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.

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.

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.

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.

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.

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 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.

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.

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.

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.