The customary school method of dealing with logarithms (i.e. the initial definition : if ax =N then …) has one great defect : it gives no lead whatever towards the higher theory of logarithms, nor even towards methods for calculating them. “How shall we introduce our sixth form girls to the theory?” a mistress recently asked; “it seems to come like a bolt from the blue.” Klein has some true and forcible things to say on the subject [Elementar Mathematik vom höheren Standpunkte aus, (3 ed. 1924) I, p. 155 et seq.] from among which a sentence or two may be quoted.

The formulæ for the powers and the reversion of any given series are not so readily accessible to those who are dependent upon modern mathematical text-books as they should be considering the frequent and manifold uses to which such formulæ can be put. I thought therefore that the following compendium of such formulae would be useful to readers of the Mathematical Gazette. The chief formulae in this paper are given in De Morgan’s Differential and Integral Calculus, but this book must be getting increasingly unavailable to ordinary readers except in reference libraries.

One of the oldest problems in the theory of Fourier series is that of looking for a criterion that a Fourier series shall converge. No one, however, has been able to find a simple, necessary and sufficient condition for this. Thus, for instance, bounded variation of the function is sufficient but not necessary. Continuity is neither necessary nor sufficient. That is to say, there are functions whose Fourier series converge at points of discontinuity, and others whose Fourier series diverge at points of continuity. If we consider the same problem for Cesàro summability of any particular order, similar difficulties arise.