1. Whence these “products”? Why one a scalar (actually a real number—see below) and the other a vector? These are interesting questions, if once we shake ourselves free from inherited knowledge of the facts and become curious as to the why and the wherefore.

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.

The interest of pure mathematicians is nowadays shifting from the study of special functions to general function theory. In fact the study of special functions might ultimately be forsaken, as for example in geometry the study of the triangle is now virtually a closed chapter, were it not for one important consideration. That is, that these functions must always be in constant use in mathematical physics. Hence their properties tend to become questions of practical application rather than of theoretical interest. It follows that there is a demand for some method, even if only a formal one, which will derive the required properties in an expeditious manner, without always calling for the services of the professional pure mathematician.

It does not seem to be generally known that a sundial, which has been correctly constructed and graduated for a particular latitude, can be used in any other latitude if it is set up at the proper angle. This appears from the very simple observation that if a sundial is moved in any way without rotation (of course within a range small compared with the sun’s distance), the shadow of the style will continue to fall upon the same figure of the dial, and if it remains on the same half-meridian of the Earth, where the local time is the same, the dial time will be still the same; when transported to another meridian, without change of latitude, it will give the correct local solar time for that meridian. (In transporting to another meridian, the dial will of course be rotated, so that its inclination and orientation relative to the earth’s surface shall be unaltered.) Thus a horizontal sundial constructed for north latitude ø will be accurate when erected as a prime-vertical sundial, facing north, in south latitude 90° – ø (Fig. 1).

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 this article an attempt has been made to indicate the ordinary methods of integration of the total differential equation

Then the condition for integrability of (1) , i.e, the condition that (1) may possess a single integral equivalent of the form ϕ(x, y, z)= c, is

Considering the pertinent remarks on “Approximations” in The Teaching of Arithmetic in Schools, the steady change of treatment of the subject in text-books and the recent article by Mr. Inman in the Gazette, it seems that considerable independent investigation of this matter has been going on. Also few people will disagree with Mr. Inman when he says that the teaching of the subject leaves room for improvement. That here is matter worth spending time on in the classroom is clear—but the same may be said of many more topics than we can ever have time to deal with. It is in the belief that the subject of the accuracy of figures, or approximations, is one we really ought not to omit that I venture further remarks.

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.

The connection between the fact that 212 – 1 is a multiple of 45 and the angles of pedal triangles may not be obvious but it is there, as Euclid may have remarked when chaffed about his suggestion of a liaison between medial section and the angles of an isosceles triangle.

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.

In the following note, Stirling’s formula for a factorial is first found by a very simple process. The expression so obtained has a meaning for non-integral as well as integral values of the variable. The function it defines is easily seen (§§ 3 ff.) to possess the fundamental properties of the Gamma function, and provides an easy and natural avenue of approach to its study.