The formulae

where γ = ½ (α + β + l), and

where γ = 1 + β, were given by Gauss (Ges. Werke, iii, pp. 225, 226). It is here proposed to find the corresponding expressions for the hypergeometric function when γ has general values (not zero or negative integral). These will be derived in section 2 by applying Lagrange's expansion

where

and that root of equation (4) in x is taken which is equal to λ when w = 0. Two generalisations of Whipple's Transformation will be obtained in section 3.

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 The work is an extension of unpublished researches by Sir G. I. Taylor on the corresponding problem for a thin wire, and a summary of his results is included.

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.

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.

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.

§1. Introductory. The formula

where w is zero or a positive integer and | ζ | > 1, was given by F. E. Neumann “Crelle's Journal, XXXVII (1848), p. 24”. In § 2 of this paper some related formulae are given; the extension to the case when n is not integral is dealt with in § 3; while in § 4 the corresponding formulae for the Associated Legendre Functions when the sum of the degree and the order is a positive integer are established.

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.

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.

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 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:

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

The formula

where αp+1 = 1/2(m + n), αp+2 = 1/2(m-n), R(m±n)>0 and x is real and positive, was given by MacRobert (Phil. Mag., Ser. 7, XXXI, p. 258). From it the formula (6) below will be deduced.

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.

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.

In this note we derive some integrals involving confluent hypergeometric functions and analogous to Lommel's integrals for Bessel functions. Although the method of derivation is straightforward, the integrals do not seem to be mentioned in the literature.

If the right-hand side of the expansion

is integrated m times from 1 to µ, it becomes

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.