A Banach space which is not reflexive may or may not be equivalent (in Banach's sense) to an adjoint space. For example, it is an elementary fact that the space (l), though not reflexive, is equivalent to (co)*, where (co) is the space of all sequences that converge to zero, normea in the usual way. On the other hand, (co) itself is not equivalent to any adjoint space : this can be proved by means of the Krein-Milman theorem, but here we obtain the result by an elementary argument which is scarcely more complicated than the standard proof that (co) is not reflexive.

Whittaker's studies of the relations between physics and philosophy are contained in about forty books and papers, nearly all of which were published during the last fifteen years of his life. His approach is mainly historical and is a natural sequence to his History of the Theories of Aether and Electricity. Many of the publications were based on lectures delivered to various groups and there is naturally a certain amount of repetition.

In this note generalisations of certain integrals involving Legendre functions including the Mehler-Dirichlet integral for Legendre functions of the first kind are given, these new results expressing associated Legendre functions of the first or second kinds as integrals involving corresponding functions of the same degree but different order. These integrals appear to be analogous to Sonine's integral in the theory of Bessel functions.

Certain functions of infinite matrices are known to exist.† This gives rise to the following questions:

1. Whether the power series of matrices

has a zero in the field ‡ of infinite matrices, and

2. If f(A) exists for a certain infinite matrix A, is there an infinite matrix B such that

In other words, is there a matrix period for f(A)?

In this paper theorems concerning zeros and periodicity of functions of semi block infinite matrices § (defined below) are established.

A special meeting of the Edinburgh Mathematical Society was held in the Mathematical Institute, Edinburgh, on Friday, 17th May 1957, to commemorate Sir Edmund Whittaker's work. Present at this meeting were Sir Edmund's widow, Lady Whittaker, and members of her family.

Dr J. M. Whittaker, F.E.S., on behalf of Lady Whittaker and family, asked the Society to accept the sum of £500 to be used to provide a prize which would be awarded for meritorious work in mathematics by young mathematicians in Scotland.

The application of the hodograph method in problems in fluid dynamics dates back to the time of Helmholtz and Kirchhoff. The underlying principle is simple. It is in effect to rewrite the governing differential equations with the roles of the original dependent and independent variables reversed. Such a procedure is not uncommon in problems depending upon ordinary differential equations. For example, if the velocity of a particle in rectilinear motion is prescribed as a function of distance from a fixed point, the problem of finding the relation between its position and the time t can be solved by one quadrature if t is regarded as the dependent variable.

One of the outstanding theorems in the theory of continued fractions is the result described by O. Perron, Die Lehre von den Kettenbrüchen (1912, 1929, 1954–7) as the transformation of Bauer and Muir (for brevity I shall call it the BM theorem); this theorem and limiting cases of it give rise to numerous extremely interesting consequences.

Making use of properties of doubly-stochastic matrices, I recently gave a simple proof (4) of a theorem of Ky Fan (Theorem 2b below) on symmetric gauge functions. I now propose to show that the same idea can be employed to derive a whole series of results on convex functions ; in particular, certain well-known inequalities of Hardy-Littlewood-Pólya and of Pólya will emerge as specìal cases.

Agnew (1) has defined a binary transformation T(α), with α real, as one which takes the sequence {Si}, i = 0, 1,…, into the sequence {si(1,α)} where

The classic application of dual integral equations occurs in connexion with the potential of a circular disc (e.g. Titchmarsh (9), p. 334). Suppose that the disc lies in z = 0, 0≤ρ≤1, where we use cylindrical coordinates (p, z). Then it is required to find a solution of

such that on z = 0

Separation of variables in conjunction with the conditions that ø is finite on the axis and ø tends to zero as z tends to plus infinity yields the particular solution .

In the full linear theory of thermoelasticity there is a coupling between the thermal and the purely mechanical effects so that not only does a nonuniform distribution of temperature in the solid produce a state of stress but dynamical body forces or applied surface tractions produce variations in temperature throughout the body. In a recent paper (Eason and Sneddon, (2)) an account was given of the calculation of the dynamic stresses produced in elastic bodies, both infinite and semi-infinite, by uneven heating. In this paper we shall consider the propagation of thermal stresses in an infinite medium when, in addition to heat sources, there are present body forces which vary with the time.

Let A = [aij] be an n-th order irreducible non-negative matrix. As is very well-known, the matrix A has a positive characteristic root ρ (provided that n>l), which is simple and maximal in the sense that every characteristic root λ satisfies |λ| ≤ρ, and the characteristic vector x belonging to ρ may be chosen positive. These results, originally due to Frobenius, have been proved by Wielandt (4) by means of a strikingly simple basic idea. Recently, a variant of Wielandt's proof has been given by Householder (2).

Let K be a valued field, let v denote its valuation and B its valuation ring. Let P denote the valuation ideal. For each a in B, let ā denote the residue class a + P in the field B/P; for f(x)=∑arxr in B[x], let f(x) denote ∑ārxr in B/P[x]. Let Λp denote the leading coefficient of a polynomial p, and ∂p the degree of a non-zero polynomial.