172 J. W. Gibbs — Equilibrium of Heterogenous Substances. 



an equation, when tlie fundamental eqnation is known, may be 

 reduced to the form of an equation between the independent variables 

 of the fundamental equation. 



Again, as the determinant (IVS) is equal to the product of the 

 differential coefficients obtained by writing d for A in the first 

 members of (166)-(169), the equation of the limit of stability may be 

 expressed by setting this determinant equal to zero. The form of 

 the differential equation as thus expressed will not be altered by the 

 interchange of the expressions ;/, «?.j, . . . »?„, but it will be altered 

 by the substitution of v for any one of these expressions, which will 

 be allowable whenever the quantity for which it is substituted has 

 not the value zero in any of the phases to which the formula is to be 

 applied. 



The condition formed by setting the expression (182) equal to zero 

 is evidently equivalent to this, that 



that is, that 





3=0, (183) 



or by (98), if we regard ^, //j, ... /^„ as the independent variables, 



(It?) = '"^ <'««> 



In like manner we may obtain 



(186) 



d^p d^p d^p 



^-"' ^? = "'- • • diAZ7' = '^- 



Any one of these equations, (185), (186), may be regarded, in gen- 

 eral, as the equation of the limit of stability. We may be certain 

 that at every phase at that limit one at least of these equations will 

 hold true. 



GEOMETRICAL ILLITSTRATIONS. 



Surfaces in tchich the Composition of the Body represented is 



Constant. 

 In vol. ii, p. 382, of the Trans. Conn. Acad., a method is described of 

 representing the thermodynamic properties of substances of invariable 

 composition by means of surf^xces. The volume, entropy, and energy 



