This functional equation for Dedekind's

has been established in many different ways; however, it seems that the following proof, a straightforward application of the theorem of residues, has not been observed before. Since

it suffices to prove that

It is well known that every irrational number θ possesses an infinity of rational approximations p/q satisfying

It is also well known that there is a wide class of irrational numbers which admit of no approximations which are essentially better, namely those θ whose continued fractions have bounded partial quotients. For any such θ there is a positive number c such that all rational approximations satisfy

The following theorem is typical of a group of results which have been found to be of importance in the theory of Cesàro summability.

Theorem A. If 0 ≤ ρ ≤ κ (κ, ρ integer), p > −1,p+q −1 and

and ifsn = 0(np) (C, κ), thensn εn = 0(np+q) (C, ρ).

That these two subjects—the History of Science and the Psychology of Invention—are intimately connected with one another, is immediately evident and needs no explanation. Perhaps, however, it has not always been sufficiently appreciated. The recent Congress for the History of Science (Jerusalem, 1953) has given me an opportunity of trying to apply to the latter the data of the former.

Let q(x1; …, xn) be a positive definite quadratic form in n variables with real coefficients. Minkowski defined the successive minima of q as follows. Let S1 denote the least value assumed by q for integers x1 …, xn, not all zero, and let be a point at which this value is attained. Let S2 denote the least value assumed by q at integral points which are not multiples of x(1), and let x(2) be such a point at which this value is attained. Let S3 be the least value of q at integral points which are not linearly dependent on x(1) and x(2), and so on. We have

and it is easy to see that these numbers are uniquely defined, even though there may be several choices for the points x(1), …, x(n). The determinant N of the coordinates of the points x(1), …, x(n) is a non-zero integer. We denote by N (q) the least value of this integer (taken positively) for all permissible choices of the n minimal points, and by N′(q) its greatest value. Plainly N(q) and N′(q) are arithmetical invariants of q, that is, they are the same for two forms which are equivalent under a linear substitution with integral coefficients and determinant ±1.

The problem of the stability of a fluid rotating about an axis to an axisymmetric disturbance has been examined in the inviscid case by Rayleigh [1], who derived a simple criterion based on an analogy with the stability of plane stratified fluid of variable density. Later a complete discussion of the stability of viscous motion between rotating cylinders for small axisymmetric disturbances was given by G. I. Taylor [2]. More recently, the problem of magneto-hydrodynamic stability has claimed the attention of several workers, and, amongst other problems, the stability of a rotating fluid, when a constant magnetic field is applied in the direction of the axis of rotation, has been examined by Chandrasekhar [3]

As is well known the stability of viscous flow between two concentric rotating cylinders was first successfully treated both experimentally and theoretically by G. I. Taylor [1]. The mathematical problem underlying this classical investigation in hydrodynamic stability is the following:

The hydrodynamical equations allow the stationary solution

for the rotational velocity at a distance r from the axis of rotation, where A and B are constants related to the angular velocities Ω1 and Ω2 with which the inner and outer cylinders (of radii R1 and R2, R2 > R1) are rotated. Thus

where

In §§2, 3 a simple expression in finite terms is found for the small transverse displacement of a thin plane elastic plate due to a transverse force applied at an arbitrary point of the plate. The plate is clamped along, and is bounded internally by, the parabola CDE shown in Fig. 1.

In a previous paper [5] the equivalence of randomisation and normal theory distributions of linear combinations was discussed. In the present paper we discuss the asymptotic randomisation distributions of statistics used in analysis of variance and in a closely related problem which includes, in particular, the “problem of m-rankings“. Kruskal [4] has studied the first of these questions in the case where observations are replaced by ranks.

There are some simple facts which distinguish Lie-algebras over fields of prime characteristic from Lie-algebras over fields of characteristic zero. These are

(1) The degrees of the absolutely irreducible representations of a Lie-algebra of prime characteristic are bounded whereas, according to a theorem of H. Weyl, the degrees of the absolutely irreducible representations of a semi-simple Lie-algebra over a field of characteristic zero can be arbitrarily high.

(2) For each Lie-algebra of prime characteristic there are indecomposable representations which are not irreducible, whereas every indecomposable representation of a semi-simple Liealgebra over a field of characteristic zero is irreducible (cf. [4]).

(3) The quotient ring of the embedding algebra of a Lie-algebra over a field of prime characteristic is a division algebra of finite dimension over its center, whereas this is not the case for characteristic zero. (cf. [4]).

(4) There are faithful fully reducible representations of every Lie-algebra of prime characteristic, whereas for characteristic zero only ring sums of semi-simple Lie-algebras and abelian Lie-algebras admit faithful fully reducible representations (cf. [6], [2], [4]).

Let μ be a homomorphic mapping of some subgroup A of the group G onto a subgroup Ḃ (not necessarily distinct from A) of G; then we call μ a partial endomorphism of G. If A coincides with G, that is, if the homomorphism is defined on the whole of G, we speak of a total endomorphism; this is what is usually called an endomorphism of G. A partial (or total) endomorphism μ*extends or continues a partial endomorphism μ if the domain of μ* contains the domain of μ, that is, μ* is defined for (at least) all those elements for which μ. is defined, and moreover μ* coincides with μ where μ is defined.

The two following formulae are to be established.

If R (m ± n) > 0, | amp z | < π,

If p ≧ q + 1, R(k ± n + 2αr) > 0, r = 1, 2, …, p, | amp z | < π,

For other values of p and q the formula holds if the integral is convergent.

A modified form of the centroid method used in factor analysis is described. Various large sample results are obtained, including a test of significance of the residuals. The method is compared with the corresponding form of maximum likelihood estimation and its efficiency is investigated. A numerical illustration is given of some of the foregoing theory.

The Bragg two-hologram method for eliminating the unwanted image in Diffraction Microscopy is here thoroughly examined. It is shown to be of general validity for objects containing both amplitude and phase-contrast terms.

Other two-hologram methods are briefly discussed. In principle any two sufficiently distinct holograms may be used, but the Bragg 2:1 focal ratio is the simplest. The pair of holograms at ±f is also of interest, since it allows moderate degrees of phase-contrast and amplitude-contrast terms in the original object to be separated. A “three-wavelength” technique for obtaining the Bragg records is briefly outlined.

A theorem, due to Dr Grace, about a configuration of lines in [3], has been shown by him to be a rewording of a theorem about a configuration of points and spheres in [4]; in this six spheres pass through each point. The present paper discusses the analogous configuration in which seven spheres pass through each point.

The two regular representations of quaternions give rise to a classical set of sixteen 4 × 4 matrices that have fairly recently reappeared in a paper by S. R. Milner. He uses them as the basis of a calculus of “Ɛ-numbers”, which he develops for the purpose of making physical applications. The covariantive nature of his calculus is, however, not always fully apparent, and raises some points of interest of which an examination is made in the present paper in terms of 3-dimensional projective geometry. The theory that emerges is the classical one of the collineations of projective 3-space that transform a quadric into itself, but the formulation is different from that of existing theories based on the same set of matrices and having the same or a similar geometrical background. For example, the present theory is quite different from that of 4-component spinors. The constants of multiplication γijk of quaternion algebra make their appearance in a generalized form and in a geometrical setting. In the final section an indication is given of possible generalizations to Riemannian geometry, and of the connection of the present work with the theory of Kähler manifolds of two complex dimensions.

The writer's theory of unimolecular dissociation rates, based on the treatment of the molecule as a harmonically vibrating system, is put in a form which covers quantum as well as classical mechanics. The classical rate formulæ are as before, and are also the high-temperature limits of the new quantum formulæ. The high-pressure first-order rate k∞ is found first from the Gaussian distribution of co-ordinates and momenta of harmonic systems, and is justified for the quantum-mechanical case by Bartlett and Moyal's phase-space distributions. This leads to a re-formulation of k∞ as a molecular dissociation probability averaged over a continuum of states, and to a general rate for any pressure of the gas.

The high-pressure rate k∞ is of the form ve-F/kT, where v and F depend, in the quantum case, on the temperature T; but v is always between the highest and lowest fundamental vibration frequencies of the molecule. Concerning the decline of the general rate k with pressure at fixed temperature, k/k∞ is to a certain approximation the same function of as was tabulated earlier for the classical case, apart from a constant factor changing the pressure scale in the quantum case.