258 WITHDEAWN" FROM THE ACTION OF GRAVITY. 



§ 4. Tlie reasoning of § 2 necessarily supposes that the film rests by its actual 

 border upon the plane surface of the liquid in the vessel, and that the portion of 

 that surface circumscribed by the film preserves its plane shape and its level ; 

 now, these conditions, being, as has been just seen, never all entirely fulfilled, 

 it fellows that the reasoning in question can only be considered sufficiently rig- 

 orous when the variance from the imaginary conditions on which it rests is but 

 inconsiderable. To be more precise : If we fill with glyceric liquid, and some- 

 what above the edge, a large porcelain salver, previously levelled and placed on 

 a table opposite a window, and, after having deposited thereon a bubble, station 

 ourselves so as to see the film projected on a dark ground, and, closing one eye, 

 keep the other at the level of the little annular mass, we shall distinguish per- 

 fectly well the two meridian lines of this little mass, as well that which looks 

 towards the exterior of the figure, as the commencement, proceeding from the 

 summit of the crest, of that which fronts the interior. We therefore clearly 

 perceive this summit, and can estimate, approximately, its vertical height above 

 the exterior plane surface. We shall thus recognize that, for large bubbles, this 

 height scarcely exceeds 2™™, and is less still for small ones. Oh the other hand, 

 when the film has large dimensions, when, for example, its diarheter is a deci- 

 metre, the portion of the surface of the liquid circumscribed within its interior 

 may be regarded as exactly plane through almost its whole extent. It results, 

 in fine, from the experiments of the preceding paragraph, that with such a film, 

 the depression of this surface, though still quite sensible, is yet very minute. 

 From the results of § 28 of the 5th series, it follows that if the film, assumed to 

 be hemispherical and of the diameter of a decimetre, were, although formed of 

 glyceric liquid, deposited on pure water, the depression in question would be 



22.6 

 but rwrr= 0™™,226 ; and consequently in order to obtain the value of the de- 

 pression in the present case, that is, when the film is deposited on the glyceric 

 liquid, it suffices to divide the preceding quantity by the density 1.1065 of that 

 liquid, which reduces the depression definitively to 0™™.204. With such a vol- 

 ume of air and a hemispherical film, a state of things would exist therefore in 

 close conformity with the conditions of the reasoning in question, and we may 

 conclude that the film would, in effect, take that form, or that, at least, the de- 

 viation would be inappreciable. 



But it is easy to show that, with a volume of air sufficiently small, the film 

 will be far from constituting a hemisphere. Let us imagine, for instance, that 

 the bubble of air is but one millimetre in diameter, and suppose, for an instant, 

 that the film is hemispherical. Upon this hypothesis, the portion of the surface 

 of the liquid circumscribed by the film and reckoned from the border of the 

 latter, or, if you will, from the crest of the small annular mass, would neces- 

 sarily constitute, by reason of its minute dimensions, a concave hemisphere, so 

 that the bubble of air would continue to form an entire sphere of one millimetre 

 in diameter. That being the case, let us remember that the pressure exerted by 

 a spherical film in virtue of its curvature is (5th series, § 23) the sum of the 

 actions separately due to the curvatures of each of its two faces; or, since these 

 two actions are equal, the double of one of them. Now, the action of the interior 

 face of our little hemispherical film would, as regards its effort to cause the 

 bubble of air to descend, be counterbalanced by the opposite action of the con- 

 cave hemisphere, which would, as I have said, limit the bubble beneath, and 

 there would remain, on the one hand, the action due to the exterior face of the 

 film, an action which would impel the bubble from above downward, and, on the 

 other hand, a slight hydrostatic pressure which would impel this bubble from 

 below upward if the lower point of it were below the level of the liquid. But, 

 in the case of the glyceric liquid, it further follows, from the results of § 28 of 

 the 5th series, by taking, agreeably to the remark above made, half the value 



