exact numerical solution to this problem for the case where μ2/ρ2 = μ1/ρ1 The results of the approximate theory, by which not all the boundary conditions can be satisfied, and the exact theory are in good agreement. Expressions for the wave-length and growth rate of the mode of maximum instability in the unstable case (ρ2 > ρ1) are found. The stable case (ρ2 > ρ1) is also discussed in detail; the formulae obtained for the phase and group velocities of gravity waves in deep viscous fluids are especially interesting. There is an upper limit to the natural frequency of oscillation of the system which would be well within the range of experimental investigation.

The second problem considered is that of two superposed fluids of small depth. Formulae for the phase and group velocities of ‘long gravity waves’ (which are not equal when viscous effects are included) are obtained.

Finally, a continuously stratified fluid of finite depth d in which ρ(z) = ρ0eβz is investigated. The character of the equilibrium depends on a Grashof number G = (gβd4)/(π4ν2) where g is the acceleration of gravity and ν is the coefficient of kinematical viscosity, assumed for simplicity to be constant. Expressions for the wave-length and growth rate of the mode of maximum instability in the unstable case (G > 0) are obtained. In the stable case (G < 0) all modes are aperiodically damped if – G < . Otherwise, the possibility of a disturbance being propagated as a horizontal wave does arise, but only within a limited wave-length range.

1. In a previous paper (1) the authors used the theory of the moment generating function to deduce the known random sampling distributions of the estimated variance and co-variance in a system of variables following the normal law of frequency. In this paper we shall go much further. By means of the same general method the simultaneous distribution of the ½p (p + 1) second order moment statistics in a normal system of p mutually correlated variables will be deduced. It will be shown to be completely independent of that of the p sample means, and incidentally the method of proof to be developed will be found, in the special cases p = 1 and 2, to be an improvement on that given in the earlier paper, being independent of previous results reached by other authors.

Expressions are found for the progress of slow selection in a Mendelian population where generations overlap. The changes are very similar to those which occur when generations are separate.