The question has frequently been raised, whether the laws of ordinary Geometrical Optics, especially with regard to the reflexion and refraction at the confines of two media, are equally applicable when the transition is accomplished in a gradual manner. The problem has an important bearing on physical phenomena of a widely varying character, such as the propagation of sound and light in the atmosphere, with occurrence of mirage or zones of silence, the transmission of seismic disturbances through the earth, tidal waves in a canal of varying cross-section, and others.

One method of measuring frequency depends upon resonance. If a body of unit mass, subject to elastic constraints and experiencing a damping resistance proportional to its speed, be acted on by a periodic force F sin pt, the equation of motion may be written

The amplitude of the steady vibration ultimately set up is

the maximum value occurring when p2 = q2 − 2s2. If the motion, in the absence of the dri ving force F sin pt, be only slightly damped, s2 is very small compared with q2, and little error will be made if the value of p giving maximum amplitude be taken as equal to q. When s/q is small, the resonance is “sharp,” i.e. a small discrepancy between p and q, or, more strictly, between p and (q2 − 2s2)½, causes the amphtude to fall far below its maxìmum value. Thus, when the damping is small, maximum amplitude, f or variation of p, indicates that p is very nearly equal to q.

Some previous work on the theory of the electrical counter is extended by a detailed consideration of the ionic process involved. It is shown that with a positive point the total number of ions produced should be proportional to the initial ionisation. Evidence is adduced to show that with a negative point extinction of the discharge is probably brought about by a local increase of pressure, accompanied by an electrical wind along the sides of the point. The latter would bring about the discharge of a definite quantity of electricity, independent of the initial ionisation, provided the initial pressure was not too low. It is shown that these considerations account for a number of other phenomena connected with discharges from small electrodes, and that the quantities involved are of the right order of magnitude.

The experimental work in connection with this paper has been carried out in the Cavendish Laboratory. The author desires to thank Professor Sir E. Rutherford for his continual interest in its progress, and further to thank Professor Appleton, and Dr Chadwick for much helpful discussion of several of the points involved.

If a gas, each of whose molecules is capable of dissociating into two similar molecules, be kept at a constant temperature, it would attain an equilibrium state in which the rate at which double molecules dissociate is equal to the rate at which they are formed by recombination of single molecules, there being a different equilibrium state for each different temperature. If, now, the gas be subjected to a temperature gradient, the concentrations necessary for equilibrium would be different at different points, so there would be diffusion between the two kinds of molecules, a steady state being attained when the rate of diffusion of double molecules into any region is equal to their excess rate of decomposition over their rate of formation within that region.

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.

The sensitivity of photographic plates to mass rays bears practically no relation whatever to their sensitivity to light. A short account is given of tests on the relative merits of different brands of plates since the first experiments on positive ray analysis. The process of improving the qualities of plates by schumannising is discusseci and details are given of the process found likely to give the best results. Success cannot be guaranteed even with the most careful manipulation, but if achieved yields plates of outstanding value for use in such methods of analysis as the mass-spectrograph.

1. Geodesic curves, even of simple surfaces such as quadrics, are almost always transcendental: occasionally they are algebraic: still more rarely do they belong to simple and familiar types. The form of the differential equation referred to proves conclusively that even such integrals as are expressible by elementary functions must be quite exceptional.

In a recent series of papers analytical methods have been introduced which allow of a mathematically simple treatment of the theorems of statistical mechanics for the usual assemblies of isolated or effectively isolated systems. By this we mean that the individual component systems may be treated for energy content as if they were never interfered with. It is only then that energy can be assigned to systems rather than to the assembly as a whole, and it is on this partition of energy among the systems that the analysis is based. When this independence, for example between separate atoms, breaks down as in a molecule, and still more in a crystal, we can take the whole complex to be a system. The analysis will still apply, and if we can formulate the dynamical motions of the complex system, we can still make progress. The essential step for any system is to construct its partition function. Examples of such constructions for molecules and crystals will be found in the papers quoted, and are of course otherwise well known.

1. The kinetic theory of metallic conduction is usually supposed to lead to formulae for the conductivity coefficients which in some respects are very much at variance with the actual phenomena. In spite of the important successes achieved by the theory, which in themselves are almost sufficient evidence that the fundamental basis is correct, these discrepancies are regarded in some quarters as so serious as to warrant the complete rejection of the theory as beine inadequate to account properly for the facts.

The theorem of Wallace that if four arbitrary lines he taken in a plane the circumcircles of the four triangles formed by them in threes meet in a point—in which for circles we may equally well take conies drawn through two arbitrary points I, J of the plane —may be proved very easily from three dimensions. Through the four given lines draw four arbitrary planes, forming a tetrahedron X, Y, Z, T. A unique cubic curve can be drawn through I, J and the four vertices. Projecting this curve from the point T upon the original plane, we get a conic which passes through I, J and the vertices of the triangle formed by the lines in which the plane meets the planes TYZ, TZX, TXY, that is, by three.of the given lines. This conic, which is one of the conics of the theorem, also passes through the third point K, other than I, J, in which the original plane meets the cubic curve. Projecting in turn from the other three vertices of the tetrahedron, we get the other three conics of the theorem, which also pass through K.

§ 1. The groundwork of the present paper is a general transformation of Maxwell's equations. As relativist speculation had its origin in a simple transformation, it was to be expected that something more would result when the general transformation was explored: the determining influence has proved stronger than I had anticipated.

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.

The following paper is in two parts.

In Part 1 it is shown that Kronecker's theorem can be extended in the form

If is greater than ηfor all sets of integers l1, l2, … l3 less in absolute value than K/σ and not all zero, l also being an integer, then, for any x1, x2 … x3, an integer q less than l/(ησ8) can be found such that qv1–x1, qv2–x2…qv3–x4 all differ from integers by less than σ;. K, L depend only on s.

It is an immediate corollary that if

is greator than for all sets of integers l1, l2… l3, l less in absolute value than K/σ and not all zero while F(v1, v2… v3), l less in absolute value than K/σ and not all zero while F(v1, v2… v3, v) is periodic period 1, in v1, v2 … v3,v, then T can be found between 0 and such that

Where K, L depend on s, and N on s and the bounds of

The problem of finding the number of r-dimensional regions which are situated in a space Sn of n dimensions, and which satisfy a suitable number of conditions of certain assigned types (called “ground-conditions”) has been investigated by Schubert. A special class of such problems arises when the r-dimensional region is merely required to intersect k regions Pλ of rλ dimensions (λ = 1, 2 … k) situated in general position in Sn where for the finiteness of the sought number we must have

Landé has indicated a connection between the “relativity” doublets of X-ray spectra, and the doublets and triplets of optical spectra, and so has obtained a formula for the separation of optical doublets and triplets. This formula can only be expected to hold if the orbit of the series electron penetrates into the core, so it may be possible to obtain evidence on the question whether an orbit penetrates from the doublet or triplet separation of the corresponding term.

From this evidence it is concluded that, except for lithium-like atoms, p terms of all known spectra correspond to penetrating orbits; this disagrees with Bohr's assignment of quantum numbers for those terms of the spectra of neutral atoms of the Cu and Zn sub-groups. For d terms the evidence is not so definite, but it seems possible that for Cs I and Tl I the d terms correspond to penetrating orbits, in disagreement with Bohr's assignment. It is shown that the Landé formula seems to hold approximately for separations in multiplet spectra when the terms belong to a sequence of the Rydberg type.

Ideas suggested by Heisenberg in a recent paper would involve some modifications of the interpretation of the magnitudes of doublet separations, but at least the conclusions as to the p orbits penetrating still stand.

In a previous paper it has been shown that the corrosion of zinc, cadmium, iron and lead is connected with electric currents set up through unequal distribution of oxygen; the currents have been measured and compared with the weight of metal corroded away. The “unaerated” areas are anodic, and thus the attack tends to become concentrated upon the points relatively inaccessible to oxygen. When a drop of salt water is placed on an iron sheet, the central part of the drop suffers anodic corrosion, giving rise to ferrous salts, whilst the marginal ring over which oxygen has ready access to the metal becomes cathodic, alkali being produced there. Where the ferrous salt and alkali come into contact, ferrous hydroxide is precipitated, which oxidises to ferric hydroxide; after a few hours we get a membrane of brown ferric hydroxide extending right over the drop, but surrounded by a ring of clear alkaline liquid. All the actual corrosion of the metal is confined to the area within the membrane; the metal below the external ring of clear liquid, to which oxygen has access, suffers no attack.

This problem is in general treated in connection with the division of the periods of the elliptic functions. It is the object of the present note to shew, from a purely algebraical point of view, that the condition of closing of the polygon depends solely on a difference equation of the form

In the Transactions of the American Math. Society, vol. I, 1900, and vol. IV, 1904, Mr E. Kasner treats exhaustively the (2, 2) double binary form and discusses the theory connected with double binary forms and multiple binary forms in general terms. He shows the relations between the systems of multiple binary forms with digredient variables and the forms with cogredient variables. Hitherto nothing seems to have been written on systems of triple binary forms (with regard to higher forms see a paper on the (1, 1, 1, 1) form by C. Segre); so here I propose to discuss the complete system of a (1, 1, 1) binary form which consists of six forms connected by one syzygy. When two of the variables are the same we naturally get the (2, 1) form and when the three are the same We get the (3) or cubic binary form.

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

(1) A method has been devised and applied for measuring the decay Constant of radium emanation, the method depending on careful comparisons with a simple electroscope under the most favourable conditions.

(2) A mean value for the Constant of radium emanation has been obtained, with an error unlikely to exceed 0·2 per cent.: λ = 0·1808 day−1, or half-period 3·833 days.