EQUILIBKIUM OF HETEROGENEOUS SUBSTANCES. 171 



suppositions otherwise the same), we shall have for infinitesimal 

 values of t an infinitesimal value either of m l or m 2 , and for infinite 

 values of t finite or infinitesimal values of ra 3 according as /3 1 + # 2 

 is equal to or greater than unity. 



The case which we have considered is that of a ternary gas-mixture, 

 but our results may easily be generalized in this respect. In fact, 

 whatever the number of component gases in a gas-mixture, if there 

 are relations of equivalence in ultimate analysis between these com- 

 ponents, such relations may be expressed by one or more equations of 



the form 



A^+A^+As^g+etc^O, (311) 



where ($ 1} 2 , etc. denote the units of the various component gases, 

 and A 15 A 2 , etc. denote positive or negative constants such that 

 2^ = 0. From (311) with (86) we may derive for phases of dis- 

 sipated energy, 



A!//! + A 2 yu 2 + A 3 // 3 + etc. = 0, 



or 2 1 (A 1 // 1 ) = 0. (312) 



Hence, by (276), 



(313) 



where A, B and C are constants determined by the equations 



A = ^(AA - \fr - A^), (314) 



1 ), (315) 



1 ). (316) 

 Also, since pv = 2 1 (a l m l )t, 



2j (A^ log m^ 2^04) log S^Ojmj) 



+ 2(\ l a l )logp = A+B'\ogt-j, (317) 



where -B^S^X^+X^ (318) 



If there is more than one equation of the form (311), we shall have 

 more than one of each of the forms (313) and (317), which will hold 

 true simultaneously for phases of dissipated energy. 



It will be observed that the relations necessary for a phase of dis- 

 sipated energy between the volume and temperature of an ideal gas- 

 mixture, and the quantities of the components which take part in the 

 chemical processes, and the pressure due to these components, are not 

 affected by the presence of neutral gases in the gas-mixture. 



From equations (312) and (234) it follows that if there is a phase of 

 dissipated energy at any point in an ideal gas-mixture in equilibrium 

 under the influence of gravity, the whole gas-mixture must consist of 

 such phases. 



