490 J. W. Gihbs — Equilibriimi of Heterogeneous Substances. 



able. If the crystal grows on one side a distance SJV, withoiU other 

 change, the increment of energy in the vicinity of the surface will be 



(fy'- fy") s 6J\^+ ^^'(^8(1)' i' cosec ca' — fs(i) Z' cot co') dJST, 

 where fy' and fy" denote the volume-densities of energy in the crystal 

 and fluid respectively, s the area of the side on which the crystal 

 grows, %,) the surface-density of energy on that side, fg^j)' the surface- 

 density of energy on an adjacent side, oj' the external angle of these 

 two sides, l' their common edge, and the symbol 2' a summation 

 with respect to the difl'erent sides adjacent to the first. The incre- 

 ments of entropy and of the quantities of the several components will 

 be represented by analogous formulae, and if we deduce as on pages 485, 

 486 the expression for the increase of energy in the whole system due 

 to the growth of the crystal without change of the total entropy or 

 volume, and set this expression equal to zero, we shall obtain for the 

 condition of equilibrium 



{e,'—t,/y'-pi,"r^'+p")sdN' 



+ 2' ( ff' I' cosec Go'—o- r cot co') SJSr= 0, (664) 



where C and a' relate respectively to the same sides as fs(i) ^^^^ ^sd; "i 

 the preceding formula. This gives 



fy'-— t ??v+p" 2'i c' I' cosec d — aV cot &?') 



^1 - y^ ■ sy, 



It will be observed that unless the side especially considered is 

 small or narrow, we may neglect the second fraction in this equation, 

 which will then give the same value of ///' as equation (387), or as 

 equation (661) applied to a j)lane surface. 



Since a similar equation must hold true with respect to every other 

 side of the crystal of which the equilibrium is not affected by meet- 

 ing some other body, the condition of eqixilibrium for the crystalline 

 form (when unaffected by gravity) is that the expression 



^'((t'/' cosec (i^'—c/' cot oj') ^r6r^ 



s 

 shall have the same value for each side of the crystal. (By the value 

 of this expression for any side of the crystal is meant its value when 

 a and s are determined by that side and the other quantities by the 

 surrounding sides in succession in connection with the first side.) 

 This condition will not be affected by a change in the size of a crys- 

 tal while its proportions remain the same. But the tendencies of 

 similar crystals toward the form required by this condition, as mea- 

 sured by the inequalities in the composition or the temperature of 



