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of spherules. This figure refers to the case of a very slightly viscid liquid, such as water, alcohol, &c., and where the convex surface of the cylinder is perfectly free; consequently, in accordance with the probable conclusion with which § 60 terminates, the proportion of the length of the divisions to the diameter has been taken as equal to 4.
The phenomenon of the formation of lines and their resolution into spherules is not confined to the case of the rupture of the equilibrium of liquid cylinders; it is always manifested when one of our liquid masses, wbatever may be its figure, is divided into partial masses. This is the manner in which, for instance, in 29 of the preceding memoir the minute masses which were then compared to satellites are formed.* The phenomenon under consideration is also produced when liquids are submitted to the free action of gravity, although it is then less easily shown. For instance, if the rounded end of a glass rod be dipped in ether, and then withdrawn carefully in a perpendicular direction, at the instant at which the small quantity of liquid remaining adherent to the rod separates from the mass, an extremely minute spherule is seen to roll upon the surface of the latter. Lastly, the phenomenon in question is of the same nature as that which occurs when very viscid bodies are drawn into threads, as glass softened by leat, except that in this case the great viscidity of the substance, and moreover the action of cold, which solidifies the thread formed, maintains the cylindrical form of the latter and allows of its acquiring an indefinite length.
63. To complete the study of the transformation of liquid cylinders into isolated spheres, it still remains for us to discover the law according to which the duration of the phenomenon varies with the diameter of the cylinder, and to endeavor to obtain at least some indications relative to the absolute value of this duration in the case of a cylinder of a given diameter, composed of a given liquid, and placed in given circumstances.
We can understand, à priori, that when the liquid and the external circumstances are the same, and supposing the length of the cylinder to be always such that the divisions assume exactly their normal length, ($ 53,) the duration of the phenomenon must increase with the diameter; for the greater this is, the greater the mass of cach of the divisions, and, on the other hand, the less the curvatures upon which the intensities of the configuring forces depend. It is true that the surface of each of the divisions increases also with the diameter of the cylinder; consequently it is the same with the number of the elementary configuring forces; but this augmentation takes place in a less proportion than tha' of the mass. This we shall proceed to show more distinctly. Under the above conditions two cylinders, the diameters of which are different, will become divided in the same manner; i.c., the proportion of the length of a division to the diameter will be the same in both parts, (s 55.) Now, it may be considered as evident that the similitude in figure will be maintained in all the phases of the transformation; ihis is, moreover, confirmed by experiment, as we shall soon see. Hence it follows at each homologous instant of the transformations of the two cylinders the respective surfaces of the divisions will always be to each other as ihe squares of the diameters of these cylinders, whilst the masses, which evidently remain invariable throughout the entire duration of the phenomena, will always be to each other as the cubes of these diameters. Thus, at each homologous instant of the respective transformations, the extent of the superficial layer of a division, consequently the number of the configuring forces which emanate from each of the elements of this layer, change from one figure to the other only in the proportion of the squares of the primitive diameters of
* It is clear that this mode of formation is entirely foreign to La Place's cosmogonic hypothesis ; therefore we have had no idea of deducing from this little experiment, which only refers to the effects of molecular attraction, and not to those of gravitation, any argument in favor of the hypothesis in question-an hypothesis which, in other respects, we do not adopt.
these figures; whilst the mass of a division, all the parts of which mass receive, under the action of the forces in question, the movements constituting the transformation, changes in the proportion of the cubes of these diameters. As regards the intensities of the configuring forces, we must remember, first, that the measure of that which corresponds to one element of the superficial layer
A (1 has ($ 59) for its expression G + ). This granted, if, at an homologous instant in the transformations of the two figures, we take upon one of the divi. sions of each of the latter any point similarly placed, it is clear from the similitude of these figures that the principal radii of curvature corresponding to the point taken upou the second will be to those corresponding to the point taken upon the first in the proportion of the diameters of the original cylinders, so that if this proportion be n, and the radii relating to the point of the first figure be R and R', those belonging to the point of the second will be nR and nR'; whence it follows that the measure of the two configuring forces corresponding to these points will be respectively
G+ ). and Cr+)= 1A/1 1 - 3G t ). Thus, in passing from the first to the second figure, the intensities of the elementary configuring forces in all the phases of the transformation will be to each other in the inverse proportion of the diameters of the cylinders.
I have convinced myself, by means of cylinders of mercury 1.05 millimeters and 2.1 millimeters in diameter, ($ 54 and 55,) that the duration of the phenomenon increases, in fact, with the diameter: although the transformation of these cylinders is effected very rapidly, yet we have no difficulty in recognizing that the duration relating to the greater diameter is greater than that which refers to the least.
64. As regards the law which governs this increase in the duration, it would undoubtedly be almost impossible to arrive at its experimental determination in a direct manner, i. e., by measuring the times which the accomplishment of the phenomenon would require in the case of two cylinders of sufficient length to allow of their being respectively converted into several complete isolated spherules, and of their satisfying the conditions indicated at the commencement of the preceding section. In fact I can hardly see any method of realizing such cylinders without giving them very minute diameters, like those of our cylinders of mercury, and then their duration is too short to allow of our obtaining the proportion with sufficient exactness.
But we may be able to arrive at the same result, but with certain restrictions, which we shall mention presently, by means of two short cylinders of oil formed between two disks, (§ 46 ;) there is nothing to prevent these cylinders from being obtained of such diameters as to render the exact measure of the durations easy. In the transformation of a cylinder of this kind, only a single constriction and a single dilatation are produced; but as in the transformation of cylinders which are sufficiently long to furnish several complete isolated spheres, the phases through which the constrictions and the dilatations pass are the same for all, we need only consider one constriction and one dilatation. We can understand that the relative dimensions of the two solid systems ought to be such, that the relation between the distance of the disks and their diameters is the same in both parts, in order that similitude may exist between the two liquid figures at their origin and at each homologous instant of their transformations.
Before giving an account of the employment of these figures of oil for the determination of the law of the durations, we shall take this opportunity of making several important remarks. We shall only require to make use of the law in question in that case, which in other respects is the most simple, where
the cylinders are formed in vacuo or in air, and are free from all external resistance, or, in other words, free upon the whole of their convex surface. Now our short cylinders of oil are formed in the alcoholic liquid, and it might be asked whether this circumstance does not exert some influence upon the proportion of the durations corresponding to a given proportion between the diameters of these cylinders. At first, a greater or less portion of the alcoholic liquid must be displaced by the modifications of the figures, so that the total mass to be moved in a transformation is composed of the mass of oil and this portion of the alcoholic liquid; but it is clear that in virtue of the similitude of the two figures of oil and that of their movements, the quantities of surrounding liquid respectively displaced will be to each other exactly, or at least apparently, as the two masses of oil; so that the relation of the two entire masses will not be altered by this circumstance. Hence it is very probable that this circumstance will no longer exert any influence upon the proportion of the durations, except that the absolute values of these durations will be greater. On the other hand, the mutual attraction of the two liquids in contact diminishes the intensities of the pressures, (§ 8,) and consequently the configuring forces; but it is easy to see that this diminution does not alter the relation of these intensities in the two figures. For let us imagine that at an homologous instant of the two transformations the alcoholic liquid becomes suddenly replaced by the oil, and let us conceive in the latter the surfaces of the two figures as they were at that instant. The configuring forces will then be completely destroyed by the attraction of the oil outside these surfaces, or, in other words, the external attraction will be at each point equal and opposite to the internal configuring force. If we now replace the alcoholic liquid, the intensities of the external attractions will change, but they will evidently retain the same relations to each other; whence it follows that those corresponding to two homologous points taken upon both the figures will still be to each other as the configuring forces commencing at these points; so that in fact the respective resultants of the external and internal actions at these two same points will be to each other in the same proportion as the two internal forces alone. Thus the attractions exerted upon the oil by the surrounding alcoholic liquid will certainly diminish the absolute intensities of the configuring forces, but they will not change the relations of these intensities, consequently they may be considered as not exerting any influence upon the durations. But it is clear that this cause will nevertheless greatly increase the absolute values of the latter. For the two reasons which we have explained, the presence of the alcoholic liquid will then increase the absolute values of the two durations to a considerable extent; but we may admit that it will not alter the relation of these values, so that this proportion will be the same whether the phenomenon take place in racuo or in air. We shall, therefore, consider the law which we deduce from our experiments upon short o of oil as independent of the presence of the surrounding alcoholic liquid, and this will be found to be supported by the nature of the law itself. But the exact formation of our short cylinders of oil requires (§ 46) that in these cylinders the proportion between the length and the diameter, or what comes to the same thing, between the sum of the lengths of the constriction and the dilatation and the diameter, exceeds but little the limit of stability. Now, in the transformation of cylinders sufficiently long to furnish several spheres, which would be formed in vacuo or in the air, and free upon their entire convex surface, and the divisions of which have their normal length, the proportion of the sums of the lengths of one constriction and one dilatation to the diameter, which proportion is the same as that of the length of one division to the diameter, would vary with the nature of the liquid, (§ 59.) and we are ignorant whether the law of the durations is independent of the value of this proportion. The law which we shall obtain in regard to short cylinders of oil can only there
fore be legitimately applied to cylinders of sufficient length to furnish several spheres supposed to be in the above conditions, in the case where these latter cylinders are formed of such a liquid that they would give for the proportion in question a value but little greater than that of the limit of stability
Now this is the case of mercury, ($ 60,) and it is also very probable that of all other very slightly viscid liquids, (60.) Thus the law given by the short cylinders of oil will be exactly or apparently that which would apply to cylinders of mercury of sufficient length to furnish several spheres, supposing the latter to be produced in vacuo or in air, free at the whole of their convex surface, and of such length that the divisions in each of them would assume their normal length. Moreover, the same law would be undoubtedly applicable to cylinders formed of any other very slightly viscid liquid, and supposed to be in the same conditions as the preceding.
The law may possibly be completely general, i. e., it may apply to cylinders. formed, always under the same circumstances, of any liquid whatever; but our experiments do not furnish us with the elements necessary to decide this question. Lastly, the transformation of our short cylinders presents a peculiarity which entails another restriction. The two final masses into which a cylinder of this kind resolves itself being unequal, the smallest acquires its form of equilibrium considerably before the other, so that the duration of the phenomenon is not the same. Hence we can only determine its duration up to the moment of the rupture of the line; consequently the proportion which we thus obtain for both cylinders will only be that of the durations of two homologous portions of the entire transformations. Moreover, the proportion of these partial durations is exactly that of which we shall have hereafter to make use.
65. I made the experiments in question by employing two systems of disks, the respective dimensions of which were to each other as one to two; in the former, the diameter of the disks was 15 milliméters, and they were 54 millimeters a part; and in the second their diameter was 30 millimeters, and their distance apart 108 millimeters. The cylinders formed respectively in these two systems were therefore alike, and, as I have previously stated, ($ 63,) these two figures exactly maintained their similarity, as far as the eye was capable of judging, in all the phases of their transformations. It sometimes happened that the cylinder, when apparently well formed, was not at all persistent and immediately began to alter; this circumstance being attributable to some slight remaining irregularity in the figure, I immediately re-established the cylindrical form,* and the time was only taken into account when the figure appeared to maintain this form for a few moments. Another anomaly then sometimes presented itself, which consisted in the simultaneous formation of two constrictions with an intermediate dilatation; this modification ceased when it had attained a very slightly marked degree, and the figure appeared to remain in the same state for a considerable period;t then one of the constrictions became gradually more marked, whilst the other disappeared, and the transformation afterwards went on in the usual manner. As this peculiarity constituted an exception to the regular course of the phenomenon, I ceased to reckon as soon as it showed itself, and I again re-established the cylindrical form. The estimation of the time was only definitively continued in those cases in which, after some persistence in the cylindrical form, a single constriction only was produced.
I repeated the experiment upon each of the two cylinders twenty times, in order to obtain a mean result. As soon as one transformation was completed,
* See the second note to paragraph 46.
+ We shall see, in the following series, to what this singular modification in the figure is orring.
I reunited the two masses to which it had given rise, and again formed the cylinder,* in order to proceed to a new measure of the time.
The number of seconds are given below; each expresses the time which elapsed from the moment of the transformation of the cylinder to that of the rupture of the line. These periods were determined by means of a watch, which beat the fths of a second. Cylinder
15 millions, in diameter.
30 millims, in diameter.
Mean 605.38. It is evident that the numbers relating to the same diameter do not diffe sufficiently from each other to prevent our regarding the proportion of the two means as closely approximating to the true proportion of the durations. Now the proportion of these two means is 2.04, i. e., almost exactly equal to that of the two diameters. Moreover, it is evident that in the case of each of the latter the greatest of the numbers obtained must correspond to that case where the cylinder is formed in the most perfect manner; consequently it is probable that the proportion of these two greatest numbers also closely approximates to the true proportion of the durations. Now, these two numbers are, on the one hand 36.Ā, and on the other 73.6, and their proportion is 2.02, which number differs still less from 2, or from the proportion of the diameters.
We may, therefore, admit that the durations relating to these two cylinders are to each other as their diameters; whence we deduce this law, that the partial duration of the transformation of a cylinder of the same kind is in proportion to its diameter.
I have said ($ 64) that the law thus obtained would of itself furnish a new motive for believing that it would not change if our short cylinders of oil were produced in vacuo or in air. In fact the proportionality to the diameter is the simplest possible law; and, on the other hand, the circumstances under which the phenomenon is produced are less simple in the case of the presence of the alcoholic liquid than they would be in that of its absence; consequently, if the law changed from the first to the second, it would follow that a simplification in the circumstances would, on the contrary, induce a complication of the law, which is not very probable.
* This was effected by conducting the large mass towards the small one, by means of the ring of which I spoke in the first note to paragraph 46. But care must be taken to prevent the ring, on separating from the liquid figure, from carrying away with it any perceptible quantity of oil; for this purpose, instead of making the entire ring adhere to the great mass, I left a small portion of the latter free, and, as its action was then insufficient to make the large mass reach the other, I aided it by gently pushing the oil with the extremity of the point of the syringe. On withdrawing the ring after the reunion of the two masses, only & very small spherule of oil separated from it in the alcoholic liquid, which in the next experi. ment I again united to the rest of the oil by means of the ring itself, as also the largest of the spherules arising from the transformation of the line.