An exact solution in finite terms is derived within the limitations of the classical thin-plate theory, for the problem of a thin circular plate acted upon normally by hydrostatic pressure distributed over the area of a concentric ellipse, and subject to boundary conditions covering the usual rigidly clamped and simply supported boundaries.

Hamiltonian circuits with associated word an n-cycle, in Schreier right coset graphs for symmetric groups Sn mod cyclic groups Zn, correspond to change ringing principles on n bells for which the plain course is the extent; that is, neither bobs nor singles are required. This connection is made explicit for the general case, and then specialized to the cases n = 4 (minimus) and n= 5 (doubles). In particular, all 102 no-call doubles principles on three generators are found and catalogued.

1. The theoretical study of the transport phenomena in metals requires the solution of a certain integral equation, which is formed by equating the rate of change of the distribution function of the conduction electrons due to the applied fields and temperature gradients to the rate of change due to the mechanism responsible for the scattering. This integral equation has so far been solved only in the simplest case where the electrons are assumed to be quasi-free, the energy being proportional to the square of the wave vector, and this assumption will be made throughout the present paper. The scattering of the electrons is due to two causes: the thermal vibration of the crystal lattice and the presence of impurities or strains. The scattering due to the impurities can always be described in terms of a free path l (or more conveniently a time of relaxation τ, were l = τν, ν being the average velocity of the conduction electrons), whereas for the scattering due to the thermal vibration this is possible only at high temperatures such that (Θ/T)2 can be neglected, where Θ is the Debye temperature (1). At very low temperatures the scattering due to the thermal vibration can be neglected compared with that due to the impurities. When a time of relaxation exists, the integral equation for the distribution function reduces to an ordinary equation, and its solution is then a comparatively simple matter. In the general case the problem is much more difficult, and no rigorous and generally valid solution has so far been obtained.

R. A. Millikan and C. F. Eyring have shown that the auto-electronic emission is independent of the temperature of the cathode up to 1000° K. At higher temperatures a thermionic emission takes place and this masks the auto-emission. W. S. Pforte and the author have recently shown that the thermionic emission is increased by the field in accordance with a relation derived by Schottky, so that when both thermionic and auto-electronic emissions are present it is only necessary to subtract the thermionic current, as given by the Schottky relation, from the total measured current to obtain the value of the auto-emission at the temperature considered. In this way the thermionic and auto-emissions can be separated.

Let σs(n) denote the sum of the sth powers of the divisors of n,

and let , where ζ(s) is the Riemann ζfunction. Ramanujan, in his paper “On certain arithmetical functions*”, proves that, if

and

then , whenever r and s are odd positive integers. He conjectures that the error term on the right of (1·1) is of the form

for every ε > 0, and that it is not of the form . He further conjectures that

for all positive values of r and s; and this conjecture has recently been proved to be correct.

Although the classification of affine cubic curves was undertaken by Newton(4), in one of the first major exercises ever in coordinate geometry (see Cayley(2) for a fuller account), a parallel study of cubic functions seems not to have been contemplated till recently. The essential difference is that a function f defines a pencil of curves – its level curves – which have to be considered simultaneously. The author's interest in the subject arose from problems in singularity theory (concerning canonical stratifications), and a later paper in the series will have applications of this kind. Here we study the simplest case as an introduction. Our techniques are entirely classical, but the results are hard to find elsewhere. In later papers we intend to study cubic functions on ℝ2, and on ℂ3.

The two-dimensional flow in a jet, falling under gravity from a slot in a horizontal plane, is studied. The fluid is considered to be incompressible and inviscid; the flow is taken to be irrotational; and the reciprocal ε of the Froude number is considered to be small. By taking the complex potential as the independent variable we overcome the difficulty that the boundary geometry is not known in advance. The method of matched asymptotic expansions is applied. The first two terms of an inner asymptotic expansion and the first three of an outer one are found: the inner expansion is valid above and near the slot, but is inappropriate far downstream, while the outer expansion is valid far downstream, but fails to satisfy the conditions upstream. The two expansions are matched and ‘composite’ approximations, covering the whole flow field, are derived.

The physical significance of the parameters which appear in the combinatory factors g(Ni) and g(Ni, Xij) in the statistical mechanics of mixtures is examined. Hitherto, closed ring molecules have been excluded from the jurisdiction of the formulae for these combinatory factors. The means by which this restriction on the validity of the formulae can be removed is indicated. It is concluded that the equations given by Guggenheim are formally applicable in all cases provided that appropriate values, which differ according as the molecules are simple and branched chain or closed ring molecules, are substituted for certain parameters which occur in the equations.

In a note in the April 1942 issue of these Proceedings, Hodge gives a formula for the number of independent terms in what he calls a k-connex, but while conjecturing its general validity he proves it only for very restricted cases. Its truth is here demonstrated in the general case.

1. The results. Let l be an odd prime, r ≥ 1, and let

be the quaternion group of order 4lr, as given by generators and relations. Throughout N is a tamely ramified normal number field with Galois group Gal (N/Q) = H (a ‘quaternion field’), and its ring of integers. We are interested in the structure of as a module over the integral group ring ZH. Deriving, first, certain classgroup invariants for locally free ZH-modules, we shall then determine those for the module in terms of the arithmetic invariants of N/Q. When 1 ≡ – 1 (mod 4), this yields again a Galois module interpretation of Artin root numbers quite analogous to that in (2). On the other hand for l ≡ 1 (mode 4), we shall get a weak ‘normal integral basis theorem’. The original impetus for this work came from computations of J. Queyrut, who – in different language – obtained these results in the case l = 3, r = 1 (cf. (7)). The tools, we are using, come from the general theory developed in recent years with such concrete applications in mind, and it is perhaps of interest to see how the various ‘strands’, on root numbers (cf. (3), (4)), on locally free modules (cf. (5)), and on Galois module structure (cf. (6)) are here pulled together. For technical reasons, we shall impose on N the slight further restriction, that l be non-ramified, although our results would remain true without this. Both the statements and the proofs of the theorem depend on ideas contained in (5) and (6). The reader who is prepared to take them for granted should, however, be able to read the present paper independently of those papers.

We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.

In recent years there has been an interest in using elliptic logarithms to find integral points on elliptic curves defined over the rationals, see [23], [17], [6] and [12]. This has been partly due to work of David [5], who gave an explicit lower bound for linear forms in elliptic logarithms. Previously, integral points on elliptic curves had been found by Siegel's method; that is, a reduction to a set of Thue equations which could be solved, in principle, by the methods in [19]. For examples of this method see [3], [7], [16], [18], [21], [22] and [8]. Other techniques can be used to find all integral points in some special cases, see, for instance, [14].

This paper is in two parts. In Part I we are concerned with one or more linear series on an algebraic curve; we consider a set of points on the curve which are contained with assigned multiplicities in a set of each of the linear series and, by persistent use of Severi's equivalence relation for the united points of an algebraic correspondence with valency, we derive formulae for the number of such sets of points when the constants involved are such as to make this number finite. All this is essentially a generalization of the formula for the number of points in the Jacobian set of a linear series of freedom 1, and the main result is Theorem 3.

Posner (1) proved the following result:

Theorem. Let R be a prime ring and d a derivation of R such that, for all a∈R, ad(a)–d(a) a∈Z (centre of R). Then, if d is not a zero derivation, R is commutative.

The purpose of this note is to give a short proof of this theorem. We state the following known result ((1), Lemma 1).

In a recent paper (2) Rankin has proved that for all positive integral n where As stated by Rankin, his proof is not simple although it is based on elementary ideas. Also, as noted by him, no other proof is available for any inequality of the form with λ a positive constant independent of n. It is the main purpose of this note to give a simple proof for the following result which implies the truth of (3) with λ = ¼, though not the truth of (1).

The problem of conduction of heat in a solid whose surface is in contact with fluid, which is so well stirred that its temperature is constant throughout its mass, often arises in practice, for example in calorimetry, or in the warming of a closed room. Despite their importance, problems of this type have not been much studied since the presence of the mass of fluid at the surface gives rise to a boundary condition for which the classical methods of solution do not apply without modification; the Laplace transformation method, however, has the advantage that it can be applied in the same way to all cases.

We show that every locally compact Polish group is isomorphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultrametric space as a bundle with an ultrametric base and with ultrahomogeneous fibers which are invariant under the action of the isometry group.

In a previous contribution to these Proceedings (Ramakrishnan(1)) the concept of product density was introduced to describe the statistical distribution of a discrete number of particles in a continuous space E, corresponding to a single point t, where t is the parameter with respect to which the stochastic process evolves. This is extended to densities corresponding to n points on the t axis and correlation problems associated with these density functions are studied with particular reference to electron-photon cascades.

We show that the mapping class group of a closed, connected, oriented surface of genus at least three is generated by 3 elements of order 3. Moreover, we show that the mapping class group of a closed, connected, oriented surface of genus at least three is generated and by 4 elements of order 4.

Complex variable methods are applied to obtain exact solutions for the complex potentials and deflexions of thin isotropic slabs bounded by regular curvilinear polygonal contours with n sides and subject to symmetrical loading distributed over a concentric circle. The supported boundary is either clamped or has equal boundary cross-couples. The plates taken in the z-plane are conformally mapped on the unit circle in the ζ-plane by the mapping function . Polynomial approximations to the Schwarz—Christoffel transformations are then used to discuss the bending of clamped and simply supported rectilinear plates symmetrically loaded over a concentric circle or acted upon by a central point load.