Historical. That the problems of plane kinematics should have been studied for a long period is natural; they present themselves quite easily, and in simple fashion. At the end of the seventeenth century, and the beginning of the eighteenth century, De La Hire investigated roulettes, and to him also is due the introduction of the inflexion circle, as the aggregate of the points of a plane, subject to coplanar motion, which are instantaneously at inflexions of their paths. At a much later date, Ball (1) drew attention to that one point of the moving plane—upon the inflexion circle—which, in quite general motion, has four point contact with its approximately straight line path; or, has four consecutive positions in line.

Given any skew-symmetric n x n matrix A, we have

det (A - λI)= det (A - λI)′ = det (- A - λI) = (-1)n det (A + λI),

whence we see that the non-zero eigenvalues of A can be arranged in pairs α, - α. Since the set of n eigenvalues of A2 is precisely the set of the squares of the eigenvalues of A, it follows that every non-zero eigenvalue of A2 occurs with even multiplicity, so that the characteristic function ϕ(λ) = det (A2 - λI) of A2, regarded as a polynomial in λ, is a perfect square if n is even, while, if n is odd, then we may write ϕ(λ) =λ{f(λ)}2 for a suitable polynomial f(λ).

Some years ago, under the heading “Introductory theorems in geometrical conics” (Gazette, XVIII (1934), p. 223), an elementary geometrical treatment was offered of some general properties of conics based on the focus-directrix definition

Other definitions of the conic were not considered nor was the relation of the curve to its auxiliary circle introduced.

In this note I discuss the general bilinear transformation

where a, b, c, d are any given complex numbers (with c, d not both zero), and where w, z are complex variables, whose planes are regarded as coincident.

The literature dealing with curves of constant diameter is not very extensive, and the following notes may be of general interest.

If the distance between parallel tangents to any curve is constant, whatever their common direction, then the curve is a corve of constant diameter. Thus, if p (ψ) is the perpendicular distance from the origin to the tangent to the curve, where ψ is the angle the tangent makes with the positive x-axis, and D(ψ) is the diameter of the curve, that is, the distance between parallel tangents, then

If any point P1 is taken on the side A2A3 of the triangle A1A2A3 and P1P2 is drawn parallel to A2A1 to meet A3A1 in P2,P2P3 parallel to A3A2 to meet A1A2 in P3, P3Q1 parallel to A1A3 to meet A2A3 in Q1 and then Q1Q2, Q2Q3, Q3R1 are drawn parallel to P1P2, P2P3, P3Q1 in the same way, finishing with R1 on A2A3, it is easily shown that R1 coincides with P1, as in Fig. 1.

In recent years the people concerned with admission of students to undergraduate courses reported that a good deal of difficulty and confusion had arisen from the multiplicity of applications and from students accepting places in several Universities without notifying the Departments concerned, or withdrawing at the last minute from a place already accepted. In this way places were often kept for applicants who did not turn up in the end and vacancies were left for which suitable candidates would have been available earlier.

In the discussion of this problem various remedies had been suggested, including a deposit to be paid either at the time of application or at the time when a place was offered, but it was felt that this would not be a complete remedy and in addition might, in some cases, cause hardship.

The substance of this note arose out of an illustration in the Eighteenth Yearbook of the (American) National Council of Teachers of Mathematics. On p. 299 of this work, which is full of many good things, there appears a diagram of a number of compound polyhedra. One of them is an icosahedron surmounted by an octahedron on every face, producing a solid which is simple and singly-connected (not counting the faces in common with the icosahedron itself). Since the dihedral angle of the icosahedron is 138° 11′ and that of the octahedron is 109° 28′, the total angle at each edge of the icosahedron is 357° 7′, and the solid thus has narrow fissures between each pair of octahedra. A slight deformation would close these fissures and produce a solid which consists entirely of equilateral triangular faces, which is neither regular nor Archimedean, but can be thought of as an Archimedean icosidodecahedron with pentagonal pyramids indented in its pentagonal faces.

Crawling under a very old six-cylinder car to find out why the self-starter kept jamming, I found that the large gear wheel round the outside of the fly-wheel was worn very badly in three regions spaced at 120° intervals round its outside. Presumably then the engine always stops in one of three positions, and every time the self-starter is used it crashes always into one of these three places. It is an interesting little problem to see just why the engine always stops in these places and to find out mathematically where they are.

Statistics is now obtaining a foothold in the curricula of some sixth forms in schools, and with its introduction as an option into parts of the General Certificate of Education, it is likely to be taught more in the future. One of the great difficulties which is being encountered is the amount of computation necessary in even the simplest applications of the science, and computation unfortunately of a type which is not greatly facilitated by a slide rule or table of logarithms. The bulk of the work, even in advanced applications, very frequently consists of (or may readily be reduced to) the summation of accurate squares and products of two-figure numbers such as form the raw material for applications of correlation, regression, and so forth.

It is nearly a hundred years since Boole published An Introduction to the Laws of Thought (1). An attempt to reduce the art of logical deduction to algebra was a bold venture, and perhaps not all of Boole’s original hopes have been fulfilled. Nevertheless, Boole algebra has become a well-established branch of mathematics which has given rise to theoretical masterpieces such as Principia Mathematica (2), and has found practical applications in electrical circuitry, switching problems, insurance law and statistics.

The fallacies in the traditional approach to the problem of evaluating

in a pre-calculus course have been well brought out in an interesting article by Mr. Turton where the limit is obtained by means of polynomial approximations to the area of a circle. On re-reading the article recently it occurred to me that, at very little extra cost in effort and depth of ideas, a rigorous evaluation of the limit was possible which was completely free from appeals to the physico-spatial properties of geometrical figures and assumptions of symmetry and continuity which Mr. Turton makes.