264 WITHDKAWN FEOM THE ACTION OF GRAVITY. 



but a few moments after the formation of the films ; when these were at first 

 somewhat too distant from each other, so that the time required for their spon- 

 taneous congress was rather long, they vmited without a partition, becoming 

 transformed into one large hemisphere, because doubtless they had grown too 

 thin, and the incipient partition had broken before its existence could be veri- 

 fied. 



§ 13, If a third spherical laminar cap joins itself to two others already united, 

 the system will evidently have three partitions, namely, one proceeding from the 

 union of the first two films, and two from the union of each of these films with 

 the third. These three partitions will necessarily terminate at the same arc of 

 junction, and, supposing that they still have spherical curvatures, it will result 

 that at three lines of junction of each of them with two of the films the angles 

 will still be of 120°; it will result, moreover, for reasons already given, (§ S,) 

 that at the arc of junction of the three partitions with each other the angles will 

 be also of 120°. 



This being premised, let us see by what means we can trace the base of a 

 system of this kind, as we have traced (Fig. 11) that of a system of two films. 

 After having described (Fig. 15) the bases of the first 



two films, bases having for centres c and c', and for radii ^°' , u/^ ^\ 



the lengths given which we will again designate by p X"^ \ Kn \ 

 and p' , let us take, commencing at the point s, where / / ,.-'f'--, , ) 



these two bases meet, and on the radii sc and s c', two / / ,.--''' \ ":'^ 



lengths .?/' and sf, equal to one another and to the I <^'v; V-v'4'''' ) 



radius p" of the third base, then from the points c and c' \ / \ / J 



as centres, and with the lengths cy* and c'y as radii, \ \y ^y^ 



let us trace two arcs of a circle ; their point of intersection \,, ^* 



on c" will be the centre of the base of the third film, a base 



which we will then describe with the radius p." Let us, in effect, suppose the 

 problem solved, and thi&base traced. If we draw from the point u where it termi- 

 nates at one of the former the right lines u c and u c", which will be respectively 

 equal to p and p", these lines will make between them an angle of 60°, like the 

 right lines s c and s c' ; whence it follows that the triangle cue" will be equal to 

 the triangle c sf, in which sc and *y are also respectively equal to p and p", and 

 thus c c" will be equal to cf; for the same reasons the triangle c' v c" will be equal 

 to the triangle c' sf', and consequently c' c" will be equal to c^ f . Let us propose 

 now to trace the bases of the three partitions. Those of the three films being 

 described (Fig. 17) after the preceding outline we determine, as in Fig. 11, 

 the centre d of the partition pertaining to the first two films, and commencing from 

 s, by drawing s d making with s c' an angle of 60°, until it meets, at d, with the 

 line cc' prolonged; we determine likewise the centre y of the partition pertaining 

 to the first and the third film by drawing uf, making an angle of 60° with c" u, 

 until it meets, at f, with c c" prolonged ; finally, we determine by the same 

 process the centre g of the third partition. There remains then only to describe 

 from the points d, f, and g, as centres, and with the radii d s, fu, and g v, 

 three arcs of a circle beginning, respectively, at the points s, u, and v, and directed 

 toward the middle of the figure ; these arcs will be the bases of the three par- 

 titions, on the hypothesis, however, that these partitions are portions of spheres. 

 If the figure has been constructed with care, we shall recognize, 1st, that the 

 three arcs just spoken of all terminate at the same point o; 2d, that the three 

 centres, jT, d, and g, are disposed in a right line; 3d, that if we join the point o 

 to these three centres, the anglesyo d and god are equal, and each of 60°. 



§ 14. But as it might be thought that these results of a graphic construction 

 are merely very close and not strictly exact, I proceed to establish them abso- 

 lutely, assuming, however, that the angles of the films with one another and with 

 the partitions are precisely of 120°. I am indebted for this demonstration to 

 M. Vander Mensbrugghe, a young doctor of sciences of our university. 



