An analogue in a solid of the well known Pascal's theorem (Baker, [1], p. 219) for a conic is established by Baker ([2], pp. 53–54, Ex. 15) after Chasles [6] and by Salmon ([2], p. 142). The same is discussed in detail by Court [8]. The purpose of this paper is to extend it to a projective space of n dimensions or briefly to an n-space Sn. To prove it, we introduce here once again the idea of a set of n+1 associated lines in Sn as indicated in an earlier work (Mandan, [12]) in analogy with a set of 5 associated lines in S4 (Baker, [4], p. 122), and make use of the method of induction.

The purpose of this paper is to prove that the altitudes of an n-simplex (a simplexin an n-space) S form an associated set of n+1 lines (see Baker, [4] for n = 4) such that any (n–2)-space meeting n of them meets the (n+1)th too. As an immediate consequence 2 quadrics are associated with S, one touching its primes at the respective feet of its altitudes and the other touching n(n+1) primes, n parallel to each of its altitudes and 2 through each of its (n−2)-spaces. Certain special cases are also mentioned.

In [2] we defined an irreducible B(J)-cartesian membrane, and used this to obtain a characterization of an n-sphere by generalizing the definition of simple closed curve given by Theorem 1.2 below. There B(J) is a class A(n) of (n−1)-spheres, but here it is a class of mutually homeomorphic continua. In Theorem 1.1 we give a definition of hereditarily unicoherent continua and generalize this in Section 3 by means of B(J)-cartesian membranes. To do this we paraphrase by a translation some of Wilder's work in [7]. In his Unified Topology [8: p. 674] he gives a principle: “The connectedness of a domain is a special case of the bounding properties of its i-cycles”. We substitute the element J of B(J) for the i-cycle and for “bound” we substitute that “J membrane-bases an irreducible B(J)-cartesian membrane. The very nature of an i-cycle seems to limit the complexity of the point set studied, although the restriction to “nice” manifolds is due partly to the difficulty of the subject matter treated. There are similar difficulties here, but also advantages, in the very general set-theoretic approach by means of B(J)-membranes.

The following paper is a sequel to the author's earlier paper [2]. In that paper some general results were obtained which described the motion of a fluid with a free surface subsequent to a given initial state and prescribed boundary conditions of a certain type. The analysis was based on a linearized theory but gravity effects were included. Viscosity, compressibility and surface tension effects were neglected. Among the problems treated was that of the normal symmetric entry of a thin wedge into water at rest. This Water entry problem has attracted a considerable amount of attention since the pioneer paper by Wagner [5]. Both linear and non-linear approximations have been used but all papers apart from [2] neglect gravity on the assumption that in the early stages of the penetration this is unimportant. One of the objects of [2] was to determine the solution with the gravity terms retained. A formal solution was obtained but no attempt was made to analyse this quantitatively. In the present paper we examine the extent of this effect in some detail. It will be of help to the reader to have some familiarity with the first three or four sections of [2] but in order to make the present paper self-contained we shall first reintroduce the notation used there and quote the necessary results from that paper without proof.

In a recent paper by P. D. Finch and myself [1], the solution for the limiting distribution of a moving average queueing system was obtained. In this paper the system is generalised to the case of batch arrivals in batches of size ρ > 1.

The mechanics of a system of packed spheres has relevance to several physical disciplines. A particular case has been a recent trend among engineers to use a close-packed sphere model to aid research into the strength of cohesionless granular masses, such as sand.

Let K be a bounded, open convex set in euclidean n-space Rn, symmetric in the origin 0. Further let L be a lattice in Rn containing 0 and put extended over all positive real numbers ui for which uiK contains i linearly independent points of L. Denote the Jordan content of K by V(K) and the determinant of L by d(L). Minkowski's second inequality in the geometry of numbers states that Minkowski's original proof has been simplified by Weyl [6] and Cassels [7] and a different proof hasbeen given by Davenport [1].

In this paper the theory of periodic solutions of analytic Hamiltonian systems of differential equations, which is due to Cherry [5], is specialized to systems which have one symmetry property.

Let ν be a discrete random variable taking on nonnegative integer values and set P{ν = κ} = Pk, κ = 0, 1, hellip. Suppose that the binomial moments are finite. Frequently the problem arises under what conditions the probabilities Pk, k = 0, 1,…, can be determined uniquely by the sequence of moments Br, r = 0, 1,…, and how it can be done.

Let X be an non-empty set and denote by PX the set algebra consisting of all sunsets of X. An operator i: PX → PX is said to be Pre-interior if (i) iX = X; (ii) i(A∩B) = iA ∩ iB for all A, B in PX.

Let G be a locally compact group with left invariant Haar measure m. Le H be a closed subgroup of G and K a compact group of G. Let R be the equivalence relation in G defined by (a, b)∈R if and if a = kbh for some k in K and h in H. We call E =G/R the double coset space of G modulo K and H. Donote by a the canonical mapping of G onto E. It can be shown that E is a locally compact space and α is continous and open Let N be the normalizer of K in G, i. e. .

The groups whose 2-generator subgroups are all nilpotent of class at most 2 are nilpotent of class at most 3 (see Levi [6]). Heineken [3] generalized Levi's result by proving that for n ≧ 3, if the n-generator subgroups of a group are all nilpotent of class at most n, then the group itself is nilpotent of class at most n. Other related problems have been considered by Bruck [1].