Page:Scientific Papers of Josiah Willard Gibbs.djvu/250

Moreover, since ${\displaystyle p}$ must vanish in (452) when ${\displaystyle v=r_{0}^{3}}$, we have

${\displaystyle 2e+4fr_{0}^{3}+hr_{0}=0.}$

(456)

From the three last equations may be obtained the values of ${\displaystyle e,f,h}$ in terms of ${\displaystyle r_{0},V}$, and ${\displaystyle R}$; viz.,

${\displaystyle e={\tfrac {1}{3}}r_{0}R-{\tfrac {1}{2}}r_{0}V,}$${\displaystyle f={\frac {R+3V}{6r_{0}}}}$${\displaystyle h=-{\tfrac {4}{3}}R-V.}$

(457)

The quantity ${\displaystyle r_{0}}$, like ${\displaystyle R}$ and ${\displaystyle V}$, is a function of the temperature, the differential coefficient ${\displaystyle {\frac {d\log r_{0}}{dt}}}$ representing the rate of linear expansion of the solid when without stress.

It will not be necessary to discuss equation (443) at length, as the case is entirely analogous to that which has just been treated. (It must be remembered that ${\displaystyle \eta _{V'}}$, in the discussion of (443), will take the place everywhere of the temperature in the discussion of (444).) If we denote by ${\displaystyle V'}$ and ${\displaystyle R'}$ the elasticity of volume and the rigidity, both determined under the condition of constant entropy, (i.e., of no transmission of heat,) and for states of vanishing stress, we shall have the equations:—

${\displaystyle V'=-{\frac {2e'}{3r_{0}}}+{\tfrac {4}{3}}f'r_{0},}$

(458)

${\displaystyle R'={\frac {2e'}{r_{0}}}+2f'r_{0},}$

(459)

${\displaystyle 2e'+4f'r_{0}^{2}+h'r_{0}=0.}$

(460)

Whence

${\displaystyle e'={\tfrac {1}{3}}r_{0}R'-{\tfrac {1}{2}}r_{0}V',}$${\displaystyle f'={\frac {R'+3V'}{6r_{0}}},}$${\displaystyle h'=-{\tfrac {4}{3}}R'-V'.}$

(461)

In these equations ${\displaystyle r_{0},R'}$, and ${\displaystyle V'}$ are to be regarded as functions of the quantity ${\displaystyle \eta _{V'}}$. If we wish to change from one state of reference to another (also isotropic), the changes required in the fundamental equation are easily made. If a denotes the length of any line of the solid in the second state of reference divided by its length in the first, it is evident that when we change from the first state of reference to the second the values of the symbols ${\displaystyle \epsilon _{V'},\eta _{V'},\psi _{V'},H}$ are divided by ${\displaystyle a^{3}}$, that of ${\displaystyle E}$ by ${\displaystyle a^{2}}$, and that of ${\displaystyle F}$ by ${\displaystyle a^{4}}$. In making the change of the state of reference, we must therefore substitute in the fundamental equation of the form (444) ${\displaystyle a^{3}\psi _{V'},a^{2}E,a^{4}F,a^{3}H}$ for ${\displaystyle \psi _{V'},E,F}$, and ${\displaystyle H}$, respectively. In the fundamental equation of the form (443), we must make the analogous substitutions, and also substitute ${\displaystyle a^{3}\eta _{V'}}$ for ${\displaystyle \eta _{V'}}$ (It will be remembered that ${\displaystyle i',e',f'}$, and ${\displaystyle h'}$ represent functions of ${\displaystyle \eta _{V'}}$, and that it is only when their values in terms of ${\displaystyle \eta _{V'}}$ are substituted, that equation (443) becomes a fundamental equation.)