378 J. TK Gibbs — Equilibrium of Heterogeneous Substances. 



those which have just been represented by F^^ /^j', etc., and which 

 relate to the fluid components of the body, as well as the correspond- 

 ing quantities relating to its solid components. Again, if we sup- 

 pose the solid matter of the body to remain without variation in 

 quantity or position, it will easily aj^pear that the potentials for the 

 substances which form the fluid components of the solid body must 

 satisfy the same conditions in the solid body and in the fluids in con- 

 tact with it, as in the case of entirely fluid masses. See eqs. (22). 



The above conditions must however be slightly modified in order 

 to make them sufiicient for equilibrium. It is evident that if the 

 solid is dissolved at its surface, the fluid components which are set 

 free may be absorbed by the solid as well as by the fluid mass, and 

 in like manner if the quantity of the solid is increased, the fluid com- 

 ponents of the new portion may be taken from the previously exist- 

 ing solid mass. Hence, whenever the solid components of the solid 

 body are actual components of the fluid mass, (whether the case is 

 the same with the fluid components of the solid body or not,) an 

 equation of the form (383) must be satisfied, in which the potentials 

 A'a, j^b-, etc., contained implicitlj^ in the second member of the equa- 

 tion are determined from the solid body. Also if the solid compon- 

 ents of the solid body are all possible but not all actual components 

 of the fluid mass, a condition of the form (384) must be satisfied, the 

 values of the potentials in the second member being determined as in 

 the preceding case. 



The quantities 



t, A^x,, . . . -^z., ;^«, /'i, etc., (469) 



being difierential coeflicients of fy* with respect to the variables 



"- £'•••!' ^•' ^*'' ^'«- f""' 



will of course satisfy the necessary relations 



dt _dX^ ^^^ 



This result may be generalized as follows. Not only is the second 

 member of equation (468) a complete differential in its present form, 

 but it will remain such if we transfer the sign of differentiation {d) 

 from one factor to the other of any term (the sum indicated by the 

 symbol ^ 2' is here supposed to be expanded into nine terms), and 

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



