EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 283 



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

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

 unless y" y 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 2 z / dz^\d 2 z 



dy 2 Jdx 2 dxdydxdy \ dx 2 /dy 2 <y(y"_y') 



~ -* (620) 



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 are given in the 

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

 of chemical equilibrium between the different parts of the system, 

 since they relate to the separate components.* But such a designation 

 is not entirely appropriate unless the number of components is greater 

 than one. In no case are the conditions of mechanical equilibrium 

 entirely independent of those which relate to temperature and the 

 potentials. For the conditions (612) and (614) may be regarded as 

 consequences of (605) and (617) in virtue of the necessary relations 

 (98) and (508). t 



The mechanical conditions of equilibrium, however, have an especial 

 importance, since we may always regard them as satisfied in any 

 liquid (and not decidedly viscous) mass in which no sensible motions 

 are observable. In such a mass, when isolated, the attainment of 

 mechanical equilibrium will take place very soon; thermal and chemical 

 equilibrium will follow more slowly. The thermal equilibrium will 

 generally require less time for its approximate attainment than the 

 chemical; but the processes by which the latter is produced will 

 generally cause certain inequalities of temperature until a state of 

 complete equilibrium is reached. 



* 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, see pages 138-144. 



t Compare page 146, where a similar problem is treated without regard to the influence 

 of the surfaces of discontinuity. 



