SeeTheory of Sound, § 226.

[1899. In general the bending of a square plate with free edges takes place in both planes; but when the material is such that Poisson's ratio vanishes, i.e. when longitudinal extension entails no lateral contraction, the bending may be in one plane only, so that the surface remains cylindrical. In this case the bending follows the law of a simple bar, and the nodal system consists of two straight lines parallel to one pair of edges.

By superposing, after Wheatstone, two such modes of vibration, the phases being the same and the amplitudes equal or opposite, we obtain other nodal systems easily constructed from the known functions expressing the free vibrations of a bar. In the latter case symmetry suffices to shew that the nodal lines are the diagonals of the plate.]

The papers of the present Collection are reprinted very nearly as they originally appeared, and with a few partial exceptions in order of date. Obvious misprints have been corrected, in several cases with the aid of the original manuscript. Other alterations of the slightest significance are indicated by the use of square brackets [], while additional matter is introduced with the proper date in the form of footnotes or at the end of a memoir. In a few cases, where it has not been thought worth while to reproduce a paper in full, a brief statement of the principal results is given.

Some short papers of a rather slender character have been included. These may serve to mitigate the general severity. In consulting similar collections I have usually felt even more grateful for the reproduction of short and often rather inaccessible notes than for the larger and better known memoirs.

TBRLING PLACE, WITHAM,

October1899.

There is an important class of functions, often occurring in physical investigations, whose numerical calculation is easy when the argument is either small or great. In the first case the function is readily calculated from an ascending series, which is always convergent and might be employed whatever the value of the variable may be, were it not for the length to which the calculations would run. When the argument is great, a series proceeding by descending powers is employed, whose character is quite different. In this case the series is of the kind called semi-convergent, though strictly speaking it is not convergent at all; for, when carried sufficiently far, the sum of the series may be made to exceed any assignable quantity. But, though ultimately divergent, it begins by converging, and when a certain point is reached the terms become very small. It can be proved that, if we stop here, the sum of the terms already obtained represents the required value of the functions, subject to an error which in general cannot exceed the last term included. Calculations founded on this series are therefore only approximate; and the degree of the approximation cannot be carried beyond a certain point. If more terms are included, the result is made worse instead of better. In the class of functions referred to, the descending series is abundantly adequate when the argument is large, but there will usually be a region—often the most interesting part of the whole— where neither series is very convenient. The object of the present note is to point out how a part of the difficulty thence arising may sometimes be met.

It is now, I believe, generally admitted that the light which we receive from the clear sky is due in one way or another to small suspended particles which divert the light from its regular course. On this point the experiments of Tyndall with precipitated clouds seem quite decisive. Whenever the particles of the foreign matter are sufficiently fine, the light emitted laterally is blue in colour, and, in a direction perpendicular to that of the incident beam, is completely polarized.

About the colour there is no prima facie difficulty; for as soon as the question is raised, it is seen that the standard of linear dimension, with reference to which the particles are called small, is the wave-length of light, and that a given set of particles would (on any conceivable view as to their mode of action) produce a continually increasing disturbance as we pass along the spectrum towards the more refrangible end; and there seems no reason why the colour of the compound light thus scattered laterally should not agree with that of the sky.

On the other hand, the direction of polarization (perpendicular to the path of the primary light) seems to have been felt as a difficulty.

It is, I believe, the common opinion, that a satisfactory mechanical theory of the reflection of light from metallic surfaces has been given by Cauchy, and that his formulæ agree very well with observation. The result, however, of a recent examination of the subject has been to convince me that, at least in the case of vibrations performed in the plane of incidence, his theory is erroneous, and that the correspondence with fact claimed for it is illusory, and rests on the assumption of inadmissible values for the arbitrary constants. Cauchy, after his manner, never published any investigation of his formulae, but contented himself with a statement of the results and of the principles from which he started. The intermediate steps, however, have been given very concisely and with a command of analysis by Eisenlohr (Pogg. Ann, vol. civ. p. 368), who has also endeavoured to determine the constants by a comparison with measurements made by Jamin. I propose in the present communication to examine the theory of reflection from thick metallic plates, and then to make some remarks on the action on light of a thin metallic layer, a subject which has been treated experimentally by Quincke.

The peculiarity in the behaviour of metals towards light is supposed by Cauchy to lie in their opacity, which has the effect of stopping a train of waves before they can proceed for more than a few wave-lengths within the medium.

The claims of the theory propounded by Maxwell, according to which light consists of a disturbance in a medium susceptible of dielectric polarization, are so considerable that it is desirable to extend its application as far as possible to various optical phenomena. The question of the velocity of propagation in vacuum and in singly or doubly refracting transparent dielectrics was considered by Maxwell himself; and the agreement with experiment, though far from perfect, is sufficiently encouraging. More recently it has been shown by Helmholtz, Lorentz, Fitzgerald, and J. J. Thomson, that the same theory leads to Fresnel's expressions for the intensity of light reflected and refracted at the surface of separation of transparent media, and that the auxiliary hypotheses necessary in this part of the subject agree with those required to explain the laws of double refraction. In this respect the electromagnetic theory has a marked advantage over the older view, which assimilated luminous vibrations to the ordinary transverse vibrations of elastic solids. According to the latter, Fresnel's laws of double refraction, fully confirmed by modern observation, require us to suppose that in a doubly-refracting crystal the rigidity of the medium varies with the direction of the strain; while, in order to explain the facts relating to the intensities of reflected light, we have to make the inconsistent assumption that the rigidity does not vary in passing from one medium to another. A further discussion of this subject will be found in papers published in the Philosophical Magazine during the year 1871. [Arts. VIII. IX. x. XI.]

It is known that a large part of the radiation from terrestrial sources is non-luminous. Even in the case of the electric arc the obscure radiation amounts, according to Tyndall, to eight-ninths of the whole, and of the remainder probably no inconsiderable part is to be found in the extreme red rays of feeble luminosity. For practical purposes this obscure radiation is useless; and the question forces itself upon us, whether or no there is any necessity, absolutely inherent in the case, for so large a proportion of waste. The following arrangement, not of course proposed as practical, seems to prove that the question should be answered in the negative.

Conceive a small spherical body of infusible material, to which energy can be communicated by electricity or otherwise, to be surrounded by a concentric reflecting spherical shell. Under these circumstances no energy can escape; but if a small hole be pierced in the shell, radiation will pass through it. In virtue of the suppositions which we have made, the emergent beam will be of small angle, and may be completely dealt with at a moderate distance by a prism and lens. Let us suppose then that a spectrum of the hole is formed and is received upon a reflecting plate so held at the focus as to return the rays upon the lens and prism. These rays will re-enter the hole, and impinge upon the radiating body, which is thus again as completely isolated as if the shell were unperforated. We have now only to suppose a portion of the focal plate to be cut away in order to have an apparatus from which only one kind of radiation can escape. Whatever energy is communicated to the internal body must ultimately undergo transformation into radiation of the selected kind.

Coronas are formed whenever, between the eye and the source of light, a large number of small opaque bodies, regular in size but irregular in distribution, are to be found. By a principle known as Babinet's, it is permitted to substitute for the layer of obstructing globules an opaque screen having apertures, which admit the light precisely where before it was stopped. From each aperture diverges a secondary wave, whose phase is arbitrarily related to those of the other waves on account of the irregularity in the distribution of the centres from which they emanate. In the theory of coronas, the intensities of the partial waves are supposed to be equal.

Any vibration may be represented by a straight line drawn from a given point, whose length is proportional to the amplitude, and angular position depends on the phase. When several vibrations affect the same point, the resultant may be found from the representative lines, according to the ordinary rules for compounding directed magnitudes. We have, therefore, to consider the resultant of a great number of lines radiating from an origin whose lengths are all equal and directions thoroughly irregular. From this statement, it appears at once that the question is perfectly indeterminate, and that no conclusion whatever can be drawn as to the probable value of the phase of the resultant, while all that could be inferred as to the amplitude relates merely to the order of its magnitude. Nevertheless, Verdet (vol. i., p. 297) comes to the conclusion that, the number of partial vibrations being n, the resultant is definitely √n times greater than each component, provided, of course, that n is a large number.

It is well known that the sudden development of a current in a conductor is opposed by an influence analogous to the inertia of ordinary matter. A powerful movement of electricity cannot be suddenly produced; neither can it be suddenly stopped. One consequence is that a periodic interruption of a circuit in which a constant electromotive force acts is sufficient, when the self-induction is great, to stop all sensible current, even although the interruptions themselves may be of very short duration. Before any copious flow can be produced the circuit is broken, and the work has to be begun over again. Whether in any particular case the influence of self-induction is paramount, or not, will depend also on the resistance of the circuit, and on the rapidity of the intermittence. The magnitudes which really come into direct comparison are the interval between the breaks, and the time which would elapse while a current, generated in the circuit and then left to itself, falls to a specific fraction (such as one half) of its original magnitude. In ordinary cases the duration of transient currents is but a small part of a second of time, so that, in order to bring out the effects of self-induction, the breaks must recur with considerable rapidity.

There is, however, one remarkable exception to the general rule, which occurs when, alongside of the principal coil to which the sluggishness is due, there exists an independent course along which the electricity can circulate. For instance, suppose that a coil with two wires, such as is often used for electro-magnets, is so arranged that one wire is included in the principal circuit, while the ends of the other are joined.

Many, it may even be said most, of the still unexplained phenomena of Acoustics are connected with the instability of jets of fluid. For this instability there are two causes; the first is operative in the case of jets of heavy liquids, e.g., water, projected into air (whose relative density is negligible), and has been investigated by Plateau in his admirable researches on the figures of a liquid mass withdrawn from the action of gravity. It consists in the operation of the capillary force, whose effect is to render the infinite cylinder an unstable form of equilibrium, and to favour its disintegration into detached masses whose aggregate surface is less than that of the cylinder. The other cause of instability, which is operative even when the jet and its environment are of the same material, is of a more dynamical character.

With respect to instability due to capillary force, the principal problem is the determination, as far as possible, of the mode of disintegration of an infinite cylinder, and in particular of the number of masses into which a given length of cylinder may be expected to distribute itself. It must, however, be observed that this problem is not so definite as Plateau seems to think it; the mode of falling away from unstable equilibrium necessarily depends upon the peculiarities of the small displacements to which a system is subjected, and without which the position of equilibrium, however unstable, could not be departed from. Nevertheless, in practice, the latitude is not very great, because some kinds of disturbance produce their effect much more rapidly than others.

Theory of Sound, §§ 210, 211.

[1899. This is an application of a theorem of Art. xxi, to the effect that an increase in the potential energy of a vibrating system is attended by a rise in pitch, extending to all the fundamental modes. If the system consist of a uniformly stretched membrane with a fixed edge, it follows that any contraction of the boundary must cause an elevation of pitch, because the new state of things may be conceived to differ from the old merely by the introduction of an additional constraint. Springs (without inertia) are supposed to urge the line of the proposed boundary towards its equilibrium position, and gradually to become stiffer. At each step the vibrations become more rapid, until they approach a limit, corresponding to infinite stiffness of the springs and absolute fixity of their points of application.

Use is also made of the theorem that if the area of a membrane be given, the pitch is gravest when the form is circular.]

The well-known fact that hydrogen tends to escape through fine apertures more rapidly than air enters to supply its place, even although the advantage of the greater pressure may be on the side of the air, proves that the operation of mixing the two gases has a certain mechanical value. In a common form of the experiment a tube containing hydrogen and closed at the upper end with a porous plug of plaster of Paris stands over water. In a short time the escape of hydrogen creates a partial vacuum in the tube, and the water rises accordingly. Whenever then two gases are allowed to mix without the performance of work, there is dissipation of energy, and an opportunity of doing work at the expense of low temperature heat has been for ever lost. The present paper is an attempt to calculate this amount of work.

The result at which I have arrived is extremely simple. It appears that the work that may be done during the mixing of the volumes v1 and 2 of two different gases is the same as that which would be gained during the expansion of the first gas from volume v1 to volume v1 + v2, together with the work gained during the expansion of the second gas from v2 to v1 + v2, the expansions being supposed to be made into vacuum.