230 THE FIGURES OF EQUILIBRIUM OF A LIQUID MASS 



this number to be infiuitely great, or, iu other words, if the successive positions 

 of the .system of the two plaues arc iufiuitely approximated, aud consequently 

 if this same system turns around the normal in such a manner as to determine 

 all the curvatures which belong to the surface around the point in question. 



The quantity - | - H ^^ represents, then, the mean of all the curvatures of 



the surface at the same point, or the mean curvature at this point. Now if, in 



passing from one point of the surface to another, the quantity - -1 -^ retains 



the same value, i. c, if for the whole surface we have — | : = C, this sur- 



P P 

 face is such that its mean curvature is constant. 



Considered in this purely mathematical point of view, the equation (4) has 

 formed the object of the researches of several geometriciaud, aud we shall profit 

 by these researches iu the subsequent parts of this memoir. 



Thus our liquid surfaces should satisfy this condition, that the mean curve 

 must be the same everywhere. We can understand that if this occurs, the 

 mean efiect of the curvatures at each point upon the pressure corresponding to 

 this point also remains the same, and that this gives rise to equilibrium. Hence 

 we now see more clearly the nature of the surfaces we shall have to consider, 

 and why they constitute surfaces of equilibrium. 



6*. We must now call attention to an immediate consequence of the theo- 

 retical principles which have led us to the general condition of equilibrium. 

 According to these principles, each of the linos of molecules exerting upon the 

 mass the pressures upon which its form depends, commences at the surface and 

 terminates at a depth equal to the radius of the sensible activity of the mole- 

 cular attraction, so that these lines collectively constitute a superficial layer, 

 the thickness of which is equal to the radius itself, and we know that this is 

 of extreme minuteness. It results from this that the formative forces exerted 

 by the liquid upon itself emanate solely from an excessively thin superficial 

 layer. We shall denominate this consequence the princ/j^le of the superficial 

 layer. 



7. A spherical surface evidently satisfies the condition of equilibrium, because 

 all the curvatures in it are the same at each point; also when our mass is per- 

 fectly free, i. e., when it is not adherent to any solid which obliges its surface 

 to assume some other curve, it in fact takes the form of the sphere, as shown 

 in the preceding memoir. 



8. Before proceeding further, we ought to elucidate one point of great im- 

 portance in regard to the experimental part of our investigations. The liquid 

 mass in our experiments being immersed in another liquid, the qiiestion may 

 be asked whether the molecular actions exerted by the latter exert no infiuence 

 upon the iigure produced; or, in other words, whether the figure of equilibrium 

 of a li([uid mass adherent to a solid system, and withdrawn from the action 

 of gravity by its immersion in another liquid of the same density as itself, is 

 exactly the same as if the mass adherent to the solid system were really de- 

 prived of gravity and Avere placed in vacuo. Now, we shall show ithat this 

 really is the case. The molecular actions resulting from the presence of the 

 surrounding liquid are of two kinds, viz., those resulting from the attraction of 

 this li([uid for itself, and those resulting from the mutual attraction of the two 

 liquids. Let us first consider the former, imagining for an instant that the 

 otlu'rs do not exist. The surrounding liquid being applied to the i'ree surface 

 of the immersed mass, the former presents in i^ifngHo the same figure as the 

 latter mass presents in relief. Those molecules of this same liquid which are 

 near the common surface of the tAVO media must then exert pressures of the same 



