In this paper the Clifford groups PCT(pm), p > 2, PCG(pm) and CS'(pm), and the factor groups ½CS' (pm), which were defined in Paper I of this series (Bolt, Room and Wall [1]), are considered as transformations of projective [pm–1] over the complex field, C. We note that the geometrical results are the same if any of the corresponding groups CT, CG or GG and CS, respectively, are considered instead.

Papers I, II of this projected series lay the algebraic foundations of the theory of the Clifford groups; I deals with the case p > 2, II with the case p = 2. The present introduction refers to both papers. Our theory has applications in group theory, geometry and number theory.

Determinants [1] of 5th and 7th orders have already been discussed in connection with the Theorems 4 and 5 of Atkin and Swinnerton-Dyer [2]. The determinant under consideration occurs in the investigation of Dyson's rank function for q = 11 given by Atkin and Hussain [3]. As regards the notation it may be mentioned that the author has adopted the same as that of Atkin and Hussain [3].

The stochastic birth-death process considered in this paper provides an approximate model for phage reproduction in a bacterium. In a recent paper, Hershey [1] has discussed reproduction and recombination in phage crosses, and a deterministic model for the reproductive process has been the subject of a previous note by the author [2]. A very readable account of the process is given by Sanders [3] in his recent article, “The life of viruses”.

The present paper deals with the Clifford groups in the case p = 2. For the most part, it runs parallel to the previous paper I ([1]) on the case p > 2, and a number of proofs are therefore either given in outline or omitted. A general introduction to both papers is given in I, § 1.

We are interested in the motion of non linear systems. In this paper we use a variation principle and an iteration procedure in order to treat the stability of free oscillations against small disturbances of the initial conditions. It is found that approximations to the low lying stability lines can be obtained using the Rayleigh-Ritz variation principle and that these approximations can be consistently improved using an iteration procedure. These approximations are compared with the tabulated values in the special case of the Mathieu Equation. The results are in general a considerable improvement on those obtained using the usual Perturbation Theory, and have a much wider range of validity.

The continuous-time behaviour of a model which represents certain queues and infinite dams with correlated inputs is considered. It is shown how the transient behaviour may be investigated, and the asymptotic behaviour is obtained. Finally the methods are illustrated for a queue whose input consists of two superimposed renewal processes.

As an illustration of the use of his identity [10], Spitzer [11] obtained the Pollaczek-Khintchine formula for the waiting time distribution of the queue M/G/1. The present paper develops this approach, using a generalised form of Spitzer's identity applied to a three-demensional random walk. This yields a number of results for the general queue GI/G/1, including Smith' solution for the stationary waiting time, which is established under less restrictive conditions that hitherto (§ 5). A soultion is obtained for the busy period distribution in GI/G/1 (§ 7) which can be evaluated when either of the distributions concerned has a rational characteristic function. This solution contains some recent results of Conolly on the quene GI/En/1, as well as well-known results for M/G/1.

The familiar variation-iteration method for solving the eigenvalue equation Cψ = λBψ (C and B are Hermitean operators), is applied to a case in which the operator C, and hence also the eigenvalues λ, depend on a continuous parameter a. It is shown that certain exact properties of the functions λ = λ(a) can be deduced from low-order results in the variation-iteration scheme.

Contrary to the general impression that variation principles are of purely theoretical interest, we show by means of examples that they can lead to considerable practical advantages in the solution of non-linear vibration problems. In this paper, we develop a variation principle for the period of a free oscillation, as a function of the average value of the Lagrangian over one period. Even extremely simple-minded approximations to the true motion lead to excellent values for the period. The stability of such free oscillations against small disturbances of the initial conditions is treated in a previous paper.

A similarity method is used to develop a solution of the wave equation within a sector with mixed boundary conditions. In this manner the field which results from the diffraction of an incident pulse of step function time dependence is found.

This paper considers a generalisation of the queueing system M/G/I, where customers arriving at empty and non-empty queues have different service time distributions. The characteristic function (c.f.) of the stationary waiting time distribution and the probability generating function (p.g.f.) of the queue size are obtained. The busy period distribution is found; the results are generalised to an Erlangian inter-arrival distribution; the time-dependent problem is considered, and finally a special case of server absenteeism is discussed.

The Siegel conjecture on the rational approximation to algebraic numbers was proved a few years ago by K. F. Roth [1] with the following theorem: Let α be any algebraic number, not rational. If has an infinity of solutions in integers h and q (q > 0) tehn k ≤ 2.

For a theory of the flow about an aerofoil equipped with a split flap to be satisfactory, account must be taken of the wake which is formed downstream of the combination. In this paper the wing and the flap are each represented by a flat plate and it is assumed that the pressure in the wake is constant and equal to the free stream value, i.e. (it is assumed that the wake may be represented by the region between two free stream-line which extend downstream from the trailing edges of the wi ng and the flap1) (see Fig. 1).

The spinor analysis of Infeld and van der Waerden [1] is particularly well suited to the transcription of given flat space wave equations into forms which constitute possible generalizations appropriate to Riemann spaces [2]. The basic elements of this calculus are (i) the skew-symmetric spinor Γμν, (ii) the hermitian tensor-spinor σκ(img)ν(generalized Pauli matrices), and (iii) the curvature spinor Ρμνκι. When one deals with wave equations in Riemann spaces V4 one is apt to be confronted with expressions of somewhat bewildering appearance in so far as they may involve products of a large number of σ-symbols many of the indices of which may be paired in all sorts of ways either with each other or with the indices of the components of the curvature spinors. Such expressions are generally capable of great simplification, but how the latter may be achieved is often far from obvious. It is the purpose of this paper to present a number of useful relations between basic tensors and spinors, commonly known relations being taken for granted [3], [4], [5]. That some of these new relations appear as more or less trivial consequences of elementary identities is largely the result of a diligent search for their simplest derivation, once they had been obtained in more roundabout ways.

The theory of relativity shows that the times measured by two observers will in general be different if they are in relative motion, so that their respective times between any two coincidences will differ. Bergmann [1] has investigated the problem of a particle moving in a small simple harmonic motion in a static gravitational field, and has found that the time difference for this particle and an observer at rest becomes zero whenever the particle passes through the centre and limits of its swing. This problem will now be dealt with in a different manner, using Schwarzschild's interior solution of the gravitational equations. The exterior solution for a point mass is not suitable in the present case, due to the singularity of the field at a point in the path of the particle.

In the controversy which has been continuing over the “clock paradox”, it has been claimed by supporters ([1]) of the orthodox interpretation that the difference in behaviour of clocks carried by observers in motion relative to each other can be traced to differencesin the motion relative to the universe. Whilst A remains unaccelerated relative to the universe, B for a portion of his journey does not. Professor Dingle, in reply ([2]), remarked that he would “not dispute a possible slight effect” but could not understand why, if the acceleration of B were so important, it did not appear explicitly in the answer.