1. Let P = 0, Q = 0, U = 0, V = 0 represent four independent quadric surfaces, and consider the equation

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:

Let us consider a discontinuous bivariate distribution. That is, let us consider N × Q non-negative values pνq (ν = 1, 2, …, N; q = 1, 2, …, Q), being the theoretical probabilities of the νth value of a variate Xμ (μ = 1, 2, …, N) concurring with the qth value of a second variate Ys (s = 1, 2, …, Q).

Let p (x) be an essentially positive function defined in the interval 0 ≤ x ≤ π. We consider the non-linear partial differential equation

for the boundary conditions u (x, t) = 0,

for x = 0 and x = π,

The successful development in recent years of the topology of general closed sets is largely due to the application of combinatory methods, which have led to an elaborate theory of approximation of closed sets by infinite cycles, to the generalization of duality theorems for closed sets, and to a geometrical theory of dimensions. Corresponding to the combinatory invariants involved the main results of these theories concern for the most part properties of the set as a whole, which cannot possibly express fully its internal and in particular its local structure. Still less is our knowledge of the eventual relations between the local structure of a set and its properties in the large.

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.

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.

The general solution of Born's new field equations is found for the two-dimensional electrostatic case, by which the coordinates are expressed as functions of the field vectors, Conditions for inversion are discussed. Special cases are worked out, namely: singnle charge, two charges, charge in a constant field. Expressions are given for forces acting on the charges. A singular solution is also discussed, with reference to the neutron. The implication of the solutions on the general theory and the equations of motion is discussed in the conclusion.

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 theoretical method for computing the grade of a curve on an algebraic surface is well known. In practice difficulties arise which are not considered in the theory; so that it seems worth while to describe a practical method. This is done in § 1 of this paper. The method is then applied to some examples with the object of discovering whether Noether exceptional curves are necessarily exceptional curves†. In particular, a certain quintic surface with three tacnodes is studied, and our examination leads us to results which differ from those which have been accepted up till now. Another example illustrates the limitations of a practical method for computing grades, because of the possible presence of infinitesimal curves, and leads to the transformation of the quintic surface with two tacnodes into a double plane of order ten of a certain type, which has the singularity known as a (5, 5) point on its branch curve. Light is thrown on the Noether composition of this singularity by the transformation, which also shows the relation between two well-known types of surface for which the Noether relation p(2) = p(1) − 1 does not hold.

Some theoretical results on the effect of non-normality on the t test of significance, though it is stressed that they are incomplete and not perhaps of much quantitative value, nevertheless tend to agree with the result of sampling investigations in showing that for moderate departures from normality this test may still be used with confidence, particularly for testing differences in means of equal numbers of observations.

1. This paper is supplementary to an article by M. H. A. Newman on the superposition of manifolds. Guided by his main idea, I extend Newman's theorem that two equivalent manifolds have a common subdivision from equivalent manifolds to equivalent complexes of an arbitrary nature. The theorem is also sharpened by the possibility of leaving a certain subcomplex unaltered, and by substituting a partition for a general subdivision.

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 object of this paper is to extend the ordinary theory of integral equations of the second kind to certain classes of more general kernels and functions, with special reference to equations of the Volterra type.

A period of a function f(z) is defined to be a number ω (≠ 0) such that

is identically zero; and it can be shown that an integral function may either have no periods or else a single sequence kλ (k = ± 1, ± 2, …).

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.

Owing to a chance observation some interesting phenomena were observed during the splashing of water in an electric field. If a water drop falls into water, there is a characteristic splash of which the details were investigated photographically by Professor A. M. Worthington in the latter years of the last century. The final account of this work was given in two papers (1) and also in a book entitled A Study of Splashes (2). The splash cannot be examined with the unaided eye because the various stages of the splash follow one another too quickly. For the moment it may be said that an important stage of the splash consists of the rising up of a cylindrical column of water to a considerable height. The radius of the column is comparable with the radius of the drop.

Recent discoveries have shown that two light quanta with total energy not less than 2mc2 can create a pair of electrons (inverse process to the ordinary annihilation). Hence it follows that ordinary black-body radiation no longer represents an equilibrium state. In thermodynamic equilibrium a certain number of electron pairs must be present, the number of which will depend upon the temperature. It can be calculated as follows.

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.

The probability relations which can occur between two separated physical systems are discussed, on the assumption that their state is known by a representative in common. The two families of observables, relating to the first and to the second system respectively, are linked by at least one match between two definite members, one of either family. The word match is short for stating that the values of the two observables in question determine each other uniquely and therefore (since the actual labelling is irrelevant) can be taken to be equal. In general there is but one match, but there can be more. If, in addition to the first match, there is a second one between canonical conjugates of the first mates, then there are infinitely many matches, every function of the first canonical pair matching with the same function of the second canonical pair. Thus there is a complete one-to-one correspondence between those two branches (of the two families of observables) which relate to the two degrees of freedom in question. If there are no others, the one-to-one correspondence persists as time advances, but the observables of the first system (say) change their mates in the way that the latter, i.e. the observables of the second system, undergo a certain continuous contact-transformation.