In a recent paper Dr G. C. McVittie discussed the solution with axial symmetry of Einstein's new field-equations in his Unified Field Theory of Gravitation and Electricity. Owing to an error in his calculation of the field equations, Dr McVittie did not obtain the general solution, which we discuss in the present paper.

In connection with Bertrand's algebraical exercise of 1850, Muir remarks that it is not unlikely that the divisibility of a3 + b3 + c3 − 3abc by a + b + c had been previously noted, although. there is no record of the fact. Bertrand's exercise is to the effect that the circulant of the third order repeats under multiplication or, what is the same, admits of composition; the formulae of composition are stated in the exercise precisely as they would follow from Spottiswoode's theorem on the linear factors of a circulant. The whole of this is implied as an immediate special case, and indeed as one that any reader would construct at once, in an identity due to Lagrange, reproduced by Legendre.

The classic application of dual integral equations occurs in connexion with the potential of a circular disc (e.g. Titchmarsh (9), p. 334). Suppose that the disc lies in z = 0, 0≤ρ≤1, where we use cylindrical coordinates (p, z). Then it is required to find a solution of

such that on z = 0

Separation of variables in conjunction with the conditions that ø is finite on the axis and ø tends to zero as z tends to plus infinity yields the particular solution .

It is well known (see Thomson and Tait, §§ 517, 518) that a spherical shell, whose surface-density is inversely as the cube of the distance from an external point, as well as a solid sphere whose density is inversely as the fifth power of the distance from an external point, are centrobaric. The centre of gravity is, in each case, the “image“ of the external point.

The present paper is a further contribution towards the object defined in my former paper, namely, to derive the principal known results regarding Continued Fractions, and some new theorems, by transforming the functions considered from infinite series to Continued Fractions, by use of the theory of determinants.

An inverse transversal of a regular semigroup S is an inverse subsemigroup So that contains precisely one inverse of each element of S. Here we consider the case where S is quasi-orthodox. We give natural characterisations of such semigroups and consider various properties of congruences.

§ 1. The radiation of electromagnetic charges from a moving point.

Let the electric charge associated with a moving point Q at time s be f(s). This charge is supposed to vary on account of the radiation of electric charges from Q in a variable direction, which, at time s, has direction cosines l (s), m (s), n (s) Using ξ(s), η(s), ζ(s) to denote the coordinates of Q at time s, and defining the effective time τ by means of the equation

the electromagnetic field may be specified by means of the potentials

where

denote the n-th Cesàro mean of order k for the series aΣan, that is,

where

and let

In this paper, finite travelling waves for the semilinear parabolic systems

are studied, where di > 0, ei > 0, mij ≥ 0 for all 1 ≤ i,j ≤ n, and for all 1 ≤ i ≤ n. Let M = (mij)n × n and A = I – M. It will be proved that (*) has finite travelling waves if and only if all principal minors of A are positive. Moreover, some asymptotic behaviours of finite travelling waves will be obtained.

There are various algebras which may be associated with a discrete group G. In particular we may consider the complex group ring ℂG, the convolution Banach algebra l1(G), the enveloping C*-algebra C*(G) of l1(G), and the reduced C*-algebra determined by the completion of l1(G) under the left regular representation on l2(G). There is a substantial literature on the circle of ideas associated with the embeddings

A well known lemma of Burnside is generalised, to give necessary and sufficient conditions for a finite p-group K to be normally embedded in a nilpotent group V, with K⊆ω(V). (Here, ω denotes a single word and ω(V) is the corresponding verbal subgroup.) Our main result is related to earlier work of Blackburn, Gaschütz and Hobby.

A continuous function φ on the unit circle is called badly approximable if ‖ φ − p ‖∞ ≧ ‖ φ |∞ for all polynomials p, where ‖ |∞ is the essential supremum norm. In (4), Poreda asked whether every continuous φ may bewritten φ = φW+φB, where φW is the uniform limit of polynomials (i.e. φW belongs to the disc algebra A) and φB is badly approximable. We call such a function φ decomposable. In (4), he characterised the badly approximable functions as those of constant non-zero modulus and negative winding number around the origin, i.e. ind (φ)<0. (See (3) for two new proofs of this result.) We show that the answer to Poreda's question is no in general, but give a necessary and sufficient condition for a given φ to have such a decomposition. Then we apply this criterion to solve an interpolation problem.