THE' FIGURES OF EQUILIBRIUM OF A LIQUID MASS. 289 



about, but with two vacancies, by breaking, in the system of the octahedron, 

 (Fig. 26,) first the superior quadrilateral, next the film which then replaces the 

 inferior quadrilateral. 



§ 41. We will terminate this series by a remark relative to the persistence of 

 laminar systems. All those of the polyhedral frames are less enduring than the 

 fifjures formed of a single film, such as spherical bubbles or figures of revolution, 

 (oth series, §§14 and 15.) The explanation of this fact rests on the considerations 

 stated in §§ 32 and 33 of the 2d series, in reference to laminar systems realized 

 with oil in the interior of the alcoholic liquid — considerations which I shall here 

 reproduce in a more complete manner. 



It will be remembered that a liquid film adhering to a metallic thread is 

 necessarily joined to this, or rather to the liquid layer which moistens it, by a 

 small mass with transverse curvatures strongly concave. The two surfaces of 

 this small mass exert, therefore, a capillary pressure less than those which cor- 

 respond to the two surfaces of the film, and consequently the excess of pressure 

 of the latter must continually drive the liquid towards the former. Hence a new 

 cause for the progressive attenuation of the film, and consequently of the destruc- 

 tion of the figure. This pause exists in every laminar figure adherent to a thread 

 or to wires ; but in the systems formed in the interior of polyhedral frames, there 

 is added a cause of the same nature, and still more energetic. All these systems, 

 in effect, contain liquid edges, which are united four by four in liquid points, and 

 of which several are attached by their other extremity to solid threads ; now, on 

 these edges, as well as at their extreme points, there are also, we recollect, small 

 surfaces strongly concave. Those which pertain to the edges are, like those 

 which prevail along the wires, only concave in one direction, and consequently 

 iheir action in rendering the films thinner is of the same order ; but, at the points 

 of junction of the liquid edges, and at the points where these edges terminate at 

 tite wires, the small surfaces are concave in every direction, so that their capil- 

 lary suction is about twice more strong, and hence the afflux of liquid towards 

 these same points must be much greater. In the systems in question, the films 

 must therefore become thin much more rapidly, and these systems endure for a 

 shorter time, as experience evinces they do. There is no gradual augmentation 

 of thickness, however, either in the liquid edges or in their poiuts of junction, 

 but this is because the liquid, in proportion as it flows thither, is drawn by 

 giavitation towards the base of the system. The persistence of each of these 

 systems is, indeed, very variable, as may be readily conceived, for, in conse- 

 quence of their complexity, slight accidental causes may act, sometimes more, 

 sometimes less, to induce their destruction ; whence, of all of them, that which 

 in general is longest maintained is the most simple, namely, that of the tetra- 

 hedron. 



I shall recur anew to the laminarsystems in order to consider the theory under 

 a more general point of view. In effect, as I have already shown, (§ 2 and note 

 of § 8,) the liquid films which compose them may be assimilated to stretched 

 membranes, and hence each system will be arranged in such a manner that the 

 sum of the surfaces of all its films shall be a minimum. But I reserve this 

 subject for another series. 



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