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.

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.

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.

The actual form of the set of linear substitutions on 3 symbols, which gives a representation of the simple group of order 168, has been known for many years.

In these Proceedings (vol. 20, pp. 247–9) I have determined the form of the set of linear substitutions on 5 symbols which gives a representation of the simple group of order 660.

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.

§ 1. Bohr's original (1913) theory of the stopping power of light atoms for α-particles seems to invite further study. It is well-known that in this theory, which exhibits a general agreement with experiment, the loss of energy by the α-particle is calculated on purely classical principles, as the energy transferred to electrons initially at rest; and elastically bound to atoms. A large part of the loss so calculated (about one-half) comes from transfers to electrons in distant atoms of amounts less than the energy required to transfer those atoms to their next higher stationary state. This has created a serious difficulty according to more recent views of the quantum theory in which it has been presumed that any effective interaction must always leave the atom in a stationary state, and has given rise to the investigations of Henderson, which however do-not agree with experiment. In some recent speculations Bohr has suggested a possible way of escape from this difficulty, of great theoretical interest, to which he has been led by somewhat similar difficulties in the theory of radiation. He suggests that there may be reason to expect that energy may be only statistically conserved in just such a type of interaction by the atomic switches, while the classical calculation for the α-particle ought to retain its validity.

If G is a group of finite order which contains an operation P of prime order p, permutable only with its own powers, the order of G must, by Sylow's theorem, be of the form (1 + kp) pś, where s is a factor of p – 1. The greatest subgroup of G, which contains self-conjugately {P};, the subgroup generated by P, must be a metacyclical subgroup {S, P}, where

while g is a primitive root of the congruence gs = 1 (mod. p).

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 ‖.

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.

(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.

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.

This paper completes an investigation, of which the first part has already been published, into the integrals of a Hamiltonian system which are formally developable about a singular point of the system. Let

be a system of differential equations of which the origin is a singular point of the first type, i.e. a point at which H is developable in a convergent Taylor series, but at which its first derivatives all vanish. We suppose that H does not involve t, and we consider only integrals not involving t. Let the exponents of this singular point be ± λ1, ± λ2,…±λn. In Part I, I considered the case in which the constants λ1,…λn are connected by no relation of commensurability, i.e. a relation of the form

where A1…An are integers (positive, negative or zero) not all zero, and showed that the equations (1) possess n, and only n, integrals not involving t which are formally developable as power series in the xk, yk. In this paper I consider the case in which λ1 … λn are connected by one or more relations of commensur-ability. Suppose that there are p, and only p, such relations linearly independent (p > 0): it will be shown that the equations (1) possess (n − p) independent integrals not involving t, formally developable about the origin and independent of H.

Let the eight points in space common to three arbitrary quadric surfaces be called eight associated points. These points possess the property that any one of them is uniquely determined if the other seven are given. It is interesting to give the analytical expressions for eight associated points in terms of symbols entirely free from coordinate systems. Thus we take the first eight digits 1, 2,…8 to denote the points; a group of two digits to denote the line joining these two; a group of three to denote a plane, and so on. Also a conjunction of two groups denotes the point, line or plane common to the two groups, thus (34, 678) is the point common to the line 34 and the plane 678.

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.

(1) A method has been devised and applied for measuring the decay Constant of radium emanation, the method depending on careful comparisons with a simple electroscope under the most favourable conditions.

(2) A mean value for the Constant of radium emanation has been obtained, with an error unlikely to exceed 0·2 per cent.: λ = 0·1808 day−1, or half-period 3·833 days.

One method of measuring frequency depends upon resonance. If a body of unit mass, subject to elastic constraints and experiencing a damping resistance proportional to its speed, be acted on by a periodic force F sin pt, the equation of motion may be written

The amplitude of the steady vibration ultimately set up is

the maximum value occurring when p2 = q2 − 2s2. If the motion, in the absence of the dri ving force F sin pt, be only slightly damped, s2 is very small compared with q2, and little error will be made if the value of p giving maximum amplitude be taken as equal to q. When s/q is small, the resonance is “sharp,” i.e. a small discrepancy between p and q, or, more strictly, between p and (q2 − 2s2)½, causes the amphtude to fall far below its maxìmum value. Thus, when the damping is small, maximum amplitude, f or variation of p, indicates that p is very nearly equal to q.

(1) The conditions of dissociative equilibrium in an external gravitational field have been given by Willard Gibbs, as follows: If the state of a system is such that it is in (mechanical and thermodynamic) equilibrium when the constituents are taken to be independent, and if a condition of dissociative equilibrium is satisfied at one point, then the same condition is satisfied at all points, and the state is one of equilibrium if the constituents are actually capable of dissociating.

(2) In particular, the constituents of a column of dissociating gas under gravity settle out according to Dalton's law; if the condition of dissociative equilibrium is satisfied at one height it is satisfied at all heights.

(3) The conditions of equilibrium are generalised so as to take account of external electric fields and of the possibility of the products of dissociation being charged. Result (1) is shown to be unaffected.

(4) The theory is applied to the equilibrium of an ionized gas such as a stellar atmosphere under gravity. Whatever the charge on the star, the tendency of the light electrons to diffuse away from the heavy ions is almost entirely prevented by the electrostatic forces between them, and the result is the production of a field in the interior capable of supporting half the weight of the positiveions (Pannekoek, Rosseland). For the purposes of this statement the “interior” may be taken to commence at a pressure of 10−24 atmos. It is only above the level corresponding to 10−34 atmos. that electrons and ions separate out according to Dalton's law.

(5) The theory is applied to the equilibrium of an ionized gas under an external electric field. An external applied field of the order of 40,000 volt cm.−1 cannot give rise to potential differences exceeding about 1 volt under the conditions of a typical stellar atmosphere.