./ W. (jTibhs — EquUlbrlwn of Heteroyeneoas ISuhstaHceti. 219 



the,, l^\ =iP): 



and therefore, by (98), the quantity of any component gas <t ^ in tlie 

 gas-mixture, and in the separate gas to which p^, //j, eic. relate, is 

 the same and may be denoted by the same symbol )ii ^. Also 



whence also, by (93)-(96), 



All the same relations will also hold true whenever the value of t/^ 

 for the gas-mixture is equal to the sum of the values of this func- 

 tion for the several component gases existing each by itself in 

 the same quantity as in the gas-mixture and with the temperature 

 and volume of the gas-mixture. For if ^^j, //j, fj, i/\, Xi-> ^i ? 2^21 

 etc. ; etc. are defined as relating to the components existing thus 

 by themselves, we shall have 



'I- =^ lip I, 

 whence 



\drn^ /i, V, m \dm^ ft, v 



Therefore, by (88), the potential //j has the same value in the gas- 

 mixture and in the gas G^ existing separately as supposed. More- 

 over, 



'^=^ idiJv, nT ~ ^A~df)v, ra " ^'^'^^' 



whence 



Whenever different bodies are combined without communication of 

 work or heat between them and external bodies, the energy of the 

 body formed by the combination is necessarily equal to the sum of the 

 energies of the bodies combined. In the case of ideal gas-mixtures, 

 when the initial temperatures of the gas-masses which are combined 



* A subscript m after a differential coefficient relating to a body having several 

 independently variable components is used here and elsewhere in this paper to indi- 

 cate that each of the quantities m^,m2, etc., unless its differential occurs in the 

 expression to which the suffix is applied, is to be regarded as constant in the differ- 

 entiation. 



