1. In the first paper of this series I have explained a method by which the complete irreducible system of combinants of a pencil of quadric surfaces may be obtained, and have determined explicitly the combinantal invariants and covariants. The present paper deals with combinantal contravariants. The notation used is that of I, a knowledge of which will be assumed.

By electrical analysis of the output from a hot-wire anemometer, it is possible to measure rapidly and accurately all the quantities appearing in the theoretical equations for the decay of isotropic turbulence. The technique for these measurements is described, and possible extensions and limitations of the method are discussed. Measurements have been made of the second and fourth derivatives of the longitudinal double-velocity correlation at the origin, the third derivative of the longitudinal triple-velocity correlation at the origin, the statistical distribution in time of the velocity fluctuations, and the integral scale of turbulence.

The classical account of the invariants and covariants of a pair of quadric surfaces is due to Salmon. The actual determination, in explicit form, of the complete system of concomitants of two quaternary quadratic forms is much later in date, and is due to Turnbull. This system, which includes mixed concomitants of various kinds, is complicated. As originally determined, it comprised 125 forms, three of which were later shown to be reducible. In the accounts of both these authors, however, the forms are considered as belonging to a particular pair of quadric surfaces, and the problem of determining these covariant forms which are invariant, not merely under change of coordinate system but also under change of base in the pencil denned by the two quadrics, is not alluded to. Such forms, which are of obvious geometrical interest, are called combinants. It is rather surprising to find that very little seems to be known about the combinants attaching to a pencil of quadric surfaces; the Encyklopädie scarcely mentions them, and I have been unable to trace any references in the literature except to the two invariants whose vanishing expresses the condition that the parameters of the four cones in the pencil form an equianharmonic or a harmonic set.

A new and fruitful theory of turbulent motion was published in 1941 by A. N. Kolmogoroff. It does not seem to be as widely known outside the U.S.S.R. as its importance warrants, and the present paper therefore describes the theory in some detail before presenting a number of extensions and making a comparison of experimental results with some of the theoretical predictions.

Kolmogoroff's basic notion is that at high Reynolds number all kinds of turbulent motion, of arbitrary mean-flow characteristics, show a similar structure if attention is confined to the smallest eddies. The motion due to these eddies of limited size is conceived to be isotropic and statistically steady. Within this range of eddies we recognize two limiting processes. The influence of viscosity on the larger eddies of the range is negligible if the Reynolds number is large enough, so that their motion is determined entirely by the amount of energy which they are continually passing on to smaller eddies. This quantity of energy is the local mean energy dissipation due to turbulence. On the other hand, the smaller eddies of the range dissipate through the action of viscosity a considerable proportion of the energy which they receive, and the motion of the very smallest eddies is entirely laminar. The analytical expression of this physical picture is that the motion due to eddies less than a certain limiting size in an arbitrary field of turbulence is determined uniquely by two quantities, the viscosity and the local mean energy dissipation per unit mass of the fluid.

The mathematical method of describing the motion due to eddies of a particular size is to construct correlations between the differences of parallel-velocity components at two points at an appropriate distance apart. Kinematical results analogous to those for turbulence which is isotropic in the ordinary sense are obtained, and then the scalar functions occurring in the expressions for the correlations are determined by dimensional analysis. The consequences of the theory in the case of turbulence which possesses ordinary isotropy are analysed and various predictions are made. One of these, namely that dimensionless ratios of moments of the probability distribution of the rate of extension of the fluid in any direction are universal constants, is confirmed by recent experiments, so far as the second and third moments are concerned. In several other cases it can be said that relations predicted by the theory have the correct form, but further experiments at Reynolds numbers higher than those hitherto used will be required before the theory can be regarded as fully confirmed. If valid, Kolmogoroff's theory of locally isotropic turbulence will provide a powerful tool for the analysis of problems of non-uniform turbulent flow, and for the determination of statistical characteristics of space and time derivatives of quantities influenced by the turbulence.

A general form of the covering principle and relative differentiation of additive functions. II*.

By A. S. Besicovitch

P. 5, line 7 from the bottom, after ‘F-singular’ add ‘in the class of B-sets’.

P. 5, the bottom line, after ‘function Θ(X)’ add ‘in the class of B-sets’.

P. 7, line 9 from the bottom, forG⊃Xnread.

P. 8, line 4, for.

P. 8, line 5, for.

* Vol. 42 (1946), pp. 1–10.

1. We denote by the class of all functions regular in |z| < 1, and never taking the values 0 or 1 in |z| < 1. Then Schottky's(1) theorem states:

Theorem A. Let f(z) = a0 + a1z + … belong to . Then

and hence

where

and the constantsK(a0), K(a0,ρ) depend only on the quantities indicated.

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.

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?

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 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.

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 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 .

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.

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.

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 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.