282 EQUILIBEIUM OF HETEROGENEOUS SUBSTANCES. 



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



(616) 

 etc. 



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

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

 system in which any component actually occurs are not entirely 

 separated 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 



*-M v \ 



(617) 



(M l ,M 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 suffix a ) by parts 

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

 expression ^ a -\-gz 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 (617) we may easily obtain 

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

 sense of the term), when p r , p" y and <j are known in terms of the 

 temperature and potentials. For c-t + c 2 and may be expressed in 

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

 to the horizontal co-ordinates, and p' t p", or, and T 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 horizontal 

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

 Within moderate differences of level, we may regard y ', y", and or as 

 constant. We may then integrate the equation {derived from (612)} 



d(p'-p")=g( 7 "-y)dz, 

 which will give 



p'-p"=9(y"-y)z, (618) 



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

 Substituting this value in (613), and neglecting the term containing 

 T, we have 



(619) 



