In a previous paper (Shenton, 1953) we have given an expansion for integrals of the form This expansion may be expressed as a determinantal quotient or Schweinsian series. In the present paper we state more general terms under which the expansion holds and consider the case when the limits of integration are infinite and the weight function of the form In particular we giye expansions for the Psi function, and where C (x) is a positive polynomial.

In this note I prove the following result:—

Theorem. A compact, orientable, Riemannian manifold Mn, with positive definite metric and zero Ricci curvature, is flat if the first Betti number R1 exceeds n — 4.

In this statement of the theorem it is assumed that the dimensions of Mn are not less than four. If this is not the case, the result is still valid but appears as a purely local result and is true for a metric of arbitrary signature.

Vajda's paper in this volume has suggested to the author the following problem:

Let A be a n × m matrix, m≦ n, let B be a m × n matrix, and let M = I – AB, where I is the unit matrix of order n. Given A, to find B such that of the n latent roots of M′M, k are unity, and the remaining n — k are zero.

The geometry associated with the five invariants of two quadrics is well known. There appears to be an omission, however, in the treatment of the Φ-invariant. Salmon gives the vanishing of Φ merely as a necessary condition for the possibility of the construction of a tetrahedron self-conjugate for one of the quadrics and having its six edges tangential to the other. Sommerville proves its sufficiency, and shows that this condition is poristic, giving rise to two systems of ∝ tetrahedra of the requisite type. No investigation appears to have been made of the locus of the vertices of the ∝ tetrahedra of each system and the dual problem regarding the nature of the developable surface arising from their ∝ faces.

We have shown in [1] that under certain conditions the definite integral may be approximated by a determinantal ratio. It is our object now to develop the theory when C (x) is a polynomial, showing the relation to the continued fraction form for . In particular we shall give various forms for the approximants, and an integral form for the numerator.

The diffraction of a simple harmonic wave train by a straightedged semi-infinite screen was originally discussed by Sommerfeld in 1895. The analysis is of a recondite character, involving the use of multivalued functions and Riemann surfaces (1). An alternative formulation of the problem is as an inhomogeneous Wiener-Hop integral equation, the solution of which also involves considerable difficulties (2). It is the purpose of this note to show that following Friedlander (3) it is possible by the use of parabolic co-ordinates to solve the problem by elementary methods. The method can be applied either to the case of sound or that of electromagnetism, the results being formally identical.

This note gives a proof of the result:

A necessary and sufficient condition that a trigonometrical series T (x) be the Fourier series of a function is that σn – σm = O(n-n) uniformly in [0, 2π] for all m≤n, where σn is the nth (C, l) mean of T (x).

1. This note gives an asymptotic evaluation of an integral of the form

as n tends to infinity, where is a sequence of real-valued functions. The theorem to be established is a natural extension of B. Levi's generalised Laplace-Darboux theorem (1, 341-51); it gives a rule for evaluating a wider class of asymptotic integrals.

Introduction and Notation. In this paper all the scalars are real and all matrices are, if not stated to be otherwise, p-rowed square matrices. The diagonal and superdiagonal elements of a symmetric matrix, and the superdiagonal elements of a skew-symmetric matrix, will be called the distinct elements of the respective matrices. Σ will denote both the set of all symmetric matrices and the ½p(p + 1)-dimensional space whose coordinates are the distinct elements arranged in some specific order. K will denote both the set of all skew-symmetric matrices and the ½p(p – 1)-dimensional space whose coordinates are the distinct elements arranged in some specific order. Any sub-set of Σ(K) will mean both the sub-set of symmetric (skew-symmetric) matrices and the set of points of Σ(K). Any point function defined in Σ(K) will be written as a function of a symmetric (skew-symmetric) matrix. Dα will denote the diagonal matrix whose diagonal elements are α1, α2, …, αp. The characteristic roots of a symmetric matrix will be called its roots.

A useful test for uniform convergence is that first established by Buchanan and Hildebrandt [4] which is as follows.

(A) “If a sequence fn (x) of monotonic functions converges to a continuous function f(x) in [a, b] then this convergence is uniform.”

In §1 of this paper it is shown that this test is included in a sequence of theorems, each of which establishes a type of uniform convergence. The first is a well-known topological theorem on limit sets, the second is a result on the limits of rectifiable arcs, the third is a generalisation of (A) due to Behrend [3], the fourth is (A) itself, the fifth is a one-sided version of Bendixson's test and the sixth is Bendixson's test.