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the liquid mass is not too great, the curvatures again terminate abruptly along the rim of the plate. By gradually augmenting the volume of the primitive sphere of oil, without, however, rendering it sufficiently large to allow of the mass completely enveloping the plate so as to retain the spherical form, a limit is attained at which the edge of the plate ceases to reach the superficial layer of the new figure of equilibrium except at the two summits of the ellipse. The discontinuity in the curvatures then only occurs at these two places. Figs. 4 and 5 exhibit the result of the experiment in this case. In Fig. 4 the long axis of the ellipse is presented to view, in Fig. 5 its short axis. 16. All the facts which we have hitherto detailed show that so long as the interior of the mass is modified its external shape undergoes no alteration; but that directly the superficial layer is acted upon, the mass acquires a different form. To complete the proof, by experiment alone, that the configuring actions exerted by the liquid upon itself emanate solely from the superficial layer, the only point would then be the possibility of reducing a liquid mass to its superficial layer, or at least to a thin pellicle, and to see if in this state it would assume the same figure of equilibrium as a complete mass. Now this is completely realized in soap-bubbles; for these bubbles, when detached from the tube in which they have been made, assume, as is well known, a spherical form, i. e., the same figure as that which we find a complete mass acquires in our apparatus when withdrawn from the action of gravity and perfectly free. When the mass adheres to a solid system, which modifies its figure, it is clear that the entire configurative action is composed of two parts, one of which belongs to the solid system; and we find that this system only exerts it when acting upon the superficial layer; the other belongs to the liquid, and emanates directly from the free portion of this same superficial layer. The facts which we have related show clearly what is the seat of this second part of the whole configurative action, but they do not make us acquainted with the nature of the forces of which it consists. On referring to theory, we find that these forces consist in pressures exerted upon the mass by all the elements of the superficial layer, pressures the intensity of which depend upon the curvatures of the surface at the points to which they correspond. Hence it follows that the mass is pressed upon by every part of its superficial layer, with an intensity depending in the same manner upon the curvatures of the surface. For instance, a mass the free surface of which presents a convex spherical curvature, will be pressed upon by the whole of the superficial layer belonging to this free surface, with a greater intensity than if this surface had been plane; and this intensity will be more considerable in proportion as the curvature is greater, or as the radius of the sphere to which the surface belongs is less. Let us see whether experiment will lead us to the same conclusions. 17. The solid system which we shall employ is a circular perforated plate, (Fig. 6.) It is placed vertically, and attached by a point of its circumference to the iron wire which supports it. Let the diameter of the sphere of oil be less than that of the plate, and let the latter be made to penetrate the mass by its edge in a direction which does not pass through the centre of the sphere. At first, as in the experiment at paragraph 14, the oil will form two unequal spherical segments; but matters do not remain in this state. The most convex segment is seen to diminish gradually in volume, consequently in curvature, whilst the other increases, until they have both become exactly equal. One }. of the oil then passes through the aperture in the plate, so as to be transrred from one of the segments towards the other, until the above equality is attained. Let us now examine into the consequences deducible from this experiment, judging from the preceding ones, and independently of all theoretical considerations. When the oil has once become extended over both surfaces of the plate, in such a manner that the superficial layer is applied to every part of the
margin of the latter, the action of the solid system is completed; and the movements which subsequently ensue in the liquid mass, to attain the figure of equilibrium, can only then be due to an action emanating from the free part of the superficial layer. It is, therefore, the latter which compels the liquid to pass through the aperture in the plate; and the phenomenon must necessarily result either from a pressure exerted by that portion of the superficial layer which belongs to the most convex segment, or by a traction o: by the portion of this same layer belonging to the other segment. Our experiment not being alone capable of determining our choice between these two methods of explaining the effect in question, let us provisionally adopt the first, i.e., that which attributes it to pressure. In our experiment, this pressure emanates from the superficial layer of the most curved segment; but it is easy to see that the superficial layer of the other segment also exerts a pressure which, alone, is less than the preceding. In fact, if for the most curved segment a segment less curved than the other were substituted, the oil would then be driven in the opposite direction. Hence it follows that the entire superficial layer of the mass exerts a pressure upon the liquid which it encloses, and that the intensity of this pressure depends upon the curvatures of the free surface. Moreover, as the liquid proceeds from the most curved segment to that which is least so, it is evident that in the case of a convex surface, the curvature of which is spherical, the pressure is greater in proportion as the curvature is more marked, or as the radius of the sphere to which the surface belongs is smaller. Lastly, since a plane surface may be considered as belonging to a sphere, the radius of which is infinitely great, it is evident that the pressure corresponding to a convex surface, the curvature of which is spherical, is superior to that which would correspond to a plane surface. All these results were announced by theory. They perfectly verify, then, that of the latter to which they refer, and this concordance ought now to decide in favor of the hypothesis of pressure. This same part of the theory was already verified, in its application to liquids submitted to the action of gravity, by the phenomenon of the depression presented by liquids in capillary tubes, the walls of which they do not moisten; but the series of our experiments, setting out with the elements of the theory, and following it step by step, yields far more direct and complete verification. Our last experiment leads us to still further consequences. The liquid passing from one of its segments to the other, so long as their curvatures have not become identical, and the pressures corresponding to the two portions of the superficial layer becoming equal to each other simultaneously with the two curvatures, it follows that the mass only attains its figure of equilibrium when this equality of pressure is established. We thus have a primary verification of the general theory of equilibrium which governs our liquid figures, a condition in virtue of which the pressures exerted by the superficial layer ought to be everywhere the same. Moreover, it is evident that if a superficial layer, having a spherical curvature, exerts by itself a pressure, this principle must be true, however small the extent of this layer may be supposed to be. It follows, therefore, that an extremely minute portion of the superficial layer of our mass, taken from any part of either of the two segments, ought itself to be the seat of a slight pressure; consequently, that the total pressure exerted by the superficial layer is the result of individual pressures emanating from all the elements of this layer. This was also shown by theory. Further, following the same train of reasoning, we see that the intensity of each of the minute individual pressures ought to depend upon the curvature of the corresponding element of the layer, which is also in conformity with theory. Lastly, as in a state of equilibrium the two segments belong to spheres of equal radii, the curvature is the same in all points of the surface of the mass; whence it follows that all the minute elementary pressures are equal to each other. The general condition of equilibrium (§ 5) is, therefore, perfectly verified in the instance of our experiment.
18. The principle of the superficial layer, applied to the preceding experi
ment, allows of the latter being modified in such a manner as to obtain a very
remarkable result. When the figure of equilibrium is once attained, the per
forated plate acts upon the superficial layer by its external border only. The
whole of the remainder of this plate then exerts no influence upon the figure in
question. Hence it follows that this figure would still be the same if the aperture were enlarged, only the greater the diameter of the latter the less time is required for the establishment of the equality between the two curvatures. Lastly, we ought to be able to enlarge the aperture nearly to the margin of the plate without changing the figure of equilibrium; or, in other words, to reduce the solid system to a simple ring of thin iron wire. Now, this is confirmed by experiment; but, to put it in execution, we cannot confine ourselves, as before, to making the solid system }. a sphere of oil of less diameter than that of this same system, and subsequently to allow the molecular forces to act, because the metallic wire, on account of its small extent of surface, would not exert a sufficient action upon the superficial layer to cause the liquid to extend so as to adhere to the entire surface of the ring. The mass would then remain traversed by part of the latter, and its spherical form would not be sensibly altered if the metallic wire were small; the liquid surface would merely be slightly raised upon the wire in the two small spaces at which it issued from the mass. To speak more exactly, under the circumstances in question two figures of equilibrium are possible. One of these differs but very slightly from the sphere; it is not symmetrical with regard to the ring, one part of which traverses it whilst the other part remains free. The second figure is perfectly symmetrical as regards the ring, and completely embraces its margin; its surface is composed of two equal spherical curves, the margins of which rest upon the ring; in other words, it constitutes a true doubly convex lens of equal curvatures. This is the figure which it is our object to obtain. For this purpose we first give the sphere of oil a diameter slightly greater than that of the metallic ring; we then introduce the latter into the mass so that it is completely enveloped; lastly, by means of the small glass syringe, (§ 9,) some of the liquid is gradually removed from the mass.” As this diminishes in volume, its surface is soon applied to every part of the margin of the ring, and the volume continuing to diminish, the lenticular form becomes manifest. Afterwards, by withdrawing more of the liquid, the curvatures of the two surfaces may be reduced to that degree which is considered suitable. In this way a beautiful double convex lens is obtained, which is entirely liquid except at its circumference. Moreover, in consequence of the index of refraction of the olive oil being much greater than that of the alcoholic mixture, the lens in question possesses all the P. of converging lenses; thus, it magnifies objects seen through it, and this magnifying power may be varied at pleasure by removing some of the liquid from, or adding more to, the mass. Our figure, therefore, realizes that which could not be obtained with glass lenses, i.e., it forms a lens, the curvature and magnifying power of which are variable. The diameter of that which I formed was 7 centimetres, and the thickness of the metallic wire was about # a millimetre. A much finer wire might have been used with the same success; but the apparatus would then become inconvenient on account of the facility with which it would be put out of shape. By operating with care, the curvatures of the lens may be diminished so as almost to make them vanish; thus I have been enabled to reduce the lens which I
formed, and the diameter of which, as I have stated, was 7 centimetres, to
such an extent that it was only 2 or 3 millimetres in thickness. Hence we
might presume that it would be possible to obtain, by a proper mode of pro
* The point of the instrument is introduced into the vessel through the second aperture in the lid.
ceeding, a layer of oil with plane faces. This is, in fact, confirmed by experience, as we shall see further on.
19. To render the curvatures of the liquid lens very slight, the point of the syringe must naturally be applied to the middle of the lens, because the maximum of thickness exists there. Now, when a certain limit has been attained, the mass suddenly becomes divided at that point, and a curious phenomenon is produced. The liquid rapidly retires in every direction towards the metallic circumference, and forms a beautiful liquid ring along the latter; but this ring does not last for more than one or two seconds, after which it spontaneously resolves itself into several small, almost spherical masses, adhering to various parts of the ring of iron wire, which passes through them like the beads of a necklace.
20. The reasoning which led us, at the commencement of paragraph 18, to reduce the primitive solid system to a simple metallic wire representing the line in the direction of which this system is met by the superficial layer belonging to the new figure of equilibrium, may be generalized. We may conclude that whenever a solid system introduced into the mass is not met by the superficial layer of the figure produced, excepting in the direction of small lines only, simple iron wires, representing the lines in question, may be substituted for the solid system employed. But if the volume of the primitive solid system were considerable, it would evidently be requisite to add to the mass of oil an equivalent volume of this liquid, to occupy the place of the solid parts suppressed.
There is, however, an exception to this principle; it occurs when the solid system separates the entire mass into isolated portions, as in the experiment of paragraph 14; for then these portions assume figures independent of each other, and which may correspond to different pressures. In this case the suppression of one portion of the solid system would place the figures primitively isolated in communication, and the inequality of the pressures would necessarily induce a change in the whole figure. Excluding this exception, the principle is general, and the result of it is that well-developed effects of configuration may be obtained on employing simple iron wires instead of solid systems. The experiment of the biconvex lens furnishes one instance of this, and we shall meet with a great many others hereafter. Nevertheless, to be enabled to comprehend the influence of a simple metallic wire upon the configuration of the liquid mass, it is not requisite to consider this wire as substituted for a complete solid system; it may also be considered by itself. It is, in fact, clear that the solid wire acting by attraction upon the superficial layer of the mass, the curvatures of the two portions of the surface resting upon it ought not to have any further relation of continuity with each other. The metallic wire may, therefore, determine a sudden transition between these two portions of the surface, the curyatures of which will terminate abruptly at the limit which it places to them. The principles which we have established ought undoubtedly to be considered as among the most remarkable and curious consequences of the principle of the superficial layer, and one cannot avoid being astonished when we see the liquid maintained in such different forms by an action ex- Fig. 7. erted upon the extremely minute parts of the superficial layer of the mass.
21. We have experimentally studied the influence of convex surfaces of spherical curvature; let us now ascertain what experiment is able to teach us in regard to plane surfaces and concave surfaces of splerical curvature. Let us take for the solid system a large strip of iron, curved circularly so as to form a bollow cylinder, and attached to the suspending iron wire by some point on its outer surface, (Fig. 7.) To prevent the production of accessory phenomena in the experiment, we shall suppose that the breadth of the metallic band is less than the diameter of the cylinder formed by the same band, or that it is at least equal to it. Make the mass of oil adhere to the internal surface of this
system, and let us suppose that the liquid is in sufficient quantity then to project outside the cylinder. In this case the mass will present on each side a convex surface of spherical curvature, and the curvatures of these two surfaces will be equal. This figure is a consequence of what we have previously seen; and we must not stop here, for it will serve us as a starting point in obtaining other figures which we require. Apply the point of the syringe to one of the above convex surfaces, and gradually withdraw some of the liquid; the curvatures of the two surfaces will then gradually diminish, and with care they may be rendered perfectly plane. It follows from this first result that a plane surface is also a surface of equilibrium, which is evidently in conformity with theory. Let us now apply the end of the syringe to one of these plane surfaces, and again remove a small quantity of liquid. The two surfaces will then become simultaneously hollow, and will form two concave surfaces of spherical curvature, the margins of which rest upon the metallic band, and the curvatures of which are the same. Finally, by the further removal of the liquid, the curvatures of the two surfaces become greater and greater, always remaining equal to each other. Hence it results, first, that concave surfaces of spherical curvature are still surfaces of equilibrium, which is also in accordance with theory. Moreover, as the plane surface left free sinks spontaneously as soon as that to which the instrument is applied becomes concave, it must be concluded that the superficial layer belonging to the former exerts a pressure which is counterbalanced by an equal force emanating from the opposite superficial plane layer, but which ceases to be so, and which drives away the liquid as soon as this opposite layer commences to become concave. Again, as further abstraction of the liquid determines a new rupture of equilibrium, so that the concave surface opposite to that upon which we directly act exhibits a new spontaneous depression when the curvature of the other surface increases, it follows that the concave superficial layer belonging to the former still exerted a pressure, which at first was neutralized by an equal pressure arising from the other concave layer, but which be. comes preponderant, and again drives away the liquid, when the curvature of this other layer is increased. Hence it follows, first, that a plane surface produces a pressure upon the liquid; second, that a concave surface of spherical curvature also produces a ressure; third, that the latter is inferior to that corresponding to a plane surface; fourth, that it is less in proportion as the concavity is greater, or that the radius of the sphere to which the surface belongs is smaller. These results were also F. out by theory, and had already been verified in the application of the atter to liquids submitted to the action of gravity, by the phenomenon of the elevation of a liquid column in a capillary tube, the walls of which are moistened by it. owning upon these facts, as we have done at the end of paragraph 17 in regard to convex surfaces of spherical curvature, we shall arrive at the conelusion that the entire pressure exerted by a concave superficial layer of spherical curvature is the result of minute individual pressures arising from all the elements of this layer, and that the intensity of each of these minute pressures depends upon the curvature of that element of the layer from which it emanates. Our last experiment, therefore, perfectly verifies that part of the theory which relates to plane and convex surfaces of spherical curvature. Lastly, in the state of cquilibrium of our liquid figure, the curvature being the same at all points of each of the two concave surfaces, it is again evident that all the minute elementary pressures are equal to each other, which gives a new complete verification of the general condition of equilibrium. 22. The figure we have just obtained constitutes a biconcave lens of equal curvatures, and possesses all the properties of diverging lenses, i. e., it diminishes objects scen through it, &c. Moreover, as the curvature of the two surfaces may be increased or diminished by as small degrees as is wished, it follows