218 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



symbol 22' is here supposed to be expanded into nine terms), and 

 at the same time change the sign of the term from + to . For to 

 substitute jjydt for tdq T , for example, is equivalent to subtracting 

 the complete differential d(trj T ). Therefore, if we consider the quan- 

 tities in (469) and (470) which occur in any same term in equation 

 (468) as forming a pair, we may choose as independent variables 

 either quantity of each pair, and the differential coefficient of the 

 remaining quantity of any pair with respect to the independent 

 variable of another pair will be equal to the differential coefficient 

 of the remaining quantity of the second pair with respect to the 

 independent variable of the first, taken positively, if the independent 

 variables of these pairs are both affected by the sign d in equation 

 (468), or are neither thus affected, but otherwise taken negatively. 

 Thus 



idT a 



(473) 



where in addition to the quantities indicated by the suffixes, the 

 following are to be considered as constant: either t or q v ,, either 



X T or -T-,, ... either Z z > or -^-7, either jn b or IY, etc. 



It will be observed that when the temperature is constant the 

 conditions jUL a = const., yu & = const., represent the physical condition of 

 a body in contact with a fluid of which the phase does not vary, and 

 which contains the components to which the potentials relate. Also 

 that when IY, IY, etc., are constant, the heat absorbed by the body 

 in any infinitesimal change of condition per unit of volume measured 

 in the state of reference is represented by tdq v ,. If we denote this 

 quantity by dQ T , and use the suffix Q to denote the condition of no 

 transmission of heat, we may write 



ax' /y 



where IY, IY, etc., must be regarded as constant in all the equations, 

 and either X T or -7-7, . . . either Z z > or -^ n in each equation. 



