It was in a paper bearing this title that Cayley(1) first considered the problem of representing a curve in projective space of three dimensions by means of the complex of lines which meet the curve. He took the conic given by the equations

and found that the line

with dual Grassmann coordinates (…,pij,…), where

intersects the conic if, and only if,

where F(u0, u1) is homogeneous and of degree 2 in both sets of indeterminates u0 and u1 and G(…,pij,…) is a form of degree 2 in the pij. Both F(u0, u1) and G(…,pij,…) are easily determined in this case.

Some data obtained at Leiden on relaxation effects in a diluted potassium chrome alum are discussed in the light of a paper by Temperley (10), and excellent agreement is found with the conclusions of that paper. The primary effect of diluting the salt is to reduce the magnetic interaction, and the observed effects are a big increase in the relaxation time in zero field, and a decrease of the relaxation time with increasing field instead of an increase. It is predicted that a decrease should set in with an ordinary salt at liquid helium temperatures at fields somewhat higher than have so far been used. Van Vleck's criticisms of Temperley's paper are discussed, and it is concluded that they are not valid in their present form.

The aim of this paper is to make some remarks about the theory of parallel immersions in ℝn. We show that the existence of parallels to a curve in ℝn depends on the values of the integrals of the n − 1 curvature functions along the curve. We then give examples of immersions of S1 in ℝ3 and of immersions of codimension 2 with local parallel rank zero, thereby answering certain questions in [1].

Categories of rigid/plastic response are investigated along arbitrary paths of homogeneous deformation. Successive yield surfaces are presumed to be convex and self-similar relative to certain canonical measures of stress and finite strain. After a general analysis attention is focussed on path-sensitive response in configurations where the canonical rate of hardening vanishes. The analysis is subsequently extended to other types of eigenstate, especially those associated with actual or prospective loading devices. The results are illustrated in the context of formability tests on sheet metal.

In this paper, we are concerned with the boundedness of all the solutions of the two-dimensional differential system $$ \left.\begin{array}{@{}l@{}}x'\,{=}\,{-}|y|^{p-2} y+f(t)\\ y'\,{=}\,|x|^{q-2} x+g(t) \end{array}\right\} $$ where $1\,{<}\,p$, $1\,{<}\,q\,{<}\,2$ and $5pq\,{>}\,3p+4q$, $pq/(p+q)\,{<}\,1$, $f, g$ are $C^5$ 1-periodic functions.

It is concluded that the “overshoot phenomenon” in superconductivity described by F. B. Silsbee was due to his method of mounting the specimen.

Methods are developed for the computation of the complex zeros of (½z)−νJν(z) when the index ν is an arbitrary complex number. These methods, which do not require an explicit knowledge of the Jv(z), are susceptible to rapid numerical evaluation on a computer. Beyond the interest in the zeros in their own right, these methods now make feasible the use of the infinite product representation of Jν(z) for the rapid computation of Bessel functions of complex order.

Bagnera(1) shows that when a certain primitive finite group of order 576, G576, is represented as a group of collineations in three-dimensional space, it contains twenty-four harmonic inversions with respect to points and planes, and he shows how to calculate the coordinates of the points and planes of the inversions. The purpose of this note is to discuss the configuration of these points, which we shall call vertices, to show that the whole group is generated by the twenty-four harmonic inversions, termed projections after Prof. Baker (2) and Dr J. A. Todd (4), and to enumerate the conjugate sets of the group.

In this paper wave-maker theory including the effect of surface tension is determined for three-dimensional motion of water in a semi-infinite rectangular channel with outgoing surface wave modes allowed for at infinity; the motion is generated by a harmonically oscillating vertical plane wave-maker at the end of the channel and the cases of both infinite and finite constant depth are treated. The solution of the boundary-value problem for the velocity potential is more complicated in the presence of surface tension due mainly to the additional effect of the channel walls at which the normal free surface slopes are prescribed—as also is the slope at the wave-maker—to ensure uniqueness. The simpler three-dimensional solution for a semi-infinite region—obtained long ago by Sir Thomas Havelock in the absence of surface tension for the case of infinite depth—is also noted.

A Kolmogorov-type inequality is derived from a Jackson-type inequality and a Bernstein-type inequality. Several new results follow as corollaries.

Let F:${\mathbb R}$n→${\mathbb R}$n be a polynomial local diffeomorphism and let SF denote the set of not proper points of F. The Jelonek's real Jacobian Conjecture states that if codim(SF) ≥ 2, then F is bijective. In this work we prove a weak version of such Conjecture, but for more general maps than polynomial, namely: the semi-algebraic maps.

For any category A and group G there is a category AG of A-objects with G-action. An object of AG is an A-object A together with a group morphism, θA:G → Isom (A, A), of G into the group of invertible A-morphisms A → A. A morphism of AG is an A-morphism f: A → B which commutes with the G-action; that is, for each g∈G, the following diagram commutes.

Let a(n) denote the number of non-isomorphic Abelian groups of order n. It is well-known that

for a natural number k we define

and

A spray is a second-order differential equation field on the slit tangent bundleof a differentiable manifold, which is homogeneous of degree 1 in the fibrecoordinates in an appropriate sense; two sprays which are projectivelyequivalent have the same base-integral curves up to reparametrization. We showhow, when the base manifold is two-dimensional, to construct from any projectiveequivalence class of sprays a conformal class of metrics on a four-dimensionalmanifold, of signature (2, 2); the Weyl conformal curvature of these metrics issimply related to the projective curvature of the sprays, and the geodesics ofthe sprays determine null geodesics of the metrics. The metrics in question havepreviously been obtained by Nurowski and Sparling (Classical and Quantum Gravity20 (2003) 4995–5016), by a differentmethod involving the exploitation of a close analogy between the Cartan geometryof second-order ordinary differential equations and of three-dimensionalCauchy–Riemann structures. From this perspective the derived metricsare seen to be analoguous to those defined by Fefferman in the CR theory, andare therefore said to be of Fefferman type. Our version of the constructionreveals that the Fefferman-type metrics are derivable from the scalar tripleproduct, both directly in the flat case (which we discuss in some detail) and bya simple extension in general. There is accordingly in our formulation a verysimple expression for a representative metric of the class in suitablecoordinates.

In this paper we formulate a new conjecture and introduce methods to verify it in many cases.

Two perpendicular intersecting straight lines will be said to be normal to each other. Then the Petersen-Morley theorem is: Given arbitrarily three lines a, b, c, let a′ be the unique normal to b and c, b′ that to c and a, and c′ that to a and b; then the lines x, y, z, normal respectively to the pairs of lines a and a′, b and b′, c and c′, have a common normal.

In this paper we describe solutions of the equation: F(A(z)) = G(B(z)), where A,B are polynomials and F,G are continuous functions on the Riemann sphere.

Let

be a real quadratic form in n variables (n ≥ 2) with integral coefficients and determinant D = |fij| ≠ 0. Cassels ((1),(2)) has recently proved that if the equation f = 0 is properly soluble in integers x1, …, xn, then there is a solution satisfying

where F = max | fij and cn depends only on n. An example given by Kneser (see (2)) shows that the exponent ½(n – 1) is best possible. A simpler proof of Cassels's result has since been given by Davenport(3), and the theorem has been improved in certain cases by Watson(4). Here I consider the inequality f(x1, …, xn) > 0, where f is an indefinite form, and obtain a result analogous to that of Cassels.

In this paper we study periodic functions of one and two variables that are invariant under a subgroup of the Euclidean group. Starting with a function $f$ defined on the plane we obtain a function of one variable by two methods: (1) we project the values on a strip into its edge, by integrating $f$ along the width; and (2) we restrict $f$ to a line. If the functions had been obtained by solving a partial differential equation equivariant under the Euclidean group, how do the symmetries of the function of one variable compare to those of solutions of equations formulated directly in one dimension?

Some of the symmetries of projected and of restricted functions may be obtained knowing only the symmetries of the original functions. Extra symmetries arise at special widths of the strip and at some special positions of the line used for restriction. We obtain a general description of the two types of symmetries and discuss how they arise in the wallpaper groups (crystallographic groups of the plane). We show that the projections and restrictions of solutions of differential equations in the plane may have symmetry groups larger than those of solutions of problems formulated in one dimension.

In [1], Dowbor, Ringel and Simson consider hereditary artinian rings of finite representation type; it is known that if A is an hereditary artinian algebra of finite representation type, finite-dimensional over a field, then it corresponds to a Dynkin diagram in a natural way; they show that an hereditary artinian ring of finite representation type corresponds to a Coxeter diagram. However, in order to construct an hereditary artinian ring of finite representation type corresponding to a Coxeter diagram that is not Dynkin, they show that it is necessary though not sufficient to find an extension of skew fields such that the left and right dimensions are both finite but are different. No examples of such skew fields were known at the time. In [3], I constructed such extensions, and the main aim of this paper is to extend the methods of that paper to construct an extension of skew fields having all the properties needed to construct an hereditary artinian ring of finite representation type corresponding to the Coxeter diagram I2(5).