Although the theory of groups has not proved itself the universal solvent Felix Klein supposed it would become, the group concept is fundamental in algebra and geometry and plays a steadily increasing part in modern mathematics. In recognition of this fact the University of Oxford (perhaps following the lead of her youngest neighbour) has recently (1949) added the theory of groups to the schedule of compulsory subjects for the Final Degree Examination in Mathematics.

Let an ellipse or a hyperbola be given by the equation

Then, following Vaidyanathaswamy [8, p. 124], we shall refer to a hyperbola H having its asymptotes parallel to the principal axes of S as a principal hyperbola of S. H is thus given by an equation of the type

and it is uniquely determined when we know three of its intersections with S. It is one of the purposes of this article to bring together some of the elementary properties of H which have been ignored, quite unjustifiably, by most of the textbooks on conic sections produced after Casey's Analytical Geometry [2]. Another purpose which it is hoped the article will serve is that of bringing within easy reach of students, who have had only a preliminary course in geometries of two and three dimensions, some of Vaidyanathaswamy's investigations into the concurrency of normals to a central conic [6].

The differential equation for y, the lateral displacement of an electrically-driven tuning fork, is

where a>0, c, κ small>0. (cẏ2-2κ), the coefficient in the damping term, is positive or negative according as cẏ2>2κ or cẏ2<2κ. During periodic motion, the coefficient changes sign, such that the inherent loss per period is compensated exactly by the energy supplied from the driving agent. In the language of “electronics”, loss corresponds to a “positive” resistance and energy supply to a “negative” resistance, both of which vary periodically.

Methods of approximation, and successive approximation, are often used in more advanced mathematics, but it is not widely realised that they can with advantage be introduced at quite an early stage in the teaching of school mathematics. This note is a plea that these methods should be taught earlier; indeed it is written in the belief that a wider use of these methods would have not only a salutary effect on teaching and learning alike, but would help to eliminate some of the widespread troubles with which teachers of science are generally confronted and with which they often ask us to help. How often are we told by science colleagues that pupils calculate a physical quantity, for instance, a specific heat, to four decimal places when it is likely to be in error in the secorld place! Similarly those of us who teach practical mechanics, know full well how the resources of four- or five-figure logarithmic tables are invariably used to the full, when the data are correct only to two significant figures.