A complex or system ∞3 of conics in space of four dimensions is such that a finite number of conics pass through an arbitrary point. Linear complexes are those for which this number is unity, and are such that their curves are defined by conditions of incidence with fixed surfaces, curves and points. In this paper are discussed briefly the linear complexes defined by the condition that their curves meet an irreducible curve in four points. Denoting by a curve of order m and genus p it is found that the curves in question are The complex associated with is considered in greater detail, since it is found to have an interesting connection with the well-known Weddle quartic surface of ordinary space. In fact the conics of the system touching a space (of three dimensions) do so in the points of such a surface. The main properties of this surface can be thence deduced. In addition we discuss certain results in connection with this curve . The paper closes with certain enumerative results which were obtained in the course of the researches giving the results recorded and which we believe are worth record.

The results of the present paper were developed some years ago for quite trivial purposes. It seems possible, however, to judge by a recent publication, that they may sooner or later have scientific utility. They have also a certain slight intrinsic interest.

Let Λ (n) be the arithmetic function usually so denoted, which is zero unless n is a prime power pm (m ≥ 1), when it is log p. We write as usual

and

where the dash denotes that if x is an integer the last term Λ (x) of the sum is to be taken with a factor ½. We wrute further

Beltrami's discovery that corresponding to any ruled surface there is another applicable on it having corresponding generators parallel but with the parameter of distribution of opposite signs, has brought into prominence the distinction between the applicability of two surfaces and the continuous deformation of one into the other. It is pointed out that since the parameter of distribution differs in sign, the tangent plane rotates in opposite directions as the point of contact proceeds along the two corresponding generators on the respective surfaces. One surface cannot, therefore, be deformed into the other. It is proposed in the present paper to investigate the distinction between applicability and deformability of two surfaces in general, and to connect this isolated fact with the general theory of deformation.

§ 1. In the transformation corresponding to constant acceleration, and in the resulting quadratic form, there is an arbitrary constant. In Einstein's formula for central acceleration a mass m defining the acceleration appears, but no arbitrary constant. In seeking to account for, or to repair, the deficiency I retained all constants naturally arising in the integration of the differential equations, and it then appeared that the result could be obtained by transformation. It will be convenient to state the transformation at the outset, and then to examine the differential equations with a view to a clear understanding of the assumptions made in reaching the more general result.

In two recent papers in the Philosophical Magazine the statistical theory of collisions of electrons with atoms has been considered, and applied to explain the observed features of the capture and loss of electrons by α-particles. The basis of that discussion can however be improved. Firstly, interactions with the core of any atom entered by the α-particle were explicitly ignored. Such interactions—or rather interactions with the general intra-atomic field as distinct from individual electrons—might be expected to be (and are) important, and it has been found possible to include them in this paper. Secondly, the frequency laws for the processes concerned were based on the classical Thomson-Bohr theory of ionization by collision. This theory—it is well known—does not completely reproduce experimental facts at α-particle velocities, and a less restrictive form can be given to the frequency laws which appears to be in accord with all the known facts. The application to the capture and loss of electrons by α-particles proves to be unaffected.

Let

be a system of differential equations in which X1, … Xn are analytic functions independent of t, expansible in convergent series of powers of x1, … xn. Suppose further that not all the functions X1, … Xn vanish when x1 = … = xn = 0. It will be shown in §§ 1–3 that these equations possess (n — 1) independent integrals independent of t, expansible in convergent series of powers of x1, … xn; i.e. functions

such that

identically. In § 4 it is shown how the series for ϕ1,…ϕn−1 may be most directly constructed, and in § 5 is briefly considered the corresponding problem when the origin is a singular point of the equations, i.e. when the expansions of X1,…Xn all begin with terms of the first or higher degrees.

I have collected in the present note some theorems regarding the solution of a certain system of linear equations with an infinity of unknowns. The general form of the equations is

the numbers a1, a2, … c1, c2, … being given. Equations of this type are of course well known; but in studying them it is generally assumed that the series depend for convergence on the convergence-exponent of the sequences involved, e.g. that and are convergent. No assumptions of this kind are made here, and in fact the series need not be absolutely convergent. On the other hand rather special assumptions are made with regard to the monotonic character of the sequences an and cn.

In Chapter v of his Les Méthodes Nouvelles de la Mécanique Céleste Poincaré proves a theorem which he calls The non-existence of Uniform Integrals. This theorem is as follows:

Let

be a system of equations in which the characteristic function F is expanded in a series

convergent for sufficiently small values of | μ |, for all real values of y1, … yn, and for values of x1, … xn within certain finite real intervals; while F0, F1; … are analytic functions of x1, … xn, y1, … yn, of which F0 is independent of y1, … yn and Fl, F2, … are periodic with respect to these variables with period 2π.

Let N (T) denote, as usual, the number of zeros of ζ (s) whose imaginary part γ satisfies 0 < γ < T, and N (σ, T) the number of these for which, in addition, the real part is greater than σ. In this definition we suppose, in the first place, that no zero actually lies on the line t = T: if the line contains zeros we define

In hypergeometry loci defined by parametric equations

have been frequently discussed from various points of view. Among other properties these loci possess two systems of generating spaces obtained by keeping the α's (or β's) fixed while the β's (or α's) vary, the simplest example being the generators of a quadric in S3. Another method of obtaining loci having these properties is by taking the flat spaces through the sets of a linear series of sets of points on an elliptic curve and the flat spaces through the sets of the residual series (the curve being supposed normal. In this way we shall obtain a configuration of ∞PSP's and ∞qSq's, the dual of which gives a locus included among those represented by (1).

Let

be a system of differential equations of Hamiltonian form, the characteristic function H being independent of t and expansible in a convergent series of powers of x1, … xn, y1, … yn in which the terms of lowest degree are

The problem of finding general types of solution for the steady flow of gases in two dimensions, adiabatically and without friction, does not appear to have received a great deal of attention. An interesting and suggestive treatment of hydrodynamics applied to gases is given in Riemann-Weber's Partielle Differential-Gleichungen, but the application is to pressure propagation. Reference must also be made to a paper by the late Lord Rayleigh, who gave a general differential equation which must be satisfied by the velocity potential ø, but the problem of utilizing this equation does not appear easy.

In addition to the simple thallous and thallic chlorides and bromides, a considerable number of intermediate and mixed halides have been described by various investigators. Most of the modern work on these substances has been done by Thomas, by Meyer and by Cushman. There is general agreement as regards the existence of substances of the type TlX3. 3TlX, but considerable divergence exists with regard to the mixed chlorobromides. Compounds of the type TlX3. TlX have also been described, the best known of which appears to be the dibromide TlBr3. TlBr.

Many years ago Rutherford remarked that the high energy lines of the radium C β-ray spectrum showed the existence of γ-rays of very high frequency, and later he pointed out that their frequencies must be closely connected with the β-ray energies through the quantum relation. Although the truth of Rutherford's point of view has become more and more obvious, no advance in the detailed knowledge of these lines has been made in the last seven years. This spectrum is very complicated and the lines faint.

1. Changes in the relative intensities of the lines in the fluorescent L-spectrum of Cerium excited by radiation of various wave-lengths have been observed.

2. These results imply a change in the relative absorbing powers of the three L-levels as the wave-length of the absorbed radiation diminishes from a value just below the absorption wave-length of the L-levels to a value considerably below. The absorbing power of the LI-level becomes increased relative to the absorbing powers of the other L-levels as the wave-length diminishes. The results agree with those published by H. Robinson in a recent paper.

3. These results imply a breakdown of the law that μ/λ3 is a constant (where μ is the absorption coefficient of X-rays of wave-length λ) as applied to the individual L-levels of an element.

4. A comparison is made between the above results, and some results on the relative absorbing process of the L-levels obtained by Ellis and Skinner from β-ray spectra.

1. The conductivity of liquid sulphur is found to vary with the temperature in a similar manner to that in which the viscosity varies.

2. The conduction is electrolytic in character.

3. There is a definite E.M.F. of polarisation.

Using an electrical counter, the number of α- and β-particles from a source of Radium D, Radium E and Radium F in equilibrium has been measured: it has been found that after correction for reflection at the source, their numbers were about equal. According to previous work on their absorption, the β-particles from Radium D would not be recorded under the conditions of these experiments. On this assumption the observed β-radiation was due to disintegration of Radium E, and since this was in equilibrium with Radium F, it follows that about one β-particle is emitted per disintegrating atom of Radium E.

In all methods of measuring excitation and ionisation potentials which are in common use the current-voltage characteristic curves of the apparatus are plotted. In the absence of gas these curves are smooth, but on the admission of gas at a suitable pressure sharp upward bends occur at definite voltages, which are interpreted as the critical potentials of the gas. In a variation by Franck and Hertz, the so-called inelastic impact method, these bends are downward, the critical points being represented by maxima.

It is possible to obtain approximate theoretical relations between the spectra of different atoms ionised to such an extent that they possess the same electron structure. In this paper these relations are used to extrapolate values of the terms of the spectra of various lithium-like and sodium-like atoms.

From the terms so estimated lines can be calculated, and in many cases correlated with lines observed by Millikan and Bowen in the hot spark spectrum of the corresponding element. Revised term values can then be calculated from the lines so identified.

The spectra of lithium-like atoms treated in this way include those from Be II up to O VI, and those of sodium-like atoms, the spectra P V and S VI.