It is well known that the functions x, 1 — x, 1/(1 — )x, x/(x — 1), 1 — 1/x, and 1/x form a Group, the rule of combination being substitution of one function in another. The question arises what other sets of functions there may be with a similar property. This note gives one answer to this question.

It is customary in an account of Riemann integration to give one or two examples of the evaluation of definite integrals directly from the definition of the integral in terms of Riemann (or Darboux) sums, but the method is used only to illustrate the definition, and is never fully exploited. The method is, however, capable of a more systematic treatment than one might at first expect, and in this note we show how it can be applied to some of the elementary functions of analysis.

Every migrant should realise that migration does not only mean moving from one country to another, but also assuming the ways and customs of the new country Some of these, especially those dictated by the climate, are easily assumed, others are very hard to accept.

Recent notes in the Gazette (Vol. 45 (1961), 17-23, 23-25) on the subject of change of variable in a definite (Riemann) integral have prompted me to write about the same subject for an indefinite integral, since it has often seemed to me that the conditions under which this rule is proved are, in a sense, the reverse of those that are needed when it is used in practice.

The problem I am going to discuss is the generation of the class of all mappings of the space S×S×.. × S, with any number of factors, into S itself, from a single mapping of the class, where S is a finite set. The problem has its origin in the study of truth tables and it is in this setting that the problem may be most simply presented.

In an article entitled “Two Problems on Impulsive Motion” (M. G., XLIV, p. 95) the writer (J. O’Keeffe) complains of the lack of method in elementary books on problems such as the initial motion of two jointed bars under the action of an impulse. In our view, such problems are best treated, in an elementary course, by reducing the total momentum of the system to a single linear momentum and a momentum-couple. The necessary equations are then obtained by using the principle of equivalence of this system with that of the impulses. This treatment makes it clear why and in what form it is correct to take moments about the point at which the bars are hinged. Below we give the general treatment, and also the application to this particular problem.

I have in my possession a school exercise book written by John Stewart between January 1801 and 29th June 1802, while he was a pupil at Dundee Academy, later to become the present High School of Dundee. The book runs to 140 pages covering very much the same course as that outlined by T M. Flett in the Gazette of Feb., 1961. Rules are in the identical words repeated by Edward Winder 25 years later