Experimental and Theoretical Statics of Liquids Subject to Molecular

EXPERIMENTAL AND THEORETICAL STATICS OF LIQUIDS SUBJECT TO MOLECULAR FORCES ONLY, BY J. PLATEAU Professor of the University of Ghent, Member of the Academy of Belgium, Correspondent of the Institute of France, of the Royal Society of London, of the Berlin Academy, etc. VOLUME ONE. PARIS, LONDON, GAUTHIER-VILLARS, TRÜBNER and Co, quai des Augustine, 55. Ludgate Hill, 57 and 59. GAND ET LEIPZIG: F. CLEMM. 1873. TO MONSIEUR A. QUETELET DIRECTOR OF THE OBSERVATORY OF BRUSSELS , ETC. You, who were one of the active promoters of the intellectual regeneration of Bel- gium, and whose work contributed so much to the illustriousness of this country; you, who guided my first steps in the career of science, and who taught me, by your example, to excite among young people the love of research, who did not cease being for me a devoted friend, allow me to dedicate this work to you, in testimony of recognition and constant affection. J. PLATEAU. This work is formed primarily of the contents of eleven Series of Memoires that I published, from 1843 to 1868, in the Memoires of the Academy of Belgium, under the title: Research experimental and theoretical on the equilibrium shapes of a liquid mass without gravity1. But the revision of the whole of these Series enabled me to follow, in the current work, an order a little more methodical, to rectify some passages, and to fill gaps, especially in the histories; until the end of 1869, I could extend I did of them two those new, one concerning the surface viscosity of liquids, the other with the constitution of liquid streams; I introduced several additions which appear worthy to me of interest, such as the theory of exploding laminar bubbles; on the other hand, I removed, as concerning purely dynamic phenomena, the majority of the results of my first Series; finally I have indicated, in a last paragraph, the titles of the articles which appeared after 1869 on subjects in connection with those that I treat. 1The first Series carries the a little different title: Phenomena shown by a liquid mass free and withdrawn from the action of gravity, first part. J. Plateau: Statics of Liquids... Volume I, Chapter I 4 STATICS OF LIQUIDS SUBJECT TO SOLELY MOLECULAR FORCES CHAPTER ONE. Preliminary concepts. General conditions which must be satisfied, in the state of equilibrium, free face of a liquid mass supposed without gravity. – Freeing a liquid from the action of gravity, leaving it free, either on all its surface, or on part of it to obey its own molecular attractions. Use of the first process: liquid sphere; experimental checks of the principles of the theory of capillary action. – Shapes bounded by plane surfaces: liquid polyhedra. § 1. It is known that, in ordinary circumstances, the free face of a liquid, in rest, when it has a rather large extent, is plane and horizontal, except towards its edges; but it is known, at the same time, that this form plane and horizontal is an effect of gravity, and that, between fixed boundaries, the free face can take very different forms, because then the action of gravity becomes of the same order as the molecular actions, for example, at the top of a liquid column raised or lowered in a capillary tube. One understands, accordingly, that if the liquids were deprived of gravity, their free face could have, with the at-rest state, and to an unspecified extent, forms very different than the flattened horizontal form, and it is, indeed, at what one arrives by calculation, as we will see. It results from the work of Laplace on capillary action, that the liquids exert on themselves, under the terms of mutual attraction their molecules, a normal pressure on the surface at each point; that this pressure can be viewed as emanating from a surface layer having a thickness equal to the sensitive range of activity of the molecular attraction and is, consequently, extremely small; finally that this same pressure depends on the curvature of surface at the point considered, and that if one indicates by P the pressure, the force per unit area exerted on a plane surface, by A a constant, and by R and R0 the principal radii of curvature, i.e. those of maximum and minimum curvature of surface at the same point, the pressure corresponding to this point has as a value, always with respect to the unit of area, µ ¶ A 1 1 P + + (1) 2 R R0 In this formula, the radii R and R0 are positive when they belong to convex curves, or when they are directed inside the mass, and they are negative when they belong to concave curves, when they are directed outside; as for the quantities P and A, they change only with the nature of the liquid. That being so, let us conceive a liquid mass in the absence of gravity; the pressures of the various points, its surface layer transmitting one to the others by the interior liquid, it will be necessary, so that there is equilibrium, that all these pressures are equal to each other; it will thus be necessary that the expression above has a constant value with regard to all the surface; and as the quantities P and A are themselves constant, one will express the general condition of equilibrium as simply: 1 1 + = C; (2) R R0 the quantity C being constant, and being able to be positive, negative or null. Let us specify more for people who are not very familiar with the theory of surfaces. At a point taken arbitrarily on an unspecified surface, let us imagine a normal there, and English translation °c 2005 Kenneth A. Brakke J. Plateau: Statics of Liquids... Volume I, Chapter I 5 I pass a plane through this normal; it will cut surface according to a certain line, and the radius of curvature of the latter, at the foot of the normal, will be one of those of surface in this point. Now let us turn the plan around the normal; in each one of its positions, it will determine a new line, and, consequently, a new radius of curvature; however, among all these radii, there will be, in general, one which will have a greater curvature, and one with a curvature lower than all the others; they are our two principal radii R and R0, and the quotients 1=R and 1=R0 are the two principal curvatures at the point considered. Let us add that, according to a result curious from analysis, cross-sections to which these principal curves belong are always at right angle one on the other. Thus, under the terms of formula [2], a liquid mass supposed without gravity must assume a shape such that the algebraic sum of the two principal curvatures has the same value at all the points of its surface. Each one of these principal curvatures is besides, according to what precedes, positive or negative according to whether it is convex or concave. Note furthermore that when there is, at the same point of surface, convex curveat- uer and concave curvature, the principal curvature of this last species is lowest alge- braically; it is thus that which has the greatest negative value, and consequently the smallest negative radius. Let us quote finally a second curious result from analysis: If one considers two unspecified sections at right angles containing the same normal, the radii of curvature which correspond to them, radii which we will name r and r0, will be such that the 1 1 1 1 quantity r + r0 will be equal to the quantity R + R0 . It results from it that one can substitute the first of these quantities for the second, and that, consequently, the equi- 1 1 librium equation in its greater general formulation will be r + r0 = C. If all these concepts still leave some darkness in the spirit of the reader, they will be cleared up by the many applications which we will make. Only Poisson, I think, before my research, has touched directly on the general question of the shapes of equilibrium of a liquid mass without gravity, arrived2 in 1828, at formula [2]: after having treated in all its general formulation the problem of the interior equilibrium of a fluid under the action of its own molecular forces and of exernal forces given, he seeks the equilibrium equations relating to the common surface of two fluids in contact; then he supposes that one of them is a liquid, and the other a gas exerting a uniform pressure; finally he removes the external ones, except the pressure, and thus finds, with regard to the surface of the liquid mass, the formula concerned. It shows then that there is only one shape of possible equilibrium among the spheroids of revolution very little different from a sphere, and that this shape is that of the sphere itself. § 2. It is obvious, indeed, that a spherical surface satisfies formula [2] since all its radii of curvature are equal. It is in the same way obvious for the plane, for which all the radii of curvature are infinite, which makes everywhere the first member of the formula equal to zero, only in this case the constant C is zero. I cite these two surfaces because they are immediately obvious; but, as we will learn hereafter, they are far from being alone. The constancy of the algebric sum of the two principal curves is not the only inter- pretation which formula [2] has: 1 1 In the first place, geometricians know that the quantity r + r0 is proportional to the average curvature, i.e.

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