Suppose I is a bounded plane continuum whose complement is a single domain (I) and that is a (1–1) bicontinuous transformation of the plane onto itself which leaves I invariant. Cartwright and Littlewood(1) have proved the following theorem:

Theorem A. Suppose that(I) has no prime end fixed underand the frontier of I contains a fixed point P. Ifis any prime end of(I) containing P, and C is any curve in(I) converging tosuch that PeC¯, the closure of C, then for some‡ integer N the continuum I(FN) consisting of

together with the sum B(FN) of its interior complementary domains contains all fixed points in I.

1. Let be a Lie ring in which the product of elements x and y is denoted by xy. The inner derivations of , i.e. the mappings X:a→ax for fixed elements x of , form a Lie ring under the product [X, Y] = XY – YX, and the mapping x→ X is a homo-morphism of onto . We shall say that satisfies the nth Engel condition if Xn = 0 for all X in , i.e. if

for all a, x; in . If satisfies the maximum condition on subrings, it is known (1) that this condition implies the nilpotence of ; indeed, must then be nilpotent even if the integer n is allowed to depend on the element X of . We consider here the case in which n is independent of X but does not necessarily satisfy the maximum condition, and inquire whether is then nilpotent.

Introduction. The present paper is concerned with the conformal geometry of Hermitian spaces. In the first part we find a necessary and sufficient condition for a Hermitian space to be conformally Kähler, that is, conformal to some Kähler space. The condition is that a certain conformal tensor, , vanishes identically. Then, defining a Hermitian manifold as in Hodge (3), we consider such a manifold where the restriction is made that at every point the tensor is zero. This will be called a conformally Kähler manifold, and conditions under which it may be given a Kähler metric are obtained. It is found that any conformally Kähler manifold may be given a Kähler metric provided it is simply-connected or that its fundamental group is of finite order.

1. Mahler (1) has obtained a number of valuable existence theorems for homogeneous problems in the geometry of numbers. It appears that no analogous results for inhomogeneous problems have yet been published; and although none of them present great difficulty, it is convenient for applications to have them in print.

1·1. The differential equation (E) of the title, which occurred in connexion with a problem of electrical engineering, was mentioned to the author by Miss M. L. Cart-wright. It is of some general interest in that, as will be shown, it may have a non-trivial periodic solution (p.s.), although the damping changes sign only once.

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.

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.

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.

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.

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 idea of direct action between streams is applied to a continuous charged fluid and combined with the new formulation of the electrodynamical laws of motion in terms of conservation of circulation. A simple and rigorous integrated formulation is thus obtained from the Maxwell-Lorentz differential equations, applicable to co-existing positive and negative fluids, as well as vacuum. Exact solutions are obtained, among them one which represents self-consistent, self-maintained flow in a hollow tubular region of infinite axial extent. It is hoped this tube might be bent into a torus and that an electron model will result from merely quantizing the one or two vortices around which this flow-pattern circulates.

When the boundary-layer approximations are applied to the laminar flow in an axially symmetrical jet of gas with Prandtl number unity, a particular integral of the equations of motion satisfying the required boundary conditions is found to be , where i1, is the (constant) enthalpy on the jet boundary. By a suitable transformation an equation is obtained which, to a first approximation, may be replaced by the equation for flowin an axially symmetrical jet of incompressible fluid at a sufficiently large distance from the orifice. For an ideal gas, the known solution of this equ`ation may be used as the starting point of an iterative method to build up the stream function as a power series. The next approximation is found explicitly. It shows that for subsonic Mach numbers the density at a point on the axis is greater and the temperature less than on the boundary by amounts which may be very closely represented by terms inversely proportional to the square of the distance of the point from the orifice.

A method is given for determining the stability of small sinusoidal oscillations in the steady laminar flow of a horizontal wind over the surface of a liquid at rest (with particular reference to the flow of air over water), taking into account viscosity, gravity and surface tension. It is shown that there are two fundamental types of oscillation of the system, which may be called ‘water’ waves and ‘air’ waves, and curves showing the conditions for neutral stability of these two types of wave are given for a range of wind speeds from 100 to 300 cm./sec.

1. In §§ 2–4 of this paper approximate expressions are found for the stream function and pressure in the steady two-dimensional motion of viscous incompressible liquid past a fixed parabolic cylinder; exact expressions for the stream-function and pressure in a perfect liquid are derived as limits in § 5.

The object of this paper is to discuss the one-dimensional unsteady adiabatic motion of a gas which is initially at rest with a prescribed density distribution such that the specific entropy is uniform. The contour integral methods which Copson developed recently for even analytic functions are extended to apply to general analytic initial conditions. The solution is valid in the range 1 < Υ > 3, where y is the adiabatic index of the gas. Of particular interest, in view of the hydraulic analogy, is the case Υ = 2 for which real variable methods cannot readily be adapted. The motion of the front of a water column Sowing into a dry, horizontal stream bed is discussed. A curious type of solution, corresponding to a particular choice of initial distribution, which was established by Pack for a, countable sequence of values of Υ, is verified to hold over the whole range and is interpreted in terms of the dam-break problem.

1. Whittaker (7), generalizing a result of Ferrar(3), showed that the cardinal series based on the positive and negative integers is consistent in the sense that, if

and an integral function f (x) is denned by

then provided 0 < λ > 1

In this note I show that results of Paley- Wiener, Levinson and others on biorthogonal series can be made to yield a consistency theorem for cardinal series based on sequences λn, where

that is, series

where

We use Fourier transform technique and need hypotheses, a little more restrictive than Whittaker's, which would reduce in the case .