This paper was originally intended to contain a generalization of a theorem of Banach on the extension of linear functionals. This generalized theorem now appears as a by-product of a study of a class of algebras which we believe to be of much greater interest than the theorem itself. Let X be a vector space over the real field and let π(x) be a sub-additive, positive-homogeneous functional on X. Banach ((2), pp. 27–9) proves that any real linear functional f on a subspace X0 of X which satisfies f(x) ≤ π(x) on X0 can be extended to a real linear functional F on X with F(x) ≤ π(x) on X. One of the essential differences between this theorem and the Hahn-Banach theorem is that π can take negative values.

In the classical theory (3) due to Knopp, Agnew and others, the core K(x) of a sequence x = {ξn} of complex numbers is defined by where En(x) is the smallest closed convex set containing all ξk with k ≥ n. A matrix transformation T is said to be a core-preserving transformation if

holds for all sequences x. T is core-preserving for bounded sequences if (1·1) holds for all bounded sequences x. It is readily proved that K(x) is the set of complex numbers ζ such that

for all complex numbers α (). Now is a sub-additive, positive-homogeneous real-valued functional defined on the vector space of bounded complex sequences. This suggests the construction of an abstract theory on the following lines.

The problem with which this paper is concerned is that of packing non-overlapping plane figures into a given plane figure. It is of a different type from the usual packing problem in which equal figures are used in the whole plane, and the aim is to calculate the limit of the ratio of covered area to the total area inside a large circle, as the radius of the circle tends to infinity. A closest packing in this problem is an arrangement of the figures in such a way that this limit attains its largest value.

Dvoretsky, Erdös and Kakutani (3), showed that Brownian paths in 4 dimensions have zero 2-dimensional capacity, and their method gives the same result for Brownian paths in n-space whenever n ≥ 3. However, if we apply the method indicated by Hausdorff (4) for constructing a linear set having measure 1 with respect to a given measure function h(x), the Cantor-type set we obtain when h(x) = xα log log 1/x, where 0 < α < 1, is easily seen to have zero α-capacity but infinite α-measure; and similar methods apply for other values of α. Thus the result mentioned above does not imply that Brownian paths in n-space (n ≥ 3) have zero 2-measure. The other relevant result is due to Lévy (6), who showed that Brownian paths in the plane have zero Lebesgue measure (and therefore zero Hausdorff 2-measure) with probability 1. However, his method of proof cannot be extended to deal with Brownian paths in n-space.

This note is concerned with the r-dimensional torus Tr, whose points x are r-tuples (x1, x2, …, xr), where the xi are not numbers but residue-classes modulo 1. The addition of two elements of Tr follows the vector law

1. The problem considered in this paper arose during an investigation of what Chabauty has called the ‘anomaly’ of convex bodies.

Throughout the paper denotes a closed bounded convex body in three-dimensional Euclidean space , which is symmetric in the origin O and which contains O as an interior point. Such a body determines uniquely a distance-function f(x, y, z) which is defined and finite for each point (x, y, z) of and possesses the following properties

I write X for the point (x1, x2, …, xn) of n-dimensional Euclidean space Rn. The coordinates x1, x2, …, xn are real numbers. The origin (0, 0,…, 0) is denoted by O. If t is a real number, tX denotes the point (tx1, tx2, …, txn); in particular, − X is the point (−x1, −x2,…, −xn). Also X + Y denotes the point {x1 + y1, x2 + y2, …, xn + yn).

Let

be n linear forms with real coefficients and determinant Δ = ∥ aij∥ ≠ 0; and denote by M(X) the lower bound of | X1X2 … Xn| over all integer sets (u) ≠ (0). It is well known that γn, the upper bound of M(X)/|Δ| over all sets of forms Xi, is finite, and the value of γn has been determined when n = 2 and n = 3.

In this paper I find the critical lattices of the region Kt defined by the inequalities | xy | ≤ 1, x2 + y2 ≤ t (t > 0). I deduce in §9 the arithmetic minima of a pair of binary quadratic forms

where f1 is indefinite, f2 is positive definite, and f1, f2 are harmonically related, that is to say

and

1. In statistical mechanics, we usually deal with assemblies containing a fixed number of particles in which energy is the only conserved quantity. Recently, Fermi (1) has shown that the angular distribution of the pions produced in high-energy nuclear collisions can be explained if one takes into account the conservation of angular momentum in addition to the conservation of energy. We are, therefore, led to discuss the thermodynamical properties of assemblies characterized by the conservation of two or more parameters. The simplest assumption of this type that we can make is that there are two parameters, say E and P, which are conserved, and that each particle of the assembly can occupy the levels (r, s) (r, s are non-negative integers), where the contribution of the level (r, s) to E is rε0 and to P is sη0. In order to find the entropy, and hence other thermodynamical properties of the system, we have to enumerate the distinct number of ways, p(m, n), in which an assembly of particles corresponding to given values of E = mε0 and P = nη0 can be realized. In this paper we find asymptotic expressions for p(m, n) in the following cases: (a) m is a fixed number, (b) m and n are of the same order. It is assumed here that the number of particles is greater than m and n. We deal with the case (a) in §2 and the case (b) in §4. §3 deals with the asymptotic expansions of the generating function for p(m, n) which are used in §4.

This note is concerned with automatic digital computing machines in which orders are expressed in coded numerical form and held in the same store as numbers. It is possible in such machines to modify orders by performing arithmetical operations on them. The storage locations are numbered serially, and their contents can be referred to when necessary by these numbers.

The Einstein relative-velocity formula. Milne (3) has remarked that in kinematical relativity, whereas the Lorentz transformation is valid in t-time but fails in τ-time, the Einstein relative-velocity formula is valid both in t-time and τ-time, thus implying the validity of the latter in a more general system. It was thought that some interest might be attracted to a description of the most general system under which the Einstein formula holds.

The relation between the bound states of two interacting particles and the S-matrix is discussed. The relevant terms of the S-matrix are resolved into an infinite product of operators. Capture and emission problems are discussed.

A number of methods are formulated for solving the Schrödinger equation for systems of finite extent. The methods are developed in detail for the particular case of an electron moving in a Coulomb field (e.g. hydrogen-like atom), with a boundary consisting of a sphere of given radius. In the second part of the paper these results are transformed into those for an electron moving in a uniform magnetic field, the boundary of the system being cylindrical.

The theory of collective electron ferromagnetism given by Stoner applies to a system of electrons in a single energy band; for iron, cobalt and nickel this is identified with the band of states derived from atomic 3d functions. To bring in the generally assumed overlapping of the 3d band by the wide 4s band in these metals, the theory has been extended to take account of the transfer of electrons from 3d to 4s states with change of temperature. A previous calculation of this transfer effect must be regarded as inadequate, for the part played by the exchange energy in determining the distribution of electrons between the two sets of states was omitted. The general equations are derived in § 2 and used as a basis for discussion of the properties of nickel-copper alloys at absolute zero in § 3. In §§4 and 5 numerical results are presented which show the effect of the overlapping 4s band on the magnetic properties of a system such as nickel both above and below its Curie point. Comparison with the measured paramagnetic susceptibility of pure nickel above the Curie point strongly suggests that for this metal the overlapping 4s band has only a minor influence, although in principle the effect could be very large (cf. Fig. 4). This result is not unambiguous, however, because values thus inferred for the two unknown parameters lead to inaccurate predictions below the Curie point. First, the predicted values for the spontaneous magnetization are too small. Secondly, the theory demands that the nickel-copper alloys should only be ferromagnetic below a copper content of about 20 %, whereas experimentally the limit is known to be about 60 %. In conclusion, it is suggested that the implicit assumption of Stoner's theory that the exchange integrals between all pairs of 3d states are equal to one another is a more serious restriction on the theory than the consideration of only a single energy band.

The liquid drop model of an imperfect gas in the form introduced by Wergeland is discussed by using the method of the grand ensembles and the equation of state of the system is derived. This equation of state is of the same general type as the one derived by Mayer for a more general model. It is shown that in both cases the isotherms consist of two analytically different parts in the limit where the number of particles in the system, N, goes to infinity.

This paper studies the steady motion of an electrically conducting, viscous fluid along channels in the presence of an imposed transverse magnetic field when the walls do not conduct currents. The equations which determine the velocity profile, induced currents and field are derived and solved exactly in the case of a rectangular channel. When the imposed field is sufficiently strong the velocity profile is found to degenerate into a core of uniform flow surrounded by boundary layers on each wall. The layers on the walls parallel to the imposed field are of a novel character. An analogous degenerate solution for channels of any symmetrical shape is developed. The predicted pressure gradients for given volumes of flow at various field strengths are finally compared with experimental results for square and circular pipes.

A proof is given of the existence of a slow drift associated with the passage of gravity waves over the surface of a frictionless fluid when the wave motion is irrotational. The drift is in the direction of propagation and is greatest near the surface, decreasing steadily towards the bottom, where it may be negative if the fluid is of finite depth. The particle displacement due to a solitary wave has similar properties. The proof confirms the approximate calculations of Stokes (6) and extends the proof of Rayleigh applicable only to deep water.

A theoretical examination is made of the stability of a spherical interface between two inviscid liquids accelerated radially from rest. If surface tension be neglected, it is found that the interface is unstable or stable according as the acceleration is directed towards or away from the liquid of greater density. Surface tension has a marked stabilizing effect, and numerical results are given for the minimum outward acceleration required to destroy the equilibrium of the surface of a cavity formed in water. These results are suggested as a possible explanation of the behaviour of cavitation bubbles seen to form near a fixed object under test in a water tunnel.

Let R be a set of points in n-dimensional Euclidean space, and let Δ′(R) denote the lower bound of the determinants of non-homogeneous lattices which have no point in R. For Δ′(R) to be non-zero it is necessary, as Macbeath has shown (2), that R should have infinite volume.