About twenty years ago, in a note of the same title [2], I obtained the following result.

THEOREM 1. Let u and v be relatively prime integers satisfying u> v ≥ 2 and let ε be an arbitrarily small positive number. Suppose the inequality

is satisfied by an infinite sequence of positive integers n1 n2, … Then

J. R. M. Radok [1] has applied complex variable methods to problems of dynamic plane elasticity. The object of this paper is to show that his results may be obtained in a somewhat simpler way by a more systematic use of complex variable analysis.

In fact it is shown that the problems may be reduced to a form similar to that of the static aelotropic plane strain problems considered by Green and Zerna [2].

Let λ be a random variable with the distribution function F(λ). A transform of F which has, in effect, been used in several recent papers ([1], [2], [3], [4]; see also [6]) is

defined formally by the equation

It is the main purpose of this paper to prove the inversion formulae given in the two theorems below.

Recently H.-E. Richert [10] introduced a new method of summability, for which he completely solved the “summability problem” for Dirichlet series, and which led also to an extension of our knowledge of the relations between the abscissae of ordinary and absolute Rieszian summability. This non-linear method, which may best be characterized by the notion “strong Rieszian summability” †, depends on three parameters, on the order k;, the type λ, and the index p;. While Richert's paper deals almost exclusively with the application of that method of summability in a specialized form (namely the case p = 2, λn=log n) to Dirichlet series, it is the object of the present paper, to consider the general theory of strong Rieszian summability.

If p is a prime other than 2, half of the numbers

1, 2, … p—1

are quadratic residues (mod p) and the other half are quadratic non-residues. Various questions have been proposed concerning the distribution of the quadratic residues and non-residues for large p, but as yet only very incomplete answers to these questions are known. Many of the known results are deductions from the inequality

found independently by Pólya and Vinogradov, the symbol being Legendre's symbol of quadratic character.

Let S be an ordered set, i.e. a set with a transitive irreflexive binary relation “<” such that, for any a, bεS, either a = b or a < b or b < a. By an order automorphism of S we mean a one-one mapping α of S onto itself such that

The purpose of this paper is to prove a theorem which concerns the normal subgroup of a free product Π generated by a given subset Ω. This theorem was stated in the first paper of this series (Britton [1]) and an application was made to the word problem. The present work is, however, independent.

Airy's integrals

can be expressed ([1], [2]) in terms of Bessel functions. In this paper integrals of the types

are discussed. Various subsidiary formulae are given in § 2, some integrals of the type

are evaluated in § 3, and from these the integrals of the Airy type are derived in § 4.

Let

be the rath cyclotomic polynomial, and denote by An the absolute value of the largest coefficient of Fn(x).Schur proved that

and Emma Lehmer [5] showed that An>cn1/3 for infinitely many n; in fact she proved that n can be chosen as the product of three distinct primes. I proved [3] that there exists a positive constant q such that, for infinitely many n

and Bateman [1] proved very simply that, for every ∈>0 and all n>no(∈),

This paper deals with problems of transverse displacements of thin anisotropie plates with the most general type of digonal symmetry [1]. Proofs of uniqueness of solution under certain conditions are given for problems of plates occupying both finite and infinite regions. This is a generalization to anisotropy of the uniqueness theorems given by Tiff en [2] for isotropic plates.

The problem considered here is the determination of the stresses and displacements in a semi-infinite elastic plate which contains a thin notch perpendicular to its edge, and is in a state of plane strain or generalized plane stress under the action of given loads. The axes of x and y are taken along the infinite edge and along the notch, and the scale is chosen so that the depth of the notch is unity (Fig. 1).

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