The Parseval formulae for Fourier cosine and sine transforms,

are of course most widely known in connexion with the classical theorems of Plancherel on functions of the class L2 (whose transforms are defined by mean convergence), and with their generalizations. We cannot expect to obtain anything as elegant as the ‘L2’ results when we consider (1) for functions of other kinds. Nevertheless, since the most obvious way of defining Fourier transforms is by means of Lebesgue or Cauchy integrals, we naturally wish to know how far the formulae (1) hold good for transforms obtained in this way. The two most familiar classes of functions having such transforms are:

(i) functions f(t) integrable in the Lebesgue sense in (0, ∞), whose transforms Fe(x) and Fs(x) are defined by the Lebesgue integrals respectively; and

(ii) functions f(t) which decrease in (0, ∞), tend to zero as t → ∞, and are integrable over any finite interval (0, T); in this case the transforms are defined by the Cauchy integrals .

In his tract on spaces based on the notion of area, Cartan (1) gives a theory of curves and of surfaces immersed in his 3-dimensional space. A theory of curves is possible by associating with each point of the curve the 2-dimensional orientation of the contravariant bivector (or the corresponding covariant vector density of weight − 1) of support which is normal to the curve at that point. This association is possible, since there is a unique normal 2-direction at every point of a curve in a 3-dimensional space. If the surrounding space has more than 3 dimensions, however, and the element of support is still a bivector, then a theory of curves becomes impossible since no unique normal or tangential 2-direction can be associated with the curve at every point. A theory of surfaces is still possible because a surface has a unique tangential 2-direction which can be taken to be the 2-direction of the bivector of support. It is the object of this note to consider what subspaces can be admitted by a space when the number of dimensions is n > 3.

As a partial generalization of a theorem of Hurewicz (3), Whitehead gave an important theorem on deformation retracts ((9), p. 273) which was proved only for polytopes. A demonstration of a part of Whitehead's theorem for absolute neighbourhood retracts was published by Fox ((2), p. 42) with a remark given by Samelson(8).

Wishart and Hirschfeld have considered the following problem (1). Given n points in order on a line, suppose that they may be ‘black’ or ‘white’ independently with probability p and q = 1 − p. What is the probability distribution of the number of black-white joins? They give an exact solution to this problem and prove that it tends to normality as n increases. This result is of interest in several branches of science (Mood (2)). In the present note we consider the analogous problem in more than one dimension. This has applications in physics and agriculture (compare, Ising (3), Cochran (4) and Todd (5)). Another problem of similar type occurs when we consider the number of black points to be fixed and their arrangement to be random, but we do not consider this here. A problem of arrangements of similar type has been considered by Onsager.

Suppose that events occur at random points on a line from t = −∞ to +∞, the probability of an event occurring between t and t + dt being λdt. If we select any interval of the line, say the interval [0, y], there will be a finite probability that it contains 0, 1, 2,…,r,…events; in fact, it is not difficult to show that these probabilities form a Poisson distribution, the probability that the interval contains r events being (see e.g. (1)). Consider the case when each event consists of an interval of length α (an event being characterized by its first point). What is the probability that the covered portion of the interval [0, y] lies between x and x + dx?

The drag on a fragment moving through the air at supersonic velocity tends to be proportional to the projected area of the fragment perpendicular to the line of flight and to be independent of the Reynolds number, depending only on the Mach number. Also the minimum velocity at which the fragment perforates a target is a function of the projected area at the moment of strike. Hence the probability that a fragment, with a given initial velocity, perforates the target at a given distance, depends on the mode of rotation and the distribution of projected areas of the fragment. It is customary to assume that the fragment rotates randomly so that we require to know the distribution of the projected areas when all orientations are equally likely. The distributions for fragments in the form of cylinders and rectangular parallelepipeds are derived in this paper.

The ordinary one-dimensional wave equation

has special integrals of the form

which satisfy the first-order equations

respectively, and are often called progressive waves, or progressive integrals, of (1·1). The straight lines

in an xt-plane are the characteristics of (1·1). It follows from (1·2) that progressive integrals of (1·1) are constant on some particular characteristic, and are characterized by this property.

The object of this paper is to determine all the solutions of the wave equation

which are of the simple form

where F denotes an arbitrary function. It will be shown that, in addition to the obvious cases of plane or spherical progressive waves, such solutions exist only when the wave fronts

are certain algebraic surfaces of the fourth order, the cyclides of Dupin. These include, as degenerate cases, the sphere, the plane, the cylinder, the cone, and the torus.

In this paper the two-dimensional reflection of surface waves from a vertical barrier in deep water is studied theoretically.

It can be shown that when the normal velocity is prescribed at each point of an infinite vertical plane extending from the surface, the motion on each side of the plane is completely determined, apart from a motion consisting of simple standing waves. In the cases considered here the normal velocity is prescribed on a part of the vertical plane and is taken to be unknown elsewhere. From the condition of continuity of the motion above and below the barrier an integral equation for the normal velocity can be derived, which is of a simple type, in the case of deep water. We begin by considering in detail the reflection from a fixed vertical barrier extending from depth a to some point above the mean surface.

The rheological equations of state for an isotropic Bingham solid may be written in tensor form:

where

The notation is explained fully in a previous paper (1). The frame of reference is essentially Eulerian. Primes denote deviatoric components of the stress, strain and rate of strain tensors, pik, εik and eik (for example, ). The dilatation is denoted by Δ. The rigidity and bulk moduli μ and κ are, if not constants, essentially scalar functions of the stress tensor. The variable viscosity η involves two constants, the yield value ϑ and the reciprocal mobility η1.

The velocity distribution in steady rectilinear plastic flow of a Bingham solid between coaxial circular cylinders in relative motion can be obtained exactly by simple arguments. On the basis of the general equations of plastic flow, an approximate velocity distribution is derived below for the case of eccentric circular cylinders as rigid boundaries and zero pressure gradient by making use of a conformal transformation. This approximation is valid only if a dimensionless parameter S (which is proportional to the yield value) is sufficiently small, and there is flow in the whole region between the cylinders. The differences in velocity distribution in the cases of Bingham solid (S finite) and Newtonian liquid (S zero) are illustrated by a particular example. It is shown that the effect of a finite yield value is to cause the contours of equal velocity in a normal section to become more nearly concentric with the inner boundary as the yield value increases from zero.

The complete set of matrix representations of a set of n quantities β1, β2,…,βn satisfying

is obtained for all values of n. It is found that if n = 2v or n = 2v + 1, there are v + 1 irreducible representations with the respective degrees 1, but if n = 2v + 1 there are in addition two conjugate representations of order , the symbol denoting the binomial coefficient. The explicit representations are given for n = 2, 3, 4, 5.

It is shown that the algebra of the Duffin-Kemmer matrices can be handled quite neatly by introducing an extra dimension and by using the Γ-formalism of an earlier paper (3). By this means it is possible to derive identities similar to the De Broglie-Pauli identities for the Dirac matrices. To complete the analogy a Γ-formalism is also set up for the Dirac matrices.

The physical significance of the parameters which appear in the combinatory factors g(Ni) and g(Ni, Xij) in the statistical mechanics of mixtures is examined. Hitherto, closed ring molecules have been excluded from the jurisdiction of the formulae for these combinatory factors. The means by which this restriction on the validity of the formulae can be removed is indicated. It is concluded that the equations given by Guggenheim are formally applicable in all cases provided that appropriate values, which differ according as the molecules are simple and branched chain or closed ring molecules, are substituted for certain parameters which occur in the equations.

The radioactivity of the naturally and artificially produced lutecium isotope has been investigated on a number of occasions (l, 2). A 4 hr. activity was observed when lutecium was activated by slow neutrons and later a 6-day period was observed by Hevesi and Levi (3). Heyden and Wefelmeier (4) discovered the natural radioactivity of lutecium, while Libby (5) made measurements indicating that this natural activity consisted of a negative electron spectrum of maximum energy 0·215 MeV. Mattauch and others (6) showed from mass spectrograph evidence that in addition to the main isotope of lutecium of mass 175 there was an isotope of mass 176 with an abundance of 2·5%. The second isotope had been reported previously by Gollnow (7), but not identified. Since Hf176 and Yb176 are known to be stable, whereas Hf175 and Yb175 do not exist in nature, the radioactivity of natural lutecium must be assigned to Lu176. Flammersfeld and Mattauch (8) measured the energy of the natural spectrum by the absorption method and found it to be 0·4 MeV. They observed also a γ-radiation (1–2 quanta per β-particle) of 0·26 MeV. In addition, they investigated the radiations resulting from slow neutron bombardment of lutecium. They estimate the shorter period, previously reported by Hevesi and Levi, to be 3·4 hr. and to consist of a β-spectrum of maximum energy 1·150 MeV. They estimate the longer period to be 6·6 days and to consist of β-spectrum of maximum energy 0·440 MeV. In view of the closeness of this latter energy to 0·4 MeV., Flammersfeld and Mattauch allocate the 6·6 days' activity to an isomeric state of Lu176. The short period is allocated by these authors to Lu177.