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.

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.

1. The general theory underlying sampling inspection methods introduced during the war under the name of sequential analysis, applicable in cases where the sampling units can be taken serially, has mainly been developed by Wald and is now published (5). The original purpose of the present investigation was to exhibit the main structure of the distributional theory relating to the size of sample required by noting its relationship with the classical ‘random-walk’ problem. Thus in Part I of this paper are derived the distribution and characteristic functions of the absorption time for one-dimensional random-walk theory with constant drift and absorbing barriers. Attention is confined to the asymptotic case of numerous independent displacements, for which it is well known that the unrestricted total displacement becomes Gaussian; the motion may also be treated as continuous. Since these results were obtained Wald has independently given an alternative and more general discussion of this problem((4); see also Tweedie (3)), but the extension here of the method of images used by Chandrasekhar (1) still appears of interest, and has advantages over alternative direct methods of solution for the distribution function involving Fourier expansion (cf. 2).

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.

We propose here to improve on two of the theorems given earlier, to prove some fresh theorems and to give the proof of Theorem XI stated without proof in the first part of the paper.

Let Q be a quasi-ordered set, i.e. a set on which a reflexive and transitive relation ≤ is defined. If, for every finite sequence q1, q2, … of elements of Q, there exist i and j such that i < j and qi ≤ qj then we call Q well-quasi-ordered. For any ordinal number α the set of all ordinal numbers less than α is called an initial set. A function from an initial set into Q is called a transfinite sequence on Q. If ƒ: I1 → Q, g: I2 → Q are transfinite sequences on Q, the statement ƒ ≤ g means that there is a one-to-one order-preserving function ø:I1 → I2 such that f(α) ≤ g(ø(α)) for every α ∈ I1. Milner has conjectured in (3) that, if Q is well ordered, then any set of transfinite sequences on Q is well-quasi-ordered under the quasi-ordering just defined. In this paper, we define so-called ‘better-quasi-ordered sets’, which are well-quasi-ordered sets of a particularly ‘well-behaved’ kind, and we prove that any set of transfinite sequences on a better-quasi-ordered set is better-quasi-ordered. Milner's conjecture follows a fortiori, since every well ordered set is better-quasi-ordered and every better-quasi-ordered set is well-quasi-ordered.

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.

The hypothesis of the existence of abstract structures in physics, which has been made in recent papers, is made precise. Difficulties in previous formulations are removed and insight into Maxwell electrodynamics is obtained. General relativity is related to other field theories which deal with flux from sources, and new field equations for general relativity are proposed.

Let K be a knot in the 3-sphere S3, N(K) the regular neighbourhood of K and E(K) = cl(S3−N(K)) the exterior of K. The tunnel number t(K) is the minimum number of mutually disjoint arcs properly embedded in E(K) such that the complementary space of a regular neighbourhood of the arcs is a handlebody. We call the family of arcs satisfying this condition an unknotting tunnel system for K. In particular, we call it an unknotting tunnel if the system consists of a single arc.

In an investigation which I hope to publish shortly, I think I have been able to improve my theory of the constant hc/2πe2 and to bring it at last into a precise form. No alteration is made in the value 137 obtained in the work already published. The recent advance has been mainly due to the fresh light thrown on the foundations of wave-mechanics by Dr Dirac's book. With a fuller understanding of the “theory of 137” it has been possible to discern opportunities for extension in several directions, and it is with these developments that the present paper deals. They are still in a rudimentary state; but since the theory appears to give correctly either accurate or approximate values of the masses of the electron, the helium atom, and the cosmos in terms of the mass of the proton, it would seem to be on the right lines. Moreover the principle of “ignoration of degrees of freedom” on which the numerical predictions depend is strongly suggested by the theory of the constant 137. If my view is right the only arbitrary constant of nature is the number of particles in the universe—if the number is arbitrary.

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.

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.

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.

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.

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.

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.

The exact (Navier-Stokes) equations for the flow of a viscous compressible fluid are examined in a search for simple solutions, in which the equations reduce to ordinary differential equations. Such solutions are found for the uniform shearing motion in the Couette flow between both parallel flat plates and coaxial circular cylinders in relative motion. For a compressible fluid there are no simple solutions corresponding to certain other flows (such as the Poiseuille flow between fixed parallel flat plates) described by simple solutions for an incompressible fluid. A solution is found for the flow of a gas over an infinite porous plate with uniform suction through the plate (gravity neglected), and the relation of the solution to asymptotic solutions of the boundary-layer equations is discussed. Ordinary differential equations are obtained for the problems of (i) the flow due to the rotation of a horizontal disk in a gas (dissipation neglected), and (ii) the steady circulatory flow round a porous circular cylinder with uniform suction (gravity neglected). Discussion of the gas-dynamical problems is preceded by a description of the state of a horizontally stratified gas in static and thermal equilibrium.

We consider some questions related to the signs of Hecke eigenvalues or Fourier coefficients of classical modular forms. One problem is to determine to what extent those signs, for suitable sets of primes, determine uniquely the modular form, and we give both individual and statistical results. The second problem, which has been considered by a number of authors, is to determine the size, in terms of the conductor and weight, of the first sign-change of Hecke eigenvalues. Here we improve the recent estimate of Iwaniec, Kohnen and Sengupta.

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

The notion of packing measure, introduced in [12], [13] and [10], has been used by comparison with Hausdorif measure to study the regularity and rectifiability of sets in the plane [11]. Since a few technical mistakes can be found in [10], lemma 5·11 and [11], lemma 3·2, we wish in this paper to give the corresponding exact proofs, together with a natural development of this theory in higher dimensions.