1. The notation used will be that of (2).

G is a finite group,

is a ring which is either

(1·1) a field of characteristic p, or

(1·2) the ring of valuation integers of a field of characteristic 0 with a complete discrete valuation. In this case, is the maximal ideal of .

Introduction. This paper falls into three parts.

In §§ 1–5 it is explained how, when the base field of the geometry is GF(2), there are figures of n + 2 interlocking polygons in [n], every two polygons sharing a vertex. When n is even these ½(n + 1) (n + 2) vertices lie in an [n − 1], and two of them are conjugate in a certain null polarity when, and only when, they do not belong to the same polygon.

1. Introduction. The open subsets of a topological space X form a complete Brouwerian, i.e. distributive and pseudo-complemented lattice L(X). Many topological properties of X can be formulated as properties of L(X), although, in general, X is not determined by L(X). Obvious examples are quasi-compactness and connectivity: X is quasi-compact if any set of elements of L(X) whose join is I has a finite subset whose join is I; X is connected if no element of L(X) has a complement. These properties which we shall call ‘paratopological’, can be defined for lattices that are not of the form L(X), and many topological theorems can be proved in this more general context. The purpose of this paper is to develop sections of general topology as part of the theory of complete Brouwerian lattices (CBL).

1. Objective. It is possible to define order convergence on the vector lattice of all continuous functions of compact support on a locally compact topological space. Every measure is a linear form on this vector lattice. The object of this paper is to prove that a measure is such that every set of the first category of Baire has measure zero if and only if the measure is a linear form which is continuous in the order convergence.

The irregular radial Coulomb wave function is expanded in a convergent series of Bessel functions in which the coefficients are expressed in powers of the energy and the argument of the Bessel functions depends on the radius only and not on the energy. This formulation is suitable for low-energy particles. The corresponding expansion for the regular Coulomb function can be deduced from previous work by Tricomi.

Exact tests devised by Bartlett and Williams for the adequacy of a single discriminant function in the case of several groups are applicable to a wide range of problems in multivariate analysis. The appropriate likelihood criterion is factorized into direction and collinearity factors; and the exact distribution of these factors under the conditions of only one non-zero canonical correlation was first given by Bartlett in the general case of any numbers of dependent and independent variables, making use of geometrical arguments.

In this paper the factors are expressed directly in terms of the matrices occurring in the multivariate analysis of variance table; and the results are derived by an analytical method making use of matrix transformations and rectangular coordinates.

A. J. Miller has proposed a stochastic model for the study of highway traffic in one direction along a two-lane road in which the desired speed of a vehicle is sampled from a continuous probability distribution and queues of vehicles are Poisson distributed on the highway. Miller has derived the equilibrium expression for the relation between the spacial densities of all vehicles and the spacial densities of those which are freely travelling. This relationship takes the form of a non-linear integral equation which is left unsolved. The present model differs from the original only in the assumption of a discrete distribution of desired speeds. Although the derivations of the equilibrium equations are based upon different arguments from those of Miller, these relations transform to the corresponding integral equations in the limit of a continuous speed distribution. The main body of the paper is devoted to a detailed investigation of the case of two speeds and asymptotic results are obtained for three speeds.

1. Introduction. Over the past few years, the theory of single-server queues has grown substantially. Much of the effort expended has been applied to the study of transient multivariate queuing processes in continuous time, i.e. to the study of joint time-dependent distributions of two or more system variables given some known initial system configuration. A variety of methods have been employed in these studies to accommodate general interarrival and service time distributions, among them Erlang's method of fictitious phases, the method of supplementary variables, the extended imbedded chain method, and the semi-Markov method. In the first part of this paper an informal review of these transient studies and of the methods employed is given.

It has long been known (see Lévy (3), pp. 256, 260) that the sample paths of Brownian motion in the plane form an everywhere dense set of zero Lebesgue measure, with probability 1. In (7), a capacity argument was used to show that the Hausdorff measure with respect to tα is infinite for 0 < α < 2 with probability 1 so that the dimension of the path set is known to be 2. If one considers the initial part of the sample path Cω = C(1, ω) for 0 ≤ t ≤ 1, then it becomes interesting to ask if there is a measure function ψ(t) such that, with probability 1,

for suitable positive constants c1, c2. The corresponding problem for paths in k-space (k ≥ 3) has been solved. In this case, if φ1(t) = t2 log log t−1, Lévy (4) obtained the upper bound and Ciesielski and Taylor (1) obtained the lower bound. For the planar case, the path is recurrent, and the intricate fine structure makes the measure function φ1(t) inappropriate. In (2), Erdő and Taylor showed that the measure is finite with probability 1 with respect to φ2(t) = t2 log t−1, and at that time we thought that (1) might be true with ψ(t) = φ2(t). Recently Ray (5) has obtained the lower bound in (1)with

The purpose of the present note is to obtain the upper bound in (1) for the same measure function, thus showing that (2) defines the correct measure function for measuring planar Brownian motion.

All possible symmetries of the non-degenerate Harrison space-times are found. It is shown that if such a space-time admits an r-parameter group of motions, with r > 1, then it must admit a 2-parameter Abelian subgroup.

A theory of light propagation consistent with our present cosmological views is investigated in the context of a uniformly expanding universe. It leads to the relativistic equivalence of fundamental observers and also provides a natural substratum associated with an observable fundamental reference frame for every locality of the universe. Special Relativity emerges as a theory satisfying the observable universe.

The model provides a quantitative interpretation of the recent findings in radio and optical astronomy. It has the advantage of simplicity and its approach may assist in an understanding of what is happening in the mathematics of more sophisticated systems.

A method of determining numerically to any degree of accuracy the eigen-values of Hamiltonians in the form of power series is presented. The case of a spherically symmetric potential function of the form V = ar2 + br4 + cr6 is treated in detail.

It is shown that for an axisymmetric stationary system there exists, in vacuo, a canonical coordinate system, analogous to Weyl's canonical coordinates for the static case. Some approximate solutions of the vacuum field equations are then obtained in these canonical coordinates.

In the second part of this paper, the vacuum field equations for an isolated axisymmetric stationary system are set up and approximate solutions obtained by means of a multipole expansion. The physical components of the Riemann tensor are then examined and it is found that the ‘mass term’ predominates, the ‘rotation term’ being of the next order of magnitude.

1. Introduction. This Department has made extensive observations with a proton magnetometer towed behind a ship (Hill (2)). The magnetometer consists of a bottle of water surrounded by a coil of wire through which a current flows and produces a polarizing field of about 100 oersted. A small proportion of the protons in the water are polarized along the resultant field. When the current is turned off they precess about the Earth's field and produce in the coil an e.m.f. whose frequency is a measure of the field.

In this paper, we consider the problem of diffraction of two-dimensional sound pulses by a homogeneous fluid circular cylinder contained in another homogeneous fluid. The line source is situated outside the cylinder and is parallel to its axis. It is supposed that the velocity of sound inside the cylinder is less than the velocity of sound in the surrounding medium. We investigate the problem by the method of dual transformation as developed by Friedlander. The pulse propagation modes both inside and outside the cylinder are obtained, We interpret the modes as diffracted pulses in terms of Keller's Geometrical Theory of Diffraction. The results agree with Friedlander's conjecture.

Formulae are obtained for the Laplace transforms of the first and second partial derivatives of a function of time and position which, together with its first partial derivatives, is discontinuous on a moving surface. These formulae are then used in a discussion of the application of the Laplace transform method to the solution of mixed initial and boundary-value problems in linear thermoelasticity. Particular attention is given to thermoelastic disturbances involving plane shock waves, and it is pointed out that no properly posed formulation of problems of this type has yet been found.

1. Introduction. The problem considered here derives its motivation from a paper by Friedlander (8) on the propagation of small disturbances in a compressible, conducting fluid in the presence of a uniform magnetic field (see also Courant and Hilbert (3), VI, §3a). In this the displacement current and energy dissipation by viscosity, heat conduction and Joule heat are neglected and a system of linear partial differential equations is obtained, which generalizes the equations of motion of the theory of sound. Their solution is in general the superposition of an arbitrary incompressible Alfven wave and a magneto-acoustic disturbance. This latter was considered by constructing a Green's function by means of suitable combinations of plane wave solutions and it was found that there are fast and slow wave fronts diverging from a point disturbance. The latter are conoidal in shape and have a singularity at their vertices which propagate along the field line in either direction from the source.

Our previous discussion ((4), (5)) is here extended to an ellipsoid of arbitrary conductivity. Again, the flow is ultimately cylindrical and the magnetic field undisturbed. An ellipsoidal insulator allows fluid to pass through its circumscribing cylinder, but also forces some to go around. An ellipsoidal conductor behaves like a very tall insulator, irrespective of its own height or conductivity: all the fluid goes round. If it is very wide an unexpectedly large amount of fluid is carried with it.