In the application of Elliptic Functions to the Theory of Numbers the two formulae of Jacobi

are of great importance.

The extension of Mellin's inversion formula expressed by the equations has been considered by Fowler who shows that some form of Stieltjes integral is essential to Poincaré's proof of the necessity of the quantum hypothesis. Fowler confines his discussion to a restricted type of function φ (y) which is sufficient for the physical problem. It will be proved here that the formulae hold with a general Stieltjes integral in the first equation.

The theorem is attributed to Wallace that, in a euclidean plane, the circumcircles of the triangles determined by four lines, of general position, meet at a point. It is further known that, in euclidean space of n dimensions, the circumhyperspheres of the simplices determined by n + 2 flats, of general position, meet at a point, if and only if n be even.

Expressions are found for the changes caused by slow selection in populations whose characters are determined by incompletely dominant, multiple, or polyploid factors, and for the equilibria attained in certain of these cases.

A Fourier integral will be described as of finite type if the range of integration of the integrals by means of which the coefficients in the Fourier integral are defined is a finite interval instead of the usual (−∞ ∞). Thus is of finite type (p, q), such that where ƒ(x) is the generating function of the series.

The particular form of Taylor's theorem considered states that: If ƒ(x)is n times differentiable at a thenwhere.

This correspondence has been established by Mr H. W. Turnbull, Proc. Camb. Phil. Soc. vol. XXII. (1925), pp. 694–9, a construction which leaves uncorrelated the lines of a special linear complex and the points of the quadric lying in one tangent fourfold; these have to be correlated by means of an independent construction.

In the network shown diagrammatically in Fig. 1, A0A3, A3A5,… are resistances of values a1, a3,… joined in series with one another, and A3B3, A5B5,…are resistances of values 1/a2, 1/a4,…: the points B0B3,… are all on a cable of negligible resistance. The members A0A3, A3A5,… will be called the series members of the network, and the members A3B3, A5B5,… will be called the shunt members of the network: the points A0B0, will be called the input terminals of the network. If a potential difference is maintained between the input terminals, currents will flow in the members of the network.

If it is assumed that the series electron of an atom polarises the core, then it has been shown by Born and Heisenberg that the polarisability α of the core in a given state may be calculated from the corresponding term value by means of the approximate formulae, where q is the quantum defect, δν is the difference between the term and the corresponding hydrogen term, R is the Rydberg constant in cm.−1, α1 is the radius of first hydrogen orbit, n is the principal quantum number, k is the azimuthal quantum number, For terms with small quantum defect either of the formulae (1) and may be used, but for terms with large quantum defect (1) gives a higher degree of accuracy.

For the purposes of atomic physics it has been found convenient to introduce the idea of quantities that do not in general satisfy the commutative law of multiplication, but satisfy all the other laws of ordinary algebra. These quantities are called q-numbers, and the numbers of ordinary mathematics c-numbers, while the word number alone is used to denote either a q-number or a c-number. Both q-numbers and c-numbers can occur together in the same piece of analysis, and even in the same equation, as numbers of the two kinds can be added together or multiplied. A c-number may, in fact, be regarded as a special case of the more general q-number. In the particular case when two numbers x; and y satisfy xy = yx, we shall say that x commutes with y. A c-number is assumed to commute with every number.

In the course of a long series of experiments on the scattering of positive rays I have had occasion to take some hundreds of photographs in which positive rays diverging from an approximately linear source pass through a metal slit about ·5 mm. wide parallel to the source and then strike the sensitive surface of the plate. The change caused by the actual impact of the rays can be developed like the ordinary latent image produced by light. The pattern thus formed on the negative is a rectangular strip about 1·8 mm. × 20 mm. answering in shape to the shape of the slit, and bordered by a much less intense edge (of the order 10%) due to the scattered rays whose investigation was the object of the experiment. It was found that, in a majority of the photographs, the rectangular strip, which should have been approximately uniform in blackness, at least across, had its longer edges apparently much blacker than its centre. So marked is this effect that it was supposed to be due to some cause affecting the distribution of the rays. However, careful measurements with a microphotometer showed that the effect was unreal, the edges always being less black than the centre. The effect of the positive rays is thus merely to produce a certain distribution of blackening on the plate, and the increased blackness seen at the edges is purely subjective.

The problem of the two bodies has been treated on the new mechanics by Dirac, Pauli, and Schrödinger, who have independently derived the Balmer terms. The present paper is an attempt at a more complete solution. In particular, formulae are derived for the line intensities of the hydrogen spectrum, for the photoelectric effect and its inverse, and for the continuous absorption spectrum in the ultraviolet and in the X-ray regions. Also the probabilities of transition, deflection and capture are computed for the collision of an electron and an ion. Numerical values are only obtained, however, for the simplest line intensities. It is hoped to treat the problem in greater detail.

The view which the author has expressed regarding catalytic activity as due to frozen groups of atoms with strong specific external fields is further tested by a study of the stability of preparations of copper at the instant of formation, and during the subsequent life of the catalyst.

In a recent paper, G. N. Lewis and D. F. Smith have discussed the problem of the velocity of chemical reactions, especially that of the only gas-reaction, the decomposition of N2O5, which had then been proved with sufficient accuracy to be kinetic unimolecular, i.e. unimolecular in the purely empiric sense of the word.

During the last year the homogeneous reactions of haemoglobin, and the heterogeneous reactions of the red blood corpuscle, have continued to furnish us with abundant opportunity for the application of our method for measuring the velocity of very rapid chemical reactions*. The recent types of apparatus, which we have devised, have not departed in essentials from the general principle previously employed, but a number of modifications and extensions have been introduced to meet with greater efficiency the special conditions which were imposed upon us. Since these developments would seem to be of much wider application than the special purposes for which they were in the first place designed, we think that it may be useful to workers in other fields to give a brief summary of their general features here. In the present paper we shall accordingly describe the methods we have adopted for 1. Very fast reactions. 2. Very slow reactions. 3. Very dilute solutions. 4. Small quantities of fluid. 5. Reactions following one another in rapid succession. 6. Reactions involving short-lived transient compounds. 7. The detection of concentration gradients in heterogeneous reacting solutions. Special apparatus has been devised to deal with each of these cases which will be briefly described in the following sections.

As a preliminary to an investigation of certain diffraction patterns I was led to consider, in some detail, the geometrical aberrations of a symmetrical optical system; and it appeared convenient then to classify the aberrations in orders according as they depend upon various powers of certain small quantities and to exhibit them as coefficients in the expansion of an ‘ Aberration Function.’ If aberrations of the first order only are considered, it becomes evident that one of them stands, in some sense, apart from the rest; I refer to the so-called ‘Petzval’ condition for flatness of field. It is of interest to notice that this condition was known to Coddington and to Airy before the time of Petzval—known at least as far as concerns systems of thin lenses. In the usual notation the condition is ΣΚ/μμ′ = 0; it is therefore independent of the positions of the object and pupil planes and in this respect it stands alone among the first order aberrations. But an increasing number of similar aberrations of higher orders will be found and it is of interest to examine these and to investigate their geometrical meaning. In the following note is given a proof of the Petzval condition, differing from that usually given and falling more into line with the general theory, and indicating also a general method of examining aberrations of this peculiar type.

In the famous paper in which he developed his theory of radiation, Einstein investigated two problems. He investigated first the problem of what distribution of radiant energy in the steady state would be set up by an enclosed assembly of atoms in thermodynamic equilibrium radiating and absorbing according to the assumptions of his theory; he showed that the distribution so set up was given by Planck's law. He investigated secondly the velocities set up amongst the atoms in consequence of the random variations in direction of the emissions and absorptions; provided that emissions are assumed to be directed, so giving rise to recoil momentum, he showed that the mean square velocity-component in a given direction, , is equal to its equipartition value, given by .

The assumption that the compressional waves of earthquakes follow the ordinary laws of refraction, the energy within any pencil of rays remaining permanently within that pencil, has been found to lead to too small amplitudes for the indirect waves from near earthquakes. In this paper a system consisting of two superposed compressible, but non-rigid, media is considered; the lower is supposed to transmit compressional waves with the greater velocity. It is found that ah explosion within the upper medium produces a disturbance at the upper surface involving the direct wave and all the reflected waves that might be expected; but in addition a wave is found that appears to have travelled along the interface with the velocity of sound in the lower medium. This indirect wave would have zero amplitude on the simple laws of refraction with plane boundaries. The variation of its amplitude with distance from the focus is in reasonable accordance with seismological observation, and its time of arrival agrees with that inferred from the laws of refraction for boundaries with slight curvature. But if the direct wave begins with a finite velocity, the indirect one will begin with a finite acceleration; in seismological language an iPg will be associated with an ePn. The indirect wave will also take longer than the direct one to give its maximum displacement.