EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 231 



An equation, therefore, between 



o-, t, yUj, // 2 , etc., (509) 



may be called a fundamental equation for the surface of discontinuity. 

 An equation between 



e 8 , if, s, raf, mf, etc., (510) 



or between e s , jy s , I\, F 2 , etc. (511) 



may also be called a fundamental equation in the same sense. For 

 it is evident from (501) that an equation may be regarded as sub- 

 sisting between the variables (510), and if this equation be known, 

 since 7i-t-2 of the variables may be regarded as independent (viz., 

 n+1 for the n+1 variations in the nature of the surface of dis- 

 continuity, and one for the area of the surface considered), we may 

 obtain by differentiation and comparison with (501), n + 2 additional 

 equations between the 2n + 5 quantities occurring in (502). Equation 

 (506) shows that equivalent relations can be deduced from an equation 

 between the variables (511). It is moreover quite evident that an 

 equation between the variables (510) must be reducible to the form 

 of an equation between the ratios of these variables, and therefore to 

 an equation between the variables (511). 



The same designation may be applied to any equation from which, 

 by differentiation and the aid only of general principles and relations, 

 7i+3 independent relations between the same 2n+5 quantities may 

 be obtained. 



If we set V 8 = * S -^ S > (512) 



we obtain by differentiation and comparison with (501) 



d\fs 8 = j? 8 dt + o- ds + fadm^ + /UL 2 dm% + etc. (513) 



An equation, therefore, between \[s a , t, s, mf, mf, etc., is a fundamental 

 equation, and is to be regarded as entirely equivalent to either of the 

 other fundamental equations which have been mentioned. 



The reader will not fail to notice the analogy between these funda- 

 mental equations, which relate to surfaces of discontinuity, and those 

 relating to homogeneous masses, which have been described on pages 

 85-89. 



On the Experimental Determination of Fundamental Equations for 

 Surfaces of Discontinuity between Fluid Masses. 



When all the substances which are found at a surface of discon- 

 tinuity are components of one or the other of the homogeneous 

 masses, the potentials /x 1 , yM 2 , etc., as well as the temperature, may 

 be determined from these homogeneous masses.* The tension a- may 



* It is here supposed that the thermodynamic properties of the homogeneous masses 

 have already been investigated, and that the fundamental equations of these masses 

 may be regarded as known. 



