An expression for the sum of a series of terms which takes account of the different rates of oscillation of the functions involved is of importance for series such as occur in the problems of diffraction, and a relation of this kind has been given by Poisson. This relation can be written in the form

where Sm denotes the sum of m terms of the series Σun, beginning with the term un.

§ 1. Extensions of Pascal's theorem are already known if we look at the matter from certain particular points of view – an extension of the theorem from the more general point of view is still a desideratum.

§1. Let be a binary quartic; then I propose to show symbolically that the solution of Euler's differential equation

can be written in the form

where

are otherwise arbitrary.

§1. Whittaker has shewn that a general solution of Laplace's Equation

may be put in the form

where f (v, u) denotes an arbitrary function of the two variables u and v; such a representation is valid only in the neighbourhood of a regular point.

There are certain well-known linkages for effecting the transformation of inversion, and, incidentally, by inverting a suitably situated circle, for producing straight-line motion. Reference may be made to the classic lecture by A. B. Kempe, “How to draw a straight line” (London: Macmillan, 1877). It is the object of this paper to call attention to the fact that these linkages have the corresponding property in non-euclidean geometry.

Since the publication of the memoir of Mathieu on the transverse vibrations of an elliptic membrane, the subject has been discussed by many authors from different points of view. But the corresponding problem of the plate has received but little attention. Mathieu discussed the problem as early as 1869, but the method adopted is different from that followed in the present paper, the main object of which is to apply Whittaker's solutions of Mathieu's Equation to the problem of the elliptic plate. These solutions are really better suited for numerical calculations than the evaluation of infinite determinants.

Sir T. L. Heath's translation of The Thirteen Books of Euclid's Elements, with its Introduction and Commentary, is not merely a worthy tribute to the lasting merits of Euclid's work, but is at the same time a most valuable history of elementary geometry; the language in which he describes the character of Camerer's edition of Euclid's first six books is even more applicable to his own: “No words of praise would be too warm for this veritable encyclopaedia of information.” There may, however, be room for difference of opinion on matters of detail, and I propose in this note to call attention to one or two passages in which I think he is in error in his criticism of Simson, whose edition of Euclid formed the basis of so many English text-books and kept alive the traditions of Greek geometry in this country long after Euclid's Elements had disappeared as a text-book on the Continent.

The following communication gives a summary of the opening paper by Mr G. Lawson and of the remarks by various members who afterwards took part in the discussion. The question at issue was how to treat definitions and properties of the tangent to a plane curve in teaching beginners.

It is an obvious remark that the Mathieu functions, being the harmonic functions of the elliptic cylinder, must be closely related to the Bessel functions, the harmonic functions of the circular cylinder. Reference has been made to some aspects of this relationship in two earlier communications, to which the present paper may be regarded as a sequel.

The field-equations of gravitation in Einstein's theory have been solved in the case of an empty space, giving rise to de Sitter's spherical world. In the case of homogeneous matter filling all space, the solution gives Einstein's cylindrical world. The field corresponding to an isolated particle has been obtained by Schwarzchild. He has also obtained a solution for a fluid sphere with uniform density, a problem treated also by Nordström and de Donder. A new solution of the gravitational equations has been obtained in this paper, which corresponds to the field of a heterogeneous fluid sphere, the density at any point being a certain function of the distance of the point from the centre. The law of density is quite simple and such as to give finite density at the centre and gradually diminishing values as the distance from the centre increases, as might be expected of a natural sphere of fluid of large radius. The general problem of the fluid sphere with any arbitrary law of density cannot be solved in exact terms. It will be seen, however, from a theorem obtained in this paper, that the solution depends on a linear differential equation of the second order with variable coefficients involving the density, and thus the laws of density for which the problem admits of exact solution are those for which the above coefficients satisfy the conditions of integrability of the differential equation. An approximate solution for any law of density may be obtained by the method of series.

In a recent paper Mr J. M. Whittaker has given the following Theorem:—

If ψ (x) be the indefinite Riemann integral of a bounded positive function g(x), and if f(x) be any bounded function, then the equation

is true whenever either side exists.

§ 1. The radiation of electromagnetic charges from a moving point.

Let the electric charge associated with a moving point Q at time s be f(s). This charge is supposed to vary on account of the radiation of electric charges from Q in a variable direction, which, at time s, has direction cosines l (s), m (s), n (s) Using ξ(s), η(s), ζ(s) to denote the coordinates of Q at time s, and defining the effective time τ by means of the equation

the electromagnetic field may be specified by means of the potentials

where

Certain properties of Ruled Surfaces relating to their Curves of Striction are suggested and immediately proved by the use of a particular kind of coordinates, known as Dual Coordinates, for the Generating Lines. These coordinates, were introduced by Prof. E. Study; the theory of them is fully explained and many beautiful applications are made in his treatise, Geometrie der Dynamen, Teubner, 1903, and in his article, Complexe Grössen in the Encyclopädie der math. Wissenschaften, Bd. I., pp. 147–183, and specially p. 166. In applying the method here I have ventured to regard the fundamental quantity c in a light which, if less rigorous, has the advantage of being familiar to most of us in other mathematical work.

In the majority of the processes with which one is concerned in the study of the medical sciences, one has to deal with assemblages of individuals, be they living or be they dead, which become affected according to some characteristic. They may meet and exchange ideas, the meeting may result in the transference of some infectious disease, and so forth. The life of each individual consists of a train of such incidents, one following the other. From another point of view each member of the human community consists of an assemblage of cells. These cells react and interact amongst each other, and each individual lives a life which may be again considered as a succession of events, one following the other. If one thinks of these individuals, be they human beings or be they cells, as moving in all sorts of dimensions, reversibly or irreversibly, continuously or discontinuously, by unit stages or per saltum, then the method of their movement becomes a study in kinetics, and can be approached by the methods ordinarily adopted in the study of such systems.

In regard to the algebra of binary forms and the theory of rational curves there exists a wide literature, with which I am not well acquainted. Very possibly the simple remark made in this note is found elsewhere. But the note is a grateful echo of recent delightful colloquy with persons of good will.