THE FIGURES OF EQUILIBRIUM OF A LIQUID MASS 

 WITHDRAWN FROM THE ACTION OF GRAVITY. 



BY J. PLATEAU. 



[Continued from page 435, Report for 1865.] 

 SIXTH SERIES. 



Theory of liquid jilms. Laics of films. Constitution of froth. Generation of 

 films. Production of films hy frame-work. 



§ 1. Let us prosecute the study of liquid films commenced in the second and 

 continued in the fifth series. In § 2S of the second series I regarded the gene- 

 ration of films as being due to a tendency towards a new figure of equilibrium. 

 I now abandon that opinion, at least in part ; though still believing that in the case 

 where gravity does not intervene, the system, when the films are once produced, 

 tends towards the new state of equilibrium which I have indicated. I consider 

 the formation of the films itself as a result of the cohesion and viscosity of the liquid. 

 Let us examine the matter somewhat more closely. We first take a very simple 

 example, that of the film in shape of a spherical cap developed on the surface of 

 a liquid by a bubble of air which has ascended from the interior of the liquid. 

 Let U3 consider this bubble at the moment when it is within only a few milli- 

 metres of the surface, (Fig. 10. *) In order that its summit 

 shall traverse the distance m n which intervenes, the liquid 

 molecules situated around this little right line must neces- 

 sarily be driven toward every azimuth at once, in such man- 

 ner that these molecules shall undergo relative displace- 

 ments. Let us suppose, for simplification, that the ascen- 

 sional movement of the bubble of air is uniform, so that, in equal intervals of time, 

 the bubble shall propel, between itself and the upper surface of the liquid, equal 

 quantities of this liquid. Let us further suppose that the liquid has no viscosity. 

 Then, in proportion as the distance m n diminishes, the portions of liquid pro- 

 pelled during the above intervals of time will respectively acquire velocities con- 

 stantly increasing, since they must accomplish their movements in spaces more 

 and more straitened ; thus tlie relative displacements of the liquid molecules are 

 so much the more rapid as the summit of the bubble is nearer attaining the sur- 

 face. I have supposed the ascensional movement of this bubble to be uniform ; 

 but as it is in reality accelerated, its acceleration will contribute to increase the 

 rapidity of the relative displacements in question. 



Now, we know that viscosity opposes to the relative displacements of the 

 molecules of liquids a resistance which increases considerably with the velocity 

 of those displacements. If, then, in order to pass to the actual case, we restore 

 to our liquid its viscosity, the resistance to the lateral translation of the liquid 

 molecules around m n Avill continue to augment in proportion as that right line 

 diminishes in length, and will become very great when this line shall have be- 

 come very short. Hence it necessarily results that when the summit of the 

 bubble has arrived near the surface, the portion of liquid which still separates it 

 therefrom cannot disperse itself with a rapidity equal to that of the ascensional 

 movement of this summit ; and, in order that the air which constitutes the bubble 

 shall continue to ascend and pass beyond the level of the liquid, it is evidently- 

 necessary either that the liquid shall be rent asunder, or be lifted up. But it 

 can be no longer doubted, after the iugenious researches of MM. Donnyt and 



* lu this figure the bubble is repveseuted a little flattened vertically, which does in fact 

 occur by reason of the resistance of the liquid. 



+ Mcinoircs de I' Academic, tome XVII des Mtmoircs des Savants £trangers. The memoir 

 of M. Donny was presented to the Academy in December, 1843. 



