Stability of thin Films of Liquids. 345 



the prism the value 



2r.\/3 = 2rx 1-732. 



This value, which I had arrived at hy neglecting the very slight 

 curvature of the films composing the system, is not quite accu- 

 rate ; the true value is 



2 /r '^ =2rx 1-618. 



\/3-\/5 



2. Here again, for it to be possible to produce a laminar 

 pentagonal prism at the middle of the figure, the skeleton must 

 not have too great a height in comparison with the dimensions 

 of the base. Under these conditions, and provided also that 

 the height exceeds the typical height, the laminar prism will 

 have faces of spherical curvature, if it has a certain determinate 

 volume relatively to the size of the skeleton. This laminar 

 prism will then have a height equal to 2T853 times the chord of 

 the side of its base, and the radius of the sphere to which its 

 faces belong will be equal to 23*072 times the same chord. 



3. When the height of the skeleton is only very little less 

 than the typical height, it is possible, by certain manipulations, 

 to obtain at pleasure, in the system of films resulting from a 

 single immersion, either a very small pentagonal film at the 

 middle of the figure, or the other system — that is to say, the one 

 corresponding to the typical height or to a greater height. 



V . System of the Regular Dodecahedron, 



The faces of the laminar dodecahedron produced at the middle 

 of the system have a spherical curvature ; the radius of the spheres 

 to which they belong is equal to about 23 times the chord of 

 their side. 



In all the above systems with an internal laminar polyhe- 

 dron, when the faces of the latter have a spherical curvature, all 

 the films which extend from the edges of the skeleton to the 

 edges of this laminar polyhedron are plane; and consequently 

 all the liquid edges which join the angles of the skeleton with 

 those of the same polyhedron are straight. 



VI. Systems of two Special Polyhedra. 



These polyhedra, as well as their systems of films, would 

 require for their proper comprehension either a long description 

 or figures. I will therefore confine myself to saying only a few 

 words about them. The first is composed of two equal and 

 parallel squares, one of which is turned through a quarter of a 

 revolution in relation to the other, and which are connected 

 together by eight half-regular pentagons. In each of these pen- 



