It is usually considered that the solution of isoperimetric problems is only to be achieved through the medium of the more or less advanced technique of variation calculus. The following theorem is derived using only the notions of pure geometry and elementary calculus, and is interesting since the method of proof is easily extended to several other problems of the same type.

1. This perennial question has come up, once again, in a Note (No. 1805, February 1945) by the President for the year.

The object of this contribution is to try to state something like a full case (the writer seems to have been working at this for most of his academic life) for the traditional view of the subject, for which the President contends : while at the same time indicating the complementary importance of the other view. (As in all such cases, strong difference of view by competent authorities implies important truth on both sides (which it is necessary to synthesise; and this is, in fact, a fundamental mathematical case of the kind).)

As usual we define the p-multiplied L.T. of f(t) to be (1) and we adopt the abbreviation (2) introduced in 1938. The theorem in § 2 was obtained by the author in 1940. It would have been published, together with other work on L.T., some years ago, but for war-time restrictions.

Most of the tractable problems pertaining to submarine cables and electrical transmission lines have been solved. However, no result seems to have been given for the current at any point of a very long “loaded” cable with a series capacitance between it and the sending battery. By aid of a general theorem published recently in the Mathematical Gazette, the required result may be obtained without difficulty.

What is centrifugal force? On selecting at random about a dozen elementary textbooks on Mechanics to find a definition, I was pleasantly surprised to discover (unless I have inadvertently missed it in any of them) that only four of them mention the name “centrifugal force”, and two only of these four mention “centripetal force” as well. No doubt if I had had easy access to textbooks written for and used by engineers, the results would have been different; but as I am concerned only about mathematical students, I have not troubled to pursue my search except in the kind of textbooks likely to be used in schools which teach elementary mechanics as a branch of mathematics.

The idea of the “reversed effective force” or “centrifugal force” has been widely used to facilitate the solution of certain types of dynamical problems concerning the motion of particles. But the method can also be applied, as will be shown below, to investigate the constrained motion of a rigid body.

The locus of the centres of spherical curvature of a singly infinite family of geodesics which pass through a regular point O on a surface S, one in each direction in the tangent plane there, is, in general, a twisted curve. It will be proved that the only real surfaces, at all points of which (excluding umbilics) this locus is a plane curve, are quadrics of revolution and general cylinders.

Although it is difficult to think of a rational reason for being more interested in, say, the square number 698896 than in the square 700569, or for that matter in 4 or 49, there is apparently some psychological reason, and the pursuit of square numbers of particular digital forms has a long history. The publication of the attractive collection of digital curiosities in Note No. 1823 has suggested to me that there may be some interest in further figures from discoveries I have made in analogous fields. References in a few cases indicate previous publication in scattered problem columns.