142 J. W. Gihbs —Equilibrluiii. of Heterogeneous Substances. 



be deduced we will call a fundamental equation for the substance in 

 question. We shall hereafter consider a more general form of the fun- 

 damental equation for solids, in which the pressure at any point is not 

 supposed to be the same in all directions. But for masses subject only 

 to isotropic stresses an equation between f, //, w, m^,m^, . . . m„ is 

 a fundamental equation. There are other equations which possess 

 this same property.* 

 Let 



'/'=f-^'A (87) 



then by differentiation and comparison with (86) we obtain 



d ij' =z — i/dt — pdv -f- /^j dni^ -\- m^ dm^ . . . + /.i^dm^. (88) 



If, then, y- is known as a function of t, v, m^, m.^, . . . m„, we can 

 find If, p, J-i 1, /'■>, • ■ • A'n i" terms of the same variables. If we then 

 substitute for //' in our original equation its value taken from eq. (87), 

 we shall have again 7i -\- 3 independent relations between the same 

 2n + 5 variables as before. 



Let 



X=£+pv, (89) 



then by (86), 



dx — tdi] + V dp 4-/^1 dm^ + //g dm^ ... -|- //„ drn^. (90) 



If, then, X be known as a function of }i,p, m^, m.^, . , . rn„, we can 

 find t, V, yUj, /<2» • • • /^n i" terms of the same variables. By elimi- 

 nating J, we may obtain again n + 3 independent relations between 

 the same 2?/ + 5 variables as at first. 



Let 



^ = e - ttf +pv, (91) 



then, by (86) 



di^=:. — ffdt + V dp + ;<j dm^ + 1.(2 dm „ . . . + ^^dm^. (92) 



If, then, ^ is known as a function of ^, /?, mj, mg, . . . ;;?„, we can 



* M. Massieu (Comptes Rendus, T. Ixix, 1869, p. 858 and p. 1057) has shown 

 how all the properties of a fluid " which are considered in thermodynamics" may be 

 deduced from a single function, which he calls a characteristic function of the fluid 

 considered. In the papers cited, he introduces two different functions of this kind ; 

 viz., a function of the temperature and volume, which he denotes by 1/), the value of 



— t + tn ~ f 



which in our notation would be 7 or — r— ; and a function of the temperature 



and pressure, which he denotes by V^', the value of which in our notation would be 



— e + tr/ —pv — C 



1 or -7-. In both cases he considers a constant quantity (one kilogram) 



of the fluid, which is regarded as invariable in composition. 



