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

202 $$x, y, z$$ the co-ordinates of the same points in the second state of reference, we shall have necessarily 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   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