226 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



surface of discontinuity. For let us imagine the material system to 

 remain unchanged, while the plane surface s without change of area 

 or of form moves in the direction of its normal. As this does not 

 affect the boundaries of the mass M, 



Also Ss = 0, <$0i + c 2 ) = 0, 5(c x - c 2 ) = 0, and 8v" f = - &/"'. Hence p' =p", 

 when the surface of discontinuity is plane. 



Let us now examine the effect of different positions of the surface 3 

 in the same material system upon the value of C^ + C^, supposing at 

 first that in the initial state of the system the surface of discontinuity 

 is plane. Let us give the surface S some particular position. In the 

 initial state of the system this 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 about 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 equilibrium 

 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 surface S. In the unvaried 

 system let it be parallel to its former position but removed from it 

 a distance X 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 X. 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 c". 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 



Se tSq fji! S^ fjL 2 $m 2 etc. 



in equation (496) is not affected by the position of S, being deter- 

 mined simply by the body M. The same is true of p' &v f " +p" 8v"" or 

 p'S( v '"+ v "") } v '"+<u"" 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, ^(c 1 c 2 ) = both for s' and s". If 



