388 J. W. Oihhs — Equilibrium of Heterogeneous Substances. 



initial state of the system tliis surface will of course be plane like the 

 physical surface of discontinuity, to which it is parallel. In the 

 varied state of the system, let it become a portion of a spherical 

 surface having positive curvature ; and at sensible distances from 

 this surface let the matter be homogeneous and with the same phases 

 as in the initial state of the system ; also at and aboiit the surface let 

 the state of the matter so far as possible be the same as at and about 

 the plane surface in the initial state of the system. (Such a variation 

 in the system may evidently take place negatively as well as posi- 

 tively, as the surface may be curved toward either side. But 

 whether such a variation is consistent with the maintenance of equi- 

 librium is of no consequence, since in the preceding equations only 

 the initial state is supposed to be one of equilibrium.) Let the 

 surface s, placed as supposed, whether in the initial or the varied 

 state of the surface, be distinguished by the symbol s'. Without 

 changing either the initial or the varied state of the material system, 

 let us make another supposition with respect to the imaginary sur- 

 face s. In the unvaried system let it be parallel to its former posi- 

 tion but removed from it a distance A on the side on which lie the 

 centers of positive curvature. In the varied state of the system, let 

 it be spherical and concentric with s', and separated from it by the 

 same distance A, It will of course lie on the same side of s' as in the 

 unvaried system. Let the surface s, placed in accordance with this 

 second supposition, be distinguished by the symbol §". Both in the 

 initial and the varied state, let the perimeters of s' and s" be traced 

 by a common normal. Now the value of 



6s — t 6?] — /-ii Sm^ — yUg ^^^2 ~ ^**^- 



in equation (496) is not aifected by the position of §, being deter- 

 mined simply by the body M : the same is true p' dv'" + p" Sv"" or 

 p'd{v"' -f w""), v"'-\- v"" being the volume of M. Therefore the second 

 member of (496) will have the same value whether the expressions 

 relate to s' or s". Moreover, 6{c^ — c„)-=iQ both for s' and s". If 

 we distinguish the quantities determined for s' and for s" by the 

 marks ' and ", we may therefore write 



o-'(ys'+i(C/+ C^') d{c,'+c.J)=.a"6s"-\-^{C,"+ Co") S{c," +c/). 



Now if we make 6s" =. 0, 



we shall have by geometrical necessity 



6s' = sX6{c^"+c/). 



