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face of the modified cube presents in intaglio. Then, in fact, in passing from the hollow surface to that in relief, the radii of curvature corresponding to cach point will only change their signs without changing in absolute value; consequently, ($ 8,) since the condition of equilibrium is satisfied as regards the first of these surfaces, it will be equally so with regard to the second.
Now, let us imagine a plane passing through one side of the plate, and tangentially to the surface of the liquid which adheres to it at that point. As long as this liquid is in small quantity, we should imagine and experiment bears us out that the plane in question will be strongly inclined towards the plate; but if we gradually increase the quantity of liquid, the angle comprised between the plane and the plate will also continue to increase, and instead of being acute, as before, will become obtuse. Now, so long as this angle is less than 45°, the convex surface of the liquid adhering to the plate will remain identical with the concave surfaces of the mass attached to the metallic frame, and suitably diminished; but beyond this limit, the coexistence in the frame of the six hollow identical surfaces with the surface in relief becomes evidently impossible, for these surfaces must mutually intersect each other. Thus, when the withdrawal of the liquid from the mass forming the cube is continued, a point is attained at which the figure of equilibrium ceases to be realizable in accordance with the ordinary law of pressures. We then meet with a new verification of the principle enunciated in $ 28, 2. e., that the formation of layers commences. These layers are plane; they commence at each of the wires of the frame, and connect the remainder of the mass to the latter, which continues to present six concave surfaces. In fact, we can imagine that, by this modification of the liquid figure, the existence of the whole of this in the metallic frame again becomes possible, as also the equilibrium of the system; for there is then no further impediment to the concave surfaces assuming that
form which accords with the ordinary law of pressures ; and, on the other hand, in supposing the layers to be sufficiently thin, the pressure belonging to them might be equal to that which corresponds to these same concave surfaces, ($ 25.)
On removing still further portions of the liquid, the layer will continue to enlarge, whilst the full mass which occupies the middle of the figure will diminish in volume, and this mass can thus be reduced to very minute dimensions : Fig. 12 represents the entire system in this latter state. It is even possible to make the little central mass
disappear entirely, and thus to obtain a complete laminar system ; but for this purpose certain precautions must be taken, which I shall now point out. When the central mass has become sufficiently small, the point of the syringe must first be thoroughly wiped; otherwise the oil adheres to its exterior to a certain height, and this attraction keeps a cer. tain quantity of oil around it, which the instrument cannot absorb into its interior. In the second place, the point of the syringe must be depressed to such an extent that it nearly touches the inferior surface of the little mass. During the suction this surface is then seen to become raised, so as to touch the orifice of the instrument, and the latter then absorbs as much of the alcoholic mixture as of the oil; but this is of no consequence, and the minute mass is seen to diminish by degrees, so as at last completely to disappear. The system, then, consists of twelve triangular layers, each of which commences at one of the wires of the frame, and all the summits of which unite at the centre of the figure; it is represented in Fig. 13. But this system is only formed during the action of the syringe. If, when this is complete, the point of the instrument is slowly withdrawn, an additional lamina of a square form is seen
to be developed in the centre of the figure, (Fig. 14.) This, then, is the defini. tive laminar system to which the liquid cube is reduced by the gradual diminu. tion of its mass.
32. In the preceding experiment, as in that of paragraph 23, the thickness of the layers is at first greater than that which would correspond to equilibrium. If, then, the system were left to itself whilst it still contains a central mass, we should imagine that one portion of the liquid of the layers would be slowly driven towards this mass, and that the layers would gradually become thinner. Moreover, it always happens that one or the other of the latter increases after some time, undoubtedly for the reason which we have already pointed out, ($ 26.) Hence, for the perfect success of the transformation of the cube into the laminar system, one precaution, which has not yet been spoken of, must be attended to. It consists in the circumstance that, from the instant at which the layers arise, the exhaustion of the liquid must be continued as quickly as possible until the central mass has attained a certain degree of minuteness. In fact, as soon as the formation of the layers commences, their tendency to become thinner also begins to be developed ; and if the operation is effected too slowly, the system might break before it was completed. When the central mass is sufficiently reduced-and experience soon teaches us to judge of the suitable point—the action of the syringe must be gradually slackened, and at last the other precautions which we have mentioned must be taken.
We are able, then, to explain the rupture of the layers so long as there is a large or small central mass; but when the laminar system is complete, we do not at the first glance see the reason why the thickness of the layers diminishes, and consequently why destruction of the system takes place. Nevertheless the rupture ultimately takes place in this as in the other case, and the time during which the system persists rarely extends to half an hour. In ascertaining the cause of this phenomenon, it must be remarked that the intersections of the different layers cannot occur suddenly, or be reduced to simple lines : it is evident that the free transition between two liquid surfaces could not be thus established in a discontinuous manner. These transitions must, therefore, be effected through the intermedium of minute concave surfaces, and with a little attention we can recognize that, in fact, this really takes place. We can then understand that the oil of the layers ought also to be driven towards the places of junction of the latter; and consequently the absence of the little central mass does not prevent the gradual attenuation of the layers, and the final destruction of the system.
33. If, during the action of the syringe, when the system showu in Fig. 13 has been attained, instead of slowly withdrawing the instrument, it is suddenly detached by a slight shake in a vertical direction, the additional layer is not developed; but the little mass in Fig. 12 is seen to be reproduced very rapidly. This fact confirms in a remarkable manner the explanation which we have given in the preceding paragraph. In fact, at the moment at which the point of the instrument is separated from the system, the latter may be considered as composed of hollow pyramids. Now it also follows, from causes relating to their continuity, that the summits of these pyramids should not constitute simple points, but little concave surfaces. But as the curvatures of these minute surfaces are very great in every direction, they would give rise to still far less pressure than those which establish the transitions between each pair of surfaces of the layers; for in the latter there is no curvature in one direction. The oil of the layers will, therefore, be driven with much greater force towards the centre of the figure than towards the other parts of the junctions of these layers. Again, the twelve layers terminating in this same centre, the oil flows there simultaneously from a large number of sources. These two concurrent causes ought then, in conformity with experiment, to produce the rapid reappearance of the small central mass; and we can understand why it is impossible to obtain the complete system of the pyramids otherwise than during the action of the syringe.
37. All the other polyhedric liquids become transformed, like the cube, into laminar systems when the mass of which they are composed is gradually diminished. Among these systems some are complete; the others still contain very small masses, which cannot be made to disappear entirely. Analogous considerations to those which we applied with regard to the cube would show, in each case, that the formation of layers commences as soon as the hollow surfaces which would correspond to the ordinary law of pressures cease to be able to coexist in the solid frame. Figs. 15, 16, 17, and 18 represent the Fu 15.
laminar systems resulting from the triangular prism, the hexahedral prism, the tetrahedron and the pyramid with a square base, these systems being supposed to be complete. They are all formed of plane layers, commencing at each of the metallic wires; and that of the hexahedral prism, as is shown, contains an additional layer in the centre of the figure.
35. The systom arising from the regular octohedron presents a singular exception, which I have not been able to explain. The layers of which this eystem is composed are curved, and form a fantastical group, of which it is difficult to give an exact idea by graphic representations. Fig. 19 exhibits them projected upon two rectangular vertical planes ; and it is seen that the aspects of the system observed upon two adjacent sides are inverse as regards each other. The formation of this system presents a curious peculiarity. At the commencement of the operation all the faces of the octohedron become simultaneously hollow; the layers in progress of formation are plane, and arranged symmetrically, so that the system tends towards the form represented at Fig. 20. But when a certain limit is attained, a sudden change occurs, the layers become curved, and the system tends to assume the singular form which we have mentioned. I have several times repeated the experiment, varying the circumstances as much as possible, and the same effects are always pro. duced.
In the course of this memoir I shall point out another process for obtaining laminar systems; it is an extremely simple one, and has moreover the advantage of producing all the systems in a complete state.
36. In concluding our observations upon polyhedric liquids, I shall remark that the triangular prism may be employed to produce the phenomena of dispersion. In this way a beautiful solar spectrum may be obtained by means of a prism with liquid faces. But as the effect only depends upon the excess of the refracting action of the oil above that of the alcoholic liquid, to obtain a considerably extended spectrum the angle of refraction of the prism must be obtuse; an angle of 1100 gives a very good result. Moreover, it is evidently requisite that the faces of the prism should be perfectly plane, which is obtained by using a carefully made frame; by establishing exact equilibrium between the density of the liquids; and, lastly, by arresting the action of the syringe exactly at the proper point.
Other figures of Revolution besides the Sphere. Liquid Cylinder. 37. Let us now endeavor to form some new liquid figures. Those best adapted to theoretical considerations would be figures terminated by surfaces of revolution other than the sphere and lenticular figures, which we have already studied. Surfaces of revolution enjoy simple properties in regard to the radii of the greatest and least curvature at every point; we know that one of these two radii is the radius of curvature of the meridional line, and that the other is that portion of the normal to this line which is included between the point under consideration and the axis of revolution. We shall now endeavor to obtain figures of this nature.
38. Let our solid system be composed of two rings of iron | Fig. 20:* wire, equal, parallel, and placed opposite to each other. One
of these rings rests upon the base of the vessel by three feet composed of iron wire; the other is attached, by means of an intermediate piece, to the axis traversing the central stopper, so that it may be approximated to or removed from the former by depressing or elevating this axis.* The system formed by these two rings is represented in Plate VII, Fig. 20 bis; the diameter of those which I employed was 7 centimeters.
After having raised the upper ring as much as possible, let a sphere of oil, of a slightly larger diameter than that of the rings, be formed, and conducted towards the lower ring in such a manner as to make it adhere to the entire circum
ference of the latter; then depress the upper ring until it comes into contact with the liquid mass, and the latter is uniformly attached to it. When the mass has thus become adherent to the system of the two rings,
let the upper ring be slowly raised; when the two rings are Fig. 21. at a proper distance apart, the liquid will then assume the
form the vertical projection of which is represented in Fig. 21, in which the lines a b and c d are the projections of the rings. The two portions of the surface which are respectively applied to each of the rings are convex spherical segments; and the portion included between the two rings con
stitutes a figure of revolution, the meridional curve of which, as is shown, is convex externally. We shall recur, in the following series, to this part of the liquid figure. If we now continue gradually to raise the upper ring, the curvature of the two extremities and the meridional curvature of the intermediate portion will be diminished ; and if there is exact equilibrium be
tween the density of the oil and the surrounding liquid, the Fig. 22. surface included between the two rings will be seen to assume
a perfectly cylindrical form, (Fig. 22.) The two bases of the liquid figure are still convex spherical segments, but their curvature is less than in the preceding figure. If the interval between the rings be still further increased, it is evident that the surface included between them would lose the cylindrical
form, and that a new figure would result. This is what occurs; but the consideration of the figure thus produced must be deferred.
Instead, then, of immediately increasing the distance between the rings, let us commence by adding a certain quantity of oil to the mass, which will again
* In the experiments which we are now about to describe, the short axis represented in Fig. 2 of the preceding memoir, and which has hitherto answered our purpose, must be replaced by another of about 15 centimeters in length.