1. In den folgenden Zeilen möchte ich einige Sätze mit Beweisandeutungen mitteilen, die sich teilweise auf die Besonderheiten der Konvergenz der Partial-summen zur entwickelten Funktion teilweise auf die Eigenschaften der “Legen-dreschen Polynome” einer Potenzreihe

beziehen, in welcher die Koeffizientenfolge {αn} mehrfach monoton ist. Eine Folge {αn} heisst k-fach monoton, wenn die Zahlen der k + 1 unendlichen Folgen {αn}, {δ1αn} {δ2αn}, …,{δkαn}, n = 0, 1, 2, …, durchwegs nichtnegativ ausfallen. Hier bezeichnet δvαn die v-te Differenz von αn, d.h.

1. In this note we give first our proof of a theorem (Theorem 1) which we stated in Note XIII. We then prove a new theorem (Theorem 2) which leads to another proof of the main theorem of Note XVII.

The first of these theorems requires some preliminary explanations. We are concerned with an integrable function f (θ) with the period 2π. We write

being the complex Fourier series of f (θ).

If pCε is a curve of order ε and genus p without singularities in space of three dimensions, the formula for the number of quadrisecants is well known, namely*,

Welchman has shown that the necessary reduction in this formula when the curve pCε has a point of multiplicity r with r distinct tangents, no three of which are coplanar, is

W. G. Welchman in his work on fundamental scrolls* obtains as directrix curves to such scrolls a canonical curve pK2p−2 in [p − 1] and a non-special curve pCn in [n − p]. These latter curves may not, however, be general curves regarded projectively, and it is an interesting question to find out the geometrical interpretation of their particularity. The curves C and K are, of course, in birational correspondence, and the prime sections of C correspond to the sections of K by quadrics through a contact set*, i.e. a set of points such that there is a quadric which touches K at every point of the set. For k small enough it is clear that every set of k points is a contact set and in this case the curves C are quite general. For larger k the fact that the set is a contact set simply means that, on C, the points of the bicanonical sets residual to a prime section lie themselves in primes (when k is sufficiently small the number of points in this residual set is such that they always lie in a prime).

1. A set of functions {øn (x)} is said to be closed L over an interval (a, b) if for an f (x) belonging to L

implies that f(x) = 0 almost everywhere. Here f(x) is a complex valued function of the real variable x.

1. The object of this note is to show the relation between certain results obtained by Wiener and Paley*, and by L'evinson from the theory of Fourier transforms, and a theorem which I proved in a recent paper. We require a general theorem on the function of Phragmèn and Lindelöf:

1. G. I. Taylor has lately* pointed out how his vorticity-transport theory of turbulent motion may be extended to three dimensions. Of the typical term expressing the effect of turbulence on the mean motion he remarks: “In general it is so complicated that it is of little practical use, but in certain special cases considerable simplifications may occur.” In the special cases which he himself discussed the mean velocity was in the direction of the axis of x, and its magnitude was a function of y only, (x, y, z) being rectangular Cartesian coordinates.

The considerations set out in this communication arose originally in the discussion of the refraction of surface elastic waves at the edge of a continent*. In view, however, of their general application, it is perhaps worth while to state them apart from their special context.

The stability of a rectangular plate, subjected to constant thrust over opposite pairs of edges, has been treated with some degree of completeness for various boundary conditions. The more general problem, in which the thrusts are no longer constant, has not yet received any treatment apart from the approximate method developed by E. Schwerin†, which would appear to be capable of only limited extension. The object of this paper is accordingly the detailed consideration of a simple case when the thrust is no longer constant.

Verblunsky has recently proved the following theorem:

Theorem A. Suppose that

for every positive σ. Write

and suppose that J (σ, x) is bounded in σ, x for σ > 0 and all x. Then

exists almost everywhere, and

almost everywhere.

For material stratified in parallel planes in local thermodynamical equilibrium we have Milne's well-known result that

where πF is the constant net integrated flux of radiation, τ is the optical depth and B is the “ergiebigkeit” which is related to the temperature T by the relation

σ being the Stefan-Boltzmann constant. If we define the effective temperature by the relation

we have from (1)

where T1 is the temperature ar τ = 1. In this note we establish a similar result for extended photospheres where the curvature of the outer layers is properly taken into account.

In a recent paper I developed a method for calculating the probability of the creation of an electron pair in the collision of two charged particles moving with a relative velocity very near that of light. I showed there that under certain conditions it is legitimate to treat one of the colliding particles, say the heavier one of charge Z2 and rest mass M2, which we shall call the particle 2, as fixed at the origin of coordinates, and the other, of charge Z1 and rest mass M1, which we shall call the particle 1, as moving classically along a straight line with uniform velocity V in the direction of the z-axis, passing the other particle 2 at a minimum distance of approach (impact parameter) b. I developed expressions (given by (18) to (22) of A) giving the probability of the transition of an electron from an initial state of negative energy E0 and momentum p0 lying in an element of momentum space dp0 to a final state of energy E and momentum p lying in an element of momentum space dp, under the combined perturbing influence of the two colliding particles when they pass at a minimum distance b. The initial state of the electron, left vacant after the transition, appears as a positron of momentum p+ = − p0. To get the differential effective cross-section for the creation of the above pair, we must integrate this probability over all values of the impact parameter b, and this integration can be performed easily as shown in A. The final result can be written as a sum of a finite number of doubly infinite integrals (A, (24)). The purpose of this paper is to carry through the evaluation of these integrals for certain special cases, and to consider the effects of screening. The results have already been communicated in A.

1. The purpose of this paper is to examine in detail the types of radiation emitted by a quantum mechanical system (representing a nucleus) when the radiating particle changes its azimuthal quantum number l by two, one, or nought. We assume that the radiating particle is spinless* and that it moves in a field possessing central symmetry, so that the particle may be described by a wave function ψn,l,m which factorizes into radial and angular parts,

We assume secondly that the nucleus has a finite radius aN of the order of 10−12 cm., that is to say that π(r) = 0 for r > aN.

The temperature dependence of the photo-electric effect with the image field is discussed, the result obtained agreeing with Fowler's T2 law near the threshold. Fowler's law is discussed, and shown to rest upon a mistake in the normalization of the free wave functions.

The two methods of analysing experimental results due to Fowler and Du Bridge are also discussed, and it is shown that in general the latter should give the more reliable results.

The author wishes to repeat his thanks to Prof. R. H. Fowler and to Mr A. H. Wilson for their constant interest in the work.

In previous communications* the author has discussed the recombination of gaseous ions, subsequently extending the theory to include large ions† and also to apply to the coagulation of colloids and of smokes‡. The predictions of the original theory were in excellent agreement with experiment, but discrepancies appeared when it was generalized. The author is now indebted to Dr F. J. W. Whipple for the discovery of a numerical error in the first paper, the correction of which leads on the whole to a much more satisfactory agreement with experiment.

If a simple calculation is made of the amount of heat liberated when one body is slid over another, and of the possible rate of escape of the heat by conduction and emission, it shows that the temperature of the surface may reach a high value.