Any harmonic function which is defined and positive in the half-plane η > 0 may be expressed by

where C is a non-negative number, and G(x) is a bounded non-decreasing function. For a simple proof see Loomis and Widder (2). Let us write

where w(z) is a regular function of z in η > 0, and satisfies the following conditions: (i) w(z) is real and continuous at all points of the open interval (a, b) of the real axis [the interval may be unbounded]; (ii) there exists a simply connected domain Δ, lying in the half-plane η > 0, whose frontier contains the interval (a, b) of the real axis and which is mapped ‘simply’ on the half-plane υ > 0 by the conformal transformation w = w(z).

Suleῐmanova (3) describes a geometrical technique* for discussing the conditions under which a given set of n + 1 numbers can be a set of characteristic roots of a positive stochastic matrix of order n + 1. Suleῐmanova states that, using this technique, it is possible to prove that a set of n + 1 real numbers 1, λ1, λ2, …, λn, where | λi | < 1 for i = 1, 2, …, n, is a set of characteristic roots of a positive stochastic matrix provided that the sum of the moduli of the negative numbers of the set is less than unity. No indication of the method to be employed is given in (3), and it would seem that the problem is a difficult one. In the present paper I give a full discussion of how Suleῐmanova's technique may be applied to solve the problem for third-order matrices, and simply an indication of the solution for fourth-order matrices; and in, the latter case I point out that Suleῐmanova's result may be replaced by a stronger one. For second-order matrices the problem is almost trivial. As a preliminary I prove two general theorems.

During the course of recent work (6) on the zeros of the Bessel functions Jn(x) and Yn(x), it became evident that the theory of the asymptotic expansion of Bessel functions whose arguments and orders are of comparable magnitudes was incomplete. The existing expansions for large orders are those of Debye and Meissel, detailed derivations of both of which are given by Watson ((8), pp. 237–48).

1. In a recent paper(1), Afriat has given necessary and sufficient conditions for a real quadratic form to be positive definite on a linear manifold, in terms of the dual Grassmannian coordinates of the manifold. Considerable matrix manipulations were used in Afriat's method, but most of these may be avoided by the method of the present paper, which depends on some well-known properties of the Grassmannian coordinates. We first show that the conditions may be expressed as a set of inequalities which are quadratic in the Grassmannian coordinates of the manifold. Then, by a standard theorem, these may be transformed into Afriat's conditions on the dual coordinates.

In this paper the principles for the choice of a pair of standard solutions of a homogeneous linear differential equation of the second order, described in an earlier paper (2), are applied to Weber's equation

1. The study of queues has been of interest to mathematicians and engineers for the past forty years, and a very extensive literature on the subject exists. The applications of the theory are many and varied, from Erlang's original work on telephone engineering, to present-day studies in the design of airports. The number of papers which deal with the theoretical side of the subject is, however, small, and it therefore seems desirable to attempt to develop a general theory which will cover the diverse practical requirements, so that the unity in the applications will become apparent. This paper gives such a development in the case where there is a single queue and a single server attending to it; the theory of multiple queues or many servers seems, except under simplifying assumptions which do not always correspond to reality, to be a problem of considerable difficulty. Previous work on the subject has mainly been confined to a special case where the customers arrive at random, as, for example, in a recent paper by Kendall(5);* the present theory makes no such assumption and allows the customers to join the queue in other ways, though the theory simplifies when the more restrictive assumption is made. The centre of interest in the present theory is the waiting times of the customers.

In this paper I prove the following results.

Theorem 1. Let f(x, y) = ax2 + bxy + cy2be an indefinite form. Then there exist real (x0, y0) such that

for all (x, y) ≡ (x0, y0) (mod 1). If f represents 0 there are indenumerably many such incongruent (x0, y0).

In a recent paper (2) I gave some integrals which represented general transformations of hypergeometric series of both ordinary and basic types. In this paper I give the most general form of this kind of integral, from which all the previously known general basic transformations can be deduced.

In the usual notation for basic series,

where x may be a function of n, for example,

and Π is written for ‘Idem (a; b)' means that the expression immediately preceding is to be repeated with b written in place of a and a written in place of b.

A method is given for obtaining sequential tests in the presence of nuisance parameters. It is assumed that a jointly sufficient set of estimators exists for the unknown parameters. A number of special tests are described and their properties discussed.

Abel's series(1)

may be regarded as a generalization of the Taylor expansion

This note generalizes the two-point series of Lidstone and Whittaker (see (9)) in a similar way. Alternatively, the series discussed might be regarded as generalizations of Abel's series.

S. Bernstein (see(l), pp. 198–204) has proved that if

is an entire function of genus zero, then the series

is convergent. By considering the most unfavourable case, when all the (βn) are positive real numbers, and writing

which is convergent by hypothesis, he proved that

.

1. Published tables facilitating the numerical evaluation of the definite integral

from given numerical values f(xi) of the integrand in the form of a weighted sum

essentially deal with the following situations:

(a) The arguments xi are equidistant, i.e. xi = x0+iw; i=integer and the weight coefficients ci are tabulated for all arguments xi occurring in the summation. Among these tables the most important are:

(a1) panels in which the end-points of the integration a and b are two of the xi, and in particular

(a11) panels confined to the range of the integration in which

(a12) panels extending beyond the range of integration which usually have the same number of arguments, k, below a and beyond b, so that

(a2) panels in which a and b are at distance ½w from one of the xi, and in particular

(a21) panels confined to the range of integration in which

(a22) panels extending beyond the range of integration which usually have the same number of (k) points below a and beyond b, so that

(b) The n arguments xi are not equidistant but are specially chosen so that for a given n the sum S approximates to I with highest precision. Gauss's formula (see, for example, Milne-Thomson (4)) and various formulae of approximate product integration (see, for example, Beard (l)) are examples of these. In Chebyshev's integration formula (see, for example, Runge and König (5)) the xi required for equal weights ctare used, resulting in an important formula when the ordinates f(xi) are affeected by observational errors of equal variance.

In the theory of functions of a complex variable a positive quantity g(r), denned in terms of a circle |z| = r, can often be estimated by means of another similar quantity f(R) denned in terms of a circle |z| = R with R > r. Inequalities of the form

arise in this connexion. Putting R = r + h we have for every h > 0

.

A method of calculating S-matrix elements is described. All factors corresponding to lines in Feynman-Dyson graphs are put in the form of scalar boson line factors acted on by certain operators and involving only the energy-momenta of external lines and particle masses. A general formula is given for the integral of a product of scalar boson factors, and the effect of a product of operators and on this result is described. The infinities appear in a different form, and it is shown how subtraction procedure can be applied to eliminate them.

1. General remarks. The theorem that has come to be known as Cochran's theorem in works on mathematical statistics was published in these Proceedings in 1934(1). If x1, …, xn are independently distributed standard normal deviates, and q1, …, qk are k real quadratic forms in the xi with ranks n1, …, nk respectively, and such that then Cochran's Theorem II states that a necessary and sufficient condition that q1 …, qk are independently distributed in χ2 forms with n1, …, nk degrees of freedom is that Σnj = n. The necessity of the condition is obvious. Cochxan proves its sufficiency by expressing each qj as a sum, involving nj squares of real linear forms in the xi; it follows easily that the coefficients ci are in fact + 1, and that the transformation is orthogonal. The theorem then follows immediately from the properties of orthogonal transformations in relation to independent normal deviates.

Consider the infinite system of linear ‘inequations’,

in which the coefficients pij have the two properties

We define as properly divergent (p.d.) a solution, {xi} (i = 0, 1, 2, …), such that

and such that |xi| < ∞ for i < ∞. Since {k + xi} is also a solution for any non-negative constant k, there is no loss in generality in assuming (as we always shall) in place of this last condition that

We are concerned with conditions for the existence of p.d. solutions.

A hexagonal lattice bound by central forces between nearest neighbours is always close-packed. If there is in addition an energy which, like the Fermi energy of free electrons, depends only on the volume, a close-packed equilibrium structure is still possible, but there may be another hexagonal structure of lower energy. A numerical example is given in which this occurs, the lattice spacing, binding energy and elastic constants being comparable with those of Zn, and the law of force showing no obvious peculiarities.

In considering the statistics of the ‘no-field’ square Ising lattice in which each unit is capable of two configurations and only nearest neighbours interact, Kramers and Wannier (3) were able to deduce an inversion transformation under which the partition function of the lattice is invariant when the temperature is transformed from a low to a high (‘inverted’) value. The important property of this inversion transformation is that its fixed point gives the transition point of the lattice.

An Abelian theorem is a theorem stating that a given behaviour on the part of each of several quantities entails similar behaviour for their average. A Tauberian theorem is a converse to an Abelian theorem. As a rule, a given behaviour of an average will not entail similar behaviour of the individual quantities themselves unless there is some condition imposed to secure reasonably uniform behaviour amongst the individuals. Such a condition, known as a Tauberian condition, is usually sufficient but not necessary, and it enters into the premises of the Tauberian theorem. We interpret ‘average’ in a wide sense to include any kind of smoothing process; for example, the integral of a function f(t) is an average of the values of f(t) corresponding to individual values of t; and we may seek a sufficient Tauberian condition such that a limit-behaviour of an integral-average

entails the corresponding limit-behaviour

for individual values of t.

A method is given for finding, at the end of a sequential sampling procedure, a nearly unbiased estimate of the population mean.