The idea of direct action between streams is applied to a continuous charged fluid and combined with the new formulation of the electrodynamical laws of motion in terms of conservation of circulation. A simple and rigorous integrated formulation is thus obtained from the Maxwell-Lorentz differential equations, applicable to co-existing positive and negative fluids, as well as vacuum. Exact solutions are obtained, among them one which represents self-consistent, self-maintained flow in a hollow tubular region of infinite axial extent. It is hoped this tube might be bent into a torus and that an electron model will result from merely quantizing the one or two vortices around which this flow-pattern circulates.

1. Introduction. A number of authors (2,3,7) have recently quantized the motion of an inviscid fluid. The starting point has been the variational principle of Bateman which uses the Clebsch variables. The density turns out to be the canonical conjugate to the velocity potential, and the transition to the quantum theory is then made in the usual way. If the fluid is making small vibrations, ‘phonons’, which are scalar, appear as a result of the quantization.

Suppose that f(x) is non-negative for x ≥ 0 and let

In a recent paper (4) inequalities were proved relating fn(x) with the nth derivative f(n)(x) of f(x). However, the two main results, Theorems 3 and 4, were proved only for n = 1 and 2. In the first part of this paper we shall prove the corresponding results for all positive integral n under a slightly less restrictive hypothesis (Theorem 3). We shall then give an application showing that from the hypothesis f(x) ≤ μ(x) for all x it is sometimes possible to deduce f(n)(x) = O{μ(n)(x)} for a set of values x depending only on μ(x).

Two simple laws connecting the amplitude and phase functions of a monochromatic electromagnetic wave of arbitrary form are derived, holding in the case when one of the field vectors is linearly polarized. The first is a generalized Fermat's principle which enables determination of the phase when the amplitude is known; the second expresses the propagation of the (vector) amplitude along the curves orthogonal to the co-phasal surfaces. Some other general properties of linearly polarized fields are also discussed, and illustrative examples are given.

When the boundary-layer approximations are applied to the laminar flow in an axially symmetrical jet of gas with Prandtl number unity, a particular integral of the equations of motion satisfying the required boundary conditions is found to be , where i1, is the (constant) enthalpy on the jet boundary. By a suitable transformation an equation is obtained which, to a first approximation, may be replaced by the equation for flowin an axially symmetrical jet of incompressible fluid at a sufficiently large distance from the orifice. For an ideal gas, the known solution of this equ`ation may be used as the starting point of an iterative method to build up the stream function as a power series. The next approximation is found explicitly. It shows that for subsonic Mach numbers the density at a point on the axis is greater and the temperature less than on the boundary by amounts which may be very closely represented by terms inversely proportional to the square of the distance of the point from the orifice.

1. The conformal mapping of a strip domain in the z-plane on to a parallel strip— parallel, say, to the real axis of the ζ ( = ξ + iμ)-plane—brings about a certain distortion. More precisely: consider a cross-cut on the line ℜz = c joining the two sides of the frontier of the strip domain (in these introductory remarks we suppose for simplicity that there is only one such cross-cut on that line), and denote by ξ1(c) and ξ2(c) the lower and upper bounds of ξ on the image in the ζ-plane. The theorem of Ahlfors (1), now classical, states that

provided that

where a is the width of the parallel strip and θ(c) the length of the cross-cut.

In an attempt to understand the mechanism governing separation the possibility of there being further solutions to the Falkner-Skan equation f′′′ + ff″ + β(1 − f′2) = 0, in addition to those found by Hartree, is investigated.

It is found that when β < − 0·1988 there are no solutions satisfying |f′| ⩽ 1 for all η, and that if 0>β> − 0·1988 there are two acceptable solutions, one with f″(0) < 0. The new ones are computed to three places of decimals for various values of β and tabulated. In addition, it is shown that if − 0·5 < β < 0 there is a family of solutions corresponding to boundary layers bounded on one side by free streamlines. These are also computed and graphs of f′(0) and δ1 are displayed. The impact of these solutions on the theory of separation is discussed.

1. An expression in finite terms is found in §§2,3 for the small displacement of a thin elastic plate due to the application of a transverse force at an arbitrary point; the plate is infinite and is bounded internally by an ellipse along which it is clamped. The limiting case in which the ellipse degenerates to a straight line is considered in §4, and the other limiting case of a circular boundary in §5.

A method is given for determining the stability of small sinusoidal oscillations in the steady laminar flow of a horizontal wind over the surface of a liquid at rest (with particular reference to the flow of air over water), taking into account viscosity, gravity and surface tension. It is shown that there are two fundamental types of oscillation of the system, which may be called ‘water’ waves and ‘air’ waves, and curves showing the conditions for neutral stability of these two types of wave are given for a range of wind speeds from 100 to 300 cm./sec.

In this paper I give a direct proof of a new expansion of the Whittaker function M(a; b; x) as a series of Bessel functions, and some generalizations of this result for the functions 2F2(x) and 4F4(x).

1. Introduction. The problem of determining the distribution of stress in a medium with transverse isotropy (one axis of elastic symmetry) has recently been considered by Elliott (l, 2) and Shield (3). They have obtained solutions to certain special problems both with and without rotational symmetry. Elliott (2) has employed the method of dual integral equations to solve a number of axially symmetric crack and punch problems. (This crack problem is not to be confused with the torsional crack problem treated by Weinstein (4)). Shield (3) has reduced the general crack and punch problems for a medium with transverse isotropy to corresponding isotropic problems as formulated by Green (5), and has treated a few special examples. For the axially symmetric isotropic crack and punch problems the author (6) has shown that Green's formulation may be simplified, and has obtained in each case a simple usable solution of the general axially symmetric problem without the need for dual integral equations. As one might expect these simplifications may also be realized in the case of a medium with transverse isotropy. In this paper the author, guided by the work of Green (5) and Shield (3), presents a straightforward method for solving the general axially symmetric crack and punch problems for media with transverse isotropy. This method permits one to write down immediately in an appropriate system of curvilinear coordinates the solution to the general axially symmetric problem, whereas the method given by Shield requires an individual treatment of each problem. In fact it would appear to be extremely difficult if not impossible to obtain the complete solution to the general axially symmetric crack or punch problem by the methods suggested by Shield. Since the formulation given in this paper differs from that given by Shield, his equations for transforming the problem in transverse isotropy to an isotropic problem must be altered slightly. The solutions to the general axially symmetric problems for a medium with transverse isotropy then follow immediately from the solutions to the corresponding isotropic problems in (6). We shall therefore find it necessary to repeat some of the arguments and results of (6).

Cylinder functions, (λx), of integral orders which vanish at x = 1 and x = η (where η is an assigned positive constant less than 1) occur in the solution of many problems in applied mathematics. Such functions can be expressed in terms of the Bessel functions Jn(x) and Yn(x) of the two kinds in the form

where λ is a root of the equation

Computation of integrals of the type is of frequent occurrence in applied problems. Here λ is real and i is the imaginary unit. Tables of the integrals are often required for a wide range of λ values, some of which may be quite large. Tabulation of the integrand for each λ, followed by ordinary methods of numerical integration, is slow and tedious because the number of points required increases with increasing λ. In this paper formulas are given so that once f(y) is tabulated over the range of interest, the integral in question can be easily evaluated for any λ.

It is a commonplace observation in experiments on cutaneous sensation that a constant stimulus, applied under supposedly constant conditions, evokes a varial response. In this paper it will be shown that the variability is in part a geometri consequence of the fact that the sense organ is not a continuum but an aggregati of discrete, sensitive ‘spots’. The total extent of the ‘spots’ for one modality, acti at a given time, is only a small fraction of the area of the site tested; this is illustrat in all of the classical maps, and is particularly true of the warmth-spots (Rein (8)). I purposes of theoretical discussion, the sensory spots may therefore be treated points that form an array or pattern. This pattern is unlikely to be regular and mt indeed, be highly irregular. No assumptions about its regularity are necessary the present argument.

1. In §§ 2–4 of this paper approximate expressions are found for the stream function and pressure in the steady two-dimensional motion of viscous incompressible liquid past a fixed parabolic cylinder; exact expressions for the stream-function and pressure in a perfect liquid are derived as limits in § 5.

In the course of preparing lectures on elementary quantum theory the author developed a way of presenting time-independent perturbation theory that seems to have some advantages over more familiar presentations. There is nothing fundamentally new in the method, and if any claim of greater power can be made for the new version, it is at best for situations in which high-order perturbations are of interest. Quantum-field theorists will no doubt recognize the relation of this presentation to much that has been done in their subject recently.

If E is a linear set with a positive Lebesgue measure, and E(δ) is its translation through a distance δ, it is obvious that there always exist non-zero values of δ for which E. E(δ) has a positive measure. In the present note I show that for α-dimensional linear sets (0 < α < 1), and α-dimensional measure, the analogous result is not true. This is a further illustration of the deep structural difference between α-dimensional sets for α < 1 and for α = 1.

The object of this paper is to discuss the one-dimensional unsteady adiabatic motion of a gas which is initially at rest with a prescribed density distribution such that the specific entropy is uniform. The contour integral methods which Copson developed recently for even analytic functions are extended to apply to general analytic initial conditions. The solution is valid in the range 1 < Υ > 3, where y is the adiabatic index of the gas. Of particular interest, in view of the hydraulic analogy, is the case Υ = 2 for which real variable methods cannot readily be adapted. The motion of the front of a water column Sowing into a dry, horizontal stream bed is discussed. A curious type of solution, corresponding to a particular choice of initial distribution, which was established by Pack for a, countable sequence of values of Υ, is verified to hold over the whole range and is interpreted in terms of the dam-break problem.

Even if ideas of measurement did not exist, a great deal of the simple mechanical and electromagnetic structure of the physical world could be understood. To see how this can be so, the orthodox ideas of the relationship of mathematics to physics are inadequate. In this paper a development of this fundamental non-metrical physics is made to depend on a different view of this relationship.