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.

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.

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.

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.

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. The present paper deals with an enclosed (linear) harmonic oscillator. The usual boundary condition that the wave function vanishes at infinity is here replaced by the condition that the wave function vanishes at the walls of the enclosure. The problem has been treated before (1, 2, 4). However, the present discussion goes much further than that given in the papers cited*.

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.

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 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.

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.