A Logarithm as generally understood and used by the ordinary student is the expression of a number as a power of 10. In order to multiply two numbers together he adds the logarithms of these numbers and takes as the result the anti-logarithm of this sum. Similarly for division he takes the anti-logarithm of the difference of the logarithms of the two numbers.

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.

Perhaps my title will be thought to savour too much of the nursery, but I should like to reassure you. I have no intention of boring you with stories of my own children, although they have been responsible for directing my thoughts from my former pupils in Secondary Schools to the untaught little people whose ideas are so instructive.

1. We all know that if we wish to sketch a cubic curve such as

xy (x + 2y − 1 ) = 2x − 3y + 4,

the first thing to do is to draw the three asymptotes and the satellite. These divide the plane into eleven compartments, and the simple consideration that there can be no points of the curve in a compartment throughout which xy (x + 2y − 1 ) and 2x − 3y + 4 have different signs enables us at once to rule out six of the compartments, and so to see how the curve goes to infinity.

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.