The contents of this paper will, it is hoped, be found to be of value to teachers of physics and radio principles, to their pupils, and to those engineers who prefer a full understanding of phenomena to a superficial knowledge.

It is, I think, of happy augury that it should have been fourid possible to arrange a joint meeting of members of the local branches of the Classical and Mathematical Associations, as well as members of the Yorkshire Natural Science Association, to hear a few words about Greek science. No doubt the classical or humane studies can stand by themselves and so (in a way) can the study of science based only on good English : but I venture to think that they are both much stronger when they stand side by side and support one another. The idea that thero is any necessary antagonism or opposition of interests between them is surely unfoimded, and there is room for both in our system of education. It is a mistake to suppose that in our school education there is not time for the one or the other ; the guestion is really one of using the time that thero is to the best atlvantage. No doubt schoolboys in the past have wasted much time in the perfunctory learning of Latin and Greek.

When we want to use algebraic methods to discuss metrical properties of the triangle such as Feuerbach’s theorem, Simson’s line, etc., we have the dilemma that Cartesian coordinates confuse the symmetry of the triangle, whereas in areals or other triangular coordinates the metrical formulae are not sufficiently simple. A convenient alternative is to use the method of the Argand diagram: that is, to use complex coordinates (or “affixes”) x ± iy.

If 𝜙(t) is a given function of the real variable t then the integral (in which p is constant)

if convergent, is defined to be the Laplace transform of 𝜙(t). The use of the integral in determining the solutions of certain types of differential equations is now well known; see, for example, Carslaw and Jaeger, Operational methods in applied mathematics (Oxford University Press).

In connection with a general form of the covering principle and the relative differentiation of additive functions Mr. A. S. Besicovitch has proved that given a unit circle C with centre O, then any set of circles satisfying the conditions

1. Each circle of the set meets (or touches) C;

A. 2. Each circle of the set has radius not less than 1;

3. No circle contains O or the center of any other circle of the set, has less than 22 members.

The following question came near to being set in a recent examination: The area of a triangle is calculated from the formula

where 2s = a + b + c. It is known that the measurements of the sides are correct to within 1 per cent. Find the greatest possible percentage error in the area.

The equation to be solved is

where a, b, ƒ are real parameters. (1) is non-linear by virtue of the term by3. Suppose that the free end of a restoring device or spring is fixed to a mass m resting on a frictionless horizontal plane. Let the force-displacement relationship of the spring ƒ1 = g(y), a function of the displacement y. If the mass is driven by an external force F cos ωt, the equation of motion is

or

In (l), g(y/m = ay + by3 and ƒ = F/m. If b = 0, the spring stiffness is ma, a constant, so (3) is linear. When b ≠ 0, the stiffness is not constant, but has the value mdg(y)/dy = m(a + 3by2). This increases or decreases with increase in y, according as b ≷ 0, and (1) is then non-linear. Putting a = (ω2 + λ2), (1) becomes

This equation may be considered to symbolise a simple (linear) mass-spring system of free pulsatance ω, driven by a force ƒ cos ωt – (λ2y + by3).