Henry, that the cohesion of liquids is not of the same order with that of solids; it will be understood, therefore, that when the distance m n is so far reduced that its further diminution cannot be effected with a rapidity nearly equal to that of the ascension of the summit of the bubble, the liquid will still present in m ® much too great a resistance to the disunion of its molecules to be forced asunder, and that hence it will be lifted up by the bubble under the shape of a film; and as this film, during its generation, is pressed from below upwards by the bubble of air, and adheres by its circumference to the liquid of the vessel, it must be convex towards the exterior. After the film has commenced forming, it must become still more developed: for, incessantly pressed by the bubble of air. it must continue to rise, while the liquid to which its circumference adheres cannot follow it in mass on account of its weight; this liquid must, therefore, remain behind; but, by virtue of the cohesion and viscosity, there can be no rupture between the incipient film and the environing liquid, and the film will simply increase until the action from below upward exerted on the lower part of the bubble of air shall have had its whole effect. Mr. Hagen, who has sought to prove, contrary to the principle established by Poisson in his new theory of capillary action, that the density of the superficial stratum of liquids is greater than that of their interior, cites, in support of his opinion, the fact of the forma tion of the films in question; but we see that is not at all necessary to resort to such an hypothesis in order to account for this formation.

In § 25 of the first series it was said that when a mass of oil a little less dense than the alcoholic liquid in which it is immersed rises to the surface of the latter. it is at first more or less flattened against that surface, as if encountering resist ance in traversing it; that after some time it makes its way through, and then presents a portion of plane surface more or less extended on a level with that of the alcoholic liquid. This phenomenon is now explicable in a natural manner from the considerations which precede: it fares with the sphere of oil as with the bubble of air; it can only make its way to the exterior by disuniting the molecules of the upper stratum of the ambient liquid, but this not growing thin with sufficient rapidity on account of its viscosity, resists a rupture by virtue of its cohesion. Only it is plain that, in this case, the pellicle cannot be elevated above the level.

§ 2. Let us recur to our convex film developed by the ascension of a bubble of air. When it has attained its full development, and hence remains stationary. it should assume (5 series, § 12) one of the figures of equilibrium which would correspond to the surface of a liquid mass without gravity; now this figure. which is formed by an equal action in all azimuths around the vertical axis of the air bubble, must evidently be one of revolution, and, as it is closed on the axis, it can only constitute (IV, § 2) a portion of a sphere. What, now, does theory teach us on the extent of this portion relatively to the complete sphere? As regards molecular action, the superficial stratum of a full liquid mass may, as we know, be assimilated to a stretched membrane; our liquid film, which 's obviously reduced to the superficial strata of its two faces, may therefore be likened to a stretched membrane, and consequently has a tendency to occapy the least possible extent. The question, then, if we neglect certain particulars of which I shall presently speak, and which have no sensible influence when the volume of air is somewhat large, is reduced to this: what, for a given volum is the segment of a sphere whose surface is smallest? This problem is read.lg solved by calculation, and we thus find that the segment in question is a hemisphere; but we reach the same result still more simply by the following reas ing, for the idea of which I am indebted to M. Lamarle.

Let us conceive any two spherical segments, equal as regards one another, and

Philosophical Magazine, 1845, vol. xx, p. 541.

+ Ueber die Oberflächen der Flussigkeiten (Ann. de M. Poggendorf, 1846, vol. LXVII, p. ! )

applied one against the other by their bases. In order that the convex surface of each of them should be the smallest possible with reference to the volume enclosed between itself and the common base, it suffices evidently that the whole convex surface of both segments together should be the smallest possible in relation to the total volume; now, according to a known principle, this last condition will be fulfilled if the two segments constitute together a single sphere, in which case each of the two segments will be a hemisphere. Our liquid film, if it contains a sufficient body of air, ought, therefore, to take the hemispherical form, and this is what common observation verifies.

§3. We will advert now to the particulars to which allusion was made a little while ago. In the first place, the liquid of the vessel must be slightly raised by the capillary action on both the exterior and interior face of the laminar figure, as it would be on the two faces of a solid lamina previously moistened with the same liquid and partially immersed; it must, therefore, form a small annular mass with concave meridian surfaces, and this also is confirmed by observation. Hence the border of the liquid film does not rest immediately on the plane surface of the liquid of the vessel, but on the crest of the small annular mass in question. In the second place, it will be perceived from this, that if the enclosed volume of air is so small that the space circumscribed by the border of the film shall have little diameter, the surface of the liquid in this space will be in no part plane, but will present, even in the middle, a concave curvature more or less decided, as in the interior of a tube of small extent. This result is also in accordance with experience, and I have ascertained, by means presently to be indicated, that the central portion of the surface in question ceases to appear plane when the diameter of the film, at the crest of the little annular mass, is less than about two centimetres. In the third place, even with a volume of air great enough for the surface of the liquid, in the space circumscribed by the film, to show itself absolutely plane through nearly its whole extent, this surface must be sunk below the exterior level by the pressure which the film, by reason of its curvature, exerts on the enclosed air, (V, § 21,) and that this is the case may be evinced by the following process:

In a large porcelain dish placed on a table in front of a window pour a stratum of glyceric liquid (V, § 13) about 2 centimetres in depth, and, after having inflated, by means of an earthen pipe, a bubble of the same liquid, deposit it in the middle of the surface of the stratum, when it will at once form a spherical segment. We now place ourselves in such a position as to see the sky by reflection on the surface in question, holding, at the same time, a black thread stretched horizontally at a small distance from the film in such manner that a portion of its reflected image shall be perceived in the space circumscribed by the film. The complete image of the thread now appears to be formed of three parts-two without and one within the laminar figure; the former are both incurvated near the film, in consequence of the capillary elevation before spoken of; as regards the third, if the circumscribed surface has, in its middle, a plane portion, we shall find, by suitably adjusting the thread, a position of the latter for which the middle of the image will be rectilinear. This will be the case with films whose diameter exceeds two centimetres, but within that limit the entire part of the image in the interior of the film will appear curved.

When the film has a large diameter, that part of the image of the thread is rectilinear for nearly its whole length; it curves only toward its extremities, by virtue of the capillary attraction; but its straight portion is not in the prolongation of the straight portions exterior to the film; it will be seen a little lower. This depression, which shows that the circumscribed plane surface is, as has been said, below the exterior level, becomes less decided in proportion as the diameter of the film is more considerable-a circumstance referable to the diminution of the curvature, and consequently of the pressure of the film, but which is still very perceptible for a film of a decimetre in diameter.

§ 4. The 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 follows 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 somewhat 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 2mm, and is less still for small ones. On the other hand, when the film has large dimensions, when, for example, its diameter is a decimetre, 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 but 0mm 226; and consequently in order to obtain the value of the depression 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 Omm.204. With such a volume 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 deviation would be inappreciable.

22.6

100

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 aunular mass, would necessarily 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 cancave 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. in the case of the glyceric liquid, it further follows, from the results of § 25 4 the 5th series, by taking, agreeably to the remark above made, half the valle

which they yield, and dividing by the density of the liquid, that the first of the above two actions would be equivalent to a difference of level of 10mm.21; while, even supposing the absence of the little annular mass, the second would evidently proceed only from a difference of level equal to the radius of the airbubble-that is, to 0mm.5. With our small volume of air and a hemispherical film, equilibrium, then, is impossible; that it should exist, it would be necessary that the bubble of air should remain almost entirely beneath the level of the liquid, and hence should give rise to a film scarcely at all elevated and of very feeble curvature; then, in effect, the slight hydrostatic pressure which tends to cause the bubble of air to rise will be equivalent to the minute weight of a volume of liquid a little less than that of this bubble, and the light pressure exerted by the exterior face of the film, in virtue of its feeble curvature, will suffice to counterbalance it.

Experiment again fully verifies this deduction of theory. Having poured, to a certain height, glyceric liquid into the vessel with plane glass walls which served for experiments on the masses of oil, and slightly agitated the liquid in order to produce small bubbles of air, I chose one of these about 1mm in diameter, and sufficiently near to one of the walls, and observed it by successively placing the eye a little below and then above the level of the liquid. In this way I perceived that the little bubble appeared spherical, and was so far immersed that its projection above the level was very inconsiderable.

§ 5. From this it is clear that if we form successive films on the surface of soap-water or glyceric liquid, beginning with a diameter of one decimetre, followed by others progressively smaller, a limit will be reached below which the films will exhibit a sensible depression, or appear, in other words, to constitute less than a hemisphere. In order to determine this limit approximately in regard to the glyceric liquid, I deposited the bubbles, as was indicated in the preceding paragraph, on the surface of the liquid contained in a salver a little more than full, and ascertained that they appear hemispherical only for diameters greater than about 3 centimetres; below that value the bubbles form segments sensibly less relatively to the entire sphere, and this diminution is the more decided as the diameter of their base is smaller.

§ 6. Although a film of spherical curvature thus formed at the surface of a iquid be in equilibrium of figure, still absolute repose does not exist: it slowly ecomes thinner until it bursts. The principal causes of this have been long ince indicated: they are, firstly, evaporation, in the case of liquids which are usceptible of it; and secondly, the action of gravity which causes the liquid onstantly to descend from the summit of the film towards its base. And here gain viscosity has a great influence: if this be very weak, it is plain that the liding of the molecules towards the base of the film will be effected with great pidity, and consequently the film will have scarcely any persistence; hence, hen we succeed in forming films with pure water, they scarcely subsist at all. his remark concerning the agency of viscosity in the duration of films had ready been presented, though in a somewhat different manner, by Professor enry, in regard to bubbles of soap compared with those of pure water. §7. Let us suppose now that a second bubble of air rises from the bottom of e vessel, and that at the moment when it has nearly reached the surface, it ppens to be partly under the first film; it will thus occasion the formation of ilm which will necessarily lift up the former on one side, so that the two antities of air respectively imprisoned by these two films will be separated by portion of the second, as by a liquid partition. But this partition will not serve the curvature of the rest of the second film, as I shall proceed to show. In virtue of their liquid nature, films can evidently not meet under angles with ear edges: for continuity it is necessary that, along the whole line of junc

* See the article cited in 3d note of section 1.

tion, a small mass with 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 5th 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 soapwater 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 imprisoned 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; but if the two quantities 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 convexity on the side where the elasticity of the air is least, until the effort which 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 p, 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 (5th series, §§ 22 and 29 in the inverse ratio of the diameters, and consequently of the radii, we shall have P ; but, according to what has been seen above, it is

r

and p'

r

p'

necessary, for equilibrium, that we should have q=p'—p; whence, 1=?'—!

P'

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

p

Pp'

4

a formula which gives the

solving by reference to r, there results r radius of the partition when we know those of the two films. If, for example. these two films pertain to equal spheres, we have pp', and the formula give 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 large in other terms, if we have p'p, the formula gives r=p; in this case, cest quently, the curvature of the partition will be equal to that of the larger fla

§ 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.