Ann Hirst and Keith Lloyd have derived many interesting properties of the Cassini ovals rr′ = a2, r and r′ being the distances of a variable point P from two fixed points A and B, which we shall find it convenient to take a unit distance apart. r, r′ are often referred to as bipolar coordinates and by relating them in different ways, a variety of interesting families of curves can be defined; such families can be associated with other families which are their orthogonal trajectories. The purpose of this article is to draw attention to a number of such cases.

Several societies have during the past few years considered the teaching of mathematics and of science ; since, however, they have confined their attention to the subjects separately, and in many instances to particular types of school, the Joint Committee decided not to rely upon data previously collected, but to send a circular to all the schools mentioned in the School-marters Year Book, asking for details of the conditions under which the subjects are at present taught. This has been done and replies received from about 300 schools representing all types of non-primary education: a summary of the replies will be found in Appendix B, and the Committee wish to express their thanks to those gentlemen who, by so fully answering the questions asked, have made it possible to present a report dealing with a larger number of schools than has been hitherto possible.

The Committee wish to make several recommendations ; they however recognise with pleasure that the returns prove not only that a considerable advance has been made during the past ten years in the teaching of mathematics and science, but that in many cases a definite attempt has been made to correlate this teaching.

Although the approximation of elementary functions is a well-researched area of mathematics, very little has been published on built-in routines in any particular calculator or computer. Computer manufacturers are, understandably perhaps, reluctant to publicise details about their algorithms. Hewlett-Packard have been less secretive than most and allowed an occasional glimpse behind the scene in their own Journal and various other sources. The following square root routine for one of their machines, the now extinct 2000F series, has appeared in an Open University text on numerical computation [1]. The process of evaluating √x is carried out in four steps:

The following notes arose from a talk given by Professor Halberstam to the South West Wales branch of the Association. In the marriage bureau problem we imagine that n boys and n girls are to be paired off in marriage, that we are able to establish preference lists for each boy of all the girls and vice versa. These preference lists may be displayed in a table such as that given below, which shows the rankings for four boys α, β, γ and δ, and for four girls A, B, C, and D. The boys’ rankings are the first members of the number pairs, in rows, and the girls’ are the second members, in columns.

At the time of writing, second class stamps cost 12p and first class 17p. To minimise the amount of change in my pocket I have a preference for buying these stamps in ratios that cost a whole number of pounds.

In 1683, the German mathematician Ehrenfried Walther von Tschirnhaus introduced a polynomial transformation which, he claimed, would eliminate all intermediate terms in a polynomial equation of any degree, thereby reducing it to a binomial form from which roots can easily be extracted [1]. As mathematicians at that time were struggling to solve quintic equations in radicals with no sign of any success, the Tschirnhaus transformation gave them some hope, and in 1786, Bring was able to reduce the general quintic to the form x5 + ax + b = 0, even though he didn't succeed in his primary mission of solving it. It seems Bring's work got lost in the archives of University of Lund. Unaware of Bring's work, Jerrard (1859) also arrived at the same form of the quintic using a quartic Tschirnhaus transformation [2]. From the works of Abel (1826) and Galois (1832) it is now clear that the general polynomial equation of degree higher than four cannot be solved in radicals.