224 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



as the masses M' and M" respectively. We shall then have, by 

 equation (12), if we regard the volumes as constant, 



M" = t'W + fr'tonS" + ju 2 '<$m 2 '" + etc., (485) 



&"" = tf'W" + fr" tonS'" + // 2 "<5m 2 "" + etc. ; (486) 



whence, by (482)-(484), we have for reversible variations 



(487) 

 (488) 



From these equations and (477), we have for reversible variations 

 S(e - e'" - e"") = tS(rj- if" - i'") 



+ /^(^i - m/" - m/'") + fJL 2 8(m 2 - m 2 '" - m 2 "") + etc. (489) 

 Or, if we set* 



fiB^c-e'"-^'", n* = ri-ri" f -ri"", (490) 



mf = m x m/" m/'", mf = m 2 m 2 '" m 2 "", etc., (491 ) 



we may write 



Se 8 = t8tj s + frSm* + fr&mS + etc. (492) 



This is true of reversible variations in which the surfaces which have 

 been considered are fixed. It will be observed that e s denotes the 

 excess of the energy of the actual mass which occupies the total 

 volume which we have considered over that energy which it would 

 have, if on each side of the surface S the density of energy had the 

 same uniform value quite up to that surface which it has at a sensible 

 distance from it ; and that q s , mf, mf> etc., have analogous significations. 

 It will be convenient, and need not be a source of any misconception, 

 to call e s and T/ S the energy and entropy of the surface (or the super- 



pO w& 



ficial energy and entropy), and the superficial densities of energy 



s s 



77v 77i 



and entropy, - , - , etc., the superficial densities of the several com- 

 ponents. 



Now these quantities (e s , if, mf, etc.) are determined partly by the 

 state of the physical system which we are considering, and partly by 

 the various imaginary surfaces by means of which these quantities 

 have been defined. The position of these surfaces, it will be remem- 

 bered, has been regarded as fixed in the variation of the system. It 

 is evident, however, that the form of that portion of these surfaces 

 which lies in the region of homogeneity on either side of the surface 

 of discontinuity cannot affect the values of these quantities. To 

 obtain the complete value of <Se 8 for reversible variations, we have 



*It will be understood that the 8 here used is not an algebraic exponent, but is 

 only intended as a distinguishing mark. The Roman letter S has not been used to 

 denote any quantity. 



