Page:Scientific Papers of Josiah Willard Gibbs.djvu/238

$x'',y'',z''$ the co-ordinates of the same points in the second state of reference, we shall have necessarily

${\frac {dx}{dx'}}={\frac {dx}{dx''}}{\frac {dx''}{dx'}}+{\frac {dx}{dy''}}{\frac {dy''}{dx'}}+{\frac {dx}{dz''}}{\frac {dz''}{dx'}},$ etc. (nine equations),

(412)

and if we write $R$ for the volume of an element in the state $(x'',y'',z'')$ divided by its volume in the state $(x',y',z')$ we shall have

$R={\begin{vmatrix}{\frac {dx''}{dx'}}&{\frac {dx''}{dy'}}&{\frac {dx''}{dz'}}\\{\frac {dy''}{dx'}}&{\frac {dy''}{dy'}}&{\frac {dy''}{dz'}}\\{\frac {dz''}{dx'}}&{\frac {dz''}{dy'}}&{\frac {dz''}{dz'}}\end{vmatrix}},$

(413)

$\epsilon _{V'}=R\epsilon _{V''},$$\eta _{V'}=R\eta _{V''}.$

(414)

If, then, we have ascertained by experiment the value of $\epsilon _{V'}$ in terms of $\eta _{V'},{\frac {dx}{dx'}},...{\frac {dz}{dz'}},$ and the quantities which express the composition of the body, by the substitution of the values given in (412)–(414), we shall obtain $\epsilon _{V''}$ terms of $\eta _{V''},{\frac {dx}{dx''}},...{\frac {dz}{dz''}},{\frac {dx''}{dx'}},...{\frac {dz''}{dz'}}$ and the quantities which express the composition of the body.

We may apply this to the elements of a body which may be variable from point to point in composition and state of strain in a given state of reference $(x'',y'',z'')$, and if the body is fully described in that state of reference, both in respect to its composition and to the displacement which it would be necessary to give to a homogeneous solid of the same composition, for which $\epsilon _{V'}$ is known in terms of $\eta _{V'},{\frac {dx}{dx'}},...{\frac {dz}{dz'}}$ and the quantities which express its composition, to bring it from the state of reference $(x',y',z')$ into a similar and similarly situated state of strain with that of the element of the non-homogeneous body, we may evidently regard ${\frac {dx''}{dx'}},...{\frac {dz''}{dz'}}$ as known for each element of the body, that is, as known in terms of $x'',y'',z''.$ We shall then have $\epsilon _{V''}$ in terms of $\eta _{V''},{\frac {dx}{dx''}},...{\frac {dz}{dz''}},x'',y'',z''$; and since the composition of the body is known in terms of $x'',y'',z''$, and the density, if not given directly, can be determined from the density of the homogeneous body in its state of reference $(x',y',z')$, this is sufficient for determining the equilibrium of any given state of the non-homogeneous solid.

An equation, therefore, which expresses for any kind of solid, and with reference to any determined state of reference, the relation