THE FIGURES OF EQUILICKIUM OF A LIQUID MASS 279 



is cstablisliocl when the central film has attained half the height of the frame, 

 because then the whole is symmetrical ; we have thus the system spoken of at 

 the end of § 20. 



It will be understood that in this system the oblique films, which direct 

 themselves towards the central film, can only form, two by two, angles of 120^ 

 on the condition of being convex, in the direction of their breadth, towards the 

 interior of the figure, which again implies the re-entering curvature of the sides 

 of the central film ; but, by reason of the necessity of a mean curvature null, 

 this convexity requires that the films in question should be concave in the direc- 

 tion of their height ; in this way all the laws are satisfied, and in the system 

 realized these two opposite curvatures are verified in the oblique films. As to 

 the central film and the films which proceed from the lateral solid edges, these 

 are necessarily plane, because of their sj'uimetrical position in relation to the 

 others. In tiiis system it is to be remarked that the oblique liquid edges which 

 pass to the summits of the central film do not proceed exactly from the sum- 

 mits of the two bases, but from points situated at a small distance from these 

 last summits, on the lateral solid edges ; of this we shall see the reason in the 

 sequel. 



§ 27. We pass now to the cases of the pentagonal prism with fine lateral 

 edges, the quadrangular prism or cube, and the triangular prism, cases in Avhich 

 the films proceeding from the edges of the supei'ior base take, as has been seen, a 

 re-entering direction when that base emerges from the liquid. When these films 

 begin to show themselves the small masses raised at their lower borders will 

 necessarily describe on the surface of the liquid a polygon of the same number 

 of sides with the solid base — that is to say, a pentagon, a quadrilateral, or a trian- 

 gle, according to the frame employed. But, as the films in question are vertical 

 at their lower part and are joined under angles of 120°, it is necessary that the 

 sides of the above polygons should also form between them angles of 120°, 

 which evidently requires that they be convex towards the exterior ; these sides 

 must, moreover, partake the horizontal curvature of the films, a curvature which 

 we know to be concave towards the interior of the figure, and consequently 

 convex towards the exterior. This convexity of the sides of our polygons will 

 be slight for the pentagon, more decided for the quadrilateral, and still more so 

 for the triangle. All this likewise is found to be verified by experiment. 



These first facts being established, we will follow separately the develojD- 

 ment of the laminar system in each of the three frames, in proportion as it is 

 more and more elevated. 



With the pentagonal frame, the curvilinear pentagon, which takes form at the 

 surface of the liquid, at first undergoes a slight contraction, but again enlarges 

 when a sufficiently considerable part of the height of the frame is out of the 

 liquid ; and when the inferior base attains the surface the re-entering films at- 

 tach themselves by their lower borders on the sides of that base. These films, 

 however, do not then occupy the lateral faces of the prism; they slightly re- 

 enter, in the direction of their height, towards the interior of the frame, so as to 

 be united, two by two, by liquid edges, each of which constitutes an arc of feeble 

 curvature resting by its two extremities on those of a lateral solid edge. Finally, 

 some short time after the emergence of the inferior base, the phenomena are com- 

 pleted as in the frame of the preceding paragraph — that is to say, the film pro- 

 duced between the base in question and the liquid occupies this base under a 

 pLme form, and then ascends rapidly, drawing to itself, at the same time, the 

 other fihns, so as to give, in fine, the system of Fig. 30. 



With the frame of the quadrangular prism, the curvilinear quadrilateral formed 

 at the surface of the liquid is soon annnUed, and is replaced by a small horizontal 

 liquid edge, from the extrer.-itics of which proceed two descending liquid edges, 

 which continue to diverge from one another ; these three edges limit a vertical plane 

 film parallel to two of the fiices of the prism, and attached by other fihns to the 



