EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 321 



where e v ' an d e v" denote the volume-densities of energy in the 

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

 crystal grows, e s(1) the surface-density of energy on that side, e B(1) ' 

 the surface-density of energy on an adjacent side, ' the external 

 angle of these two sides, I' their common edge, and the symbol 2' 

 a summation with respect to the different sides adjacent to the 

 first. The increments of entropy and of the quantities of the several 

 components will be represented by analogous formulae, and if we 

 deduce as on pages 316, 317 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 expres- 

 sion equal to zero, we shall obtain for the condition of equilibrium 



(e v ' _ t^ - yu/'y/ +jp")8 SN+ 2'(<rT cosec o>' - d! cot w)6N= 0, (664) 



where cr and or' relate respectively to the same sides as e s(1) and e s(1) ' 

 in the preceding formula. This gives 



2'(o- 

 1 



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 plane 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 meeting some other body, the condition of equilibrium for the 

 crystalline form (when unaffected by gravity) is that the expression 



2'(o-T cosec ft/ o-l' cot ft/) /*\ 



- - - (666) 



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 crystal while its proportions remain the same. But 

 the tendencies of similar crystals toward the form required by this 

 condition, as measured by the inequalities in the composition or the 

 temperature of the surrounding fluid which would counterbalance 

 them, will be inversely as the linear dimensions of the crystals, as 

 appears from the preceding equation. 



If we write v for the volume of a crystal, and S(o-s) for the sum 

 of the areas of all its sides multiplied each by the corresponding 

 value of o-, the numerator and denominator of the fraction (666), 



multiplied each by 8N, may be represented by <$2(<rs) and Sv 

 G.I. x 



