The formula

was stated by Bateman ([2], p. 457); a proof is sketched in [3], p. 144. Here

the Laguerre polynomial of degree

Let K be a bounded n-dimensional convex body, with its centroid at the origin o. Let ϑ denote the density of the most economical lattice covering of the whole of space by K (i.e. the lower bound of the asymptotic densities of the coverings of the whole space by a system of bodies congruent and similarly situated to K, their centroids forming the points of a lattice); and let ϑ* denote the density of the most economical covering of the whole space by K (i.e. the lower bound of the asymptotic lower densities of the coverings of the whole space by a system of bodies congruent and similarly situated to K).

Let Q be a local ring and let q be an m-primary ideal of Q, where m is the maximal ideal of Q. With q we may associate a ring F(Q, q), termed the form ring of Q relative to the ideal q. If u1, …, um is a basis of q, and if B denotes the quotient ring Q/q, there is a homomorphism of the ring B[X1, …, Xm] of polynomials over B in indeterminates X1 …, Xm onto F(Q, q). The kernel of this homomorphism is a homogeneous ideal of B[X1 …, Xm]. Finally, if a is an ideal of Q there is a homomorphism of F(Q, q) onto F(Q/a, q+a/a). The kernel of this latter homomorphism will be termed the form ideal relative to q of a and denoted by ā.

Let ƒ = ƒ(x1, …, xk) be a quadratic form in k variables, which has integral coefficients and is not degenerate. Let n ≠ 0 be any integer representable by ƒ, that is, such that the equation

is soluble in integers x1, …, xk. We shall call a solution of (1) a bounded representation of n by ƒ if it satisfies

In integrating E-functions with respect to their parameters the contours are usually of the Barnes type, deformed if necessary to separate the increasing and decreasing sequences of poles of the integrands. Also the constants are taken to be such that the integrals converge. The following formulae are required in proving the theorems given in this paper.

Let D be the discriminant of an algebraic number field F of degree n over the rational field R. The problem of finding the lowest absolute value of D as F varies over all fields of degree n with a given number of real (and consequently of imaginary) conjugate fields has not yet been solved in general. The only precise results so far given are those for n = 2, 3 and 4. The case n = 2 is trivial; n = 3 was solved in 1896 by Furtwangler, and n = 4 in 1929 by J. Mayer [6]. Reference to Furtwangler's work is given hi Mayer's paper. In this paper the results for n = 5, that is, for quintic fields, are obtained.

It is well known that the thinnest covering of the plane by equal circles (of radius 1, say) occurs when the centres of the circles are at the points of an equilateral lattice, i.e. a lattice whose fundamental cell consists of two equilateral triangles. The density of thinnest covering is

Our main object in this note is to establish (Theorem 1) a necessary and sufficient condition to be satisfied by a sequence {εn} so that a series Σ an εnmay be summable | A |whenever the series Σanis summable (C, — 1). We suppose that an and εn are complex numbers. The condition is unchanged if the an are restricted to be real, but our proof is adapted to the case where they may be complex. Theorem 1 has been quoted by Bosanquet and Chow [12] in order to fill a gap in the theory of summability factors. We also obtain some related results, which are discussed in the Appendix.

An expression is found here for the small transverse displacement of a thin elliptic plate due to a force applied at an arbitrary point of the plate. The plate is in the form of a complete ellipse and is clamped along the boundary. The displacement is expressed in terms of infinite series in §§2–4. The convergence of the series is rapid unless the eccentricity of the ellipse is nearly unity. The simplest case in which the force is applied at the centre of the plate is considered in §5; the displacement of the centre due to this force is compared in §6 with the corresponding displacements of a circular plate and of an infinite strip.

The Faroe-Shetland Channel is the threshold from the north-eastern Atlantic Ocean to north-west European seas. Through it passes the main bulk of the oceanic water-mass which is the predominant influx, among several other water-masses, to these seas.

The following research into the dynamics and general hydrography of the region is based on numerous observations of temperature and salinity, from surface to bottom, taken mainly on two vertical cross-sections of the Channel between the years 1927 and 1952 inclusive, excepting the war years 1940 to 1945.

The research reveals very large scale seasonal and long-term variations in the northeastward volume-transport of oceanic water, suggests the existence on occasions of what appear to be horizontal tortional currents within the oceanic water-mass, and demonstrates (a) the intrusion of Gulf of Gibraltar (extra-Mediterranean) water into this mass over a period of years, (b) the formation of heavy oceanic water and (c) of a sub-oceanic watermass. The last-mentioned may sometimes almost entirely displace the bottom Norwegian Sea water-mass which normally underlies the oceanic mass.

One or other, or both, of two types of Arctic water may also sometimes displace bottom Norwegian Sea water as the bottom water-mass of the region, the process, like that of the above-mentioned Gulf of Gibraltar water influx, waxing and waning over a term of years and thus exemplifying the phenomenon of marine climatic change.

A Lie group is said to be metrisable if it admits a Riemannian metric which is invariant under all translations of the group. It is shown that the study of such groups reduces to the study of what are called metrisable Lie algebras, and some necessary conditions for a Lie algebra to be metrisable are given. Various decomposition and existence theorems are also given, and it is shown that every metrisable algebra is the product of an abelian algebra and a number of non-decomposable reduced algebras. The number of independent metrics admitted by a metrisable algebra is examined, and it is shown that the metric is unique when and only when the complex extension of the algebra is simple.

A study is made of faithful representations of the free cyclic (non-associative) groupoid by means of bifurcating root-trees and by means of index polynomials in two indeterminates emphasizing the intimate connection between these representations. The properties of trees and index polynomials are investigated and the concept of lattice of trees is introduced.

In this note we study the asymptotic behaviour of a product of matrices where Pj is a matrix of transition probabilities in a non-homogeneous finite Markov chain. We give conditions that (i) the rows of P(n) tend to identity and that (ii) P(n) tends to a limit matrix with identical rows.

A number of analogues to the simple fluid compressibility equation are deduced by considering fluctuations in a binary mixture; and their simplest expressions are found to be in terms of the binary mixture direct correlation functions. The accuracy of these results is tested with the aid of the appropriate extension of the approximate Born-Green theory, which facilitates the demonstration of consistency with the first three terms of the virial expansion. The problem of the evaluation of the corresponding radial distribution functions by means of X-ray or optical scattering is taken as far as the determination, in principle, of a concentration correlation function from observations of critical opalescence.

Given a power series with real non-negative coefficients and having radius of convergence p, a summability method P is defined as follows:

The main concern of this note is to establish conditions sufficient for one such method to include another.

The temperature gradient in the lower atmosphere can be directly determined by measuring the optical refractive index of the air. This method is suitable for use on the Greenland ice sheet where errors introduced by water vapour are small, and where the strong solar radiation reflected by the snow surface makes it difficult to measure temperature differences over height differences of about I metre.

The refraction was measured by observing the apparent vertical angle of each of a set of targets at distances up to 4 km. from a theodolite. The refraction was found to vary linearly with the distance of the target. The true vertical angle to the targets was determined when a second theodolite was available and reciprocal sights could be taken with it from the site of target to the fixed theodolite. The true vertical angle varied with time due to slow descent of the theodolite as the firn slumped; a correction for this was made. The standard error of the temperature gradient measurements was about 1.5 × 10−2 C.° per metre. It is considered that the method could be developed and improved so that over a range of only 100 metres temperature gradients could be measured to an accuracy of about 0·1° C. per metre.

The National Institute of Oceanography owes its formation to a growing need for more detailed and systematic knowledge of the physics as well as the biology of the oceans, and the feeling that the United Kingdom, in co-operation with the Commonwealth countries, should play a part in marine science more in keeping with its tradition of interest in the oceans and their navigation. The Institute is controlled by the National Oceanographic Council, which was granted a Royal Charter in 1949 and operates with the aid of Government grants.

The emphasis of the programme is on long-term research under the main headings:

Interchange of energy between the atmosphere and oceans

Response of the sea surface to wind and pressure changes

General circulation of the oceans

Free and forced oscillations in the oceans

Distribution of marine organisms

Organic production in the oceans

Bionomics of whales.

There are many other activities and as much as possible is done to work out the practical implications of the research, and to help other marine scientists. The headquarters of the Institute are at Wormley, near Godalming, Surrey, and its research ship, the R.R.S. “Discovery II”, is based at Plymouth.