The results and methods of algebraic geometry, when analysed in terms of modern algebra, have revealed on several occasions algebraic principles of surprising generality. Recently it has become apparent that the geometric theory of infinitely near points has, as it were, an abstract form which forms part of the ideal theory of commutative rings, but there are many details which have yet to be worked out. Roughly speaking, one may say that what corresponds to the theory of the sequence of points on a curve branch is now known in some detail, and forms a substantial addition to our knowledge of the properties of one-dimensional local rings†; but the construction of an abstract theory similarly related to the theory of neighbourhoods in n-dimensional projective space can hardly be said to have been started. A number of necessary preliminary steps were taken by the author in (3)—in the process of providing algebraic foundations for certain applications of dilatation theory—and later some applications were made to 2-dimensional problems. However, the present paper should be regarded as an attempt to initiate a dilatation theory of regular local rings to run parallel to the general theory of infinitely near points in n-dimensional space.

Eilenberg and Steenrod(1) give a set of axioms for the homology theory of pairs of spaces and their maps, and prove that these axioms are categorical on triangulable pairs.

Here we give a set of axioms for the homology theory of single spaces and their maps, that is, for absolute rather than relative homology. This axiomatization is shown to be essentially equivalent to that of Eilenberg and Steenrod, the relative homology groups being suitably denned in terms of the absolute groups.

Suppose throughout that

and that

is an integral function. Suppose also that l, sn(n = 0,1,…) are arbitrary complex numbers and denote by ρ(ps) the radius of convergence of the series

1. Let φ(z) be regular in │z│ < 1, let

and for any real α let

where (in)−α = n−α exp(— ½απi). The function φα is an adaptation to power series of the Weyl fractional integral of order a of α function of a real variable.† It exists and is regular in │z│ < 1 for all α. Let also

Since the first introduction of the concept of a matrix, questions about functions of matrices have had the attention of many writers, starting with Cayley(i) in 1858, and Laguerre(2) in 1867. In 1883, Sylvester(3) defined a general function φ(a) of a matrix a with simple characteristic roots, by use of Lagrange's interpolation formula, and Buchheim (4), in 1886, extended his definition to the case of multiple characteristic roots. Then Weyr(5) showed in 1887 that, for a matrix a with characteristic roots lying inside the circle of convergence of a power series φ(ζ), the power series φ(a) is convergent; and in 1900 Poincaré (6) obtained the formulae

for the sum, where C is a circle lying in and concentric with the circle of convergence, and containing all the characteristic roots in its ulterior, such a formula having effectively been suggested by Frobenius(7) in 1896 for defining a general function of a matrix. Phillips (8), in 1919, discovered the analogue, for power series in matrices, of Taylor's theorem. In 1926 Hensel(9) completed the result of Weyr by showing that a necessary and sufficient condition for the convergence of φ(a) is the convergence of the derived series φ(r)(α) (0 ≼ r < mα; α) at each characteristic root α of a, of order r at most the multiplicity mα of α. In 1928 Giorgi(10) gave a definition, depending on the classical canonical decomposition of a matrix, which is equivalent to the contour integral formula, and Fantappie (11) developed the theory of this formula, and obtained the expression

for the characteristic projectors.

1. The object of this note is to deduce some self-reciprocal functions and kernels for Hankel transforms. We denote by Rμ the class of functions which are self-reciprocal for Hankel transforms of order μ; that is, those that satisfy

where Jμ(x) is a Bessel function of order μ. In particular we write R8 for R½ and Rc for R−½.

Quantum mechanics is formulated in terms of the sequences introduced in earlier papers of the series. This reformulation resolves some difficulties in the foundations of quantum theory and gives a new picture of an elementary particle.

It is well known that there is no static axisymmetric two-body solution of Einstein's gravitational field equations, if it is assumed that the bodies are separated in a certain definite sense. In this paper it is shown, by the construction of a complete physically sensible model, that static two-body solutions do exist for systems in which one body is hollow and contains the other. The stability of the particular system described is briefly discussed.

A recent investigation of the interaction between the Maxwell radiation field and neutral atoms or molecules has led us to consider the relation between the straightforward interaction Hamiltonian

between particle (k) and the radiation field, and the more physical, and explicitly gauge invariant, couplings of the form q. E and µ. H, etc., where q and µ are the electric and magnetic dipole moments of a system of particles. That these latter form of interaction energies follow from well-formulated approximations for atoms in external fields has been known for many years (Goeppert-Mayer(1); see also Rosenfeld(2)). When the Maxwell field is an internal and dynamical field further discussion is necessary, especially concerning the Coulomb interactions. Their advantage has recently been shown in dealing with dipole interactions in the long range Van der Waals force between two neutral systems where the complete coupling, in dipole approximation, is given by — q.ET for each neutral system, where ET is the transverse electric field. The short-range London limit is contained in this coupling despite the absence of longitudinal electric fields (Power and Zienau (3)). In this note we extend these arguments to higher electric multipoles and magnetic multipoles. We also demonstrate their use in the quantum theory of scattering of light, and in particular the derivations of Stephen (4) of the phase angle between emergent beams in optical double refraction phenomena are shown to follow immediately.

It has been shown recently that a fruitful method of studying the formation of waves on the sea surface is through the average intensity of reflected sunlight, as seen from different angles ((3), (4)). This gives, in effect, the statistical distribution of the components of surface slope.

Following the same line of thought, we may inquire what information could be derived from the distribution of the sizes of the reflected images of the sun. The size of an image depends essentially on the total curvature of the surface at the point of reflexion (see § 2).

This paper is concerned with the small transverse displacement of a thin circular plate elastically restrained along the circular boundary and subject to uniform normal pressure distributed over the area of an ellipse concentric with the plate. The method of complex potentials is used and the general assumptions relating to the small deflexion theory of thin plates are adopted. Explicit formulae for the bending and twisting moments and shearing forces are given at any point on the boundary and at the centre of the plate.

An exact solution in finite terms is derived within the limitations of the classical thin-plate theory, for the problem of a thin circular plate acted upon normally by hydrostatic pressure distributed over the area of a concentric ellipse, and subject to boundary conditions covering the usual rigidly clamped and simply supported boundaries.

In this paper complex variable methods are used to derive exact solutions in closed forms for the small deflexions of certain thin elastic plates due to transverse concentrated forces or couples applied at arbitrary or specified points. The isotropic plates considered are bounded by curvilinear edges of certain types along which the plates are rigidly clamped. Plates bounded by quartic curves having the forms of the inverses of an ellipse with respect to its centre or its focus are included as special cases.