132 Prof. J. Plateau on the Figures of Equilibrium 



than three films, forming equal angles with each other, would 

 meet in each liquid edge ; but eight liquid edges would come 

 together at the central point. Now a cubic skeleton never gives 

 this figure : a thirteenth film, quadrangular in form, takes the 

 place of the central point, and at each of its four corners two of 

 the oblique edges come together ; so that at none of them more 

 than four liquid edges meet each other. Here again, then, as in 

 the previous case, the system first supposed, although a figure of 

 equilibrium, and more simple than the one which is actually 

 formed, would be unstable. 



It remained to verify the equality of the angles formed at the 

 junction of three films, or of four edges. I show, in the first place, 

 that these equalities are mutual consequences of each other ; so 

 that it is sufficient to prove one of them — the second, for example. 

 Por this purpose I investigate the common value of the angles 

 contained by the liquid edges, and find it to be 1 09° and a frac- 

 tion ; I then select from my systems of films those in which all 

 the films are plane, and consequently all the edges straight lines, 

 namely, those formed by the regular tetrahedron, by the right 

 triangular prism with equilateral bases, and by the regular octa- 

 hedron. In the first system, which has only four liquid edges 

 going from the summits to the centre and joining the films 

 which start from the solid wires, the equality of the angles in 

 question is self-evident by reason of the perfect symmetry of the 

 entire figure. 



Among the liquid edges of the second system, that, namely, 

 formed by the triangular prism, there is one which extends 

 between two liquid points, and whose length can be easily mea- 

 sured with the cathetometer. Now by calculating from the 

 previously measured dimensions of my wire skeleton, and from 

 the theoretical value above given for the angles formed by liquid 

 edges, I had found that the length of the edge in question 

 ought to be 42*44 millims., while measurement with the cathe- 

 tometer gave 42*37 millims., the difference between which and 

 the calculated length may be disregarded. 



The third system, that formed by the regular octahedron, 

 contains six equal rectangles, a diagonal of any of which can be 

 easily measured with the cathetometer. In this case I found 

 for the length of this diagonal, by calculating as before from the 

 known size of the wire skeleton and the theoretical value of the 

 angles formed by the liquid edges, 23*16 millims., while direct 

 measurement gave 23*14 millims., the difference being even 

 smaller than in the last case. 



The systems of films formed by other skeletons, that is to say, 

 such as contain certain curved films and consequently curvilinear 

 edges, also afford, though not so distinctly, a verification of the 



