In 1930 Łukasiewicz (3) developed an ℵ0-valued prepositional calculus with two primitives called implication and negation. The truth-values were all rational numbers satisfying 0 ≤ x ≤ 1, 1 being the designated truth-value. If the truth-values of P, Q, NP, CPQ are x, y, n(x), c(x, y) respectively, then

Let G be a group and let us write [x, y] = x−1y−1xy. We shall say that G is an Engel group, or that it satisfies the Engel condition, if to any pair of elements x, y in G there can be assigned an integer k (which is allowed to depend on x, y) such that

1. The mathematician's tree is a collection of pairs of elements, conveniently called points. A pair of points is a link, and a collection of links is a tree if any two of the points from which the links are formed can be connected by a chain, and if there are no closed chains. A tree is determined if there is provided a complete schedule of the links composing the tree. Conversely, no description of a tree can be adequate unless a complete schedule is recoverable from the description. An arbitrary schedule of links does not usually describe a tree, and the object of this note is to give canonical schedules for the most general finite tree, and to specify these schedules as economically as possible. The fundamental result established is that the structure of a tree with n points can be encoded in a row of n − 2 symbols each of which may stand for any one of the points. Cayley's theorem, that the number of different trees with n given points is nn−2, is an immediate corollary.

The recent progress of modern algebra in analysing, from the algebraic standpoint, the foundations of algebraic geometry, has been marked by the rapid development of what may be called ‘analytic algebra’. By this we mean the topological theories of Noetherian rings that arise when one uses ideals to define neighbourhoods; this includes, for instance, the theory of power-series rings and of local rings. In the present paper some applications are made of this kind of algebra to some problems connected with the notion of a branch of a variety at a point.

It is a classical result (5) that the algebraic surfaces which admit a continuous group of birational self-transformations, or automorphisms, and which are neither rational nor scrollar, belong to one or other of two families: (i) the elliptic surfaces; these possess an elliptic group of ∞1 automorphisms; (ii) the hyperelliptic surfaces of rank 1 (which, for brevity, we shall call hyperelliptic); these possess a completely transitive permutable group of ∞2 automorphisms. The chief properties of the two classes of surface, in so far as they are required in the present work, are described in § 1.

In a note (3) published jointly with E. A. Maxwell in 1937, a set of relations was obtained connecting the arithmetic genera Ωi of the i-dimensional characteristic systems of the canonical system on an algebraic Vd. When d is odd these relations are

and when d is even they are

1. Let Λ denote any plane lattice of determinant Δ ╪ 0. Then it is well known that the region

contains a pair of generating points of Λ; and that, unless the lattice Λ is of the special form

where both α│β, γ│δ are rational, there are an infinity of such pairs in the interior of the region. Minkowski (4) obtained this result by considering the ‘tangent’ parallelogram Пλ,

inscribed in the region (1). With simple geometrical arguments, he showed that

(i) Пλ always contains a point of Λ, other than 0,

(ii) if Пλ contains two primitive points of Λ then these generate Λ, except possibly when Λ has points, other than 0, on both coordinate axes.

In this paper we consider algebraic integers, θ, such that θ and θ¯ (θ ╪ θ¯) are the only algebraic conjugates of θ that lie outside the unit circle.

Let S1 be the set of algebraic integers, ρ, necessarily real, all of whose conjugates, except ρ, satisfy | z < 1.

Let Em and En be orthogonal Euclidean spaces of dimensions m and n respectively and with the origin of each as their only common point. In a previous paper (3) I gave what was intended to be a proof of the relation

where the dimension of A, dim A, is the Besicovitch dimension, i.e. the number s such that the Hausdorff measure in any dimension greater than s is zero whilst that in any dimension less than s is infinite, where A and B are subsets of En and Em respectively and where A × B is the Cartesian product of A with B.

About twenty years ago a number of results were given expressing the sum of n terms of an ordinary hypergeometric series with unit argument in terms of an infinite series of the type 3F2. The interest in the subject was aroused by a theorem due to Ramanujan, who stated that

Spherical Bessel functions which vanish at x = 1 and x = η (where η is an assigned positive constant less than 1) occur in the solution of many problems in applied mathematics. Such functions can be expressed in terms of Bessel's functions of the half odd integral orders in the form

where λ is a root of the equation

The hypothetical model that we shall be considering in this paper is referred to as the single-server queue, and the details of this model are given in a recent paper by Lindley(5). The present treatment involves exactly the same assumptions as Lindley has given already, and we refer to his paper for a rigorous statement of them. Briefly, we shall be assuming general independent service times and general independent input or arrival times. Theoretical studies of the single-server queue are capable of wide applications, many of which are described in a paper by Kendall (4) and in the discussion to that paper.

The central limit theorem in the calculus of probability has been extensively studied in recent years. In its simplest form the theorem states that if X1, X2,… is a sequence of independent, identically distributed random variables of mean zero, then under general conditions the distribution function of Zm = (X1 + … + Xn)/√ n converges as n → ∞ to the normal or Gaussian distribution function. This form of the theorem in terms of distribution functions is the one required in statistical work, since it enables statements to be made about the limiting behaviour of prob {a ≤ Zn ≤ b}.

A class of stochastic processes associated with points randomly distributed in a line of finite extension L, is considered. A general integral equation for the function representing the probability distribution of the stochastic variable under consideration is derived and solved by using the Laplace transform technique. Examples of the above class of processes are cited. In particular, the problem of the fluctuations in brightness of the Milky Way is discussed in detail. The results of Chandrasekhar and Munch in regard to this astrophysical problem are derived in a simple and direct manner.

The isothermal flow of gas past a gravitating point-sink has been studied for four values of the undisturbed gas velocity, all of which are supersonic. From these studies, estimates have been made of the rates of accretion of an isothermal gas by a star moving with a supersonic velocity. These estimates are of the same order as those obtained by earlier workers who used simpler mathematical models.

A solution is found in closed form for the one-dimensional motion of an ideal, inviscid and isentropic gas with ratio of specific heats (2m + 3)/(2m + 1), m being any positive integer, when the gas is supposed to start from rest with an inhomogeneous temperature distribution.

An example is given which is of interest in the study of interstellar gas clouds. A cloud is assumed to be initially at rest in contact with a vacuum; the Riemann invariant r is assumed to increase linearly for a finite distance into the cloud (from zero at the surface), beyond which it remains constant. The r-characteristics from the inhomogeneous strip meet simultaneously in the distance-time plane at a point of the gas surface; the surface does not move until this happens, when it acquires instantaneously a finite non-zero velocity. A numerical estimate is made of the time during which the surface remains at rest for an actual gas cloud, under the above conditions. In the subsequent development the gas motion takes the form of a growing simple wave.

In an earlier paper of the same general title (1) the possibility that the core of the Earth, in view of its supposed liquid nature, does not partake of the rigid-body motion of the outer shell was discussed with particular reference to the secular diminution of the angular velocity. In addition to this small rate of change of the magnitude of the angular velocity vector of the shell there occur changes in its direction consisting of the precession and nutation, but all the rates of change therein involved are small. The secular retardation takes place with extreme slowness, the nutations involve deviations of the axis with small angular amplitudes, while the precession, though of large angular amplitude, is of very long period compared with the rotation period of the Earth. Accordingly, it may be supposed that the effects of these various changes in the angular velocity can be considered separately in their relation to the motion within the core, and it is the object of this paper to give an account of our investigation into what may be termed for brevity the precession problem. It should perhaps be stated at the outset that the work does not constitute a solution of the problem, which our studies have led us to believe is one of the utmost mathematical difficulty presenting features of an exceptional character in hydro-dynamic theory. After first obtaining the equations of steady motion applicable to the interior, and those applicable to the boundary layer, the solution of the latter equations has been obtained; but in respect of the former equations we have been able to carry the question of the interior motion only as far as showing that no motion representable everywhere by analytic functions and consistent with the boundary conditions is possible. The investigation strongly suggests that no steady-state motion of a permanent character is possible for the interior, though the precise nature of the motion that actually occurs poses a problem of special interest from a hydrodynamic standpoint, but it is one to which we are not able to arrive at any definite answer at present. Without making any progress with the problem thus produced, the paper nevertheless makes clear the inherent difficulties of the problem and also serves to emphasize the inadequacy of any simplified mode of attack assuming classical fluid and resembling, for example, Poincaré's method for the nutation problem adopted by Lamb (3). Thus despite its incompleteness it seemed worth while to publish some account of such progress with these highly interesting questions as we have been able to make.

The small transverse oscillations of a string excited by bowing are treated by neglecting all external forces except that due to the bow, taking the ends of the string as fixed, and replacing the bow by a concentrated force acting transversely at the ‘bowing point’, whose magnitude depends on the velocity, v, of the string at this point. Non-linear recurrence equations for v are obtained, from which some of the principal features of the motion can be inferred, and some qualitative conditions about the dependence of the transverse force of the bow on v can be drawn. A detailed examination of the case when the string is bowed at the mid-point shows that, in order to account for the stability of the periodic motion excited, it appears to be necessary to allow for the effect of dissipative forces, such as air friction; to a first approximation this can be done by postulating a small concentrated dissipative force at the bowing point hi addition to the frictional force exerted by the bow. The theory predicts that for small bowing speeds a ‘noise’ is produced instead of a note; this phenomenon does not appear to have been noted in the literature, but can easily be verified in practice.

The energy scattered when a sound wave passes through turbulent fluid flow is studied by means of the author's general theory of sound generated aerodynamically. The energy scattered per unit time from unit volume of turbulence is estimated (§3) as

where I is the intensity and ∧ the wave-length of the incident sound, and is the mean square velocity and L1 the macro-scale of the turbulence in the direction of the incident sound. This formula does not assume any particular kind of turbulence, but does assume that ∧/L1 is less than about 1. For turbulence which is isotropic and homogeneous, the energy scattered, and its directional distribution, are obtained for arbitrary values of ∧/L1. It is predicted that components of the turbulence with wave-number κ will scatter sound of wave-number K at an angle 2 sin−1 (k/2k). The statistics of multiple successive scatterings is considered (§4), and it is predicted that sound of wave-length less than the micro-scale λ of the turbulence will become uniform (i.e. quite random) in its directional distribution in a distance approximately .

The theory is extended (§5) to the case of an incident acoustic pulse. However, this extended theory cannot be applied directly to the case of a shock wave, for which it would predict infinite scattered energy. This is due to the perfect resonance between successive rays emitted forwards which would occur if the shock wave were propagated at the speed of sound. By taking into account (§6) the true speed of the shock wave (subsonic relative to the fluid behind it), the theory is improved to give a finite value, 0·8s tunes the kinetic energy of the turbulence traversed by a weak shock of strength s, for the total energy scattered. However, the greater part of this energy catches up with the shock wave, and probably is mostly reabsorbed by it, and only the remainder (tabulated as a function of s in Table 1) is freely scattered, behind the shock wave, as sound. The energy thus freely scattered when turbulence is convected through the stationary shock-wave pattern in a supersonic jet may form an important part of the sound field of the jet.

The following theorems are proved for irrotational surface waves of finite amplitude in a uniform, incompressible fluid:

(a) In any space-periodic motion (progressive or otherwise) in uniform depth, the mean square of the velocity is a decreasing function of the mean depth z below the surface. Hence the fluctuations in the mean pressure increase with z.

(b) In any space-periodic motion in infinite depth, the particle motion tends to zero exponentially as z tends to infinity. The pressure fluctuations at great depths are therefore simultaneous, but they do not in general tend to zero.

(c) In a progressive periodic wave in uniform depth the mass-transport velocity is a decreasing function of the mean depth of a particle below the free surface, and the tangent to the velocity profile is vertical at the bottom. This result conflicts with observations in wave tanks, and shows that the waves cannot be wholly irrotational.

(d) Analogous results are proved for the solitary wave.