The paper gives an exact evaluation of the sum of the integral parts of the terms of an arithmetical progression, with application to the checking of a table of rounded-off multiples of a constant.

Typical cyclic configurations of five-bar linkages are shown in Figs. 1, 2, 3, 4. It is proposed to solve the problem of finding all the cyclic configurations of a five-bar linkage of which the lengths of the bars are given. It will be shown that the solution depends upon an equation of the seventh degree which can be expressed in a very simple form.

1. In a recent issue of the American Mathematical Monthly, L. Moser proved the following result:

If an enthusiastic problemist proposes at least one problem every day, but never more than 730 in any calendar year (not even in a leap year), then, given any positive integer n, there is some set of consecutive days in which he proposes a total of exactly n problems.

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).

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.

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.

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.

In her paper (Gazette, May 1950, p. 94) on the roots of two types of determinantal equation, M. J. Moore obtained various results by elementary but somewhat complicated methods . In this note, these results are obtained by a simpler method, which moreover helps to exhibit their real significance. The essential facts are expressed by the following theorem:

THEOREM I. If a rational function of x has poles and zeros (including a possible pole or zero at infinity) which are all real, simple and strictly alternate, then its partial fraction expansion

has coefficients kr (including k∞ if non-zero) all of one sign, while its reciprocal has expansion coefficients all of the opposite sign.

In this note I give the important trigonometric relations between the Brocard and Steiner angles of the cross-section of a triangular prism and the angle which the plane of this cross-section makes with the plane cutting the prism in an equilateral triangle. To facilitate a comprehension of the whole problem, I also give a complete and brief solution of the known problem for the equilateral section of the prism, deducing the equations of the equilateral case from the equations for the general case in which the section is a triangle similar to a given triangle.