EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 225 



therefore only to regard variations in the position and form of the 

 limited surface s, as this determines all of the surfaces in question 

 lying within the region of non-homogeneity. Let us first suppose 

 the form of s to remain unvaried and only its position in space to 

 vary, either by translation or rotation. No change in (492) will be 

 necessary to make it valid in this case. For the equation is valid if 

 8 remains fixed and the material system is varied in position ; also, if 

 the material system and s are both varied in position, while their 

 relative position remains unchanged. Therefore, it will be valid if 

 the surface alone varies its position. 



But if the form of s be varied, we must add to the second member 

 of (492) terms which shall represent the value of 



Se B tSrj 8 /Zj Smf /z 2 #mf etc. 



due to such variation in the form of S. If we suppose S to be suffi- 

 ciently small to be considered uniform throughout in its curvatures- 

 and in respect to the state of the surrounding matter, the value of 

 the above expression will be determined by the variation of its area 

 $s and the variations of its principal curvatures 8c^ and 8c 2 , and 

 we may write 



raf -f etc. 



c, + <7 2 Sc 2 , (493) 



or 



Se s = tSri B + fjL 1 (5m? + /UL 2 #mf + etc. 



+<r38+l(C l + Ct)3(c l + Ci)+l(C l -C s )t(c l -c t ), (494) 



or, C\, and (7 2 denoting quantities which are determined by the initial 

 state of the system and the position and form of s. The above is 

 the complete value of the variation of e 8 for reversible variations 

 of the system. But it is always possible to give such a position to 

 the surface s that C l -\-C 2 shall vanish. 



To show this, it will be convenient to write the equation in the 

 longer form {see (490), (491)} 



deiSy fa 8^ /jL 2 3m 2 etc. 



8rf" + fr Sm^" + H 2 Sm 2 '" + etc. 



i.e., by (482X484) and (12), 



- etc. 



(496) 



From this equation it appears in the first place that the pressure 



is the same in the two homogeneous masses separated by a plane 

 G. i. p 



