J. TK Gibbs — Equilibrium of Heterogeneous Substances. 449 

 wliicli will give 



p'^p" = g{y"-/)z, (618) 



where z is to be lueasuretl from the horizontal plane for which p'-=.p" . 

 Snbstituting this valne in (<)13), and neglecting the term containing 

 / ', we have 



c, + e,=.^ir'---Z:)., (619) 



where the coefficient of 2 is to be regarded as constant. Now the value 

 of z cannot be very large, in any surface of sensible dimensions, unless 

 /"—;/' is very small. We may therefore consider this equation as 

 practically exact, unless the densities of the contiguous masses are 

 very nearly equal. If we substitute for the sum of the curvatures 

 its value in terms of the differential coefficients of z with respect to 

 the horizontal rectangular co-ordinates, x and y, we have 



/ dz^\ d'Z _ ^dz dz d^z , (-, , dz^\ d'^z 

 V, dy'^ ) dx'^ dxdydxdy \ dx'^/dy'^ 



g{7"-y') 



\ dx^ dy^l 

 With regard to the sign of the root in the denominator of the 

 fraction, it is to be observed that, if we always take the positive 

 value of the root, the value of the whole fraction will be positive or 

 negative according as the greater concavity is turned upward or 

 downward. But we wish the value of the fraction to be positive 

 when the greater concavity is turned toward the mass specified by a 

 single accent. We should therefore take the positive or negative 

 value of the root according as this mass is above or below the surface. 

 The particular conditions of equilibrium which ai-e given in the 

 last paragraph but one may be regarded in general as the conditions 

 of chemical equilibrium between the difterent parts of the system, 

 since they relate to the separate components.* But such a desio-na- 

 tion is not entirely appropriate unless the number of components is 

 greater than one. In no case are the conditions of mechanical equi- 

 librium entirely independent of those which relate to temperature 

 and the potentials. For the conditions (612) and (614) may be re- 

 garded as consequences of (605) and (617) in virtue of the necessary 

 relations (98) and (508). f 



* Concerning another kind of conditions of chemical equilibrium, which relate to 

 the molecular arrangement of the components, and not to their sensible distribution in 

 space, seepages 197-203. 



f Compare page 206, where a similar problem is treated without regard to the influ- 

 ence of the surfaces of discontinuity. 



