228 THE FIGURES OF EQUILIBRIUM OF A LIQUID MASS 



action of the molecule m upon the portion of the line comprised between the 

 base of the perpendicular and the surface will attract this portion towards the 

 interior of the mass. If afterwards we take a portion of the line equal to 

 the former from tlic other side of the perpendicular, and commencing at the base 

 of the latter, the action of the molecule m upon this second portion will be equal 

 and oppo.site to that which it exerted upon the first ; so that these two portions 

 conjointly would neither be attracted towards the interior nor the exterior of 

 the ma?s ; if beyond these two same portions another part of the line is com- 

 prised within the sphere of activity of m, this part will evidently be attracted 

 towards the exterior. The definitive action of vi upon the line Avill then be in 

 the latter direction. Ilcnce it follows that all the molecules of the space com- 

 prised between the surface and the tangent plane which are sufficiently near the 

 line to exert an effective action upon it, will attract it towards the exterior of 

 the mass. If, then, we suppress this portion of the liquid so as to reproduce 

 the convex surface, the result will be an augmentation of the pressure on the 

 pai't of the line. Thus the pressure corresponding to a convex surface is greater 

 than that corresponding to a plane surface, and its amount will evidently be 

 greater in proportion as the convexity is more marked. 



4. If the surface has a spherical curvature, it may be demonstrated that, rep- 

 resenting the pressure corresponding to a plane surface by P, the radius of the 

 sphere to which the surface belongs by r, and by A a constant, the pressure 

 exerted by a line of molecules, and reduced to unity of the surface, will have 

 the following value : P 4- ^ (\\ 



r 

 r being positive in the case of a convex, and negative in that of a concave 

 surface. 



Whatever be the form of the surface, let us imagine two spheres, the radii of 

 which are tiiosc of greatest and least curvature at the point under consideration. 

 It is evident that the pressure exerted by the line will be intermediate between 

 those corresponding to these two spheres, and calculation shows that it is ex- 

 actly their mean. Denoting the two radii in question by R and R', the press- 

 ure exerted by the line, referred to the unity of surface, would be 



--ia^r.) • •• <^' 



The radii R and R' are positive when they belong to convex curves, or, in other 

 terms, Avhen they are directed to the interior of the mass ; whilst they are nega- 

 tive when they belong to concave curves, i. e., when they are directed towards 

 the exterior. 



5. From the preceding details we can now easily deduce the condition of 

 equilibrium relative to the free surface of the mass. 



The pressures exerted by the lines of molecules which commence at the dif- 

 ferent points of the surface are transmitted to the whole mass ; consequently, 

 for the existence of equilibrium in the latter, all the pressures must be equal to 

 each other. In fact, let us imagine a minute canal running perpendicularly 

 from some point of the surface, and subsequently becoming recurved so as to 

 terminate perpendicularly at a second point of this same surface, it is evident 

 that equilibrium can only exist in this minute canal when the pressures exerted 

 by the lines which occupy its two extremities are eqtial ; and if this equality 

 exists, equilibrium will necessarily exist also. Now, the pressures exerted by 

 the different lines depend upon the curves of the surface at the point at which 

 they commence ; these curves must therefore be such, at the various points of 

 the free surface of the mass, as to determine everywhere the same pressure. 



Such is the condition which it Avas our object to arrive at, and to which in 

 each case the free surface of the mass must be subject. 



