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cardinal of a set E. The purpose of the present paper is to prove the following result, constituting the solution of an unpublished problem of Erdös, Hajnal and Milner.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 paper formulates a general solution, within the scope of classical elastostatic theory, for the problem of layered systems subjected to asymmetric surface shears. As an illustrative example the solution for the problem of an elastic layer supported on an elastic half-space is presented for the particular loading consisting of a surface shearing force uniformly distributed over a circular area. Numerical results are included indicating some displacement and stress components of interest.

It is known [1] that for a partial endomorphism μ of a group G that maps the subgroup A ⊆ G onto B ⊆ G. G to be extendable to a total endomorphism μ* of a supergroup G* ⊆ G such that μ an isomorphism on G*(μ*)m for some positive integer m, it is necessary and sufficient that there exist in G a sequence of normal subgroups

such that L1 ƞA is the kernel of μ and

for ι = 1, 2,…, m–1.

The question then arises whether these conditions could be simplified when the group G is abelian. In this paper it is shown not only that the conditions are simplified when Gis abelian but also that the extension group G*⊇G can be chosen as an abelian group.

Let {Si; i ε I} be a finite or infinite family of cancellative semigroups. Let U be a cancellative semigroup, and suppose that there exists, for each i in I, a monomorphism φi: u→ Si. We are interested in finding a semigroup T with the following properties.

(a) For each i in I, there is a monomorphism λi: Si → T such that uφiλi = uøjλi for all u ɛ U and all i, j in I. That is to say, there exists a monomorphism λ: U → T which equals øiλi for all i in I.

Siλi∩Sjλj = Uλ (i, j ε I; i ≠ j).

Let D be a bounded, closed, simply-connected domain whose boundary C consists of a finite number of analytic Jordan curves. Let γ be any analytic arc of C. Then we shall prove the following theorem.

Theorem 1. Let u(x, y) be harmonic in the interior of D and continuous on γ, and let ϱu(x, y)/ϱn=g(s) when (x, y) is on γ, where g(s) is an analytic function of arc-length s along γ. Then u(x, y) can be harmonically continued across γ.

It is shown that by using the methods developed in papers I–III of the present series it is possible to reduce the problem of deriving the solution of a certain class of dual relations involving Jacobi polynomials to that of solving an integral equation of Schlömilch kind.

For i, j = 1, 2, …, let aij be real. A matrix A = (aij) will be called positive (A>0) or non-negative (A≧0) according as, for all i and j, aij>0 or aij≧0 respectively. Correspondingly, a real vector x = (x1, x2, …) will be called positive (x>0) or non-negative (x≧0) according as, for all i, xi>0 or x≧0. A matrix A is said to be bounded if ∥ Ax ∥ ≦M ∥ x ∥ holds for some constant M, 0 ≦ M < ∞, and all x in the Hilbert space H of real vectors x = (x1, x2, …) satisfying . The least such constant M is denoted by ∥ A ∥. If x and y belong to H, then (x, y) will denote as usual the scalar product Σxiyi. Whether or not x is in H, or A is bounded, y = Ax will be considered as defined by

whenever each of the series of (1) is convergent.