Suppose that a sequence of discs

arranged in decreasing order of diameters, forms a packing within the unit plane square I2. It has been shown, by Florian(1), that the area of

is at least O(a), where a is the radius of θn. However, Gilbert (2) has produced some empirical results for the Apollonius packing 71 of discs which seem to suggest that for such a packing, the area of the set

is at least O(as) for some positive real number s, less than one. As Gilbert remarks, it is difficult to imagine that the Apollonius packing is not the extremal case, and so, that it would seem likely that there exists a positive real number s, less than one, such that for a general packing, the area of

is at least O(as). The purpose of this paper is to establish this result by showing that 0·97 is an allowable value for s.

Recently Fox ((5)) has given an inversion formula for the transform whose kernel is xνKν(x), where Kν(x) is the modified Bessel function of the second kind, by the application of fractional integration theory. In the present paper it has been shown that the integral equation

can be thrown into the form of a Laplace integral, with the help of fractional integration, which can be solved by known methods.

In the usual notation let

where .

Also, for the generalized hypergeometric function pFq(x) let us employ a contracted notation and write

Throughout the present paper i will run from 1 to p, I from 1 to P, and so on. Thus ((a) )m is to be interpreted as

,

and similar interpretations hold for ((A))m, etc.

1. Let (S, S,μ) be a measure-space, and let f(x) and ø(x) be two μ-integrable functions defined in S. We shall assume that f(x) is a ‘difficult’ function, of which all that is known is how to calculate f(x), generally by a rather complicated process, for any given x∈S; and that ø(x) is an ‘easy’ function, whose value for any x, and whose integral over any subset T of S belonging to the class S of μ-measurable sets, are relatively simple to obtain. We shall define the sets

and assume that

It will be of interest to consider those functions ø(x) which may be regarded as approximations to f(x) in S.

Corresponding to a fixed sequence {μn}, the Hausdorff method of summability (H, μn) is defined by the sequence-to-sequence transformation†

where we write

The quasi-Hausdorff method (H*, μn) is defined by the transformation

thus the matrix of the (H*, μn) transformation is the transpose of that of the (H*, μn) transformation. A method introduced by Ramanujan (9), which we will call‡ (S,μn) is given by the transformation

Thus the elements of row n of the matrix of the (S, μn) transformation are those of the corresponding row of the (H*, μn) transformation moved n places to the left.

A solution of Einstein's vacuum field equations, apparently new, is exhibited. The metric, which is homogeneous (that is, admits a three-parameter group of motions transitive on space-like hypersurfaces), belongs to Taub Type V. The canonical form of the Riemann tensor, which is of Petrov Type I, is determined.

Introduction. Agarwalin 1953 ((1)) gave the most general transformations involving bilateral cognate trigonometrical series of the hypergeometric type. Later, Shukla in 1957 ((2)) gave certain relations involving the products of two bilateral hypergeometric series. In this paper an attempt has been made to establish certain new transformations involving the products of two and more bilateral hypergeometric type of series. (For notations and definitions see Agarwal ((1)).)

The problem of diffraction of a plane wave by a semi-infinite grating of iso-tropic scatterers leads to the consideration of a non-linear integral equation. This bears a resemblance to Chandrasekhar's integral equation which arises in the study of radiative transfer through a semi-infinite atmosphere. It is shown that methods which have been used with success to solve Chandrasekhar's equation are equally useful here. The solution to the non-linear equation satisfies a more simple functional equation which may be solved by factoring (in the Wiener-Hopf sense) a given function. Subject to certain additional conditions which are dictated by physical considerations, a solution is obtained which is the unique admissible solution of the non-linear integral equation. The factors and solution are found explicitly for the case which corresponds to closely spaced scatterers.

Throughout this paper we shall denote the space Lp(− ∞, ∞) by Lp, and the numbers p and p′ will be related by 1/p + 1/p′ = 1.

The Poisson operator Pa is defined by

The propagation of magneto-elastic plane waves is a topic which has recently aroused much interest. Thus Paria ((l)), for example, and the present author ((2)), have studied in some detail the interaction between the magneto-elastic field and the thermal field. Analyses published so far, however, have neglected the effects due to the passage of the wave upon the magnetic permeability of the medium. It is the object of the present paper to consider these effects and it will be shown that although the elastic strains induced cause but a small change in the numerical values of the permeability, nevertheless the effects upon wave propagation may be very considerable. The principal result is the demonstration that the effects considered are equivalent so far as the dispersion equation is concerned, to an anisotropic rescaling of the primary magnetic field, the direction of propagation of the wave being a preferred direction. For simplicity, thermal effects are not considered here.

Two natural definitions for the distribution function of the height of an ‘arbitrary local maximum’ of a stationary process are given and shown to be equivalent. It is further shown that the distribution function so defined has the correct frequency interpretation, for an ergodic process. Explicit results are obtained in the normal case.

By representing integral operators defined by kernels involving Bessel functions as compositions of two operators, we determine their mapping properties and, in one case, derive an inversion process. The operators considered are of the form

where τ(t) is given respectively by and and in the last three cases the integrals converge in norm.

Let f = f(x1, …, xn) be a positive definite quadratic form in n ( ≤ 8) variables; then we consider the old problem of estimating the minimum of f (its least value for integers xi not all 0) in terms of the determinant Δ(f). Normalizing by supposing the minimum to be 1, the known results may be stated as

each inequality being best possible. Further, for each n, all forms for which equality holds in (1) have integral coefficients and are equivalent to each other.

This paper contains the derivation of a metric for flat space-time associated with the supertranslations of the Bondi-Metzner group, and a description of the geometry of the null hypersurfaces on which the metric is based.

Introduction. The subject of the mixtures of probability distributions has aroused a fresh interest in the fields of probability and statistics. Several authors, including Blischke, Rider, Robbins, Teicher, Patil and Seshadri, have discussed a variety of the associated probabilistic and statistical problems in varying details in the recent past. This paper attempts to solve by using the methods of infinite matrices some of the problems of the identifiability of countable mixtures of discrete probability distributions. It appears that here is still another area where Mathematics and Statistics can help one another grow. We are extremely thankful to Dr P. Vermes of Birkbeck College of the University of London for developing some theory of infinite matrices relevant to a couple of problems discussed in this paper. While most of the results discussed below are believed to be new, a few of them can be shown to be immediate consequences of a powerful result due to Teicher ((7)) obtained by using a different approach.

This note supplements our paper ((4)) on the hydromagnetics of an ellipsoid moving in a cross-field. Familiarity with this paper will be assumed. Together they account for creeping motion in an arbitrary direction when the applied magnetic field is parallel to a principal axis.

A well-known result in the interpolation theory of integral functions (see Whittaker (16, 17), Pólya (14), Iyer (2), Pfluger (10)) states that an integral function of at most type K < ½π of order 2 bounded at the lattice points m + in (m, n = 0, ± 1, ± 2, … ) is necessarily constant. That the value ½π cannot be increased is shown by the Weierstrass σ-function. The result has, however, been generalized in several ways; the lattice points being replaced by more general sets and the bounded-ness condition by one of restricted rate of growth.

We describe some properties of a stationary, isolated, axially symmetric, rotating body of perfect fluid, according to general relativity. We first specialize to the case of constant specific entropy and constant angular velocity. The latter condition is equivalent to rigidity in the Born sense; both conditions are consequences of a simple variational principle. The hydrodynamic equations can then be integrated completely. Analogous first integrals are given also for the case of differential rotation. No use is made of the full field equations.

The effect of restricting the displacements of a quantum mechanical harmonic oscillator to semi-infinite and finite regions is studied. It is shown how the eigenvalue problem may be solved, both graphically, and from certain series expansions.

Within the limitations of the classical small deflexion theory of thin plates and using complex variable methods, exact expressions are obtained in series form for the deflexion at any point of a thin isotropic circular plate simply supported along a concentric circle and subject to loading symmetrically distributed over an eccentric circular patch which lies inside the circle of support. In special and limiting cases the solutions reduce to those obtained before.