WITHDRAWN FROM THE ACTION OF GRAVITY. 591 



planes. If then we divide the above equivalent quantity by 2n, 

 the result o ( ~ "^ "7 ) ^'^^^ represent the mean of all these curva- 

 tures. Now as this result is independent of the value of n, or 

 of the number of positions occupied by the system of the two 

 planes, it will be equally true if we suppose this number to be 

 infinitely great, or, in other words, if the successive positions of 

 the system of the two planes are infinitely approximated, and 

 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 



-(- + —) represents then the mean of all the curvatui-es 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 — + — retains the same value, i. e. if for 



the whole surface we have — + -7 =C, this surface is such that 



P P 

 its mean curvature is constant. 



Considered in this purely mathematical point of view, the 

 equation (4.) has formed the object of the I'esearches of several 

 geometricians, and we shall profit by these researches in the 

 subsequent parts of this memoii'. 



Thus our liquid surfaces should satisfy this condition, that the 

 mean cm've must be the same everywhere. We can understand 

 that if this occurs, the mean effect 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 theoretical principles which have led us to the general 

 condition of equilibrium. According to these principles, each 

 of the lines of molecules exerting upon the mass the pressures 

 upon which its form depends, commences at the surface and ter- 

 minates at a depth equal to the radius of the sensible activity of 

 the molecular attraction ; so that these lines collectively con- 

 stitute 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 



