Let r, s be two fixed integers greater than 1. A positive integer will be called r-free if it is not divisible by the rth power of any prime.

In a series of papers ((l)–(5)) Evelyn and Linfoot considered the problem of determining an asymptotic formula for the number Qr, s(n) of representations of a large positive integer n as the sum of s r-free integers; for s ≥ 4 their results were subsequently sharpened by Barham and Estermann(6).

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 was in a paper bearing this title that Cayley(1) first considered the problem of representing a curve in projective space of three dimensions by means of the complex of lines which meet the curve. He took the conic given by the equations

and found that the line

with dual Grassmann coordinates (…,pij,…), where

intersects the conic if, and only if,

where F(u0, u1) is homogeneous and of degree 2 in both sets of indeterminates u0 and u1 and G(…,pij,…) is a form of degree 2 in the pij. Both F(u0, u1) and G(…,pij,…) are easily determined in this case.

We begin with some definitions.

A cubical graph is a simplicial 1-complex in which each 0-simplex is incident with just three 1-simplexes.

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.

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.

An account is given of the transformation of coordinates and of absolute axial frames in Euclidean space and Galilean space-time. The connexion with Eddington's group of E-numbers is shown. The geometrical properties of Dirac's wave equation and of analogous equations in three dimensions are discussed in terms of absolute axial frames.

After solving the equations of motion of Dirac's self-accelerating electron, a physical picture of it is formed by plotting graphically the surfaces of constant scalar potential when the electron has built up a velocity close to the velocity of light.

In relativistic field theories derived by a variation principle from a Lagrangian, the problem arises of finding a symmetric tensor of rank 2 which has vanishing divergence in virtue of the field equations and is such that taken over a space-like section is equal to the corresponding integral of the so-called canonical energy-momentum tensor. It is well known that the latter condition is satisfied if the difference between the two tensors is the divergence of an antisymmetric tensor of rank 3.

A short discussion of steady rectilinear plastic flow of Bingham material between rigid cylinders of general shape in relative motion parallel to their generators was given in a paper of the same main title(1). It was shown that, for sufficiently great relative velocity V of the boundaries, plastic flow in the whole region would be expected, as for a Newtonian liquid. But for slower relative motion, part of the material might remain elastically deformed (i.e. stressed below the yield point) with a limited region of plastic flow surrounding the inner boundary. In the paper referred to, the problem was illustrated by the case of eccentric circular cylindrical boundaries. The choice of circular boundaries introduced a great simplification when the region of plastic flow did not extend to the outer boundary, namely axial symmetry in the flow pattern in the plastic region. It is the main purpose of the present paper to illustrate how the methods already given for determining velocity distributions may be modified to suit more general problems, by consideration of confocal elliptic cylinders as boundaries.

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.

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.

1. The theoretical study of the transport phenomena in metals requires the solution of a certain integral equation, which is formed by equating the rate of change of the distribution function of the conduction electrons due to the applied fields and temperature gradients to the rate of change due to the mechanism responsible for the scattering. This integral equation has so far been solved only in the simplest case where the electrons are assumed to be quasi-free, the energy being proportional to the square of the wave vector, and this assumption will be made throughout the present paper. The scattering of the electrons is due to two causes: the thermal vibration of the crystal lattice and the presence of impurities or strains. The scattering due to the impurities can always be described in terms of a free path l (or more conveniently a time of relaxation τ, were l = τν, ν being the average velocity of the conduction electrons), whereas for the scattering due to the thermal vibration this is possible only at high temperatures such that (Θ/T)2 can be neglected, where Θ is the Debye temperature (1). At very low temperatures the scattering due to the thermal vibration can be neglected compared with that due to the impurities. When a time of relaxation exists, the integral equation for the distribution function reduces to an ordinary equation, and its solution is then a comparatively simple matter. In the general case the problem is much more difficult, and no rigorous and generally valid solution has so far been obtained.

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.