When independent random events are scaled down by a factor n, the number of counts in a given time interval no longer obeys the Poisson Law. The correct distribution function is derived together with its moment-generating function and explicit formulae (n = 2, 4) for the first four moments about the mean. The theory is in agreement with experiments carried out with a scale-of-four counter. A more general treatment is given and is applied to find the variance for all even values of n.

1. Let Jn be a sphere with volume V(Jn) in the n-dimensional Euclidean space Rn. Let Λ be a lattice of determinant d(Λ) such that every point in Rn lies in one at least of the bodies obtained from Jn by applying to it all possible lattice translations. Then Λ is called a covering lattice for Jn and V(Jn)/d(Λ) is called the density of the lattice covering by Jn provided by Λ. The lower bound θn of V(Jn)/d(Λ) taken over all covering lattices Λ for Jn is called the density of the thinnest lattice covering by Jn.

Let

where the frequencies vr are strictly positive and the amplitudes ar are non-zero; and let

be the number of zeros of f(t) – a for 0 ≤ t < T. In physical problems, where f(t) may represent, for example, a general coordinate in a vibrating dynamical system or a set of alternating currents in a cable, formulae are required for the frequency with which f(t) passes through a given value a; that is, for asymptotic or average values of Ga/T. The purpose here is to collect such formulae, and to sketch their background and relations in a way which may suggest extensions. The writer is particularly indebted to Prof. P. Stein for a manuscript containing new results (SI-V in §§2 and 5 below) and for his permission to publish them without his original proofs; these proofs were extensions of the method he gave (6) for the case n = 2.

In this paper, I prove some new results which are identities connecting groups of general infinite products, and I give two tables, calculated using EDSAC, of the function 1/II(1 − aqn−1) over the range a = − 0·90(0·05) + 0·95, q = 0·0(0·05) + 1·00, and of the function 1/n(1 − qn) over the range q = 0(0·005)+ 0·890.

The paper is concerned with the construction of polynomials in two variables, which form a complete orthogonal set for the interior of the unit circle and which are ‘invariant in form’ with respect to rotations of axes about the origin of coordinates. It is found that though there exist an infinity of such sets there is only one set which in addition has certain simple properties strictly analogous to that of Legendre polynomials. This set is found to be identical with the set of the circle polynomials of Zernike which play an important part in the theory of phase contrast and in the Nijboer-Zernike diffraction theory of optical aberrations.

The results make it possible to derive explicit expressions for the Zernike polynomials in a simple, systematic manner. The method employed may also be used to derive other orthogonal sets. One new set is investigated, and the generating functions for this set and for the Zernike polynomials are also given.

As Armitage(1) has noted, this is a problem of some importance in dust particle counting where overlapping particles are usually counted as single particles owing to their small size and the difficulty of detecting overlapping. In his paper Armitage(1) obtained approximate formulae and also expansions for the expected number of clumps (as he called a self-overlapping group) (i) when the particles were all circular laminae (though of different sizes) and (ii) when the particles were similar rectangular laminae randomly oriented. His expansions were accurate to the first two terms, but he only gave approximate values for the third terms. However, by using a method similar to that of the author (5,6) in treating aggregates in random placings of points and to that of Hammersley (4) in treating one-dimensional problems analogous to those dealt with here, this paper gives exact formulae for any mixture of laminae of different shapes and sizes, the only restrictions being (i) that they are all convex (though they need not be symmetrical), (ii) that they are randomly oriented with no preferred direction, and (iii) that there are a large number of each kind. With these restrictions we shall show in addition that, surprisingly, the only relevant geometrical properties of the laminae are (i) their areas and (ii) their perimeter lengths, their shapes apart from this being immaterial.

1. Let K1, K2, K3 be three conjugate* cubic algebraic number fields, of which we shall take K1 to be real and K2, K3 to be complex conjugate. Let ωi1, ωi2, ωi3 the conjugate basis for the integers of Ki. Write

so that ξi runs through the integers of Ki as the xj run independently through the rational integers, and the ξi (i = 1, 2, 3) are conjugate. The determinant of the ξi, regarded as linear forms in the xj, is ± √d, where d is the common discriminant of the fields.

This paper contains all those formulae which are necessary for the numerical evaluation of the basic confluent hypergeometric functions, including asymptotic formulae, and three skeleton tables which can be used as a basis for the study of the numerical behaviour of these functions over a wide field.

Estimators are given for the solution by Monte Carlo methods of the Fredholm integral equation of the second kind and the variances of the estimators are compared. Two integral equations arising in sequential analysis are considered in detail; some numerical examples are given.

The paper shows that the calculation of transients by Tricomi's method using Laguerre functions is a practical alternative to the use of partial fractions. By taking numerical examples of Laplace transforms ranging from quadratic to sixth-power polynomials it shows that the approximation offered by this method would be acceptable for many practical uses. It analyses the composition of the coefficients of the Laguerre series, thus disclosing the conditions for quick convergence; for some transforms these conditions are not attainable.

If R is any n-dimensional convex region, symmetric about the origin, and V(R), Δ(R) denote the content and critical determinant of R respectively, then

Consider a random-walk problem on a simple lattice, the probabilities of the walker taking any direction in the lattice at each lattice point being equal. Then Polya (6) has shown that if a walker starts at the origin and continues to walk indefinitely, the probability of his passing through his starting point is unity in one and two dimensions, but less than unity in three or more dimensions. Recently, a generalization of this problem has been considered (1),(3) in which the walker is allowed to jump several lattice points with assigned probabilities. F. G. Foster and I. J. Good (3) have shown that if the assigned probabilities satisfy certain conditions, Polya's result still holds, and K. L. Chung and W. H. J. Fuchs (1) have shown that the result is valid under far less restrictive conditions. The above authors were primarily concerned with the question whether return is almost certain or not, and did not consider a detailed calculation of the probability at any stage. It is the purpose of the present paper to show that the use of contour integrals allied with the method of steepest descents (4) enables one to perform this calculation very simply.

Introduction. Many automatic computers employ magnetic drums or ultrasonic delay lines as their main store. A part of the time taken by a computation on such a machine is spent in waiting for words to become available from the store. A number of methods and devices have been used or suggested to reduce this waste of time. This paper attempts to evaluate these methods by determining on the one hand the amount of waiting time eliminated and on the other hand the design or programming complications introduced. No quantitative method for determining the amount of eliminated waiting time was known. Available estimates were based on the feeling of the programmer and varied accordingly. It was therefore necessary in the first place to develop a process by which the amount of waiting time eliminated by any given method could be determined. This process is set out while being applied to the methods discussed. The methods themselves are not new, except that closer analysis has shown that the method adopted with the ACE pilot model and known as ‘optimum programming’ is in fact not a single method but a combination of several methods with different features. Also, it appeared that, contrary to an opinion expressed elsewhere (2), there is a simple way of staggering locations in the store even with words of two lengths.

Sets of rules are obtained for writing down directly the exact integral equations which are satisfied by certain functions of Heisenberg operators in quantum field theory. Three kinds of function are considered: the direct product, the chronological product and the M-product. The matrix elements of the M-product are equal to the Feynman amplitudes studied by Matthews & Salam (1) and the corresponding integral equation is called here the Matthews-Salam (M.-S.) equation. These authors have given a symbolic form of the M.-S. equation and a method of repeated differentiation and integration which can be used to obtain the explicit form of the integral equation in any particular example. In practice their method involves an immense amount of calculation even in quite simple examples. The rules obtained in the present paper make it possible to write down directly the M.-S. equation without any of the tedious calculations implied by the M.-S. method.

So long as the exact theory is used, the three sets of equations (for direct, chronological and M-products) are completely equivalent. When bound state theory is considered by an approximation based on a power series in the coupling constant different results are obtained. The approximation is inapplicable to the direct product equations, and leads to different approximate equations for the amplitudes obtained from the chronological and the M-products even when these amplitudes are identical. This paradox is explained and it is shown that the equation coming from the M-product corresponds to the Bethe-Salpeter equation.

In a recent paper (3) Diananda has given an extension of the central limit theorem to stationary sequences of random variables {Yt} (t = 1,2,3,…), which are m–dependent, i.e. are such that m + 1 is the smallest integer r having the property that two sets of the Y's are independent whenever the suffix of any member of one set differs from that of any member of the other set by at least r. As Diananda points out, this can be used to show that for linear autoregressive processes with m–dependent residuals, the joint distribution of any finite number of the forms which occur in Bartlett and Diananda's goodness of fit test (see (1)) is asymptotically multivariate normal, provided also that the fourth moment of the residuals is finite. These forms are linear functions of the sample serial correlations rt, and if the autoregressive process is defined by

they may be denoted by , where

Et being the usual shift operator such that for any function ft defined for integral t, In particular, when , as for a moving average process, the joint distribution of any finite number of the rt, is asymptotically multivariate normal.

It is the object of this paper to obtain an asymptotic formula for the number of partitions pm(n) of a large positive integer n into m parts λr, where the number m becomes large with n and the numbers λ1, λ2,… form a sequence of positive integers. The formula is proved by using the classical method of contour integration due to Hardy, Ramanujan and Littlewood. It will be necessary to assume certain conditions on the sequence λr, but these conditions are satisfied in most of the cases of interest. In particular, we shall be able to prove the asymptotic formula in the cases of partitions into positive integers, primes and kth powers for any positive integer k.

This paper contains a proof that the description of the phenomenon of Bose-Einstein condensation is the same whether (1) an open system is contemplated and treated on the basis of the grand canonical ensemble, or (2) a closed system is contemplated and treated on the basis of the canonical ensemble without recourse to the method of steepest descents, or (3) a closed system is contemplated and treated on the basis of the canonical ensemble using the method of steepest descents. Contrary to what is usually believed, it is shown that the crucial factor governing the incidence of the condensation phenomenon of a system (open or closed) having an infinity of energy levels is the density of states N(E) ∝ En for high quantum numbers, a condition for condensation being n > 0. These results are obtained on the basis of the following assumptions: (i) For large volumes V (a) all energy levels behave like V−θ, and (b) there exists a finite integer M such that it is justifiable to put for the jth energy level Ej= c V−θand to use the continuous spectrum approximation, whenever j ≥ M c θ τ are positive constants, (ii) All results are evaluated in the limit in which the volume of the gas is allowed to tend to infinity, keeping the volume density of particles a finite and non-zero constant. The present paper also serves to coordinate much of previously published work, and corrects a current misconception regarding the method of steepest descents.

The theory of a previous paper is used to discuss Eddington's work in fundamental physics.

1. Introduction. A previous paper (2), referred to here as I, described a theory depending on an unorthodox view of the relation of mathematics to physics. This theory was begun from the study of the work of Eddington which we take to be summarized in Fundamental Theory (5). In the present paper is shown the extent to which, and the way in which, the two theories are connected.

The reciprocity theorem for electrical systems which include a radiation link was believed to be true only when the media within the system have dielectric constants which are symmetric tensors. This condition is not fulfilled by the ionosphere, so that the reciprocity theorem is not generally applicable when the radiation link includes one or more reflexions from the ionosphere. It is here proved that, when the ionosphere is horizontally stratified, and when the path from transmitter to receiver is in the magnetic meridian (north-south and south-north transmission), the reciprocity theorem applies (a) when the transmitting and receiving aerials both radiate or receive waves whose electric vector is in the plane of incidence, and (b) when both aerials radiate or receive waves whose electric vector is horizontal. Further, (c) if the electric vector radiated or received is horizontal for one aerial and in the plane of incidence for the other, then there is reciprocity in signal amplitude, but the phase changes for transmission in the two directions differ by 180°. These results are valid for any law of variation of electron density and collision frequency with height. They are based on a ‘full-wave’ theory, and therefore apply to all frequencies. They are unaffected if the path includes multiple reflexions, and if allowance is made for the curvature of the earth.

In a previous paper (2) of the author, there was given a treatment of Tauberian theorems for Laplace transforms with values in an arbitrary Banach space. Now, in § 2 of the present paper, this kind of technique is applied to the more special case of Laplace transforms with values in a Banach lattice, and investigations are made on what additional results can be obtained by taking into account the existence of an ordering relation in the space. The general argument is again based on Widder (5) to which frequent references are made.