THE FIGURES OF EQUILIBRIUM OF A LIQUID MASS 271 



the solid edges, directing itself towards the interior of the frame, and attached 

 to the other films by the liquid edges which the drawings represent in fine lines 

 In the first the equality in question is evident, the whole being perfectly sym 

 metrical in every direction ; it will further be seen that the four liquid edges 

 occupy precisely the positions of the right lines oa, oh, oc, ad, of Fig. 23. In 

 the second, as the angle between the verticle edge mn, and the oblique edge np, 

 for example, must have — ^ for cosine, it follows that the vertical height of 

 the point n, above the plane of the base, is the third of np, whence it is easily 

 deduced that, designating by a the edge^g- of the base, this vertical height is 



equal to . ■■ Accordingly, if h designate the lateral edge ]}r of the frame, 



we have mn z^ b j—. In the frame which I employed, the edge h was 



70™'".70, and the edge a 69""".23 ; * by substituting these numbers in the above 

 formula we find mn z=. 42""". 44; now the measurement by the cathetometer 

 gave nm ^= 42°'™. 37, a value which differs from the preceding by only 0™"'.07, 

 that is, by less than two thousandths of.the one or the other. 



As to the system of the regular octahedron, it is composed of plane films 

 when obtained with glyceric liquid, as I have already stated in the fifth series, 

 while it is formed of curved films when realized with oil in the interior of the 

 alcoholic liquid (2d series, § 35;) further on I shall indicate the cause of this 

 difference, but at present I shall occupy myself only with the system of plane 

 films. This system is both elegant and remarkable, consisting, as is shown 

 by Fig. 26, of the following parts: 1. Six quadrilaterals elongated in the direc- 

 tion of one of their angles, having each the summit of that angle at one of those 

 of the frame and the summit opposite to the centre of the figures ; in each couple 



* It is here necessary to explain in what manner these valuations have been obtained. A 

 frame of iron wire being incapable of geometric exactness, each of the lateral edges was 

 measured separately and the mean of the results taken ; the same thing was then done in 

 regard to the edges of the bases ; but these measurements were not made directly on the 

 edges, because their terminations were more or less masked by the soldering. The mode 

 adopted was as follows : the frame having been placed vertically, and one of its lateral faces 

 in front of the cathetometer, the distance comprised between the two horizontal edges Avas 

 determined, by taking sight as near the vertical edge of the right as the soldering would per- 

 mit, and then doing the same near the vertical edge of the left ; afterwards the prism was 

 turned on its axis, so as to present successively to the cathetometer the two other lateral 

 faces, and the same operations were repeated on each of them ; thus were obtained, for the 

 length of the lateral edges, six valuations diifering but little, the mean of which was found 

 to be t)9'n">.83. 



This being done, the frame was placed horizontally on suitable supports in such manner 

 thftt, while one of its lateral faces still looked toward the cathetometer, the two edges of the 

 bases which bounded it to the right and left were vertical, and the same operations were con- 

 ducted as in the former case ; then turning the prism on its axis, the two other faces were 

 examined, so that, for the edges of the bases also, six values were obtained, the mean of 

 which was 68™™.'36. But these two means must undergo a slight correction; the liquid 

 films do not in fact terminate at the solid wires according to the generating lines of which 

 the distances have to be measured, but at other generating lines still more within the frame, 

 whence it follows that the two above numbers are a little too small. To make the correction 

 in a simple manner, it should be remarked that when a liquid film rests on a solid surface 

 moistened, it is necessarily through the intervention of a small mass with concave transverse 

 curvatures, and that, if the film be plane, it is necessary, for equilibrium, that these curva- 

 tures should be identically the same on both sides, which evidently requires that the plane 

 of the film should be perpendicular to the solid surface. It thence results that if we prolong 

 in thought the films of our system, each of them will pass by the axis of the wire on which 

 it rests, and that consequently the prolonged liquid edges will alike terminate at the points 

 .where these axes intersect; we may, therefore, without altering in any manner the laminar 

 system, and consequently the length of the liquid edge mn, substitute, in idea, for our frame, 

 the assemblage of axes of the wires which compose it, which evidently amounts to the addi- 

 tion of the diameter of these wires to each of our two numbers ; now this diameter, determined 

 by means of the cathetometer, was found, as the mean of ten measurements taken on the 

 difterent wires, equal to 0'"™.87 ; it is by the addition of this quantity to our two numbers 

 that we arrived finally at the values of b and of a given in the text. 



