In teaching algebra, it is obviously desirable to prove the Binomial Theorem for a real (or at any rate for a rational) index without assuming certain theorems on convergence which are used in Euler’s proof. So far as I know, no one has thought it worth while to devise such a proof.

The game of Banker’s Clock provides an interesting question in mathematical probability In this game the banker turns up in sequence the first twelve cards of a well-shuffled ordinary pack of 52 cards. He backs himself to turn up at least one card of which the face value corresponds to its position in the sequence, an Ace ranking as one, a Jack as eleven and a Queen as twelve. The interest in the question from the mathematical point of view is in finding the probability that the event will happen.

“ … And so the impulse of the blow is equal to the change of momentum of the body.”

This phrase, coming at the conclusion of a more or less standard lesson in Elementary Mechanics, must surely have struck many teachers with a feeling of shame for having participated in a piece of not entirely straightforward work, and their pupils with an idea that they have witnessed some downright “wangling”. The usual impulse-momentum treatment cannot but be unsatisfactory to all concerned in the lesson, for it follows on the introduction of a new quantity of measurement of which, as so many of the text-books plaintively remark, “no special name has been given to the unit”.

The problem under discussion is the relation between the behaviour of a function ɸ (t) near a particular point t = x and the behaviour of the partial sums sn of its Fourier series at that point. We start from Hardy and Littlewood’s theorem † that if ɸ(t) tends to a limit in the Ceaáro sense as t → x, then the Fourier series at that point is summable in the Cesàro sense, and conversely. The exact order of the Cesàro means in each case is unspecified.

The purpose of this article is to describe briefly some investigations of the properties of certain integrals associated with the Riemannian manifold of an algebraic variety, which, besides having an interest of their own, appear to be a fruitful source of new properties of the more classical Abelian integrals usually considered in connection with algebraic varieties. The investigations are still at a very early stage, and up to the present only deal with the Abelian integrals of the first kind (those which are without singularities), and in what follows we shall limit our attention to these.

At a summer school for teachers held at Oxford in 1924 Professor Neville delivered a lecture with the above title. Some explanation of the publication of a ten-year-old manuscript may be desirable. The address deals with a topic of enduring interest, and, in addition, the editor feels that, since members of the Association have been asked to consider this subject in view of the approaching revision of the Report of 1923, the opinions of the Chairman of the Committee which drew up that Report will be particularly valuable at this time.

In connection with n regular polygon of N sides, many neat and interesting problems cnn be set, and it is the object of the present article to focus attention on theie in order that the complete solution of all these problems in the case of an even-sided polygon when N is a mllltiple of 3 may be completely determined.

Trigonometry , like most abstract sciences, has both an operational side and a quantitative side.

The operational side is dominated by the hypothesis that a function f(x) exists which has the property

It is from this that the usual rules for operating with the trigonometric functions are deduced.