Theorem. If a he any positive quantity not equal to 1, and x, y, z be any three rational quantities in descending order of magnitude, then

The objects of the present note are to exhibit an interesting property of some groups of subtraction and division, and to present a method of using such groups in the study of some elementary properties of the hyperbolic functions. The four finite groups of subtraction and division which transform every rational number into a rational number are of orders 4, 6, 8, and 12 respectively. With respect to each of these groups all the numbers, with the exception of two more than the order of the group, are divided into sets of conjugates, each set containing as many distinct numbers as the order of the group. The property to which we referred in the first sentence is that each of these sets contains two and only two negative numbers whenever the set is composed of real numbers and the number from which we subtract is positive.

The Council has to deplore the removal by death of three of the members of the Association during the past year. Mr. E. Budclen was a useful member whose name will be familiar to readers of the Gazette, to which he was a regular contributor of notes and reviews. He was ever ready to give up part of the leisure of a busy life to work for the Gazette; indeed, at the time of his painfully sudden and unexpected death he had reviews for the Gazette in hand. Mr. J. C. Palmer was for some years a helpful member of the Council, and attended its meetings and the general meetings of the Association with unfailing regularity It is many years since the late Lord Kelvin conferred distinction upon the Association by accepting its honorary membership. Although he took no active part in the special work of the Association, yet its endeavours to promote better methods of teaching elementary mathematics had his silent sympathy and approval.

THERE is a perfectly natural way of arriving at the differential coefficient of ax, and the corresponding integration of dy/y, provided that the ordinary inadequate bookwork of indices is properly supplemented to include irrational indices. (This is really necessary for logarithms to be treated rigorously.) The graph of y = ax, x any real number and a positive,

In the first place, the present paper refers to conditions prevailing in England only, to the exclusion of Scotland, Ireland and Wales.

Secondly, the term " public school" has, in England, a rather special meaning. A public school is generally understood to be an endowed secondary school of fair size, educating boys of the upper and middle classes, independent of state control except in so far as its income is supplemented from public funds. Public schools may be either boarding schools or day schools, but most of them belong to the former category.

The following practical suggestions are offered as an answer to the question of ever-increasing interest, “How, when, where, shall the child begin arithmetic?”

The importance of the science of number, as a great teacher has shown, is enormous, not only in its relation to the study of abstract mathematics and in its dominance over the prosperity of nations, but in the fact that it is the “conclusive science which we have to apply all our days to all our affairs.”

Writers on mensuration do not seem to call attention to the convenience of quadrature-formulae for determining positions of centres of gravity, magnitudes of moments of inertia, etc.

In the teaching of mathematics as of other subjects the primary determining factor ought to be, but seldom has been, the consideration of the pupil himself—his interests, aptitudes, and capabilities. The teacher should be free to devise the best means of securing his fullest co-operation, and examinations should allow the greatest possible elasticity. It is the reversal of the rational order that has wrought so much harm in the past and that so long delayed and still impedes reform.

Mr. Grant, in the last issue of this Gazette, having made several references to “Arithmetical Types and Examples,” perhaps a few remarks in reply will not be altogether out of place.

1. Multiplication. The only difference between the two methods first given is in the position of the various digits of the multiplicand.

We are told that the first rule (that “there are the same number of figures to the right of the decimal point in the first partial product as there are in the multiplicand,” and that therefore the first figure set down must be under the right-hand figure of the multiplicand) is mechanical; whereas the rule of “putting each first figure of a partial product underneath the figure to which that partial product is due” is sound and fundamental. It is not obvious to the writer that one is more mechanical than the other.

When Algebra, fair Daughter of King Arithmetic, is nearing her age of adolescence her eyes fall on Prince Limit, and there cometh a change upon her. Her tender heart goes forth to him, and in her lithe and nimble frame she feels a torture from end to end as it would fall to pieces. She is in love, and the Prince reciprocates her love.

Happily they are wedded. The King gave away his Daughter, the Angels witnessed it, and all was joy. The marriage was performed with the Hymn of the Transcendent Number and Ratio, preceded by the Song of Limits, and followed by the Chorus of Continuity and Convergence.

1. It is very desirable that the subjects of examination for Mathematical Entrance Scholarships at Cambridge and Oxford should admit of the same course of preparation.

2. In order to allow of a broader syllabus the exercises and riders should be simplified.

3. Questions on elementary bookwork ought not to be entirely omitted.

Restricting bookwork questions to the higher bookwork would encourage the neglect of the more elementary parts, which are of at least equal importance.