410 J. W. Gihhs — JSquilihrium of Heterogeneous Substances. 



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

 sui-faces of tension has been solved by considering the case of neutral 

 e(j[uilibrium, — a condition of neutral equilibrium affording the equa- 

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

 as any to the result, when that consists in the determination 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 

 requii'e 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 occun-ing in the system are 

 supposed to be known. In applying the general criteria of stal)ility 

 which are given on page 110, we encounter the following difticulty. 



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

 the consideration of states of the system which are slightly varied 

 from that of which the stability is in question. These varied states 

 of the system are not in general states of equilibrium, and the rela- 

 tions expressed by the fundamental equations may not hold true of 

 them. More than this, — if we attempt to describe a varied state of 

 the system by varied values of the quantities which describe the 

 initial state, if these varied values are such as are inconsistent with 

 equilibrium, they may fiiil to determine with precision any state of 

 the system. Thus, when the phases of two contiguous homogeneous 

 masses are specified, if these phases are such as satisfy all the condi- 

 tions of equilibrium, the nature of the surface of discontinuity (if with- 

 out additional components) is entirely determined ; but if the phases 

 do not satisfy all the conditions of equilibrium, the nature of the sur- 

 face of discontinuity is not only undetermined, but incapable of deter- 

 mination by specified values of such quantities as we have employed 

 to express the nature of surfaces of discontinuity in equilibrium. For 

 example, if the temj)eratures in contiguous homogeneous masses are 

 different, we cannot specify the thermal state of the surface of discon- 

 tinuity by assigning to it any particular temperature. It would be 



