It is well known that in non-euclidean geometry a plane triangle has four circumcircles, and that each of these circles touches each of four other circles. The latter theorem, which is an extension of that of Feuerbach, is essentially due to Hart.

In this note we obtain a system of characteristics on which depend the main enumerative properties of complexes, or systems ∞3, of conics in space of four dimensions. The method used is applicable to a large number of problems of this nature, and we select this illustration as being analogous to congruences in ordinary space. In particular a finite number of conics pass through an arbitrary point, thereby defining the order, n, of the system. Linear complexes are those for which this order is unity. The conics of a complex satisfy eight simple conditions, in general conditions of contact with a fixed form or forms, but they may include conditions of incidence with a surface, each such counting once, with a curve, each such counting twice, or with fixed points each such counting thrice. In particular only incidence conditions can occur when the system is linear, for otherwise more than one conic would pass through certain ∞3 points of space. Points through which more than a finite number of conics pass are termed singular, as well as their loci. Directrix constructs are necessarily singular, but they do not necessarily exhaust the singular system, for the complex may possess ∞2 curves lying on a surface. In the linear case through a point of a singular curve pass ∞2 conics, and through a point of a singular surface ∞1 conics. The possibilities in the nonlinear cases are too numerous to be detailed. Similarly the system of planes of the conies may possess a singular curve, through which ∞2 of the planes pass.

The Lorentz group is denned by the equations

where the coefficients are subject to the conditions

It is an intransitive group possessing the single invariant

I consider functions of numerical magnitudes ast, where s, t may each assume the values 1,2, … n.

Symmetric functions are those which are unaltered by the whole of the substitutions of the n integers, when they are applied simultaneously to both suffixes.

(1) The simplex in a space Sn of order n (or n − 1 dimensions) is a set of n points a1 a2 …an which do not all lie in a subspace of Sn.

The elements of the simplex a1 a2 …an are the n points a1 a2 …an whieh are elements of order one, the lines ar as which are elements or faces of order two, etc., etc., and finally the n primesa1 a2 … ar−1 ar+1 … an which are elements or faces of order n – 1.

The following is a simple example of what Poincaré calls probabilité du continu.

Two points are marked at random on a unit line and it is assumed that all positions of either point are equally probable. What is the average or probable positive distance between the two points?

In the course of a series of signal intensity measurements made at Cambridge on the electric waves received from broadcasting stations a marked difference between day and night conditions has been observed. The day-time signals are sensibly constant but marked fluctuations of intensity become apparent about sunset and continue throughout the night. These variations are detectable at Cambridge on the signals from London, where they represent a change in strength of about 5 per cent.

Some previous work on the theory of the electrical counter is extended by a detailed consideration of the ionic process involved. It is shown that with a positive point the total number of ions produced should be proportional to the initial ionisation. Evidence is adduced to show that with a negative point extinction of the discharge is probably brought about by a local increase of pressure, accompanied by an electrical wind along the sides of the point. The latter would bring about the discharge of a definite quantity of electricity, independent of the initial ionisation, provided the initial pressure was not too low. It is shown that these considerations account for a number of other phenomena connected with discharges from small electrodes, and that the quantities involved are of the right order of magnitude.

The experimental work in connection with this paper has been carried out in the Cavendish Laboratory. The author desires to thank Professor Sir E. Rutherford for his continual interest in its progress, and further to thank Professor Appleton, and Dr Chadwick for much helpful discussion of several of the points involved.

The theorem of Wallace that if four arbitrary lines he taken in a plane the circumcircles of the four triangles formed by them in threes meet in a point—in which for circles we may equally well take conies drawn through two arbitrary points I, J of the plane —may be proved very easily from three dimensions. Through the four given lines draw four arbitrary planes, forming a tetrahedron X, Y, Z, T. A unique cubic curve can be drawn through I, J and the four vertices. Projecting this curve from the point T upon the original plane, we get a conic which passes through I, J and the vertices of the triangle formed by the lines in which the plane meets the planes TYZ, TZX, TXY, that is, by three.of the given lines. This conic, which is one of the conics of the theorem, also passes through the third point K, other than I, J, in which the original plane meets the cubic curve. Projecting in turn from the other three vertices of the tetrahedron, we get the other three conics of the theorem, which also pass through K.

In the Transactions of the American Math. Society, vol. I, 1900, and vol. IV, 1904, Mr E. Kasner treats exhaustively the (2, 2) double binary form and discusses the theory connected with double binary forms and multiple binary forms in general terms. He shows the relations between the systems of multiple binary forms with digredient variables and the forms with cogredient variables. Hitherto nothing seems to have been written on systems of triple binary forms (with regard to higher forms see a paper on the (1, 1, 1, 1) form by C. Segre); so here I propose to discuss the complete system of a (1, 1, 1) binary form which consists of six forms connected by one syzygy. When two of the variables are the same we naturally get the (2, 1) form and when the three are the same We get the (3) or cubic binary form.

§ 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.

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.

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.

The stability of a charged raindrop has been discussed mathematically by Lord Rayleigh. The case of an uncharged drop in a uniform electric field is perhaps of more meteorological importance but a mathematical discussion of the conditions for stability turns out to be very much more difficult in this case, owing to the fact that the drop ceases to be spherical before it bursts. Moreover it does not seem possible to express its geometrical shape by means of any simple mathematical expressions. On the other hand, by using a soap bubble instead of a water drop it was found possible to carry out experiments under well-defined conditions in this case, whereas experiments with Rayleigh's charged drop would be difficult.

The velocities of electrons emerging through an aperture in an anode to which they were accelerated have been investigated.

The velocity distribution of electrons passing through holes ranging from 0·13 mm. to 3·24 mm. in diameter in a thin copper sheet is shown in Fig. 2. In this case the accelerating field between filament and anode was 204 volts.

Corresponding results for a capillary tube 0·40 mm. in diameter and 16 mm. long are shown in Fig. 3, where the curves are plotted for different accelerating potentials. Experiments were also made with shorter lengths, down to 5 mm., and with a tube 1·25 mm. in diameter. All the tubes investigated gave the same distribution of velocities of the electrons in the emergent beam, within the limits of error of the experiments.

In conclusion the authors wish to express their thanks to Sir Ernest Rutherford for his interest and advice during the course of the work.

(1) An apparatus has been devised in which the composition of the vapour phase passing over the catalyst can be varied at will, or two vapours can be used alternately as reactants.

(2) By using ethyl alcohol, and amyl alcohol from fusel oil alternately as reactants, the poisoning of a very active copper gauze catalyst has been followed and the temperature coefficient of the dehydrogenation of ethyl alcohol by the copper has been measured during the poisoning.

(3) The activity of the catalyst, poisoned by amyl alcohol, fell in accordance with the simple logarithmic law

(4) Confirmation was obtained that the rates of dehydrogenation of ethyl and amyl alcohols were equal.

(5) The temperature coefficient of dehydrogenation on the catalyst was unchanged throughout the poisoning, though the reaction velocity was reduced 30 times.

(6) The temperature coefficient was also shown to be unchanged during sintering, the reaction-velocity having been reduced 10 times.

(7) Equations are derived for the rate of poisoning of an exponential distribution of centres of activity when the poison attacks the more active centres selectively.

(8) The relation deduced

cannot usually be distinguished (except with special relations between the constants) from the simple logarithmic law of poisoning.

(9) Thus the exponential distribution normally poisons like a surface composed of homogeneous centres, the temperature coefficient being unchanged by the poisoning.

(10) The equation

describes the reaction-velocity as a function of time, and temperature, and gives adequate explanation of previous experimental results on the forms of the heating and cooling curves with “poisonous” alcohols.

(11) Frenkel has shown , where τ0 is the vibration frequency of the molecule perpendicular to the surface, and μ0 is the heat of desorption of one gram molecule. Thus if the heat of desorption is not small compared with the energy necessary to cause chemical disruption of the molecule, the previous equation becomes

the form of the equation being unaltered.

1. The following are well-known theorems of elementary geometry: Given any euclidean plane triangle, A0 A1 A2, and any pair of points, X, Y, isogonally conjugate q. A0 A1 A2; then the orthogonal projections of X, Y on the sides of A0 A1 A2 lie on a circle, the pedal circle of the point-pair. If either of the points X,- Y describe a (straight) line, m, then the other describes a conic circumscribing A0 A1 A2, and the pedal circle remains orthogonal to a fixed circle, J; thus the pedal circles in question are members of an ∞2 linear system of circles of which the circle J and the line at infinity constitute the Jacobian. In particular, if the line m meet Aj Ak at Lt (i, j, k = 0, 1, 2), then the circles on Ai Li as diameter, which are the pedal circles of the point-pairs Ai, Li, are coaxial; the remaining circles of the coaxial system being the director circles of the conics, inscribed in the triangle A0 A1 A2, which touch the line m. If Mi denote the orthogonal projection on m of Ai, and Ni the orthogonal projection on Aj Ak of Mi, then the three lines Mi Ni meet at a point (Neuberg's theorem), viz. the centre of the circle J. Analogues for three-dimensional space of most of these theorems are also known ‖.

The problem of the temperature distribution in transformer or other laminated cores in which heat is generated at a uniform rate, does not appear to have been treated by exact mathematical methods. A good deal of work of a semi-empirical nature has been done by Vidmar, Kuchler and others, the object of which has been to enable designers of transformers to obtain an estimate of the maximum temperature likely to prevail in the core.

The purpose of this note is to calculate, on the basis of the assumptions which underlie the theory of the stopping of α-particles by matter, the influence of space quantization on this phenomenon. The results arrived at may be capable of an experimental test and will serve to illuminate these assumptions and to indicate their limitations.

1. The extended Riesz-Fischer theorem, in the special case of the trigonometrical orthogonal system, states that, if a and β are real numbers such that

then

implies, whenever α ≤ 2, the existence of a solution of the system of integral equations,

such that

and further

When a = 2, this reduces to the original Riesz-Fischer theorem.