I Had some difficulty in choosing a title for this paper. I might have called it “A Plea for More Practical Mathematics”, but there would have been a doubt as to whether “more “ qualified “practical “ or “mathematics ”. And I am not very fond of the word “ practical” De Morgan's remark that the self-styled practical man could always be relied on to produce some false logic is too often true. “Practical “ is invariably used as the antithesis of “theoretical “ The boy often regards theoretical chemistry as an unneces¬sary interruption of the practical work ; he does not always regard them as complementary, like bread and butter.

It is not my purpose to expound a new Theory of Differentials in the sense used by Professor G. Temple in his paper to the Mathematical Association in 1936 (G. Temple, “The Rehabilitation of Differentials,” M.G., Vol. XX, p. 120), but merely to embark on what he called the “comparatively trivial business” of giving an interpretation to the differential notation which pre-supposes the definition of derivatives, and to do so from the point of view of School rather than University teaching. A study of elementary textbooks on the Calculus shows two main schools of thought. The first, and older, of these regards differentials as infinitesimals, that is to say as variables which tend simultaneously to zero and whose ultimate ratio is their only interesting property. This presents great difficulties to beginners and savours too much of the early days when the theory of limits had not been developed. It has been largely ousted by the second school, which regards the differentials of independent variables as actual increments and defines the differential of a dependent variable (in terms of them) as what I like to call a “bogus increment”. It seems to me that this second interpretation is not as easy as it pretends to be, and that it suffers from serious drawbacks ; and the purpose of this article is to put forward a third interpretation which so far as I know is original and which is free from the objectionable features of the other two. Of what follows, the earlier part is a criticism of the second interpretation referred to, while the later part is an attempt to show how differentials can be more easily taught, and to demonstrate the adequacy of my interpretation.

1. Certain steps in the solution of linear differential equations involving the operatoṙ nearly always cause the student some confusion. It may therefore sometimes be helpful to have an alternative way of proceeding. None of the results of this note is original, but some of the familiar difficulties are presented in a new light, which may prove useful, especially in a revision course.

While a full discussion of nomograms requires the use of determinants and other ideas beyond the School Certificate stage, many practical needs can be met using two simple types. These can be understood and applied with only a knowledge of similar triangles; and, besides being of interest in themselves, produce as a by-product a working knowledge of logarithmic and other scales.

Before considering the method of construction, use of one or two types should be made by the pupil. This will help him to appreciate their simplicity. Their practical value, too, becomes apparent by using types (e.g. the focal length of a lens) that tie up with his work in the laboratory. Construction on squared paper can be done quickly and with fair accuracy. One or two carefully done examples with subdivisions inserted should be attempted also.

BY a simplifying assumption we express amplitude, reaction, frequency, and change of plate current in triode oscillations in terms of the fraction of the time during which plate current is flowing. The mathematical methods are simple, and were used by E. V. Appleton and B. van der Pol, Phil. Mag., 43, 177, 1922.

A triode is an exhausted glass vessel containing three electrodes: a filament or heater, a grid, and a plate. Electrons emitted by the heater are caught on the grid and plate in numbers depending on their potentials, and constitute currents from the leading-in wires towards those electrodes, called the grid current and plate current. If Vgis the potential of the grid relative to the heater, and Vpthat of the plate, the plate current Ipis theoretically and experimentally a function f(Vp + μVg) of Vp + μVgover a large range, where μ is a constant called the amplification. Thus the curves of Ipfor different plate potentials Vpare obtained from each other by translation parallel to the axis of Vgby an amount proportional to the change of Vp. The grid current is much smaller if the grid potential is not too high.

BROWSING for amusement in a book that I never saw at school, Casey's Sequel to Euclid, I was introduced to something new to me. It was my first encounter with harmonic polygons, and it was interesting to trace how the painstaking geometers built up a series of propositions applicable to such figures. What annoyed me, however, was the attempt to derive some of their properties by “inversion” of a regular polygon. Again it seemed illogical to build up a proof of general properties by proving a succession of mensurations. This feeling was intensified when it came to deal with the “Brocard” properties and I started to build up the theory from a new angle.

The series

is the simplest known series * which could be used to compute π, but it converges far too slowly to be of any practical use for this purpose. In this paper we prove an asymptotic formula for the remainder of the series (1) after a finite number of terms. The formula is

where

The polar equation to any curve can be written r=f(θ).

If P and Q are inverse points with respect to a circle whose centre O is at the pole and whose radius is k, then, since OP OQ = k2, it follows that the polar equation to the inverse curve is k2/r=f(θ).

It is proposed to find by inversion some properties of the limaçon and the cardioid that follow from certain well-known properties of the conies : others will suggest themselves. For convenience the same letter is used to denote both a point on the original figure and the corresponding point on the figure obtained by inversion.

The Theorem that, if y3 + 3xy + 2x3 = 0, then has been the topic of four recent notes in the Gazette, namely 1542 (not 1546) by E. H. N., 1610 by Mr. F. Bowman, 1684 by Mr. F. M. Goldner, and 1801 by Mr. H. V, Lowry. It seems worth while to supplement these notes by some comments which were for the most part suggested by the concluding lines of the note by E. H. N.

The writers of the four notes have looked at the theorem from the point of view of a candidate in an examination ; if it were set in a Higher Certificate Distinction Paper, I have little doubt that a fair proportion of the candidates would recognise that the cubic is a unicursal cubic, they would write x/y = u (or make some equivalent substitution), they would express x and y as rational functions of u, and they would realise that the question must inevitably come out by calculating accurately the differential coefficients of y with respect to x and then substituting in the left-hand side of the differential equation. In fact, they would reproduce the substance of the work of E. H. N. with the omission of his refinement of taking the reciprocal of y as an auxiliary variable to abbreviate the analysis.