J. W. Gihhi< — Equilihrmia of Heterogeneous Substances. 161 



or, if we substitute the value of de taken from equation (86), usinj^ sub- 

 script a . . . g for the quantities rehiting to the actual components of 

 the body, and subscript h . . . k for those relating to the possible, 



t dt] 4- //„ dm, . . . + M, dm^ -\- j.i^ dm^ . . . -+- jm dm.^ 



— Tdtf - 31^ dm^ — Jfs f^^'h • • • - M„dm„^ 0. (137) 



That is, the temperature of the body must be equal to T, and the 

 potentials of its components must satisfy the same conditions as if it 

 were in contact and in equilibrium with a body having potentials 

 M^, M2, . . . M„. Therefore the same relations must subsist betAveeu 

 //„... //,„ and M^ . . . Jf„ as between the units of the corresponding 

 substances, so that 



m,/.i, . . . ■j-m^ju„ = m^ TJf^ . . . + m„ Jf„; (138) 



and as we have by (93) 



£ = t}]^p V -h IX, m„ . . . -\- pij m„ (139) 



the expression (133) will reduce (for the body or bodies for which it 

 has the least value per unit of volume) to 



{F-p)v, (140) 



the value of which will be positive, null, or negative, according as 

 the value of 



P — jo (141) 



is positive, null, or negative. 



Hence, the conditions in regard to the stability of a fluid of which all 

 the ultimate components are independently variable admit a very sim- 

 ple expression. If the pressure of the fluid is greater than that of any 

 other phase of the same components which has the same temperature 

 and the same values of the potentials for its actual components, the 

 fluid is stable without coexistent phases ; if its pressure is not as great 

 as that of some other such phase, it will be unstable ; if its pressure is 

 as great as that of any other such phase, but not greater than that 

 of every other, the fluid will certainly not be unstable, and in all 

 probability it will be stable (when enclosed in a rigid envelop which 

 is impermeable to heat and to all kinds of matter), but it will be one 

 of a set of coexistent phases of which the others are the phases which 

 have the same pressure. 



The considerations of the last two pages, by which the tests 

 relating to the stability of a fluid are simplified, apply to such bodies 

 as actually exist. But if we should form arbitrarily any equation as 

 a fundamental equation, and ask whether a fluid of which the proper- 



Tbans. Conn. Acad., Vol. III. 21 January, 1876. 



