It is shown that in a crystal there exist states in which the electron is bound to a surface of the crystal and has an energy lying within a forbidden band. The wave functions and energies of these states are calculated, on the nearly free electron approximation, in terms of the constants of the crystalline potential field, which is represented by a triple Fourier series having the periodicity of the lattice. The method is shown to be applicable to a general crystal having a surface parallel to any one of the crystal planes.

The theory of holes in liquids, suggested in a previous paper, is developed by means of classical statistical mechanics, and it is shown that the principal thermodynamic properties of the liquid state can be derived in this way and that they are in numerical agreement with the experiments.

A new and simple proof is given of the known theorem that, if T1, T2,… is an infinite sequence of finite trees, then there exist i and j such that i < j and Ti is homeomorphic to a subtree of Tj.

In a previous joint paper (‘The dissection of rectangles into squares’, by R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte, Duke Math. J. 7 (1940), 312–40), hereafter referred to as (A) for brevity, it was shown that it is possible to dissect a square into smaller unequal squares in an infinite number of ways. The basis of the theory was the association with any rectangle or square dissected into squares of an electrical network obeying Kirchhoff's laws. The present paper is concerned with the similar problem of dissecting a figure into equilateral triangles. We make use of an analogue of the electrical network in which the ‘currents’ obey laws similar to but not identical with those of Kirchhoff. As a generalization of topological duality in the sphere we find that these networks occur in triplets of ‘trial networks’ N1, N2, N3. We find that it is impossible to dissect a triangle into unequal equilateral triangles but that a dissection is possible into triangles and rhombuses so that no two of these figures have equal sides. Most of the theorems of paper (A) are special cases of those proved below.

Let f be a function transcendental and meromorphic in the plane, and define g(z) by g(z) = Δf(z) = f(z + 1) − f(z). A number of results are proved concerning the existence of zeros of g(z) or g(z)/f(z), in terms of the growth and the poles of f. The results may be viewed as discrete analogues of existing theorems on the zeros of f' and f'/f.

The energy scattered when a sound wave passes through turbulent fluid flow is studied by means of the author's general theory of sound generated aerodynamically. The energy scattered per unit time from unit volume of turbulence is estimated (§3) as

where I is the intensity and ∧ the wave-length of the incident sound, and is the mean square velocity and L1 the macro-scale of the turbulence in the direction of the incident sound. This formula does not assume any particular kind of turbulence, but does assume that ∧/L1 is less than about 1. For turbulence which is isotropic and homogeneous, the energy scattered, and its directional distribution, are obtained for arbitrary values of ∧/L1. It is predicted that components of the turbulence with wave-number κ will scatter sound of wave-number K at an angle 2 sin−1 (k/2k). The statistics of multiple successive scatterings is considered (§4), and it is predicted that sound of wave-length less than the micro-scale λ of the turbulence will become uniform (i.e. quite random) in its directional distribution in a distance approximately .

The theory is extended (§5) to the case of an incident acoustic pulse. However, this extended theory cannot be applied directly to the case of a shock wave, for which it would predict infinite scattered energy. This is due to the perfect resonance between successive rays emitted forwards which would occur if the shock wave were propagated at the speed of sound. By taking into account (§6) the true speed of the shock wave (subsonic relative to the fluid behind it), the theory is improved to give a finite value, 0·8s tunes the kinetic energy of the turbulence traversed by a weak shock of strength s, for the total energy scattered. However, the greater part of this energy catches up with the shock wave, and probably is mostly reabsorbed by it, and only the remainder (tabulated as a function of s in Table 1) is freely scattered, behind the shock wave, as sound. The energy thus freely scattered when turbulence is convected through the stationary shock-wave pattern in a supersonic jet may form an important part of the sound field of the jet.

In this paper an n-stepped self-avoiding walk is defined to be an ordered sequence of n + 1 mutually distinct points, each with (positive, negative, or zero) integer coordinates in d-dimensional Euclidean space (where d is fixed and d ≥ 2), such that any two successive points in the sequence are neighbours, i.e. are unit distance apart. If further the first and last points of such a sequence are neighbours, the sequence is called an (n + 1)-sided self-avoiding polygon. Clearly, under this definition a polygon must have an even number of sides. Let f(n) and g(n) denote the numbers of n-stepped self-avoiding walks and of n-sided self-avoiding polygons having a prescribed first point. In a previous paper (3), I proved that there exists a connective constant K such that

Here I shall prove the truth of the long-standing conjecture that

I shall also show that (2) is a particular case of an expression for the number of n-stepped self-avoiding walks with prescribed end-points, a distance o(n) apart, this being another old and popular conjecture.

The statistical theory of isotropic turbulence, initiated by Taylor (3) and extended by de Kármán and Howarth (2), has proved of value in attacking problems associated with the decay of turbulence. In its application to such hydro-dynamical problems, the theory falls into two parts, a kinematical part and a dynamical part. The kinematical aspect consists in setting up correlations between velocity components, or their derivatives, at two arbitrary points in the fluid, and reducing the form of the tensor thus obtained in accordance with the severely restrictive assumption of isotropic turbulence; the success of de Kármán and Howarth's investigations is largely attributable to their improved treatment of this purely kinematical problem. The dynamical part then consists in applying the implications of the equations of continuity and motion to the functions defining the correlation tensors, in order to obtain information concerning their functional dependence on time and on the displacement between the two points for which the correlations are computed.

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.

The importance of principal ideal domains (PIDs), both in algebra itself and elsewhere in mathematics is undisputed. By contrast, Bezout rings, ‡ although they represent a natural generalization of PIDs, play a much smaller role and are far less well known. It is true that many of the properties of PIDs are shared by Bezout rings, but the practical value of this observation is questioned by many on the grounds that most of the Bezout rings occuring naturally are in fact PIDs. However, there are several fairly natural methods of constructing Bezout rings from other rings, leading to wide classes of Bezout rings which are not PIDs, and it is the object of this paper to discuss some of these methods.

The spectrum of −∇2 (and of −∇2 + b) is investigated when the boundary conditions are given on surfaces which extend to infinity. Simple criteria are obtained for determining whether point-eigenvalues are present in the lower part of the spectrum.

Semi-infinite domains which are conical at infinity are found to possess purely continuous spectra when the boundary condition is u = 0 or ∂u/∂v = 0; the radiation condition ensures a unique solution. A counter-example shows that this is not true in general for the boundary condition ∂u/∂v + σu = 0.

Semi-infinite domains which are cylindrical at infinity have a continuous spectrum with a discrete spectrum embedded in it. An example is given.

The results are applied to the theory of surface waves. It is shown that Ursell's ‘trapping modes’ can occur in a canal of finite width when the bed has a protrusion over a finite longth but is otherwise of uniform depth. Trapping modes can also occur when the canal contains a submerged cylinder (not necessarily small in cross-section).

The ‘phase-space’ method in quantum theory is used to derive exact expressions for the transition probabilities of a perturbed oscillator. Comparison with the approximate results obtained by perturbation methods shows that the latter must be multiplied by an exponential factor exp (− ∊/ℏω), where ∊ is the non-fluctuating part of the work done by the perturbing forces; as long as ∊ is small, exp (− ∊/ℏω) ˜ 1 and only dipole transitions have an appreciable probability. As the perturbation energy increases, however, this is no longer true, and multipole transitions become progressively more probable, the most probable ones being those for which the change in energy is approximately equal to the work done by the perturbing forces.

1. In a recent paper(1) expressions were found for the elastic stresses produced in a semi-infinite elastic medium when its boundary is deformed by the pressure against it of a perfectly rigid body. In deriving the solution of this problem—the ‘Boussinesq’ problem—it was assumed that the normal displacement of a point within the area of contact between the elastic medium and the rigid body is prescribed and that the distribution of pressure over that area is determined subsequently. The solutions for the special cases in which the free surface was indented by a cone, a sphere and a flat-ended cylindrical punch were derived, but no attempt was made to give a full account of the distribution of stress in the interior of the medium in any of these cases.

The random pure radiation field postulated in an earlier paper is set up in a relativistically invariant manner. The requirement that this field be isotropic in three dimensions makes much of the formalism identical with the theory of isotropic turbulence, as has been noted by a previous author. It is found, however, that the more stringent requirement of invariance under the Lorentz group, together with the fact that the field components satisfy Maxwell's equations, mean that, to within a multiplicative constant, the entire random process is uniquely specified.

In accordance with results obtained previously, the constant is effectively identified as Planck's constant. There then results a striking resemblance between the formalism of the present theory, and that of the quantized pure radiation field, with random variables taking the place of operators, and correlations between these random variables taking the place of commutator brackets.

The infinite field energy, which arises also in quantum field theory, where, however, it is usually considered to be purely formal, remains a central difficulty of the random theory. It can be shown to be a natural consequence of assuming Maxwell's equations, and it therefore becomes necessary to consider non-linear modifications of these.

The connexion between the present results and the inconsistencies of classical and quantum electrodynamics is discussed. Also some new information is obtained concerning the behaviour of thermodynamic quantities under a Lorentz transformation.

This problem, in the particular case of the hydrogen molecular ion, is so well known that a detailed reference to the large number of papers dealing with it is unnecessary. The investigations to which chief reference is made will be found at the end of this paper.

The homeotopy group Λx of a space X is the group of all homeomorphisms of X to itself, modulo the subgroup of those homeomorphisms that are isotopic to the identity. In this paper X will be taken to be a closed oriented 2-manifold, together with a polyhedral structure, and the definition of Λx is then restricted to the consideration of piecewise-linear homeomorphisms and isotopies. Although this restriction to the polyhedral category is not really essential to what follows, it does tend to simplify some of the arguments. In (2) a homeomorphism of X was associated with every simple closed (polyhedral) curve c in X in the following way. First, let A be an annulus in the Euclidean plane parametrized by (r, θ) where 1 ≤ r ≤ 2 and θ is a real number mod 2 π. We define a homeomorphism H: A → A by

H is then fixed on the boundary of A. If now e: A → X is an orientation-preserving embedding, and eA is a neighbourhood of c in X, then eHe−1|eA can be extended by the identity on X − eA to a homeomorphism h:X → X. Any piecewise linear homeomorphism hc which is isotopic to h will be called a twist about c or, if c is not specified, just a twist.

In a recent paper [5], the classical Bernoulli and Euler polynomials were expressed as finite sums involving the Hurwitz zeta function. The object of this sequel is first to give several remarkably shorter proofs of each of these summation formulas. Various generalizations and analogues, which are relevant to the present investigation, are also considered.

The observation of faint radio sources is often limited by the angular resolving power of the radio telescope, and not by its sensitivity. Continuous recording of the output from the receiving apparatus then produces a trace which always shows the sum of the effects of a number of sources in the reception pattern, instead of showing sources passing through the reception pattern one at a time. The apparent intensities of sources found on the records are affected by the presence of several other, fainter, sources in the reception pattern, and even the existence of many of the faint sources is doubtful. Counts of numbers of sources at given intensity levels are therefore unreliable.

Any assumed number-intensity law for radio sources can, however, be tested by a method free from the uncertainties due to confusion between adjacent sources, using a direct relation between the number-intensity law and the frequency distribution of deflexions of various sizes on the observed records. The present paper shows the derivation of the relation between M (I) dI, the mean number of radio sources per steradian in the intensity interval dI, and P(D)dD, the probability that the deflexion D recorded at an arbitrary instant lies in the interval dD. Expressions for P(D) are worked out for some specific models of M (I); a comparison with observation has been given in an earlier paper (6).

Let G be a locally finite connected graph and c be a positive real-valued function defined on its edges. Let D(ξ) denote the sum of the values of c on the edges incident with a vertex ξ. A particle starts at some vertex α and performs an infinite random walk

in which (i) the ξj are vertices of G, (ii), λj. is an edge joining ξj–1 to ξj (j = 1, 2, 3, …), (iii) if λ is any edge incident with ξj, then

Let υ be a set of vertices of G such that the complementary set of vertices is finite and includes α. A geometrical characterization is given of the probability (τ, say) that the particle will visit some element of υ without first returning to α. An essentially equivalent problem is obtained by regarding G as an electrical network and c(λ) as the conductance of an edge λ; the current flowing through the network from α to υ when an external agency maintains α at potential I and all elements of υ at potential 0 is found to be τD(α).

A necessary and sufficient condition (of a geometrical character) for the particle to be certain to return to α. is obtained; and, as an application, a new proof is given of a conjecture of Gillis (3) regarding centrally biased random walk on an n–dimensional lattice.

On the assumption that the potential energy of the three cubic lattices of the Bravais type consists of two terms, an attractive one proportional to r−m and a repulsive one proportional to r−n, n > m, stability conditions are expressed in the form that two functions of the number n should be monotonically increasing. These functions have been calculated numerically for n = 4 to 15, and are represented as curves with the abscissa n. The result is that the face-centred lattice is completely stable, that the body-centred lattice is unstable for large exponents in the law of force, and that the simple lattice is always unstable,—in complete agreement with the results of Part I.