﻿Theory of Surface Forces. 



563 



Now I will demonstrate that the capillary layer of the 

 liquid is a true elastic membrane of which the tension is H. 



Let us imagine (fig. 1) a liquid membrane, the breadth 

 being equal to the unit, lying between two solid drops sup- 

 ported on the right and left by cords stretched within a space 

 holding the vapour only of the liquid. Fig. 1 represents a 



Fiir. 1. 



L 



. . 





^-■^ a 



Cy 



cu fi p 





c 





ir 













E 



F 



<x 





d' 







ti 







ii>- 







& 





hi* 





y^ — s 







D' 



section of the membrane normally at its surface. This mem- 

 brane is held in equilibrium by the action of the vapour- 

 pressure and the tensions of the cords which must be equal 

 to 2H, H denoting the superficial tension. Intersect the mem- 

 brane by the plane AB, normal to the surface and the plane 

 of the figure, and let us consider the system ApCC'D'DrBA. 

 The external pressures of the vapour opposed to ApCC'9 and 

 BrDDV are in equilibrium ; it only remains for us to 

 consider the pressure on <js and the influence exercised by 

 the liquid situated to the left of AB. Construct the planes vh 

 and wg at distances from the free surfaces respectively equal to 

 the thickness of the capillary layers of the two surfaces of the 

 membrane. Between these two planes there is no departure from 

 the law of Pascal, and we have in every direction the hydrostatic 

 pressure p x , equivalent to the external pressure or the vapour- 

 pressure. The pressure on EF, created by the liquid on the 

 left, is therefore annulled by the external pressure p u opposed 

 to em. It remains thus for us to consider the two masses 

 AE^ and BFms. Allow p 2 to represent the hydrostatic 

 pressure per unit of surface in a direction parallel to the 

 surface ; the force exercised by the mass to the left of AE 



C 2 

 is 1 p 2 dh, where h indicates the direction normal to the 



J 1 



surface, the indices 1 and 2 corresponding respectively to the 



interior liquid- and vapour-phase. It is the same with the 



mass to the left of BF. The external pressures opposed to 



qe and wis are pi 1 dh. The equilibrium of the system 



AEFBrDD'C'CpA therefore demands 



2 ( \ Pl -p,)dh = m or H = C\ pi -p. ? )dh. 



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