260 WITHDRAWN FROM THE ACTION OF GRAVITY. 



tioD, a small mass witli surfaces strongly concave in a perpendicular direction to 

 this line should take form; it is what is realized, as has been seen (2d series, 

 §§31 and 32) when, in the interior of the alcoholic liquid, a laminar figure of 

 oil is produced by the gradual exhaustion of a polyhedron. Let us recall, in 

 this respect, the experiment of § 2 of the 5 oh series — an experiment in which 

 a similar mass, though a thick one, establishes the transition between a plane 

 film and two curved films, as is seen in meridian section in Fig. 1 of this 5th 

 series. It will be understood, therefore, that in the case of our films of soap- 

 water or of glyceric liquid, a mass of this kind exists, though too minute to be 

 distinguished, along the whole length of the arc of junction of the partition and 

 the two other films ; now, the surfaces of the latter and that of the partition 

 being thus united by small surfaces having their own curvatures, it is plain that 

 these small surfaces establish an entire independence between the respective 

 curvatures of the other surfaces. It is thus, for example, that in the experiment 

 above recalled, a plane film is connected with two films which are portions of 

 catenoids. In this experiment, it is true, the junction takes place by a thick 

 mass ; but the result must evidently be the same as regards the independence 

 of the curvatures, however minute be the transverse dimensions of the mass 

 serving as an intermedium. 



This being the case, let us remark that the partition must also constitute a 

 portion of a sphere, for it falls within the same conditions as the two other films ; 

 that is to say, it has, like the latter, for limits the small mass of junction and 

 the water of the vessel. As regards its curvature, this evidently depends on 

 the difference of the action exerted on its two faces by the two portions of im- 

 prisoned air. If these two portions of air are equal, the two films will pertain 

 to equal spheres, which will press the two volumes of air with the same inten- 

 sity, and consequently the partition, exposed on its two faces to equal actions, 

 will have no curvature, or, in other words, will be plane; bat if the two quan- 

 tities of air are unequal, in which case the two films will pertain to spheres of 

 different diameters, and will therefore press these two quantities of air unequally, 

 the partition subjected on its two faces to unequal actions will acquire convex- 

 ity on the side where the elasticity of the air is least, until the effort Avhich it 

 exerts, in virtue of its curvature, on the side of its concave face, counterbalances 

 the excess of elasticity of the air which is in contact with that face. 



Let p, p', and r be the radii of the spheres to which respectively appertain 

 the larger film, the smaller and the partition, and let ^j>, p', and q be the respect- 

 ive pressures which they exert, in virtue of their curvatures, on the air which 

 bathes their concave faces. These pressures being (5 th series, §§ 22 and 28) 

 in the inverse ratio of the diameters, and consequently of the radii, we shall 



have — =1 — , and — =: — ; but, according to what has been seen above, it is 

 IP <1 P 



v' v 



necessary^ for equilibrium, that we should haveg'izzp' — f, whence, 1:^ . 



rp' J) 



Transferring to this last equation the above values of — and of — , and re- 



q q 



pp 



solving by reference to r, there results r sz: •, a formula which gives the 



P — P 

 radius of the partition when we know those of the two films. If, for example, 

 these two films pertain to equal spheres, we have pz^p', and the formula gives 

 r = infinity ; that is to say, the partition is then plane, as we have already found 

 it to be. If the radius of the smaller of the two films is half that of the larger, 

 in other terms, if we have p' :^^jO, the formula gives r^=p ; in this case, conse- 

 quently, the curvature of the partition will be equal to that of the larger film. 



§ 8. In order to complete the study of our laminar system it remains only to 

 inquire under what angles the two films and the partition intersect one another. 



