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.

Before entering upon the matters that I had intended to lay before you, it is fitting that I should refer to the loss we have sustained within the last few days in the death of Sir William Crookes, a former President of the Society during several years from 1896–1899, and a man of world-wide scientific reputation. During his long and active life he made many discoveries in Physics and Chemistry of the first importance. In quite early days his attention was attracted by an unknown and brilliant green line in the spectrum, which he succeeded in tracing to a new element named Thallium, after its appearance. Later he was able so to improve vacua as to open up fresh lines of inquiry with remarkable results in more than one direction. The radiometer, a little instrument in which light, even candlelight, or ordinary day-light, causes the rotation of delicately suspended vanes, presents problems even yet only partially solved. And his discoveries relating to electric discharge in high vacua lie near the foundation of the modern theories of electricity as due to minute charged particles called electrons, capable of separation from ordinary chemical atoms, and of moving with speeds of the order of the speed of light. One is struck not only by the technical skill displayed in experiments more difficult at the time they were made than the younger generation of workers can easily understand, but also by the extraordinary instinct whieh directed Crookes’ choice of subjects. In several cases their importance was hardly realized at the time, and only later became apparent.

In a short paper “On the Diffraction of Light by Particles Comparable with the Wave-length,” Keen and Porter describe curious observations upon the intensity and colour of the light transmitted through small particles of precipitated sulphur, while still in a state of suspension, when the size of the particles is comparable with, or decidedly larger than, the wave-length of the light. The particles principally concerned in their experiments appear to have decidedly exceeded those dealt with in a recent paper, where the calculations were pushed only to the point where the circumference of the sphere is 2.25 λ. The authors cited give as the size of the particles, when the intensity of the light passing through was a minimum, 6 μ to 10 μ, that is over 10 wave-lengths of yellow light, and they point out the desirability of extending the theory to larger spheres.

The calculations referred to related to the particular case where the (relative) refractive index of the spherical obstacles is 1.5. This value was chosen in order to bring out the peculiar polarisation phenomena observed in the diffracted light at angles in the neighbourhood of 90°, and as not inappropriate to experiments upon particles of high index suspended in water. I remarked that the extension of the calculations to greater particles would be of interest, but that the arithmetical work would rapidly become heavy.

In copying a subject by photography the procedure usually involves two distinct steps. The first yields a so-called negative, from which, by the same or another process, a second operation gives the desired positive. Since ordinary photography affords pictures in monochrome, the reproduction can be complete only when the original is of the same colour. We may suppose, for simplicity of statement, that the original is itself a transparency, e.g. a lantern-slide.

The character of the original is regarded as given by specifying the transparency (t) at every point, i.e. the ratio of light transmitted to light incident. But here an ambiguity should be noticed. It may be a question of the place at which the transmitted light is observed. When light penetrates a stained glass, or a layer of coloured liquid contained in a tank, the direction of propagation is unaltered. If the incident rays are normal, so also are the rays transmitted. The action of the photographic image, constituted by an imperfectly aggregated deposit, differs somewhat. Rays incident normally are more or less diffused after transmission. The effective transparency in the half-tones of a negative used for contact printing may thus be sensibly greater than when a camera and lens is employed. In the first case all the transmitted light is effective; in the second most of that diffused through a finite angle fails to reach the lens. In defining t—the transparency at any place—account must in strictness be taken of the manner in which the picture is to be viewed. There is also another point to be considered. The transparency may not be the same for different kinds of light.

Some two years ago I asked for suggestions as to the formation of an artificial hiss, and I remarked that the best I had then been able to do was by blowing through a rubber tube nipped at about half an inch from the open end with a screw clamp, but that the sound so obtained was perhaps more like an f than an s. “There is reason to think that the ear, at any rate of elderly people, tires rapidly to a maintained hiss. The pitch is of the order of 10,000 per second.” The last remark was founded upon experiments already briefly described under the head “Pitch of Sibilants.”

“Doubtless this may vary over a considerable range. In my experiments the method was that of nodes and loops (Phil. Mag. Vol. VII. p. 149 (1879); Scientific Papers, Vol. I. p. 406), executed with a sensitive flame and sliding reflector. A hiss given by Mr Enock, which to me seemed very high and not over audible, gave a wave-length (λ) equal to 25 mm., with good agreement on repetition. A hiss which I gave was graver and less definite, corresponding to λ = 32 mm. The frequency would be of the order of 10,000 per second, more than 5 octaves above middle C.”

The manner in which aqueous vapour condenses upon ordinarily clean surfaces of glass or metal is familiar to all. Examination with a magnifier shows that the condensed water is in the form of small lenses, often in pretty close juxtaposition. The number and thickness of these lenses depend upon the cleanness of the glass and the amount of water deposited. In the days of wet collodion every photographer judged of the success of the cleaning process by the uniformity of the dew deposited from the breath.

Information as to the character of the deposit is obtained by looking through it at a candle or small gas flame. The diameter of the halo measures the angle at which the drops meet the glass, an angle which diminishes as the dew evaporates. That the flame is seen at all in good definition is a proof that some of the glass is uncovered. Even when both sides of a plate are dewed the flame is still seen distinctly though with much diminished intensity.

The process of formation may be followed to some extent under the microscope, the breath being led through a tube. The first deposit occurs very suddenly. As the condensation progresses, the drops grow, and many of the smaller ones coalesce: During evaporation there are two sorts of behaviour. Sometimes the boundaries of the drops contract, leaving the glass bare. In other cases the boundary of a drop remains fixed, while the thickness of the lens diminishes until all that remains is a thin lamina.

In a former paper I gave solutions applicable to the passage of light through very narrow slits in infinitely thin perfectly opaque screens, for the two principal cases where the polarisation is either parallel or perpendicular to the length of the slit. It appeared that if the width (2b) of the slit is very small in comparison with the wave-length (λ), there is a much more free passage when the electric vector is perpendicular to the slit than when it is parallel to the slit, so that unpolarised light incident upon the screen will, after passage, appear polarised in the former manner. This conclusion is in accordance with the observations of Fizeau upon the very narrowest slits. Fizeau found, however, that somewhat wider slits (scratches upon silvered glass) gave the opposite polarisation ; and I have long wished to extend the calculations to slits of width comparable with λ. The subject has also a practical interest in connection with observations upon the Zeeman effect.

The analysis appropriate to problems of this sort would appear to be by use of elliptic coordinates; but I have not seen my way to a solution on these lines, which would, in any case, be rather complicated. In default of such a solution, I have fallen back upon the approximate methods of my former paper. Apart from the intended application, some of the problems which present themselves have an interest of their own. It will be convenient to repeat the general argument almost in the words formerly employed Plane waves of simple type impinge upon a parallel screen.

In an early paper Stokes showed “that in the case of a homogeneous incompressible fluid, whenever udx + vdy + wdz is an exact differential, not only are the ordinary equations of fluid motion satisfied, but the equations obtained when friction is taken into account are satisfied likewise. It is only the equations of condition which belong to the boundaries of the fluid that are violated.” In order to satisfy these also, it is only necessary to suppose that every part of the solid boundaries is made to move with the velocity which the fluid in irrotational motion would there assume. There is no difficulty in the supposition itself; but the only case in which it could readily be carried into effect with tolerable completeness is for the two-dimensional motion of fluid between coaxal cylinders, themselves made to rotate in the same direction with circumferential velocities which are inversely as the radii. Experiments upon these lines, but not I think quite satisfying the above conditions, have been made by Couette and Mallock. It would appear that, except at low velocities, the simple steady motion becomes unstable.

But the point of greatest interest is not touched in the above example. It arises when fluid passing along a uniform or contracting pipe, or channel, arrives at a place where the pipe expands. It is known that if the expansion be sufficiently gradual, the fluid generally speaking follows the walls, or, as it is often expressed, the pipe flows full; and the loss of velocity accompanying the increased section is represented by an augmentation of pressure, approximately according to Bernoulli's law.