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nomena are manifested. The liquid sphere first takes rapidly its maximum Fig. 3.
of flattening, then becomes hollow above and below around the axis of rotation, stretching out continually in a horizontal direction, and, finally abandoning the o: transformed into a perfectly regular ring, g. 3. ( This ring is rounded transversely, and appears to have a circle for its generatrix. At the moment of its formation its diameter increases rapidly up to a certain limit; when this is reached the movement of the disc must be stopped. The ring now remains for some seconds in the same state. Then, the resistance of the ambient liquid weakening its movement of rotation, it returns upon itself and changes back into a sphere around the disc and its axis. The velocity of the handle most suitable for producing a beautiful ring, is about three turns per second. The ring thus obtained has a mean diameter of 9 to 10 centimetres. 12. When, at the instant of the formation of the ring, the mass of oil which constitutes it separates from the disc, a singular circumstance is observable; the ring remains united to the disc by an extremely thin pellicle or film of oil, which fills all the space between them. But at the instant that, thering having reached its greatest extent, we stop the motion of the disc, this pellicle breaks and disappears of itself, and the ring then remains perfectly isolated. It may be conceived that this pellicle is not a circumstance essential to the phenomenon of the formation of the ring; and we shall see, in another part of these experiments, that it is probably connected with an order of facts wholly different. 13. The heavens exhibit to us also a body of a form analogous to our liquid ring. I allude to Saturn's ring. That, indeed, is flattened, whilst the transverse contour of ours appears altogether round; but I do not think that this difference is so great as it appears at first. In fact, the centrifugal force, which goes on increasing from the inner circumference of the ring of oil up to its outer circumference, necessarily tends to stretch this ring in the direction of its breadth, or, in other words, to flatten it. But the flattening must be of very small amount; for, on account of the inconsiderable dimensions of the ring, and the slowness of its angular movement, the kind of traction which results from the variation of centrifugal force must be very trifling in comparison with the forces developed by molecular attrac. tion. 14. It appears to me, then, that we may reasonably admit that our ring of oil is in reality slightly flattened, and that in consequence it only differs from that of Saturn, with regard to general form, in the less quantity of flattening." But further, in the system of Saturn, the flattening of the ring is in part deter. mined by the attraction of the central planet. Now, at the first moment of the formation of the ring of oil, the latter is submitted to a particular force, which plays a part analogous to that of the above attraction. In fact, this attraction acts with the greatest intensity at the inner circumference of Saturn's ring, and thence decreases rapidly in the rest of this body. Now, at the first moment of the formation of the ring of oil, we have seen (§ 12) that the latter remains united to the disc by a thin film of the same liquid, and we may convince ourselves that this film exerts, on the inner circumference of the ring, a considerable force of traction. In fact, if we stop the movement of the disc a little too soon, that is to say a little before the ring has reached its maximum
* I leave out of the question here the subdivision of the ring of Saturn. This subdivision, as is known, is not essentially connected with the conditions of equilibrium of the ring.
of diameter, the film of oil does not break, and the ring then returns upon itself (§ 11) with a much greater rapidity than when the film of oil is broken, and the ring remains isolated. The traction which the film of oil exerts on the inner circumference of the ring ought therefore to produce an effect analogous to that of the attraction of Saturn, that is to say, contribute to increase the flattening. Well, the ring of oil before the rupture of the film presents a very marked flattening. In order to obtain it perfectly, care must be taken that the sphere be well centred in relation to the disc, before beginning the experiment; and it is useful to turn the handle with a velocity somewhat less than that indicated at § 11; the most suitable velocity has appeared to me to be about two turns in a second. As soon as the film of oil breaks the flattening disappears, and the generatrix of the ring becomes, as we have seen, sensibly circular." 15. Geometricians, who have investigated the figure of equilibrium of a liquid mass in rotation, have only regarded the case in which the attraction which counteracts the centrifugal force is that of universal gravitation, and they have demonstrated that elliptical figures in that case satisfy this equilibrium. Are we thence to conclude that the annular form developed by the rotation of our mass of oil results from the different law which governs molecular attraction, (§ 10,) and that, in the instance of the heavenly bodies, the figure of an isolated ring could not be produced by the sole combination of centrifugal force and of the mutual attractions of the different parts of the mass 1 1 am not of that opinion, and I think it, on the contrary, very probable that if calculation could approach the general solution of this great problem, and lead directly to the determination of all the possible figures of equilibrium, the annular figure would be included among them. This general and direct solution presenting very great difficulties, geometricians have contented themselves with trying whether elliptical figures could satisfy the equilibrium, and with proving that they in fact do satisfy it; but they leave the question in doubt, whether other figures would not fulfil the same conditions. In truth, M. Liouville, in his last researches on this subject,f appears at first view to have nearly solved the question, by introducing the consideration of the stability of the figure of equilibrium, and showing that for each value of the moment of rotation, or, in other words, for any initial movement, whatever, of the mass, there is always an elliptical figure, either of revolution or of three uncolual axes, according to
* I had thought that it would be possible to obtain rings isolated and greatly flattened by operating upon larger masses of oil, for then, the ring laying a larger volume, the influence of the molecular attraction should be less. Iłut I have found that, in operating on larger masses, it was necessary, in order to obtain the ring in a regular manner, to employ a more feeble velocity of rotation, so that, if the influence of the molecular attraction was diminished, that of the centrifugal force was so equally. The flattening, then, did not become more sensible; or, if I have sometimes imagined that I observed any. I have not becn able to reproduce it at will. I have operated thus on spheres which were, successively, about 10, 11, 12, and 14 vontimetres in diameter, with discs of a diameter of from seven to nine centimetres, and in a vessel with plane surfaces, having a bottom 33 centimetres square, and a depth of 25 centimetres. The effects, however, thus obtained are very beautiful. The rings are magnificent; present a considerable diameter, and remain sometimes for eight to ten seconds before returning on themselves. With a sphere of ten centimetres diameter, a disc of seven, and a velocity a little less than one turn of the disc per second, we obtain, in a very beautiful and very marked manner, the flattening resulting from the traction of the film of oil. These experiments, however, are inconvenient and difficult, on account of the large dimensions of the vessel, and the great quantity of alcoholic liquid necessary to fill it. It may be conceived, moreover, why a larger mass of oil requires a less velocity of rotation to produce a regular ring. It is precisely because the molecular attraction has less influence; whence, it results that, if we attempt to employ the same velocity of rotation which would give a beautiful ring with a less quantity of oil, the mass disunites, and is scattered into herules. •r The memoir of M. Liouville was communicated to the Academy of Sciences in the sitting of the 13th of February in this year. An analysis of it may be found in the Journal L'Institut, No. 477.
the circumstances, which constitutes a form of stable equilibrium. It appears, in effect, natural to admit that, for a given disturbance of a liquid mass, there is but one single final state admissible; and, in this case, this state must necessarily possess stability. . However, I do not deem the conclusion which may be drawn from these results so general as it appears at first sight. Without doubt, for a primitive disturbance given, there is only one final state possible, and that state must be stable. But the condition of stability of a found figure of equi. librium does not necessarily involve the consequence that this figure will con stitute the final state in question, for it may happen that several figures of equilibrium, corresponding to the same primitive disturbance, might equally possess stability, and that the choice of the mass for one of these figures may have been determined by other circumstances; for example, by the modifications which its movement experiences in the first moments of rotation. In fact, it is by examining these modifications, to which the attention of geometricians has not been directed, that I shall attempt to arrive at the mode of generation of annular figures. 16. When the mass begins to revolve upon itself, the angular velocity of the portions remote from the axis, which are carried off by their centrifugal force, necessarily goes on diminishing. This diminution is especially apparent on the equator of the mass, and it is the more considerable in proportion as the initial movement of rotation was more rapid. It thence results that, in the first instants of a sufficiently rapid rotation, there will be a great difference of angular velocity between the portions which are near the axis and those which are near the equator. Nevertheless, if we admit for a moment that, in virtue of the adherence of the liquid for itself, and of the friction of its several parts, the ortions which turn most rapidly communicate by degrees a part of their veocity to the others, so that in the end the result is a mean angular velocity, corresponding to the same moment of rotation, and equal in all the points of the mass, this may take an ellipsoidal figure. But long before the feeble forces, of which we have just spoken, can bring about this mean result, another order of phenomena would be manifested, which may impede the development of the elliptical figure and give rise to an annular form. In fact, it follows necessarily from the preceding considerations that, in the first instants of a rotation sufficiently rapid, the centrifugal force at the equator of the mass will be much less than that which would correspond to the above mean velocity; and that, on the other hand, the centrifugal force of the portions near the axis will be by much superior to that which would correspond to the same mean velocity. The liquid next the axis will, therefore, be driven towards the liquid of the equator, whence there will necessarily result the formation of a sort of circular cushion, (bourrelet, ) more or less marked. In other words, the mass will soon become hollow in the middle, and will swell out all around. Now as soon as this phenomenon takes place, it will be conceived that the attraction exerted by this bourrelet on the liquid remaining around the axis must be an addition to the action of the centrifugal force, and contribute to increase the volume of the hourrelet at the expense of the central liquid. Hence, therefore, it may evidently result that all the liquid will leave the axis for the bourreles, and the latter become in a manner a veritable ring. This generation of the annular figures would therefore be independent of the law which the attraction follows, and would be, in consequence, the same in the case of universal attraction and in that of molecular attraction. 17. It is easy to verify this mode of generation upon our mass of oil, or at least to assure ourselves that during the formation of the bourrelet and of the ! ring, the angular velocity is much less at the equator of the mass than towards the axis. For this purpose I shall first point out that when a certain number of experiments have been performed upon the same mass of oil, and this has been several times disunited and reformed into a single sphere and into a ring, it always holds within it a multitude of small bubbles of alcoholic liquor, which borne along by the oil that surrounds them, render the movements of the different points of the mass perfectly observable. Now, if the experiments which we have described be repeated with the aid of a sphere of oil thus filled with alcoholic bubbles, the following results are observed. So kong as we give to the disc such slight velocities only as are sufficient to produce a simple flattening, there is not a great difference of angular velocity between the portions next to the axis and the portions adjoining the equator; but this difference becomes very considerable when the disc turns more rapidly, and the bourrelet and the ring are developed. We may thus prove, by means of the small alcoholic bubbles, that the mean angular velocity is established in the ring once formed, and that all the points of the latter perform their revolutions in the same time. Furthermore, in our experiments upon the masses of oil, there are two foreign forces which act, in addition to the causes which we have noticed, to facilitate the development of the bourrelet and of the ring. One is the resistance of the ambient liquid, which contributes to weaken the angular velocity of the equator of the mass; the other is the action of the hand which keeps up the motion of rotation of the disc, and consequently hinders the central portions of the mass from participating gradually in the slackening of the equatorial portions. But that which is produced by these two foreign forces would be equally produced by a greater initial velocity of rotation if we could annul them. 18. When, by the aid of a moderate velocity of the disc, we limit ourselves to producing the flattening of the mass, the two foreign forces of which we have just spoken necessarily hinder the latter from attaining an angular velocity equal in all its points, even though we keep turning the disc. The result is, that the mass cannot take exactly the figure which would correspond to that equality of angular velocity. That which it adopts is a figure of revolution; but on placing the eye at the height of the centre of the mass, it is easily recognized that it is not an ellipsoid. The curvature at the equator is too small, and this is the more evident in proportion as the flattening is more considerable. Now, is this difference between the figure thus produced and that which would correspond to the case of universal gravitation solely the result of the action of the two foreign forces in question, or is it in part caused by the difference of the laws which the two kinds of attraction follow . In other words, if we could eliminate or render insensible the differences of angular velocity of the several parts of the mass of oil, would the figure produced be an ellipsoid or not? Now, we should render these differences of angular velocity insensible if we could impress a movement of rotation on a mass of oil suspended in an isolated manner, without interior system, in the alcoholic liquid, and then leave it to itself. In this case the resistance of the ambient liquid would be exercised, indeed, on the exterior of the mass; but nothing maintaining the constancy of velocity of the central parts, these, by virtue of the strong self-adherence of the oil, would participate eventually in the slackening of the exterior portions, and we might consider the mass as having each instant an angular velocity equal throughout. Now, it is very easy to realize the above by availing ourselves of the fact that, when the ring of oil is formed, it returns, after some time, upon itself, (§ 11.) At the instant when the ring is well developed, and when we have just stopped the disc, we lift the latter cautiously by means of the metallic stopper which bears its axis. Then the mass of oil, which is again formed by the return of the ring upon itself, continues still to revolve for some time, completely isolated the ambient liquid. Its figure is then, as well as the eye can judge of it, a perfect ellipsoid of revolution, which gradually approximates to a sphere in proportion as the rotatory motion becomes weaker." Thus, the difference of the laws which govern the two sorts of attraction appears not to influence the nature of the figure taken by the mass that turns upon itself. 19. A liquid mass can only assume and preserve an annular form under the influence of a sufficient centrifugal force. Thus, as we have seen, when the resistance of the alcoholic liquid has diminished below a certain limit the velocity of rotation of the ring of oil, the latter, obeying the preponderating action of the molecular attraction, returns upon itself, loses its annular form, and reconstitutes itself into an entire mass, first ellipsoidal and then spherical. But if, by a method which I shall describe, we prevent the ring from agglomerating thus, and still leave the action of its centrifugal force to diminish, we then witness the appearance of other phenomena well meriting interest. In order to produce them perfectly, in place of the disc of 35 millimetres, a disc of about 5 centime
* I had expected to be able to obtain a revolving isolated mass o means of another process, viz: by forming a splore of oil in the middle of a cylindrical flask so arranged as to be able to turn upon its axis; then causing this flask thus to turn with rapidity, until all the liquid within, alcoholic mixture and mass of oil, had taken the same motion; then suddenly stopping the flask. In effect, it seems that then the alcoholic liquor being the first to lose its rotatory motion by the friction against the stationary sides of the flask, a moment must occur when the mass of oil maintains an excess of angular velocity over the ambient liquid, and that then the effects of centrifugal force upon that mass may manifest themselves. But the experiment gives few results. First, it is extremely difficult to keep a mass of oil in the middle of the flask. We keep it tolerably in the axis of the latter, because, if we have succeeded in placing it so that its centre is litle removed from that axis, the rotation of the ambient liquid brings it there, and then retains it there very well. But it is not the same in the direction of the height of the flask. If a homogeneous alcoholic mixture be employed, and the sphere of oil is placed, before turning the flask, a little higher or lower than the middle of the height of the latter, it quits its place when the flask turns to ascend, in the first case, or to descend, in the second, until it comes to be dispersed against one of the two bases of the flask. This effect is attributable, I think, to the fact that the two bases exercising upon the sections of liquid which touch them a motive action much greater than that to which the parallel sections of the interior of the mass are subjected, there ensues near these bases, at the commencement of the rotation, an excess of centrifugal force, which determines a tendency upwards and downwards of the liquid near the axis. It is therefore necessary to endeavor to place the sphere of oil in a position very near to the middle of the height of the vessel. Unfortunately we cannot use for this purpose the process of superposition of the alcoholic layers of unequal density, (§ 9:) for then, in the rotation of the flask, the denser inferior layers come necessarily, by the excess of centrifugal force which results from their excess of density, to rise against thesides, causing the less dense liquid to occupy the axis; and in this movement the mass of oil is drawn downwards, and is also dispersed upon the bottom of the vessel. By employing a homogeneous alcoholic mixture and a sphere of oil of only about three centimetres diameter, I however succeeded several times, by dint of patience, in giving to this sphere a sufficiently exact position in the flask to be able to keep it at the same height until it had itself taken the rotatory movement of the whole system. But then, when I stopped the flask, a violent internal agitation was produced, which almost always dispersed the oil in innumerable spherules throughout the alcoholic liquid, or at least destroyed its form in a completely irregular manner. I attribute these effects to the following cause. When the flask is stopped, the portions of the alcoholic liquid which touch the sides and bases, losing first their centrifugal force, the more internal portions, which still retain theirs, make their way through them, dividing them, and this confusion is soon propagated to the axis, where it gives rise either to the dispersion or to the irregular disfiguring of the mass of oil. n the cases in which I have been able to give a suitable position to the sphere of oil, I have observed a curious effect; namely, that in the first instance of the rotation of the vessel the mass of oil quits the spherical form, and becomes elongated in the direction of the axis of rotation. This elongation is easy explained: the movement of rotation is communicated to the portions of the mixture which are nearest the axis above and below the mass of oil, before being able to communicate itself with the same intensity to the latter: hence, in the different points of this mass, there must result a less centrifugal force than in the points of the alcoholic mixture situated at the same distances from the axis of rotation. Thence a rush of the oil to the axis, and an elongation of the mass of the latter in the direction of this same axis. But, on continuing the rotation, the oil comes to receive the same movement as the surrounding liquid, and it also resumes so the spherical form. On stopping the flask, not suddenly, but in a rather rapid manner, I succeeded once in obtaining a result sufficiently regular, and I observed, as I expected, the sphere become flattened considerably in the direction of the axis of rotation.