J. TT^ GihJm — Equillhrin))} of Heterogeneous Substances. A5'^ 



dE= -^fp dDv -fclp IJv +/(r cWs ^fda Ds 



= - d/p Bv + dfff Ds. (622) 



If we now integrate with respect to d, commencing at the given 

 state of the system, we obtain 



AE= - Jfp Ihi -f AfG Bs, (623) 



where J denotes the vahie of a (jnantity in a varied state of the sys- 

 tem diminished by its vahie in the given state. Tliis is trne for finite 

 variations, and is therefore true for infinitesimal variations without 

 neglect of the infinitesimals of the higher orders. The condition of 

 stability is therefore that 



///^^7>y-///o-7>.s<0, (624) 



or that the quantity 



fpDv—J'GDs (625) 



has a maximum value, the values of /> and o', for each diiferent mass 

 or surface, being regarded as determined functions of z. (In ordin- 

 ary cases (7 may be regarded as constant in each sui-face of discon- 

 tinuity, and jo as a linear function of z in each diflerent mass.) It 

 may easily be shown (compare page 416) that this condition is always 

 sufficient for stability with reference to motion of surfaces of discon- 

 tinuity, even when the variations of t, J/j, if/g? 6tc. cannot be neg- 

 lected in the determination of the 7iecessary condition of stability 

 with respect to such changes. 



On the Possibility of the Forniatio7i of a New Surface of Discon- 

 tinuity where several Surfaces of Discontinuity tneet. 



When more than three surfaces of discontinuity between homo- 

 geneous masses meet along a line, we may conceive of a new surface 

 being formed between any two of the masses which do not meet in a 

 surface in the original state of the system. The condition of stability 

 with respect to the formation of such a surface may be easily obtained 

 by the consideration of the limit between stability and instability, as 

 exemplified by a system which is in equilibrium when a very small 

 surface of the kind is formed. 



To fix our ideas, let us suppose that there are four homogeneous 

 masses A, B, C, and D, which meet one another in four surfaces, 

 which we may call A-B, B-C, C-D, and D-A, these surfaces all meeting 

 along a line L. This is indicated in figure 11 by a section of the sur- 



