1. Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, and

where γ is Euler's constant. Thus P(x) is the error term in the problem of the lattice points of a circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of a rectangular hyperbola.

This paper is a continuation of four others under the same title†. The paragraphs are numbered following on to those of the fourth paper of the series. In § XXIV we show that, if λ(A) is a linear functional, then there exists a resolution Eμ such that λ(A) = ∫μdr(AEμ), and if B = ∫μdEμ is bounded, then λ(A) = τ(AB)‡ for all A, where τ is the trace. This implies that τ(A) is a linear functional, and that the conjugate space ℒ, i.e. the space of the linear functionals, has a subset ℒ′ which is in (1, 1) correspondence with the original set of operators, and that in this correspondence the linear functional τ(A) is associated with the unit operator.

Suppose that is an integral modular form of dimensions − κ, where κ > 0, and Stufe N, which vanishes at all the rational cusps of the fundamental region, and which is absolutely convergent for The purpose of this note is to prove that

The notation employed is that of my second paper under the same general title* I refer to this paper as II.

1. Any number x can be expressed uniquely in the form

where the xr's are positive integers and where xr < r. In the present paper we consider the set of numbers Eb for which the xr's are bounded, so that 0 ≤ xr < b say, where b also is an integer. We prove that this set has dimension function

h(t) = b−u,

where t = euu−u−½(b − 1)(2π)−½, in the sense of Hausdorff.

The stability of lattices is discussed from the standpoint of the method of small vibrations. It is shown that it is not necessary to determine the whole vibrational spectrum, but only its long wave part. The stability conditions are nothing but the positive definiteness of the macroscopic deformation energy, and can be expressed in the form of inequalities for the elastic constants. A new method is explained for calculating these as lattice sums, and this method is applied to the three monatomic lattice types assuming central forces. In this way one obtains a simple explanation of the fact that the face-centred lattice is stable, whereas the simple lattice is always unstable and the body-centred also except for small exponents of the attractive forces. It is indicated that this method might be used for an improvement of the, at present, rather unsatisfactory theory of strength.

On the assumption that the potential energy of the three cubic lattices of the Bravais type consists of two terms, an attractive one proportional to r−m and a repulsive one proportional to r−n, n > m, stability conditions are expressed in the form that two functions of the number n should be monotonically increasing. These functions have been calculated numerically for n = 4 to 15, and are represented as curves with the abscissa n. The result is that the face-centred lattice is completely stable, that the body-centred lattice is unstable for large exponents in the law of force, and that the simple lattice is always unstable,—in complete agreement with the results of Part I.

The penetrating component of cosmic radiation is now believed to consist of mesons—particles with a mass about one-tenth of that of a proton—while the soft component is attributed to electrons. By collisions with atomic electrons a fast meson occasionally produces fast secondary electrons, “knock-on” electrons, thus giving rise to a certain amount of the soft component. It was first suggested by Bhabha(1) that fast electrons produced in this way may be responsible for the electron showers known to be associated with the penetrating particles. In the present note we calculate the ratio, which we denote by R, of the average number of electrons which accompany a meson due to the knock-on process, making use of the simplification which results from the fact that the rate of loss of energy by fast electrons in ionization and excitation and also the emission of soft quanta are nearly independent of their energy. In virtue of this, the total length of electron track, and therefore the magnitude of R, is only slightly affected by the cascade process of shower production, a process which causes the energy of a fast electron to be shared among a number of electrons. The difficulties associated with a treatment of the cascade process may therefore be avoided in calculating R.

It is shown that the usual molecular orbital treatment for the mobile electrons in unsaturated hydrocarbon molecules is not always satisfactory, since it does not yield a self-consistent field. The conditions under which the field is self-consistent are analysed, and are shown to be satisfied in many cases.

It frequently happens that we wish to evaluate the total energy of the mobile, or π, electrons in an unsaturated hydrocarbon molecule. According to the method of molecular orbitals(1) this energy is the sum of individual electronic energies ∊r, and the summation is over all the occupied orbitals. The energies ∊r are the roots of the secular determinant, and, if we write

and suppose that there are n unsaturated carbon atoms, then the equation that gives the energies zr is

The paper describes a method of integrating differential equations by means of the Mallock linear equation machine. The method, which is based on one described by Hartree, is applicable to pairs of second order simultaneous linear ordinary differential equations in two unknowns. Any required degree of accuracy in the solution can be obtained.

The statistical theory of isotropic turbulence, initiated by Taylor (3) and extended by de Kármán and Howarth (2), has proved of value in attacking problems associated with the decay of turbulence. In its application to such hydro-dynamical problems, the theory falls into two parts, a kinematical part and a dynamical part. The kinematical aspect consists in setting up correlations between velocity components, or their derivatives, at two arbitrary points in the fluid, and reducing the form of the tensor thus obtained in accordance with the severely restrictive assumption of isotropic turbulence; the success of de Kármán and Howarth's investigations is largely attributable to their improved treatment of this purely kinematical problem. The dynamical part then consists in applying the implications of the equations of continuity and motion to the functions defining the correlation tensors, in order to obtain information concerning their functional dependence on time and on the displacement between the two points for which the correlations are computed.

First-order calculations are given concerning the possibility of designing a β-particle spectrograph in which the correlation between the photoelectrons of the stronger “natural β-ray lines” and the disintegration particles of the “continuous spectrum” might be investigated. It has been shown that, with a coincidence counting circuit of resolving time 10−6 sec., and with a source of suitable strength, this investigation should be possible in the case of any natural β-ray line the absolute intensity of which is sensibly greater than 0.01. A successful investigation of this kind would provide important data for the analysis of a complex continuous spectrum into the “partial spectra” corresponding to different modes of transformation (and might furnish direct information regarding the internal conversion coefficients appropriate to the stronger γ-radiations involved).

The unusual physical properties of liquid He ii have led to considerable speculation concerning its internal structure. Since the entropy difference between the liquid and solid states tends to zero as the temperature approaches absolute zero, it appears that in the neighbourhood of zero temperature the liquid must have some kind of ordered structure, and London's theoretical investigations(1) indicated that the best agreement could be obtained with the observed energy and molecular volume by supposing the liquid to have a diamond lattice. On the basis of this work Fröhlich(2) then suggested that in the ordered state He ii possessed a diamond lattice, and that the disordering process consisted in an interchange of the atoms between this lattice and the vacant points of the body-centred cubic lattice from which the diamond lattice may be considered as derived.

The following correction has to be made to my recent paper on the adsorption of dipoles. Equation (10) and the sentence immediately preceding it are to be replaced by the following paragraph:

In the polynomial expressions for f1(x) and f2(x) the coefficients of the powers of x depend on η2 and η3 which are functions of T.

The transition 20F → 20Ne is shown to occur with the emission of a β-particle of maximum energy 5·0 × 106e. V., followed by a γ-quantum of energy 2·2 × 106e.V.

The transition from the excitation level of 20Ne at 2·2 × 106 e.V. sometimes occurs in two stages with the emission of two quanta of smaller energy.