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 primecharacteristic 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 primecharacteristic, 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.

By examining certain connections between the derivatives and the powers of a Lie algebra, bounds are obtained for the indices of nilpotent Lie algebras over an arbitrary field. The results are used to obtain bounds for the indices of solvable Lie algebras over a field of characteristic zero.

Certain types of 2n-dimensional Riemannian spaces admitting parallel fields of null n-planes are studied. These are known as Riemann extensions of conformal, projective or other classes of spaces of affine connection. The circumstances under which a 2n-dimensional Riemannian space admits two non-intersecting parallel fields of null n-planes are also discussed. Such spaces satisfy a condition similar to Kähler's condition in the theory of complex manifolds, and hence are called Kähler spaces. Necessary and sufficient conditions are found for a Kähler space to be a Riemann extension with respect to one of the parallel fields of null n-planes, and canonical forms are found for the metrics in the cases of Riemann extensions of conformal and projective spaces.

The logarithmetic L of a non-associative algebra or class of algebras S has been previously defined as the arithmetic of the indices of powers of the general element when indices are added (non-associatively) and multiplied by certain conventions similar to those of ordinary algebra. With respect to addition, L is a homomorphic image of the “most general” logarithmetic B, the free additive groupoid with one generator 1, and in the case of algebras of one operation is essentially the same as the free algebra in one variable on S. The definition is now extended so that L is defined when S is any subset of an algebra or class of subsets of algebras, with the result that every homomorph of B is a logarithmetic ; but a distinction has then to be drawn between closed logarithmetics in which as before both addition and multiplication are defined, and other logarithmetics in which there is only addition. L is its own logarithmetic (taken with respect to addition) only if L is closed. For subsets of palintropic algebras, L is necessarily closed.

The methods of S. N. Lin (1943) and B. Friedman (1949) for approximating to the factors of a polynomial by iterated division are studied from the point of view of convergence. The general theory, hitherto lacking, is supplied. The matrices which transform the errors in coefficients from one iterate to the next are explicitly found, and the criterion of convergence derived. Numerical examples are given. The tentative conclusion is that the methods are less simple in theory and less adaptable than the method of penultimate remainder, which admits of accelerative devices.

For a given sequence {am} and p≠0, Schur (2) defined

In particular if p is a prime, a an integer and , then by Fermat's theorem

is integral. Schur proved that if p † a, then all the derivatives

are integral. Zorn (3) using p-adic methods proved Schur's results and also found the residue of Xm (mod pm), where and x = 1 (mod p). The writer (1) proved Zorn's congruences by elementary methods as well as certain additional results of a similar sort.

The formula to be proved is

The formulae required in the proof are the Barnes Integral

and Whipple's Formula (1)

§ 2. Proof of the Formula. From (2) the E-function on the left of (1) is equal to