316 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



With these preliminary notions, we now proceed to discuss the 

 condition of equilibrium which relates to the dissolving of a solid at 

 the surface where it meets a fluid, when the thermal and mechanical 

 conditions of equilibrium are satisfied. It will be necessary for us to 

 consider the case of isotropic and of crystallized bodies separately, 

 since in the former the value of tr is independent of the direction of 

 the surface, except so far as it may be influenced by the state of strain 

 of the solid, while in the latter the value of or varies greatly with the 

 direction of the surface with respect to the axes of crystallization, and 

 in such a manner as to have a large number of sharply defined 

 minima.* This may be inferred from the phenomena which crystal- 

 line bodies present, as will appear more distinctly in the following 

 discussion. Accordingly, while a variation in the direction of an 

 element of the surface may be neglected (with respect to its effect on 

 the value of <r) in the case of isotropic solids, it is quite otherwise 

 with crystals. Also, while the surfaces of equilibrium between fluids 

 and soluble isotropic solids are without discontinuities of direction, 

 being in general curved, a crystal in a state of equilibrium with a 

 fluid in which it can dissolve is bounded in general by a broken 

 surface consisting of sensibly plane portions. 



For isotropic solids, the conditions of equilibrium may be deduced 

 as follows. If we suppose that the solid is unchanged, except that an 

 infinitesimal portion is dissolved at the surface where it meets the 

 fluid, and that the fluid is considerable in quantity and remains 

 homogeneous, the increment of energy in the vicinity of the surface 

 will be represented by the expression 



/[e v '-e v "+( Cl + c 2 )e 8(1) ] SNDs 



where Ds denotes an element of the surface, SN the variation in its 

 position (measured normally, and regarded as negative when the solid 

 is dissolved), c x and c 2 its principal curvatures (positive when their 

 centers lie on the same side as the solid), e s(1) the surface-density of 

 energy, e v ' an d e v" the volume-densities of energy in the solid and 

 fluid respectively, and the sign of integration relates to the elements 

 Ds. In like manner, the increments of entropy and of the quantities 

 of the several components in the vicinity of the surface will be 



r' - >/v" + (c, 4- c,)fc (1) ] SNDs, 



etc. 

 The entropy and the matter of different kinds representd by these 



* The differential coefficients of <r with respect to the direction-cosines of the surface 

 appear to be discontinuous functions of the latter quantities. 



