1. In this paper I establish in a certain sense all the relativistically invariant sets of linear partial homogeneous differential equations in which every unknown function u satisfies being the relativistic Laplace operator. Analogous questions for any orthogonal group are dealt with in a previous paper, but it was thought that an independent and not too technical treatment for the Lorentz group might be of interest.

The Schrödinger equation for a ‘hole’ in the hole-theory of liquids is constructed and solved by the B.W.K. method. The energy levels are found to be discrete, and the eigenvalues are obtained in terms of the density and surface tension of the liquid. The relation between the energy of the ground state and the temperature of melting is considered.

In recent years, the theory of integral transforms—often in the guise of ‘operator calculus'—has been employed to obtain the solutions of a wide variety of problems in mathematical physics, particularly in the theory of vibrations. The most commonly employed transforms are those of Fourier, Laplace and Mellin; in problems where the range of one of the variables is (0, ∞) or (− ∞, ∞) these transforms have often been used with success to reduce a partial differential equation in n independent variables to one in n − 1 variables and hence to simplify the boundary value problem involved. In this paper we consider some simple problems in the theory of elastic vibrations in which the Bessel, or Hankel, transform is more useful than those mentioned above and is consequently employed throughout. In the first instance we consider problems in which the variable to be eliminated from the differential equation ranges from 0 to ∞ so that the theory of Hankel transforms is directly applicable; the application of finite intervals is then extended to the Hankel transform by a method similar to that used by Doetsch in his treatment of the Fourier transform.

Let Rc(n) be the number of representations of any non-negative integer n as the sum of c squares; i.e. Rc(n) is the number of different solutions (x1, x2, …, xc) of the equation

where the xμ are integers, and may be positive, negative, or zero. Two such solutions, (x1, x2, …, xc) and are only considered to be the same solution if

1. In Dirac's classical theory of radiating electrons, the relativistic equations of motion of a point-electron in an electromagnetic field are

where (x0, x1, x2, x3) denote the electron's coordinates in flat space-time, dots denote differentiation with respect to the proper time , and the external electromagnetic field is described by the usual field quantities Fμν. The units are chosen so that the velocity of light is unity. These equations, which are derived from the principles of conservation of energy and of momentum, are the same as those obtained by Lorentz, when he used the spherical model of the electron and included radiation damping in an approximate way. But Dirac's method of derivation suggests that this treatment of radiation damping, and therefore the resulting scheme of equations, is exact within the limits of the classical theory.

If O is an ordinary point of a plane curve and the tangent and inward normal at O are taken as axes, then, under certain conditions, the coordinates x and y of a neighbouring point P on the curve can be expanded in powers of s, the arc OP, or of Ψ, the angle between the tangents at O and P. Lamb(3) gives these expansions as far as the terms in s4 and Ψ3. Dockeray(1) goes as far as the terms in s6, but there seem to be three errors in his results. No one, I believe, has given the general terms. I obtain these for both pairs of expansions. Fowler(2) gives the term in s2r+1 in the expansion of x in the special case where, owing to the vanishing of the curvature and of its first r − 2 derivatives, the expansion of y begins with the term in sr+1. I shall give reasons for disagreeing with this result.

1. A few months ago, in the course of teaching an elementary class. I had occasion to discuss problems of the familiar type, ‘if α, β, γ are the roots of the equation x3 + qx + r = 0, form the equation whose roots are α2 + βγ, β2 + γα, γ2 + αβ'. After I had explained the device of replacing βγ by αβγ/α = −r/α, so that the roots of the new equation are the same rational function of the separate roots of the original equation, I was asked ‘whether such a transformation was always possible’. On investigation the answer to the question proved to be a surprising one.

It has been shown that orthodox probability theory may consistently be extended to include probability numbers outside the conventional range, and in particular negative probabilities. Random variables are correspondingly generalized to include extraordiary random variables; these have been defined in general, however, only through their characteristic functions.

This generalized theory implies redundancy, and its use is a matter of convenience. Eddington(3) has employed it in this sense to introduce a correction to the fluctuation in number of particles within a given volume.

Negative probabilities must always be combined with positive ones to give an ordinary probability before a physical interpretation is admissible. This suggests that where negative probabilities have appeared spontaneously in quantum theory it is due to the mathematical segregation of systems or states which physically only exist in combination.

If f is a real, indefinite, binary quadratic form of discriminant d and if κ(f) is the minimum of | f | taken over all integer values of x, y, not both zero, then it is well known that and that this is a ‘best possible’ result.

Any particular form of mechanics (e.g. classical or quantum) makes use of a particular ‘formalism’ or set of rules governing the use of the symbols representing dynamical variables and the symbols (such as +, =) representing relations between dynamical variables. In the present paper an attempt is made to examine the physical content of this formalism, particularly that of quantum mechanics. This is done by building up a formalism as the direct expression of a number of physical postulates having direct operational meaning. It is shown that, in a simple case, with any two observables A and B can be associated a unique sum observable A + B and a unique (real) symmetric product observable A.B (= B.A).

It is next shown that, if, like a classical Hamiltonian system, the system has the property that the time rate of change of an observable depends linearly on some observable H when the environment in which the system moves is varied, then a (real) skew product observable A × B (= − B × A) can be defined.

Finally, if the equations of motion are ‘of second order', it is possible to express the second rate of change of any observable as an algebraic function of that observable and H. This leads to algebraic identities from which it follows that ‘complex multiplication’, defined by AB = A.B + iA × B, is associative (but not commutative). Observables are thus shown to possess the properties usually ascribed to them in quantum mechanics. These properties make possible a representation by Hermitian matrices.

It was shown by Mehler (1866) that

where Hk(x) denotes the Hermite polynomial

(Hermite, 1864a, b), which can be expressed in terms of Weber's parabolic cylinder function (Whittaker, 1903). The series is convergent if | ρ | < 1, and divergent if | ρ | > 1. If ρ = 1 and x = y = 0 the series is divergent, and Hille's work (1938) shows that it will therefore be divergent for all real or complex values, except possibly real positive values, of x and y.

The problem of conduction of heat in a solid whose surface is in contact with fluid, which is so well stirred that its temperature is constant throughout its mass, often arises in practice, for example in calorimetry, or in the warming of a closed room. Despite their importance, problems of this type have not been much studied since the presence of the mass of fluid at the surface gives rise to a boundary condition for which the classical methods of solution do not apply without modification; the Laplace transformation method, however, has the advantage that it can be applied in the same way to all cases.

In this note I consider the Abelian integrals of the first kind on an algebraic curve Γ which is a normal multiple of a curve C, as defined in Note I*.

In the present note, which is introductory to the following paper, closed expressions, suitable for computational purposes, are found for the sums of the series

where α > 1, t = 1, 2, 3, …, and n is a positive integer. In each case a recurrent relation is found giving the values of and for t > 2 in terms of and the series Θκ(α) (κ = 1, 2, …, t), where

When κ is even the last series is expressed in closed form in terms of the Bernoullian polynomial φκ(l/α) and, when κ is odd and α is rational, a closed form is found involving the polygamma function Ψ(κ)(z), where The general expressions for and involve Ψ(z) and Ψ′(z) when α is rational, but for special values of α they reduce to a form independent of the Ψ-function. and are independent of n and are expressible as simple rational functions of α.

Let γ be an irreducible variety of dimension d, and let In denote an involution of order n on Γ. The image of In is a variety C, also irreducible and of dimension d, and Γ is said to be a multiple of C. Among the possible multiples of a given variety C there is a class of particular interest; this arises when the involution In possesses the following properties:

(1) it is unramified, that is, in every set of the involution the n points are all distinct;

(2) it is generated by an Abelian group G of order n of birational transformations of Γ into itself. Given any point P of Γ the n transformations of G transform P into n points P1 = P, P2, …, Pn which constitute a set of In. The two conditions together imply that the function field κ of Γ is an unramified Abelian Galois extension of the function field k of C, and this shows that the study of the particular class of multiple varieties—which we may call normal multiple varieties when it is necessary to give a name to them—is analogous to the study of a well-known branch of the theory of numbers.

In this paper, using the results of the preceding paper, I develop a modification of Hahn's method (1) for calculating the electromagnetic field inside a type of axially symmetric resonator. A process of successive approximation is given for solving the equation which leads to the resonant wave-length λ, both in the case where all the resonator parameters are given and we wish to find λ, and also in the case where λ is given and one of the parameters (the outer radius) is to be found. Some numerical results are given.

In the solution of certain boundary-value problems by means of complex integrals or by means of the Laplace transformation, transcendental equations of the form

arise, and it is then usually necessary to show that this equation has no roots in the right half of the complex z-plane.

Attempts have been made to extend to higher dimensional varieties the theory of correspondences developed for algebraic curves. So far efforts have been concentrated on ‘point-point’ correspondences (i.e. between two varieties of the same dimension such that a generic point of one corresponds to a 0-dimensional variety of the other), and even in the case of surfaces important problems, such as the base number for the correspondences, are still unsolved (cf. Hodge(3)). The purpose of these papers is to draw attention to, and study in the simpler cases, another class of correspondences, the ‘point-primal’ type, in which to a generic point of one variety corresponds a primal of the other. These correspondences provide at least as natural a generalization of the theory for curves as point-point correspondences, but have so far only been touched on by Severi(7, 8) in two simple cases. In this paper we give a number of general results for point-primal correspondences (mostly immediate generalizations of Severi's results), and embark on a detailed discussion of such correspondences, which we naturally call point-curve correspondences, between two surfaces, the chief result being the determination of the base in that case. Further results on the transformation of the cycles of a surface by a point-curve correspondence, the correspondences induced between curves of the two surfaces and, in the case where the surfaces coincide, the ‘united curve’ of the correspondence, as well as the question of correspondences of non-zero valency, will be dealt with in a later paper, where we shall also consider the case of correspondences between a curve and a surface discussed in Severi's paper(7).

The Vitali covering principle is a powerful method in a wide class of problems of the theory of functions of a real variable and of the theory of sets.