In practice it is generally as inverse functions of elliptic integrals that elliptic functions are needed; and so a simple demonstration of this important property of the functions is to be desired. The important step in such a demonstration is a proof that the inverse of an elliptic integral is a one-valued function, which for every value of the variable is regular or has at worst a pole—that is, is meromorphic over the complete plane (save possibly at ∞).

The author of this paper has, for many years, used a plane mirror at 45 degrees with the table, to demonstrate the properties of rays and beams of light. A horizontal projection of rays must be viewed from above in the absence of a mirror. If a mirror is used, the horizontal layout is seen in a vertical plane in the mirror.

It is only recently, as the result of an A.R.P. demonstration on incendiary bombs, that the effectiveness of the mirror method of demonstration has become apparent as well as the frequent and diverse occasions in which it is useful. In the demonstration referred to, some houses, farm buildings, hay stacks, and the like were exhibited as models on a table. The point of striking of an incendiary bomb was registered by the lighting up of a small flash lamp. Scarcely anyone in the large audience could see what was happening. A mirror at 45 degrees would have rendered the whole proceedings visible to everyone.

The textbook demonstrations of the well-known properties of two conics, the common self-polar triangle and the conic-envelope of lines cut in harmonic conjugates by them, usually suffer from the defects that the general case only is given and that the methods are not always adaptable to special cases. As the author feels the habit of contentment with general cases only a dangerous one and encouraging of loose thinking, he proposes to list the special cases, indicating for the sake of brevity proofs in outline only or, where very trivial, omitting them entirely. A third property—that of the conic-locus of points at which the conics subtend harmonic conjugates, as the immediate reciprocal of the second, has not been listed.

Attention was called to the graphical representation of a geometric progression by W. J. Dobbs, Gazette, IX, p. 245, as in Fig. 1. Such illustrations,

together with numerical examples, are sufficient for School Certificate examination purposes. The following considerations are intended to fill up time after the examination.

In Note 1568 (V for Victory) very brief reference was made to Generalised Chess, that is, an expansion of the ordinary game as played on the 8 × 8 board. In that note attention was called to the existence of the 5-leaper, whose moves are from (0, 0) to (5, 0); (0, 5); (3, 4); (4, 3). Here it is proposed to give some account of chess in three and four dimensions. In anticipation it must be pointed out that fractions do not exist in chess geometry—a special kind of lattice geometry; but that lengths such as √(a2 + b2) and √(a2 + b2 + c2), which are not integral, but where a, b, c are, do exist, since they can be constructed from integral lattices.

During the last year or two I have set in various places, from points of view varying with the examination, questions on cyclic quadrilaterals formed by two pairs of tangents to a conic. As this use of the configuration seems to have reached temporary saturation, I pass on an amplified account of the work to a larger, and perhaps more willing, audience.