The article by C. C. Puckette in a recent issue of the Gazette (Vol. 37, No. 322, pp. 256–260) prompts me to give some results I obtained when investigating the pursuit of a bomber by an attacking fighter.

In this paper, we consider a sequence lr(x) (r = 1, 2, ..) of rational functions of a positive rational variable x, which approach log x as r→∞. Some properties of these functions are analogous to properties of the natural logarithm, and go over into the latter, when the limit r→∞ is taken. Except for pointing out such limiting results, we shall avoid transcendental methods.

Recently a good deal of interest has been taken in the properties of numbers which can be expressed and proved without going beyond the number 1 and the Four Rules, in other words by using only rational numbers. Professor T. S. Broderick has established various theorems about the number and distribution of primes, using this “sub-language”: (Journal of the London Mathematical Society, Vol. 14, pp. 303–310; Proceedings of the Royal Irish Academy, 1940, pp. 17–24).

To attach, in relation to a curve f(x, y) = 0, a geometrical significance to the value of f(x, y) at an arbitrary point of the plane, it is necessary to impose a condition which has the effect of standardising the function f(x, y) by means of some divisor independent of the variables. This article, another outcome of Mr. Durell’s query on the semi-cubical parabola, is concerned with normalisation of the equation of an algebraic curve, and with Laguerre’s interpretation of a standard form of f(x, y). The axes are rectangular throughout.

Cyclically repeating patterns are a common feature of mathematical films. The study of the symmetries which arise in such configurations has a theoretical interest to mathematicians watching the film, and it also has a practical interest to the film maker, because the symmetries present in a diagram can often be used to reduce the labour of drawing. It does not seem that these symmetries have been commented on before, and in this article we attempt a complete enumeration of the cases which can arise.

We describe a variant of the game with digits discussed in a previous article. There we considered a transformation of any number in a scale s + 1 in which each digit is replaced by a digit equal, modulo s, to the sum of the remaining digits. Starting for instance with the number 16427, in the scale of 10, the first digit remains 1, since 1 = 6 + 4 + 2 + 7 (mod 9), the second digit is replaced by 5, since 5 = 1 + 4 + 2 + 7 (mod 9), and so on, forming the number 15794; repeating the transformation we obtain in turn 73184, 72461, 49751, 48137, 16427, the original value recurring after six transformations. We showed that under this transformation any number x, with m digits in the scale s + 1 (none equal to s) recurs unless m − 1 has a common factor (greater than unity) with the quotient of sm by the highest common factor of sm with the sum of the digits of x.

It was conjectured by David Gregory that a sphere can touch thirteen non-overlapping spheres equal to it. In a recent paper proofs have been given that this in fact is impossible. In the present paper I outline an independent proof of this impossibility, certain details which are tedious rather than difficult being omitted.

A series of investigations is being planned by a section of the Board of Studies for Mathematics in the Institute of Education of the University of Birmingham, with the object of gaining information about the types of difficulty experienced by children in various branches of grammar school mathematics, and the types of error made. An analysis of these errors, both from the mathematical and psychological points of view, is highly informative and relevant to a discussion of teaching methods in the subject. It is hoped that later it will be possible to prepare standardised tests which would be of diagnostic value to teachers. The co-operation of other teachers would be welcomed in furthering this investigation, either in the form of comments or criticisms in the light of their own experience, or in helping to widen the sample of children tested in future work.

Through the initiative of the senior mathematics master, Mr. R. D. Knight, a simple Bush Differential Analyser, was built almost entirely from “Meccano” parts at Kings School, Worcester in the last year.

The Farey series Fn consists of all the proper fractions, in their lowest terms and in order of magnitude from 0/1 to 1/1, whose denominators do not exceed n. The series Fn may be formed from Fn–1 by inserting the mediant (a + c)/(b + d) between two adjacent terms a/b and c/d of Fn–1 provided b + d ≤ n. The process requires inspection of terms and rewriting of earlier series. Professor Neville showed (Proc. London Math. Soc., Series 2, Vol. 51), how the order n may be increased approximately in the ratio 3 to 2 at one time, but inspection and repetition will still be necessary.

In a recent book review in the Mathematical Gazette it was stated that the idea of a unified syllabus in mathematics had proved less helpful than had been hoped, and it was implied that many teachers were abandoning the idea of a unified syllabus, or were compromising with a system of partial unification. This was only one of a number of articles and reviews in various periodicals, all noticeably tepid in their enthusiasm for the new ways, which must cause some dismay to all those who have hoped for much good from this newest phase in the development of mathematical teaching. It is, of course, hardly surprising that so momentous a step should have given rise to a certain amount of controversy: the dethronement of Euclid was the cause of a great deal of fierce argument as well as of the birth of the Mathematical Association, so that the establishment of a unified syllabus, another and greater step along the same road, must be expected to receive an equally mixed reception. Nevertheless, if the unified syllabus has indeed been found unsatisfactory by some teachers, it would be well to try to find out what is wrong.