THERMODYNAMIC AL SYSTEM OF GIBBS 107 



fore unaltered. When a component is neither an actual nor a 

 possible component of some part of the system, the total 

 number of variations of the phases, considered separately, is 

 one less than {n -\- l)r and, since there is no condition as to the 

 potential of this component in the part of the system of which it 

 is not a possible component, the number of conditions is also 

 reduced by one. Finally we may consider the case in which 

 some of the components can be formed out of others. Let n, 

 as before, be the number of independently variable components 

 of the system as a whole, and let n + /i be the total number of 

 substances which are regarded as components in various parts of 

 the system. If all these latter components were independent, 

 the number of degrees of freedom of the system would be 

 n + A — r + 2. But, since they are not independent, there are 

 h additional equations between their potentials similar to (84), 

 corresponding to h equations representing the relations between 

 the units of these substances. The number of independent 

 variations of the system, therefore, is still n — r -{- 2. 



Gibbs deduced the phase rule more concisely by the following 

 considerations, "A system of r coexistent phases, each of 

 which has the same n independently variable components is 

 capable of n + 2 — r variations of phase. For the temperature, 

 the pressure, and the potentials for the actual components have 

 the same values in the different phases, and the variations of 

 these quantities are by [97] subject to as many conditions as 

 there are different phases. Therefore, .... the number of inde- 

 pendent variations of phase of the system, will he n -\- 2 — r. 



"Or, when the r bodies considered have not the same independ- 

 ently variable components, if we still denote by n the number of 

 independently variable components of the r bodies taken as a 

 whole, the number of independent variations of phase of which 

 the system is capable wUl still he n -\- 2 — r. In this case, it 

 will be necessary to consider the potentials for more than n 

 component substances. Let the number of these potentials be 

 n -\- h. We shall have by [97], as before, r relations between 

 the variations of the temperature, of the pressure, and of these 

 n -{• h potentials, and we shall also have . . . . h relations 

 between these potentials, of the same form as the relations 



