THE FIGURES OF EQUILIBRIUM OF A LIQUID MASS 267 



aJi n: 



Ti-. 17 



ba.ac 

 ba+ac 



This result applied to the triangle 



fog (Fig. 17) gives therefore do=z 



fogo 



, and, 



after substitutions, do =z 



PP 



t'—l'' 



fo+go 

 Now, this is 



^ \ 



\ 



i precisely (§7) the value of the radius d s oi 

 • the third partition. It results, then, from the 

 J,d above, as we had proposed to demonstrate, that 

 ! the centres of the three partitions are in a right 

 i line; that the bases of these partitions meet 

 I at one and the same point; finally that the 

 I radii of these bases, commencing from the point 

 I in question, make with one another angles of 

 ! 60°, .and that hence these bases unite under 

 i angles of 120°. 



i Now, since we can consider the three parti- 



I tions as really constituting portions of spheres, 



j it remains to prove that these three portions 



I intersect one another by a single arc; but this 



j is what evidently follows from the centi-es f, d, 



% \ I ^ of these three spheres being in a right line, 



W j and from these same spheres having a com- 



\\ i mon point at o. Thus all the theoretic con- 



\\\ ditions are satisfied by three partitions of 



ll spherical curvature disposed, with the three 



V caps, so as to form a system having for its base 



that which was traced by the construction of the preceding paragraph ; it must 



then be regarded as extremely probable that the system will really take that 



form. 



§ 15. This, in effect, I have experimentally verified by the means heretofore de- 

 scribed, (§ 11,) by tracing, namely, in broad lines on paper, in the manner indi- 

 cated, the base of a system of three spherical caps with their partitions. The 

 base of the laminar system thus realized has been found to be exactly superposed 

 on the drawing. The graphic constructions given in preceding pages for the 

 bases of systems of spherical caps implicitly suppose the law of the inverse 

 ratio of the pressure to the diameter; the exact coincidence of the bases of the 

 systems realized with the bases traced furnishes therefore a new verification of 

 that law, in addition to the direct verification obtained in § 28 of the fifth series. 

 It was to the present experiments that I had allusion in the paragraph just cited. 

 § 16. If we imagine that a fourth spherical cap unites itself with the system 

 of the thi-ee preceding ones, we can conceive two different arrangements of the 

 assemblage besides that in which the fourth cap should so place itself as to be 

 united with but one of the others. One of these arrangements would contain 

 four partitions uniting by a single edge, and the other would contain five uniting 

 by two edges. To simplify the question and the graphic 

 constructions, I will suppose the four caps to be equal in 

 diameter, in which case all the partitions will evidently 

 be plane. Then, it may be conceived, in the first place, 

 that the four caps unite in such way that their centres 

 shall be placed like the four summits of a square, which 

 will give the system whose base is represented by 

 Fig. 18, where there are four partitions terminating at 

 the same edge under right angles ; this system is evi- 

 dently one of equilibrium, since everything in it is sym- 

 metrical. It may be conceived, in the second place, that, 



Fis. 18 



