§ 1. 1 have ventured to bring the formulae of this paper before the Society, as I have been unable to find reference to them in any text-book or any original contribution to mathematical literature which I have come across. I confine my attention completely to the central surface, as the corresponding formulae for the paraboloids are very readily deduced by a similar process.

The hypergeometric function1F(a, b; c; z) is analytic in the domain |arg(−z)| < π, and, when |z| < 1, may be represented by the series

When |z| = 1 in the domain |arg(−z)| <π, this series converges2 to F(a; b; c; z) if R(a+b−c) < 0 (integral values of a, b and c are excluded in the present paper).

The construction of all irreducible modules of the symmetric groups over an arbitrary field which reduce to Specht modules in the case of fields of characteristic zero is given by G. D. James. Halicioğlu and Morris describe a possible extension of James' work for Weyl groups in general, where Young tableux are interpreted in terms of root systems. In this paper we show how to construct submodules of Specht modules for Weyl groups.

If X is a Tychonoff topological space, and if βX is the Stone-Cech compactification of X, then βX\X will denote the complement of X in βX. If A is a subset of X, then cl [A: X] will denote the closure of A in X, and int [A: X] will denote the interior of A in X. In Isbell ((3), p. 119) a property of βX\X is called a property which X has at infinity, and it is the aim of this paper to give necessary and sufficient conditions for X to be finite at infinity. Since βX is T1 we can say that if X is finite at infinity, then βX\X is closed in βX. So we lose nothing by restricting our attention to locally compact, Tychonoff spaces, and for the remainder of the paper X will denote such a space.

Some Hilbert spaces of continuous functions satisfying a mean value property are studied in which the generalised eigenfunctions of any selfadjoint operator again satisfy the same mean value property. Applications are made to nullspaces of some differential operators. The classes of functions involved in these applications are less general than those studied by K. Maurin (6); however, the Hilbert space norms may be arbitrary, while Maurin only considered L2-norms.

1. Theorem I. If , where a and n are positive integers, and if f(n) is a finite single-valued function of n for values of n蠇a, then

In a previous paper we studied the asymptotic distribution of the multiparameter eigenvalues of uniformly right definite multiparameter Sturm–Liouville eigenvalue problems. In this paper we extend the analysis to deal with multiparameter Sturm–Liouville problems satisfying uniform left definiteness, and non-uniform left and right definiteness.

In this paper a characterization of the regular ω-semigroups whose congruence lattice is modular is given. The characterization obtained for such semigroups generalizes the one given by Munn for bisimple ω-semigroups and completes a result of Baird dealing with the modularity of the sublattice of the congruence lattice of a simple regular ω-semigroup consisting of congruences which are either idempotent separating or group congruences.

In an interesting appendix to a letter written by John Collins to James Gregory on August 3, 1675, but not published until a few months ago, appear some formulae, given without proof, for expressing the roots of an equation of any degree from the 2nd to the 9th in terms of the coefficients, under the assumption that these roots are in arithmetical progression. The formulae were discovered by the well known contemporary of Leibniz, Baron W. von Tschirnhaus. It is evident that in the case of an equation of degree n this particular assumption imposes n − 2 conditions on the coefficients; so that two of these coefficients can be chosen ad libitum. Tschirnhaus did not go to the trouble of obtaining these relations explicitly, in fact he makes no mention of them, but he gives expressions, in the cases indicated above, for the roots as functions of the first two coefficients of the equation in question, and these coefficients, as we have observed, are arbitrary. It is not known by what approach he arrived at his formulae; it seems likely to us, however, that he expressed the desired roots in terms of two arbitrary unknowns, that he evaluated the sum of these, and the sum of their products two at a time, and that, finally, he equated the results to the first two coefficients of the equation. In this way two equations are obtained, sufficient to determine the two auxiliary unknowns; and the problem can be considered as solved. Without seeming to imply that this procedure was the same as that adopted by the eminent German mathematician, we shall show that by its means one can not only derive his results, but also solve the question in the case of an algebraic equation of any degree.

Definition. If x, y, z and ξ, η, ζ be the perpendiculars on the sides BC, CA, AB of the Δ ABC from points O and O′, then O and 0′ are antireciprocal points if xξ η, zζ:: tanA : tanA : tanB: tanC.

I. Construction to find a point antireciprocal to O (Fig. 4).

Draw through O a line MN antiparallel to BC. Draw OY perpendicular to AC, and OZ perpendicular to AB. Draw lines parallel to AB and AC, and at distances from them respectively equal to YN and MZ, and let them cut in P. Join AP. Find a similar line BQ, and let AP and BQ cut in O′.

It is common property in the theory of transformation semigroups that the presence of all the constant maps ensures that automorphisms are induced by a permutation of the underlying set. Essentially, this goes back to Malcev (2); it has been extensively generalised by Sullivan in (4). For semigroups which do not contain the constants (for example, all surjective transformations of a set, or all injections) there is, as yet, no similar result. The purpose of this note is to provide one.

Let (S, Σ, μ) and (T, Θ, v) be two measure spaces of finite measure where we assume S, T are compact Hausdorff spaces and μ, v are regular Borel measures. We construct the product measure space (T x S, >, Φ σ) in the usual way. Let G = [gl, g2, …, gp] and H = [hl, h1, …, hm be finite dimensional subspaces of C(S) and C(T) respectivelywhere G and H are also Chebyshev with respect to the L1-norm. Note that a subspace Y of a normed linear space X is Chebyshev if each x ∈X possesses exactly one best approximation y ∈Y. For example, in C(S) with the L1-norm, the subspace of polynomials of degree at most n is a Chebyshev subspace. This is an old theorem of Jackson. Now set

A sequence x = {si} has been defined to be summable-(Z, p) to s if

Where p is a positive integer and