416 THE FIGURES OF EQUILIBEIUM OF A LIQUID MASS 



bting also that of the equilibrinm of full masses, it results tliat tlie films assume, 

 as I said in advance, the same figures with those masses. 



Thus, under the circumstances of my experiments, it must be possible to form 

 with films of oil all the figures which I have obtained with full masses of oil, and 

 it has, in effect, been seen (§ 7) that a film of oil which is not adherent to any 

 solid system, takes a spherical figure, as does a full mass placed iu the sama 

 conditions. 



§ 10. I should here present an important remark in reference to the sign of 

 the constant C and the signification of that sign. From the manner in which I 

 have just arrived at the general equation of the equilibrium of laminar figures, 



it is clear that, in this equation, the quantity ^4-p-/ 1^37> ^s to its absolute value, 



be referred indifferently to one or to the other* of the two surfaces of the filnj. 

 If we agi-ee to refer it to that one of the two which regards the exterior of the 

 figure, then when this same quantity, or, what amounts to the same, the con- 

 stant C is positive, the pressure corresponding to 4;he surface in question will be 

 superior to P — that is to say, to that of a plane surface, and the pressure corre- 

 sponding to the other surface will be less than that of a plane, and consequently 

 less than the former; wherefore the resultant, which necessarily acts in the 

 direction of the greatest of the two forces, will be directed, like that, towards 

 the interior of the figure. On the other hand, when the interior surface is con- 

 sidered and is negative, the greatest of the two pressures wdl pertain to thai 

 surface, whence it follows that the resultant will be directed towards the exte- 

 rior. When C, therefore, is positive, the laminar figure will exert a pressure on 

 the alcoholic mass which it encloses ; and when C is negative, the laminar figure 

 will, on the contrary, exert a traction on the mass in question ; in both cases 

 the action will be destroyed by the resistance of this mass; finally, when C is 

 null, the laminar figure will exert neither pressure nor traction. 



§ 11. When the laminar figure is closed, the condition of equilibrium has con- 

 sequently its entire generality, since may be positive, negative, or null ; but if 

 the figure is not closed, equilibrium can evidently subsist only for G zzz O. 

 Hence it follows, for example, that a single film in a solid ring will be in equi- 

 librium if it is plane, and we have seen, in effect, in the experiment of § 3, the 

 two separate films take respectively, in each of the rings, a plane form. For the 

 same reason, a single film attached to two parallel and opposite rings, like that 

 which is formed in the experiment just recalled, and wliich is represented. Fig. 3, 

 must constitute a portion of a catenoid, as announced by its aspect, and as the 

 value of the maximum separation of the rings substantiates. Finally, in the 

 combination of films of Fig. 1 the two short films are necessarily also two por- 

 tions of a catenoid, but taken sufficiently far from their respective cercles de 

 gorge for their meridian curvature to be little perceptible, and for these films to 

 seem to appertain to cones. 



§ 12. It would be easy to imagine proper means for realizing with our films of 

 oil all the other figures of equifibrium which have been considered in the second 

 and fourth series; but this, we shall see, would be useless, inasmuch as we are 

 directly led by what has been premised to a more simple mode of producing 

 laminar figures of equilibrium. Let us suppose that we could form, in the air, 

 liquid films without weight; these films would necessarily take the same figures 

 with the films of oil formed in the alcoholic mixture. In effect, if the system is 

 closed and exerts pressures directed towards its interior, the enclosed mass of 

 air will be compressed until its elasticity neutralizes those pressures, and equi- 

 librium will then evidently exist if these same pressures are all equal as regards 



1 1 



one another; in other words, if the figure is such that the equation — +— rz: 0, 



an equation in which is positive, be satisfied. If again the system be closed, 



