330 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



dissolution. The general condition of mechanical equilibrium would 

 be of the form 



-fp SDv +fgy Sz Dv+fa- SDs +fgT Sz Da 



+fg Sz Dm +/ 9 SDs +fg(T)te Ds = 0, (680) 



where the first four integrals relate to the fluid masses and the 

 surfaces which divide them, and have the same signification as in 

 equation (606), the fifth integral relates to the movable solid masses, 

 and the sixth and seventh to the surfaces between the solids and 

 fluids, (F) denoting the sum of the quantities (r 2 ), (F 3 ), etc. It should 

 be observed that at the surface where a fluid meets a solid Sz and Sz, 

 which indicate respectively the displacements of the solid and the 

 fluid, may have different values, but the components of these dis- 

 placements which are normal to the surface must be equal. 



From this equation, among other particular conditions of equili- 

 brium, we may derive the following : 



df=g(T)dz (681) 



(compare (614)), which expresses the law governing the distribution 

 of a thin fluid film on the surface of a solid, when there are no passive 

 resistances to its motion. 



By applying equation (680) to the case of a vertical cylindrical tube 

 containing two different fluids, we may easily obtain the well-known 

 theorem that the product of the perimeter of the internal surface by 

 the difference 9' 9" of the superficial tensions of the upper and lower 

 fluids in contact with the tube is equal to the excess of weight of the 

 matter in the tube above that which would be there, if the boundary 

 between the fluids were in the horizontal plane at which their pres- 

 sures would be equal. In this theorem, we may either include or 

 exclude the weight of a film of fluid matter adhering to the tube. 

 The proposition is usually applied to the column of fluid in moss 

 between the horizontal plane for which p' =p" and the actual boundary 

 between the two fluids. The superficial tensions 9' and 9" are then to 

 be measured in the vicinity of this column. But we may also include 

 the weight of a film adhering to the internal surface of the tube. 

 For example, in the case of water in equilibrium with its own vapor 

 in a tube, the weight of all the water-substance in the tube above the 

 plane p'=p", diminished by that of the water- vapor which would fill 

 the same space, is equal to the perimeter multiplied by the difference 

 in the values of 9 at the top of the tube and at the plane p' =p". If 

 the height of the tube is infinite, the value of 9 at the top vanishes, 

 and the weight of the film of water adhering to the tube and of the 

 mass of liquid water above the plane p' =p" diminished by the weight 

 of vapor which would fill the same space is equal in numerical value 

 but of opposite sign to the product of the perimeter of the internal 



