In the year 1815 an anonymous article appeared in Thomas Thomson's Annals of Philosophy (I) entitled “On the Relation between the Specific Gravities of Bodies in their Gaseous State and the Weights of their Atoms ”. Its introductory paragraph illustrates the hesitancy of the writer in its exposition: “The author of the following essay submits it to the public with the greatest diffidence; for although he has taken the utmost pains to arrive at the truth, yet he has not that confidence in his abilities as an experimentalist as to induce him to dictate to others far superior to himself in chemical acquirements and fame. He trusts, however, that its importance will be seen, and that some one will undertake to examine it, and thus verify or refute its conclusions. If these should be proved erroneous, still new facts may be brought to light, or old ones better established, by the investigation; but if they should be verified, a new and interesting light will be thrown upon the whole science of chemistry.”

Poincaré, Liapounoff, Perron and others have proved theorems about the order of smallness, as the independent variable tends to + ∞, of solutions of differential equations with non-linear perturbation terms. A similar theory exists for difference equations. By a simple use of transforms, we here extend the theorems, with suitable modifications, to difference-differential equations. The results are an essential step in the development of a general theory of non-linear equations of this type.

A differential equation of the second order, arising in problems of disturbed oscillation, such as occur in frequency modulation, is considered. The nature of its solutions is examined by the method of continued fractions. The cases in which the solutions are periodic, and the regions of stability and instability (lability), are determined according to the values taken by the two parameters involved.

The convergence of customary processes of iteration for solving linear equations, in particular simple and Seidelian iteration, is studied from the standpoint of matrices. A new variant of Seidelian iteration is introduced. In the positive definite case it always converges, the characteristic roots of its operator being real and positive and less than unity.

With the aim of establishing, under wide conditions, the ergodic theorem of G. D. Birkhoff, the author extends the class of asymptotically almost-periodic functions, considering now not only continuous functions, as he had already done in 1943, but discontinuous functions. Definitions and properties of the extended class of functions are set out, some comparisons being made with almost-periodic functions in the sense of Bohr, Stepanoff, Weyl and Besicovitch. Applications to the ergodic theorem are adumbrated.

The problem considered is that of the estimation of a statistical parameter from a sample of values of the variate or variates concerned. Reference is made to the method of unbiased statistics with minimum variance, developed by Aitken and Silverstone. The principal result obtained by these authors is generalized, and an inequality involving the variances of unbiased statistics is obtained. Several examples illustrating the theory are appended.

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.

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.