If we consider, by the method of small oscillations, the stability of a viscous fluid flow in which the undisturbed velocity is parallel to the axis of x and its magnitude U is a function of y only (x, y, z being rectangular Cartesian co-ordinates), and if we assume that any possible disturbance may be analysed into a number (usually infinite) of principal disturbances, each of which involves the time only through a single exponential factor, then it has been proved by Squire, by supposing the disturbance analysed also into constituents which are simple harmonic functions of x and z, and considering only a single constituent, that if instability occurs at all, it will occur for the lowest Reynolds number for a disturbance which is two-dimensional, in the x, y plane. Hence only two-dimensional disturbances need be considered. The velocity components in the disturbed motion will be denoted by (U + u, v). Since only infinitesimal disturbances are considered, all terms in the equations of motion which are quadratic in u and v are neglected. When u and v are taken to be functions of y multiplied by ei(αx−βi), the equation of continuity becomes

and the result of eliminating the pressure in the equations of motion then gives the following equation for v, where ν is the kinematic viscosity of the fluid:

It is generally agreed that instants are mathematical constructions, not physical entities. If, therefore, there are instants, they must be classes of events having certain properties. For reasons explained in Our Knowledge of the External World, pp. 116–20, an instant is most naturally defined as a group of events having the following two properties:

(1) Any two members of the group overlap in time, i.e. neither is wholly before the other.

As particular cases of the general problem of finding the most general limiting tangential form of a degenerate manifold, we discuss here the degeneration of a curve on a surface into one or more multiple components, and the degeneration of a surface on a threefold into a single surface counting multiply, the method used consisting essentially in finding the approximate form of the curve or surface as the limit is approached. The approach system is supposed in all cases to be contained in a linear system of curves or surfaces, on the surface or threefold considered.

When a probability distribution is specified by more than one parameter, the statistical information in a sample on an unknown α will usually depend on the values of the other unknowns. For large samples where the joint distribution of the estimates tends to normality, and the efficiency of the estimation to a maximum, little theoretical difficulty exists. In a discussion(1) of the more general theory of small samples, I have stressed the importance of properties of sufficiency.

J. A. Todd has enunciated the following theorem:

“If the canonical adjoints ∑ of an irreducible curve Γ contain fixed parts, then, if Γ is not elliptic, all the fixed curves are simple, rational, determined uniquely by the base points, which present independent conditions for them, and have no free intersections with each other or with the variable part of the system ∑.”

1. This note contains a second proof of a remarkable theorem, announced by W. Killing and first proved by Eugenio-Elia Levi, according to which any infinitesimal group G, in the sense of Lie, contains a subgroup which is simply isomorphic to the factor group G/Γ, where Γ is the greatest integrable invariant subgroup in G§. Also the theorem is sharpened in that it is proved for groups depending on real as well as on complex parameters. In the final section it is seen that the methods used in proving this theorem can be applied to the problem of solving certain equations which are stated in terms of any semi-simple infinitesimal group of linear operators.

It is proved that if there exists a sufficient statistic for the estimation of an unknown parameter of a population, the frequency function of the population must be of a certain type.

It is shown that some modification of previous theory of the intrinsic accuracy of statistics is necessary when the range of the population sampled is a function of the parameter to be estimated.

Finally, the theory is extended to sufficient sets of statistics, i.e. sets of statistics which together contain all the information provided by a sample about an unknown parameter.

The paper deals with the problem of the effect of two-dimensional first order disturbances on the linear and parabolic flows of a viscous fluid. If the laminar flow is in the direction x and is bounded by the planes y = ±h, and if the stream function of the disturbance is assumed to be of the form , it is found that, for small Reynolds numbers R (= Uh/v), C can be developed in the power series

The first term in (1) corresponds to a disturbance in still water and is known from Rayleigh's investigation of that problem; the succeeding C's are real and can be obtained by an application of Schrödinger's perturbation theory of wave mechanics. In the case of the parabolic flow, expressions for C0 and C1 were thus derived for both the symmetrical and anti-symmetrical types of disturbance.

The imaginary part of C, as represented by in this approximation, is found to be positive. When α is of the order of unity, the value of R which makes the second term of the first is about 50 for the symmetrical disturbance and 100 for the antisymmetrical disturbance. C0, the first term in the expression for the phase velocity, is less than 1 and increases with α, which implies a group velocity greater than the phase velocity.

In the case of the linear flow it is found that the C's with even subscripts vanish, implying a vanishing phase velocity. This verifies the assumption to that effect made by Southwell and Chitty, and it also suggests that the region of convergence of (A) is limited to the range of values of R covered by these authors. Expressions are derived for C1 and C3 and are evaluated for α=1. The formulae thus obtained for the damping constant give an initial decrease with R, which is not shown in the corresponding curve of Southwell and Chitty.

An alternative method of determining C is outlined in the last section. Use is made of the orthogonal system of the characteristic functions for the vorticity of a disturbance in still water. The equation determining C is reduced to an infinite determinant in which C occurs only in the diagonal elements. When R→0, the determinant can be evaluated and the roots C are found to coincide with the first two terms in (A).

Finally it is shown that the two-dimensional laminar flow is stable for values of R less than the smaller of the quantities 2α and α3.

In the general systems of vortices represented by (1) the mean variation in pressure is

where K is a number which varies between 1 and √2. When the vortices are confined to cubical partitions, the case most nearly analogous to that of free turbulence, K = 1·06, so that the conjecture which I made some years ago, that would be equal to is probably nearly correct.

7. We have discussed the gliding of a plate of finite length on a stream of finite depth. Numerical calculations have been made for the case when the angle of incidence of the plate to the stream was 30°, the results for any other angle being similar.

It was found that, for a given depth of stream and a given height of the trailing edge above the bed of the stream, the value of the lift increases with the length of the plate, until finally, when the plate is infinitely long, the lift assumes a maximum value. Further, for a given depth of stream, the total normal lift on the plate is independent of its height above the bed of the stream, when the length of the plate is small, except when the trailing edge of the plate is above the surface of the stream. Finally, when the depth of the stream is very large and the plate is near the middle of the stream, then our solution approximates to the classical Rayleigh flow past a plate in an infinite fluid.

In this paper I prove some new Pflastersätze for r−dimensional sets in the n−dimensional Euclidean space Rn belonging to one and the same family, in which Lebesgue's fundamental lemma represents a “0-dimensional” case out of r + 1 possible cases of dimensions 0, 1, 2, …, r.

In the study of random events and associated fluctuations such as occur in the shot effect, a theorem first stated and discussed by Dr N. R. Campbell can often be employed. It applies on any occasion when there occur at random a number of events whose effects are additive. Let us suppose that a single event occurring at time tr causes at time t an effect f(t − tr) in some part of the observed system, and that the effects of different events are additive, so that the total effect or output is ϑ(t), given by

We may suppose that the same events cause another set of effects g(t − tr) with output ϑ(t), where

Both the functions are assumed to be bounded and integrable in the Riemann sense, as are all the functions studied in physics.

1. The velocity distribution in fluid of small viscosity, or at large Reynolds number, in the laminar boundary layer associated with a thin flat plate along the main direction of flow, has been worked out by Blasius on the basis of the Prandtl theory. These original investigations covered the region from the forward edge of the plate to the downstream edge. In a recent paper Goldstein investigated the velocity distribution in the wake immediately behind the downstream edge over a region within about 0·5l from it, l being the length of the plate. Later

Tollmien obtained a first approximation to the flow at a great distance down-stream. Goldstein § extended Tollmien's solution by finding a second approximation and by joining it up with his previous work on the flow very near the plate.

1. The gliding of bodies on the surface of water has received mathematical treatment by Wagner for both two- and three-dimensional cases. In this note, attention is confined to the two-dimensional gliding of a plane plate on a stream of infinite depth. Wagner's method is approximate and only applies when the angle of incidence of the gliding plate to the stream is small. He expresses such quantities as the total lift on the plate, not in terms of the total length of the plate, but in terms of the length of the pressure surface, which is the distance between the trailing edge of the plate and the point of stagnation. Further, his solution requires that the direction of the jet or spray which is formed should be parallel to the length of the plate. If Lis the total normal lift on the plate, β the small angle of incidence of the plate to the stream, V the velocity of the plate, δ the thickness of the spray and c the length of the pressure surface, then Wagner gives

It is well known that the boundary value problem for the non-linear differential equation

can be reduced with help of a Green's function K (x, ξ) to a non-linear integral equation of the type

In this paper we consider the slow two-dimensional motion of viscous liquid past a sharp edge projecting into and normal to the undisturbed direction of the stream. The liquid is supposed bounded by rigid planes represented by ABCDE in Fig. 1, and, apart from the disturbance caused by the projection, is assumed to be in uniform shearing motion. The stream function is then a bi-harmonic function that must vanish together with its normal derivative at all points of the boundary, and must be proportional to y2 at a great distance from the projection.

1. There is a lemma given by Severi which is of importance because it is used by him in his proof that the number of finite Picard integrals belonging to an algebraic surface is equal to the irregularity of the surface; it is also used by Castelnuovo † in his proof of the same result. The lemma is: If upon an algebraic curve there is an irreducible algebraic series, ∞1, of sets of s points, of index r; and if the sets of sr points which consist of all the r sets which contain a given point (this point taken r times) move in a linear series as this point varies, then any one of the r sets of s points separately moves in a linear series. The proof given by Severi was held satisfactory by Castelnuovot, but I find difficulty in stating it with precision; and Castelnuovo gives an entirely different proof, founded on an enumerative formula due to Schubert (loc. cit. p. 341); this is also the course adopted by Enriques-Chisini.

When relative motion of a viscous incompressible fluid of constant density and an immersed solid body is started impulsively from rest, the initial motion of the fluid is irrotational, without circulation. This is shown by observation, and may be seen in many of the published photographs of fluid flow. The theoretical proof is exactly the same as that given, for inviscid fluids, in treatises on hydrodynamics; for it may be assumed that the viscous stresses remain finite. The fluid in contact with the solid body is, however, at rest relative to the boundary, whilst the adjacent layer of fluid is slipping past the boundary with a velocity determined from the theory of the velocity potential. There is thus initially a surface of slip, or vortex sheet, in the fluid, coincident with the surface of the solid body. In other words, there is a “boundary layer” of zero thickness. The vorticity in the sheet diffuses from the boundary further into the fluid, and is convected by the stream. The boundary layer grows in thickness. (The same results follow from a consideration of the equations for the vorticity components in a viscous incompressible fluid, or of the equation for the circulation in a circuit moving with the fluid.)

The object of this paper is to make some remarks about a minimal problem suggested by the theory of the buckling of plates. A question left open in my previous paper on this subject is thereby solved.

It is shown that the eigenvalues of the angular momentum of the electromagnetic field containing a number of charged particles, apart from spin angular momentum, are only the integral multiples of ħ. This is shown by using cylindrical polar variables, and taking a particular choice of the vector potential in which the radial component is zero, defined explicitly in terms of the magnetic field strengths. By expanding in terms of the Fourier functions einθ, the angular momentum is separated out into terms independent of one another, each taking on only integral values in units of ħ.

The arguments all apply equally well to a modified field theory such as that of Born and Infeld.