In an article, published some time since, the author of the present paper deduced an asymptotic expression for the functions of the elliptic cylinder, which expression took the following form:

Here P and Q are certain asymptotic series, and C and a arbitrary constants. General expressions for these constants were not determined in the aforementioned article on account of the difficulties there set forth, though it was pointed out that their numerical calculation for any particular problem was simply a matter of arithmetic computation. It is the object of the present paper to deduce general expressions for these constants C and a in terms of the two parameters which appear in the defining equation for U.

In his recent book A History of the Conceptions of Limits and Fluxions in Great Britain from Neuton to Woodhouse (Chicago and London: The Open Court Publishing Company, 1919), as well as in a series of articles in the American Mathematical Monthly for 1915 on The History of Zeno's Arguments on Motion, Mr Cajori discusses certain aspects of the conception of a limit, and treats in considerable detail the controversy between Jurin and Robins that arose out of the publication of Berkeley's Analyst. I gave an account of the controversy in a paper that appears in Volume XVII. of our Proceedings, and as Mr Cajori's estimate of the respective merits of the contributions by Jurin and Robins differs greatly from mine, and as the conception of a limit is fundamental in modern mathematics I venture to draw the attention of the Society to the matter.

Writing of his now well-known Interpolation Formula, Professor Everett said, “The only novelty in the formula is the simplicity of its form.… The best known formulae for interpolation by central differences are difficult to carry in the memory on account of their unsymmetrical aspect, one law being applicable to the odd and another to the even terms. … This disadvantage does not apply to the formula proposed,” viz., that which now goes by Everett's name.

The definition and properties of Mathieu (or elliptic-cylinder) functions are well known to the members of this Society, owing to the appearance in its Proceedings from time to time of various papers by different authors, wherein these functions are discussed. The object of the present paper is to introduce a new kind of function which can be considered as a generalisation of Mathieu functions, and for which we propose the name of “Mathieu Functions of Higher Order.”

In a previous paper, entitled the “Vibrations of a Particle about a Position of Equilibrium,” by the author in collaboration with Professor E. B. Ross (Proc. Edin. Math. Soc., XXXIX, 1921, pp. 34–57), a particular dynamical system having two degrees of freedom was chosen and solutions of the corresponding differential equations were obtained in terms of periodic series and also in terms of elliptic functions. It was shown that for certain values of the frequencies of the principal vibrations, the periodic series become divergent, whereas the elliptic function solution continues to give finite results.

Hardly had Edinburgh University entered upon its present academic year than it was deprived by the hand of Death of the services of Cargill G. Knott, Reader in Applied Mathematics. The news of his death came as a great shock to all his friends, as there was no illness or other sign which might have foreshadowed the summons that was coming to him. Apparently in his usual robust health, he was engaged in his various duties until the very end, which came to him with startling suddenness.

Consider any two algebraic equations, which for simplicity we shall take to be a cubic

whose roots are y1, y2, y3, and a quadratic

whose roots are x1, x2.