A meeting of this branch was held on February 20th at the Friars’ School, Bangor—the President, Dr. Bryan, in the chair. Mr. R. W. Jones, Headmaster of Glanadda Elementary School, opened a discussion on the teaching of the foundations of arithmetic to children of 6 or 8 years old. After referring to the difficulty of forming any ideas as to a child’s conception of number, and mentioning instances of the vague answers given by a child who is asked to estimate some large number of objects, Mr. Jones proceeded to show how the four fundamental laws of arithmetic could be gradually and insensibly taught to children by means of experiments involving simple operations in weighing and measuring. He emphasised the necessity for using familiar illustrations in expounding the four rules and brought out two points as worthy of special notice: (1) that in teaching the elements to children it is absolutely necessary to confine one’s attention to small numbers; (2) that the time spent in teaching the multiplication tables to children of 6 or 8 years was not well spent, since for small numbers at any rate, they could be learnt insensibly, and the higher tables could be learnt with much greater facility in the child’s third year at school. Most of those present took part in the interesting discussion which followed, generally expressing their agreement with Mr. Jones’s views.

To establish the order of the locus we must note that besides the above mentioned n intersections with the circumference, the point D also belongs to the curve, so that the latter cuts the circle in n+1 points. The order of the curve is therefore Let n be even.

The ratio is not an integer, so that a complete rotation of the straight line BC must take place before the two generating straight lines can be in their initial mutual position; the line revolving round D will be, after a semi-rotation of BC, parallel to BC in DE, and we may say that all the loci for n = 2p are asymptotic in the direction BC, on account of the initial position of BC and DA.

Professor Bryan’s paradox on Fourier’s theorem (Gazette, vol. iv. p. 390) seems to call for a few words of explanation.

In the first place a function f(θ) is constructed of the form

and the function is arranged so as to have the same values as an arbitrary function F(θ) at the places θ = 0, a, 2a, …, (n-1)a, where α=2Π/n. It will be observed here that we have only n data from which to determine (2n-1) coefficients, and consequently the form of f(θ) is largely at our disposal ; but the particular form selected by Professor Bryan leads to the formulae for the coefficients,

My object is to indicate what seem to be the fundamental ideas connected with homography which would be readily understood and appreciated by a beginner.

If a pencil of concurrent rays is cut by any transversal, the ratios between the parts into which it is divided are the same as for any parallel transversal. This is not the case when the two transversals are not parallel, and it becomes an important problem to find out what property of the segments is common to both transversals in such a case. To the Greeks, who so exhaustively studied the properties of the conic sections as sections of a cone, this problem must have presented itself quite early, and it appears that not only Pappus, but also Euclid, and therefore probably Apollonius, knew the solution.