WITHDRAWN FROM THE ACTION OF GRAVITY. 589 



that it is exactly their mean. Denoting the two radii in question 

 by R and R', the pressure exerted by the line, referred to the 

 unity of surface, would be 



p4(^w) f^-) 



The radii R and R' are positive when they belong to convex 

 curves, or, in other terms, when they are directed to the interior 

 of the mass; whilst they are negative when they belong to con- 

 cave 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 w hich com- 

 mence at the different 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 pex'pendicularly at a second point of this same 

 surface, it is evident that equilibiium can only exist in this 

 minute canal when the pressures exerted by the lines which 

 occupy its two extremities are equal ; and if this equality ex- 

 ists, 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 was our object to arrive at, and 

 to which in each case the free surface of the mass must be 

 subject, 



The analytical expression of this condition is directly deducible 

 from the general value of the pressure given in the preceding 

 pai'agraph ; we only require to equalize this value to a constant, 

 and as the quantities P and A are themselves constant, it is in 

 fact sufficient to make 



E+&-'=C. («•) 



the quantity C being constant for the same figure of equilibrium. 



This equation is the same as those which are given by geome- 

 tricians for capillary surfaces, when, in the latter equations, the 

 quantity representing gravity is supposed to be 0. 



R and R' may be replaced by their analytical values; we are 



