CAPILLARY ACTION. 559 



If x is the potential energy of unit of mass of the substance in vapour, 

 then at a distance z from the plane surface of the liquid 



x=x' 



At the surface 



X = x' 

 At a distance z within the surface 



If the liquid forms a stratum of thickness c, then 



(z) + 2irp0 (z - c). 



The surface-density of this stratum is <r = cp. The energy per unit of 

 area is 



e = xP dz > 



J 



(c fc 



= c/j{x'-4irp0(0)} + 27r/9 a d(z)dz + 2Trp*\ 0(c-z)dz. 



Jo Jo 



Since the two sides of the stratum are similar the last two terms are 

 equal, and 



e = cp \x - 4irpO (0)} + 477y> 2 ( e 6 (z) dz. 



J o 



Differentiating with respect to c, we find 



da- 

 dc =P > 



Hence the surface-tension 



de 



T=e - (r cfc> 



= 47r / ) 2 |f C e(z)dz-c6(c)\. 

 Integrating the first term within brackets by parts, it becomes 



