This paper shows that the coherence function associated with a band-limited optical spectrum is expressible as an infinite product involving its zeros. Only a finite number of zeros are physically significant and these can be determined from measurements of the visibility of interference fringes; but an ambiguity remains in the sign of the imaginary part of each zero. If the spectrum is to be recovered using the visibility measurements, an auxiliary experiment is needed to supply the necessary signs, but it need not be especially accurate. If the signs are not known there is only a limited number of different spectra that are compatible with the visibility measurements. Finally it is suggested that wavelength measurements made on an asymmetric spectral line with a Michelson interferometer may yield differing results when used with long and short delays.

Segrè in his recent book [1] on Fermi gives an interesting account of the discovery of slow neutrons at the Institute of Physics of the University of Rome on October 22, 1934. He quotes (p. 80) a statement Fermi made many years later in a conversation with S. Chandrasekhar on the process of discovery in physics:

‘I will tell you how I came to make the discovery which I suppose is the most important one I have made. We were working very hard on the neutron-induced radioactivity and the results we were obtaining made no sense. One day, as I came to the laboratory, it occurred to me that I should examine the effect of placing a piece of lead before the incident neutrons. Instead of my usual custom, I took great pains to have the piece of lead precisely machined.

Norman Feather was one of the first physicists to realise the power of the coincidence method for elucidating nuclear level schemes. He observed coincidences occurring during the sensitive time of a Wilson cloud chamber in the 1920s. This note reports an application of some recent advances in coincidence technique in an attempt to set upper limits to the intensities of some unconfirmed γ-rays of 144Pr, which are indicated by broken lines in text-fig. 1.

The state of knowledge of fast neutron polarisation due to elastic scattering is reviewed in relation to the optical model of the process. There is clearly a substantial lack of polarisation data at neutron energies and concerning nuclei appropriate to the optical model description. The situation for the extremes of small and of large angle scattering is also considered.

The invitation, made to me, to write the introduction to this volume of the Proceedings of the Royal Society of Edinburgh, which is dedicated to Norman Feather, suggested that I should place on record some of my reminiscences which might convey a picture of his character and personality. I feel, however, that first I should make some reference, albeit brief, to the distinction of his contributions to science over the 45 years which have elapsed since he first started to research in the Cavendish Laboratory at Cambridge.

The asymptotic behaviour of double integrals over a portion of the plane is investigated when the integrand contains an exponential factor with a large parameter. The exponent can have a stationary point which may or may not be close to the boundary of the domain of integration. Results are first derived for a rectangular region with a particularly simple exponent in the integrand and shown to be uniformly valid under certain conditions. In some circumstances the asymptotic terms can be evaluated by means of a universal function. The theory is then generalised to cover more complicated exponents and arbitrary wedge-shaped domains; it is found that the asymptotic behaviour can still be expressed in terms of the same universal function.

It is a great pleasure to join in this tribute to Professor Norman Feather: My recollections of him as teacher, friend and colleague extend back nearly forty years. I recall attending, as a Cambridge undergraduate, his lectures on ‘Properties of Matter’ (what an archaic and nostalgic flavour that title has now!); and hearing him describe his pioneering experiments on the properties of the neutron. He was my doctoral thesis examiner; and later when he was Editor of the Cambridge monographs, it was at his suggestion, and with his help and encouragement, that I made my first essay as an author. I was privileged to succeed my former teachers, Professor C. D. Ellis and Professor Feather, at Trinity College; and in the 1950s I was regularly and warmly welcomed at Edinburgh, the ambivalent benevolence of my role as ‘External Examiner’ notwithstanding.

This paper is concerned with the asymptotic properties of the eigenvalues and eigenfunctions of the boundary value problem

With suitable restrictions placed on the real-valued coefficient q the spectrum of this problem, with respect to the eigenvalue parameter λ, is discrete; let {λn; n = 1, 2, …} and {ψn; n = 1, 2, …} be the eigenvalues and associated eigenfunctions. Asymptotic formulae are obtained for N(λ), the number of eigenvalues not exceeding the real number λ, and for ψn(x) as n→∞ where x is a fixed, positive real number.

The curve Γ common to two quadric surfaces has 24 principal chords; they are the sides of six skew quadrilaterals each of which has for its two diagonals a pair of opposite edges of that tetrahedron S which is self-polar for both quadrics. These three pairs of quadrilaterals serve to identify the three pairs of quadrics through Γ that are mutually apolar.

The vertices of the quadrilaterals lie four on each edge of S. Both nodes of the plane projection of Γ from such a vertex are biflecnodes.

Let P be a simplicial d-polytope, and, for – 1 ≤ j < d, let fj(P) denote the number of j-faces of P (with f_1 (P) = 1). For k = 0, ..., [½d] – 1, we define

and conjecture that

gk(d + 1)(P) ≥ 0,

with equality in the k-th relation if and only if P can be subdivided into a simplicial complex, all of whose simplices of dimension at most d – k – 1 are faces of P. This conjecture is compared with the usual lower-bound conjecture, evidence in support of the conjecture is given, and it is proved that any linear inequality satisfied by the numbers fj(P) is a consequence of the linear inequalities given above.

An example will be given of a compact linear set K which is of zero or non-σ-finite measure for every translation-invariant Borel measure. This answers a question asked me by C. Dellacherie.

Let D denote the unit disc in the complex plane. If p and q are two complex numbers, let x(p, q) denote the chordal distance between p and q on the Riemann sphere. In particular, we have the formula

and

The following problem was posed by Paul Gauthier: if f(z) and g(z) are meromorphic functions in D such that Clunie [4] has answered this problem in the negative by constructing different meromorphic functions f(z) and g(z) with the desired property. However, the functions constructed by Clunie both have an infinity of poles in D. It is the purpose of this note to give an example of two analytic functions―which thus have no poles―which also give a negative answer to the problem of Gauthier.

In [5] Knowles proved the following.

Every compact Hausdorff space with no isolated points admits a non-atomic measure.

This note is concerned with the converse problem in a more general set up. Here we deal with certain properties of the family of completely regular spaces admitting no continuous measures. In §3 it is shown that this family contains spaces with no isolated points, thus theorem (1.1) does not generalize to completely regular spaces. In §4 a canonical decomposition of the compact members of the above family into discrete subspaces is obtained, and it is shown that these spaces are metrizable whenever they satisfy the first axiom of countability.

The method of the large sieve has played a very important role in number theory. It turns out that estimates of exponential sums are of basic importance for large sieve inequalities. Let

be an exponential sum with complex coefficients c(n). It follows from Theorem 1 of Bombieri and Davenport [1] that

A section of the non-negative orthant by an affine subspace is a polyhedral set. A technique, analogous to that of Gale diagrams, is described which enables one to determine the facial structure of such a polyhedral set.

Both S. Bergman [1] and I. N. Vekua [13] have constructed integral operators which map ordered pairs of analytic functions of one complex variable onto solutions of fourth order elliptic equations in two independent variables. Such operators play an important role in the investigation of the analytic properties of solutions to higher order elliptic equations and in the approximation of solutions to boundary value problems associated with these equations. Unfortunately, little progress has been made in developing an analogous theory for elliptic equations in more than two independent variables. Recently, however, Colton and Gilbert [7] constructed integral operators for a class of fourth order elliptic equations with spherically symmetric coefficients in p + 2 (p ≥ 0) independent variables, and at present Dean Kukral [11], a student of R. P. Gilbert, is in the process of trying to extend some recent results of Colton [3, 4, 5] for second order equations in three independent variables to the fourth order case.