Modern views respecting mechanical lubrication are founded mainly on the experiments of B. Tower, conducted upon journal bearings. He insisted upon the importance of a complete film of oil between the opposed solid surfaces, and he showed how in this case the maintenance of the film may be attained by the dragging action of the surfaces themselves, playing the part of a pump. To this end it is “necessary that the layer should be thicker on the ingoing than on the outgoing side,” which involves a slight displacement of the centre of the journal from that of the bearing. The theory was afterwards developed by 0. Reynolds, whose important memoir includes most of what is now known upon the subject. In a later paper Sommerfeld has improved considerably upon the mathematics, especially in the case where the bearing completely envelops the journal, and his exposition to be recommended to those who wish to follow the details of the investigation. Reference may also be made to Harrison, who includes the consideration of compressible lubricants (air).

In all these investigations the question is treated as two-dimensional. For instance, in the case of the journal the width—axial dimension—of the bearing must be large in comparison with the arc of contact, a condition not usually fulfilled in practice. But Michell has succeeded in solving the problem for a plane rectangular block, moving at a slight inclination over another plane surface, free from this limitation, and he has developed a system of pivoted bearings with valuable practical results.

It is singular that the explanation of some of the most striking and beautiful of optical phenomena should be still matters of controversy. I allude to the brilliant colours displayed by many birds (e.g. hummingbirds), butterflies, and beetles, colours which vary greatly with the incidence of the light, and so cannot well be referred to the ordinary operation of dyes. In an early paper, being occupied at the time with the remarkable coloured reflexions from certain crystals of chlorate of potash described by Stokes, and which I attributed to a periodic twinning, I accepted, perhaps too hastily, the view generally current among naturalists that these colours were “structurecolours,” more or less like those of thin plates, as in the soap-bubble. Among the supporters of this view in more recent times may be especially mentioned Poulton and Hodgkinson. In Poulton's paper the main purpose was to examine the history of the very remarkable connexion between the metallic colours of certain pupse (especially Vanessa urticce) and the character of the light to which the larvae are exposed before pupation. In a passage describing the metallic colour itself he remarks:

“The Nature of Effects Produced.—The gilded appearance is one of the most metal-like appearances in any non-metallic substance. The optical explanation has never been understood. It has, however, been long known that it depends upon the cuticle, and needs the presence of moisture, and that it can be renewed when the dry cuticle is moistened. Hence it can be preserved for any time in spirit.

That this work should have already reached a fourth edition speaks well for the study of mathematical physics. By far the greater part of it is entirely beyond the range of the books available a generation ago. And the improvement in the style is as conspicuous as the extension of the matter. My thoughts naturally go back to the books in current use at Cambridge in the early sixties. With rare exceptions, such as the notable one of Salmon's Conic Sections and one or two of Boole's books, they were arid in the extreme, with scarcely a reference to the history of the subject treated, or an indication to the reader of how he might pursue his study of it. At the present time we have excellent books in English on most branches of mathematical physics and certainly on many relating to pure mathematics.

The progressive development of his subject is often an embarrassment to the writer of a text-book. Prof. Lamb remarks that his “work has less pretensions than ever to be regarded as a complete account of the science with which it deals. The subject has of late attracted increased attention in various countries, and it has become correspondingly difficult to do justice to the growing literature. Some memoirs deal chiefly with questions of mathematical method and so fall outside the scope of this book; others though physically important hardly admit of a condensed analysis; others, again, owing to the multiplicity of publications, may unfortunately have been overlooked. And there is, I am afraid, the inevitable personal equation of the author, which leads him to take a greater interest in some branches of the subject than in others.”

Dear Prof. Nernst,

Having been honoured with an invitation to attend the Conference at Brussels, I feel that the least that I can do is to communicate my views, though I am afraid I can add but little to what has been already said upon the subject.

I wish to emphasize the difficulty mentioned in my paper of 1900 with respect to the use of generalized coordinates. The possibility of representing the state of a body by a finite number of such (short at any rate of the whole number of molecules) depends upon the assumption that a body may be treated as rigid, or incompressible, or in some other way simplified. The justification, and in many cases the sufficient justification, is that a departure from the simplified condition would involve such large amounts of potential energy as could not occur under the operation of the forces concerned. But the law of equi-partition lays it down that every mode is to have its share of kinetic energy. If we begin by supposing an elastic body to be rather stiff, the vibrations have their full share and this share cannot be diminished by increasing the stiffness. For this purpose the simplification fails, which is as much as to say that the method of generalized coordinates cannot be applied. The argument becomes, in fact, self-contradictory.

Perhaps this failure might be invoked in support of the views of Planck and his school that the laws of dynamics (as hitherto understood) cannot be applied to the smallest parts of bodies. But I must confess that I do not like this solution of the puzzle.

In a paper with the above title, Ehrenfest (Zeitsch. physikal. Chem. 1911, 77, 2) has shown that, as usually formulated, the principle is entirely ambiguous, and that nothing definite can be stated without a discrimination among the parameters by which the condition of a system may be defined. The typical example is that of a gas, the expansions and contractions of which may be either (α) isothermal or (β) adiabatic, and the question is a comparison of the contractions in the two cases due to an increment of pressure δp. It is known, of course, that if δp be given, the contraction |δv| is less in case (β) than in case (α). The response of the system is said to be less in case (β), where the temperature changes spontaneously. But we need not go far to encounter an ambiguity. For if we regard δv as given instead of δp, the effect δp is now greater in (β) than in (α). Why are we to choose the one rather than the other as the independent variable?

When we attempt to answer this question, we are led to recognise that the treatment should commence, with purely mechanical systems. The equilibrium of such a system depends on the potential energy function, and the investigation of its character presents no difficulty. Afterwards we may endeavour to extend our results to systems dependent on other, for example, thermodynamic, potentials.

In the problems here considered the fluid is regarded as incompressible, and the motion is supposed to take place in two dimensions.

Potential and Kinetic Energies of Wave Motion.

When there is no dispersion, the energy of a progressive wave of any form is half potential and half kinetic. Thus in the case of a long wave in shallow water, “if we suppose that initially the surface is displaced, but that the particles have no velocity, we shall evidently obtain (as in the case of sound) two equal waves travelling in opposite directions, whose total energies are equal, and together make up the potential energy of the original displacement. Now the elevation of the derived waves must be half of that of the original displacement, and accordingly the potential energies less in the ratio of 4 : 1. Since therefore the potential energy of each derived wave is one quarter, and the total energy one half that of the original displacement, it follows that in the derived wave the potential and kinetic energies are equal”.

The assumption that the displacement in each derived wave, when separated, is similar to the original displacement fails when the medium is dispersive. The equality of the two kinds of energy in an infinite progressive train of simple waves may, however, be established as follows.

Consider first an infinite series of simple stationary waves, of which the energy is at one moment wholly potential and [a quarter of] a period later wholly kinetic.

[Note.—The concluding paragraphs of this paper were dictated by my father only five days before his death. The proofs therefore were not revised by him. The figure was unfortunately lost in the post, and I have redrawn it from the indications given in the text.—Rayleigh.]

One of the most important questions in meteorology is the constitution of the travelling cyclone, for cyclones usually travel. Sir N. Shaw says that “a velocity of 20 metres/second [44 miles per hour] for the centre of a cyclonic depression is large but not unknown, a velocity of less than 10 metres/second may be regarded as smaller than the average. A tropical revolving storm usually travels at about 4 metres/second.” He treats in detail the comparatively simple case where the motion (relative to the ground) is that of a solid body, whether a simple rotation, Or such a rotation combined with a uniform translation; and he draws important conclusions which must find approximate application to travelling cyclones in general. One objection to regarding this case as typical is that, unless the rotating area is infinite, a discontinuity is involved at the distance from the centre where it terminates. A more general treatment is desirable, which shall allow us to suppose a gradual falling off of rotation as the distance from the centre increases; and I propose to take up the general problem in two dimensions, starting from the usual Eulerian equations as referred to uniformly rotating axes. The density (ρ) is supposed to be constant, and gravity can be disregarded.

In a short paper “On the Electrical Vibrations associated with thin terminated Conducting Rods” I endeavoured to show that the difference between the half wave-length of the gravest vibration and the length (l) of the rod (of uniform section) tends to vanish relatively when the section is reduced without limit, in opposition to the theory of Macdonald which makes λ = 2·53 l. Understanding that the argument there put forward is not considered conclusive, I have tried to treat the question more rigorously, but the difficulties in the way are rather formidable. And this is not surprising in view of the discontinuities presented at the edges where the flat ends meet the cylindrical surface.

The problem assumes a shape simpler in some respects if we suppose that the rod of length l and radius a surrounded by a cylindrical coaxial conducting case of radius b extending to infinity in both directions. One advantage is that the vibrations are now permanently maintained, for no waves can escape to infinity along the tunnel, seeing that l is supposed great compared with b. The greatness of l secures also the independence of the two ends, so that the whole correction to the length, whatever it is, may be regarded as simply the double of that due to the end of a rod infinitely long.

At an interior node of an infinitely long rod the electric forces, giving rise (we may suppose) to potential energy, are a maximum, while the magnetic forces representing kinetic energy are evanescent. The end of a terminated rod corresponds, approximately at any rate, to a node.

Dans les Comptes rendus du 30 décembre 1912, M. Eiffel donne des résultats très intéressants pour la résistance rencontrée, à vitesse variable, par trois sphères de 16·2, 24·4 et 33 cm. de diamètre. Dans la première figure, ces résultats sont exprimés par les valeurs d'un coefficient Κ, égal à R/SV2, où R est la résistance totale, S la surface diamétrale et V la vitesse. En chaque cas, il y a une vitesse critique, et M. Eiffel fait remarquer que la loi de similitude n'est pas toujours vraie; en effet, les trois sphères donnent des vitesses critiques tout à fait différentes.

D'après la loi de similitude dynamique, précisée par Stokes et Reynolds pour les liquides visqueux, Κ est une fonction d'une seule variable v/VL, où v est la viscosité cinématique, constante pour un liquide donné, et L est la dimension linéaire, proportionnelle à S½. Ainsi les vitesses critiques ne doivent pas être les mêmes dans les trois cas, mais inversement proportionnelles a L. En vérité, si nous changeons l'échelle des vitesses suivant cette loi, nous trouvons les courbes de M. Eiffel presque identiques, au moins que ces vitesses ne sont pas très petites.

Je ne sais si les écarts résiduels sont réels ou non. La théorie simple admet que les sphères sont polies, sinon que les inégalités sont proportionnelles aux diametres, que la compressibilité de l'air est négligeable et que la viscosité cinématique est absolument constante. Si les résultats de l'expérience ne sont pas complètement d'accord avec la théorie, on devra examiner ces hypothèses de plus près.

J'ai traité d'autre part et plus en détail de la question dont il s'agit ici.

The action of a cone in collecting sound coming in the direction of the axis may be investigated theoretically. If the diameter of the mouth be small compared with the wave-length (λ) of the sound, the cone may operate as a resonator, and the effect will vary greatly with the precise relation between λ and the length of the cone. On the other hand, the effect will depend very little upon the direction of the sound. It is probably more useful to consider the opposite extreme, where the diameter of the mouth is a large, or at any rate a moderate, multiple of λ, when the effect may be expected to fall off with rapidity as the obliquity of the sound increases.

A simple way of regarding the matter is to suppose the sound, incident axially, to be a pulse, e.g. a condensation confined to a narrow stratum bounded by parallel planes. If the angle of the cone be small, the pulse may be supposed to enter without much modification and afterwards to be propagated along. As the area diminishes, the condensation within the pulse must be supposed to increase. Finally the pulse would be reflected, and after emergence from the mouth would retrace its course. But the argument is not satisfactory, seeing that the condition for a progressive wave, i.e. of a wave propagated without reflection, is different in a cylindrical and in a conical tube.

I have often been impressed by the scanty attention paid even by original workers in physics to the great principle of similitude. It happens not infrequently that results in the form of “laws “are put forward as novelties on the basis of elaborate experiments, which might have been predicted a priori after a few minutes′ consideration. However useful verification may be, whether to solve doubts or to exercise students, this seems to be an inversion of the natural order. One reason for the neglect of the principle may be that, at any rate in its applications to particular cases, it does not much interest mathematicians. On the other hand, engineers, who might make much more use of it than they have done, employ a notation which tends to obscure it. I refer to the manner in which gravity is treated. When the question under consideration depends essentially upon gravity, the symbol of gravity (g) makes no appearance, but when gravity does not enter into the question at all, g obtrudes itself conspicuously.

I have thought that a few examples, chosen almost at random from various fields, may help to direct the attention of workers and teachers to the great importance of the principle. The statement made is brief and in some cases inadequate, but may perhaps suffice for the purpose. Some foreign considerations of a more or less obvious character have been invoked in aid. In using the method practically, two cautions should be borne in mind.

At an early date my attention was called to the problem of the stability of fluid motion in connexion with the acoustical phenomena of sensitive jets, which may be ignited or unignited. In the former case they are usually referred to as sensitive flames. These are naturally the more conspicuous experimentally, but the theoretical conditions are simpler when the jets are unignited, or at any rate not ignited until the question of stability has been decided.

The instability of a surface of separation in a non-viscous liquid, i.e. of a surface where the velocity is discontinuous, had already been remarked by Helmholtz, and in 1879 I applied a method, due to Kelvin, to investigate the character of the instability more precisely. But nothing very practical can be arrived at so long as the original steady motion is treated as discontinuous, for in consequence of viscosity such a discontinuity in a real fluid must instantly disappear. A nearer approach to actuality is to suppose that while the velocity in a laminated steady motion is continuous, the rotation or vorticity changes suddenly in passing from one layer of finite thickness to another. Several problems of this sort have been treated in various papers. The most general conclusion may be thus stated. The steady motion of a non-viscous liquid in two dimensions between fixed parallel plane walls is stable provided that the velocity U, everywhere parallel to the walls and a function of y only, is such that d2U/dy2 is of one sign throughout, y being the coordinate measured perpendicularly to the walls. It is here assumed that the disturbance is in two dimensions and infinitesimal.