An equation with real coefficients and given degree n being selected at random, about how many real roots may it be expected to have? The present series of papers is concerned with this question and matters arising out of it. The results we have arrived at were stated without proof in our paper I (with the same general title), which contains also some introductory remarks to which we may refer the interested reader. Here we summarize as follows.

In a previous paper I proved that the density of the positive integers of the form where the letters p, q, and later P, Q, r, denote primes, is positive. As indicated in the Introduction of I, I now give proofs of the following results:

The density of each of the sets of integers

is positive.

If F is a free linear system of surfaces in an algebraic threefold V which is either non-singular or possesses only normal singularities, then F has Jacobian and adjoint surfaces, J2(F) and A2(F), and Jacobian and adjoint curve systems, J1(F) and A1(F), such that

where X2, X2 are the canonical systems of surfaces and curves on V, and X1(F) is the canonical system of curves of F. The imposition of base elements (points or curves) Ei, of assigned multiplicities λi, on F defines a system F1 which we may represent formally by the equation

and it is natural to enquire how the Jacobian systems of F1 differ from those of F, and how we may define adjoint systems A2(F1) and A1(F1) which cut on F1 its canonical curves and sets respectively.

Multivariate generalizations. In multivariate statistical analysis, common terms such as variances and correlation coefficients have received certain generalizations. Wilks (7) has called the determinant |V|, where V is the matrix of variances and covariances between several variates, a generalized variance; certain ratios of such determinants have been called by Hotelling(5) vector correlation coefficients and vector alienation coefficients. While these determinantal functions have properties which justify to some extent this kind of generalization, it sometimes seems more reasonable to leave any generalized parameters, or corresponding sample statistics, in the form of matrices of elementary quantities. This is stressed by the formal analogy which then often exists between the generalized and the elementary formulae.

The formal conditions of Lorentz-invariance apply to coordinates x1, x2, x3, x4 which form a 4-vector, and to expressions associated with them. In this paper attention is called to the frequent disregard in quantum investigations of the proviso that the coordinates form a 4-vector.

In most quantum investigations with a practical application the coordinates employed are relative space coordinates ξ, η, ζ coupled with a progressive time coordinate, so that (ξ, η, ζ, it) is not a 4-vector. Nevertheless, conditions of Lorentz-invariance are applied by many authors. Alternatively, they attempt to base the investigation on wave functions of non-relative coordinates x, y, z, t. It is here pointed out that such wave functions give no information about the eigenstates, and that there is no means of deriving wave functions of ξ, η, ζ from those of x, y, z.

Special attention is directed to the misapplication of Lorentz-invariance in Dirac's theory of the hydrogen atom, and in the calculation of degeneracy energy of ionized material.

It is shown that, if account is taken of the uncertainty of position and momentum of the reference origin with respect to which observations are made, the self-energy and transverse self-energy of the electron become finite and the quantum theory is considerably modified for high-energy electrons and quanta.

It is shown that in a crystal there exist states in which the electron is bound to a surface of the crystal and has an energy lying within a forbidden band. The wave functions and energies of these states are calculated, on the nearly free electron approximation, in terms of the constants of the crystalline potential field, which is represented by a triple Fourier series having the periodicity of the lattice. The method is shown to be applicable to a general crystal having a surface parallel to any one of the crystal planes.

The wave functions and energies of both the normal and surface electronic states of a finite linear chain are determined in terms of the overlap integrals by the approximation of tight binding, it being assumed that the interaction of atomic s-states with p-states can be neglected. It is shown that the existence of surface states depends on the ratio of two overlap integrals being greater than unity, and reasons are given for expecting this to be so.

The method of the previous paper is extended to determine the surface states of a simple cubic crystal on the approximation of tight binding. It is also applied to the case of a semi-infinite linear chain when the atomic s- and p-states are to be regarded as degenerate, the existence of surface states being again predicted.

By means of purely qualitative arguments which do not depend on any particular model, the general scheme of stable nuclei and the isotopic breadth of nuclei with odd charge number are explained.

The breadth of the isobaric region can be obtained if the numerical values of certain energies are known. Though these can be estimated only very roughly, the values for the breadth of the isobaric region obtained in this way are in good agreement with the experimental values. The increase in the breadth of the isobaric region from light nuclei to heavier nuclei can be explained, but no plausible explanation has been found for the fact that the breadth decreases again for the heaviest nuclei.

In consequence of the magnetic and exchange interactions between the ions of a paramagnetic salt, a new mechanism of interaction between ion and lattice is possible, involving the simultaneous transition of two or more ions with the emission or absorption of only one elastic quantum. This new mechanism seems to be capable of removing the discrepancy between theory and experiment for iron alum, and considerably reduces the discrepancy for caesium titanium alum.

We consider an ideal problem of adsorption of single and double particles upon a solid surface which has its sites of accommodation regularly arranged, and by comparing the equilibrium properties obtained by Bethe's method with the ordinary statistical formulae, we obtain approximate expressions for:

(1) g(N, n, X), the number of ways of arranging n particles upon N sites of a lattice so that the number of neighbouring sites occupied by the particles is X.

(2) g2(N, n, X), the number of ways of arranging n double particles upon N sites so that each of the double particles takes up two adjacent sites and the number of neighbouring sites occupied by two different particles is X.

Both these expressions are found to agree with the exact values when the N sites lie on a straight line. When we use the first expression to construct the configurational partition functions of certain physical assemblies and expand them in powers of 1/kT, they are found to agree with the corresponding rigorous expressions as far as (1/kT)3, which is the highest power which we can find rigorously at present. With the help of the first expression, formal equations for superlattice formation in an alloy with the composition 1: 1 and equations for the separation into phases of regular liquids are given. Lastly we show that atoms and molecules in a regular liquid may dissociate or recombine suddenly accompanied by a latent heat. This is a new cooperative phenomenon, which may bear some resemblance to the melting process between the solid and liquid states.

Bethe's method is applied to determine statistically the rate of condensation of diatomic molecules which are adsorbed with dissociation to form an immobile film, the two atoms of one adsorbed molecule occupying two neighbouring sites. A lower limit to the number of gaps, or isolated unoccupied sites, in the final film is also obtained. The results calculated statistically agree with those obtained earlier by using models.

The disintegration of the separated isotopes of boron, under proton and deuteron bombardment, has been investigated, and it is shown that the groups of disintegration particles previously discovered had been attributed to the transmutation of the correct isotopes. In addition, experimental evidence has been obtained for a previously unknown excitation level of 8Be at about 7·5 × 106 e.V. above the ground state.

The decay period of the α-activity produced when boron is bombarded with neutrons from a (Li + D) source has been measured, and found to agree with the known decay period of 8Li. It has been shown that no activity is produced when (Be + D) neutrons (maximum energy 4·5 × 106 e.V.) are used. The results are explained by postulating the formation of radioactive 8Li according to the process

A short investigation of the number-range distribution for the α-particles emitted by 8Li indicated that a maximum occurred in the distribution at a range of about 6 mm.

The technique adopted in some experiments employing Geiger-Mueller counters is described. The behaviour of a counter in which the extinction of the discharge is external is contrasted with behaviour of a counter in which extinction is internal and dependent on the addition to the gas of some organic vapour such as alcohol. It is shown that the internally extinguished type of counter may be used with a resistance of the order 50,000 ohms in series with it while still retaining all its useful features.

Methods of construction of internally extinguished counters are described, particular attention being paid to the question of developing a thin-walled type suitable for experiments employing the method of coincidences. The effect of using different wall materials and gases is investigated with a view to finding those most suitable.

The measurement of the quantum energy of hard γ-radiation by the method of coincidences is discussed and details of a reliable two-stage amplifying circuit with a resolving time T of 10−6 sec. are given. Various factors which may influence the value of T are noted.

The absorption of the harder γ-rays from ionium has been studied, the first, of energy 68 ± 1 k.e.V., by the method of bracketing, making use of the K absorption discontinuities of tantalum and tungsten; the second, of energy 190 ± 20k.e.V., by analysis of the absorption curves. The intensity of these γ-rays is estimated to be 1 quantum in 10−3 disintegration for each γ-ray.

The irreversible heating of iron ammonium alum by an alternating field is shown to be proportional to the frequency of the field. The heating is therefore similar to that arising from hysteresis in ferromagnetics, but no remanence could be found by a ballistic method at the lowest temperature obtained, about 0·075° K. The rate of heating was much increased by the superposition of a steady magnetic field of a few hundred gauss.