This article contains a discussion of the problem considered by Routh and other writers under the general heading “Rocking Stones”, but is restricted to cases which can be treated as problems in two dimensions. Thus Routh writes: “A perfectly rough heavy body rests in equilibrium on a fixed surface; it is required to determine whether the equilibrium is stable or unstable. We shall suppose the body to be displaced in a plane of symmetry so that the problem may be considered to be one in two dimensions.”

Mr. K. S. Snell (Harrow), in opening the discussion on the syllabuses suggested by the Cambridge Advisory Committee, said : Schoolmasters are apt to criticise heavily examinations and examination boards. We now have an opportunity of doing the reverse. A committee has been formed with the object of producing, and keeping up to date, an agreed syllabus for Sixth form work. This was brought about by the initiative of the Cambridge Syndicate, and in particular of Mr. Brereton, whom we are glad to welcome here to-day. I am sure I am expressing your opinion when I say that this Association welcomes the first report of this Committee, and is grateful for being offered permanent representation on the Committee. We must hope that other examining boards will consider the report with care, and will be prepared to come in line with it, for the advantage of any syllabus is so greatly enhanced if it can be one that is agreed by all boards.

BY a simplifying assumption we express amplitude, reaction, frequency, and change of plate current in triode oscillations in terms of the fraction of the time during which plate current is flowing. The mathematical methods are simple, and were used by E. V. Appleton and B. van der Pol, Phil. Mag., 43, 177, 1922.

A triode is an exhausted glass vessel containing three electrodes: a filament or heater, a grid, and a plate. Electrons emitted by the heater are caught on the grid and plate in numbers depending on their potentials, and constitute currents from the leading-in wires towards those electrodes, called the grid current and plate current. If Vgis the potential of the grid relative to the heater, and Vpthat of the plate, the plate current Ipis theoretically and experimentally a function f(Vp + μVg) of Vp + μVgover a large range, where μ is a constant called the amplification. Thus the curves of Ipfor different plate potentials Vpare obtained from each other by translation parallel to the axis of Vgby an amount proportional to the change of Vp. The grid current is much smaller if the grid potential is not too high.

1. The discussion of the motion of bodies of varying mass recently revived by Mr. A. S. Ramsey's article in the Gazette for July 1941 has produced many sidelights on the momentum method of treating these problems, but not one of your correspondents has suggested the alternative approach by the method of energy. This method was shown to me (and, if I am not greatly mistaken, also to Mr. Ramsey) by one of the ablest, if least known, of the last generation of Cambridge teachers of mathematics, William Welsh of Jesus, but it does not yet seem to have reached the textbooks. It has one great advantage over the momentum approach in that it forces attention on the physical assumptions involved, the slurring of which is the main cause of the difficulty experienced by many students in applying the equations of momentum.

The urgent necessity for examining thoroughly the content of the Mathematical Curriculum in all types of schools at the present time is my reason for introducing a subject which is becoming a well-worn one, and I hope that in the course of the afternoon I may be successful in showing you what I believe to be a fresh and more fundamental approach to the problems involved, and that by being provocative I may invite a discussion on a very important question.

In the preparation of mathematical diagrams for reproduction, it is often necessary to draw an ellipse or parabola with considerable accuracy. The methods described below need only a ruler, compasses and a set of French curves ; they produce very satisfactory results with a minimum of construction.

The process consists in setting out, by simple geometrical constructions, short circular arcs of curvature and then filling in the spaces between them with the aid of French curves. It is best to do the actual constructions in pencil and to ink the curves in afterwards with a drawing pen and French curves.

BROWSING for amusement in a book that I never saw at school, Casey's Sequel to Euclid, I was introduced to something new to me. It was my first encounter with harmonic polygons, and it was interesting to trace how the painstaking geometers built up a series of propositions applicable to such figures. What annoyed me, however, was the attempt to derive some of their properties by “inversion” of a regular polygon. Again it seemed illogical to build up a proof of general properties by proving a succession of mensurations. This feeling was intensified when it came to deal with the “Brocard” properties and I started to build up the theory from a new angle.

The series

is the simplest known series * which could be used to compute π, but it converges far too slowly to be of any practical use for this purpose. In this paper we prove an asymptotic formula for the remainder of the series (1) after a finite number of terms. The formula is

where

If white light falls on a soap bubble or on a film of oil on a pool of water, the reflected light is very often coloured, due to interference at the surface of the film. Within the last few years fresh interest has been added to the phenomenon by the work of John Strong, Katherine B. Blodgett and others, who have shown that interference films can be deposited on glass and that these films reduce the reflection from the surface and increase the transmission through it, a matter of great importance in optical instruments. In many of the writings on the subject, the intensity of the reflected beam only has been considered, the transmitted beam being dismissed with the statement that, by the Conservation of Energy, light which is not reflected is transmitted. In some of the writings, only perpendicular incidence has been considered. In the discussion below expressions are obtained for the reflected and transmitted beams without the use of the Conservation of Energy for all angles of incidence.

The polar equation to any curve can be written r=f(θ).

If P and Q are inverse points with respect to a circle whose centre O is at the pole and whose radius is k, then, since OP OQ = k2, it follows that the polar equation to the inverse curve is k2/r=f(θ).

It is proposed to find by inversion some properties of the limaçon and the cardioid that follow from certain well-known properties of the conies : others will suggest themselves. For convenience the same letter is used to denote both a point on the original figure and the corresponding point on the figure obtained by inversion.