448 J. W. Gibhs — JEquilihriutn of Heterogeiieojis tiubstances. 



Avhich that siihstance is not an actual component.* The same is true 

 with respect to the equations of condition, which are of the form 



fSDml -^fdBml = 0, t (616) 



etc. ) 



(It is here supposed that the various components are independent, L e., 

 that none can be formed out of others, and that the parts of the sys- 

 tem in which any component actually occurs are not entirely sepa- 

 rated by parts in which it does not occur.) To satisfy the condition 

 (603), subject to these equations of condition, it is necessary and 

 sufficient that the conditions 



IA,+ gz = M^, \ (617) 



etc, ) 



(Tl/j, J/2, etc. denoting constants,) shall each hold true in those parts 

 of the system in which the substance specified is an actual component. 

 We may here add the condition of equilibrium relative to the possible 

 absorption of any substance (to be specified by the siiffix „) by parts 

 of the system of which it is not an actual component, viz., that the 

 expression //„-|-r/z must not have a less value in such parts of the 

 system than in a contiguous part in which the substance is an actual 

 component. 



From equation (613) with (605) and (61*7) we may easily obtain 

 the difierential equation of a surface of tension (in the geometrical 

 sense of the term), when ^^', p\ and G are known in terras of the 

 temperature and potentials. For Cj + ^3 and S may be expressed in 

 terms of the first and second differential coefficients of z with respect 

 to the horizontal co-ordinates, and ^>', p" , ff, and F in terms of the 

 temperature and potentials. But the temperature is constant, and for 

 each of the potentials we may substitute — gz increased by a constant. 

 We thus obtain an equation in which the only variables are z and its 

 first and second differential coefficients with respect to the hoi'izontal 

 co-ordinates. But it will rarely be necessary to use so exact a method. 

 Within moderate differences of level, we niiay regard y', y\ and g as 

 constant. We may then integrate the equation [derived from (612)] 

 d{p'-p") = g{y"^y')(lz, 



* The term actual component has been defined for homogeneous masses on page 117, 

 and the definition may be extended to surfaces of discontinuity. It will be observed 

 that if a substance is an actual component of either of the masses separated by a sur- 

 face of discontinuity, it must lie regarded as an actual component for that surface, as 

 well as when it occurs at the surface but not in either of the contiguous masses. 



