Nearly two years ago I was given permission to form a School Mathematical Society, hoping thereby to create a live interest in mathematics and to show that there is a limitless treasure untouched by the demands of the academic courses of the School Certificates. Naturally topics have to be chosen which require little more than a knowledge of School Certificate and Higher School Certificate mathematics, and the Mathematical Society has been limited, at the present, to the Fifth and Sixth forms.

1. This is a sequel to a paper with similar title (Gazette, Vol. XXXI, pp. 270-8), which will be referred to as C.T. (I). It concerns Case I (§§ 8-11). For clearness of diagrams, a new shape of triangle has been used; and, to avoid confusion with previous results, and to suit the present development, different letters and suffixes are introduced.

As H. G. Wells pointed out a long time ago, in the country of the blind the two-eyed man tends to misjudge his neighbours. Relying almost entirely upon his power of vision, he ignores the other senses, and thinks that to be blind is to be mentally defective. In fact, a blind person sometimes has a more balanced mind than the man with keen edges, ten thumbs and ingrowing ears!

The method usually employed for the extraction of square roots involves a series of successive approximations, one for each significant decimal figure of the root. A binomial series may be found which converges quicker than Σ10-n, but this involves a number of separate divisions. Iteration by dividing an approximate value of the root into its square and taking the mean nearly doubles the number of significant figures at each operation. The methods here given converge rather faster than this type of iteration, and enables a root to be expressed as a simple infinite series. They arose from a question put to me by Mr. K. A. Kermack, as to how an ancient Roman could have obtained a good approximation to a square root as a rational fraction. I believe them to be novel, at least in part. It will clearly be sufficient to give methods for finding the square root of a positive integer k.

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.