It seems to me that, no matter what doubts and perplexities may have assailed me in the selection of the subject and the contents of this Address, my primary duty on this occasion is obvious and inevitable ; it is to express my appreciation of the impartiality of the Council in so arranging this afternoon’s proceedings that my Address is subsequent to the discussion on Professor Neville’s motion, so that supporters of the motion were unable to use the Address as an argument in favour of limiting each President to one year of office.

In Fig. 1, AA1A2A3 … is a rectangular hyperbola, whose asymptotes are OT, OT', and whose vertex is A. We make the following construction

At A1, a point on the hyperbola, a tangent is drawn, cutting OA at P The tangent at A is drawn, cutting OA1 at P1. Let the other tangent from P1 touch the hyperbola at A2. Produce PA1 to cut OA2 at P2. Let the other tangent from P2 touch the hyperbola at A3, and let P1A2 produced cut OA3 at P3.

Consider this process to be continued indefinitely and to be repeated on the other side of the axis OA, A′1 being the point of contact of the other tangent from P We obtain a pencil of rays OAn, OA′n which gradually lie closer together, becoming infinite in number as the asymptotes, which are the ultimate rays of the pencil, are approached. The system is obviously symmetrical about OA, and I shall show that this pencil has further remarkable properties, which I have named “pansymmetry” These properties can be stated thus

If a tangent to the hyperbola at the point An be produced both ways, it will be symmetrically divided by all the rays of the pencil, the p th section on the one side of An being equal to the p th section on the other side.

One service which Euclid rendered to the school-teacher in the days when his supremacy was unquestioned was, that he set a logical standard.

Euclid’s logic has been attacked by many writers, and by none more severely than by a philosopher whom I have worshipped this side idolatry as fervently as any But I doubt whether even Russell, in his less controversial moods, would deny that broadly speaking Euclid pitches the standard reasonably When he is not proving the obvious, what he takes for granted is usually what we should take for granted. This is the more interesting because it is to some extent accidental. Since Euclid wrote down such axioms as “ Things which are equal to the same thing are equal to one another ” and “ Magnitudes which coincide are equal ” he seems to have thought that he had made explicit every assumption involred in his work. We know that this opinion was mistaken, but we know also that the ideal of exposing every assumption is not appropriate to school geometry—I would go so far as to say not appropriate to any subject except that which it dominates, namely, formal logic as expounded by Peano and in Principia Mathematica.

Referring to Mr Freeman's article in the July number of the Gazette, there can be no doubt as to the desirability of introducing these operators early in algebra. But they should be used only for the transformation of finite series. For this purpose it is never necessary to use negative powers of E and Δ. Some of Mr. Freeman's examples are considered below from this point of view

So far as addition, subtraction and multiplication are concerned, E and Δ, operating on a single definite function, combine with themselves, with each other and with constants according to the laws of algebra.

Notation. By (x/y, X/Y) = a1.a2a3 … are let it be understood that after a vulgar fraction X/Y has been converted into a continued fraction (CF.), a1.a2a3 … are the partial quotients, and x/y is the penultimate convergent. The point placed after a1 is intended to show that, in the case of a proper fraction, the first partial quotient must be taken as zero.

Given the equation s = at or its equivalent t = logas, where the base a is a specified positive number other than unity, the problem is to find first ds/dt and dt/ds and then the power series for s(t) and a series for s(t).

The method given in Smith's Conic Sections (1910), p. 258, fails, like other methods, to correlate foci, directrices, eccentricities and axes , the following method, based on the equations there, shows how that can be done. If the conic is an ellipse, the real eccentricity goes with the real foci and the imaginary eccentricity with the imaginary foci, but if the conic is a hyperbola there is no such relation, as both values of the eccentricity are real.