EQUILIBRIUM, OF HETEROGENEOUS SUBSTANCES. 243 



equilibrium will be stable in respect to change in one direction and 

 unstable in respect to change in the opposite direction, and is therefore 

 to be considered unstable. In general, therefore, we may call (523) 

 the condition of stability. 



When the interior mass and the surface of discontinuity are formed 

 entirely of substances which are components of the external mass, p' 

 and cr cannot vary, and condition (524) being satisfied the equilibrium 

 is unstable. 



But if either the interior homogeneous mass or the surface of dis- 

 continuity contains substances which are not components of the 

 enveloping mass, the equilibrium may be stable. If there is but one 

 such substance, and we denote its densities and potential by y\, Y v 

 and juL lt the condition of stability (523) will reduce to the form 



or, by (98) and (508), 



(526) 



In these equations and in all which follow in the discussion of this 

 case, the temperature and the potentials ju. 2> /* 3 , etc. are to be regarded 

 as constant. But 



which represents the total quantity of the component specified by the 

 suffix, must be constant. It is evidently equal to 



Dividing by 4?r and differentiating, we obtain 



(r*yi' + ^lydr +4** d yi '+r 2 eO\ = 0, 

 or, since y x ' and I\ are functions of JUL V 



0. (527) 



By means of this equation, the condition of stability is brought to 

 the form 



"- 



3 



If we eliminate r by equation (522), we have 



VlL+Ii) 2 



1 dT >l - (529) 



H 



2o- a/*! 



If p' and o- are known in terms of t, fa, // 2 , etc., we may express the 

 first member of this condition in terms of the same variables and p". 



