The following is an extension of the note by Mr W. M. Roberts in the Gazette (215 [v. 4, c], 1906); or of Mr A. H. Barker's discussion on p. 163 of his “Graphical Calculus.”

a, (a + d)r, (a + 2d)r2,…

The following method of summing the mixed series is an extension of the well-known method for the G.P. given by Dr Dougall in the April issue of the Notes for 1909.

Having given the sides of a rational right-angled triangle, to find from them the sides of other rational right-angled triangles.

The object of this note is to suggest finding the formula for centrifugal force as an exercise on space-rate of change of energy.

In Vol. XXII., p. 64, of the Proceedings of the Society, Professor Gibson called attention to difficulties in the treatment of tangents by the method of limits. The aim of the following remarks is to show how young students may become accustomed to regard a tangent as the limit of a secant, and at the same time to furnish a type of exercise which might encourage initiative and research in the early study of Geometry.

Dr Stokes has already shown (Mathematical Notes, No. 13, p. 143) how the summation of series may be put to its proper use in elementary mathematics, namely, to provide a sequence of approximations to a limit. The method may be used to give a sequence of approximations to sinx and cosx.