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SE COND SERIES.
Preface.

AT the lo when attacked by the disease which has entirely deprived me of sight, I had terminated the greater part of the experiments relating to this series, as well as the following. M. Duprez, correspondent of the Brussels Academy, and M. Donny, had the kindness to undertake those .#. were still wanting. I constantly directed their execution; nearly all were made in my presence, and I followed all the details. I have therefore considered myself justified, in order to simplify the description, in expressing myself in the course of this investigation as if I had made the experiments,

With respect to the theoretical portions, I am indebted to the able assistance of one of my colleagues, M. Lamarle, who has most kindly devoted many long hours to listening to the details of my investigations, and to aiding me in the explanation of several difficult points. I am also indebted to another of my colleagues, M. Manderlier, for the execution of a part of the calculations.

May I be permitted to express in this place my gratitude to these devoted friends? Thanks to their generous help, science is still an open field for me: notwithstanding the infirmity with which I am afflicted, I am able to put in order the materials I have collected, and even to undertake fresh researches.

Preliminary considerations and theoretical principles. General condition to be satisfied by the free surface of a liquid mass withdrawn from the action of gravity, and in a state of equilibrium. Liquid sphere.

1. The process described in the previous memoir enabled us to destroy the action of gravity upon a liquid mass of considerable volume, leaving the mass completely at liberty to assume the figure assigned to it by the other forces to which it is subject. This process consists essentially in introducing a mass of oil into a mixture of water and alcohol, the density of which is exactly equal to that of the oil employed. The mass then remains suspended in the surrounding liquid, and behaves as if withdrawn from gravity. By this means we have j a series of phenomena of configuration, dependent either simply upon the proper molecular attraction of the mass, or upon the combination of this force with the centrifugal force. We shall now abandon the latter force, and introduce another of a different kind, the molecular attraction exerted be. tween liquids and solids; in other words, we shall cause the liquid mass to adhere to solid systems, and study the various forms assumed ol. these circumstances by those portions of the surface which remain free. In this way we shall have the curious spectacle presented by the figures of equilibrium appertaining to a liquid mass, absolutely devoid of gravity and adherent to a given solid system. But the figures which we shall obtain present another kind of interest. The free portions of their surface belong, as we shall show, to more extended figures, which may be conceived by the imagination, and which, in the same condition of total absence of gravity, would belong to a perfectly free liquid mass; thus our processes will partially realize the figures of equilibrium of a mass of this kind. The latter are far from being confined to the sphere; but among them the sphere alone is capable of being completely formed, the others presenting either infinite dimensions in certain directions, or other peculiarities which we shall point out, and which equally render their realization in the complete state impossible. Moreover, the results at which we shall arrive will constitute so many new and unexpected confirmations of the theory of the pressures exerted by liquids upon themselves in virtue of the mutual attraction of their molecules, a theory upon which the explanation of the phenomena of capillarity is based. Lastly, in our liquid figures we shall discover remarkable properties, which will lead us to some important applications.

2. In order to guide us in our experiments, and also to enable us to comprehend their bearing, we shall first consider the question in a purely theoretic {. of view. The action of gravity being eliminated and the liquid mass eing at rest, the only forces upon which the figure of equilibrium will depend will be the molecular attraction of the liquid for itself, and that exerted between the liquid and the solid system to which we cause it to adhere. The action of the latter force ceases at an excessively minute distance from the solid; hence, in regard to any point of the surface of the liquid situated at a sensible distance from the solid, we have only to consider the first of the two above forces, i.e., the molecular attraction of the liquid for itself. The general effect of the adhesive force exerted between the liquid and the solid is to oblige the surface of the former to pass certain lines; for instance, if a liquid mass of suitable volume be caused to adhere to an elliptic plate, the surface of the mass will pass the elliptic outline of the plate. At every point of this surface, situated at a sensible distance from this margin, the molecular attraction of the liquid for itself alone is in action. Let us now examine into the fundamental condition which all points of the free surface of the mass must satisfy, in virtue of the latter force. The determination of this condition and its analytical expression are comprised in the beautiful theories upon which the explanation of the phenomena of capillarity is based, although geometricians have not specially studied the problem of the figure of a liquid mass void of gravity adherent to a given solid system. We shall, therefore, now resume the principles and the results of the theories in question, at least those which relate directly to our subject. 3. Within the interior of a liquid mass, at any notable distance from its surface, each molecule is equally attracted in every direction; but this is not the case at or very near the surface. In fact, let us consider a molecule situated at a distance from the surface less than the radius of the sphere of sensible activity of the molecular attraction, and let us imagine this molecule to be the centre of a small sphere having this same radius. It is evident that one portion of this sphere being outside the liquid, the central molecule is no longer equally attracted in every direction, and that a preponderating attraction is directed towards the interior of the mass. If we now imagine a rectilinear canal, the diameter of which is very minute, to exist in the liquid, commencing at some point of the surface in a direction perpendicular to the latter, and extending to a depth equal to the above radius of activity, the molecules contained in this minute canal, in accordance with what we have stated, will be attracted towards the interior of the mass, and the sum of all these actions will constitute a pressure in the same direction. Now, the intensity of this pressure depends upon the curves of the surface at that point at which the minute canal commences. In fact, let us first suppose the surface to be concave, and let us pass a tangent plane through the point in question. All the molecules situated externally to this plane, and which are sufficiently near the minute canal. for the latter to penetrate within their sphere of activity, will evidently attract the line of molecules which it contains from the interior towards the exterior of the mass. If, therefore, we suppressed that portion of the liquid situated externally to the plane, the pressure exerted by the line would be augmented. Hence it follows that the pressure corresponding to a concave surface is less than that which corresponds to a plane surface, and we may conceive that it will be less in proportion as the concavity is more marked. If the surface is convex, the pressure is, on the contrary, greater than when the surface is plane. To render this evident, let us again draw a tangent plane at that point at which the line of molecules commences, and let us imagine for a moment that the space included between the convex surface and this plane is filled with liquid. Let us then consider a molecule, m, of this space sufficiently near, and from this point let fall a perpendicular upon the minute canal. The action of the molecule m upon the portion of the line comprised between the base of the perpendicular and the surface will attract this portion towards the interior of the mass. If afterwards we take a portion of the line equal to the former from the other side of the perpendicular, and commencing at the base of the latter, the action of the molecule m upon this second portion will be equal and opposite to that which it exerted upon the first; so that these two portions conjointly would neither be attracted towards the interior nor the exterior of the mass ; if beyond these two same portions another part of the line is comprised within the sphere of activity of m, this part will evidently be attracted towards the exterior. The definitive action of m upon the line will then be in the latter direction. Hence it follows that all the molecules of the space comprised between the surface and the tangent plane which are sufficiently near the line to exert an effective action upon it, will attract it towards the exterior of the mass. If, then, we suppress this portion of the liquid so as to reproduce the convex surface, the result will be an augmentation of the pressure on the part of the line. Thus the pressure corresponding to a convex surface is greater than that corresponding to a plane surface, and its amount will evidently be greater in proportion as the convexity is more marked.

4. If the surface has a spherical curvature, it may be demonstrated that, rep. resenting the pressure corresponding to a plane surface by P, the radius of the sphere to which the surface belongs by r, and by A a constant, the pressure exerted by a line of molecules, and reduced to unity of the surface, will have the following value:

.............. (1.)

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gs being positive in the case of a convex, and negative in that of a concave surface.

Whatever be the form of the surface, let us imagine two spheres, the radii of which are those of greatest and least curvature at the point under consideration. It is evident that the pressure exerted by the line will be intermediate between those corresponding to these two spheres, and calculation shows that it is exactly their mean. Denoting the two radii in question by R and R', the pressure exerted by the line, referred to the unity of surface, would be

DA/1 1
al R+R)

.. (2.) The radii R and R' are positive when they belong to convex curves, or, in other terms, when they are directed to the interior of the mass; whilst they are uegative when they belong to concave curves, i. e., when they are directed towards the exterior.

5. From the preceding details we can now easily deduce the condition of equilibrium relative to the free surface of the mass.

The pressures exerted by the lines of molecules which commence at the different points of the surface are transmitted to the whole mass ; consequently, for the existence of equilibrium in the latter, all the pressures must be equal to each other. In fact, let us imagine a minute canal running perpendicularly from some point of the surface, and subsequently becoming recurved so as to terminate perpendicularly at a second point of this same surfare, it is evident that equilibrium can only exist in this minute canal when the pressures exerted by the lines which occupy its two extremities are equal; and if this equality exists, equilibrium will necessarily exist also. Now, the pressures exerted by the different lines depend upon the curves of the surface at the point at which they commence; these curves must therefore be such, at the various points of the free surface of the mass, as to determine everywhere the same pressure.

Such is the condition which it was our object to arrive at, and to which in cach case the free surface of the mass must be subject.

The analytical expression of this condition is directly deducible from the general value of the pressure given in the preceding paragraph; we only require to equalize this value to a constant, and, as the quantities P and A are themselves constant, it is in fact sufficient to make

1, 1 RTRT

=C, ......................... (3.) the quantity C being constant for the same figure of equilibrium.

This equation is the same as those which are given by geometricians for capillary surfaces, when, in the latter equations, the quantity representing gravity is supposed to be 0.

R and R' may be replaced by their analytical values;. we are thus led to a complicated differential equation, which only appears susceptible of integration in particular cases. Yet the equation (3) will be useful to us in the above simple form. Now we know that the normal plane sections which correspond to the greatest and the least curvature at the same point of any surface form a right angle with each other. Geometricians have shown, moreover, that if any two other rectangular planes be made to pass through the same normal, the radii of curvature, p and p', corresponding to the two sections thus deter

11. mined, will be such that the quantity - + will be equal to the quantity

1 1 iti Hence the first of these two quantities may be substituted for the second; and, consequently, the equation of equilibrium, in its most general expression, will be

1+1=C, ........................ (4.) in which equation p and p' denote the radii of curvature of any two rectangular sections passing through the same normal.

6. These geometric properties lead to another signification of the equation (4.) We know that unity divided by the radius of curvature corresponding to any point of a curve is the measure of the curvature at this point. The quantity 1 1 -+-, represents, then, the sum of the curvatures of two normal rectangular pp' sections at the point of the surface under consideration. This being admitted, if we imagine that the system of the two planes occupies successively different positions in turning around the same normal, a sum of curvatures 1 1 1 1 1 1 -+ Jothie itt &c., will correspond to each of these positions ; and, according to the property noticed in the preceding paragraph, all these sums will have the same value. Consequently, if we add them together, and let n denote the number of positions of the system of the two planes, the total sum will be equal to n times the value of one of the partial sums,

11 1 or ton ( - + - Now, this total sum is that of all the curvatures

1 1 1 1

, &c., in number 2n, corresponding to all the sections determined by the two planes. If, then, we divide the above equivalent quantity by 2n, the result - C + will represent the mean of all these curvatures. Now, as this result is independent of the value of n, or of the number of positions occupied by the system of the two planes, it will be equally true if we supposo

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this number to be infinitely great, or, in other words, if the successive positions of the system of the two planes are infinitely approximated, and consequently if this same system turns around the normal in such a manner as to determine all the curvatures which belong to the surface around the point in question. The quantity

represents, then, the mean of all the curvatures of the surface at the same point, or the mean curvature at this point. Now if, in passing from one point of the surface to another, the quantity -- + retains

the same value, i. e., "if for the whole surface we have -- +-=C, this surface is such that its mean curvature is constant.

Considered in this purely mathematical point of view, the equation (4) has formed the object of the researches of several geometricians, and we shall profit by these researches in the subscquent parts of this memoir.

Thus our liquid surfaces should satisfy this condition, that the mean curve must be the same everywhere. We can understand that if this occurs, the mean effect of the curvatures at each point upon the pressure corresponding to this point also remains the same, and that this gives rise to equilibrium. Hence we now see more clearly the nature of the surfaces we shall have to consider, and why they constitute surfaces of equilibrium.

6*. We must now call attention to an immediate consequence of the theoretical principles which have led us to the general condition of equilibrium. According to these principles, each of the lines of molecules exerting upon the mass the pressures upon which its form depends, commences at the surface and terminates at a depth equal to the radius of the sensible activity of the molecular attraction, so that these lines collectively constitute a superficial layer, the thickness of which is equal to the radius itself, and we know that this is of extreme minuteness. It results from this that the formative forces exerted by the liquid upon itself emanate solely from an excessively thin superficial layer. We shall denominate this consequence the principle of the superficial layer.

7. A spherical surface evidently satisfies the condition of equilibrium, because all the curvatures in it are the same at each point; also when our mass is perfectly free, i. e., when it is not adherent to any solid which obliges its surface to assume some other curve, it in fact takes the form of the sphere, as shown in the preceding memoir.

8. Before proceeding further, we ought to elucidate one point of great importance in regard to the experimental part of our investigations. The liquid mass in our experiments being immersed in another liquid, the question may be asked whether the molecular actions exerted by the latter exert no infinence upon the figure produced; or, in other words, whether the figure of equilibrium of a liquid mass adherent to a solid system, and withdrawn from the action of gravity by its immersion in another liquid of the same density as itself, is exactly the same as if the mass adherent to the solid system were really de. prived of gravity and were placed in racuo. Now, we shall show that this really is the case. The molecular actions resulting from the presence of the surrounding liquid are of two kinds, viz., those resulting from the attraction of this liquid for itself, and those resulting from the mutual attraction of the two liquids. Let us first consider the former, imagining for an instant that the others do not exist. The surrounding liquid being applied to the free surface of the immersed mass, the former presents in intaglio the samc figure as the latter mass presents in relief. Those molecules of this same liquid which are near the common surface of the two media must then exert pressures of the same

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