660 PLATEAU ON THE PHENOMENA OF A FREE LIQUID MASS 



As we are now acquainted with the entire course which the 

 transformation of a liquid cylinder into isolated spheres must 

 take, we can represent it graphically ; fig. 30 represents the 

 successive forms through which the liquid figure passes, com- 

 mencing with the cylinder up to the system of isolated spheres 



thus form with these masses a system in equilibrio, notwithstanding tlie in- 

 compatibilitj' of this equilibrium with the ordinary law of pressures. Now we 

 shall show that this equilibrium is in reality possible, at least theoi-etically. 

 Let us always take as example, the resolution of our unstable cylinder into 

 partial masses. When the cylindrical lines form, their diameter is even 

 then very small in comparison with the dimensions of the thick masses, con- 

 sequently their curvature in the direction peiijendiciilar to the axis is very 

 great in comparison with the curvature of these masses. The pressure corre- 

 sponding to the lines is then originally much greater than those corresponding 

 to the thick masses, whence it follows that the liquid must be driven from the 

 interior of the lines towards these same masses, and that the lines, like the 

 layers, ought to continue diminishing. Moreover, their curvatures, and con- 

 sequently their pressure augmenting in proportion as they become more atte- 

 nuated, their tendency to diminish in thickness will go on increasing, and con- 

 sequently if we disregard the instability of the cylindrical form, we see that 

 they must become of an excessive tenuity. But I say that the augmentation of 

 the pressure will have a limit, beyond which this pressure will progressively 

 diminish, so that it may become equal to that which corresponds to the thick 

 parts of the liquid system. 



In fact, without having recourse to theoretical developments, it is readily 

 seen that if the diameter of the line becomes less than that of the sphere of the 

 sensible activity of the molecular attraction, the law of the pressure must be- 

 come modified, and the diameter continuing to decrease, the pressure must 

 finish by also progressively diminishing, notwithstanding the increase of the 

 curvatures, in consequence of the diminution in the nimiber of attracting mole- 

 cules. Hence the pi-essure may diminish indefinitely ; for it is clear that it 

 would entirely vanish if the diameter of the line became reduced to the thick- 

 ness of a single molecule. Those geometricians who study the theory of capil- 

 lary action know, that the formulae of this theory cease to be applicaMe in the 

 case of very great curvatures, or those the radii of which are comparable to that 

 of molecular attraction. Now it follows from what has been stated, that we may 

 always suppose the thinness of the line to be such that the corresponding pres- 

 sure may be equal to that existing in thick masses which have attained a state 

 of equilibrium. In this case, admitting that the lines are mathematically re- 

 gular, so that the pressure there may be everywhere rigorously the same, con- 

 sequently that they have no tendency to resolve themselves into small partial 

 masses, equilibrium will necessarily exist in the system. In this case, the form 

 of the thick masses will not be mathematically spherical ; for their surface must 

 become slightly raised at the junctures with the lines, by presenting concave 

 curvatures in the meridional direction. This form will be the same as that of 

 an isolated mass, traversed diametrically by an excessively minute solid line 

 (§ 10). This system, like those into the composition of which layers enter, is 

 composed of surfaces of a different nature ; but this heterogeneity of form be- 

 comes possible herr, as in the case of the layers, in consequence of the change 

 which the law of pressures undergoes in passing from one to another kind of 

 surface. 



We can moreover understand, that the equilibrium in question, although 

 possible theoretically, as we have shown, can never be realized, in consequence 

 of the cylindrical form of the lines. The same does not apply to the case of the 

 plane layers; for as we shall show in the following scries, the j)lane surfaces 

 are always surfaces of stable equilibrium, whatever may be their extent. 



