200 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



constant, or when the pressures on all sides are normal and equal, 

 vanishes only when the density of the fluid is equal to that of the 

 solid. 



The case is nearly the same when the fluid is not identical in 

 substance with the solid, if we suppose the composition* of the fluid to 

 remain unchanged. We have necessarily with respect to the fluid 



flu \< F > 



dt+W dp* (408) 



dt/ p , m \dpJ tt 



where the index (F) is used to indicate that the expression to which 

 it is affixed relates to the fluid. But by equation (92) 



F) 



i -TV r/ \:j ) -j 



\ at / Pt m \dm l / tl Pim \dp/t, m lt Pt m 



Substituting these values in the preceding equation, transposing 

 terms, and multiplying by m, we obtain 



dp+mdu^O. (410) 



.m ' j ' 



By subtracting this equation from (404) we may obtainfan equation 

 similar to (406), except that in place of rj f and V F we shall have the 

 expressions 



dv V F) 



The discussion of equation (406) will therefore apply mutatis Mutandis 

 to this case. 



We may also wish to find the variations in the composition of the 

 fluid which will be necessary for equilibrium when the pressure p or 



.... dx dx dy . , ., 



the quantities T -T-?, -grp are varied, the temperature remaining 



constant. If we know the value for the fluid of the quantity repre- 

 sented by f on page 87 in terms of t, p, and the quantities of the 

 several components m^ m 2 , m 3 , etc., the first of which relates to the 

 substance of which the solid is formed, we can easily find the value 

 of //! in terms of the same variables. Now in considering variations 

 in the composition of the fluid, it will be sufficient if we make all but 

 one of the components variable. We may therefore give to r m l 

 constant value, and making t also constant, we shall have 



o-fetc. 



* A suffixed m stands here, as elsewhere in this paper, for all the symbols m lt m. 2 , etc., 

 except such as may occur in the differential coefficient. 



