Throughout this note the series is a rearrangement (in the order of its terms) of the series of complex terms a1 + a2 + ….

Wright (14), Feller (2) and others have proposed various stochastic models of genetics to study the fluctuations of gene frequency under the influence of mutation and selection. One of their simplest models has the following structure. There are a fixed number N of gametes each of which may be of two types a or A. The process X(n), n = 0, 1, 2, … which is assumed to have stationary transitions, is said to be in state j when there are j gametes of type a, and N − j of type A. Let γ1 denote the probability that immediately after formation an a gamete mutates into an A gamete and let γ2 denote the probability of an A gamete mutating into an a gamete. Each of the gametes of the next generation is independently formed by making a random selection from the gametes of the present generation. The probability of a particular gamete in the next generation being of type a is then and of type A where j represents the state of the process. Roughly speaking, the chance of a mating resulting in a gamete of a prescribed kind for the next generation is proportional to the fraction of these gametes present in this generation allowing for mutation effects.

The behaviour of a system of two coupled disk dynamos is studied in considerable detail. It is shown that in several cases the stability criteria and general behaviour are closely related to those for the single dynamo studied by Bullard, which are resumed briefly. Extended numerical investigations of the non-linear behaviour of the system are presented: these show that a true ‘reversal of field’ may occur.

The second method of Liapunov is a useful technique for investigating the stability of linear and non-linear ordinary differential equations. It is well known that the second method of Liapunov, when applied to linear differential equations with real constant coefficients, gives rise to sets of necessary and sufficient stability conditions which are alternatives to the well-known Routh-Hurwitz conditions. In this paper a direct proof of the Routh-Hurwitz conditions themselves is given using Liapunov's second method. The new proof is ‘elementary’ in that it depends on the fundamental concept of stability associated with Liapunov's second method, and not on theorems in the complex integral calculus which are required in the usual proofs. A useful by-product of this new proof is a method of determining the coefficients of a linear differential equation with real constant coefficients in terms of its Hurwitz determinants.

MacMahon (2) makes much use of the following result, which he describes as the “Master Theorem”.

Let

and let m1, m2, … mn be non-integers. Then the coefficient ofinis equal to the coefficient ofin

Equations are derived for the process of catching-up and overtaking which occurs in one lane of traffic for a model in which vehicles travel in random bunches and overtake at random times. The model allows vehicle velocities to be distributed and allows for several vehicles at a time to overtake. At first only stationary states are considered, that is states in which the rate of catching-up equals, on average, the rate of overtaking.

We continue below the note ‘Some cyclic and other inequalities’ (3).

MacMahon (5) used his “Master Theorem”, which is stated in the preceding note (4), in order to obtain an elegant proof of Dixon's identity

Let be, for a set of n real continuous parameters the probability density function of a random variable x with respect to a σ-finite measure μ on a σ-algebra of subsets of the sample space . If x; is a continuous random variable, μ will be Lebesgue measure on the Borel sets of a Euclidean sample space and, if x is discrete, μ will be counting measure on the class of all sets of a countable sample space. The parameters αi are said to be orthogonal (Jeffreys (3), pp. 158,184) if .

For a certain class of differential equations it is shown that the Stokes phenomenon can be related to a basic change in the character of the differential equation.

The case of a cylindrically symmetric cluster of particles moving in circles perpendicular to the axis of symmetry is discussed on the basis of the equations of general relativity. A number of special solutions are obtained, and two interesting results on the relation between the gravitational mass and the sum of the free particle masses for these clusters are deduced.

It is usual in the theory of queues to assume that customers are served in the order of their arrival. In some applications (e.g. telephone engineering), however, other forms of queue discipline are more realistic. The precise effect of any such change on the waiting-time distribution of a customer will depend on the procedure envisaged (random service, “last come, first served”, etc.), but it is possible to make certain general statements. Thus it is well known that, under certain conditions, the mean is independent of the queue discipline. The purpose of the present note is to consider the variance of waiting time, and we shall prove that this is a minimum when the customers are served in order of arrival. Thus this is, in a sense, the “fairest” queue discipline. This does not, of course, mean that other procedures may not be justified when different criteria are taken into account.

We consider a random walk defined in the following way. We have a set of states indexed by n where n takes on all negative and positive integral values and zero. When we are at state n, there is a probability per unit time λ of going to n + 1, and a probability per unit time λ of going to n − l. Let us start out at n = 0, and study Wn(t), the probability of being at n at time t. Continuity of probability requires that whence since G(s, 0) = 1, we have It follows from the well-known result .

Four-index tensors with all possible terms either (a) quadratic in the Riemann curvature tensor, or (b) linear in its second derivatives, with coefficients constructed from products of metric tensors, are enumerated for the case of Riemannian space-time. All linear combinations of these which have zero divergence, and which might therefore be associated with the super-energy in gravitational theory, are found by elementary methods. The only independent such combination reduces in empty space-time to the tensor previously found by Bel and Robinson.

In the course of researches on the formation of galaxies one meets the following idealized problem. What is the form of the collapse under gravitational forces of a uniformly rotating spheroidal gas cloud? In the special case where initially the gas is absolutely cold and of uniform density within the spheroid, we show that the collapse proceeds through a series of uniform, uniformly rotating spheroids until a disk is formed. This demonstration is followed by some remarks and conjectures as to what will happen in the physical case when the initial temperature is not necessarily zero.

In (l) Khuri has proved the validity of a dispersion relation for non-relativistic potential scattering. More precisely, he has shown that if the potential V(r) is central and satisfies: then the scattering amplitude f(k, τ) = M(E, τ) (where E = k2 is the energy) satisfies the following dispersion relation for fixed momentum transfer τ ≤ 2α: In (2), Rj(τ) is the (real) residue of the scattering amplitude at the bound state Ej and is the Fourier transform of the potential .