All points in an n-space equidistant from a fixed point (the centre) constitute what may be called a spherical continuum of the nth order,—the continuum being of n — 1 dimensions ((n — l)-dimensional spread) and of the 2nd degree. Any region of this spherical continuum bounded by n (n — l)-dimensional linear continua or, primes (spaces of n — 1 dimensions), passing through the centre shall be called a spherical simplex of the nth order. This spherical simplex is bounded by n faces, spherical simplexes of the (n — l)th order, each of which in turn is bounded by n — 1 spherical simplexes of the (n — 2)th order, and so on till we reach spherical triangles, arcs and lastly points, the vertices. The total number of spherical simplexes of different orders connected with one of the nth order is 2n — 2. The n spherical continua of the (n — l)th order which contain the faces of the spherical simplex of the nth order determine a set of 2n spherical simplexes of the same order, 2n–1 pairs, the two spherical simplexes of a pair being symmetrically situated with respect to the centre and therefore congruent.

In a paper “On the Theory of Long Waves” recently published, I drew attention to a point of some interest in the theory of the solution of partial differential equations by what is usually termed the method of reciprocation. As the subject was apart from the main object of the paper, however, I there noticed the peculiarity but briefly, and so I have thought that it might be of some use to enter into the matter here somewhat more fully. On first adverting to the point, it was my opinion that it could hardly have escaped the notice of those who had made much application of the method of solution by reciprocation, though I had not myself seen any reference made to it; but as I have not had my attention drawn to any such notice since the publication of the paper last March, it seems probable that it has not previously been discussed.

The problem of finding a suitable representation for a fractional power of an operator defined in a Banach space X has, in recent years, attracted much attention. In particular, Balakrishnan [1], Hovel and Westphal [3] and Komatsu [4] have examined the problem of defining the fractionalpower (–A)α for closed densely-defined operators A such that

If H (Fig. 1) be the middle point of a straight bar QP and if a straight bar OH of length one-half of QP be pin-jointed to QP at H, a simple linkage is formed, which may be called a T -linkage.

A largely untouched problem in the theory of inverse semigroups has been that of finding to what extent an inverse semigroup is determined by its lattice of inverse subsemigroups. In this paper we discover various properties preserved by lattice isomorphisms, and use these results to show that a free inverse semigroup ℱℐx is determined by its lattice of inverse subsemigroups, in the strong sense that every lattice isomorphism of ℱℐx upon an inverse semigroup T is induced by a unique isomorphism of ℱℐx upon T. (A similar result for free groups was proved by Sadovski (12) in 1941. An account of this may be found in Suzuki's monograph on the subject of subgroup lattices (14)).

We are concerned with the following classical version of the Borsuk–Ulam theorem: Let f:Sn→Rk be a map and let Af = {x∈Sn|fx= f(−x)}. Then, if k≦n, Af≠φ. In fact, theorems due to Yang [17] give an estimation of the size of Af in terms of the cohomology index. This classical theorem concerns the antipodal action of the group G=ℤ2 on Sn. It has been generalized and extended in many ways (see a comprehensive expository article by Steinlein [16]). This author ([9, 10)] and Nakaoka [14] proved “continuous” or “parameterized” versions of the theorem. Analogous theorems for actions of the groups G=S1 or S3 have been proved in [11], and [12]; compare also [4, 5, 6].

The following is an easy method of representing geometrically the Sextic Covariant of a Binary Quartic which I have not seen given elsewhere. It was suggested to me by my work on Bi-Circular Quartics. The usual method of representing this Covariant is to regard the Quartic as four points on a conic, when the Sextic is represented by the intersections of the conic and the sides of the Harmonic Triangle of the Quadrangle whose vertices are the points of the given Quartic. The present method has the advantage of regarding the given Binary Quartic as a Tetrad of points on a straight line.

Let R be an associative ring with 1 and G a finitely generated torsion-free abelian group. In this note, we classify all R-automorphisms of the group ring RG. The special case where G is infinite cyclic was previously settled in [8], and our interest in this problem was rekindled by the recent paper of Mehrvarz and Wallace [7], who carried out the classification in the case where R contains a nilpotent prime ideal.

A short proof of Chow's generalization of Widder's inequality and generalizations of some related Hardy's results are given.

The differential equation in question is of the second order and has three regular singular points. It is usually denoted by

where a, b, c are the singular points, a, a′ etc. the exponents at those singularities. The solution when no one of the numbers α – α′, β – β′, γ – γ′ is an integer or zero is well known; all types of solution are expressible in terms of the hypergeometric functions.

Let M and N be simply connected space forms, and U an open and connected subset of M. Further let π: U → N be a horizontally homothetic harmonic morphism. In this paper we show that if π has totally geodesic fibres and integrable horizontal distribution, then the horizontal foliation of U is totally umbilic and isoparametric. This leads to a classification of such maps. We also show that horizontally homothetic harmonic morphisms of codimension one are either Riemannian submersions modulo a constant, or up to isometrics of M and N one of six well known examples.

The following arrangement of the proof of this theorem could, I think, be given at a comparatively early stage, even if the necessary case of De Moivre's theorem had to be proved as an introductory lemma.

Let u, v be two rational integral algebraic functions of x, y with real coefficients, and let c be a simple closed contour in the plane. As the point (x, y) travels round c let those changes in the sign of u that take place when v is positive be marked and let (u, v; c) denote the excess in number among these of changes from + to − over changes from − to + *.