Introduction. If we call the number pa + qb the Tp,q transform of the number n = 10a + b, then in general the Tp,q transform of n will differ from n; it easily seen, however, that given any number n, we can always find a pair p, such that the Tp.q transform of n is n itself. In fact, if n = 10a + b and p = 10 − b q = a + 1 then the Tp,q transform of n is (10 − b) a + (a + 1)b = 10a + b = n.

There is a problem which is much in favour at present of which one version is this : a man has 12 pennies among which there may be a counterfeit coin, which can only be told apart by its weight being different from the others. How can one tell in not more than three weighings whether there is a counterfeit penny, if so which one it is, and whether it is heavier or lighter than a normal penny? (Math. Gaz., XXIX (1945), pp. 227–229; XXX (1946), pp. 231–234.) This puzzle seems to have originated in America. The purpose of the present note is to make clear what is the best possible solution in the most general case of any version of the problem. No originality is claimed for these solutions, as no doubt many of them have been obtained independently many times before : but I am not aware that any complete and systematic account has yet been published.

Much progress in Science and Technology has been made from applications of the linear differential equation with constant coefficients. Extensions of the theory have made it necessary to consider cases when the coefficients are no longer constant, but are variable. The present inquiry is to investigate one of these applications.

I WOULD like first to thank the Association for the honour they have done me in inviting me to lecture at this meeting. It is, however, an honour which carries with it a certain danger. When a mathematician comes to lecture on a subject in which he has specialised, he is always liable to suffer from the delusion that what has gradually become plain and straightforward to him will at once appear plain and straightforward to others. It may be that I am suffering from this delusion in thinking that I can present the main idea of the geometry of numbers in one lecture.

It is the intention of this paper broadly to survey present possibilities of work for honours graduates in mathematics. Detailed information, which obscures the picture and soon becomes out of date, will not be given.

Mary Rose was busy with her homework. John, her brother, had gone the Pictures with some friends, and her parents were out as well. So Mary we quite alone. The shades of evening were falling, and the fire glowed cheerful in the grate. But the growing gloom and the soothing warmth made Ma feel rather drowsy. She looked at her book with heavy eyes and ree “Multiply 2–763 by 100”. Then she put down her pen with a sigh, as rested her head wearily on her hands.

Consider a rational (unicursal) plane curve of degree n given by

x : y : z = f(t) : Φ(t) :ψ(t)

In this account of the incorporation of a new subject into the school curriculum I shall assume that it is agreed that the teaching of statistics in schools is desirable. This aspect of the subject has already been fully siscussed at previous meetings of this Association, and in other places. I am one, and I hope there are others present in the audience, who believes not only that it is desirable to teach statistics in schools, but also that far more is gained than is lost by its introduction. In this paper I therefore present the story of one attempt to put into practise what has been preached so long.

Prologue. This article began in an attempt, for purposes which shall be nameless to find the result of inverting a triangle and associated lines and circles with respect to its nine-points circle. The work outran its aim, and I now present it from a quite different starting-point.