If(X1, …, Xn), n ≥ 3, is a non-singular quadratic form with rational integral coefficients whose greatest common divisor is 1, then G. L. Watson [1] showed that f(x1, …, xn) = 1, for suitable algebraic integers x1, …, xn. In the present paper we extend this result to forms of arbitrary degree, with algebraic integers as coefficients (see Theorem 3). In fact we prove the stronger result (Theorem 2) that, if f(X1, …, Xn) is any polynomial with relatively prime algebraic integers as coefficients, then f(x1, …, xn) is a unit, for suitable algebraic integers x1, …, xn. Unfortunately, our result is just an existence theorem. We cannot limit the size of the field which x1, …, xn generate, as Watson could.

Following A. S. Besicovitch [1] and L. J. Mordell [3], we say that a polygon is rational if the lengths of all its sides and diagonals are rational. Besicovitch proved that the set of all rational right-angled triangles is dense in the set of all right-angled triangles and that the set of all rational parallelograms is dense in the set of all parallelograms. Then Mordell showed that the set of all rational quadrilaterajs is dense in the set of all quadrilaterals.

Our purpose is to develop a new and simple procedure for embedding graphs into orientable surfaces. This will involve the identification of the oriented edges of two oriented polygons, subject to certain rules.

A steady motion of incompressible viscous liquid caused by the slow rotation of a rigid sphere of radius a is considered. The medium is bounded by an infinite rigid plane and the axis of rotation is parallel to, and at a distance d from, this plane. To complete the analysis the solution by successive approximation of an infinite set of linear equations is required. Satisfactory solutions have been found numerically for four values of d/a, of which 1·13 is the smallest; we gratefully acknowledge valuable help from Miss S. M. Burrough in this part of the work.

Denote by |E| the cardinal of a set E. The purpose of the present paper is to prove the following result, constituting the solution of an unpublished problem of Erdös, Hajnal and Milner.

Recent papers [1, 2, 3[ have considered dual series equations in Legendre and associated Legendre functions and have given applications of these series to various potential and diffraction problems. This note gives a further application to the problem of the axisymmetric Stokes flow of a viscous fluid past a spherical cap. The stream-function of the flow is found by solving two pairs of dual series equations in associated Legendre functions, these equations being of a form considered previously [1, henceforth referred to as DSE]. As an example a uniform flow past the cap is considered and the drag of the cap calculated. This flow has previously been investigated by Payne and Pell [4], who by a suitable limiting process derive the stream-function for the flow past the cap from the stream-function for the flow past a lens-shaped body. Their method, however, involves the use of peripolar coordinates, besides much complicated algebra, and results are given only for a cap whose semi-angle is π/2. Further, their value for the drag of this cap is incorrect.

Let be a set of points on a sphere, centre O, radius R, in (n+1)– dimensional space. Suppose a spherical cap of angular radius α≤½π is centred at each point of . Let k be a positive integer and suppose that no point of the sphere is an inner point of more than k caps. We say that provides a k–fold packing for caps of radius α.

It is shown that in a certain sense, inversion transforms biharmonic functions into biharmonic functions. The first boundary value problem of elastostatics is also largely unchanged by this transformation, and known solutions can be used to obtain new results for inverse regions. As an example, the problem of a stress free dumb-bell shaped hole in an infinite plate is solved.

Minkowski [1] first proved that the surface of a convex body in E3 can be approximated by a level surface of a convex analytic function. His proof is strikingly simple. His proof is also presented for En by Bonnesen-Fenchel [2, pp. 10–12[. We here prove that the same kind of result is achieved using level surfaces of convex non-negative polynomials. We give two types of approximation, one based on finite sums as Minkowski did, and the other using integration. Since these approximations may be used for other applications we also extend them and give special formulae when the surface is centrally symmetric.

The theorem proved in this paper is basic for a general intersection theory applicable to multigraded polynomial rings. When the polynomial ring is graded by the non-negative integers the facts, in one form or another, are well known, but on passing to more general gradings fresh complications appear. These are not wholly trivial and, as the author was unable to find an account of these matters in the literature, it may be of interest if one is given here.

The first and second boundary value problems of plane elastostatics are solved for the interior of a parabola. A conformal transformation is used to map the interior of the parabola onto an infinite strip. An analytic continuation technique reduces the boundary value problem to the solution of a form of differential-difference equation. This is solved by a Fourier integral method. The resulting integrals are evaluated by residues to give eigenfunction expansions for the complex potentials.

Erdös, Kestelman and Rogers [1[ showed that, if A1, A2,… is any sequence of Lebesgue measurable subsets of the unit interval [0, 1] each of Lebesgue measure at least η > 0, then there is a subsequence {Ani} (i = 1, 2,…) such that the intersection contains a perfect subset (and is therefore of power ). They asked for what Hausdorff measure functions φ(t) is it possible to choose the subsequence to make the intersection set ∩Ani of positive φ-measure. In the present note we show that the strongest possible result in this direction is true. This is given by the following theorem.

Based on the method of analytic continuation of Buchwald and Davies [1[, [2], the first boundary value problem of an elliptic plate is solved. The results are in agreement with the more complicated solution of Muskhelishvili [3]. The solution of the second boundary value problem is also obtained.

The following problem was proposed by Professor N. J. Fine: to prove that there do not exist rational functions F1, F2, F3 of x1, x2, x3, x4, with real coefficients, such that

identically. In the present note I give a proof.†

This note is concerned with an inconsistency in the assumptions made in the Reissner theory of elastic plates. An expression is derived for the stress component t33 and a form is suggested for the shear stress components tα3 which form a suitable basis for an approximate theory of elastic plates.

During the past seven years Northcott has published several papers (see, for example, [6, 7, 8, 9]) in which he has investigated the local aspect of the theory of dilatations. In a similar manner we shall develop in a later paper a local theory of monoidal transformations of which the global analogue appears in [2]. The present note is concerned with such a theory in the one-dimensional case and closely follows the development given in [8] for local dilatations. Indeed the theorems of the present note are all natural generalizations of theorems which have previously been given by Northcott, and for the most part the proofs are essentially Northcott's proofs.

Let ℋ be the system of all continuous increasing functions h(t), denned for t ≥ 0, with h(0) = 0 and h(t)>0 for t > 0. Let Ω be a separable metric space. Then, for each h of ℋ, we may introduce a Hausdorff measure into Ω, by taking

where d(Fi) denotes the diameter of Fi, and where the infimum is taken over all sequences {Fi} of closed sets, covering E and having diameters less than δ. We introduce a natural partial order in the system of these Hausdorff measures by writing j < h, if j, h are functions of ℋ and

A graph is said to be k-chromatic if its vertices can be split into k classes so that two vertices of the same class are not connected (by an edge) and such a splitting is not possible for k−1 classes. Tutte was the first to show that for every k there is a k-chromatic graph which contains no triangle [1].