Stokes's stream-function for an axisymmetrical irrotational motion of perfect incompressible liquid, bounded by a plane [1], or by a sphere [2], has been expressed in terms of the stream-function for the motion in an unbounded liquid. Corresponding results are found in this note for the slow steady motion of a viscous liquid, when the motion in unbounded liquid is irrotational.

In 1935 van der Corput, in connection with his work on distribution functions, was led to the following conjecture which expresses the fact that no sequence can, in a certain sense, be too evenly distributed.

Let f = f(x, y, z) be a positive definite form of the type

where x, y, z are integral valued variables, and the coefficients a, …, t are integers whose highest common factor is 1. As the determinant of such a form may be fractional, I define

and

thus — C is the discriminant of the binary form f(x, y, 0), and the necessary and sufficient condition for f to be positive definite is that a > 0, C > 0, and d > 0.

Among Schubert's many experiments in the application of a symbolic calculus to problems of enumerative geometry, some special attention is due to his long memoir entitled “Anzahlgeometrische Behandlung des Dreiecks” [1]. For one thing, he is dealing here with a simple, though not elementary, kind of geometric variable, the triangle in a fixed plane, so that the paper gives a clear insight into his general method; and, for another, there is contained in this paper, as was recently suggested by Freudenthal ([2], p. 19), an apparently miraculous device, the introduction of “infinitesimal triangles”, which we can now recognize (§4) as having had the effect of desingularizing the triangle domain in which the calculus was to operate. The principal target of Schubert's investigations was the discovery of Bézout-type formulae for the number of triangles common to two algebraic systems Σr and Σ6-r (r = 1, 2, 3) of complementary dimensions, the systems being supposed to intersect in only a finite number of triangles, and the multiplicities of these triangles being assumed to be suitably defined. His systems, also, had to be “normal” i.e. they could only contain such sub-systems of degenerate triangles as were of the dimensions he regarded as normal. He found, by his methods, that “normal” system Σ1 and Σ5 are each characterized (in so far as intersection numbers are concerned) by 7 projective characters, systems Σ2 and Σ4 by 17 such characters, and systems Σ3 by 22 such characters.

The object of this paper is to solve the Saint-Venant torsion problem for those cross-sections with inclusions, which are such that the z-plane boundaries involved can be mapped into concentric circles in a complex ζ-plane by the transformation

with z´(ζ) ≠ 0 or ∞ within the cross-section. We shall consider both solid and hollow inclusions having different elastic rigidities μ. In the case of the solid inclusion we have to restrict the coefficients as to be zero for all negative s, but it is an advantage to leave this restriction to the end of the analysis, since the forms of certain coefficients in the two cases differ only in this respect.

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.

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.