Let n be a positive integer. Then we know that, if m>– 1,

Consider the integral

which is equal to

Attention has recently been drawn to the obscurity of the usual presentations of Mayer's method of solution of the total differential equation

This method has the practical advantage that only a single integration is required, but its theoretical discussion is usually based on the validity of some other method of solution. Mayer's method gives a result even when the equation (1) is not integrable, but this cannot of course be a solution. An examination of the conditions under which the result is actually an integral of equation (1) leads to a proof of the existence theorem for (1) which is related to Mayer's method of solution in a natural way, and which moreover appears to be novel and of value in the presentation of the subject.

Professor Hemraj has given a proof of a part of a theorem of Gauss without using the theory of quadratic residues. Proceeding on similar lines, I have obtained a complete proof which is rather simpler and certainly more concise.

Consider a plane curve C of order n and class X; it is to be supposed throughout that C has only ordinary Plücker singularities, i.e. nodes, cusps, inflections and bitangents. Through any point P1 of C there pass, apart from the tangent at P1 itself, X – 2 lines which touch C; let T12 be the point of contact of any one of these tangents and P2 any one of the n – 3 further intersections of P1T12 with C. Through P2 there pass, apart from the tangent at P2 itself and the line P2P1, X — 3 lines which touch C; let T23 be the point of contact of any one of these with C and P3 any one of its n — 3 further intersections with C.

If x1, x2, …., xk, …. are independent random variables each of which is subjected to a distribution law σ = σ(x) independent of k and having a finite positive dispersion, then x1 + x2 + …. + xn is known to obey the Gauss law as n→ + ∞, no matter how σ (x) be chosen. There arises, however, the question whether it is nevertheless possible to determine the elementary law σ (x) from the asymptotic behaviour of the distribution law of x1 + x2 + …. + xn for very large but finite values of n. It will be shown that the answer is affirmative under very general conditions.

In classical mechanics Gauss' Theorem for a gravitational field states that, if S is a closed surface and N the component of gravitational force along the outward normal, then

where β is the Newtonian constant of gravitation and M is the total mass inside S. This result has recently been extended to general relativity by E. T. Whittaker,1 who, however, considered only the case of a statical gravitational field, the line-element of which is given by2

where the coefficients U and αμν are independent of t. It is not immediately clear from his work whether the results are extensible to more general space-times.

With any twisted curve of order six is associated a system of planes, usually finite in number, which touch the curve at three distinct points. The curve with its system of tritangent planes possesses properties which recall the properties of a plane quartic curve and its system of bitangent lines; and this is specially true of the sextic which is the intersection of a cubic and a quadric surface. But whereas the properties of the plane curve were discovered by geometrical methods, such methods have only recently been applied with success to the space-curve; the earliest properties were obtained by Clebsch from his Theory of Abelian Functions. In the absence of any one place to which reference can conveniently be made, an account of these properties in their geometrical aspect will be useful.

The object of this note is to derive a form of Poisson's equation from general relativistic mechanics, without assuming the field to be either static or “weak”. The problem is essentially a “local” problem, all observations being made by one observer; this observer determines the apparent gravitational field in his vicinity by observing the motions of free (isolated) particles. Defining gravitational mass by means of Poisson's equation, we find the relation between the densities of gravitational and inertial mass relative to any observer. We also find what may be called the non-rotating frame of reference belonging to any observer.

The study of the topological properties of algebraic surfaces, considered as continua of four real dimensions, has thrown much light on the theory of the birational invariants of such loci. The results obtained for surfaces have been generalised to varieties of higher dimension by Hodge, and, particularly, by Lefschetz. Apart from this, little seems to be known about the general topological properties of algebraic loci of three (or more) dimensions, the detailed study of which seems to present considerable difficulty. In particular, apart from the general theorems of Lefschetz, nothing seems to be known about the cycles of three dimensions of an algebraic V3. The object of the present paper is to study these cycles on certain quite special V3, in the hope that some insight may be gained into the general theory.