In this paper, all numbers are real and all radicals are positive.

Let f(x, y) = ax2 + bxy + cy2 be an indefinite quadratic form, and let d = Δ2 = b2 − 4ac, where Δ > 0. A well-known theorem of Minkowski states that, if (x0, y0) is any pair of numbers, then there exists a pair (x, y), x ≡ x0 (mod 1), y ≡ y0 (mod 1), say (x, y) ≡ (x0, y0) (mod 1), such that

for k ≥ ¼, and Davenport has shown how this result may be improved if we know a value assumed by f(x, y) for coprime integral values, (x, y) = (m, n) ≠ (0, 0). In this paper, we discuss the more general inequality

where R and S are constants, and use the method developed in the first paper of this series to obtain a sharper and more general theorem than Davenport's. We give an application to the theory of real Euclidean quadratic fields and to a problem in Diophantine approximation discussed by Khintchine.

If h(r, θ) is harmonic in the unit circle | r | < 1 and satisfies the condition | h | ≤ 1, then there is a function u(ø) which satisfies | u | ≤ 1 such that

and conversely. Hence, any properties of such harmonic functions should be deducible from equation (1). A number of such properties have been proved by Koebe (Math. Z. 6 (1920), 52–84, 69), using Schwarz's lemma and the geometry of simple conformal transformations. They can be deduced from (1) together with an elementary lemma on the rearrangement of a function (Lemma 1 below). As, however, students of this subject will regard Koebe's method as the one best adapted to establish his theorems, we shall illustrate the alternative method by considering two new problems, namely to find max ∂h/∂r, max ∂h/∂θ, where the maximum in each case is taken for all harmonic functions h which satisfy

In this paper we investigate the following problem.

We suppose given a sequence of complex values wn, defined for n = 0, 1, 2, …, and for n = ∞, and such that

while at least one wn differs from zero and ∞. We consider functions f(z), which are regular in | z | < 1, and take none of the sequence of values wn, and we investigate the effect of this restriction on the rate of growth of the function, as given by the maximum modulus

The general linear difference-differential equation takes the form

where x is a real variable, ν(x) and Aμν(x) are known functions and

The two previous papers of this series dealt with the determination of the complete system of combinantal covariants and contravariants of a pencil of quadric surfaces. The present paper is concerned with combinantal forms involving line-complexes, and with certain syzygies which exist between these forms. It is essentially a preparation for the determination of the complete system of combinantal line-complexes of the pencil, which will be carried out explicitly in the next paper of the series.

In this paper we determine the complete system of combinantal line-complexes associated with a pencil of quadric surfaces. The method employed is that already used in (I) and (II) to obtain the complete systems of combinantal covariants and contravariants.

It is apparent from the two previous papers of the same main title (1, 2) that velocity distributions in steady rectilinear plastic flow of a Bingham solid between moving boundaries are not easy to determine, even when the boundaries are of simple shape. In the familiar case of a Newtonian liquid (which can be regarded as a special case of a Bingham solid with zero yield value) the velocity ω is a harmonic function and can be obtained by a conformal transformation of the region of flow of the type

In order to generalize and include materials of finite yield value, in which ω is not harmonic, one must first regard the Newtonian case from a slightly different point of view. The transformation

must be regarded as defining a change of coordinates, from Cartesian coordinates x, y to the natural curvilinear coordinates for a particular problem ω, W, one of which is the velocity. For a Bingham solid in general, it is shown in the present paper that natural coordinates ω, W exist, but are not obtainable by a conformal transformation from Cartesian coordinates, except in the limit when the yield value tends to zero. In practice it may be difficult to determine the natural coordinate system in a given problem, but when it is found the velocity distribution is automatically known.

1. The general problem. In the first three papers under the same main title (1, 2, 3), attention has been confined almost entirely to a discussion of steady flow. The equations to be solved in order to determine velocity distributions in non-steady rectilinear plastic flow were given in § 1 of the third paper (3). On the assumptions of isotropy and incompressibility, the conditions of the general problem are as follows.

Several authors (1, 2, 3) have discussed the formulation of quantum electrodynamics in the configuration space of the light quanta; they conclude that no meaning can be given to the position of a light quantum, but it is not made clear whether this situation is due to the vanishing of the charge or of the rest mass. However, since the previous discussion (2, 3), various neutral particles of non-vanishing rest mass have been postulated, and it is of interest to consider what meaning can be given to the position of such particles. We shall at first consider integral spin and Bose statistics; the case of spin ½ and Fermi statistics is simpler and is briefly discussed in the last section.

In the following I discuss the properties, in particular the completeness of the set of eigenfunctions, of an eigenvalue problem which differs from the well-known Sturm-Liouville problem by the boundary condition being of a rather unusual type.

The problem arises in the theory of nuclear collisions, and for our present purpose we take it in the simplified form

where 0 ≤ x ≤ 1. V(x) is a given real function, which we assume to be integrable and to remain between the bounds ± M, and W is an eigenvalue. The eigenfunction ψ(x) is subject to the boundary conditions

and

The methods of a previous paper (9) are improved and extended. It is assumed that the eigenfunctions and eigenvalues of an eigenvalue problem given by an elliptic differential equation are known subject to given boundary conditions on a finite boundary. It is shown how the corresponding quantities can be obtained for a similar problem in which the original differential equation, boundary and boundary conditions are simultaneously perturbed. The introduction of a surface displacement vector allows of a Taylor expansion of all quantities and a subsequent separation of orders. The problem of finding the perturbed eigenfunctions for each order then reduces to the solution of an inhomogeneous differential equation subject to known boundary conditions. These equations are solved by a variational method. An application of Green's theorem at each stage enables us to find the perturbed eigenvalues. The method is applied to a problem of which an exact solution is known and good agreement is obtained.

The author is greatly indebted to Dr H. Fröhlich for many interesting discussions and some valuable suggestions.

The grand partition function of any statistical assembly may be defined by the equation

where E denotes any value of the energy of the assembly, k is Boltzmann's constant, T the thermodynamic absolute temperature and λi a parameter which is later to be connected with the chemical potential, μi, of the ith species in the assembly by the simple formula

It is well known (1) that for an unsaturated hydrocarbon of the type CnHn+2 (where n is the number of carbon atoms in the linear chain), the theory of molecular orbitals applied to calculate the binding energy of the mobile π-electrons leads to the secular determinant, of continuant type,

where β is the resonance integral between adjacent carbon centres and x stands for the quantity generally written E0 − E, E being the energy of a mobile electron, which we seek to calculate. The equation for the allowed energies E is, of course, simply Pn = 0.

A method is described for measuring thermal conductivities of metals at liquid helium temperatures. It is shown that the method of determining the temperature gradient by directly measuring the temperature at two points along a rod is probably free from systematic error, but that the end of a heated rod in direct contact with a helium bath is not at the temperature of the surface of the bath, even in the He ii region. A description is also given of a technique for using mercury as a low-temperature solder, tight to He ii. The results for copper are in reasonable agreement with Makinson's theory of metallic conduction. Tentative extrapolation to very low temperatures in the presence of fairly large magnetic fields shows that pure copper is a reasonably good material for thermal transfer in demagnetization experiments. The results for german silver seem to indicate that only the lattice conductivity is important since it follows a T3 law, and that scattering takes place with a mean free path of the order of 10−3 cm.

If H(ξ, η) is a harmonic function which is defined and positive in η > 0, then there is a non-negative number D and a bounded non-decreasing function G(x) such that

(For a proof, see Loomis and Widder, Duke Math. J. 9 (1942), 643–5.) If we write

where λ > 1, then the equation

defines a harmonic function h which is positive in υ > 0. Hence there is a non-negative number d and a bounded non-decreasing function g(x) such that

The problem of finding the connexion between the functions G(x) and g(x) has been mentioned by Loomis (Trans. American Math. Soc. 53 (1943), 239–50, 244).

It was proved by J. Williamson (1) that when C is any circulant matrix of order n and Ω is a certain diagonal matrix then

where ∥ C ∥ is the determinant of C and I is the unit matrix of order n. A simpler proof was given by U. Wegner (2), and later the characteristic equation of ΩC was discussed independently by L. Toscano (3). In the present note we prove the more general theorem that there corresponds to any matrix A of order n with simple elementary divisors a solution X of the equation (AX)n = ∥ A ∥ I. The theorem has a simple geometrical interpretation; the proof is almost immediate.

Let ξ and η be two independent and normal random variables, with zero means and with standard deviations each equal to ½ Put

The joint distribution of X, Y, Z is a particular case of the Wishart distribution (1). It may be defined by the generating function of its cumulants (c.g.f.)

The flow past a wedge which is moving with supersonic velocity into a gas at rest has been treated by a number of investigators. In the analysis of the problem given in the present note, the condition for the detachment of the shock wave is examined, and an error in a previous treatment by Epstein is corrected.

The purpose of this note is to show that Peng's (1) method of approximation is not practicable in the case of quadrilinear interaction (i.e. Fermi's (2) original theory of β-decay), and that it does not remove the infinite self-energy of a nucleon in interaction with a charged scalar meson field. The two examples do not, however, provide a serious argument against Peng's theory, since both refer to field theories which have been abandoned for physical reasons. The principal failure of Peng's method to cope with quadrilinear interactions could even be interpreted in such a way as to mean that interactions of this sort do not exist—a statement which would be equivalent to the assumption that Fermi particles can only interact by means of a Bose field. (2n + 1) linear interaction terms must contain one Bose quantity at least, since Fermi field quantities are not observable and therefore can only enter the interaction term in bilinear form.