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.

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.

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

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.

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.

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.

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. 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. 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).

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).

It is well known that the motion of a dynamical system can be pictured in two distinct ways, which Dirac names the Heisenberg picture and the Schrödinger picture. The equations of motion take different forms in the two pictures, but of course have identical physical consequences, since the motion of the system does not depend in any way on which picture we choose. It should therefore be possible to express the equations in an invariant form (independent, that is, of the picture used). It will be shown in this note that not only can this be done (equation (3)), but it can be done in such a way that it is not even necessary to introduce a picture at all (equation (3′)).

Let F be a regular surface in ordinary space and let Z be a cut on F. Then there is on Z a point z which is the limit of a decreasing sequence of 2-dimensional Cantor manifolds lying on F.

1. It is known that the distribution of the vanishing coefficients in the power series of an integral function is closely connected with the analytic properties of the function. But the following theorem has apparently not yet been noticed, although simple and elegant both in statement and in proof.

The motion of a fluid due to the rotation of an immersed plane disc about an axis perpendicular to itself is turbulent if ωa2/ν is greater than about 105, where ω is the angular velocity of the disc, a its radius, and ν the kinematic viscosity of the fluid. The torque opposing the rotation, due to the surface friction of the fluid, was calculated originally by Kármán, both when the motion of the fluid is steady and when it is turbulent.

The statistical method described in 21·4 of Fowler's Statistical Mechanics was devised to deal with assemblies in which the total numbers of elementary physical systems remained constant. The same method is applicable, with no formal changes, to assemblies in which elementary physical systems of two different sorts can be created and annihilated in pairs, like the negative and positive electrons in the theory of Dirac.

It often happens that when two sets of data obtained by observation give slightly different estimates of the true value we wish to know whether the difference is significant. The usual procedure is to say that it is significant if it exceeds a certain rather arbitrary multiple of the standard error; but this is not very satisfactory, and it seems worth while to see whether any precise criterion can be obtained by a thorough application of the theory of probability.