J. W. (Tihhs—I'JqKMlhriviii of Ueterof/eneoKS Substances. 131 



bounding surface. The homogeneity of the new parts is of no con- 

 sequence, as we have made no assumption in that respect. It may 

 l)e doubtful whether we can consider the new parts, as thus hounded, 

 to be infinitely small even in tlieir earliest stages of development. But 

 if they are not infinitely small, the only way in which this can aftect 

 the validity of our formuhe will be that in virtue of the equations of 

 condition, i. e., in virtue of the evident necessities of the case, finite 

 variations of the energy, entropy, volume, etc., of the original parts 

 will be caused, to which it might seem that equation (12) would not 

 apply. But if the nature and state of the mass be not varied, equa- 

 tion (12) will hold true of finite dift'erences. (This appears at once, 

 if we integrate the equation under the above limitation.) Hence, 

 the equation will hold true for finite diiferences, provided that the 

 nature and state of the mass be infinitely little varied. For the dif- 

 ferences may be considered as made up of two parts, of which the 

 first are for a constant nature and state of the mass, and the second 

 are infinitely small. We may therefore regard the new parts to be 

 bounded as supposed without prejudice to the validity of any of our 

 results. 



The condition (52) understood in either of these ways (or in 

 others which will suggest themselves to the reader) will have a per- 

 fectly definite meaning, and will be valid as the necessary and sufii- 

 cient condition of equilibi-ium in regard to the formation of new 

 parts, when the conditions of equilibrium in regard to tlie original 

 parts, (50), (51), and (43), are satisfied. 



In regard tf) the condition (53), it may be shown that with (50), 

 (51), and (43) it is always suflicient for equilibrium. To prove this, 

 it is only necessary to show that when (50), (51), and (43) are satis- 

 fied, and (52) is not, (53) will also not be satisfied. 



We will first observe that an expression of the form 



_ e+ Tij- Pv^ J/, m^ + J/, "^2 • • • + -K i'^n (54) 



denotes the work olnainable V)y the formation (by a reversible pro- 

 cess) of a body of which f, ;/, v, m^, in.^, . . . m„ are the energy, 

 entropy, volume, and the quantities of the components, within a 

 medium having the pressure P, the temperature 7] and tlie potentials 

 31 , J/2, . . . M„. (The medium is supposed so large that its prop- 

 erties are not sensibly altered in any part by the formation of the 

 body.) For f is the energy of the body formed, and the remaining 

 terms represent (as may be seen by applying equation (12) to the 

 medium) the decrease of the energy of the medium, if, after the 



