EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 267 



same phases but without any surface of discontinuity. (A mass thus 

 existing without any surface of discontinuity must of course be 

 entirely surrounded by matter of the same phase.)* 



The form in which the values of f-rrj and (-*-] are given in 



\dt/ p \dp/t 



equation (580) is adapted to give a clear idea of the relations of 

 these quantities to the particular state of the system for which they 

 are to be determined, but not to show how they vary with the state 

 of the system. For this purpose it will be convenient to have the 

 values of these differential coefficients expressed with reference to 

 ordinary components. Let these be specified as usual by 1 and 2 . 

 If we eliminate d^ and djuL 2 from the equations 



da- = r] B dt + 1\ d^ + F 2 dfa, 

 dp = rjv'dt + y/d//! + y 2 'dyK 2 , 



dp = ri^'dt + y/dy 

 we obtain 



C 



* If we set 



and in like manner 



r r 



E e ' t ' " *, " lt>\ 



s e s- / *v --j/fv > 



we may easily obtain, by means of equations (93) and (507), 



E a = tH s + <r-pV. (d) 



Now equation (580) may be written 



dff=-IL s dt+Vdp. (e) 



Differentiating (d), and comparing the result with (e), we obtain 



The quantities E 8 and H 8 might be called the superficial densities of energy and 

 entropy quite as properly as those which we denote by e 8 and i) S . In fact, when the 

 composition of both of the homogeneous masses is invariable, the quantities E 8 and Hg 

 are much more simple in their definition than e s and r) S , and would probably be more 

 naturally suggested by the terms superficial density of energy and of entropy. It would 

 also be natural in this case to regard the quantities of the homogeneous masses as 

 determined by the total quantities of matter, and not by the surface of tension or any 

 other dividing surface. But such a nomenclature and method could not readily be 

 extended so as to treat cases of more than two components with entire generality. 



In the treatment of surfaces of discontinuity in this paper, the definitions and 

 nomenclature which have been adopted will be strictly adhered to. The object of this 

 note is to suggest to the reader how a different method might be used in some cases 

 with advantage, and to show the precise relations between the quantities which are 

 used in this paper and others which might be confounded with them, and which may 

 be made more prominent when the subject is treated differently. 



