242 THE FIGURES OP EQUILIBRIUM OF A LIQUID MASS 



being situated at tlie same diiftance from the surfince ; for in the latter case the 

 litth' sphere Avould only be cut on one side, and its opposite part would be per- 

 fectly full of liquid. It might also happen that the little sphere belonging to 

 tlie molecule in question in the thin layer is only cut on one side ; the molecule 

 will then still exert a pressure in the same direction, but its intensity will then 

 be as great as in the case of a thick mass. It is easy to see that if the thick- 

 ness of the layer is less than the simple length of the radius of the molecular 

 attraction, the; little spheres will all he cut on both sides; Avhilst if the thickness 

 in question is. comprised between the length of the above radius and twice this 

 same length, a portion of the minute spheres will be cut on one side only. In 

 both cases the pressure exerted by any molecule being always directed towards 

 the middle of the thickness of the layer, it is evident that the integral pressure 

 corresponding to any point of either of the two surfaces will be the result of the 

 pressures individually exerted by each of those molecules, which, commencing 

 at the point in question, are arranged upon half the length of the small perpen- 

 dicular. Now each of the two halv(;s of the small perpendicular beiug less than 

 the radius of the sphere of activity of the molecular attraction, it follows that 

 the number of molecules composing the line which exerts the integral pressure 

 is less than in the case of a thick mass. Thus, on the one hand, the intensities 

 of part or the whole of the elementary pressures composing the integral pres- 

 sure will be less than in the case of a thick mass, and, on the other hand, the 

 number of these elementary pressures will be less ; from this it cvid'ently follows 

 that the integral pressure will be inferior to that which would occur in the case 

 of a thick mass. P always denoting the pressure corresponding to any point of 

 a plane surface belonging to a thick mass, (§ 4,) the pressure corresponding 

 to any point of either of the surfeces of an extremely thin plane layer will there- 

 fore be less than P. Moreover, this pressure will be less in proportion as the 

 layer is thinner, and it may thus diminish indefinitely ; for it is clear that it 

 would be reduced to zero if we supposed that the thickness of the layer was 

 equal to no more than that of a simple molecule. 



We can obtain liquid layers with curved surfaces ; soap-bubbles furnish an 

 example of these, and we shall meet with others in the progi'ess of this investi- 

 gation. Now by supposing the thickness of such a layer to be less than twice 

 the radius of the molecular attraction, we should thus evidently arrive at the 

 conclusion that the corresponding pressures at either of its two surfaces would 

 be inferior in intensity to those given by paragraph 4, and that, moreover, these 

 intensities are less in proportion as the layer is smaller. We thus arrive at the 

 following neAV principle ; 



In the case of every liquid layer, the thichness of which is less than ticice the 

 radius of the sphere of activity of the molecular attraction., the pressure will not 

 depevd solely uj)on the curvatures of the surfaces, hit will vary with tlie thick- 

 ness of the layer. 



25. We thus sec that an extremely thin plane liquid layer, adhering by its 

 edge to a thick mass the surfeccs of which are concave, may form with this mass 

 a system in a state of equilibrium ; for we may always suppose the thickness 

 of the layer to be of such value that the pressure corresponding to the plane 

 surfaces of this layer is equal to that corresponding to the concave surfaces of 

 the thick mass. Such a system is also very remarkable in respect to its form, 

 inasmuch as surfaces of different nature, as concave and plane surfaces, suc- 

 ceed each other. This heterogeneity of form is, moreover, a natural consequence 

 of the change which the law of pressures undergoes in passing f?bm the thick 

 to the thin part. • 



26. As we liave already seen, theory demonstrates the possibility of the ex- 

 istence of such a system in a state of equilibrium. As regards the experiment 

 which has led us to these considerations, although the result presented by it 

 tends to realize in an absolute manner the theoretical result, there is one circum- 



