282 WITHDEAAYN FEOM THE ACTION OF GEAVITY. 



duriBg, and have increased in proportion to the elevation of the frame; now 

 this is impossible, for the air which must fill it would have had no passage by 

 which it could have entered ; for the same reason the system, at this period of 

 its generation, could present no space closed in part by films and in part by the 

 surface of the liquid; in fine, when the rapid modification is effected, the film 

 or films which then ascend into the system finding no space of the second kind 

 to complete its laminar closure, the complete system will necessarily satisfy the 

 law in question. 



The last law is the following: when the conditions of other laws can be 

 fulfilled by plane surfaces, the films take that form ; when this cannot be, all 

 the films or several of them become more or less curved, but always in such 

 manner as to constitute surfaces of mean curvature null. The first part of this 

 law is a matter rather of evidence than demonstration ; the plane being the 

 most simple surface, nature, which ahvays proceeds by the least complicated 

 methods, will not unnecessarily give curved forms to the films. The second 

 part of the law follows from the first, and I have already made application of it 

 in what 1 have previously stated respecting the laminar systems of prismatic 

 frames. 



It is proper that a reason should be here given for the special fact that the 

 laminar system of the regular octahedron is formed of curved films when it is 

 obtained with oil in the alcoholic liquid. We have seen (§ 16) that in this 

 mode of producing laminar systems the thickness of small remaining masses 

 has great influence. Now, in the octahedral frame, when, by the gradual with- 

 drawal of oil, the point has been reached at which the system is spontaneously 

 modified, since the masses of junction have still a considerable thickness', and 

 the oil which composes them accumulates, in the definitive system, around the 

 points where four liquid edges would terminate, so as to form masses much 

 thicker than the edges in question, and since, in fine, several of these edges are 

 sufficiently short for the masses which occupy their extremities to be in commu- 

 nication of curvature with one another, it will be readily conceived that there 

 must result from thence an influence on the form of the edges and of the films ; 

 and it cannot be doubted, that if we might, without occasioning the rupture of 

 the system, sufiiciently reduce the thickness of the masses in question, all the ' 

 films would become plane. 



§ 30. To return to the systems of the prismatic frames : Besides the facts al- 

 ready stated, these systems have presented others equally curious, which I here 

 propose to notice. 



The system obtained with the pentagonal frame of Fig. 30 is, as was seen, 

 (§ 20,) composed of films obviously plane. Now, if we consider the oblique 

 films which proceed from two homolos^ous sides of the two bases to unite at one 

 of the sides of the central pentagonal film, and recollect that these two oblique 

 films must form between them an angle of 120°, it will be evidently seen that, 

 for bases of given dimensions, an augmentation in the height of the prism in- 

 volves a diminution in the extent of the central pentagonal film, and that there 

 is a limit of height beyond which the existence of this film is impossible. It 

 will be found, without difiiculy, that the limit in question corresponds to the 

 case in which the ratio between the height of the prism and the diameter of the 

 circle which might be inscribed at the base would be equal to V 3, that is to 

 say, to 1.732. 



It might naturally be asked. What becomes of the laminar system wh>=-n this 

 limit is overpassed ? In order to know this, I had a frame constructed in 

 which the height was about 2J times the diameter of the inscribed circle, and it 

 yielded me a singular resrJt. When it is withdrawn from th^ glyceric liquid, 

 as the lateral edges are of ordinary iron wire, all the lat-^ral faces are at first oc- 

 cupied by plane films, and after complete emergence, a plane film is f,>rmed ali-o 

 in the inferior base, then ascends among the others, forming a pentagon which 



