The object of this paper is to give a method of finding the units of a set of algebraical numbers that shall be convenient practically. Little attention is given to theoretical considerations. The case of numbers composed with the cube root of an integer is treated in § 2, and a few theoretical remarks are added in § 3. In § 4 we show how to test whether the unit found is a fundamental one or not. In § 5 the method is extended to other cases.

1. The subject-matter of this communication I believe to be new, but after Lemma 1 the method is classical; Lemma 1 is itself a particular case of a theorem which I have given elsewhere, and is a straightforward extension of a well-known result.

1. In his book, A Theory of Time and Space, the writer showed that, contrary to the generally accepted view, the ideas of Congruence can be built up from purely ordinal considerations, provided that we do not confine ourselves to linear order, but admit what, for convenience, he has called Conical Order. This Conical Order is built up from the purely abstract asymmetrical relations of before and after and does not pre-suppose the existence of the cones which are used to illustrate it.

An Archimedean solid (in three dimensions) may be defined as a polyhedron whose faces are regular polygons of two or more kinds and whose vertices are all surrounded in the same way. For example, the “great rhombicosidodecahedron” is bounded by squares, hexagons and decagons, one of each occurring at each vertex. Thus any Archimedean solid is determined by the faces which meet at one vertex, and therefore by the shape and size of the “vertex figure,” which may be defined as follows. Suppose, for simplicity, that the length of each edge of the solid is unity. The further extremities of all the edges which meet at a particular vertex lie on a sphere of unit radius, and also on the circumscribing sphere of the solid, and therefore on a circle. These points form a polygon, called the “vertex figure,” whose sides correspond to the faces at a vertex and are of length 2 cos π/n for an n-gonal face. Thus the vertex figure of the great rhombicosi-dodecahedron is a scalene triangle of sides .

Given four lines in a plane, we know that the circumcircles of the four triangles formed by them meet in a point. Kühne's theorem states the analogous property for Euclidean hyperspace, viz., that if from n + 2 primes in space of n dimensions we obtain n + 2 simplexes by omitting each prime in turn, the hyperspheres circumscribing the simplexes meet in a point when n is even, but not when n is odd. Mr Grace, in a recent reference to the property, gives a short and interesting proof by help of the rational curves which osculate the given primes and the prime at infinity. It is the purpose of the following note to obtain the result by considering the points of intersection of certain cubic loci.

The limiting distribution, when n is large, of the greatest or least of a sample of n, must satisfy a functional equation which limits its form to one of two main types. Of these one has, apart from size and position, a single parameter h, while the other is the limit to which it tends when h tends to zero.

The appropriate limiting distribution in any case may be found from the manner in which the probability of exceeding any value x tends to zero as x is increased. For the normal distribution the limiting distribution has h = 0.

From the normal distribution the limiting distribution is approached with extreme slowness; the final series of forms passed through as the ultimate form is approached may be represented by the series of limiting distributions in which h tends to zero in a definite manner as n increases to infinity.

Numerical values are given for the comparison of the actual with the penultimate distributions for samples of 60 to 1000, and of the penultimate with the ultimate distributions for larger samples.

The fascinating chain of theorems due to Clifford and recently extended by Mr F. P. White originates in the elementary fact that the circumscribing circles of the triangles formed by four lines meet in a point. From a like simple germ, namely the fact that the centres of the above-mentioned circles lie on a circle, an infinite chain of theorems was first evolved by Pesci and may be enunciated as follows:

(i) The centres of the circumcircles of the triangles formed by four lines lie on a circle; thus from four lines we derive a point, namely the centre of the latter circle.

(ii) Five lines give five sets of four and the five derived points lie on a circle; thus from five lines we derive a point, namely the centre of the latter circle.

(iii) Six lines give six sets of five and the six derived points lie on a circle.

The following derivation of the equations of viscous motion, and, by analogy, the equations of motion of an isotropic elastic solid, appears to have an advantage in brevity over those in general use.

The success of the kinetic theory in accounting for the properties of gases, and that of the ionic lattice theory in accounting for those of crystals, have been such as to warrant the belief that the main features of the constitution of gases and crystals are now understood, and that future development will proceed mainly on the lines already laid down. For liquids, on the other hand, we have hardly the beginnings of a theory; amorphous solids are in a slightly better state. The present article does not claim to have gone far towards the construction of such a theory, but it is hoped that it will at any rate indicate what are some of the chief problems to be faced when one is constructed.

Consider the pencil of [4]'s through the [3] which contains l15, l25, l34, any [4] of the pencil meets the V25 again in a conic, q.

The solution for an infinite cylinder filled with viscous fluid, rotating about its axis under the action of a constant couple, has already been obtained by the present writer by the operational method of Heaviside and Bromwich, and previously by other means by T. H. Havelock.

By a twisted quartic of the first kind is intended the curve in space of three dimensions defined by two quadratic equations, i.e. the complete intersection of two quadric surfaces. If this intersection consist of the aggregate of two or more curves of lower order than the fourth, it will be called a degenerate quartic. In general, a twisted quartic of the first kind will be referred to for brevity as a quartic.

Suppose that ƒ1, ƒ2, ƒ3 are three quadratics in the complex variable z with complex coefficients, say

and consider the equation

the α's being real quantities. It determines two values ζ, ζ′ of z and thus two points P, P′ in the complex plane. Clearly when P is given the ratios α1: α2: α3 are in general determinate and accordingly P′ also. This is the involutory transformation I propose to discuss.

The problem was suggested by Professor G. I. Taylor as being of interest and importance in the phenomena of rupture in a solid, due to the presence of a crack, and the slip in a crystal. In experiments on the distortion of a crystal of aluminium under a tensile stress the conclusion is reached that “…as far as these experiments go, the distortion of a crystal of aluminium under compression is of the same nature as the distortion which occurs when a uniform single-crystal bar is stretched. The distortion is due to slipping parallel to a certain crystal plane and in a certain crystallographic direction, and the choice of which of twelve possible crystallographically similar types of slipping actually occurs depends only on the components of shear stress in the material and not at all on whether the stress normal to the slip plane is a pressure or a tension.”

In the “special” or “restricted” theory of relativity, for which the line-element ds of the “world” of space-time is given by , the geodesics of the world are straight lines, and the null geodesics (i.e. the geodesics for which ds vanishes) are the tracks of rays of light. When Einstein discovered the “general“ theory of relativity, in which the effects of gravitation are taken into account, he carried over this principle by analogy, and asserted its truth for gravitational fields; it was, in fact, the basis of his famous calculation of the deviation of light at the sun. The law was, however, not proved at the time: and indeed there is the obvious difficulty in proving it, that strictly speaking there are no “rays” of light—that is, electromagnetic disturbances which are filiform, or drawn out like a thread—except in the limit when the frequency of the light is infinitely great: in all other cases, diffraction causes the “ray” to spread out over a three-dimensional region.

If a bar is vibrating under water it will set up circulating currents which possess kinetic energy and so in effect add to its inertia: consequently the frequency under water will be less than the frequency in air. The effect of the water is precisely the same as if the density of the bar had been increased: the apparent increase of mass is usually called the “added mass” of the bar and the total apparent mass the “virtual mass”, and these terms will be used subsequently.

When the inner integrals in a pair of repeated integrals are absolutely convergent, inversion of the order of integration is a not too difficult problem and there are a number of general theorems which, under comparatively unrestricted appropriate conditions, can be applied to give a solution. When, on the other hand, the inner integrals are only conditionally convergent, the problem is of a very different nature and the solution, if it exists at all, is usually only to be obtained by special devices.

The lemma I propose to discuss may be stated as follows;

If ξ, η, ζ are three quantities whose sum is zero and λ, μ, ν three positive integers (≤n) satisfying the condition

then any polynomial of order n in ξ, η, ζ can be expressed in the form

where P, Q, R are polynomials of orders n − λ, n − μ, n − ν.

The potential of an electric charge in a gravitational field has been discussed very recently by Professor Whittaker and by E. T. Copson. The latter gives the solution

for a point centre at “distance” a from a mass of gravitational radius α. The coordinates used are such that the metric is given by

In two recent papers Dirac has shown how the “duplexity” phenomena of the atom can be accounted for without recourse to the hypothesis of the spinning electron. The investigation is carried out by the methods of non-commutative algebra, the wave function ψ being a matrix of the fourth order. An alternative presentation of the theory, using the methods of wave mechanics, has been given by Darwin. The four-rowed matrix ψ is replaced by four wave functions ψ1, ψ2, ψ3, ψ4 satisfying four linear differential equations of the first order. These functions are related to one particular direction, and the work can only be given invariance of form at the expense of much additional complication, the four wave functions being replaced by sixteen.