208 On the Conditions of Stability of thin Films ofLiq ids. 



are spherical polygons having angles of 120° exclusively. M. 

 Lamarle points out in the first place that these polygons can 

 only be either triangles, four-sided figures, or pentagons, and 

 hence he derives one analytical relation between the respective 

 numbers of these several kinds of polygons and the total number 

 of films ; a second relation he finds in the fact that the sum of 

 the surfaces of all the polygons must make up the whole surface 

 of the sphere ; lastly, these polygons must all be in simple juxta- 

 position, without encroaching upon each other in some places 

 and leaving empty spaces at other places. From these three 

 conditions, M. Lamarle deduces that there are only seven possi- 

 ble combinations of films starting from the same point, and 

 joined together three by three under equal angles. 



If, in each of these combinations, the sides of the spherical 

 polygons are replaced by their chords, we have the edges of a 

 complete polyhedron, and the seven polyhedrons thus formed 

 are, — the regular tetrahedron ; the right triangular prism with 

 equilateral base, the height being in a determinate ratio to the 

 length of the side of the base ; the cube ; the right pentagonal 

 prism with regular base, the height being in a determinate ratio 

 to the length of the side of the base ; two peculiar polyhedrons 

 made up of four- sided figures and pentagons; and, lastly, the 

 regular dodecahedron. In these polyhedrons the numbers of 

 liquid edges are 4, 6, 8, 10, 12, 16, and 20 respectively. 



Now M. Lamarle proves that for each of these systems of 

 films, except that of the regular tetrahedron, it is possible to 

 conceive such a mode of displacement that, from its commence- 

 ment up to a certain limit, a diminution of the sum of the areas 

 of the films will result from it : the system of the regular tetra- 

 hedron, in which not more than four liquid edges meet at the 

 same liquid point, and make equal angles with each other, is 

 therefore the only one of these which can be stable. Hence, 

 when the films are plane, the liquid edges which meet at any one 

 liquid point are necessarily four in number and make equal 

 angles with each other. Lastly, M, Lamarle shows that the same 

 conclusion applies to curved films, and, by consequence, to curved 

 edges; there is, in fact, no limit to the smafln ess - which the 

 above-mentioned sphere may be conceived to have, and hence we 

 are free to conceive of it as so minute that the portions of the 

 films contained within it may be regarded as plane. 



My second law — namely, that in every stable system of liquid 

 films the number of liquid edges meeting in any one liquid point 

 is always four, and that they make equal angles with each other 

 — is thus demonstrated byM. Lamarle as completely as the first, 

 and like it, is deduced from the principle of the minimum. 



It may be added that the modes of displacement supposed by 



