The methods developed in I, II of this series of papers are applied to a solution of a variety of dual series relations involving trigonometric series. In general the problem is reduced to one of solving (usually by numerical methods) a Fredholm integral equation of the second kind for an auxiliary function g(t), but for certain values of the parameters it is possible to obtain analytical solutions of the integral equations and these cases are considered in detail.

In this paper we examine the general paraboloidal co-ordinate system, in which the normal surfaces are elliptic or hyperbolic paraboloids, including as special cases the “parabolic plate” and the “plate with a parabolic hole”. We then show that normal solutions of Laplace's equation in these co-ordinates are given as products of three Mathieu functions, and apply this to the solution of boundary-value problems for Laplace's equation in these co-ordinates. In a subsequent paper the corresponding treatment of the wave equation will be given.

The solution of the dual series relations

where {λn) is the sequence of positive zeros of the Bessel function Jν(αλ), arranged in order of increasing magnitude, þ and ν are real numbers (−1 <þ < 1, ν >0), the functions, f1(ρ), f2(ρ) being prescribed, is obtained by giving an integral representation of {αn} in terms of a single function g(t). The problem is reduced to that of solving a Fredholm integral equation of the second kind for the auxiliary function g(t).

It is generally agreed that the long-range alpha particles of fission are set free before the fragment nuclei have acquired more than a small fraction of their final energy of separation, but whether the alpha particle is liberated before the instant of scission, at that instant, or from one of the fragment nuclei very shortly thereafter, has remained an open question. Each of these views has been seriously advocated. These various hypotheses are examined in relation to recently published information regarding the distribution of mass in low-eneigy ternary fission, and other considerations, and it is suggested that the hypothesis having the strongest claim to attention is that which assumes that the alpha particles originate in the heavy fragments exclusively, being liberated, very shortly after the instant of scission, with probability not much less than unity, from fragment nuclei of low yield and small neutron excess. Conclusions which would follow, if this hypothesis were accepted, are indicated, and possible experimental tests of these conclusions are suggested.

This note is concerned with an inconsistency in the assumptions made in the Reissner theory of elastic plates. An expression is derived for the stress component t33 and a form is suggested for the shear stress components tα3 which form a suitable basis for an approximate theory of elastic plates.

During the past seven years Northcott has published several papers (see, for example, [6, 7, 8, 9]) in which he has investigated the local aspect of the theory of dilatations. In a similar manner we shall develop in a later paper a local theory of monoidal transformations of which the global analogue appears in [2]. The present note is concerned with such a theory in the one-dimensional case and closely follows the development given in [8] for local dilatations. Indeed the theorems of the present note are all natural generalizations of theorems which have previously been given by Northcott, and for the most part the proofs are essentially Northcott's proofs.

Let ℋ be the system of all continuous increasing functions h(t), denned for t ≥ 0, with h(0) = 0 and h(t)>0 for t > 0. Let Ω be a separable metric space. Then, for each h of ℋ, we may introduce a Hausdorff measure into Ω, by taking

where d(Fi) denotes the diameter of Fi, and where the infimum is taken over all sequences {Fi} of closed sets, covering E and having diameters less than δ. We introduce a natural partial order in the system of these Hausdorff measures by writing j < h, if j, h are functions of ℋ and

A graph is said to be k-chromatic if its vertices can be split into k classes so that two vertices of the same class are not connected (by an edge) and such a splitting is not possible for k−1 classes. Tutte was the first to show that for every k there is a k-chromatic graph which contains no triangle [1].

Let X be a metric space and τ a non-negative function on the subsets of X. By the well-known Carathéodory process, we generate outer measures μδ(τ), for δ > 0, and (see §3). When, for every A ⊂ X, τA = (diamA)s for s ≥ 0, μ(τ) is the Hausdorff s-dimensional measure, and, if τA = h(diam A) for a monotone continuous function h with h(0) = 0, μ(τ) is the Hausdorff h-measure. In both of these cases, μ(τ) has been extensively studied.

The Local Uniformisation Theorem was proved by O. Zariski [5] in 1940, and, for the general case, it is so far the only existing proof of the theorem.

Let V be an irreducible manifold defined over a ground field of characteristic zero, and let Σ be its function field. Suppose V lies in an affine space An, and D is any subvariety of V not at infinity. Let J be the integral domain of V and ρ be the prime ideal in J defining D. Then we denote the quotient ring of D by Q(D|V), and by this we shall mean the quotient ring Jρ [1; p. 99]. Thus when we deal with subvarieties of two birationally equivalent manifolds V and V', then the quotient rings will always be subrings of the same representation of the function field of V and V'. Let B be any valuation of Σ whose centre on V is C. The Local Uniformisation Theorem states that there exists a birational transform V' of V such that the centre C' of B on V' is simple and Q(C/V) ⊆Q(C'/V').

The results given here represent an extension of previous work [1, 2] in which the author considered the oscillations of a plane current-vortex sheet in an ideal perfectly conducting fluid. In this paper we consider the effects of curvature of the sheet in a direction transverse to the velocity and magnetic field direction. This problem may be regarded as that of finding longitudinal small oscillations on a jet of fluid which moves along the lines of force of an impressed magnetic field. For oscillations, whose wavelength is small by comparison with the radius of curvature of the section of the jet, it is to be expected that the criterion for stable or unstable oscillations will be the same as for the plane case examined previously, and this is verified. When one considers the other extreme, in which the wavelength of the oscillations is large, the analysis shows that the magnetic field aligned to the jet has the effect of stabilising the jet, irrespective of the magnetic field strength. The magnetic field thus behaves for large wavelengths in the same way as a surface tension does for small wavelengths. For values of the applied magnetic field which would make the current-vortex sheet without curvature unstable, it is seen that there is a single transition from instability to stability as the wavelength increases. It is shown also that when small wavelengths are stable, in addition to large wavelengths, it does not necessarily mean that the jet will be stable for all wavelengths. Criteria are deduced to distinguish this case from another in which the jet remains unstable for a simple bounded range of intermediate wavelengths.

Let k be a finite field of q elements. The equation f(x, y) = 0, where f(x, y) is a polynomial with coefficients in k, may be construed to represent a curve, C, in a plane in which x, y are affine coordinates. On the other hand, this equation can be thought of as denning y as an algebraic function of x, where x is transcendental over k. The purpose of this paper is to show that, for a certain class of curves, corresponding in the classical case to curves having n distinct branches at x = ∞, if the degree, n (in y), of the polynomial f is large compared with q, then the genus† of C cannot be too small. We infer this result from a theorem about the genus of a function field; for we can think of C as being a model of such a field.

In this note we prove:

Theorem. Let Λ be a normal number field of absolute degree n. Then there exist infinitely many normal number fields K with the following properties.

(i) The Galois group of K over the rational field Q is the symmetric group Sn on n symbols.

Siegel defined an E-function to be a function which admits a power series expansion

with complex coefficients βn, belonging to some algebraic number field K (of finite degree over the rationals Q), satisfying the following conditions:

(i) Given ε > 0, the maximum of the absolute values of the conjugates of βn, satisfies

Let I be a homogeneous ideal in the polynomial ring R = Λ[X1, X2, …, XN], where Λ is a field or, more generally, an Artin ring†. Then R|I has an induced structure as a graded R-module and its homogeneous elements of degree n form a Λ-module of finite length. If this length is denoted by H(n, R|I), then H(n, R|I), considered as a function of n, is often known as the Hilbert function of the ideal I although, in other contexts, it is called the Hilbert function of the graded module R|I. We shall adopt the latter terminology.

Suppose we have a set of n points P1, …, Pn in a unit square. Let

and let Sn be the upper bound of d1+…dn taken over all positions of P1, …, Pn in the square. We prove that