The infinite products for sinx and cosx are most conveniently obtained in a rigorous way from the well-known factorial expressions for sinnθ and cosnθ which, when n is an even integer, take the forms

θ being put equal to and n made infinitely great.

Throughout this paper, we work in the category of (p-localized) spaces having the homotopy type of connected CW-complexes of finite type with base point. We consider a principal bundle

where Gn = SU(n), U(n) or Sp(n) and d = 1, 1 or 2 respectively. In this case, the bundle is obtained as an induced bundle by a mapping f of base space S2dn−1 from the classical group extension as follows:

The surfaces here considered were first discussed by Monge as surfaces whose normals are tangents to given developables. Under the name general surfaces moulures, Darboux treats them as the surfaces traced out by a fixed curve on a plane which rolls on a developable. When the developable is a cylinder he gives the general coordinates in his Leçons sur la Theorie Generale des Surfaces (Volume I., page 105). This led me to take up the more general case, but I later found that Darboux had also considered this in an ingenious and elegant manner in his Leçons sur les Systèmes Orthogonaux et les Coordonnées Curvilignes (Tome 1, pages 26–34). Perhaps this quite difFerent discussion will present some interesting points in analysis.

It has been shown in the previous three parts of this work, that the whole question of the convergence of the series solution, for the particular dynamical system under consideration, has turned upon the cubic equation

Before proceeding to generalize the results it will be shown how this cubic equation may be derived in a slightly different fashion.

Suppose λ is a positive number, and let , x∈Rd, denote the d-dimensional Gaussian. Basic theory of cardinal interpolation asserts the existence of a unique function , x∈Rd, satisfying the interpolatory conditions , k∈Zd, and decaying exponentially for large argument. In particular, the Gaussian cardinal-interpolation operator , given by , x∈Rd, , is a well-defīned linear map from ℓ2(Zd) into L2(Rd). It is shown here that its associated operator-norm is , implying, in particular, that is contractive. Some sidelights are also presented.

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.