322 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



respectively. The value of the fraction is therefore equal to that 



of the differential coefficient 



dl,(a-s) 

 dv 



as determined by the displacement of a particular side while the 

 other sides are fixed. The condition of equilibrium for the form 

 of a crystal (when the influence of gravity may be neglected) is 

 that the value of this differential coefficient must be independent 

 of the particular side which is supposed to be displaced. For a 

 constant volume of the crystal, 2(o-s) has therefore a minimum value 

 when the condition of equilibrium is satisfied, as may easily be 

 proved more directly. 



When there are no foreign substances at the surfaces of the 

 crystal, and the surrounding fluid is indefinitely extended, the 

 quantity 2(o-s) represents the work required to form the surfaces 

 of the crystal, and the coefficient of sSN in (664) with its sign 

 reversed represents the work gained in forming a mass of volume 

 unity like the crystal but regarded as without surfaces. We may 

 denote the work required to form the crystal by 



W B -W V , 



W s denoting the work required to form the surfaces {i.e., Z(o-s)}, 

 and W^ the work gained in forming the mass as distinguished from 

 the surfaces. Equation (664) may then be written 



-($Fv + Z(er<te) = 0. (667) 



Now (664) would evidently continue to hold true if the crystal 

 were diminished in size, remaining similar to itself in form and 

 in nature, if the values of a- in all the sides were supposed to 

 diminish in the same ratio as the linear dimensions of the crystal. 

 The variation of W s would then be determined by the relation 



d W 8 = d2(<rs) = f 2(<r ds), ' 

 and that of F v by (667). Hence, 



and, since W B and TF V vanish together, 



8 V 3 8 2 V' \ / 



the same relation which we have before seen to subsist with 

 respect to a spherical mass of fluid as well as in other cases. (See 

 pages 257, 261, 298.) 



The equilibrium of the crystal is unstable with respect to variations 

 in size when the surrounding fluid is indefinitely extended, but it 

 may be made stable by limiting the quantity of the fluid. 



