There are some problems in elementary mathematics that the average student is apt to consider as matter for memorising without any great effort being either useful or advisable at understanding them. Experience in teaching elementary statics shows that the theory of stability is an example of such a problem. The average student often does not appreciate the principle of the method generally adopted to investigate the stability of the equilibrium of one body resting on another. He is not generally in a position to grasp the significance of an “infinitesimal displacement,” or the meaning of “proceeding to the second order of small quantities” when the equilibrium is “neutral” to the first order. He is further puzzled by the fact that what appears to be neutral equilibrium becomes, when higher orders of small quantities are taken into account, stable in one case, unstable in another, whilst remaining neutral in a third. These difficulties can be overcome if we adopt a slight modification of the treatment given in the text-books. The object of this article is to explain the method used by the author for lecturing purposes, and not to obtain new results.

OH, wad some power the giftie gi’e us
To see oursels as others see us,
It wad frae mony a blunder free us,
And foolish notion;

in which case it is probable that wars would cease. During the last few years we have had glimpses of ourselves through the spectacles of the Central Powers; occasionally through the telescopes of the French, the Italians, or the Russians, and in the near future we are likely to get a good many pictures of ourselves as we appear to the last-named people. We are apt to think that our own view of ourselves is the only true view, and all other images are in error just as they differ from our own standard. Naturally other nations adopt the same course, and we are free to believe that in many cases their vision of themselves is far more at fault than our own self image. In this respect a study of Alice and her standards might be not without profit to the nations. When we want to see ourselves, or, at least, our faces, we usually employ a mirror, and what we see depends very much on the mirror we use. Sir George Greenhill was the first professor of Applied Mathematics at the Eoyal Indian Civil Engineering College, Coopers’ Hill. He told us that at the end of the first session a reception was held and the science departments were on view. A young lady entering the physical laboratory and seeing an inverted image of herself in a large concave mirror naively remarked to her companion, “They have hung that looking glass upside down.” But those who fixed the mirror were not to blame, as the lady would have discovered had she advanced past the centre of the curvature. The reply of modern mathematics to the prayer of Eobert Burns is the Theory of Relativity I have some doubt whether Burns would have appreciated the answer. As a rule, we use a plane mirror to inspect ourselves. This reverses certain spatial relations which can be expressed best, perhaps, by stating that it presents a right-handed screw as a left-handed screw and vice versû, or changes a wine screw into a beer screw, as Clerk Maxwell would say, referring to the tendrils of the vine and the hop.

In the Report of the Education Reform Council, published last year under the title “Education Reform,” a paragraph occurs on page 84 with reference to the Mathematics Course suitable for boys of 16 to 18 years of age attending a Secondary School. There it is suggested that such boys who take up Mathematics and Science may pursue the study of differential and integral calculus, including the easy parts of differential equations, and apply these with advantage to higher algebra, trigonometry, coordinate geometry, solid geometry and mechanics; that the course should include some introduction to infinite series; that too much time has been devoted hitherto to the binomial series; and that the thing to aim at is Taylor’s series.