An investigation by G. L. Clark is found to provide exceptionally instructive illustrations of the calculation of the variously defined ‘densities’ and ‘masses’ occurring in general relativity theory and of their physical significance. In particular, the effects of motion and stress upon gravitational mass are elucidated. An explicit example shows how kinetic energy is included in the gravitational mass.

Many measurements have been made of the complete absorption curves for β-rays in various substances, and it has been found that, for homogeneous β-rays, the curve is not exponential, and, in particular, when aluminium is the absorber, the curve is linear over a considerable portion of its range (e.g. Schonland, etc.).

If P is a point in space-time, and P′ is a point whose space coordinates are continuous functions of time and whose speed in space is always less than the speed of light, then the time, T, at which the square of the interval between P and P′ is R, is a two-valued real function of R and the space-time coordinates of P for a range of R that includes R = 0. For fixed R in this range, each branch of T defines a scalar field over the whole of space-time, and, as R → 0, these fields reduce to the retarded and advanced times of P′ at P. Any function of T is also a field that reduces to a retarded or advanced field as R → 0. An interesting property of these fields is that there are identical relations between their derivatives with respect to R and the space-time variables, which enable the space-time derivatives to be expressed in terms of the R-derivatives; moreover, these relations are linear in the derivatives and take the same form for every such field.

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.

We often wish to test the difference of the means of two series of measures of a quantity for a systematic difference. A suitable test, in terms of the theory of probability, can be obtained as follows. We suppose that in each series the probability of error is normally distributed with unknown standard error. Then if we have only one observation in each series, the difference of the two could equally well be interpreted as entirely due to the random error in one series, the other being right, or entirely due to systematic difference, both observations being free from random error. This suggests that we should take as one of our fundamental quantities the expectation of the difference of a single measure in one series from one in the other; the expectation of the square of the difference, however, appears still better.

The spontaneous disappearance of an electron and proton in the form of radiation has often been suggested as a possible source of energy to maintain the emission of radiation from hot stars. It can be calculated that the energy of a single quantum formed in this way should be about 1000 million electron-volts. In order to account for the absorption of the ultra-penetrating radiation observed by Regener in deep lakes, and interpreted on the basis of the Klein-Nishina formula, a quantum of still higher energy is required. Sir James Jeans has recently calculated that the transformation of a helium atom into a single quantum would provide a radiation of about the required penetrating power.

Ramanujan's Continued Fraction may be stated as follows: Let where there are eight gamma functions in each product and the ambiguous signs are so chosen that the argument of each gamma function contains one of the specified number of minus signs. Then where the products and the sums on the right range over the numbers α, β, γ, δ, ε: provided that one of the numbers β, γ, δ, ε is equal to ± ±n, where n is a positive integer. In 1935, Watson (3) proved the theorem by induction and also gave a basic analogue. In this paper we give a new proof of Ramanujan's Continued Fraction by using the transformation of Bauer and Muir in the theory of continued fractions (Perron (1), §7;(2), §2).

We show that the abelianization of the kernel of the Magnus representation of the automorphism group of a free group is not finitely generated.

We give a constructive proof of a Theorem of Izmestiev and Joswig. Namely, given $(L,\omega)$ where $L$ is a link in $S^{3}$ and $\omega$ a simple (not necessarily transitive) representation of $\pi_{1}(S^{3}\backslash L)$ onto the symmetric group $\Sigma_{4}$ of four elements $\{1,2,3,4\}$ we construct a triangulation of $S^{3}$ giving rise to $(L,\omega)$ in a natural way.

In studying the structure of perfect sets, an important method would be to establish homoeomorphisms in general classes of sets. For then the general structural features of sets could be more easily seen by considering the simplest sets of a homoeomorphic class.

In this paper we establish the homoeomorphism in two general classes of plane perfect sets (we consider only bounded sets).

1. In the complete algebraic system of concomitants of a pair of quaternary quadrics given by Turnbull† there appear sixteen covariant line-complexes, eight of the second order and eight of the third. In a later paper‡, the same author gave some geometrical interpretations of these and other concomitants. None the less, no systematic geometrical account of the relations of these complexes seems to have been made. The purpose of the present paper is to discuss the geometrical relations systematically, and make their origin clearer. Incidentally, the syzygies which exist between the complexes are obtained explicitly.

Let H(t, θ) be the hyperplane in Rn (n ≥ 2) which is perpendicular to the unit vector θ, and distant t from the origin; that is H(t, θ) = {x ε Rn: x.θ = t}. (Note that H(t, θ) and H(−t, − θ) are the same hyperplane.) If f(x) εℒ1(Rn), we will denote the integral of f with respect to (n − 1)-dimensional Lebesgue measure over H(t, θ) by F(t, θ), termed the projection or sectional integral of f over H(t,θ). By Fubini's theorem, F(t, θ) exists for almost all t for any θ. Throughout this paper we will assume that f(x) has support in X, a compact convex subset of Rn. In Section 2 we examine some of the topologies that may be defined on functions on X in terms of the F(t, θ), and in the remainder of the paper we examine the extremal problem suggested by Croft (4), that of maximizing the integral of f over the set X with the constraint that the F(t, θ) are uniformly bounded above. We examine in particular how the extremal values depend on the convex set X. In the final section the extremal problem is related to a generalization of Bang's plank theorem and the theory of capacities, and several conjectures are proposed.

The successful development in recent years of the topology of general closed sets is largely due to the application of combinatory methods, which have led to an elaborate theory of approximation of closed sets by infinite cycles, to the generalization of duality theorems for closed sets, and to a geometrical theory of dimensions. Corresponding to the combinatory invariants involved the main results of these theories concern for the most part properties of the set as a whole, which cannot possibly express fully its internal and in particular its local structure. Still less is our knowledge of the eventual relations between the local structure of a set and its properties in the large.

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].

We consider fibrewise singly generated Fell bundles over étale groupoids. Given a continuous real-valued 1-cocycle on the groupoid, there is a natural dynamics on the cross-sectional algebra of the Fell bundle. We study the Kubo–Martin–Schwinger equilibrium states for this dynamics. Following work of Neshveyev on equilibrium states on groupoid C*-algebras, we describe the equilibrium states of the cross-sectional algebra in terms of measurable fields of states on the C*-algebras of the restrictions of the Fell bundle to the isotropy subgroups of the groupoid. As a special case, we obtain a description of the trace space of the cross-sectional algebra. We apply our result to generalise Neshveyev’s main theorem to twisted groupoid C*-algebras, and then apply this to twisted C*-algebras of strongly connected finite k-graphs.

The theorem, due to Miquel, that the foci of the five parabolas which touch fours of five straight lines lie on a circle, when generalised projectively and dualised becomes the theorem: If six arbitrary points 1, 2, 3, 4, 5, 6 be taken and the five conics passing respectively through the five points obtained by omitting in turn 1, 2, 3, 4, 5, then there exists a conic touching two arbitrary lines through the point 6 and triangularly inscribed to these five conics. It appears, however, that the relation is symmetrical and that the conic obtained is also triangularly inscribed to the conic passing through the points 1, 2, 3, 4, 5. If the condition of touching the two arbitrary straight lines through the point 6 be omitted, we have a doubly infinite system of conics triangularly inscribed to the six conics passing through fives of six points. It does not immediately appear how this family of conics depends upon the two parameters involved, and the following direct analytical investigation of the general symmetrical figure was undertaken with a view to deciding this point.

Let k be a positive integer and F(x, k) denote the number of integers n < x which have a divisor in every residue class prime to k. Erdös (1) proved that for every fixed ε > 0, we have F(x, k) ˜ x when

and conjectured the following result, which we prove in this paper.

Landé has indicated a connection between the “relativity” doublets of X-ray spectra, and the doublets and triplets of optical spectra, and so has obtained a formula for the separation of optical doublets and triplets. This formula can only be expected to hold if the orbit of the series electron penetrates into the core, so it may be possible to obtain evidence on the question whether an orbit penetrates from the doublet or triplet separation of the corresponding term.

From this evidence it is concluded that, except for lithium-like atoms, p terms of all known spectra correspond to penetrating orbits; this disagrees with Bohr's assignment of quantum numbers for those terms of the spectra of neutral atoms of the Cu and Zn sub-groups. For d terms the evidence is not so definite, but it seems possible that for Cs I and Tl I the d terms correspond to penetrating orbits, in disagreement with Bohr's assignment. It is shown that the Landé formula seems to hold approximately for separations in multiplet spectra when the terms belong to a sequence of the Rydberg type.

Ideas suggested by Heisenberg in a recent paper would involve some modifications of the interpretation of the magnitudes of doublet separations, but at least the conclusions as to the p orbits penetrating still stand.

R. H. Fowler and L. Nordheim have presented a theory which seems to offer a satisfactory explanation of the phenomenon commonly known as the auto-electronic discharge, in which electrons are extracted from cold metallic cathodes by intense electric fields. Their theory considers the passage of electrons through a potential “hill” of the type shown in Fig. 1, where V is the potential energy of an electron at a point distant x units from the surface.

If E and F are ordered spaces, a map T: E → F is positive if

The map T is bi-positive if f ≥ 0 ⇔ Tf ≥ 0. It is a natural problem to seek information about the positive and bi-positive linear maps between pairs of ordered vector spaces. The present study is an attempt to elucidate one facet of this general problem (see, for example, (2), vol. II, § 16·7·1, p. 278).