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.

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.

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.

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 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 Mathieu functions of period π and 2π have recently been constructed by the help of analysis similar to that developed by Laplace, Kelvin, Darwin and Hough to find the free tides symmetrical about the axis of a rotating globe. The purpose of this note is to show that a similar construction can be carried out for the second solution of the Mathieu equation, when one solution is periodic in π or 2π, by the help of analysis similar to that used for forced tides. The construction is effected in a form suitable for numerical computation.

Most of the results of this paper have been given before by Professor T. H. Havelock, who obtained them by the solution of a difficult integral equation. The method used here is, however, much easier to handle than that given by Professor Havelock.

Gyroscopic effects furnish important instances of dynamical actions, but their general discussion is outside the range of most students reading for Part I of the Natural Sciences Tripos. If, however, we limit ourselves to the case in which the axis of the wheel makes a constant angle with the vertical, the theory becomes elementary and simple devices suffice for recording the movements of the revolving wheel. The recording gyroscope was designed to give students at the Cavendish Laboratory, Cambridge, an opportunity of making practical measurements to verify the theoretical results.

A very great deal has been published upon the subject of the electrical counter and its mode of action. Most explanations are incomplete in that they neglect some of the factors involved. The question of the self-restoring property of the system has also received attention, but this property has not been properly related to the general problem of intermittency in discharge tubes.

Experiments have been conducted by Gutton, and later by Kirchner, and by Gill and Donaldson upon electrical discharges through gases under the influence of high-frequency oscillations of the order of 107 cycles per second. It was found that the peak voltages required to maintain bright luminous discharges were of the order of 100 volts even when the pressure was as low as that in a soft X-ray tube. The present paper deals with some further studies of these phenomena.

In the light of the hypothesis that the helium atom can exist in a hydrogen-like form in which one electron is relatively far removed from the nucleus with respect to the other, a series of experiments have been carried out with a view to determine whether or not helium could be obtained in the form of a compound similar to certain known hydrides. The mass of evidence points to a positive result. A gaseous compound was shown to form when excited helium was passed over a strong source of the active deposit of radium. Compound formation was proved by counting scintillations caused by the transportation of the radioactive material from the discharge tube into a bulb containing a zinc sulphide screen. That the effect observed in the bulb was due to the presence of a gaseous radioactive helide and not to radium emanation or solid particles in suspension was definitely proved; that it was due to hydrogen impurities in the helium was shown to be very unlikely. The compound in hydrogen was more efficiently condensed by a carbon dioxide-ether cooling mixture than that in helium. The catalytic effects of tap grease and mercury vapour on the reaction were noted.

In conclusion the writer wishes to thank Professor Sir Ernest Rutherford, O.M., P.R.S., for his interest in the work; Dr J. Chadwick, F.R.S., for advice and helpful suggestions; Mr J. A. Ratcliffe, for the wireless set used in producing the electrodeless discharge; Mr Crowe, for the preparation of the radioactive sources; and Mr Brown, for help in counting scintillations.

There are two general methods of measuring the elastic constants of bodies; one involves a study of the static deformation produced by the appropriate kind of stress, and the other a measurement of the period of oscillation of a system of known inertia under the elastic forces.

In the theory of the specific heats of gases of diatomic molecules the function

plays a well-known and important part. The rotational specific heat Crot of a diatomic molecule, which is susceptible of representation as a rigid body with two equal principal moments of inertia, without spin about the other principal axis, is given by

where R is the gram-molecular gas-constant and

the pair of equal moments of inertia being equal to A. Whether such a model is or is not an adequate representation is a matter for determination by a detailed study of the structure of the band spectrum, particularly of the nature of the normal electronic and vibrational state. It is known to be applicable to normal molecules of the halogen hydrides, to CO and other molecules which have a normal state of type 1S including (but for a certain special feature) H2.

In a recent paper in these Proceedings the writer suggested the possibility of a transition from one molecular form to another in CO2. The suggestion is embodied in the equation (10) and the resulting specific heats for low temperatures given. He greatly regrets that it was not till after those results were published that he found they gave a high and altogether impossible maximum in the specific heat curve for higher temperatures before it returns to the neighbourhood of the unmodified curve Cv′.

The great increase in the speed of chemical change at the surfaces of specific substances has always been one of the most mysterious of all chemical phenomena, and the collection of evidence which will throw light on the nature of the process is research which lies at the foundation of chemical science, because it concerns the ultimate nature of chemical combination and dissociation.

To the conditions of reaction in homogeneous systems is added yet another condition, the adsorption of the reactants on the centres of activity of the surface. The surface is treated as if it were homogeneous, reaction taking place as if those centres on which the heat of activation is smallest were alone responsible for chemical change.

The general equation is worked out in terms of the rate of bombardment, and the mean lives of the molecules on the surface. It is incidentally shown that the same areas associated with the forward reaction must inevitablycatalyse the backward reaction.

The general solution is impracticable, so the special cases of irreversible synthesis and decomposition are considered and results are obtained.

Wilson's cloud method, provides a means of measuring the ranges of β-rays which is free from any uncertainties arising from the scattering of the rays, and for this reason it is an invaluable method for obtaining the true rate of loss of energy of β-rays as they traverse matter. Thetrue rate of loss of energy or ‘stopping power’ is defined as the quantity , where is the mean energy lost by β-rays of energy T in travelling a distance dx measured along their paths. The cloud method of course enables one to observe only the total ranges of β-particles, and in deducing the stopping power from the mean range it is necessary to bear in mind that owing to ‘straggling,’ which in the case of β-rays is large, the differential coefficient , where is the mean range of particles of energy T, is not exactly the same as the stopping power . is however the quantity which the cloud method enables us to observe. It is the purpose of this note to consider the exact relation between these two quantities so that the information provided by the cloud method may be rigorously interpreted.

The absorption curves and the photo-ionisation experiments agree to give a value for the molecular photo-electric threshold of λ 2555 ± 20 a.u. If our view of the ionisation process is correct, the energy of dissociation is 0·505 ∓ ·01 volts. If the picture of the ionisation process is wrong, the energy of dissociation may be a little lower (a few hundredths of a volt). The corresponding heat of dissociation is 11400 calories. In the range of temperature (200° C. to 500° C.) generally used in experiments on potassium vapour, the fraction associated varies from 10−4 to 10−2. An equation for calculating the fraction associated at any temperature is given by the writer in an earlier paper.

In a recent communication to this Society Dr Hartree has put forward a method for calculating the field of anatom containing many electrons. Each orbit—to borrow a metaphor from the old quantum theory—is related to a wave-function Ψ which obeys Schrödinger's equation. The potential energy used in this equation is due partly to the field of the nucleus, and partly to the fields of the electrons in the other orbits. The latter are calculated upon Schrödinger's interpretation of the wave-function, that |Ψ|2 is the density of charge, measured in electronic charges per unit volume. It is not the purpose of this paper to discuss the practical methods of obtaining wave-functions which reproduce the fields from which they are derived; but to relate these wave-functions and their energy parameters to those of the accepted theory.

Duane's quantum theory of diffraction is applied to the reflexion of electrons by crystals and to the spatial distribution of photoelectrons and fluorescent radiation from a crystal.

Two alternative criteria for coherence are given. According to the second of these there is coherence provided that the momentum imparted to the components of a system during the process concerned is insufficient, owing to quantum restrictions to their motion, to change their energy. Calculations made on this supposition show that in the case of the scattering of radiation by a crystal there is complete coherence for the lower orders, while for higher orders this ceases to be the case and the reflected intensity is reduced as the result of incoherent scattering. The specular reflexion by gases is also considered.

Duane's theory of diffraction has not been placed on a rigorous basis and the solutions which have been proposed here for various problems may be incorrect. Even so the theory has at least served to bring to light several points of theoretical and experimental interest which deserve consideration.