The subject of this lecture has received the attention of several generations of mathematicians and experimenters. Over a part of the field their labours have been rewarded with a considerable degree of success. In all that concerns small vibrations, whether of air, as in sound, or of water, as in waves and tides, we have a large body of systematized knowledge, though in the case of the tides the question is seriously complicated by the fact that the rotation of the globe is actual and not merely relative to the sun and moon, as well as by the irregular outlines and depths of the various oceans. And even when the disturbance constituting the vibration is not small, some progress has been made, as in the theory of sound waves in one dimension, and of the tidal bores, which are such a remarkable feature of certain estuaries and rivers.

The general equations of fluid motion, when friction or viscosity is neglected, were laid down in quite early days by Euler and Lagrange, and in a sense they should contain the whole theory. But, as Whewell remarked, it soon appeared that these equations by themselves take us a surprisingly little way, and much mathematical and physical talent had to be expended before the truths hidden in them could be brought to light and exhibited in a practical shape. What was still more disconcerting, some of the general propositions so arrived at were found to be in flagrant contradiction with observation, even in cases where at first sight it would not seem that viscosity was likely to be important.

It is known that “when a thin transparent film is backed by a perfect reflector, no colours should be visible, all the light being ultimately reflected, whatever the wave-length may be. The experiment may be tried with a thin layer of gelatine on a polished silver plate.” An apparent exception has been described by R. W. Wood: “A thin film of collodion deposited on a bright surface of silver shows brilliant colours in reflected light. It, moreover, scatters light of a colour complementary to the colour of the directly reflected light. This is apparently due to the fact that the collodion film “frills,” the mesh, however, being so small that it can be detected only with the highest powers of the microscope. Commercial ether and collodion should be used. If chemically pure ether obtained by distillation is used, the filmdoes not frill, and no trace of colour is exhibited. Still more remarkable is the fact that if sunlight be thrown down upon the plate at normal incidence, brilliant colours are seen at grazing emergence, if a Nicol prism is held before the eye. These colours change to the complementary tints if the Nicol is rotated through 90°, i.e. in the scattered light, one half of the spectrum is polarized in one plane, and the remainder in a plane perpendicular to it.”

I have lately come across an entirely forgotten letter from Rowland in which he describes a similar observation. Writing to me in March 1893, he says:—“While one of my students was working with light reflected from a metal, it occurred to me to try a thin collodion film on the metal.

This volume completes the collection of my Father's published papers. The two last papers (Nos. 445 and 446) were left ready for the press but were not sent to any channel of publication until after the Author's death.

Mr W. F. Sedgwick, late Scholar of Trinity College, Cambridge, who had done valuable service in sending corrections of my Father's writings during his lifetime, kindly consented to examine the proofs of the later papers of this volume [No. 399 onwards] which had not been printed off at the time of the Author's death. He has done this very thoroughly, checking the numerical calculations other than those embodied in tables, and supplying footnotes to elucidate doubtful or obscure points in the text. These notes are enclosed in square brackets [] and signed W. F. S. It has not been thought necessary to notice minor corrections.

Prolonged experience seems to show that, no matter how much power may be employed in the production of sound-in-air signals, their audibility cannot be relied upon much beyond a mile. At a less distance than two miles the most powerful signals may be lost in certain directions when the atmospheric conditions are unfavourable. There is every reason to surmise that in these circumstances the sound goes over the head of the observer, but, so far as I know, there is little direct confirmation of this. It would clear up the question very much could it be proved that when a signal is prematurely lost at the surface of the sea it could still be heard by an observer at a considerable elevation. In these days of airships it might be possible to get a decision.

But for practical purposes the not infrequent failure of sound-in-air signals must be admitted to be without remedy, and the question arises what alternatives are open. I am not well informed as to the success or otherwise of submarine signals, viz. of sounds propagated through water, over long distances. What I wish at present to draw attention to is the probable advantage of socalled “wireless” signals. The waves constituting these signals are indeed for the most part propagated through air, but they are far more nearly independent of atmospheric conditions—temperature and wind—than are ordinary sound waves. With very moderate appliances they can be sent and observed with certainty at distances such as 10 or 20 miles.

As to how they should be employed, it may be remarked that the mere reception of a signal is in itself of no use.

A recent paper by Richards and Coombs discusses in some detail the determination of surface-tension by the rise of the liquid in capillary tubes, and reflects mildly upon the inadequate assistance afforded by mathematics. It is true that no complete analytical solution of the problem can be obtained, even when the tube is accurately cylindrical. We may have recourse to graphical constructions, or to numerical calculations by the method of Runge, who took an example from this very problem. But for experimental purposes all that is really needed is a sufficiently approximate treatment of the two extreme cases of a narrow and of a wide tube. The former question was successfully attacked by Poisson, whose final formula [(18) below] would meet all ordinary requirements. Unfortunately doubts have been thrown upon the correctness of Poisson's results, especially by Mathieu, who rejects them altogether in the only case of much importance, i.e. when the liquid wets the walls of the tube—a matter which will be further considered later on. Mathieu also reproaches Poisson's investigation as implying two different values of h, of which the second is really only an improvement upon the first, arising from a further approximation. It must be admitted, however, that the problem is a delicate one, and that Poisson's explanation at a critical point leaves something to be desired. In the investigation which follows I hope to have succeeded in carrying the approximation a stage beyond that reached by Poisson.

In the theory of narrow tubes the lower level from which the height of the meniscus is reckoned is the free plane level.

The application of a reflector to pass light back through a prism, or prisms, is usually ascribed to Littrow. Thus Kayser writes (Handbuch der Spectroscopie, Bd. i. p. 513), “Der Erste, der Rückkehr der Strahlen zur Steigerung der Dispersion verwandte, war Littrow” (O. v. Littrow, Wien. Ber. XLVII. ii. pp. 26–32, 1863). But this was certainly not the first use of the method. I learned it myself from Maxwell (Phil. Trans. Vol. CL. p. 78, 1860), who says,” The principle of reflecting light, so as to pass twice through the same prism, was employed by me in an instrument for combining colours made in 1856, and a reflecting instrument for observing the spectrum has been constructed by M. Porro.”

I have not been able to find the reference to Porro; but it would seem that both Maxwell and Porro antedated Littrow. As to the advantages of the method there can be no doubt.

The present is an attempt to examine how for the interesting results obtained by Bénard in his careful and skilful experiments can be explained theoretically. Bénard worked with very thin layers, only about 1 mm. deep, standing on a levelled metallic plate which was maintained at a uniform temperature. The upper surface was usually free, and being in contact with the air was at a lower temperature. Various liquids were employed—some, indeed, which would be solids under ordinary conditions.

The layer rapidly resolves itself into a number of cells, the motion being an ascension in the middle of a cell and a descension at the common boundary between a cell and its neighbours. Two phases are distinguished, of unequal duration, the first being relatively very short. The limit of the first phase is described as the “semi-regular cellular regime”; in this state all the cells have already acquired surfaces nearly identical, their forms being nearly regular convex polygons of, in general, 4 to 7 sides. The boundaries are vertical, and the circulation in each cell approximates to that already indicated. This phase is brief (1 or 2 seconds) for the less viscous liquids (alcohol, benzine, etc.) at ordinary temperatures. Even for paraffin or spermaceti, melted at 100° C, 10 seconds suffice; but in the case of very viscous liquids (oils, etc.), if the flux of heat is small, the deformations are extremely slow and the first phase may last several minutes or more.

The second phase has for its limit a permanent regime of regular hexagons. During this period the cells become equal and regular and align themselves.

I suppose that everyone is familiar with the beautifully graded illumination of a paraffin candle, extending downwards from the flame to a distance of several inches. The thing is seen at its best when there is but one candle in an otherwise dark room, and when the eye is protected from the direct light of the flame. And it must often be noticed when a candle is broken across, so that the two portions are held together merely by the wick, that the part below the fracture is much darker than it would otherwise be, and the part above brighter, the contrast between the two being very marked. This effect is naturally attributed to reflection, but it does not at first appear that the cause is adequate, seeing that at perpendicular incidence the reflection at the common surface of wax and air is only about 4 per cent.

A little consideration shows that the efficacy of the reflection depends upon the incidence not being limited to the neighbourhood of the perpendicular. In consequence of diffusion the propagation of light within the wax is not specially along the length of the candle, but somewhat approximately equal in all directions. Accordingly at a fracture there is a good deal of “total reflection.” The general attenuation downwards is doubtless partly due to defect of transparency, but also, and perhaps more, to the lateral escape of light at the surface of the candle, thereby rendered visible. By hindering this escape the brightly illuminated length may be much increased.

The experiment may be tried by enclosing the candle in a reflecting tubular envelope.

I do not think that Helmholtz's theory of audition, whatever difficulties there may be in it, breaks down so completely as Dr Perrett represents. According to him, one consequence of the theory would be that “when a tuning-fork is made to vibrate, no note can be heard, but only an unimaginable din.” I cannot admit this inference. It is true that Helmholtz's theory contemplates the response in greater or less degree of a rather large number of “resonators” with their associated nerves, the natural pitch of the resonators ranging over a certain interval. But there would be no dissonance, for in Helmholtz's view dissonance depends upon intermittent excitation of nerves, and this would not occur. So long as the vibration is maintained, every nerve would be uniformly excited. Neither is there any difficulty in attributing a simple perception to a rather complicated nervous excitation. Something of this kind is involved in the simple perception of yellow, resulting from a combination of excitations which would severally cause perceptions of red and green.

The fundamental question would appear to be the truth or otherwise of the theory associated with the name of J. Müller. Whatever may be the difficulty of deciding it, the issue itself is simple enough. Can more than one kind of message be conveyed by a single nerve? Does the nature of the message depend upon how the nerve is excited? In the case of sound—say from a fork of frequency 256—is there anything periodic of this frequency going on in the nerve, or nerves, which carry the message?

[Note.—This paper, written in 1919, was left by the Author ready for press except that the first two pages were missing. The preliminary sentences, taken from a separate rough sheet, were perhaps meant to be expanded.

Prof. Wood had observed highly coloured effects in the reflexion from a granular film of sodium or potassium, which he attributed to resonance from the cavities of a serrated structure of rod-like crystals.]

This investigation was intended to illustrate some points discussed with Prof. R. W. Wood. But it does not seem to have much application to the transverse vibrations of light. Electric resonators could be got from thin conducting rods ½λ long; but it would seem that these must be disposed with their lengths perpendicular to the direction of propagation, not apparently leading to any probable structure.

The case of sound might perhaps be dealt with experimentally with birdcall and sensitive flame. A sort of wire brush would be used.

The investigation follows the same lines as in Theory of Sound, 2nd ed. § 351 (1896), where the effect of porosity of walls on the reflecting power for sound is considered. In the complete absence of dissipative influences, what is not transmitted must be reflected, whatever may be the irregularities in the structure of the wall. In the paragraph referred to, the dissipation regarded is that due to gaseous viscosity and heat conduction, both of which causes act with exaggerated power in narrow channels. For the present purpose it seems sufficient to employ a simpler law of dissipation.