J. W. Glhlta — EqidlihriNiii of Heterogeneous Substances. 303 



of these quantities are imlependently variable. Or, we may consider 

 that we liave n+'i independent equations between the 2 /i-f 5 quan- 

 tities occurring in equation (5i '2), of which n-{-2 are independently 

 variable. 



An equation, therefore, between 



^, t, Ml, M2, ^'tc, (5()i)) 



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

 An equation between 



f^, t/^, s, Ni\ iul^ etc., (5i0) 



or between fs? '/s? ^\, ^2? ^^.c, (51i) 



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

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

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

 n + 2 of the variables may be regarded as independent (viz., n -(- 1 

 for the ii -\- 1 vai"iations in the nature of the surface of discontinuity, 

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

 diffei'entiation and comparison with (501), ?/ + 2 additional equations 

 between the 2/i -j- 5 quantities occurring in (502). Equation (506) 

 shows that equivalent relations can be deduced from an equation 

 between the vaiiables (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 difterentiation and the aid only of general principles and relations, 

 n -\- S independent relations between the same 27i -f- 5 quantities 

 may be obtained. 



If we set tp"" = f', -t7f, (512) 



we obtain by differentiation and comparison with (501) 



dip^ =z — t/^ dt + a ds-{- ^^ d)n\ + yUg ^^^'4 + ^tc. (513) 



An equation, therefore, between iff, t, s, m\, ni\, 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 

 140-144. 



