Of all the problems which have perplexed teachers of mathematics in this generation, probably none has been more irritating and insistent than the choice of assumptions which must be made in each branch of the science. In geometry, in analysis, in mechanics, one and the same difficulty arises. Are we to prove that any two sides of a triangle are greater than the third? That the limit of the sum of a finite number of functions is equal to the sum of their limits? That the total momentum of two bodies is uninfluenced by their mutual action? And in every such case, on what is the proof to depend? A clear understanding of the answers to such questions, or better still, a clear understanding of principles by which answers may be found, would go far to co-ordinate and simplify elementary teaching; the object of this paper is to state such principles, and indicate their application.

Some time ago, when attempting to generalise certain methods of solving differential equations by operators, the writer came across the following mode of deducing Fourier’s Theorem symbolically.

The method is so simple that it seems improbable that it should be unknown. The nearest approach to it known to the author, however, occurs in Oliver Heaviside’s Electromagnetic Theory, Vol. II. §§ 275 et seq., but the investigation there proceeds on somewhat different lines, and the interpretation of the operators used requires to be carefully defined.

Suppose that a curve Г of constant shape moves in space so as to trace out a surface Σ. The problem has suggested itself to various authors whether it is possible for Г to be always a line of curvature, geodesic, asymptotic line, etc., on Σ. The following elementary method of solving such problems may be of interest.

In the Greenwich meteorological observations, Table XIII, gives the excess of the arithmetic mean of the daily maximum and minimum over the true mean daily temperature for each day in the year.

zn − pz + q = 0. .....(1)

The usual method of solution by means of a fixed curve and varying line is to draw the curve y = zn, and find where it is cut by the line y=pz − q.

The following method is quicker, and has some aspects of theoretic interest to recommend it. It consists in taking p, q as coordinates, and in drawing the curve given by the parametric equation

2

which is the condition that the given equation (1) shall have equal roots.