A procedure is described to determine phase velocities of a circular helix. Computations are made for three helix configurations using the method of internal constraints. The results so obtained are compared with the results obtained by using the elementary theory proposed by J. H. Michell, and the extended elementary theory proposed by A. B. Basset.

Introduction. Let Mn+1 denote a closed, orientable, combinatorial (n + 1)-manifold. Let Kn denote an n-dimensional subcomplex of Mn+1. If Mn+1 – Kn is connected, then by ‘cutting’ Mn+1 along Kn we define in a natural way a fundaments domain of a certain regular covering of Mn+1. In case Kn is a spine ((5)) of Mn+1 the any covering space of Mn+1 may be constructed. This construction is quite precis and is combinatorial in spirit.

Clifford's well-known chain of theorems concerning circles in a plane may be stated as follows (the first two links being trivial):

(C2) Given any point A, and through it two circles C1 and C2, these meet again in a point A12, say.

(C3) Given three circles C1, C2, C3 through A, the three points A23, A13, An lie on a circle C123, say.

(C4) Given four circles C1,…,C4 through A, the four circles C234,…,C123 meet in a point A1234 (Wallace(1)).

(C5) Given five circles C1,…,C5 through A, the five points A2345,…,A1234 lie on a circle C12345 (Miquel (2)).

And so on (Clifford(3)).

Grace (4) showed that in three dimensions [3], there is no direct analogue to Clifford's chain, but in [4] he noted the following:

(G4) Given any point A in [4], and through it four hyperspheres S1,…, S4, these meet again in a point A1234, say.

(G5) Given five hyperspheres Sl,…,S5 through a point A, the five points A2345, A1234 lie on a hypersphere S12345, say.

(G6) Given six hyperspheres Sl,…, S7 through A, the six hyperspheres S23456,…, A12345 have in common a point A123456.

This leads naturally to the conjecture

(G7) Given seven hyperspheres Sl,…, S7 through A, the seven points A234567,…, A123456 lie on a hypersphere S1234567.

1. Definition. Let λ ≡ λ(ω) be continuous, differentiable and monotonic increasing in (0, ∞) and let it tend to infinity as ω → ∞. A series is summable |R, λ, r|, where r > 0, if

where A is a fixed positive number(3).

A square matrix A, of order n, having complex coefficients can be inverted without the aid of any operations involving complex numbers. This can be done if the coefficients ars + iαrs are replaced by their matrix equivalents and the resulting 2n × 2n real matrix A1 is inverted. The inverse will be a 2n × 2n matrix of similar form in which the subsidiary 2 × 2 matrices can be replaced by the equivalent complex numbers, thus yielding the inverse of A. It is the purpose of this note to show that a similar technique can be employed to evaluate det A.

For a given locally convex space, it is always of interest to find conditions for its nuclearity. Well known results of this kind – by now already familiar – involve the use of tensor products, diametral dimension, bilinear forms, generalized sequence spaces and a host of other devices for the characterization of nuclear spaces (see [9]). However, it turns out, these nuclearity criteria are amenable to a particularly simple formulation in the setting of certain sequence spaces; an elegant example is provided by the so-called Grothendieck–Pietsch (GP, for short) criterion for nuclearity of a sequence space (in its normal topology) in terms of the summability of certain numerical sequences.

1. Considerable advantage has resulted from the postulation of unreal elements in projective geometry. In the first place these unreal elements were defined in terms of points represented by complex coordinates, and their use in purely geometrical reasoning had become well established before any serious attempt was made to justify this use, independently of algebraic considerations, by providing real representations of the unreal elements. The first successful attempt was that of von Staudt, who represented an unreal element by an elliptic involution associated with an order. In this system an ordered set of four real points is required to specify an unreal point. The system is comparatively simple to deal with in a single real plane, more complicated in a single real [3], and rapidly increases in complexity as the number of dimensions of the real field is increased.

1. The following investigations were undertaken in connection with Cayley's problem of seven lines on a quartic surface.

Theorem 1. Let M3be a compact 3-manifold and let M3 = E3 ∪ R be a standard decomposition where R is a subcomplex in a triangulation of M3((2)). If M3is a homology 3-sphere while R contains no arc that is wild in M3then M3 = S3

Introduction. Hall ((3), (4)) introduced the concept of a Sylow system and its normalizer into the theory of finite soluble groups. In (4) he showed that system normalizers may be characterized as those subgroups D of G minimal subject to the existence of a chain of subgroups from D up to G in which each subgroup is maximal and non-normal in the next; he also showed that a system normalizer covers all the central chief factors and avoids all the eccentric chief factors of G (for definitions of covering and avoidance, and an account of their elementary properties), the reader is referred to Taunt ((5)). This note arises out of an investigation into the question to what extent this covering/ avoidance property characterizes system normalizers; it provides a partial answer by means of two elementary counter-examples given in section 3 which seem to indicate that the property ceases to characterize system normalizers as soon as the ‘non-commutativity’ of the group is increased beyond a certain threshold. For the sake of completeness we include proofs in Theorems 1 and 2 of generalizations of two known results communicated to me by Dr Taunt and which as far as we know have not been published elsewhere. Theorem 1 shows the covering/avoidance property to be characteristic for the class of soluble groups with self-normalizing system normalizers introduced by Carter in (1), while Theorem 2 shows the same is true for A -groups (soluble groups with Abelian Sylow subgroups investigated by Taunt in (5)).

1. Introduction. The present paper is chiefly concerned with a generalization, to be known as a ‘D-interval’, of the notion of interval or segment in an arbitrary partially ordered group. This idea is originally due to Duthie (2), but was developed by him only in a lattice. In analogy with the use of the interval in the normal sense, notions of ‘D-distributivity’ and ‘D-modularity’ are defined in terms of the D-interval, and analogues of known properties of lattice-groups or ‘l– groups’ can be formulated which might be valid when a lattice structure is no longer assumed to exist; in particular, an attempt is made to provide such a generalization of the result of Freudenthal (3) that every Z-group is a distributive lattice, but, for an arbitrary partially ordered group, it is shown that only an ‘approximation’ (in terms of non-Archimedean elements) to the desired result actually holds, although any Archimedean partially ordered group is necessarily D-distributive.

This paper is a study of a new method of enumeration of the partitions of multipartite numbers.

Incidentally an algebraic function, which is derived from the repetitional exponents of partitions of unipartite numbers, presents itself. The generating function which enumerates the partitions of unipartite numbers is expressible in terms of these functions and finds in such expression its fullest connection with the divisors of numbers. There are also similarly derived functions connected directly with bipartite, tripartite, etc. numbers. It has not been necessary to study these for the purposes of this paper.

A simple solution of a general problem involving a bifurcated wave guide is presented. The purpose of the work is to explain a new and simple method of solving such problems and to exhibit an organic connexion between Sturm–Liouville theory and the theory of two-part boundary-value problems.

In [2] the author proposed the study of the geometry and arithmetic of the general curve

of genus 2 over a general groundfield such as the rationals in order to attack Diophantine problems. He gave the form of the Jacobian, and Flynn[4, 5] determined the group law on it. This paper is preparatory to a planned study of heights under duality. It extends to a general groundfield the description of the Jacobian as the variety of lines on the intersection of two quadric surfaces in P5 given over the complexes in the last chapter of the textbook [7] of Griffiths and Harris. This description is a generalization of that of a curve of genus 1 as the intersection of two quadric surfaces in P3 and is a special case of a more general result [3]: its first appearances seem to be in [12] and [13].

The purpose of this note is to call attention to an error in the paper [1]. The main result was a new proof of the main theorem in [2]. In line 1 of page 446 of [1] one finds the following remark: “This is the result obtained in [2]”. However, a comparison of the two results shows that the generator which is called $d_1 d^{-1}_2$ in the notation of [1] appears in [2] but is missing from the list in [1].

We discuss several methods which yield spanning sets for the irreducible polynomial $GL(n,K)$-modules, where $K$ is an infinite field. We show that the spanning set which arises from our first method coincides, up to sign, with the spanning set produced by a method due to Pittaluga and Strickland. We prove that these spanning sets are related to the spanning set produced by a third method via the Désarménien matrix.

A double series solution of the biharmonic equation is derived in the rectangular domain. The solution utilizes the finite double Fourier sine transform and the analysis shows how the transform may be evaluated for quite arbitrary boundary conditions, the solution then being found from the corresponding inversion formula.

The method is applied to some classical boundary value problems which have been investigated by other means by earlier workers. It is felt that some of the series solutions given here are more amenable to computation than some of the earlier methods.

In some of the problems considered, notably the elastostatics of the rectangular plate and the bending of clamped plates, the method reduces to solving an infinite set of linear equations. In such cases, a finite leading matrix of the coefficients has to be considered. In the examples considered, this technique produced results which converged to known solutions obtained by other methods. However, the mathematical criteria which permit the infinite system to be curtailed, have not been established.

1. Consider the Cauchy problem for the classical wave equation in ℝn:

with initial conditions u(0, x) = uo(x) and ∂tu(0, x)= u1(x), where uj (j = 0,1) are compactly supported C∞ functions on ℝn. In this paper we shall establish the following result.