Page:Scientific Papers of Josiah Willard Gibbs.djvu/237

Substituting this value in equation (404), and cancelling the term containing $dt$, we obtain

$\left\{m\left({\frac {d\mu _{1}}{dp}}\right)_{t,m}^{(F)}-v\right\}\quad dp+m\left({\frac {d\mu _{1}}{dm_{2}}}\right)_{t,p,m}^{(F)}dm_{2}+m\left({\frac {d\mu _{1}}{dm_{3}}}\right)_{t,p,m}^{(F)}dm_{3}+{\text{etc.}}=\left(X_{X'}+p{\frac {dy}{dy'}}{\frac {dz}{dz'}}\right)d{\frac {dx}{dz'}}+X_{Y'}d{\frac {dx}{dy'}}+\left(Y_{Y'}+p{\frac {dz}{dz'}}{\frac {dx}{dx'}}\right)d{\frac {dy}{dy'}}\cdot$

(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 ${\frac {dx}{dx'}},{\frac {dx}{dy'}},{\frac {dz}{dz'}}$ with 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 have been represented by $\epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},{\frac {dx}{dy'}},...{\frac {dz}{dz'}},x',y',$ 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 $\epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},{\frac {dx}{dy'}},...{\frac {dz}{dz'}}$ will be independent of $x',y',z'.$ 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 $\epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},{\frac {dx}{dy'}},...{\frac {dz}{dz'}}$ 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