Introduction. Dans cet article, K désigne un corps commutatif quelconque. Toutes les algèbres de Lie considérées sont des algèbres de Lie de dimension finie sur K. Si g est une algèbre de Lie, on désigne par (D0g, D1g, D2g, …) la suite des algèbres dérivées de g (D0g = g; Dig = [Di−1g, Di−1g]). Si h est un idéal de g, et si x ↦ g, on désigne par adh, x l'application y → [x, y] de h dans h.

The so-called AF + BΦ theorem of Max Noether refers to a whole collection of results. In most of these, the object is to determine conditions under which a form H(x, y, z) will belong to the ideal generated by two given forms F(x, y, z) and Φ(x, y, z). This problem is of obvious importance in the theory of plane curves, and therefore it is customary to state the conditions in geometric language. To give one example, a particularly important form of the theorem may be stated as follows.*

1. This note takes its origin in a remark by Brauer (1) and Perfect (5): Let A be a square complex matrix of order n whose characteristic roots are α1,…, αn. If X1 is a characteristic column vector with associated root α and k is any row vector, then the characteristic roots of A + X1 k are α1 + KX1, α2, …, αn. Recently, Goddard (2) extended this result as follows: If x1; …, xr are linearly independent characteristic column vectors associated with the characteristic roots α1, …, αr of the matrix A, whose elements lie in any algebraically closed field, then any characteristic root of Λ + KX is also a characteristic root of A + XK, where K is an arbitrary r × n matrix, X = (x1, …, xr) and Λ = diag (α1, …, αr). We shall prove some theorems of which these and other well-known results are special cases.

Let Z denote the extended complex plane with the natural topology. That is, Z consists of the finite complex plane Zf together with a point, denoted by ∞, any neighbourhood of which consists of the union of this point and the complement of a bounded set of ZF. Let Z* = [z │ z ↦ Z, z ╪ 0, z ╪ ∞]. Thus Z is homoeomorphic to a 2-sphere and Z* is homoeomorphic to a cylinder.

The Minkowski–Hlawka theorem† asserts that, if S is any n-dimensional star body, with the origin o as centre, and with volume less than 2ζ(n), then there is a lattice of determinant 1 which has no point other than o in S. One of the methods used to prove this theorem splits up into three stages, (a) A function ρ(x) is considered, and it is shown that some suitably defined mean value of the sum

taken over a suitable set of lattices Λ of determinant 1, is equal, or approximately equal, to the integral

over the whole space. (b) By taking ρ(x) to be equal, or approximately equal, to

where σ(x) is the characteristic function of S, and μ(r) is the Möbius function, it is shown that a corresponding mean value of the sum

where Λ* is the set of primitive points of the lattice Λ, is equal, or approximately equal, to

The Mellin transforms of generalized hypergeometric functions are discussed in this paper, and it is shown how some of the most general integrals of the Mellin type can be deduced from them. Four general theorems are considered and a number of special cases are given in detail.

The first 2N + 1 Fourier coefficients of an unknown, non-negative function f(θ) are given, and it is required to find bounds for ∫Ef(θ) dθ, where E is some given region of integration. We also wish to find the interval E for which the bounds are most strict, when the width of E is specified. f(θ) may represent a distribution of energy in the interval 0 ≤ θ ≤ 2π; the object is to determine where the energy is chiefly located.

In the present paper we show that if the energy is located mainly in the neighbourhood of not more than M distinct points, significant lower bounds for ∫Ef(θ) dθ can be found in terms of the first 2M + 1 Fourier coefficients. The effectiveness of the method is illustrated by applying the inequalities to some known functions.

The results have application in determining the direction of propagation of ocean waves and other forms of energy.

1·1. A general problem in the theory of non-linear differential equations of the second order is: Given a non-linear differential equation of the second order uniformly almost periodic (u.a.p.) in the independent variable and with certain disposable constants (parameters), to find: (i) the non-trivial relations between these parameters such that the given differential equation has a non-periodic u.a.p. solution; (ii) the number of periodic and non-periodic u.a.p. solutions which correspond to each such relation; and (iii) explicit analytical expressions for the u.a.p. solutions when they exist.

The advantages of using polynomial approximations for the purpose of constructing interpolable numerical tables on punched cards have been pointed out by Sadler (7). The object of this paper is to demonstrate the value of this method for ordinary published tables.

Let X1, X2, X3,… be a sequence of independent, identically distributed, absolutely continuous random variables whose first moment is μ1. Let Sk = X1 + X2 + … + Xk, and let fk(x) be the frequency function of Sk, defined as the k-fold convolution of f1(x). When f1(x) has been defined for all x, fk(x) is uniquely defined for all k, x. Write

Random walk on a d-dimensional lattice is investigated such that, at any stage, the probabilities of the step being in the various possible directions depend upon the direction of the previous step. The motion may be characterized by a generating function which is here derived. The generating function is then used to obtain some general properties of the walk. Certain special cases are considered in greater detail. The existence of recurrent points is investigated in particular, and the probability of returning to the origin after 2n steps. This latter function is evaluated asymptotically for the cases d = 1 and d = an even integer.

Isotropic Gaussian processes, of which we shall give a formal definition presently, arise in various practical problems. The present inquiry arose from the consideration of the variability found in the yields of plots in agricultural field experiments. Samples of such patterns of variability can be obtained from uniformity trials, whereby a piece of land is treated uniformly throughout and the crop is harvested in small units or plots. The results from such trials have been widely used to determine optimum plot sizes for future experiments with the crop concerned, but it has never been clear how valid is the generalization from a uniformity trial to future experiments on other sites. One difficulty arises from the lack of a suitable model to express the variability; another difficulty arises from the formidable analytical problems besetting any attempts to apply deductive reasoning to even simple models. We may mention two approaches that have been made to the uniformity trial problem: Fairfield Smith (3) has supplied an empirical law connecting the variance of contiguous groups of plots with the size of the group, and Quenouille (6) and Whittle (8) have considered the fitting of two-dimensional isotropic Gaussian processes to uniformity trial data. Whittle has pointed out that a satisfactory model may have to incorporate an additional random element at each point, and has outlined the difficulties of estimation which arise when such an element is introduced. The connexion between Fairfield Smith's law and the approach via two-dimensional stochastic processes, if any, seems to be quite unknown.

The work of which this paper is an account began as a study of differential equations for functions whose values are random variables of finite variance. It was intended that all questions of convergence should be treated from the standpoint of strong convergence in Hilbert space—familiar to probabilists from the writings of Karhunen(11) and Loève(13) as mean-square convergence. The more general Banach-space approach now adopted was made possible by the discovery of a theorem (Theorem 1 of this paper) which Mr D. G. Kendall, its apparent author, kindly communicated to us.

Many models of the universe have been proposed, by de Sitter, Milne, Bondi and Gold, Hoyle and others. The observed data being insufficient, the models are usually based on some simple hypothesis. The simplest is the cosmological principle, namely, that apart from local irregularities the universe presents the same general aspect at every point. Milne (5) has used a restricted form of the principle, namely, that the aspect is independent of spatial position but is dependent on the observed time from some fixed epoch in the past. Bondi and Gold(i) have proposed the ‘perfect cosmological principle’ that the aspect is completely independent of space and time.

If r̄nl is the mean radius for the radial wave function of a complete (nl) group in an atom of atomic number N, the variation of 1/r̄nl with N is nearly linear. Further the variation of a given (nl) radial wave function with N is such that for a given value of (r/r̄nl), the variation of the quantity (r̄nl)½P(nl; r) with r̄nl is nearly linear. These relations between the radial wave functions for different atoms are examined from the point of view of using them as a means of interpolating, with respect to atomic number, between results for atoms for which solutions of Fock's equations have been carried out.

Z(nl; r) is the contribution to Z(r) from an electron in the (nl) wave function. The Z(nl; r) vary systematically with atomic number and, as N becomes large, tend to the corresponding hydrogen-like functions, ZH(nl; r). A two-parameter method of fitting the Z(nl; r) to the ZH(nl; r) is described. This involves a ‘screening constant’ and a ‘slope constant’, both of which are defined. From published data, the two parameters have been obtained as functions of atomic number. The parameters for an unsolved atom can then be found by interpolation and approximate Z(nl; r) derived by appropriate adjustment of the functions for the nearest atom in the periodic table for which they are known. The method has been tested by interpolating for the (3d) function between Cu+ and Rb+ and by preparing estimates of the Z(nl; r) for the unknown structure Mo+. The results were good for all but Z(4d; r) for Mo+, where the number of values of the screening and slope constants already known was insufficient for reliable interpolation.

The results of a self-consistent field calculation without exchange for Mo+ are presented.

The behaviour of the conduction electrons in a vibrating metallic lattice is treated by the Born-Oppenheimer method. Non-adiabatic processes, due to certain terms usually neglected, are shown to correspond to scattering of electrons by the lattice vibrations, as in Bloch's theory of electrical resistivity, but the matrix elements are not the same as Bloch's except when energy is conserved in the process. This may have important consequences in Fröhlich's theory of superconductivity.

The well-known Stokes theory (9, 10) of waves of permanent form in water of finite depth has been extended to the fifth order of approximation. The solutions have been first obtained in the form of equations for the space coordinates x and y as functions of the velocity potential Φ and stream function ψ. Expressions for the complex potential W in terms of the complex variable z ( = x + iy), the form of the wave profile, and the square of the wave velocity have been obtained to the fifth order.

Expressions for the three physical quantities Q, R and S, where Q is the volume flow rate per unit span, R is the energy per unit mass (i.e. g times the total head, measuring heights from the bottom and pressures from atmospheric) and S is the momentum flow rate per unit spaa, corrected for pressure forces and divided by density, have been obtained to the fifth order. The values for the dimensionless quantities r = R/Rc and s = S/Sc, where Rc and Sc refer to the values of R and S for a critical stream of volume flow Q, are tabulated for certain values of the ratios mean depth to wavelength and amplitude to wavelength. The values of r and s thus obtained have been used to calculate the ratios of mean depth to wavelength and of wave height to wavelength according to the cnoidal wave theory as recently presented by Benjamin and Light-hill(1), and the results are found to be in satisfactory agreement with that from Stokes's theory for waves longer than six times the depth.

The (r, s) diagram introduced in the recent work of Benjamin and Lighthill(1) has been further considered, and the unshaded part of the diagram referred to in that paper has been mapped with a network of curves for constant values of the ratios of mean depth to wavelength and of wave height to wavelength (Fig. 2). The third barrier to the existence of steady flows, corresponding to ‘waves of greatest height’ referred to in that paper, has also been indicated in Fig. 2.

The appearance of a component of vorticity in the direction of flow materially alters the pattern of flow of a fluid in three dimensions. Expressions are obtained for this secondary vorticity in an inviscid compressible fluid flowing under the action of body forces. They are applied to examples such as a liquid under gravity and gas flow behind a curved shock. In compressible gas flow with varying temperature but constant stagnation pressure no secondary circulation appears. In a perfect gas atmosphere it is shown that secondary circulation may appear because of nou-adiabatic lapse rates as well as wind-speed gradients. It is also shown that in a liquid with a density gradient, gravitational effects can give rise to secondary vorticity components.