246 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



But y i v' -}- y , "v" -f- 17, 8 



(the total quantity of the component specified by the suffix) must be 

 constant ; therefore, since 



2 



dv" dv', and ds = - dv' 



r 



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



x r 



When the substance specified by the suffix is a component of either 

 of the homogeneous masses, the terms - and s -r- may generally 

 be neglected. When it is not a component of either, the terms y/, 

 Vi"> V 'T v " j may of course be cancelled, but we must not 



j 



CvjJL-, 



apply the formula to cases in which the substance spreads over the 

 diaphragm separating the homogeneous masses. 



In the cases just discussed, the problem of the stability of certain 

 surfaces of tension has been solved by considering the case of neutral 

 equilibrium, a condition of neutral equilibrium affording the equation 

 of the limit of stability. This method probably leads as directly as 

 any to the result, when that consists in the determination 4 of the 

 value of a certain quantity at the limit of stability, or of the relation 

 which exists at that limit between certain quantities specifying the 

 state of the system. But problems of a more general character may 

 require a more general treatment. 



Let it be required to ascertain the stability or instability of a fluid 

 system in a given state of equilibrium with respect to motion of the 

 surfaces of tension and accompanying changes. It is supposed that 

 the conditions of internal stability for the separate homogeneous 

 masses are satisfied, as well as those conditions of stability for the 

 surfaces of discontinuity which relate to small portions of these 

 surfaces with the adjacent masses. (The conditions of stability which 

 are here supposed to be satisfied have been already discussed in part 

 and will be farther discussed hereafter.) The fundamental equations 

 for all the masses and surfaces occurring in the system are supposed 

 to be known. In applying the general criteria of stability which are 

 given on page 57, we encounter the following difficulty. 



The question of the stability of the system is to be determined by 

 the consideration of states of the system which are slightly varied 



