This paper, while it does not merit the attention of workers in the field of summability, may be of some interest to readers of the Gazette who are using Bromwich's classical work on infinite series as a textbook. Its main purpose is to show that Bromwich's proof of the Hardy-Landau converse of Cauchy's limit theorem can be made to yield, without any modification of the underlying idea (due to A. E. Jolliffe), all the familiar converse theorems on Rieszian summability of the first order. The paper also shows that after having obtained these theorems, one can pass on, by a natural transition suggested by Szász, to the corresponding results in the theory of summability by Dirichlet's series

Before the end of the first year of the H.S.C. course, a boy has dealt with the elementary theory of indices and logarithms and has been introduced to the exponential and logarithmic functions in Calculus. This introduction, the writer has suggested in a previous Note (1896), may well be intuitive and may be based upon the fact that if y = ax, then dy/dx = ky, the fact being established by reading off gradients and ordinates from a suitable graph. It is now suggested that when the H.S.C. course in Calculus is completed, a mathematically-minded boy may learn a great deal of mathematical technique from a treatment of the exponential and logarithmic functions of the kind outlined below.

(a) The foundation of calculus operations on the circular functions is

The traditional steps by which the inequality

One of the most striking results in the theory of uniform convergence asserts that for a monotonic sequence of continuous functions, continuity of the limit function is a necessary and sufficient condition for uniform convergence. Much has been written about this theorem, but the fact that the proofs given in such works as Bromwich’s Infinite Series and De La Vallée Poussin’s Cours d’Analyse are incomplete, seems to justify the hope that this account may be of some assistance to students and teachers of convergence.

Most of the tractable problems pertaining to submarine cables and electrical transmission lines have been solved. However, no result seems to have been given for the current at any point of a very long “loaded” cable with a series capacitance between it and the sending battery. By aid of a general theorem published recently in the Mathematical Gazette, the required result may be obtained without difficulty.