EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 279 



It will be observed that this condition has the same form as if 

 the different fluids were separated by heavy and elastic membranes 

 without rigidity and having at every point a tension uniform in 

 all directions in the plane of the surface. The variations in this 

 formula, beside their necessary geometrical relations, are subject to 

 the conditions that the external surface of the system, and the lines 

 in which the surfaces of discontinuity meet it, are fixed. The formula 

 may be reduced by any of the usual methods, so as to give the 

 particular conditions of mechanical equilibrium. Perhaps the following 

 method will lead as directly as any to the desired result. 



It will be observed the quantities affected by S in (606) relate 

 exclusively to the position and size of the elements of volume and 

 surface into which the system is divided, and that the variations Sp 

 and So- do not enter into the formula either explicitly or implicitly. 

 The equations of condition which concern this formula also relate 

 exclusively to the variations of the system of geometrical elements, 

 and do not contain either Sp or Sar. Hence, in determining whether 

 the first member of the formula has the value zero for every possible 

 variation of the system of geometrical elements, we may assign to 

 Sp and So- any values whatever which may simplify the solution of 

 the problem, without inquiring whether such values are physically 

 possible. 



Now when the system is in its initial state, the pressure p, in each 

 of the parts into which the system is divided by the surfaces of 

 tension, is a function of the co-ordinates which determine the position 

 of the element Dv, to which the pressure relates. In the varied state 

 of the system, the element Dv will in general have a different position. 

 Let the variation Sp be determined solely by the change in position 

 of the element Dv. This may be expressed by the equation 



(607) 



in which --, --, -f- are determined by the function mentioned, 

 dx ay dz 



and Sx, Sy, Sz by the variation of the position of the element Dv. 



Again, in the initial state of the system the tension a; in each of 

 the different surfaces of discontinuity, is a function of two co-ordinates 

 o) l , ft> 2 , which determine the position of the element Ds. In the varied 

 state of the system, this element will in general have a different 

 position. The change of position may be resolved into a component 

 lying in the surface and another normal to it. Let the variation So- 

 be determined solely by the first of these components of the motion of 

 Ds. This may be expressed by the equation 



*-**+** (608) 



