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included between the two rings constitutes one portion only of the complete figure of equilibrium. Hence also, if the liquid mass were free, it could not assume the cylindrical form as the figure of equilibrium; for the volume of this mass being limited, it would be necessary that the cylinder should be termimated on both sides by portions of the surface presenting other curvatures, which would not admit of the law of continuity. But this heterogeneity of curvature, which is impossible when the mass is free, becomes realizable, as our experiments show, through the medium of solid rings. As each of these renders the curvatures of the portions of the surface resting upon it (§ 20) independent of each other, the surface comprised between the two rings may then be of cylindrical curvature, whilst the two bases of the figure may present spherical curvatures. We therefore arrive at the very remarkable result, that with a liquid mass of a limited volume we may obtain isolated portions of figures of equilibrium, which in their complete state would be extended indefinitely. 44. With the view of obtaining a cylinder in which the proportion between the height and the diameter was still greater than that in Fig. 23, I replaced the rings previously employed by two others, the diameter of which was only 2 centimeters. I first tried to make a cylinder 6 centimeters in height, i.e., the height of which was thrice the diameter; and in this operation I adopted a
slightly different process from that of paragraph 38. The uniformity in the
density of the two liquids being accurately established, I first gave the mass of oil a somewhat larger volume than that which the cylinder would contain; having then attached the mass to the two rings, I elevated the upper ring until it was at a distance of 6 centimeters from the other; this distance was measured by a scale introduced into the vessel and kept in a vertical position by the side of the liquid figure. In consequence of the excess of oil, the meridional line of the figure was convex externally; and as there was still a slight difference between the densities, this convexity was not symmetrical in regard to the two rings. I corrected this irregularity by successive additions of pure alcohol and alcohol of 16°, an operation which requires great circumspection, and
towards the end of which these liquids could only be added in single A&23. drops. The figure being at last perfectly symmetrical, I carefully removed the excess of oil by applying the point of the syringe to a point at the equator of the mass, and in this manner I obtained a perfect cylinder. Subsequently, after having added some oil to the mass, I | increased the distance between the rings until it was equal to 8 centimeters, i. e., to four times their diameter. The oil was in sufficient quantity to allow of the meridional line of the figure being convex ex| ternally; but the curvature was not perfectly symmetrical, and I encountered still greater difficulties in regulating it than in the preceding case. The defect in the symmetry being ultimately corrected, the meriF’ dional convexity presented aversed sine of about 3 millimeters, (Fig.25.) I then proceeded to the removal of the excess of oil; but before the versed sine was reduced to 2 millimeters, the figure appeared to have a tendency to become
thin at its lower part and to swell out at the upper part, as if the oil Afz 26, had suddenly become slightly increased in density. At this moment I withdrew the syringe, so as to be enabled to observe the effect in question better; the change in form then became more and more pronounced; the lower part of the figure soon presented a true strangulation, the neck of which was situated nearly at a fourth part of the distance be| tween the rings, (Fig.26;) the constricted portion continued to narrow gradually, whilst the upper part of the figure became swollen; finally, the liquid separated into two unequal masses, which remained respectA ively adherent to the two rings; the upper mass formed a complete &A sphere, and the lower mass a doubly convex lens. The whole of these
phenomena lasted a very short time only.
With a vicw to determine whether any particular cause had in reality produced the alteration of the densities, I approximated the rings; then, after having reunited the two liquid masses, I again carefully raised the upper ring, ceasing at the height of 73 centimeters, so that the versed sine of the meridional convexity was slightly greater than when this was 8 centimeters. The figure was then found to be perfectly symmetrical, and it did not exhibit any tendency to deformity; whence it follows that the uniformity in the densities had not experienced any appreciable alteration. I recommenced, with still more care, the experiment with that figure which was 8 centimeters in height; and I was enabled to approach the cylindrical form still more nearly; but before it was attained, the same phenomena again presented themselves, except that the alteration in form was effected in an inverted manner, i. e., the figure became narrow at the upper part and dilated at the base; so that after the separation into two masses, the perfect sphere existed in the lower ring and the lens in the upper ring. On subsequently uniting, as before, the two masses, and placing the rings at a distance of 74 centimeters apart, the figure was again obtained in a regular and permanent form. Thus when we try to obtain between two solids rings a liquid cylinder the height of which is four times the diameter, the figure always breaks up spontaneously, without any apparent cause, even before it has attained the exactly cylindrical form. Now as the cylinder is necessarily a figure of equilibrium, whatever may be the proportion of the height to the diameter, we must conclude that the equilibrium of a cylinder the height of which is four times the diameter is unstable. As the shorter cylinders which I had obtained did not present analogous effects, I was anxious to satisfy myself whether the cylinders were really stable. I therefore again formed a cylinder 6 centimeters in height with the same rings; but this, when left to itself for a full half hour, presented a trace only of alteration in form, and this trace appeared about a quarter of an hour after the formation of the cylinder, and did not subsequently increase, which shows that it was due to some slight accidental cause.
The above facts lead us then to the following conclusions: 1st, that the cylinder constitutes a figure the equilibrium of which is stable when the proportion between its height and its diameter is equal to 3, and with still greater reason when this proportion is less than 3; 2d, the cylinder constitutes a figure the equilibrium of which is unstable when the proportion of its height to its diameter is equal to 4, and with still greater reason when it exceeds 4; 3d, consequently there exists an intermediate relation, which corresponds to the passage from stability to instability. We shall denominate this latter proportion the limit of the stability of the cylinder.
45. These conclusions, however, are liable to a well-founded objection. Our liquid figure is complex, because its entire surface is composed of a cylindrical portion and of two portions which present a spherical curvature. Now we cannot affirm that these latter portions exert no influence upon the stability or the instability of the intermediate portion, and consequently upon the value of the proportion which constitutes the limit between these two states. To allow of the preceding conclusions being rigorously applicable to the cylinder, it would be requisite that the figure should present no other free surface than the cylindrical surface, which is easily managed by replacing the rings by entire disks. I effected this substitution by employing disks of the same diameter as the preceding rings, but the results were not changed; the cylinder, 6 centimeters in height, was well formed, and was found to be stable; whilst the figure 8 centimeters in height began to change before becoming perfectly cylindrical, and was rapidly destroyed. The final result of this destruction did not, however, consist, as in the case of the rings, of a perfect sphere and a double convex lens, but, as evidently ought to have been the case, of two unequal portions of spheres, respectively adherent to the two opposite solid surfaces. The limit of the stability of the cylinder, therefore, really lies between 3 and 4. The experiments which we have just related are very delicate, and require some skill. In this, as in all other cases of measurements, the oil must be allowed to remain in the alcoholic mixture for two or three days, then the pellicle must be removed from it, (note to p. 254;) afterwards, when the mass, after having been again introduced into the vessel, has been attached to the two solid disks, some time must be allowed to elapse in order that the two liquids may be exactly at the same temperature; moreover, it must be understood that the experiments should be . in an apartment the temperature of which remains as constant as possible. Lastly, it is scarcely necessary to add, that when the alcoholic liquid is mixed, after having added small quantities of pure alcohol or alcohol at 16°, the movements of the spatula .# be very slow, so as to avoid the communication of too much agitation to the mass of oil; we are even sometimes compelled momentarily to depress the upper disk, so as to give greater stability to the mass, and thus to prevent the movements in question from producing the disunion. 46. It might be asked whether the want of symmetry, which is constantly seen in the spontaneous modification of the above unstable figures, is the result of a law which governs these figures; or whether it simply arises, as we should be led to believe at first sight, from imperceptible differences still existing be. tween the densities of the two liquids, which differences acting upon unstable figures might produce this want of symmetry, notwithstanding their extreme minuteness After having concluded the preceding experiments, I imagined that to solve the question in point, all that would be requisite would be to arrange matters so that the axis of the figure, instead of being vertical, as in the above experiments, should have a horizontal direction. • In fact, in the latter case, the slightest difference between the densities ought to have the effect of slightly curving the figure, but evidently cannot give the liquid any tendency to move in greater ão towards one extremity of the figure than the other; whence it follows, that if the spontaneous alteration of the figure still occurs unsymmetrically, this can only be owing to a peculiar law. On the other hand, if the figure really tends of itself to change its form unsymmetrically, it is clear that, in the case of the vertical position of the axis, the effect of a trace of difference between the densities ought to concur with that of the instability, and thus to accelerate the moment at which the figure commences to alter spontaneously. Consequently, on avoiding this extraneous cause by the horizontal direction of the axis of the figure, we may hope to approximate more nearly to the cylindrical form, or even to attain it exactly; we can, moreover, understand that the difficulty in the operations will be found to be considerably diminished. I therefore constructed a solid system, presenting two vertical disks of the same diameter, placed parallel with each other, at the same height, and opposite each other. Each of these disks is su ported by an iron wire fixed normally to its centre, then bent vertically downwards, and the lower extremities of these two wires are attached to a horizontal axis furnished with four small feet. This system is represented in perspective in Fig. 27. The diameter of the disks is 30 millimeters, but the distance which separates them is not four times this diameter. I thought that by approximating the figure more to the limit of stability, the operations would require still less trouble; the distance in question is only 108 millimeters, so that the relation between the length . the diameter of the liquid cylinder which would extend between the two disks would be equal to 3.6. We shall now detail the results obtained by the employment of this system. In the first place, the operations were much more easily performed.* In the second place, the figure still had a tendency to deformity before it had been rendered perfectly cylindrical; but this tendency always exhibited itself unsymmetrically, as in the vertical figures; from which circumstance alone we might conclude that the unsymmetrical nature of the phenomenon is not occasioned by a difference between the densities of the two liquids. In the third place, by a little management, I have pursued the experiment further, and succeeded in forming an exact co This lasted for a moment; it then began to be narrowed at one part of its length, becoming dilated at the other, like the vertical figures; and the phenomenon of disunion was completed in the same manner, giving rise ultimately to two masses of different volumes. . . I repeated the experiment several times, and always with the same results, except that the separation occurred sometimes on one, sometimes on the other side of the middle of the length of the figure. However, although the phenomenon is produced in an unsymmetrical manner with regard to the middle of the length of the figure, whether horizontal or vertical, on the contrary there is always symmetry with regard to the axis; in other words, throughout the duration of the phenomenon the figure remains constantly a figure of revolution. We may add here, that in the horizontal figure the respective lengths of the constricted and dilated portions appear to be equal; we shall show, in the following series, that this equality is rigorously exact, at least at the commencement of the phenomenon. It is now evident that the alteration in the form of these cylinders is really the result of a property which is inherent in them. We shall hereafter deduce this property as a necessary consequence of the laws which govern a more general phenomenon. It moreover results from the above experiment that the proportion 3.6 is still greater than the limit of stability, so that the exact value of the latter must lie between the numbers 3 and 3.6. It is obvious that this method of experiment might be employed to obtain a closely approximative determination of the value in question; I propose doing this hereafter, and I shall give an account of the . result in the following series, when I shall have to return to the question of the limit of stability of the cylinder. 47. In the unstable cylinders which we have just formed, the proportion of the length to the diameter was inconsiderable; but what would be the case if we were to obtain cylinders of great length relatively to their diameter | Now, under certain circumstances, figures of this kind, more or less exactly cylindrical, may be realized, and we shall proceed to see what the results of the spontaneous rupture of equilibrium are.
* The two disks in this solid system being placed at an invariable distance from each other, it is necessary, in making a mass of oil, the volume of which is not too great, adhere to them, to employ an extra piece consisting of a ring of iron wire of the same diameter as the disks, supported by a straight wire of the same metal, the free extremity of which is held in the hand. By means of this ring the mass, which has been previously attached to one of the disks, is drawn out until it is equally attached to the other; the ring is then removed. The latter removes a small portion of the mass at the same time; but on leaving the vessel it leaves this portion in the alcoholic liquid. It is then removed by means of the syringe.
+ To effect this the following proceeding must be adopted for the removal of the excess of oil. The operation is at first carried on with a suitable rapidity until the figure begins to alter in form; the end of the point of the syringe is then drawn gently along the upper part of the mass, proceeding from the thickest to the other portion. This slight action is sufficient to move a minute quantity of oil towards the latter, and thus to re-establish the symmet of the figure; a new absorption is then made, the figure again regulated, and these o ings are continued until the exactly cylindrical form is attained.
A fact which I described in paragraph 20 of the preceding memoir, and which I shall now describe more in detail, affords us the means of obtaining a cylinder of this kind, and of observing its spontaneous destruction. When some oil is introduced by means of a small funnel into an alcoholic mixture containing a slight excess of alcohol, and the oil is poured in sufficiently quick to keep the funnel full, the liquid forms, between the point of the funnel and the bottom of the vessel where the mass collects, a long train, the diameter of which continues to increase slightly from the upper to the lower part, so as to form a kind of very elongated cone, which does not differ much from a cylinder.” This nearly cylindrical figure, the height of which is considerable in proportion to the diameter, remains without undergoing any perceptible alteration so long as the oil of which it consists has sufficient rapidity of transference; but when the oil is no longer poured into the funnel, and consequently the motion of transference is retarded, the cylinder is soon seen to resolve itself rapidly into a series of spheres, which are perfectly equal in diameter, equally distributed, and with their centres arranged upon the right line forming the axis of the cylinder. To obtain perfect success, the elements of the experiment should be in certain roportions. The orifice of the funnel which I used was about 3 millimeters in §. and 11 centimeters in height. It rested upon the neck of a large bottle containing the alcoholic mixture, and its orifice was plunged a few millimeters only beneath the surface of the liquid. Lastly, the length of the cylinder of oil, or the distance between the orifice and the lower mass, was nearly 20 centimeters. Under these circumstances, three spheres were constantly formed, the upper of which remained adherent to the point of the funnel; the latter was therefore incomplete. We may add, that the excess of alcohol contained in the mixture should neither be too great nor too small; the proper quantity is found by means of a few preliminary trials. 48. The constancy and regularity of the result of this experiment complete then the proof that the phenomena to which the spontaneous rupture of equilibrium of an unstable liquid cylinder gives rise are governed by determinate laws. In this same experiment, the transformation ensues too rapidly to allow of its phases being well observed; but the phenomena presented to us by larger and less elongated cylinders, i. e., the formation of a dilatation and constriction in juxtaposition, and equal or nearly so in length, the gradual increase in thickness of the dilated portion and the simultaneous narrowing of the constricted portion, &c., authorize us to conclude that in the case of a cylinder the length of which is considerable in proportion to the diameter, the following order of things takes place: the figure |. at first so modified as to present a regular and uniform succession of dilated portions, separated by constricted portions of the same length as the former, or nearly so. This alteration, the indications of which are very slight, gradually becomes more and more marked, the constricted portions gradually becoming narrower, whilst the dilated portions increase in thickness, the figure remaining a figure of revolution; at last the constrictions break, and each of the various parts of the figure, which are thus completely isolated from each other, acquire the spherical form. We must add, that the termination of the phenomenon is accompanied by a remarkable peculiarity, of which we have not yet spoken; but as it only constitutes, so to speak, an accessory portion of the general phenomenon, we shall transfer the description of it to a subsequent part of this memoir, (see § 62.) 49. It might be asked why, in the experiment which we have last described, the cylinder is only resolved into spheres when the rapidity of the transference of liquid of which it is composed is diminished. In fact, we cannot understand how a motion of transference could give stability to a liquid figure which in a
* The slight increase in diameter depends upon the retardation which the resistance of the surrounding liquid occasions in the movement of the oil.