There is no need to recommend to teachers in this country, familiar with Dobbs’ School Course and with later books of the same tendency, the free use of the ideas of displacement, rotation, and reflection. In the course of a paper read at the Zürich Congress last year, Prof. Thomsen, of Rostock, gave some striking examples of the effective introduction of the language and notation of the theory of groups into work of this kind, and afterwards he agreed willingly to contribute to the Gazette an article on the subject. For applications to their own problems we must refer readers elsewhere, since Prof. Thomsen has paid them the compliment of explaining a point of view that would be quite unsuitable in the school. To some of our readers we must apologise for replacing the original article by a translation, but since there are no familiar symbols or recognisable formulae to facilitate the grasping of a novel theory, we think that a substantial minority will be grateful for this assistance. In making his version, Prof. Neville has taken one or two liberties with the author’s notation.

I. Let BOY and COZ be the bisectors of the angles B and C of the triangle ABC

Draw OH, CH parallel to AB, OB respectively; OK, BK parallel to AC, OC respectively

Statics is founded upon two outstanding principles the Parallelogram of Forces and the Principle of Moments. The Parallelogram Law is universally acknowledged as fundamental and is given its rightful place in both analysis and graphics; in the former it furnishes two equations connecting the forces keeping a body at rest while the Principle of Moments furnishes a third equation. In graphical statics the Funicular Polygon is used as the equivalent of the third equation derived from the Principle of Moments.

The transformation y=f'(x) on the roots of

is intersting to study on account of the fact that it leads to a transformed equation whose constant term is, the discriminant of f(x)=0. It is easy to see that this property holds, for, writing f(x) in the factored form

1. An article by my friend, Mr. R. J A. Barnard, of this university, in the October number of the Gazette, impels me to join in the discussion of which it is a part. For, while the article in itself is an interesting application, to “constant coefficient” equations, of methods of wider applicability, the thesis of “avoiding” what is plainly the right method for such equations is, I feel sure, a mistaken one: it recalls the fin de siècle game of “dodging the calculus” (from which we seem now to be swinging to the opposite extreme—of teaching infinitesimal calculus without sufficient regard to the requisite maturity of mental development of the pupil).

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.