846 On the Conditions of Stability of thin Films of Liquids. 



tagons, the plane containing the side common to one of the 

 squares, and the two adjacent sides, makes a reentering angle of 

 about 168° with the plane which contains the other two sides. 

 This skeleton affords five different systems of films which can be 

 obtained at pleasure; and, in addition, eight different laminar 

 polyhedra can be produced at the middle of them. 



The second polyhedric skeleton is still more difficult to de- 

 scribe. It consists of four rectangles and four half-regular 

 pentagons; each of these pentagons is also formed of two parts, 

 of which the planes make a reentering angle with each other. 

 This skeleton gives two different systems of films ; and four 

 internal laminar polyhedra can be produced upon it. 



M. Lamarle then leaves the typical polyhedra, and developes 

 some general considerations upon the systems of films belonging 

 to other skeletons. 



He takes, as an example for investigation in detail, the regular 

 octahedron. By aid of an artifice of reasoning, he arrives a priori, 

 and solely by theory, at five different systems of films, which 

 are very probably the only systems possible in the skeleton in 

 question. The first is formed entirely of plane films, and its 

 centre is occupied by the common summit of six spearhead- 

 shaped quadrilateral films, having their sharp points respectively 

 at the six summits of the skeleton ; the second is formed by 

 curved films, and has an hexagonal film at the middle. I had 

 previously indicated these two systems : the three new systems 

 found by M. Lamarle, and which are formed of curved films, 

 contain at the middle, one of them a pentagonal film, another a 

 trapezoid four-sided film, and the last an equilateral four-sided 

 film. M. Lamarle is able to obtain these systems at pleasure, 

 and also to cause them to change at pleasure one into another. 



He dwells specially upon the first of these five systems — that 

 is to say, upon the one which has plane films only — and shows 

 that the dimensions of its parts are related to each other and to 

 the dimensions of the skeleton in very simple numerical ratios. 



As to the internal laminar polyhedra, M. Lamarle obtains 

 fifteen different ones in this same system. He has succeeded in 

 deducing all of them from the theory, and in effecting their 

 derivation, both theoretically and experimentally, from one 

 another. 



Lastly, M. Lamarle discusses, still in reference to the same 

 skeleton, systems of a kind which he calls incomplete, inasmuch 

 as they are formed by breaking certain films in a system already 

 formed, and they consequently present vacant spaces. When 

 one of the above-mentioned laminar polyhedra is obtained in the 

 middle of a system of plane films, six small triangular films are 

 formed abutting on this polyhedron ; and it is by breaking either 



