J. TfT Gibbs — Eqiiilibrmm of Heterogeneous Substances. 389 



Hence 



o-'sX6{c,''+c,'') + \{C,'+ t\J)S{c,'-\.e^')=\{C\''-V C./)6{c,"+c/). 



But 6{c,'+c^') = d{c,'+c/). 



Therefore, (7, ' + C'g ' + 2 o' s\=t\'+ C^ ". 



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

 to C/4- C2" 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, therefoi-e, whether the surface of 

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

 so that Cj + 6'g in equation (494) shall vanish. 



NoAV we may easily convince ourselves by equation (493) that if g 

 is placed within the non-homogeneous film, and s= 1, the quantity 

 is of the same order of magnitude as the values of £^, ?/^, m^j, v>4, etc., 

 while the values of C ^ and C^ 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 C, and C^ 

 must m general be very small relatively to o'. And hence, if s' be 

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

 make C^"-\-C./ vanish must he very small (of the same order of 

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

 tion of s, therefore, which will make (7, -\- (J.^ in (494) vanish, will 

 in general be sensibly coincident with the physical surface of 

 discontinuity. 



We shall hereafter suppose, when the contrai-y is not distinctly 

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

 such a position as to make C, -f Cg := 0. It will be remembered tliat 

 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 

 is evidently consistent with the suppositions made on page 380 with 

 regard to this surface. 



