J. Wi Gibhs — Equilihrmin of Heterogeneous Substances. 491 



the surrounding fluid wliich would counterbalance them, will be 

 inversely as the linear dimensions of the crystals, as appears from the 

 preceding equation. 



If Ave write v for the volume of a crystal, and 2{(y s) for the sum 

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

 value of (T, the numerator and denominator of the fraction (666), 

 multiplied each by SJV, may be rei)resented by S^yffs) and 6v 

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

 the diiferential coefficient 



d2{as) 

 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 crys- 

 tal (when the influence of gravity may be neglected) is that the 

 value of this differential coefficient must be independent of tlie partic- 

 iilar side which is supposed to be displaced. For a constant volume 

 of the crystal, 2{(J 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 suiToimding fluid is indefinitely extended, the quantity 

 2(ffs) represents the work required to form the surfaces of the 

 crystal, and the coefficient of s 6JV\n (664) with its sign reversed rep- 

 resents the work gained in forming a mass of volume unity like the 

 crystal but regarded as without surfaces. We may denote tlie work 

 required to form the crystal by 



TFs denoting the work required to form the surfaces [/.<?., ^'(cs)], 

 and TFy the work gained in forming the mass as distinguished from 

 the surfaces. Equation (664) may then be written 



-SWy+ 2{aids) = 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 ff in all the sides were supposed to diminish in the 

 same ratio as the linear dimensions of the crystal. The variation of 

 Ws would then be determined by the relation 



d TFs = d:S(() s) = I 2 {a ds), 

 and that of Wy by (667). Hence, 



dWs=§dWy, 



