106 BUTLER ART. D 



variations, e.g., the quantities m,i,...m„, r?, v may be varied 

 independently of each other. But if they are all varied in the 

 same proportion, the result is a change in the amount of the 

 body, while its state and composition remain unchanged. A 

 variation of the state or composition of the body involves a 

 change in at least one of the ratios of these quantities. There 

 are n + 1 independent ratios of these n -\- 2 quantities 

 (e.g., the ratios mi/v, m^/v,. . .m„/v, rj/v) so that the number 

 of independent variations of state and composition of a homo- 

 geneous body is n + 1. 



Gibbs calls a variation of the thermodynamic state or com- 

 position of a body, as distinguished from a variation of its 

 amount, a variation of the phase of the body. In a heterogene- 

 ous system, such bodies as differ in composition or state are 

 regarded as different phases of the matter of the system, and all 

 bodies which differ only in quantity or form as different examples 

 of the same phase. Thus we may say that the number of inde- 

 pendent variations of the phase of a homogeneous body which 

 contains n independent components is n + 1. 



Consider a system of r phases each of which has the same 

 v. independently variable components. The total number of 

 independent variations of the r phases, considered separately, 

 is (n + l)r. When the r phases are coexistent these variations 

 are subject to the conditions (68), (69) and (70), i.e., to 

 (r — 1) (n 4- 2) conditions. The number of independent vari- 

 ations of phase of which the system is capable is therefore 



% = (n + l)r - (n + 2) {r - 1) = n - r + 2. (96) 



The integer ^5 has been called the number of degrees of freedom 

 of the system. 



This relation, which is now known as the phase rule, holds 

 even if each phase has not the same n independently variable 

 components. For if a component is a possible, but not an 

 actual, component of some part of the system, the variation, 

 bm, of its quantity in that part, can only be positive, whereas 

 in the previous case it can be either positive or negative, and 

 instead of the equality /x = Af , we have the condition n ^ M. 

 The number of independent variations of the system is there- 



