Experiments have been performed, using purely optical methods, to verify and extend the theory of Gabor's diffraction microscope. An elementary theory of the process is first given, from which certain generalizations are provisionally drawn. In particular, a focal length is attributed to any Fresnel diffraction pattern and the hologram derived from it by photography. The variation of this focal length with wavelength and scale factor is postulated by analogy with a zone-plate, and the power-rate for a hologram is denned. These deductions are then verified by experiment, and a summary is given at the end of § 10. Various other confirmatory experiments are then described.

Adequate information is given about apparatus and technique to enable new entrants into this field to obtain satisfactory results with the minimum of preliminary trial.

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.”

Experiments in diffraction microscopy, previously described, are here continued. Special emphasis is now laid on verifying the theory by the production of an “artificial” hologram, by non-diffractive means, from data calculated for a relatively simple object. The assumed object is then reconstructed in the usual apparatus.

A type II linear zone plate of limited width is studied as a particular case of an artificial hologram. It gives rise to an unexpected black artefact, which is explained by a detailed analysis of this particular zone plate, and is shown to be due to its limited extent.

Experiments on twisting the linear zone plate skew to the reconstructing beam show that the effective focal length is affected astigmatically by a factor proportional to cos2θ, where θ is the angle of twist, for lines parallel to the axis of twist. Lines perpendicular to the axis of twist are unaffected.

The production of a hologram in an astigmatic pencil and its subsequent reconstruction while skew to a parallel beam is described. It is found that the focal length differences can be corrected in this way, but that the lateral scale factors are only partially rectified.

This paper represents the application of the Principle of Reciprocity, formulated in a previous communication, to the outstanding problems of classical and quantum electrodynamics.

The first step consists in the formulation of a reciprocally invariant Lagrangian function for a system of electrons in interaction with the electromagnetic field. A study is made of the unaccelerated motion of an electron, and this is subsequently extended to embrace the problem of an electron in arbitrary motion. It is found that the usual difficulties of classical electrodynamics do not appear. The methods of the earlier paper are applied to the derivation of the Hamiltonian energy of electron and field, and this enables a quantized formulation of the theory to be given, which also does not lead to the usual divergence difficulties.

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.

The relaxation technique of R. V. Southwell is shown to be applicable in certain cases to transonic problems. For a uniform stream with a low subsonic velocity impinging on a symmetrical 2-dimensional double wedge, an asymmetrical supersonic region can be isolated in the neighbourhood of the corner of the wedge, and the streamlines and the values of the Mach number within this supersonic region can be determined with the aid of relaxation methods. Difficulties must be expected to occur in the neighbourhood of the sonic line, but in the present problem these have been surmounted.

Since taking up a temporary appointment at the Sir John Cass College, the author has had the opportunity of checking the artificial hologram discussed in the main paper on a non-recording microphotometer. The result is interesting in demonstrating both the points of agreement between the actual and theoretical holograms, and the points where the technique of production has failed.

In this paper we discuss the Abel series for a function F(z) which is regular in an angle | arg z | ≤ α and at the origin. We investigate conditions under which the series converges and conditions under which its sum is asymptotically equivalent to the function F(z) in the half-plane R(z) > 0.

Some of the formulae obtained in this paper are likely to find application in problems concerning a rectangular lattice of “atoms”, each of which is under the influence of its near neighbours. Some of the determinants considered apply to cases in which both the nearest and the next nearest neighbours are operative. The inverses of certain types of matrices are found, and these may prove to be of value either in solving systems of linear equations such as arise in relaxation problems, or in determining the latent roots of matrices which may occur in problems in applied mathematics.

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.

A square matrix A = (aij) is expressed symbolically in terms of Clebsch-Aronhold equivalent symbols aij = aiaj = βibj = …, and the symbolic expressions for symmetric functions of the latent roots of A are considered, the relation between these functions and projective invariants of the bilinear form uAx being noted. The Newton and Brioschi relations between the symmetric functions are obtained by reduction of symbolic determinants and permanents respectively, and the Wronskian relations are shown to be equivalent to certain identities between determinants and permanents due to Muir. Also the fundamental theorem of symmetric functions is obtained symbolically as a consequence of the first fundamental theorem of invariants. The paper concludes with a note on the symbolization of the h-bialternants, that is of the traces of irreducible invariant matrices of A.

The division of one polynomial by another is studied with the object of ascertaining the errors produced in the coefficients of successive remainders by small errors in the coefficients of the divisor. It is shown that the matrix which effects this transformation of errors is a polynomial in the rational canonical matrix for which the divisor polynomial is characteristic. The theory gives rise to a numerous class of iterative processes for finding an exact factor, such as the extant method based on the penultimate remainder, Bairstow's iterative method of finding a quadratic factor, and many others. Some new suggestions are made for accelerating convergence.

The method of iteration of penultimate remainders, introduced by S. N. Lin for approximating by stages to the exact factors of a polynomial, is subjected to theoretical analysis. The matrix governing the iterative process is obtained, and its latent roots and latent vectors are found. Incidental theorems yielding further factorizations are proved, and processes are developed for accelerating convergence. Numerical examples illustrate varying situations likely to arise in practice.

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.

Analytic solutions of the functional equation f[z, φ{g(z)}] = φ(z), in which f(z, w) and g(z) are given analytic functions and φ(z) is the unknown function, are investigated in the neighbourhood of points ζ such that g(ζ) = ζ. Conditions are established under which each solution φ(z) may be given as the limit of a sequence of functions φn(z), defined by the recurrence relation φn+1(Z) = ƒ[z, φn{g(z)}], the function φn(z) being to a large extent arbitrary.

A Sargent diagram is presented containing 12 plotted points relative to capture-active species in the range of atomic number (Z) from 89 to 98 inclusive. Arguments are adduced to show that the “allowed” line of the diagram is located as theory predicts, and the capture transformations of other heavy capture-active species are discussed with the aid of the diagram. In particular, values are deduced for the energies of capture transformation of 17 species for which 79 ≤ Z ≤ 85, and, taking count of these values, the energies of β-disintegration of 150 species having 76 ≤ Z ≤ 98 are assumed known, and are suitably plotted against neutron number N. Discontinuities are found, for certain values of isotopie number, in the region of N = 126 (and Z = 82). Values of α-disintegration energy are also deduced for certain isotopes of bismuth and lead.

In very olden days chemists did not forgather merely as chemists; they merged themselves in broader organizations such as the Royal Society. The “chemical revolution”, which had its real beginning with the work of Joseph Black and which culminated in the overthrow of the phlogiston theory by Lavoisier, aroused for the first time a popular interest in the special science of Chemistry. Until recently, world priority among the chemical societies that resulted therefrom was by general agreement conceded to the Chemical Society of Philadelphia, founded by James Woodhouse in 1792. The distinguished chemical historian Edgar F. Smith (I), late Provost of the University of Pennsylvania, may be quoted in this connection:

The author presents a modification of the recently discovered “elementary” proof of the Prime Number Theorem. Nothing is assumed from the theory of numbers except the Fundamental Theorem of Arithmetic. In the second part of the proof the elements of the integral calculus are used to make clearer the basic ideas on which this part depends.

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.