The effect on the boundary-layer equations of a weak shock wave of strength ∈ has been investigated, and it is shown that if R is the Reynolds number of the boundary layer, separation occurs when ∈ = o(R−i). The boundary-layer assumptions are then investigated and shown to be consistent. It is inferred that separation will occur if a shock wave meets a boundary and the above condition is satisfied.

In § 1 a Markoff chain is defined, and a theorem of Kolmogoroff relating to its asymptotic behaviour is stated. Its stable distributions are examined in § 2 and some further results are obtained. These are then applied in §§ 3, 4 to a certain generalization of the cascade process, regarded as a Markoff chain with a special kind of matrix.

We consider n − 1 points xj (j = 1, …, n − 1) selected independently at random from the interval (0, 1), the distribution of any xj being the rectangular distribution

The points xj divide the interval (0,1) into n intervals. Let λ be the sum of the lengths of the k largest of these n intervals. It is the purpose of this paper to give the exact sampling distribution of λ for any given n and k (see equation (5)). Fisher (1) in 1929 solved the problem for k = 1. We first investigate the distribution of μ = 1 − λ.

1.1. This paper is a theoretical investigation in the real domain of the existence of subharmonic solutions of non-linear differential equations of the form

where F is analytic and of least period 2π/ω in t; ε = (ε1, …, εn) is small; and F(x, ẋ, 0, t) is not linear in x and ẋ.

During the Oxford Conference of the Econometric Society in September 1936, Ragnar Frisch proposed a problem in regression theory. A partial solution was found in 1938 by Miss H. V. Allen (1). A more complete solution was given by C. R. Rao (6) in 1947, and in the same year the present author (5) obtained a solution as a particular case of a more general result. These last two papers contained a flaw, and a correct solution was provided by Miss E. Fix (2). This last solution still leaves a part of the problem unanswered, and in the present paper a result of P. Lévy's (4), is used to complete the solution. At the same time further generalizations of the problem are considered and, in the cases of most practical importance, complete solutions are obtained. It is advisable, both from the point of view of rigour and simplicity of analysis, to use a general definition of the conditional expectation of a random variable. Accordingly, the paper begins with a summary of the relevant definitions. These notions were introduced by Kolmogoroff (3). It has been thought worth while giving the definitions here, in forms which are slightly different from Kolmogoroff's and seem more suitable for applications, in order to explain the notation and nomenclature used. The relevant consequences of these definitions are also stated in the form in which they are used.

The requirements of Kolmogoroff's theory of the equilibrium spectrum are satisfied only at very high Reynolds numbers, higher than any at which experiments have yet been done. In particular, when the theory holds, the rate of decay of the mean-square vorticity ω must be negligible compared with either its rate of increase due to the stretching of the vortex filaments or the rate of dissipation due to viscosity.

An extended version of Kolmogoroff's hypothesis may be proposed, in which the statistical properties of the turbulence in a range of wave-numbers (range of eddy sizes) depend not only on the rate of dissipation ∈ per unit volume and the viscosity ν, but also on the time rate of change d∈/dt of ∈. The result is to introduce a dependence on the Reynolds number R of the turbulence into quantities and constants which, on Kolmogoroff's original hypothesis, were independent of R. The Reynolds number R is defined from the decay law; u2t, with an origin of time suitably chosen, is a function of t, finite when t = 0, and R is defined as (u2t)t=0/ν. Lin's decay law follows logically from the extended hypothesis, according to which the rates of change of ω−1, due to the causes mentioned above, are constant during the decay; Lin's decay law would also follow if this less general part only of the extended hypothesis be assumed. The same decay law is also obtained if the similarity spectrum of Heisenberg is taken to apply not to the whole of the energy-bearing eddies, but only to the energy-dissipating eddies. But it is suggested that further generalization of the theory of the similarity spectrum, and of the decay law, is necessary; that the similarity spectrum is probably only asymptotically correct for a range of large wave-numbers, the range depending on the initial conditions and decreasing as the decay proceeds; that the general decay law is u2t = μRd(t), where d(t) is an integral function of t, such that d(0) = 1, and with an asymptotic value for large t to give correctly the law of decay in the final period; and that d(t), and the number of constants needed to specify it approximately, depend on the initial conditions. An experiment is suggested to test the dependence of the law of decay on the initial conditions. It is also suggested that the recently observed constancy of u2t in the initial period in the turbulence behind a single grid is only approximate, and this approximate constancy still needs explanation. Remarks are also included on the range of application of the equilibrium spectrum, for which some formulae are given when there is a definite cut-off in the spectrum.

1. Introduction. Following recent developments in the theory of stochastic processes, a beginning has been made by various authors with the associated problems of sampling and statistical inference as these are concerned with dependent and ordered observations, they are distinct from and usually more difficult than the corresponding problems for independent and unordered observations.

1·1. Let van der Pol's equation be taken in the form

where ε1, ε2, k1 and k2 are small, and ω ≠ 0 is a constant, rational or irrational, independent of them.

The use of a coordinate system convected with the moving medium for describing its mechanics, first proposed by Hencky (5), has since been extended by several authors, and has several advantages over the more conventional use of a coordinate system fixed in space; Brillouin(1) has shown that the relation between the strain-energy function for an ideally elastic solid and the stress tensor takes a very simple form when the latter is referred to a convected coordinate system; Oldroyd(8) has given a very general discussion of the formulation of rheological equations of state and has shown that the right invariance properties are most readily recognized when the equations are referred to a convected coordinate system; Green and Zerna (4) have similarly expressed the equations of motion and boundary conditions; and Gleyzal (2), and Green and Shield (3) have applied the formalism to certain problems in elasticity theory.

1. The present paper is concerned with a class of finite Markoff chains. It is thought that there are some new points of interest here, although the literature dealing with finite Markoff chains is large. (For bibliographies see (2) and (7).)

It is advantageous in automatic computers to employ methods of integration which do not require preceding function values to be known. From a general theory given by Kutta, one such process is chosen giving fourth-order accuracy and requiring the minimum number of storage registers. It is developed into a form which gives the highest attainable accuracy and can be carried out by comparatively few instructions. The errors are studied and a simple example is given.

It is shown that a mass of fluid bounded by fixed surfaces and by a free surface of infinite extent may be capable of vibrating under gravity in a mode (called a trapping mode) containing finite total energy. Trapping modes appear to be peculiar to the theory of surface waves; it is known that there are no trapping modes in the theory of sound. Two trapping modes are constructed: (1) a mode on a sloping beach in a semi-infinite canal of finite width, (2) a mode near a submerged circular cylinder in an infinite canal of finite width. The existence of trapping modes shows that in general a radiation condition for the waves at infinity is insufficient for uniqueness.

The geometrical configuration consisting of five straight lines tangent to the stream line (ψ = const.), isobar (p = const.), isocline (θ = const.), isovel (q = const.) and isopycnic (ρ = const.) at a point P of the physical plane in the plane flow of a fluid has a number of interesting properties. The study of this geometric configuration for a wide class of ideal fluids (which includes polytropic gases and incompressible fluids), in particular its properties at sonic points, is the purpose of this paper.

The concept of the Green's vibrational function given in an earlier paper by the author is used to obtain a general expression for the disturbance from a point source. The potential due to transient sources of sound moving with subsonic and supersonic velocities is derived from this. It is found that the Doppler effect for a supersonic source differs from that for a subsonic source. In the former case it is found that two frequencies are heard simultaneously from a source emitting a note of one frequency.

The theory is applied to determine some solutions of the two dimensional equation of supersonic, irrotational compressible flow, corresponding to the flow around an aerofoil taking into consideration the entropy changes at the shock wave.

The following matrix problems are well known in quantum mechanics:

(a) The one-dimensional harmonic oscillator. Given

determine the eigenvalues hjj of H, and the matrix elements of X, P if H is diagonal. It is found (Wigner (4)) that

This paper considers the correlation P(r) between the fluctuating pressures at two different points distance r apart in a field of homogeneous isotropic turbulence. P(r) can be expressed in terms of the fourth moment of the velocity fluctuation, which is evaluated with the aid of the hypothesis that fourth moments are related to second moments in the same way as for a normal joint distribution of the velocities at any two points. The experimental evidence relevant to this hypothesis (which cannot be exactly true since it gives zero odd-order moments) is examined. The alternative hypothesis made by Heisenberg, that the Fourier coefficients of the velocity distribution are statistically independent, has identical consequences for the fourth moments of the velocity, although it does not lead to such convenient results.

The pressure correlation is worked out in detail for the important special case of very large Reynolds numbers of turbulence; the mean-square pressure fluctuation is found to be . The mean-square pressure gradient is evaluated, from the available data concerning the doublevelocity correlation, for the cases of very small and very large Reynolds numbers, and a simple interpolation between these results is suggested for the general case. Finally, the relation between the mean-square pressure gradient and rate of diffusion of marked fluid particles from a fixed source is established without the neglect of the viscosity effect, and the available observations of diffusion are used to obtain estimates of which are compared with the theoretical values.

The nature of the turbulent motion in a boundary layer with zero longitudinal pressure gradient has been investigated with the techniques of hot-wire anemometry which have been developed for the study of shear flow in wakes. Measurements have been made of the intensities of the turbulent velocity components, the turbulent shear stresses, the rates of transport of turbulent energy by diffusive movements, the intensities and flattening factors of the down-stream spatial derivatives of the three velocity components, and spectra of the down-stream component of the velocity fluctuation, at traverses through the boundary layer at three stations where the Reynolds numbers (based on the displacement thickness and free stream velocity) were respectively 3630, 4360 and 5080. Over the range of measurement, which did not include the laminar sublayer, the turbulent motion was similar at all three stations, and could be described in terms of universal functions. By considering the turbulent energy balance, it is shown that, except in the outer part of the layer where the turbulence resembles strongly the turbulence in a wake, there is a strong flow of energy directed toward the wall and transported by the action of turbulent pressure gradients. It is concluded that, most probably in contact with the laminar sublayer, there must be a ‘dissipative layer’ within which most of the turbulent energy dissipation takes place, and that the bulk of the eddies are, in a sense, attached to the wall and have very high rates of shear in that region. In agreement with this view of the structure of the turbulence, length scales derived from the apparent eddy viscosity and the local turbulent intensity are found to be comparable with distance from the wall. Length scales in the direction of the mean stream are much larger, and it is believed that the typical eddy is very elongated in this direction, and has its vorticity directed roughly parallel to the direction of maximum positive mean rate of strain.

Dirac has shown that in quantum theory free magnetic poles are possible only if their pole strength g is an integral multiple of ℏc/(2e) = 137e/2. The ionization produced by such particles is calculated by the classical impact parameter method and also by quantal perturbation theory; at sufficiently large velocities the rate of loss of energy of poles of unit strength is g2/e2 ≑ 5000 times as large as that of an electron of the same velocity; at low velocities the energy loss of a monopole decreases steadily without rising to a maximum as in the case of electric particles.

The close similarity between the basic problems in statistical thermodynamics and the partition theory of numbers is now well recognized. In either case one is concerned with partitioning a large integer, under certain restrictions, which in effect means that the ‘Zustandsumme’ of a thermodynamic assembly is identical with the generating function of partitions appropriate to that assembly. The thermodynamic approach to the partition problem is of considerable interest as it has led to generalizations which so far have not yielded to the methods of the analytic theory of numbers. An interesting example is provided in a recent paper of Agarwala and Auluck (1) where the Hardy Ramanujan formula for partitions into integral powers of integers is shown to be valid for non-integral powers as well.

1. A slow two-dimensional steady motion of liquid caused by a pressure gradient in a semi-infinite channel is considered. The medium is bounded by two parallel semi-infinite planes represented in Fig. 1 by the straight lines AB, DE. The stream-function ψ is a biharmonic function of x, y which exactly satisfies the condition that AB, DE must be stream-lines, but the condition that there must be no velocity of slip on these boundaries is satisfied only approximately, and the calculated velocity of slip gives a measure of the accuracy of the solution.