The theory of distributions of L. Schwartz (3) provides a unified and rigorous foundation for special methods used in various branches of mathematics. Schwartz's treatment is on the most general level, and presupposes an understanding of modern abstract analysis. Several alternative approaches to distributions have been developed, all of them ‘elementary’ in one sense or another. We follow here the approach of Mikusiński (2) and Temple (4), in which distributions are defined as generalized limits of sequences of continuous functions. We find that, with this approach, it is possible to prove the basic theorem: every distribution is (locally) a derivative. The property of continuity of a distribution does not enter into the arguments establishing this result, but instead follows from it. Hence we are able to reduce the ‘regular sequence’ definition of a distribution to its simplest form. In a later paper we shall study convolution products of distributions, defined in the natural manner by regular sequences.

It is often necessary to obtain expansions in spherical harmonics that belong to a given irreducible representation of a symmetry group. This is the case, for instance, when the cellular method is applied to investigate the band structure of a metal, where expansions are required that reproduce the symmetry of the group of the k vector (see Bouckaert, Smoluchowski and Wigner(4); von der Lage and Bethe (9)). Another instance where such expansions are necessary appears when hybrid orbitals are obtained for a central atom in a molecule of given symmetry (Kimball (8)). In this case lower order spherical harmonics are considered and tables for them up to l = 2 (functions s, p and d in real form) are given in the literature (cf. for example Eyring, Walter and Kimball (5)). However, interest has recently arisen in hybrids that include f functions (Shirmazan and Dyatkina (12)) and an extension of these tables appears to be desirable.

The paper studies, in a general way, how the random properties of a ‘medium’ influence the percolation of a ‘fluid’ through it. The treatment diifers from conventional diffusion theory, in which it is the random properties of the fluid that matter. Fluid and medium bear general interpretations: for example, solute diffusing through solvent, electrons migrating over an atomic lattice, molecules penetrating a porous solid, disease infecting a community, etc.

In (3), some of us proved that Brownian paths in n-space have double points with probability 1 if n = 2 or 3; but, for n ≥ 4, there are no double points with probability 1. The question naturally arises as to whether or not Brownian paths in n-space (n = 2 or 3) have triple points. The case of paths in the plane is settled by (4), where it is shown that, with probability 1, Brownian paths in the plane have points of multiplicity k (k = 2, 3,4, …). The purpose of the present paper is to settle the remaining case, n = 3. We prove that, with probability 1, Brownian paths in 3-dimensional space have no triple points. The general idea behind our proof is to show that there are not too many double points. That is, we show that the set of double points has sigma-fmite linear measure, and therefore zero capacity, with probability 1.

A simple solution of a general problem involving a bifurcated wave guide is presented. The purpose of the work is to explain a new and simple method of solving such problems and to exhibit an organic connexion between Sturm–Liouville theory and the theory of two-part boundary-value problems.

The Ising model of a two-dimensional ferromagnetic crystal containing defects is considered. It is shown that the combinatorial method is particularly appropriate to the investigation of the two-dimensional defect crystal and how it may be readily modified to calculate the partition function for such a crystal. It is found that the cooperative behaviour of the infinite perfect crystal, as indicated by the occurrence of a logarithmic singularity in the specific heat as a function of temperature, is not destroyed in the defect crystal; the singularity persists, though at a progressively lower temperature as the density of defects is increased, until a high density of defects is reached. The value of this limiting density of defects, above which the cooperative behaviour is destroyed, is calculated.

Let f (z) be meromorphic and non-rational in the domain |z| < R ≤ ∞, and let a be an arbitrary complex number, which may be infinite. The deficiency δ(a) of the value a is defined by

where m(r, a), N(r, a) and T(r) are defined as usual (cf. (10), pp. 156 ff.). For the class of functions considered in this paper the characteristic function T(r) is unbounded, and this will be assumed throughout. The upper (or Valiron) deficiency (16) of the value a is denned by

from which it follows that 0 ≤ δ(a) ≤ Δ(a) ≤ 1. A value a for which Δ(a) > 0 is called exceptional or deficient, and a value for which Δ(a) = 0 is called normal. We shall denote by G[a, σ] the open set of all values z in | z | < R for which | f(z) – a | < σ, where σ is a given positive number; we shall say that a component Gn[a, σ] of G[a, a] is bounded if the closure G¯n[a, σ] is contained in | z | < R, otherwise Gn[a, σ] will be called unbounded. In the case a = ∞, it is natural to define Gn[∞, σ] as the set of all z for which | f(z) | > 1/σ.

The previous paper (1) omitted the proof of the theorem on the connective constant of a crystal. This paper supplies the proof. For ease of reference, we first recall the relevant nomenclature and enunciate the theorem; to this extent the present paper is self-contained.

The formula to be established is

where α > ½, β > ½and b is real and positive. In § 3 we shall generalize this by proving

where α > (p−l)/p, β > (p−l)/p, b > 0 and p = 2,3,….

A brief discussion is given of attempts that have been made to justify, in terms of electrostatic principles, the conjecture that surface charge on a conductor tends to infinity towards a convex sharp point and to zero towards a concave point, and it is concluded that the problem has not hitherto been solved. A new attempt is then made using potential-theoretic methods, and the problem is solved with a fair degree of generality. The limitations are of a kind familiar in this type of analysis and, roughly speaking, concern the degree of convexity of the conducting surface.

An experiment is suggested to settle the question whether two clocks which are synchronized and subsequently separated must necessarily show identical readings on being brought together again. Charged π-mesons play the role of a clock travelling with a velocity comparable with c, this clock being read by passing the mesons through a scintillation counter or counters. It is pointed out that the experiment can be designed in a way which makes it technically feasible for a laboratory possessing an accelerating machine capable of producing mesons in the 100 Me V. energy range.

Some considerations suggested by the proposed experiment are examined, and it is concluded that only the orthodox result—that in general the two clocks will record the passage of different time intervals between successive coincidences—is consistent with the result of an experiment which has already been made, or with the special theory of relativity.

Asymptotic solutions of the differential equation

for large positive values of u, are examined; z is a complex variable in a domain Dz in which P1(z) and z2q(z) are regular and p1(z) does not vanish. In this paper it is shown that there exist Airy-type expansions of the solutions of this equation which are valid uniformly with respect to z in a domain in which z = 0 and z = z0 are interior points. If Dz is unbounded and the equation has a regular singularity at infinity, Airy-type expansions exist which are valid at z = 0, z = z0 and z = δ. If p(z) = constant + O (│z│-1) as │ z │ → ∞ in Dz, similar expansions also exist. The results given here are new.

In Feynman's ‘space-time’ formulation of quantum mechanics the Green's function for the Schrödinger equation is defined by an integral over all histories of the system. By integrating over one-parameter sets of functions, one gets the same Green's function as by integrating over a Fourier series, in simple cases. The method may be useful for estimating the result in cases when the integration over all histories cannot be performed exactly.

A new set of stability criteria for linear systems is derived. This shows that about half of the Hurwitz criteria are redundant when certain of the coefficients of the characteristic equation are known to be positive. The theory is applied to obtain a very short derivation of the known aperiodicity criteria. The conditions for realizability of RC networks are shown to be closely related to the stability and aperiodicity criteria, and are stated as sets of criteria in terms of the polynomial coefficients. Two basic theorems are involved which give the necessary and sufficient conditions for the roots of two polynomial equations to be real and separated.

1. Introduction. The object of this investigation is to obtain by means of contour integrals exact solutions of certain nth-order differential equations, together with their n independent power-series solutions, their asymptotic solutions and the relationship between these two types of solution. The Stokes phenomenon associated with the changing constants in these asymptotic solutions will then be investigated by various methods. Paper I is concerned with the properties of the various contour-integral solutions of the equations under consideration. The manner in which the arbitrary constants in the general asymptotic solution must change as the argument of the independent variable z varies is dealt with in paper II. This phenomenon is considered in a way that has proved profitable for the approximate solution of more general linear differential equations of the nth order that are approximately identical with those considered here hi certain regions of the complex z–plane. In later publications, these approximations will be described together with their application to explicit physical problems.

A new analysis is made of the coefficients appearing in periodic Mathieu functions, and this is applied to establish the limiting properties of certain solutions of the equation

The paper is concerned with the solution of the algebraic problems arising in the partial wave treatment of electron-hydrogen atom collisions. Explicitly antisymmetrized wave functions are used throughout. The boundary conditions are written in S-matrix notation and expressions for total and differential cross-sections obtained. The algebraic coefficients fλ and gλ occurring in the continuous state Hartree-Fock equations are expressed in terms of Racah coefficients, and tabulated as functions of the total angular momentum for atomic s, p and d electrons and all angular momenta of the scattered electron. Expressions are given for the calculation of first-order corrections to the results obtained using approximate wave functions.

The theory of time-periodic wave problems falls into two parts. On the one hand there is the rigorous formulation in terms of differential wave equations, on the other there are approximate theories like geometrical optics. It should be possible, in principle, to deduce the latter from the former by a logical process; but this has been done only for a few simple configurations, e.g. the circle. A possible approach to the solution of the general problem is suggested here, and is applied to a typical two-dimensional acoustical example. An arbitrary closed convex curve (satisfying certain regularity conditions) is emitting short sound waves towards infinity, the normal velocity V(s) exp (− iωt) is prescribed on the curve as a function of the arc-length s, and the potential is to be found, first on the curve and then at any point in the sound field. (Only the first part of the problem is treated in detail.) The potential ø(s) exp (− iωt) on the curve satisfies all the integral equations

where G(s, s′) is any Green's function of the problem, and V(a) is prescribed. All the equations corresponding to different Green's functions have the same solution. An asymptotic and convergent short-wave solution can be found by iteration if G can be chosen explicitly so that the integral equation has a small kernel for high frequencies. At any point of the curve draw the local circle of curvature; then the explicit known solution for a source on this circle is (with slight modifications) a possible Green's function, and the equation formed with it has a small kernel and can be solved rigorously by iteration. If V(a) is independent of the frequency, the leading term in the resulting asymptotic expansion is

where c is the velocity of sound and 2πk−1 is the (short) wavelength corresponding to the frequency ω/2π. If V(a) varies rapidly, as in diffraction theory, the iterative solution still gives a convergent asymptotic expansion, but the first approximation is then practically useless in the shadow region. Diffraction problems are not treated in the present paper.

The present work appears to be the first practical and rigorous solution of a short-wave problem in optics or acoustics when a solution in closed form is not available. It is suggested that the technique (suitably combined with formal expansions) may be applicable to a wider class of radiation and diffraction problems.

Consideration of the direction of the normal at any point on a continuous sheet of a surface yields a sufficient condition for the existence of parabolic points on that sheet. This condition has been used to derive some simple inequalities between elastic constants, whose fulfilment determines the existence of parabolic points upon the inverse surfaces of media of orthorhombic, tetragonal, cubic or hexagonal symmetry; in virtue of the polar reciprocal relation between inverse and wave surfaces, the existence of cusp points on the latter is thereby simultaneously established. It is also pointed out that conditions for external conical refraction may prevail in hexagonal media.

1. Introduction. In paper I(2), solutions have been found for the nth-order differential equations (I, 3), (I, 9), (I, 18), namely,

where ϑ = wd/dw, m is any rational fraction and a any constant. Power-series solutions have been found for these equations, together with integral representations and their asymptotic expressions valid for restricted ranges of arg z. The object of this second paper is to consider these asymptotic solutions in more detail, and to extend these expressions to all values of arg z. The Stokes phenomenon will be manifest throughout, and this will be treated in a manner suitable for further application.