EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 201 



Substituting this value in equation (404), and cancelling the term 

 containing dt, we obtain 



du,\ () i f v dy dz\ ^dx 



-j) dm+etc. = (X x >+p-jZ-, -j-Adj, 

 dm 3 / tip , m ^ dy dz/ dx 



(411) 



This equation shows the variation in the quantity of any one of the 

 components of the fluid (other than the substance which forms the 



solid) which will balance a variation of p. or of -^ ft -, ,, -r^,, with 



dx dy dy 



respect to the tendency of the solid to dissolve. 



Fundamental Equations for Solids. 



The principles developed in the preceding pages show that the 

 solution of problems relating to the equilibrium of a solid, or at least 

 their reduction to purely analytical processes, may be made to depend 

 upon our knowledge of the composition and density of the solid at 

 every point in some particular state, which we have called the state 

 of reference, and of the relation existing between the quantities which 



. i , i d/x ctoG az , f i / 



have been represented by e V '> ?7v'> j~> j /> - ~j~. '<* %> y> and z. 



When the solid is in contact with fluids, a certain knowledge of the 

 properties of the fluids is also requisite, but only such as is necessary 

 for the solution of problems relating to the equilibrium of fluids 

 among themselves. 



If in any state of which a solid is capable, it is homogeneous in its 

 nature and in its state of strain, we may choose this state as the state 



of reference, and the relation between e V '> flv> -T~/ T-/> will be 



dx dz 



independent of a? 7 , y', z'. But it is not always possible, even in the 

 case of bodies which are homogeneous in nature, to bring all the 

 elements simultaneously into the same state of strain. It would not 

 be possible, for example, in the case of a Prince Rupert's drop. 



If, however, we know the relation between e V ', flv'> ;/"" -T"" 



for any kind of homogeneous solid, with respect to any given state of 

 reference, we may derive from it a similar relation with respect to 

 any other state as a state of reference. For if x', y', z* denote the 

 co-ordinates of points of the solid in the first state of reference, and 



