270 



WITHDRAWN FROM THE ACTION OF GRAVITY. 



when the four right lines terminate at the same point under equal angles, each 

 of these angles has for its cosine — ^ ; "^e find thence that this angle is 109° 28' 

 16", that is to say, very nearly 109 degrees and a half. 



Such, then, in laminar assemblages, is the value of the angles under which 

 four liquid edges terminate at a liquid point. Each of these edges evidently 

 unites three films, and, at least in the case of rectilinear edges, it necessarily 

 results from the symmetry of ihe assemblage that these films must form be- 

 tween them equal angles. 



Conversely, if the films which unite three by three along each of the four 

 liquid edges form between them equal angles, these four edges also necessarily 

 form between them equal angles. In effect, supposing the films to be plane, 

 the system evidently constitutes an assemblage of four trihedral angles, in each 

 of which the three dihedral angles are equal ; now, in virtue of a known 

 theorem, this equality implies, for any one of these trihedral angles, that of 

 the plane angles ; but each of these last being common to two of the trihedral 

 angles, it follows that, in the system, all the plane angles, that is to say, ttte 

 angles which the four edges form between them, are equal. If the films, and 

 consequently also the edges, are curved, it is clear that we may, at the point 

 common to these edges, replace the films by their tangential planes and the 

 edges by their tangents, which will be intersections of these planes; whence it 

 results that, in all cases, the equality of the angles under which the films termi- 

 nate at their common point, and the equality of the angles under which three 

 films unite at the same edge, are, at least in the immediate vicinity of the point 

 in question, necessary consequences one of the other. 



§ 18. Let us apply this principle to the laminar assemblage, the base of which 

 is represented by Fig. 17. The graphic construction of this base is founded 

 on the equality of the angles under which the films and the partitions terminate 

 three by three at their common edges, and has been found (§ 15) to be fully 

 verified by experiment ; the four edges of the system meet then necessarily at 

 their common point under equal angles. The principle evidently applies alike 

 to the two points of the assemblage of Fig. 19, at which four edges terminate. 



§ 19. We will recur now to the systems formed in frames of iron wire. Those 

 of them which are composed of plane films enable us to verify this equality of 

 the angles between the three edges which terminate at the same liquid point, 

 for we can generally apply direct measurement; they consequently enable 

 us also (§ 17) to verify the equality of the angles between the three films which 

 terminate at the same liquid edge. The systems of this nature which were 

 realized in my frames are three, namely : that of the regular tetrahedron, that of 

 the equilateral triangular prism, and that of the regular octahedron. I here re- 

 produce the drawings (Figs. 24, 25; and 26.) It must not be forgotten, for the 



Fig. 25 



Fisr. 24 



Vig. 26 



understanding of these delineations and the greater part of those which follow, 

 that, in the laminar systems which they represent, a film proceeds from each of 



