Let σs(n) denote the sum of the sth powers of the divisors of n,

and let , where ζ(s) is the Riemann ζfunction. Ramanujan, in his paper “On certain arithmetical functions*”, proves that, if

and

then , whenever r and s are odd positive integers. He conjectures that the error term on the right of (1·1) is of the form

for every ε > 0, and that it is not of the form . He further conjectures that

for all positive values of r and s; and this conjecture has recently been proved to be correct.

In this short note my chief object is to prove the following theorem:

If

(1) f(z) is a regular function of z(= ρeiψ) in the angle |ψ| ≤ α, where ;

(2) f(z)= 0(eαρ), where K is a positive constant, throughout this angle;

(3) f(z) is not identically zero;

then

(4)

is a continuous function of ψ for |ψ| < α.

The surfaces considered in this note are Jacobians of four quadric forms (“quadrics”) in [4], where one or more of the quadrics are space-pairs. It is believed that, with one exception, they have not been noticed previously; and so little is known of non-rational surfaces in higher space that it seems worth while putting them on record.

1. The power possessed by water and air in motion of lifting and carrying solids considerably denser than themselves is familiar to all; but it does not seem to have been pointed out that classical hydrodynamics provides a simple explanation. If a solid rests on the bottom of a stream, the points of contact are points of zero velocity; and the velocity just above the solid, by the equation of continuity, must be greater than the general velocity. Hence the velocities produce high pressures under the solid and low ones above it, and the difference tends to lift the solid up. If the resulting thrust exceeds the weight of the solid in the liquid, the solid will be raised, and will be unable to rest in equilibrium on the floor of the stream.

The stability and the stream lines of the symmetrical and unsymmetrical double rows in a channel have already been discussed by the author. It is the object of this paper to investigate the possibility of the existence of a double row whose axis is parallel to, and not necessarily coincident with, the axis of the channel. It will be seen that, if the system of vortices is to move forward as a whole, then the only possible types of double row are those that have been discussed previously, that is, the symmetrical and unsymmetrical systems, and in both of these cases the centre lines of the row and channel are coincident. It was shown previously that the symmetrical row is always unstable and that the unsymmetrical system is stable within a particular domain of stability.

Hobson has given a proof of this theorem in its fullest generality. The present note gives an alternative for part of Hobson's argument. The theorem may be stated in two forms. If f(x) is a function of x, monotone when a ≤ x ≤ b, and φ(x) is integrable over the same range, then

where a ≤ X ≤ b,

(ii) the same holds with a < X < b except in some trivial cases where f(x) is constant in the open interval a < x < b. The form (ii) is not mentioned by Hobson.

Miquel's Theorem, as generalised by Clifford, states that the four circles through the intersections of four straight lines, taken in triads, meet in a point; that the five such points derived from five lines, taken in sets of four, lie on a circle; that the six such circles, determined by six lines, meet in a point……. The process continues indefinitely.

The solution of problems in diffraction by an elementary application of Huyghens's principle is discussed. The obliquity function is investigated, using the criterion that the formula used must give the right result when integrated for the case of an undiffracted plane wave. It is shown that this is satisfied for distant points by any function which makes the integrals converge, but that to satisfy it completely a constant obliquity function is necessary. This makes a consideration of the distant boundary essential, as the integrals do not converge in this case. It is shown that a boundary distributed in a Gaussian way is completely satisfactory. The integral for the case of diffraction by a straight edge is solved exactly, leading to the usual result. Finally, the results are discussed in relation to the teaching of the subject.

I am indebted to Dr H. Jeffreys and to Mr F. P. White for their interest in this paper; also to Mr N. F. Mott, who has contributed much to the course of its development.

The problem of the collision between an electron and an atom was first considered on the Quantum theory by Born, who has worked out in detail the case of atomic hydrogen, and has obtained formulae giving the variation of scattering with angle both for elastic and inelastic collisions. Born's solution is only approximate. The purpose of this note is to discuss the physical nature of the approximations used by him, and also to extend his results to Helium, for which experimental evidence is now available. It is found that the theoretical curve agrees with the experimental as well as the approximations used would lead one to expect. We shall confine ourselves to elastic collisions.

The purpose of this note is to extend in two respects some previous work of the writer on the wave equation of a particle in a central field of force which is a Coulomb field for sufficiently large r.

The object of this paper is to make certain calculations which will be required in the estimation of the quantum defect. Hartree has defined certain solutions Gi(σ, ζ), Hi(σ, ζ) of the wave equation for an electron in a Coulomb field, viz.

In this paper we use Dirac's relativity quantum mechanics to derive the well-known Kramers-Heisenberg dispersion formula for an atom with one electron. The treatment is not limited to the case of a central field, but is quite general. An expression is also obtained for the dipole moment to which is due the incoherent scattering. The formulae obtained are similar to those obtained by O. Klein for the case of a central field. We find in this way an explicit expression for f, the number of dispersion electrons for any line of the optical spectrum (being a measure of the intensity of the line), in terms of the solutions of the four wave equations of Dirac's theory. It is further shown that the sum of the number of dispersion electrons for any state of the atom is not exactly unity, but differs from it by an amount of the order of 10−4. The result Σf = 1 has been shown by London to hold exactly for the simple wave equation as originally given by Schrödinger. It is here shown that the exact relativity treatment has a very small effect.

A method of obtaining approximately the high-velocity limits of continuous β-ray spectra by electroscope measurements is established and used for the β-ray bodies of thorium active deposit.

The absence of thorium C γ-rays is confirmed and leads to some interesting results.

The work was carried out in the Cavendish Laboratory, Cambridge. Thanks and gratitude are due to Professor Sir Ernest Rutherford, P.R.S., for his interest in the problem; to Dr C. D. Ellis, F.R.S., who suggested the problem, for continual help and advice; to Mr G. A. R. Crowe for help in preparation of the sources; and to the Department of Scientific and Industrial Research for a Grant during the time the work was being carried out.

In the measurement of small ionisations it is usual and desirable to employ an ionisation-chamber whose central collecting electrode serves also as an electroscope, of the gold-leaf or the quartz-fibre type. As an instrument for the measurement of quantity of electricity even the most sensitive of these is inferior, by a factor of about 100, to the combination of an ionisation-chamber and a Compton electrometer. The new type of ionisation electroscope here described combines a high voltage-sensitivity with a small capacity and has a sensitivity for quantity of electricity of the same order as the combination mentioned.

During my visit to Cambridge last year I was given the opportunity of carrying out in the Cavendish Laboratory some experiments with a view to studying the reflection of atoms from a solid surface. For this purpose the atoms of radioactive substances offer the advantage of extreme sensitivity and also the possibility of quantitative measurements by means of their radiations. While the main object of the experiments could not be achieved, some phenomena have been observed which seem to deserve brief mention.

While investigating the effect of a caesium film on the auto-electronic emission from a tungsten surface I noticed certain phenomena which seemed worth further investigation. Langmuir and Kingdon have shown that the emission from a tungsten filament in caesium vapour varies with the temperature of the filament in the manner shown in Fig. 1. The initial rise in the emission at low temperatures is due to the existence of a caesium layer on the filament. At higher temperatures the caesium evaporates and the emission drops until eventually the temperature becomes high enough to produce a thermionic emission from the clean tungsten surface.

It is shown that a quartic equation may be found which expresses the grid characteristic of a valve accurately over a considerable range, and that for modern dull filament valves the grid current curve is exponential over only a very restricted length.

Expressions are given for calculating rectified current and the amplitude of harmonics produced by a simple grid rectifier when voltages of the following wave-forms are applied: E sin pt, (A + B sin nt), and (A + B cos mt + C cos nt)sin pt. Further, a formula is given from which we may calculate the rectified current produced by a cumulative grid rectifier when a voltage E sin pt is applied.

All these formulae are applicable if the grid characteristic, is parabolic, cubic, quartic or quintic in form.

It is also shown that when an acoustically modulated radio frequency voltage is applied to a cumulative grid rectifier, even if the valve has a parabolic grid characteristic, an infinite number of harmonics of the modulation frequency are produced. These harmonics are due to the. action of the grid leak and condenser. It appears, however, that in practice the amplitude of these harmonics will be very small compared with those produced, primarily, by the curvature of the grid characteristic.