Since about 1939 my researches have been, in collaboration with Professor J E. Littlewood, mainly in a field of mathematics sometimes described as non-linear mechanics, but as most of the equations in rnechanics are non-linear, it is better described as non-linear vibrations or oscillations. It is rather a curious branch of mathematics developed by different people from different standpoints, straight mechanics, radio oscillations, pure mathematics and servo-mechanisms of automatic control theory The notation varies from equations in v for voltage to θ for the angle of the pendulum, R for the radius of a gas-filled cavity, and x and y of pure mathematicians, with parameters ranging through the Greek and Roman alphabets.

When the A.I.G.T. was founded in 1871 its committee was really a teaching committee. By 1878 sub-committees had been appointed to consider different branches of Geometry; in 1884 sub-committees were appointed to consider Arithmetic and Mechanics. But it was in 1902 that the Mathematical Association first appointed a committee described as a “Teaching Committee”. What was it that led up to the appointment of this teaching committee?

Finite Galois arithmetics are well-known; finite geometries however, though more interesting to the amateur, have not really acquired professional status and do not appear to any great extent in standard works. The following example arose from a chance remark in Mathematics for T. C. Mits, by L. R. and H. G. Lieber; from this I deduce that a full theory has been worked out, but I have not seen it, and as far as I am concerned what follows is original, and I hope readers of the Gazette may find it new and stimulating.

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.

Introduction, (a) Let M, M’ be two isogonal conjugate points with respect to a triangle (T) = ABC: MP, M’P’; MQ, M’Q’; MR, M’R’ the perpendiculars from M, M’ upon the sides BC, CA, AB of (T), respectively. The triangles (M) = PQR, (M’) = P’Q’R’ are the pedal triangles of the points M, M’ for (T), and the common circumcircle (L) of those two triangles is the pedal circle of M. M’ for (T) [1, pp. 238–242].

If one is concerned merely with the magnitude of the bending moment or the shear, the sign convention to be adopted is of minor importance. When, however, the question of deflection arises, it is essential to obtain the correct sign for the moment and it is therefore desirable to use a convention which not only inspires confidence in obtaining the correct sign but which is also simple to apply.

Among many engineers at the present time the hogging moment is taken as the positive moment. There is much to be said for adopting this convention. It fulfils the conditions mentioned above and provides an easy, and, from the students’ point of view, a natural approach to the subject.

It is well known that, as regards moment of inertia about any line in its plane, a uniform triangular lamina may be replaced by a set of three particles, each with mass equal to one-third of the mass of the triangle, and placed at the midpoints of the sides. The usual proofs given for this elegant result are rather cumbersome. Now it takes only a few seconds to verify that the set of particles has the same moment of inertia as the triangle about each of the three sides (using the h2/6 formula for the square of the radius of gyration). It is tempting to ask whether the equivalence of particles and lamina may not be deducible from this. In other words, starting from the fact that the equivalence holds for three particular lines in the plane, can it be deduced for all lines in the plane?

The centenary of the birth of Oliver Heaviside last year has been the occasion of celebration by electrical engineers and physicists in this and other countries. In the discussions of his work much has been said about the Operational Calculus ; and as the versions of its history which have been given both in these commemorative celebrations and in most of the textbooks of the subject are seriously incorrect, this may serve as an occasion to recount that history more correctly. The story in widest circulation is that the Operational Calculus was discovered by Heaviside (Boole being sometimes—and incorrectly—named as the discoverer of its applications to ordinary differential equations) and rejected by British mathematicians because of Heaviside’s lack of rigour. The facts, as I shall show, are that the Calculus was well known in Britain and France before Heaviside’s birth, and that the rejection of his paper had nothing to do with his use of symbolic methods.

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.

Consider a rigid body having one point O fixed, which is acted upon by a set of forces Fs (s = 1, 2, ..., n). Then the equation of motion of the body is given by

where H is the angular momentum of the body with respect to the point O and rs = OPs, Ps being the point of application of Fs. A similar equation holds in the general motion of a rigid body if H denotes the angular momentum with respect to G, its centre of gravity.

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.