EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 227 



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

 marks ' and ", we may therefore write 



<r'#+i(0/+<V)*(V+^><^^ 



Now if we make 8s" = 0, 



we shall have by geometrical necessity 



Hence 



</*x ^"+0+ ^1'+ tf 2 ^ 



But 8(Ci + c 2 ') = S(ci + c 2 ") . 



Therefore, <Y + <7 2 ' + 2o-'sX = <?/' + C 2 ". 



This equation shows that we may give a positive or negative value 

 to C^'H-Cg" by placing s" a sufficient distance on one or on the other 

 side of s'. Since this is true when the (unvaried) surface is plane, 

 it must also be true when the surface is nearly plane. And for this 

 purpose a surface may be regarded as nearly plane, when the radii 

 of curvature are very large in proportion to the thickness of the 

 non-homogeneous film. This is the case when the radii of curvature 

 have any sensible size. In general, therefore, whether the surface of 

 discontinuity is plane or curved it is possible to place the surface 8 

 so that C^-hCg in equation (494) shall vanish. 



Now we may easily convince ourselves by equation (493) that if S 

 is placed within the non-homogeneous film, and s = l, the quantity or 

 is of the same order of magnitude as the values of e 8 , if, m 8 , mf, etc., 

 while the values of C l and C 2 are of the same order of magnitude 

 as the changes in the values of the former quantities caused by 

 increasing the curvature of S by unity. Hence, on account of the 

 thinness of the non-homogeneous film, since it can be very little 

 affected by such a change of curvature in s, the values of G l and C 2 

 must in general be very small relatively to cr. And hence, if s' be 

 placed within the non-homogeneous film, the value of \ which will 

 make C/' + C^" vanish must be very small (of the same order of 

 magnitude as the thickness of the non-homogeneous film). The 

 position of s, therefore, which will make Oj + Cg in (494) vanish, 

 will in general be sensibly coincident with the physical surface of 

 discontinuity. 



We shall hereafter suppose, when the contrary is not distinctly 

 indicated, that the surface S, in the unvaried state of the system, has 

 such a position as to make (7 1 + 2 = 0. It will be remembered that 

 the surface s is a part of a larger surface S, which we have called the 

 dividing surface, and which is coextensive with the physical surface 

 of discontinuity. We may suppose that the position of the dividing 

 surface is everywhere determined by similar considerations. This 



