J. W. Gihbs — Equilibrium of Heterogeneous Substances. 145 



We may extend this definition to any material system whatever 

 which has a nniform temperature throughout. 



If we compare two states of the system of the same temperature, 

 we have 



f-f' = f'- 6" -?;(;/ -,/'). (106) 



If we suppose the system brought from the first to the second of 

 these states without change of temperature and by a reversible pro- 

 cess in which W is the work done and Q the heat received by the 

 system, then 



£'-f":rrTF- Q, (107) 



and t{if' ^ i/)=Q. (108) 



Hence 



//''-//'" = IF; (109) 



and for an infinitely small reversible change in the state of the 

 system, in which the temperature remains constant, we may write 



~dip = dW. (110) 



Therefore, — //' is the force function of the system for constant 

 temperature, just as — £ is the force function for constant entropy. 

 That is, if we consider if: as a function of the temperatm-e and the 

 variables which express the distribution of the matter in space, for 

 every different value of the temperature — ?/' is the different force 

 function required by the system if maintained at that special 

 temperature. 



From this we may conclude that when a system has a uniform 

 temperature throughout, the additional conditions which are necessary 

 and sufficient for eqiiilibrium may be expressed by 



(d^')<^0-* (111) 



* This general condition of equilibrium might be used instead of (2) in such prob- 

 lems of equilibrium as we have considered and others which we shall consider here- 

 after with evident advantage in respect to the brevity of the formulas, as the limitation 

 expressed by the subscript i in (111) applies to every part of the system taken sepa- 

 rately, and diminishes by one the number of independent variations in the state of 

 these parts which we have to consider. The more cumbersome course adopted in this 

 paper has been chosen, among other reasons, for the sake of deducing all the particular 

 conditions of equilibrium from one general condition, and of having the quantities 

 mentioned in this general condition such as are most generally used and most simply 

 defined ; and because in the longer formulae as given, the reader will easily see in each 

 case the form which they would take if we should adopt (111) as the general condi- 

 tion of equilibrium, which would be in effect to take the thermal condition of equilibrium 

 for granted, and to seek only the remaining conditions. For example, in the problem 

 treated on pages 116 fE., we would obtain from (111) by (88) a condition precisely like 

 (15), except that the terms 16?]', tSrj"^ etc. would be wanting. 



Trans. Conn. Acad., Vol. III. 19 January, 1876. 



