1. Poncelet’s well-known theorem states that if two conies S and S′ in a plane are such that an n-gon exists whose vertices lie on S′ and whose sides touch S, then an infinite number of such polygons exist. When this is the case, a relation holds between the invariants of the two conies. The explicit determination of this relation is due to Cayley, whose result is quoted by Salmon in a footnote on p. 342 of his Conic Sections (reprint of 1929). With a slight change of notation, Cayley’s result takes the following form.

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.

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.