J. TK G'lhhs — EquUlhrkun of Heterogeneous Substances. 370 



substitute — t]y,dt for tdt/y,, for exampk', is equivalent to subtract- 

 ing the complete clitterential d{ti/y,). Therefore, if we consider the 

 quantities in (469) and (470) which occur in any same term in equa- 

 tion (468) as forming a pair, Ave 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 inde- 

 pendent 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 



\drj/,U> \d'l^/^-' \<'f-'aJl7, \t^Ua ^ ' 



dx' dx' 



jdx jdx 



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

 following are to be considered as constant : either t or //y, , either 



flic (J/X 



JTy, or -;—, , . . . either Zy, or -^-^ , either //,, or / ',/, etc. 

 dy dz 



It will be observed that when the tempei-ature is constant the con- 

 ditions yw,, = const., //,, ■=. 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 i^^', /",/, 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 t dy]y, . If we denote this 

 quantity by dQy,, and use the suffix q to denote the condition of no 

 transmission of heat, we may write 



dx 



/dlog t\ _ /^^YxA {'j}o^t\ _ _ /^^\ avi^ 



"dx' 



\dx:Jt - VHog tjx^: \ /± Jt - \d\og t)'~ ' ^ '^' 



'hx' 



where FJ, Ff,', etc., must be regarded as constant in all the equations, 



, . , ,^ d.r . , ^ dz . . 



and either Ay, or -j-, , . . . eitlier Z^, or — , , in each equation. 



