THE FIGURES OF EQUILIBRIUM OF A LIQUID MASS 2G1 



With this view, let us remark that the small mass of the junction which pre- 

 vails along the entire common edge of these angles, and which was spoken of 

 in the preceding paragraph, must of itself have its equilibrium of figure. Now, 

 as it has three surfaces it is necessary that the curvatures of these should have 

 to one another a ratio which permits this equilibrium. The small mass has, 

 longitudinally, curvatures which are not of the same direction for its three sur- 

 faces, biit in consequence of its extreme tenuity in thickness, its transverse cur- 

 vatures are enormous, relatively to the longitudinal curvatures in question, so 

 that the influence of the latter may be overlooked in view of that of the former. 

 On the other hand, the two small surfaces which face the interior of the sys- 

 tem are pressed upon by the respective elasticities of the two portions of im- 

 prisoned air, elasticities which are generally unequal, and which exceed the 

 atmospheric pressure, while the small surface which faces the exterior is only 

 subjected to this latter pressure ; but as the differences between these three actions 

 of the air result from the curvatures of the two films, curvatures which are ex- 

 tremely feeble compared with the transverse curvatures of the small mass, the 

 influence of these differences may also be overlooked by the side of that of the 

 transverse curvatures in question. Hence it is evidently requisite, for the equi- 

 librium of the small mass, and consequently for that of the whole system, that 

 if we conceive this small mass cut by a plane perpendicular to its axis, the 

 three concave arcs which will limit the section shall be closely identical. Now, 

 from this near identity it necessarily results that the two films and the partition 

 terminate at the small mass under angles either strictly equal, or very nearly 

 so — angles, consequently, each of 120°, or which will diff'er from this value by 

 an unappreciable quantity.* We shall presently see this result and those of 

 the preceding paragraph verified by experiment. 



§ 9. If the bubble of air which gives rise to the second film (§ 7) arrives at 

 the surface of the liquid sufficiently far from the first film for the spherical caps 

 to be complete and isolated, or, what amounts to the same thing, if we deposit 

 on the liquid two bubbles at a suitable distance from each other, the two liquid 

 caps, drawn by capillary action in the same way with light floating bodies, will 

 by degrees approach ixntil they touch one another. To understand what would 

 then happen, let us recall a fact manifested by full spheres of oil in the interior 

 of my alcoholic mixture. When two such spheres have come in contact, the 

 system which they form is not in a state of equilibrium ; the two spheres unite 

 to form but a single one, and the reason of this is easily apprehended. This 

 contact cannot take place at one sole point ; it must necessarily' occur through- 

 out a small surface, so that the two masses constitute in reality only one ; but 

 since this is finite and entirely free, and is a system of revolution, the sole figure 

 of equilibrium that it can assume (4th series, §§ 2 and 38) is that of a single 

 sphere. Now, it is visible that the same thing must have a tendency to occur 

 in regard to our two spherical laminar caps when the two small annular masses 

 which exist along their bases (§ 3) have united at the place of their contact ; 

 that is to say, the two caps will tend to form but a single one ; but in order that 

 this tendency may have its full efiect, it would be necessary that the two liquid 

 films should open at the place of contact ; and as cohesion resists this, we per- 

 ceive that the opening will be replaced by a partition, and that the same co- 

 ordination will obtain here as in the system of the two preceding paragraphs. 

 These results, also, will be verified by experiment. 



§ 10. We have seen (§ 7) that the radius r of the partition is determined, 

 when we know the radii p and (>' of the two films, by considering the relative 

 value of the pressures respectively exerted by these three portions of spherical 

 caps on the two quantities of included air. On the other hand, the considera- 



* By considering liquid films as stretched membranes, {^ 2) we should equally arrive at 

 the equality of the angles between three films which join one another by. the same liquid 

 edge. 



