The object of the following paper is to illustrate how readily some problems, which frequently occur in physical work, may be solved by the application of certain methods which are not very generally employed in mathematical investigations. The method on which the following proof is based is that of contour lines. These are a device enabling us to represent the third dimension on a plane, and are such that all the points on one contour line are at one and the same height above the level of the base plane. Probably the best known application of the principle of contours is to be found in the Survey Maps, where successive (closed) curves pass through all points at heights of 100, 200, 300, &c., feet above the sea-level; but they are of equal value in many physical diagrams, where they appear as equipotential lines, isothermals, lines of equal illumination, lines of equal force in a magnetic field, as well as in many other forms. In this paper they are employed as representing the third dimension merely. When contour lines are placed so as to indicate an equal rise in the intervals between each successive pair [or, in fact, according to any definite system], we can, with their aid, tell two things about a surface; the one—what is the steepness, or gradient, in any direction from a given point; the other—which is the direction of steepest slope at any point. The absolute steepness is measured by the amount of vertical motion per unit of horizontal motion of a point; i.e., by the tangent of the angle between the horizontal plane and the line in which the point moves. This is a fact with which everyone is acquainted ; every person knows what is meant by saying that the gradient along a line of railway is 1 in 50 or 1 in 500, as the case may be. To find the steepness at any point we merely need to know, in terms of the length of the line taken to represent unit steepness, the length of the line passing through the point in the given direction, and terminated by the two contours on either side of that point. The length of the line will then be inversely as the steepness.

This paper is designed to call attention to certain extensions of this method, which do not seem to be generally recognised. Most writers consider the ellipse alone as amenable to orthogonal projection, whereas all the conic sections are so.

With the hyperbola we may use this method in three ways—

(1) By projecting a given diameter in the figure to be a transverse axis of the projected figure, when the result will follow by symmetry.

(2) By projecting the hyperbola into a rectangular one.

(3) By an imaginary orthogonal projection from the circle.

Throughout the paper, the symbols G1 and G2 will denote two locally compact abelian groups with character groups X1 and X2, respectively. Haar measures on Gj are denoted by μj; the ones on Xj are denoted by θj (j=1,2). The measures μj and θj are normalized so that the Plancherel Theorem holds (see [7, p. 226, Theorem 31.18]).

The inverse scattering problem we consider is to determine the surface impedance of a three-dimensional obstacle of known shape from a knowledge of the far-field patterns of the scattered fields corresponding to many incident time-harmonic plane acoustic waves. We solve this problem by using both the methods of Kirsch-Kress and Colton-Monk.

If a circle meets the sides BC, CA, AB of a triangle (Fig. 6) in the pairs of points L and l, M and m, N and n respectively, it is obvious that the pairs of connectors Mn and mN, Nl and nL, Lm and lM are antiparallel with respect to the angles A, B, C respectively.

A near-ring N is defined to be left bipotent if Na = Na2 for each a in N. Many properties of such near-rings are proved in Section 1, and results of Chandran (4) are generalised. Most of the results are different from, and contrary to, the ring case. Necessary and sufficient conditions have also been obtained under which such near-rings become regular. Section 2 deals with left bipotent near-rings without zero divisors. Some structure theorems for direct sum decompositions and J(N) = (0) are proved and it is shown that for a left bipotent S-near-ring, the singular ‘set’ S(N) = 0. Necessary examples and counter examples are supplied.

Whittaker's contributions to algebra are not numerous and are confined to special problems ; in one or two cases they are rediscoveries, of the kind that add a certain illumination to the original. His outlook on algebra was that of his time ; one might characterise it by referring to the spirit and content of Perron's Algebra, as contrasted with the Moderne Algebra of van der Waerden. Bred in the Cayley-Sylvester tradition of matrix algebra and the invariant theory of forms, he was expert also in the manipulation of continued fractions and determinants. His lectures to honours classes in Edinburgh always included a course on matrices, vector analysis and invariants, historical reference being made to Grassmann, Cayley and Sylvester; it is to such a course that the writer owes his first acquaintance with matrix algebra. In later years, from about 1925, Whittaker acquired an adequate knowledge, though never a marked taste, for those parts of modern abstract algebra, in particular the representation of groups and algebras, that serve a purpose in mathematical physics; his courses of research lectures, which he kept up until a late period in his tenure of the chair, bore witness of this.

George Alexander Gibson was born at Greenlaw, Berwickshire, in 1858. His father was a man of rare capacity and character who wrote a history of Greenlaw and taught himself Latin so as to be able to obtain the material for his history from original documents.

After attending an elementary school at Greenlaw, Gibson enrolled in 1874 as a student at the University of Glasgow, where he gained the highest places in all the classes of the Arts curriculum. At this period of his life his health was not robust and it threatened to terminate his university studies. Consequently, in 1881, he took the degree of M.A. without honours. In the following year he was able to sit the honours examination in Mathematics and Natural Philosophy, and he so distinguished himself that he was awarded the Ewing Fellowship. When the degree of D.Sc. was afterwards insti tuted and the regulation required that a candidate for it should have an honours degree in Arts or Science, his ordinary degree precluded Gibson from becoming a candidate for the doctorate which his published work would doubtless have gained for him.

Throughout, A denotes an associative ring with identity and “module” means “left, unitary A-module”. In (3), it is proved that A is semi-simple, Artinian if A is a semi-prime ring such that every left ideal is a left annihilator. A natural question is whether a similar result holds for a (von Neumann) regular ring. The first proposition of this short note is that if A contains no non-zero nilpotent element, then A is regular iff every principal left ideal is the left annihilator of an element of A. It is well-known that a commutative ring is regular iff every simple module is injective (I. Kaplansky, see (2, p. 130)). The second proposition here is a partial generalisation of that result.

Let K be a field of characteristic p>0, G a finite p-solvable group, P a p-Sylow subgroup of G of order pa, KG the group algebra of G over K, and J(KG) the Jacobson radical of KG. In the present paper we study the nilpotency index t(G) of J(KG), which is the least positive integer t with J(KG)t= 0. Since J(EG) = E⊗KJ(KG) for any extension field E of K (cf. [7, Proposition 12.11]), we may assume that K is algebraically closed.

For R ∈ {bv0, c0, ℓ∞} a multiplier of FK spaces, the classical sectional convergence theorems permit the reduction of R to any of its dense barrelled subspaces as a simple consequence of the Closed Graph Theorem. (Cf. the Bachelis/Rosenthal reduction of R = ℓ∞ to its dense barrelled subspace m0.) A natural modern setting permits the reduction of R to any of the larger class of dense βφ subspaces. Bennett and Kalton's FK setting remarkably reduced R = ℓ∞ to any of its dense subspaces. This extreme reduction also obtains in the modern βφ setting since, surprisingly, every dense subspace of ℓ∞ is a βφ subspace. Moreover, the standard results, including the Bennett/Kalton reduction, easily follow from their βφ versions and the Closed Graph Theorem. Our two supporting papers find relevant “Non-barrelled dense βφ subspaces” and study “Generalized sectional convergence and multipliers”. Here we specialize the βφ approach to ordinary, particularly unconditional, sectional convergence.