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.

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?

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.