Page:Scientific Papers of Josiah Willard Gibbs.djvu/249

will be of the same order of magnitude as the squares of the differences in (446). The same will be true with respect to $X_{Y'},Y_{Z'},Z_{X'}$, etc., etc.

It will be interesting to see how the quantities $e,f,$ and $h$ are related to those which most simply represent the elastic properties of isotropic solids. If we denote by $V$ and $R$ the elasticity of volume and the rigidity^[1] (both determined under the condition of constant temperature and for states of vanishing stress), we shall have as definitions

$V=-v\left({\frac {dp}{dv}}\right)_{t},$ when $v=r_{0}^{3}v',$

(448)

where $p$ denotes a uniform pressure to which the solid is subjected, $v$ its volume, and $v'$ its volume in the state of reference; and

$r_{0}R={\frac {dX_{Y'}}{d{\frac {dx}{dy'}}}}={\frac {d^{2}\psi _{V'}}{\left(d{\frac {dx}{dy'}}\right)^{2}}},$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ (449)
${\frac {dx}{dx'}}={\frac {dx}{dy'}}={\frac {dz}{dz'}}=r_{0},$
${\frac {dx}{dy'}}={\frac {dx}{dz'}}={\frac {dy}{dz'}}={\frac {dy}{dx'}}={\frac {dz}{dx'}}={\frac {dz}{dy'}}=0.$

Now when the solid is subject to uniform pressure on all sides, if we consider so much of it as has the volume unity in the state of reference, we shall have

$r_{1}=r_{2}=r_{3}=v^{\tfrac {1}{3}},$

(450)

and by (444) and (439),

$\psi _{V'}=i+3ev^{\tfrac {2}{3}}+3fv^{\tfrac {4}{3}}+hv.$

(451)

Hence, by equation (88), since $\psi _{V'}$, is equivalent to $\psi$,

$-p=\left({\frac {d\psi }{dt}}\right)_{t}=2ev^{-{\tfrac {1}{3}}}+4fv^{\tfrac {1}{3}}+h,$

(452)

$-v\left({\frac {dp}{dv}}\right)_{t}=-{\tfrac {2}{3}}ev^{-{\tfrac {1}{3}}}+{\tfrac {4}{3}}fv^{\tfrac {1}{3}};$

(453)

and by (448),

$V=-{\tfrac {2}{3}}{\frac {e}{r_{0}}}+{\tfrac {4}{3}}fr_{0}.$

(454)

To obtain the value of $R$ in accordance with the definition (449), we may suppose the values of $E,F$, and $H$ given by equations (432), (434), and (437) to be substituted in equation (444). This will give for the value of R

$R={\frac {2e}{r_{0}}}+2fr_{0}.$

(455)

1. See Thomson and Tait's Natural Philosophy, vol. i, p. 711.