A study is made of the propagation of elastic and plastic deformation in a thin plate, initially unstressed, and of infinite extent, when it is penetrated normally by a cone moving with uniform velocity. The work is an extension of unpublished researches by Sir G. I. Taylor on the corresponding problem for a thin wire, and a summary of his results is included.

Suggested by the analogy between the classical one-dimensional random-walk and the approximate (diffusion) theory of Brownian motion, a generalization of the random-walk is proposed to serve as a model for the more accurate description of the phenomenon. Using the methods of the calculus of finite differences, some general results are obtained concerning averages based on a time-varying bivariate discrete probability distribution in which the variates stand in the particular relation of “position” and “velocity.” These are applied to the special cases of Brownian motion from initial thermal equilibrium, and from arbitrary initial kinetic energy. In the latter case the model describes accurately quantized Brownian motion of two energy states, one of zero energy.

The relaxation technique of R. V. Southwell is developed to evaluate mixed subsonic-supersonic flow regions with axial symmetry, changes of entropy being taken into account. In the problem of a parallel supersonic flow of Mach number I·8 impinging on a blunt-nosed axially symmetric obstacle, the new technique is used to determine the complete field downstream of the bow shock wave formed. Lines of constant vorticity and Mach number are shown in the field, and where possible a comparison is made with the corresponding 2-dimensional problem.

In an earlier paper a description was given, in terms of classical projective geometry, of some of the properties of parallel fields of vector spaces (parallel planes) in a Riemannian Vn, and a detailed analysis was made of the case n = 4. The present paper contains the corresponding formulae for any n, though omits their projective interpretation. A parallel þ-plane is said to be of nullity q when the þ vectors of any normal basis contain q null and þ − q non-null vectors. The conditions of parallelism, namely that the co-variant derivatives of the basis-vectors should depend linearly upon these vectors, are examined for any þ and any q(<þ), and attention is thereafter mainly confined to the cases (i) n even, q = ½n − 1, p = ½n − 1 or ½n; (ii) n odd, q = ½(n − 3)) ,p = ½(n − 1), which possess exceptional features. In the former of these cases light is thrown upon the curious circumstance, noted in the previous paper, that the existence in a V4 of a null parallel i-plane necessitates the existence of parallel planes other than its conjugate. For a general n similar situations arise in the cases indicated.

Suppose we have a number of independent pairs of observations (Xi, Yi) on two correlated variates (X, Y), which have constant variances and covariance, and whose expected values are of known linear form, with unknown coefficients: say respectively. The pij and the qij are known, the aj and the bj are unknown. The paper discusses the estimation of the coefficients, and of the variances and the covariance, and evaluates the sampling variances of the estimates. The argument is entirely free of distributional assumptions.

A derivation is given of a sum rule which may be useful in the discussion of line intensities of molecular spectra.

The paper is concerned with the distributional properties of Markoff chains in two and three dimensions where the transition probability for the length of a step and its orientation relative to that of the previous step is specified.

The discrete two-dimensional chain of n steps is first discussed, and by the use of moving axes an equation relating characteristic functions of the end-point distribution for successive values of n is obtained. The corresponding differential equation for the limiting chain with continuous first derivatives is given and asymptotic solutions for long chains are found.

The three-dimensional chain is similarly treated in terms of moving axes, and the limiting continuous chain is again discussed. Finally the same methods are applied to the discrete chain of equal steps to obtain the asymptotic form of the end-point distribution for long chains.

In a recent paper by the author several unfortunate omissions and misstatements occurred which it seems desirable to correct. I am indebted to Professor T. W. Anderson for pointing these out to me.

The author has already described a new method of measuring the velocity of light, which replaces Fizeau's toothed wheel by a piezo-quartz. This acts as an intermittent diffraction grating, and it interrupts the beam 200 times more rapidly than Fizeau's wheel did.

The present paper describes the application of the method to the measurement of the velocity of light in air. The total length of path was 78 metres. The frequency of interruption was measured by comparison with the Droitwich radio station. The result reduced to vacuum is 299,775 kilometres per second and is in agreement with other recent determinations, but, as a result of the experience gained, it will be possible to increase the accuracy at least ten times.

Since the delivery of my presidential address (1) in July I have assembled an amount of supplementary information regarding “the Chemical Society instituted in the beginning of the Year 1785”. This, together with a brief description of some other chemical societies of the revolutionary period, forms the basis of the present paper.

First of all, it will be expedient to furnish a complete list of the dissertations read before the Society during 1785–86 and included in the first volume of its Proceedings, appending short comments with respect to the communicators or their topics when anything of special interest arises.