362 Prof. J. Plateau on the Figures of Equilibrium 



always resumes exactly its previous shape. A soap-bubble isolated 

 in the air exhibits equally a permanent and stable form. The 

 sphere, then, has no limit of stability; that is to say, whatever 

 may be the extent, relatively to a complete sphere, of the portion 

 of a sphere actually produced, this portion is necessarily in a 

 state of stable equilibrium. Thus, for instance, a perfectly per- 

 manent double convex lens of oil can be produced in the alcoholic 

 mixture upon a ring of iron wire. 



This result, being independent of the radius, and consequently 

 of the curvature of the sphere, is equally true when the radius 

 becomes infinite, or, in other terms, when the surface of the 

 sphere becomes a plane. Accordingly the plane also has no limit 

 of stability ; that is, it can be produced within a solid outline of 

 any extent without ceasing to be stable, as can be verified by the 

 formation, for instance, of a film of the glycerine-solution within 

 a plane outline of iron wire of any shape and of any size. 



My first experiments upon liquid cylinders proved that such a 

 cylinder is unstable when the ratio of its length to its diameter 

 exceeds a value comprised between 3 and 3*8, which I have called 

 the limit of stability of the cylinder. I had arrived at this result 

 by means of cylinders of oil formed within the alcoholic mixture 

 between two solid ring|or disks. In the present Series I attack 

 the theoretical investigation of the precise value of the limit in 

 question. Suppose one of our cylinders of oil to have been pro- 

 duced between two disks, and short enough to be stable. If, 

 by gently pushing the oil towards one of the disks with a glass 

 rod, we cause the artificial formation of a bulging and a constric- 

 tion, and if this modification of the figure does not go beyond a 

 certain limit, the mass, when afterwards left to itself, resumes 

 spontaneously the original cylindrical form ; but if the change 

 of shape exceeds this limit, it increases spontaneously, and the 

 transformation becomes complete — that is to say, tbe mass sepa- 

 rates into two unequal portions. 



Now, if we were to produce the precise degree of alteration which 

 separates these tendencies to two opposite effects, it is evident 

 that the mass would be indifferent to either one or the other; there 

 would therefore in that case result a condition of equilibrium, 

 although of unstable equilibrium. And since the figure would still 

 be a figure of revolution, made up of an enlargement and a contrac- 

 tion, it would necessarily form a portion of an undulating surface 

 or unduloid (ondulo'ide). In the second place, since this partial 

 unduloid would constitute the degree of alteration at which there 

 would begin a spontaneous tendency to a more profound alteration, 

 it would differ less from the initial form (namely the cylinder) in 

 proportion as the latter was nearer to its limit of stability ; and 

 this is confirmed by experiment. Lastly, when the cylinder is 



