state of repose was unstable. In explaining this apparent peculiarity, we must remark that, as the spontaneous transformation of an unstable cylinder is effected under the action of continued forces, the rapidity with which the phenomenon occurs ought to be accelerated; this may be, moreover, casily verified in experiments relating to larger and less elongated cylinders; this same rapidity ought, therefore, always to be very minute at the commencement of the phenomenon. Now, in the case in question, as the changes in figure occur in the liquid of the cylinder whilst this liquid is animated by a movement of transference, it is evident, from what we have stated, that if this movement of transference is sufficiently rapid, the changes of form could only acquire a very slightly marked development during the passage of the point of the funnel to the mass accumulated at the bottom of the vessel; so that, the liquid being continually renewed, there will be no time for any alteration in form to become very perceptible to the eye. Hence, so long as the rapidity of the flow is sufficiently great, the liquid figure will appear to retain its almost cylindrical form, although its length is considerable in comparison with its diameter. On the other hand, when the velocity of the transference is sufficiently small, there will be time for the alterations in form to take place in a perfect manner, and we shall be able to see the cylinder resolve itself into spheres throughout the whole of its length.

50. We shall now describe another method of experimenting, which allows us to observe the result of the transformation under less restrained and more regular conditions in some respects than those of the preceding experiment, and which will, moreover, lead us to new consequences as regards the laws of the phenomenon. We shall first succinctly describe the apparatus and the operations, and afterwards add the necessary details.

The principal parts of which the apparatus consists are: 1st, a rectangular plate of plate-glass, 25 centimeters in length, and 20 in breadth; 2d, two strips of the same glass, 13 centimeters in length, and millimeters in thickness, perfectly prepared and polished at the edges; 3d, two ends of copper wire, about 1 millimeter in thickness, and 5 centimeters in length; these wires should be perfectly straight, and one extremity of each of them should be cut very accurately, then carefully amalgamated. The plate being placed horizontally, the two strips are laid flat upon its surface and parallel with its long sides, so as to leave an interval of about a centimeter between them; the two copper wires are then introduced into this, placing them in a right line in the direction of the length of the strips, and in such a manner that the amalgamated extremities are opposite to, and a few centimeters distant from, each other. A globule of very pure mercury, from 5 to 6 centimeters in diameter, is next placed between the same extremities; the two strips of glass are then approximated until they touch the wires, so as only to leave between them an interval equal in width to the diameter of these wires. The little mass of mercury, being thus compressed laterally, necessarily becomes elongated, and extends on both. sides towards the amalgamated surfaces. If it does not reach them, the wires are made to slide towards them until contact and adhesion are established. The wires are then moved in opposite directions, so as to separate them from each other, which again produces elongation of the little liquid mass and diminution of its vertical dimensions. By proceeding carefully, and accompanying the operation with slight blows given with the finger upon the apparatus to facilitate the movements of the mercury, we succeed in extending the little mass until its vertical thickness is every where equal to its horizontal thickness, i. e., to that of the copper wires. Thus the mercury forms a liquid wire of the same diameter as the solid wires to which it is attached, and from 8 to 10 centimeters in length. This wire, considering the small size of its diameter, which renders the action of gravitation insensible in comparison with that of molecular attraction, may be considered as exactly cylindrical; so that in this manner we obtain a liquid cylinder, the length of which is from 80 to 100 times its diame

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ter, and attached by its extremities to solid parts, which cylinder preserves its form so long as it remains imprisoned between the strips of glass. Weights are next placed upon the parts of the two copper wires which project beyond the extremities of the bands, so as to maintain these wires in firm positions; lastly, by means which we shall point out presently, the two strips of glass are raised vertically. At the same instant, the liquid cylinder, being liberated from its shackles, becomes transformed into a numerous series of isolated spheres, arranged in a straight line in the direction of the cylinder from which they origi nated. Ordinarily the regularity of the system of spheres thus obtained is not perfect; the spheres present differences in their respective diameters and in the distances which separate them; this undoubtedly arises from slight accidental causes, dependent upon the method of operation; but the differences are sometimes so small that the regularity may be considered as perfect. As regards the number of spheres corresponding to a cylinder of determinate length, it varies in different experiments; but these variations, which are also due to slight accidental causes, are comprised within very small limits.

51. Let us now complete the description of the apparatus, and add some details regarding the operations. As the plate of glass requires to be placed in a perfectly horizontal position, it is supported for this purpose upon four feet with screws. A small transverse strip of thin paper is glued to each of the extremities of the lower surface of the strips of glass, in such a manner that the strips of glass resting upon the plate through the medium of these small pieces of paper, their lower surface is not in contact with the surface of the plate. Without this precaution, the strips of glass might contract a certain adhesion to the plate, which would introduce an obstacle when the strips are raised vertically. Moreover, the latter are furnished, on their upper surface and at a distance of 6 millimeters from each of their extremities, with a small screw placed vertically in the glass with the point upwards, firmly fixed to it with mastic, and rising 8 millimeters above its surface. These four screws are for the purpose of receiving the nuts which fix the strips to the system by means of which they are elevated. This system is made of iron; it consists, in the first place, of two rectangular plates, 55 millimeters in length, 12 in breadth, and 3 in thickness. Each of them is pierced, perpendicularly to its large surfaces, by two holes, so situated, that on placing each of these plates transversely upon the extremities of the two strips of glass, the screws with which the latter are furnished fit into these four holes. The screws being long enough to project above the holes, nuts may then be adapted to them, so that on screwing them the strips of glass become fixed in an invariable position with regard to each other. The holes are of an elongated form in the direction of the length of the iron plates; hence, after having loosened the nuts, the distance between the two strips of glass may be increased or diminished without the necessity of removing the plates. A vertical axis, 5 centimeters in height, is implanted upon the middle of the upper surface of each of the plates; and the upper extremities of these two axes are connected by a horizontal axis, at the middle of which a third vertical axis commences; this is directed upwards, and is 15 centimeters in length. The section of the latter axis is square, and it is 5 millimeters in thickness. When the nuts are screwed up, it is evident that the strips of glass, the iron plates, and the kind of fork which connects them, constitute an invariable system. The long vertical axis serves to direct the movement of this system; with this view, it passes with very slight friction through an aperture of the same section as itself, and 5 centimeters in length, pierced in a piece which is fixed very firmly by a suitable support 10 centimeters above the plate of glass. Lastly, the perforated piece is provided laterally with a thumb-screw, which allows the axis to be screwed

* We may remark that the conversion of a metallic wire into globules by the electric discharge must undoubtedly be referred to the same order of phenomena.

into the tube. By this arrangement, if all parts of the apparatus have been carefully finished, when once the little nuts have been screwed up, the two strips of glass can only move simultaneously in a parallel direction to each other, and always identically in the same direction perpendicular to the plate of glass. When the liquid cylinder is well formed, and the weights are placed upon the free portions of the copper wires, the finger is passed under the horizontal branch of the fork, and the movable system is raised to a suitable distance above the plate of glass; it is then maintained at this height by means of the thumb-screw, so as to allow the result of the transformation of the cylinder to be observed. As the amalgamation of the copper wires always extends slightly upon their convex surface, the latter is coated with varnish, so that the amalgamation only occurs upon the small plane section. It would be almost impossible to judge by simple inspection of the exact point at which the separation of the copper wires from each other, to allow of the liquid attaining a cylindrical form, should be discontinued. To avoid this difficulty, the length of the cylinder is given beforehand, and this length is marked by two faint scratches upon the lateral surface of one of the strips of glass; the weight of the globule of mercury, which is to form a cylinder of this diameter and of the length required, is then determined by calculation from the known diameter of the wire; lastly, by means of a delicate balance, the globule to be used in the experiment is made exactly of this weight. All that then remains to be done is to extend the little mass until the extremities of the copper wires between which it is included have reached the marks traced upon the glass. Lastly, in making a series of experiments, the same mercury may be used several times if the isolated spheres are united into a single mass at the end of each observation. However, after a certain number of experiments, the mercury appears to lose its fluidity, and the mass always becomes disunited at some point, in spite of all possible precautions, before it has become extended to the desired length, which phenomena arise from the solid wires imparting a small quantity of copper to the mercury. The latter must then be removed, the plates of glass and the strips cleaned, and a new globule taken. The amalgamation of the wires also sometimes requires to be renewed.

52. By means of the above apparatus and methods, I have made a series of experiments upon the transformation of the cylinders; but before relating the results, it is requisite for their interpretation that we should examine the phenomenon a little more closely.

Let us imagine a liquid cylinder of considerable length in proportion to its diameter, and attached by its extremities to two solid bases; let us suppose that it is effecting its transformation, and let us consider the figure at a period of the phenomenon anterior to the separation of the masses, i. c., when this figure is still composed of dilatations alternating with constrictions. As the surfaces of the dilatations project externally from the primitive cylindrical surface, and those of the constrictions on the contrary are internal to this same surface, we can imagine in the figure a series of plane sections perpendicular to the axis, and all having a diameter equal to that of the cylinder; these sections will evidently constitute the limits which separate the dilated from the constricted portion, so that each portion, whether constricted or dilated, will be terminated by two of them; moreover, as the two solid bases are necessarily part of the sections in question, each of these bases should occupy the very extremity of a constricted or dilated portion. This being granted, three hypotheses present themselves in regard to these two portions of the figure, i. c., to those which rest respectively upon each of the solid bases. In the first place, we may suppose that both of the portions are expanded. In this case each of the constrictions will transfer the liquid which it loses to the two dilatations immediately adjacent to it; the movements of transport of the liquid will take place in the same manner throughout the whole extent of the figure, and the

transformation will take place with perfect regularity, giving rise to isolated spheres exactly equal in diameter, and at equal distances apart. This regularity will not, however, extend to the two extreme dilatations; for as each of these is terminated on one side by a solid surface, it will only receive liquid from the constriction which is situated on the other side, and will, therefore, acquire less development than, the intermediate dilatations. Under these circumstances, then, after the termination of the phenomenon, we ought to find two portions of spheres respectively adherent to two solid bases, each presenting a slightly less diameter than that of the isolated spheres arranged between

them.

In the second place, we may admit that the terminal portions of the figure are, one a constriction and the other a dilatation. The liquid lost by the first, not being then able to traverse the solid base, will necessarily all be driven into the adjacent dilatation; so that, as the latter receives all the liquid necessary to its development on one side only, it will receive none from the opposite side; consequently all the liquid lost by the second constriction will flow in the same manner into the second dilatation, and so on up to the last dilatation. The distribution of the movements of transport will, therefore, still be regular throughout the figure, and the transformation will ensue in a perfectly regular manner. This regularity will evidently extend even to the two terminal portions, at least so long as the constrictions have not attained their greatest depth; but beyond that point this will not exactly be the case, for independence being then established between the masses, each of the dilatations, excepting that which rests upon the solid base, will enlarge simultaneously on both sides, so as to pass into the condition of the isolated sphere, by appropriating to itself the two adjacent semi-constrictions, whilst the extreme dilatation can enlarge on one side. Consequently, after the termination of the phenomenon, we should find, at one of the soild bases, a portion of a sphere of but little less diameter than that of the isolated spheres, and at the other base a much smaller portion of a sphere, arising from the semi-constriction which has remained attached to it.

Lastly, in the third place, let us suppose that the terminal portions of the figure were both constrictions, in which case, after the termination of the phenomenon, a portion of a sphere equal to the smallest of the two above would be left to each of the solid bases. In this case, to be more definite, let us start from one of these terminal constrictions; for instance, that of the left. All the liquid lost by this first constriction being driven into the contiguous dilatation, and being sufficient for its development, let us admit that all the liquid lost by the second constriction also passes into the second dilatation, and so on; then all the dilatations, excepting the last on the right, will simply acquire their normal development; but the right dilatation, which, like each of the others, receives from that part of the constriction which precedes it the quantity of liquid necessary for its development, receives in addition the same quantity of liquid from that part of the constriction which is applied to the adjacent solid, so that it will be more voluminous than the others. Hence it is evident, in the case in point, that the opposed actions of the two terminal constrictions introduce an excess of liquid into the rest of the figure. Now, whatever other hypothesis may be made respecting the distribution of the movements of transport, it must always happen either that the excess of volume is simultaneously distributed over all the dilatations, or that it only augments the dimensions of one or two of them; but the former of these suppositions is evidently inadmissible, on account of the complication which it would require in the movements of transport; hence we must admit the second, and then the isolated spheres will not all be equal. Thus this third mode of transformation would necessarily of itself induce a cause of irregularity; and, moreover, it would not allow of a uniform distribution of the movements of transport, be

cause there would be opposition in regard to these movements, at least in the terminal constrictions.

It may, therefore, be regarded as very probable that the transformation takes place according to one or the other of the two first methods, and never according to the third, i. e., that things will be so arranged that the figure which is transformed may have for its terminal portions either two dilatations, or one constriction and one dilatation, but not two constrictions. In the former case, as we have seen, the movement of the liquid of all the constrictions would ensue on both sides simultaneously; and in the second this movement would occur in all in one and the same direction. If this is really the natural arrangement of the phenomenon, we can also understand how it will be preserved even when it is disturbed in its regularity by slight extraneous causes. Now, this, as we shall see, is confirmed by the experiments relating to the mercurial cylinder. Although the transformation of this cylinder has rarely yielded a perfectly regular system of spheres, I have found in the great majority of the results either that each of the soild bases was occupied by a mass little less in diameter than the isolated spheres, or that one of the bases was occupied by a mass of this kind and the other by a much smaller mass.

53. For the sake of brevity, let us denominate divisions of the cylinder those portions of the figure each of which furnishes a sphere, whether we consider these portions in the imagination as in the cylinder itself, before the commencement of the transformation, or whether we take them during the accomplishment of the phenomenon, i. c., during the modifications which they undergo in arriving at the spherical form. The length of a division is evidently that distance which, during the transformation, is comprised between the necks of two adjacent constrictions; consequently it is equal to the sum of the lengths of a dilatation and two semi-constrictions. Let us, therefore, see how the length in question, i. e., that of a division, may be deduced from the result of an experiment. Let us suppose the transformation to be perfectly regular, and let & be the length of a division, that of the cylinder, and n the number of isolated spheres found after the termination of the phenomenon. Each of these spheres being furnished by a complete division, and each of the two terminal masses by part of a division, the length will consist of n times, plus two fractions of . To estimate the values of these fractions, we must recollect that the length of a constriction is exactly or apparently equal to that of a dilatation, (§ 46;) now, in the first of the two normal cases, (§ 52,) i. c., when the masses remaining adherent to the bases after the termination of the phenomenon are both of the large kind, each of them evidently arises from a dilatation plus half a constriction, therefore three-fourths of a division; the sum of the lengths of the two portions of the cylinder which have furnished these masses is, therefore, equal to once and a half λ, and we shall have in this case 7 = (n + 1.5) λ, whence In the second case, i. e., when the terminal masses conn+ 1.5' sist of one of the large and the other of the small kind, the latter arises from a semi-constriction, or a fourth of a division, so that the sum of the lengths of the portions of the cylinder corresponding to these two masses is equal to ; consequently we shall have λ=

As the respective denominators of these two expressions represent the number of divisions contained in the total length of the cylinder, it follows that this number will always be either simply a whole number, or a whole number and a half. On the other hand, as the phenomenon is governed by determinate laws, we can understand that for a cylinder of given diameter composed of a given liquid, and placed under given circumstances, there exists a normal length which the divisions tend to assume, and which they would rigorously assume if the total length of the cylinder were infinite. If. then, it happens