In a paper on random flight Lord Rayleigh proved the following result: A number is formed by adding together n numbers each of which is equally likely to have any value from − a to + a. Then, if f (n, s) ds is the probability that the number so formed lies between s and s + ds, and if n is sufficiently great,

This result may be stated as follows: A point moves discontinuously in a straight line. For a time τ it has a constant velocity. During the next time-interval τ it again has a constant velocity, and so on. Then if each of these velocities is equally likely to have any value from − v to + v, the probability that in the time nτ, the point moves a distance lying between s and s + ds is f (n, s) ds, with vτ written for a.

The propagation of electric currents through long cables has often been treated theoretically and the solutions, first obtained by Kelvin and by Heaviside, have moulded the development of transcontinental telegraphy and telephony. These solutions presume that the source of electrical disturbance is situated at one point in the cable and this is the condition appropriate to wired communication, but in the more recent process of wireless communication we have an analogous problem which requires different terminal conditions.

1. There seems to be little doubt that the detailed mechanism of ionisation or excitation by electron impact will not admit of description, even to a first approximation, by the system of classical mechanics. However the experimental work of Franck and Horton and their collaborators, has shown that, within the limits of the error of experiment, an electron is able to impart the whole or part of its kinetic energy to the atom it excites. When its initial energy is greater than the excitation energy, the electron, instead of being brought to rest, retains the excess as kinetic energy.

In hypergeometry loci defined by parametric equations

have been frequently discussed from various points of view. Among other properties these loci possess two systems of generating spaces obtained by keeping the α's (or β's) fixed while the β's (or α's) vary, the simplest example being the generators of a quadric in S3. Another method of obtaining loci having these properties is by taking the flat spaces through the sets of a linear series of sets of points on an elliptic curve and the flat spaces through the sets of the residual series (the curve being supposed normal. In this way we shall obtain a configuration of ∞PSP's and ∞qSq's, the dual of which gives a locus included among those represented by (1).

§ 1. The six Plücker coordinates of a straight line in three dimensional space satisfy an identical quadratic relation

which immediately shows that a one-one correspondence may be set up between lines in three dimensional space, λ, and points on a quadric manifold of four dimensions in five dimensional space, S5. For these six numbers pij may be considered to be six homogeneous coordinates of such a point.

1. The axioms necessary for the construction of a proof of Pappus' Theorem, regarded as a theorem in the geometry of projective space of three dimensions, fall into three groups:

I. Axioms of Incidence;

II. Axioms of Order, giving the properties of the relation “between”, and establishing the order type of the projective line as cyclical and dense in itself;

III. An Axiom of Continuity.

It is customary in treatises on projective geometry to adopt in Group III the Axiom of Dedekind, which states that if the points of a segment are divided into two classes, L and R, which have each at least one member, and are such that no member of L lies between two points of R, nor vice versa, then there is a point of the segment, not an end-point, which is neither between two points of L nor between two points of R. This axiom, however, assumes considerably more than is necessary for the proof of the theorem.

I have to correct the proof of the assertion on p. 471 of my paper that if f (t) is integrable over (0, a) and ν ≥ – ½, then

Thirty years ago, in a paper on continued fractions, Stieltjes published a definition of the integral which bears his name. His replacement of the variable of integration x by a more general “base function” φ(x)—a change which throws so much light upon other theories of integration—received at first little attention, but has later sprung into greater prominence; so much so that Professor Hildebrandt, in summarizing these various theories in a paper to the American Mathematical Society, makes the statement that “it [the Stieltjes Integral] seems destined to play the central rôle in the integrational and summational processes of the future.” Yet even now the integral and the allied theory of differentiation with respect to a function have been subjected to little detailed analysis, and the possibilities of extension have been only touched upon. It is the object of this present paper to establish certain results which are of some value in themselves and which prepare the way for an attack upon the integral.

The following paper aims at a more general treatment than has hitherto been given, of the integral expansions of arbitrary functions, from the point of view of integral equation theory.

Let

be a system of differential equations of Hamiltonian form, the characteristic function H being independent of t and expansible in a convergent series of powers of x1, … xn, y1, … yn in which the terms of lowest degree are

It has been pointed out to me that some of the statistical ideas employed in the following investigation have never received a strictly logical definition and analysis. The idea of a frequency curve, for example, evidently implies an infinite hypothetical population distributed in a definite manner; but equally evidently the idea of an infinite hypothetical population requires a more precise logical specification than is contained in that phrase. The same may be said of the intimately connected idea of random sampling. These ideas have grown up in the minds of practical statisticians and lie at the basis especially of recent work; there can be no question of their pragmatic value. It was no part of my original intention to deal with the logical bases of these ideas, but some comments which Dr Burnside has kindly made have convinced me that it may be desirable to set out for criticism the manner in which I believe the logical foundations of these ideas may be established.

It has been known for a very long time that, under certain conditions rather difficult to specify, the discharge through an ordinary two-electrode vacuum tube containing gas at a low pressure, even when the potential is maintained by a battery of low resistance Storage cells, is intermittent.

A statistical table is in effect a classification of a finite number, N, of objects in respect of a finite number of different classes, A, B, C, … It is assumed that unambiguous rules have been laid down by which it is possible to determine whether any particular one of the objects does or does not belong to any particular one of the classes. The application of these rules to a particular object does not depend on the fact that that object is one of a finite number of N objects, so that the process of classification may be started on a non-finite collection of objects. When the collection of objects is non-finite the process can never end. On the other hand when the collection is finite the process must end; and when completed it will determine how many of the objects belong to any particular class. If in this way it is found that N1 (≤ N) of the objects belong to class A, the proper fraction N1/N is spoken of as the frequency of class A in the collection. In particular cases it may be zero or unity. In general it is a rational proper fraction. For its determination in complicated cases it may be convenient to suppose the collection to be arranged in some special way, but its value is absolutely independent of any such particular arrangement.

It has long been realized that a Stieltjes integral can be defined and developed on the same lines as a Lebesgue integral, using in place of the measure of a set E the variation of an increasing function ø (x) upon E; but only the more obvious—and less useful—properties of such an integral seem to have been stated. The original integral defined by Stieltjes possesses the remarkable Integration by Parts property that if either of exists, so does the other, and their sum is equal to

The instrument to be described is a modification of an electrostatic oscillograph originally invented by Prof. Taylor Jones. In the original instrument a thin phosphor-bronze wire is stretched parallel to a flat metal plate towards which it is attracted when a difference of potential is established between them. A small mirror is fastened by one edge to the middle of the wire, and the other edge of the mirror is pivoted by bringing up to it a flat cork against which it presses. The motion of the wire is recorded by reflecting a beam of light from the mirror on to a rotating drum.

The problem of finding general types of solution for the steady flow of gases in two dimensions, adiabatically and without friction, does not appear to have received a great deal of attention. An interesting and suggestive treatment of hydrodynamics applied to gases is given in Riemann-Weber's Partielle Differential-Gleichungen, but the application is to pressure propagation. Reference must also be made to a paper by the late Lord Rayleigh, who gave a general differential equation which must be satisfied by the velocity potential ø, but the problem of utilizing this equation does not appear easy.

Perhaps the most peculiar fact of observation yet recorded about the wing disposition of soaring birds is the apparent large negative angle of incidence shown by the wings of vultures when in fast horizontal flight.

It is shown that, assuming conservation of energy and momentum in individuai transitions, radiation must be regarded as propagated by linearly directed quanta. Evidence is brought forward in favour of the view that these quanta are spatially localised longitudinally as well as laterally. As these conclusions seem unavoidable, an attempt is made to see if there is any possibility of accounting for interference effects in terms of quants possessing periodic properties.

The idea that a single quantum can produce interference effects is untenable. The conception of the phase of a quant is introduced, and it is supposed that atoms may absorb a quant in a given phase provided they are in a “state” corresponding to that phase. Considering radiation emanating from a point, it is shown that a tentative explanation of interference may be obtained if the quants have a constant phase at emission, and if they give atoms at any point an impulse towards the state corresponding to their phase at the point. According as these impulses reinforce or oppose each other, bright or dark parts of a fringe System are obtained. Practically the whole nomenclature and method of the wave theory may be taken over and applied to the quant phase field surrounding an emitting source. Essentially, on this view, an explanation of interference is sought in terms of different quants, and the dark and bright parte of a fringe system are to be explained, not in terms of different numbers of quante arriving at them, but in different phase relations of quante which do arrive.

Quante are characterised by a magnetic and electric vector, whose magnitudes and directions define the state of polarisation and the phase. Electron orbite may also be regarded as characterised by two vectors in terms of which it seems as if the “state” may be specified.

The relation between transition character and type of radiation emitted is finally considered and it is shown that the evidence so far available renders plausible an explanation of interference of the kind put forward. Much further experimental work, particularly in connection with the Stark and Zeeman effects, is, however, necessary, before sufficient precision can be given to the views put forward to make them satisfying. On the other hand, it seems premature to suppose that only a formal treatment is possible of the physical phenomena involved.

This paper is a study of a new method of enumeration of the partitions of multipartite numbers.

Incidentally an algebraic function, which is derived from the repetitional exponents of partitions of unipartite numbers, presents itself. The generating function which enumerates the partitions of unipartite numbers is expressible in terms of these functions and finds in such expression its fullest connection with the divisors of numbers. There are also similarly derived functions connected directly with bipartite, tripartite, etc. numbers. It has not been necessary to study these for the purposes of this paper.

In addition to the simple thallous and thallic chlorides and bromides, a considerable number of intermediate and mixed halides have been described by various investigators. Most of the modern work on these substances has been done by Thomas, by Meyer and by Cushman. There is general agreement as regards the existence of substances of the type TlX3. 3TlX, but considerable divergence exists with regard to the mixed chlorobromides. Compounds of the type TlX3. TlX have also been described, the best known of which appears to be the dibromide TlBr3. TlBr.