It may not be improper here to caution learners against some inaccuracies in style and expression which are often met with in scientific works published in this country To matters of this kind much attention is paid by the French mathematicians. By this means, as may naturally be expected, accuracy is produced in thought as well as in style, and hence it is, that many of their recent works exhibit a perspicuity and simplicity, and a logical clearness of arrangement, which enhance their value in a very considerable degree. To put the student on his guard against faults of this kind, some instances of the inaccuracies above referred to are subjoined; and he will find that, by avoiding the use of them and of similar modes of expression, he will soon acquire a comparatively correct style, instead of the careless and random one that is so common in this country Some of these instances will doubtless be found in the works of authors of the present day, but they have been purposely selected from the writings of authors no longer living.

If AC, BD, EF are the diagonals of the quadrilateral and PQR is the triangle formed by the diagonals, the circles on the diagonals as diameters form a co-axal system and the circle PQR is a circle of the complementary system. If the circles on the diagonals as diameters intersect in real points S1, S2, then S1, S2 are the limiting points of the system to which the circle PQR belongs and if S1, S2 are not real points and if the straight line joining the mid-points of the diagonals cuts the circle PQR in L1, L2 then all the circles of the system to which the circle PQR belongs pass through L1, L2 and L1, L2 are the limiting points of the system to which the circles on the diagonals as diameters belong.

It is only after considerable hesitation that I have allowed myself to be persuaded to publish this paper, for it contains nothing whatever that cannot be obtained from any of the standard text-books on the Theory of Numbers. I have, however, been influenced by the fact that to many school-masters such text-books are not easy of access, and that this is likely to be so more particularly in the case of those who were unable to attend the January meeting. Thus, although I feel that the pages of the Mathematical Gazette are hardly the place for the reproduction of selected passages from standard mathematical works, I have in this case allowed my scruples to be overcome, in the hope that the concession may prove useful to those who have neither the leisure nor the opportunity to become acquainted with the subject from the pages of Mathews or Dickson.

The following notes are the result of an attempt to gain some insight into the meaning of the Lorentz formulae, avoiding the use of technical terms such as frames of reference, coordinates, and point-events.

The hypothesis of Einstein that the velocity of light is a constant, independent of the relative velocity of the source and observer, is assumed throughout.

I Suppose that most of us have some interest in the teaching of mathematics in other countries. I have recently returned from a year spent in the United States as visiting instructor in mathematics in a large preparatory school, a boarding school of seven hundred boys with a staff of about eighty-five men, twelve of whom form the mathematical department. (The term preparatory school as applied to an American school is roughly equivalent to our public school, the word preparatory being used in the sense that the school prepares its pupils for college.)

Of all subjects which form part of a mathematical course, I believe that the average student feels most at home in his study of trigonometry There are, however, certain applications which escape him and which are of interest. Amongst these are a well-defined group of problems in heights and distances.

I am sorry that Professor Watson (this volume, p. 261) has been so puzzled by the last sentence in a recent note of mine “Query! What is the envelope of the principal axes of conics touching any four lines? ” By “Query ” I did not mean that I was delivering a seventeenth-century challenge to the learned world, but that I should be interested to see a really elementary investigation, provided it was short, though I was disinclined to seek one myself. For from a slightly higher standpoint the question is very simple. Consider the envelope of lines perpendicular to their conjugates with respect to all conics of a pencil of conic envelopes touching four lines.

This Colloquium, held at St. Andrews from July 4 to 15, was in every way as successful as its quadrennial predecessors. A hundred and ten persons attended, including about twenty wives and daughters of mathematicians, including also twenty-one professors of mathematics from the universities of Great Britain, Eire, France, Holland, Denmark, and the United States.