The study of mechanics and the solution of problems in mechanics are an intrinsic part of the subject of mathematics as taught in British schools and universities; the subject is associated with such names as those of Routh, Lamb, Loney, Love. It is perhaps a peculiarly British tradition, well worth preserving as a discipline of the first order, comparable with Latin and Euclidean geometry; many examination candidates, even perhaps theoretical physicists of distinction, can write learnedly on atomic physics, and yet find the greatest difficulty in solving dynamical problems requiring a firm grasp of principle together with facility in applying it. The present logical unsatisfactoriness of quantum mechanics in certain aspects may be due to want of interest in the foundations of mechanics amongst its exponents.

To the mathematician, at least in the field of elementary mathematics, it is immediately apparent what is true and what is false, and for the teacher of mathematics it is the truth, the correct answer, the right reasoning, that are all-important.

In this article, I propose to examine the other side of the picture, to consider the mistakes made by our pupils and to see how they can be turned to advantage in our work as teachers.

The Sturmian theory is concerned with the number of zeros that a solution of a linear differential equation of the second order may have in a given interval of the independent variable. As usually presented (e.g. Ince (1)) it appears as a very advanced theory, involving elaborate manipulations and reasoning. The purpose of this paper is to simplify the theory by an approach that makes the results intuitive and somewhat reduces the assumptions made.

Some of the properties proved in this essay can hardly have escaped notice, but judging from the lack of reference, they seem to have been forgotten; others are given for the sake of completeness, and a few are probably new. It is hoped that in any case all are of sufficient novelty to be of interest.

The locus of the point from which the tangents to two conics form a harmonic pencil is in general a conic which passes through the eight points of contact of their four common tangents; it is called the harmonic locus of the conics, and it plays an important role in projective geometry.

We say that a bilinear transformation w=f(z) leaves a given locus C in the complex plane “invariant” if C coincides, as a point set, with its image f(C) under the transformation.

The stream function, ψ, is a function specially suited for dealing with two-dimensional flow while the velocity potential, ϕ, is a function which may be used with either two- or three-dimensional flow.

In two-dimensional flow, the chief utility of ψ and ϕ occurs when they satisfy Laplace's equation ∇2ψ = 0 or ∇2ϕ = 0, for then, standard solutions of these equations are known which can be fitted to meet the needs of the particular problem under consideration. The connexion between ψ and ϕ is not clearly set out in aerodynamics books; especially is this the case when it is desired to pass from the study of incompressible flow to the study of compressible flow. In this paper, an attempt is made to make clear the connexion between ψ and ϕ.