Let Λ′ be a lattice in three dimensional space. Let G be any point and denote by Λ the “displaced lattice”, consisting of all points X + G where X ε Λ′. Suppose that a sphere of radius 1 is centred at each point of Λ and a sphere of radius R−l, where 1 < R ≤ 2, is centred at each point of Λ′. If the lattice Λ′ and the point G are such that no two spheres of the system overlap we shall call Λ a mixed packing lattice, and shall denote its determinant by d(Λ) = d(Λ′). Let Δ be the lower bound of d(Λ) taken over all mixed packing lattices Λ. It can easily be proved that this is an attained lower bound, and any mixed packing lattice Λ, having d(Λ) = Δ, will be called a critical mixed packing lattice. We prove that

and shall describe, in Lemmas 1–3, the critical mixed packing lattices.

The stability of a non-rotating column of liquid was discussed by Rayleigh [1] who showed that axisymmetric disturbances of wave-lengths greater than the circumference of the column decreased the surface area and the consequent release of surface energy enabled the disturbance to grow. Rayleigh [2] also considered the effect of viscosity on the same problem and found that for high viscosity the range of stability was unaltered. For disturbances which are the same in all planes perpendicular to the axis of the column, it is clear that any disturbance increases the length of the boundary of a plane section of the column and hence increases the surface energy. Thus the column is stable for these disturbances.

The membrane theory for calculating stresses in symmetrically loaded elastic shells of revolution was introduced in 1828 by Lamé and Clapeyron [1] who assumed that a thin shell is incapable of resisting bending. In 1892 Love [2] gave the general equations of equilibrium for an element of an elastic shell taking bending into consideration and obtained expressions for the strains in terms of the displacements as well as the stress-strain relations. Since then the problem of the elastic shell has been the subject of numerous researches. The spherical shell has, however, drawn the attention of many investigators due to its importance in structural and mechanical engineering, e.g. roof and boiler constructions.

An ordered triangle in the plane S2 is defined as a sextuple (PlP2, P3; l1, l2, l3) consisting of three points Pi and three lines lj restricted by the relations of incidence Pi Ì lj (i ¹ j) If we map unrestricted sextuples by the points of a V12—Segre product of six planes—we obtain an image-manifold Ω6 for ordered triangles as the appropriate subvariety of V12. The variety Ω6 possesses an ordinary double threefold ɸ3 whose points map the totally degenerate triangles (i.e. those for which P1 ═ P2 ═ P3 and l1 ═ l2 ═ l3); Ω6 is therefore unsuitable as a basis for the construction of an enumerative calculus for triangles, for equivalence theory is as yet developed satisfactorily only on non-singular varieties.

Ein Teil von Philip Halls bekannter Theorie der auflösbaren Gruppen läßt sich, wie im folgenden gezeigt wird, auf beliebige endiliche Gruppen G ausdehnen. Wir fragen: Was kann man über die Normalstruktur (Hauptreihen, Kompositionsfaktoren) von G aussagen, wenn man mehrere Zer legungen von G in Produkte von Untergruppen mit gegebener Normalstruktur kennt? Eine Antwort gibt der folgende Satz, das Hauptergebnis dieser Note:

In this note the theory of the integral equation is reduced to elementary matrix algebra. The Fredholm theorems are proved without the introduction of the usual parameter λ and without using the properties of the resolvent considered as an analytic function of the complex variable λ. There is an exact correspondence between the properties of the equation (1) and the properties of a finite linear system. Here the proofs of these properties are also in extact correspondence with the same for the elementary case.

In a recent paper Salem and Zygmund [1] proved the following result: Put and denote the φv(t) Rademacher function. Denote by Ln(t,θ) the unique trigonometric polynominal (in θ) of degree not exceeding n

Let ρ and d be two positive numbers. We shall be concerned in this paper with two classes of entire functions, namely: (i) the class C (ρ, d) of all entire functions of order ρ and type not exceeding d — we note that this class includes all entire functions of order less than ρ;

A mathematical theory of the separating flow past a cascade of aerofoils is developed. The flow is assumed to be inviscid, incompressible and twodimensional. The wakes are represented by regions of stationary fluid, which could, in the general case, maintain a pressure gradient, although much of the theory is developed for the case of constant wake pressure.

In this paper f(z) will always stand for an entire transcendental function of the complex variable z. For p= 1, 2, … the natural iterate fD(z) of f(z) is defined by These natural iterates are themselves entire transcendental functions; they have been studied by various writers, notably Fatou [3]. References to many papers on iterated will be found in [1].

Let represent the deviations from expectation of a set of multinomial or independent Poisson variables, and Η be a positive definite matrix. A lower bound is obtained for Pr in terms of Pr, Where is a vector of normal variables with the same mean and covariance matrix as Δ.

As in [5] a parametric n-surface in Rk (where k ≧ n) will be a pair (f, Mn), consisting of a continuous mapping f of an oriented topological manifold Mn into the euclidean k-space Rk. (f, Mn) is said to be closed if Mn is compact. The main purpose of this paper is to use the method of [4] to prove a general form of Cauchy's Integral Theorem (Theorem 5.3) for those closed parametric n-surfaces (f, Mn) in Rn+1, which have bounded variation in the sense of [5] and for which f(Mn) has a finite Hausdorff n-measure. As in [4], the proof is carried out by approximating the surface with a simpler type of surface. However, when n > 1, a difficulty arises in that there are entities, which occur in a natural way, but are not parametric surfaces. We therefore introduce a concept which we call an S-system and which forms a generalisation (see 2.2) of the type of closed parametric n-surface that was studied in [5] II, 3 in connection with a proof of a Gauss-Green Theorem. The surfaces of [5] II, 3 include those that are studied in this paper.

Introductory. This paper considers a canonical form, or rather a class of canonical forms, for three dimensional probability distributions subject to a rather mild restriction. These canonical forms are used to develop suitable tests of independence and lead to a consideration of the partition of χ2 in the analysis of complex contingency tables. Where these methods and Bartlett's are both applicable it is shown that they give comparable results; but the partitioning methods are more general.

This note explains a “numerical coincidence” observed by Professor A. C. Aitken in a recent article in these Proceedings.