1. Let .

When r is a positive integer, various writers have considered sums of the form

where ω1 and ω2 are two positive numbers whose ratio θ = ω1/ω2 is irrational and ξ is a real number satisfying 0 ≤ ξ < ω1. In particular, Hardy and Littlewood (2,3,4), Ostrowski(9), Hecke(6), Behnke(1), and Khintchine(7) have given best possible approximations for sums of this type for various classes of irrational numbers. Most writers have confined themselves to the case r = 1, in which

1. It is here expedient to ignore the continued fractions from which they are usually derived and to define the simple continuants determined by the parameters a1, a2, a3, …, an, … as given by the chain of difference equations

together with the initial values u0 = 0, u1 = 1: and these data lead successively to the functions

and

This paper is a continuation of three others under the same title. The paragraphs are numbered following on to those of the third paper.

It has been observed that the Behrens and Fisher test of the difference of the means of two samples gives a smaller percentage of significant results than might be expected on the analogy of the ordinary t test with a pooled estimate of variance. The cause of this apparent anomaly is explained, and it is shown that the criticisms of the test to which the anomaly has given rise have their origin in (a) neglect of the relevant information provided by the estimated values of the variances, and (b) an insufficient appreciation of the fiducial basis of all tests of significance (including the ordinary t test) on small samples.

It is pointed out that Sukhatme's table (constructed for the Behrens and Fisher test) also provides a test for the weighted mean of the means of two sets of observations, concerning whose relative accuracy no prior knowledge is available.

Difficulties associated with the evolution of stars by radiation alone are briefly discussed. It is clear that some other process is also affecting the stars and it is shown that the stars are capable of adding to their mass by the process of accretion of the cosmical cloud. The gravitation of a moving star causes additional collisions of the atoms of the interstellar matter and the motions become randomized to such an extent that the star probably captures all material passing within the distance at which the velocity of the star relative to the cloud is the parabolic velocity. This rate of accretion of mass of a star is 4πγ2ρM2/ν3, and is accordingly of great importance for stars of low velocity. Stars of high velocity are least affected by accretion and therefore in general remain of low mass, while stars of low velocity must attain great mass. The periods of time involved in bringing about appreciable changes in the mass of a star are of the order of 5 × 1010 years and are in agreement with independent estimates of the time scale, as deduced, for example, from the companion of Sirius. The evolution of the components and orbits of binary stars are consequences of the accretion process. The more massive component increases in mass more rapidly than the less massive component in the case of wide pairs, and may therefore in general continue to emit more ergs per gram. The orbit evolves in such a way that the total angular momentum remains constant. For equal masses the separation is proportional to the inverse cube of the mass, and the period to the inverse fifth power, so that great changes of separation and period occur. The evolution of the stars is governed almost entirely by their velocities relative to the cosmical cloud. In the case of double stars the evolution takes the form of decreasing period and decreasing separation. Such features as galactic concentration and the correlation between spectral type and velocity are direct results of accretion.

It is well known that a single neutron may cause a nuclear reaction chain of considerable magnitude, if it moves in a medium in which the number of secondary neutrons which are produced by neutron impact is, on the average, greater than the number of absorbed neutrons. From recent experiments it would appear that this condition might be satisfied in the case of uranium.

The exchange forces between two heavy particles (neutrons, protons) have been treated on the basis of the meson theory by Yukawa, Sakata and Taketani (5), and by Fröhilch, Heitler and Kemmer (2), assuming the existence of positive and negative mesons only. The calculations have been extended by Kemmer (3) for a meson field which includes in addition neutral particles (neutretto's). The first order approximation of the exchange potential, i.e. the potential due to the emission and subsequent absorption of one meson only, gives qualitatively the correct interaction between the unclear particles.

Although the existence of an electromotive force in apparently homogeneous circuits under asymmetrical temperature distributions has been questioned by only a minority of workers in this field, all attempts at explaining the effect have been little more than speculations. These speculations may be divided roughly into two types, first(2), those which agree with the contention of Benedicks that the E.M.F. is an essentially new effect not explicable in terms of the classical theory of Kelvin, and secondly(3), those which contend that the effect is essentially spurious, to be explained away as due either to hidden sources of error, or to unsuspected heterogeneities in the circuit. The somewhat fantastic claim of Benedicks' school that the effect was an inverse of the Thomson effect, which proved entirely false upon examination in the light of Sommerfeld's work(4), did much to increase scepticism. Benedicks' later claim to have found yet another inverse of his original effect(5), the “electro-thermal” effect, has also been refuted(6), and this does not inspire confidence in the genuine nature of the original effect.

For the purpose of the present discussion the magnetron will be defined as a diode thermionic tube, having concentric cylinders as electrodes, in which there is a uniform magnetic field parallel to the axis of the electrodes. An electron emitted from the cathode travels towards the anode in a path which is bent by the action of the magnetic field. If the anode has radius a and is at potential V with respect to the cathode of radius b, then it is well known that electrons emitted without velocity from the cathode will just graze the anode when the uniform and axial magnetic field has a value H such that

1. In a note published recently in the Proceedings* Goldstein suggests the truth of the following theorem.

Nunn May (1) has developed the theory of the mode of action of the Geiger-Müller counter in terms of the photoemission produced in the electron avalanches constituting the discharge of the tube. An avalanche of electrons in the counter produces excitation and ionization of atoms and molecules in the neighbourhood of the positive central wire: the photons liberated by these in their turn release from the walls of the counter the photoelectrons which initate the following avalanche. Direct experimental evidence that such a sequence of events may occur is to be found in the work of Greiner (2).