If five spheres in 3-space are such that each pair is inclined at the same non-zero angle θ, then where b1, …, b5 (the ‘bends’ (1) of the spheres) are the reciprocals of their radii. To prove this result, establish a system of rectangular cartesian coordinates (x, y, z) and let the spheres have centres (xi, yi, zi) and radii , where i = 1,…, 5. Then for x5, y5, z5, r5 we have the equations which, on subtraction, yield three linear equations and one quadratic equation. Solving the three linear equations for x5, y5, z5 and substituting, we see that the required relation is algebraic (indeed quadratic) in r5 and hence in b5. Since it is also symmetric in b1,…, b5, it follows that it can be expressed as a polynomial relation in the elementary symmetric functions p1, …, p5 in b5, …, b5.

Let [sum ]∞an denote the set of cluster points of the sequence of partial sums of a series [sum ]an with terms in ℝd. For any permutation f of the set ℕ of positive integers, [Cscr ]f (ℝd) denotes the set of all sets [sum ]∞af(n) arising from series [sum ]an with terms in ℝd and sum 0. For each f, we use the Max-Flow Min-Cut Theorem to determine all convex sets in [Cscr ]f(ℝd) which are symmetric about a point. These sets depend only on a parameter w(f) ∈ ℕ ∪ {0, ∞}, called the width of f. We show that w(f), when it is a positive integer, falls far short of completely determining [Cscr ]f(ℝd) but, for each q ∈ ℕ, we find the largest of the sets [Cscr ]f(ℝd) arising from permutations f of width q. We also describe the smallest of these sets when q = 1.

If f is meromorphic in D = {,|z| < 1}, we say that f has the asymptotic value a at ζ, ζ ∈ C(={|z| = 1}), if there exists an arc Γ such that Γ ends at ζ, (Γ − {ζ}) ⊂ D, and f has the limit a as |z| → 1 on Γ. The class (M) originally defined by G. R. MacLane ((8), p. 8), consists of those functions that are non-constant and holomorphic (meromorphic) in D and that have asymptotic values at a dense subset of C. In (8), p. 36, MacLane has shown the following three conditions are sufficient for a non-constant holomorphic function f to be in :

The subgroup structure of a direct product of two Fuchsian groups is very complicated; for example, Baumslag and Roseblade [1] have shown that a direct product Fm1×Fm2 of finitely generated free groups contains continuously many distinct isomorphism classes of finitely generated subgroups. The situation is much simpler, however, if attention is restricted to finitely generated normal subgroups of Fm1×Fm2; then on general grounds one cannot expect to get more than a countable infinity, but, in fact, the situation is almost finite; we showed, in [4], that Fm1×Fm2 contains precisely 1+min{m1, m2} orbits of maximal normal subdirect products under the natural action of its automorphism group.

In this paper we study the situation which arises if the free groups Fmi are replaced by fundamental groups of closed surfaces. This question was previously considered by Nigel Carr in the final chapter of his (unpublished) thesis [2]. By appealing to the relative invariant theory of pairs of skew bilinear forms, Carr was able to show that in a direct product Σg1×Σg2 of orientable surface groups of genus [ges ]2 the number of orbits of maximal normal subdirect products is always infinite. Here we refine and extend Carr's approach to study the manner in which the finiteness result of [4] breaks down on passing to more general Fuchsian groups.

1. A general threefold in [5] has an apparent double surface whose projection on to a [4] is a surface which contains a triple curve and which is characterised by the fact that it has no improper nodes; such surfaces have been considered in a previous paper. In the present note we consider a class of surfaces in [5] which possess triple curves and project into surfaces in [4] having improper nodes: these are the surfaces which represent the chords of a general curve of ordinary space upon a quadric primal Ω of [5]. For the representation of a line congruence of [3] by the points of a surface on Ω, reference may be made to a previous note, the results of which are used in the present work.

Let U(1, n; ℂ) be the automorphism group of the Hermitian form

for . We can regard an element of U(1, n; ℂ) as a transformation acting on , where is the closure of the complex unit ball

The non-trivial elements of U(1, n; ℂ) fall into three conjugacy types, depending on the number and the location of their fixed points. Let g be a non-trivial element of U(1, n; ℂ). We call g elliptic if it has a fixed point in Bn and g parabolic if it has exactly one fixed point and this lies on the boundary ∂Bn. An element g will be called loxodromic if it has exactly two fixed points and they lie on ∂Bn.

We refer for general background to N. Bourbaki, Intégration, chapters iv and v. We consider a locally compact Hausdorff space R and denote the set of continuous functions with compact support by The Riesz-Markov theorem shows that there is a 1−1 correspondence between the set of regular Borel measures on R and the set of positive elements of the topological dual of . Let {μn}, μ be regular Borel probability measures on R. The sequence of measures {μn} is said to converge vaguely to μ if, for all . Thus the vague topology is that of simple convergence in . We shall call a μ-measurable set μ-quarrable if its boundary is a μ-null set.

A complete and satisfying account of the classification of pencils of binary cubics over an algebraically closed field was given by Newstead (2). Extending these results to the real case is not a matter of mere routine since new questions arise, for example the separation of roots of the cubics in a pencil (as well as their reality).

The modified field equations proposed by the author can be derived from the energy density

(where is the electric displacement, the magnetic induction, the Poynting vector) with the help of the commutation laws for these vectors.

By a twisted quartic of the first kind is intended the curve in space of three dimensions defined by two quadratic equations, i.e. the complete intersection of two quadric surfaces. If this intersection consist of the aggregate of two or more curves of lower order than the fourth, it will be called a degenerate quartic. In general, a twisted quartic of the first kind will be referred to for brevity as a quartic.

It has been known for a very long time that, under certain conditions rather difficult to specify, the discharge through an ordinary two-electrode vacuum tube containing gas at a low pressure, even when the potential is maintained by a battery of low resistance Storage cells, is intermittent.

1. In the present note we obtain the postulation of a multiple surface of S4 for primals of sufficiently high order; the method is that previously used by Campedelli to determine the postulation of a multiple curve of S3.

P. M. Cohn has given a characterization of pseudovaluation on a field in (1). There he defines the notion of an integral element over a multiplicatively closed set and states that the association of the set A to the set A* of all elements that are integral over it is a closure operator. There is an error in the statement that A* = (A*)*. In this note we show, by an example, that A* and (A*)* are not equal in general. We also give an example to show that even for the special type of sets that Cohn needs, namely the gauge sets, this property fails to hold. However, since the other conditions for a closure operator (A ⊆ A* and A ⊆ B implies A* ⊆ B*) are satisfied, one obtains a Kuratowski closure operator by iterating *. If we denote by Ā the closure of A by iterating * then theorem 10·3, its corollary and theorem 10·4 in (1) that have a bearing on this error, are valid as stated with the same proof replacing only K* by . Further theorem 13·3 is valid without modification since Cohn considers only a special class of pseudovaluations namely regular pseudo valuations for which K* = .

The aim of this paper is to present a common generalization of and , the Stirling numbers of the first and the second kind ((1), ex. 186–210, pp. 42–45).

1. Introduction: Let B be a space and let E be an orthogonal (m + 1)-sphere bundle over B with projection p: E → B. Consider the fibre-preserving map T: E → E which is given by the antipodal transformation in each of the fibres. Following (4) we say that E is homotopy-symmetric if there exists a fibre-preserving homotopy of T into the identity map. Since T maps the fibres with degree (– 1)m this condition requires m to be even.

In his celebrated paper [7], Martinet showed the equivalence between the infinitesimal versality and the versality for [Ascr ]- and [Kscr ]-morphisms. By using this theorem, he obtained the following Theorem 1·1 which played one of the key roles for the classification of C∞ stable map-germs ([1, 7]).

This paper is devoted chiefly to the consideration of the surface oscillations of water contained in a vessel in the shape of a circular cylinder with its axis vertical, when the motion is slightly disturbed from a uniform rotation about the axis of the vessel. The work was undertaken with the hope of finding some indication of the effect of the depth of the water in the vessel on the period of the surface waves, and for the purpose a vessel of circular cross-section was naturally chosen. It is shown that a slight change of shape does not affect the periods of the oscillations. The solution of the corresponding problem when the surface oscillations take the form of “long waves” or “tidal waves” is well known, and the present paper deals only with “short waves,” for which the horizontal velocity is not the same at all depths.

A closed aspherical manifold is a closed manifold whose universal covering space is contractible. There is the following conjecture concerning the algebraic K-theory of such manifolds:

Conjecture. Let Γ be the fundamental group of a closed aspherical manifold. Then Whi(Γ) = 0 for i ≥ 0 where Whi(Γ) is the i-th higher Whitehead group of Γ.

Our previous discussion ((4), (5)) is here extended to an ellipsoid of arbitrary conductivity. Again, the flow is ultimately cylindrical and the magnetic field undisturbed. An ellipsoidal insulator allows fluid to pass through its circumscribing cylinder, but also forces some to go around. An ellipsoidal conductor behaves like a very tall insulator, irrespective of its own height or conductivity: all the fluid goes round. If it is very wide an unexpectedly large amount of fluid is carried with it.