Page:Scientific Papers of Josiah Willard Gibbs.djvu/248

where $i,e,f$, and $h$ denote functions of $t$. Let us first consider the second of these formulæ. Since $E,F$, and $H$ are symmetrical functions of $r_{1},r_{2},r_{3}$, if $\psi _{V'}$ is any function of $t,E,F,H$, we must have

${\frac {d\psi _{V'}}{dr_{1}}}={\frac {d\psi _{V'}}{dr_{2}}}={\frac {d\psi _{V'}}{dr_{3}}},$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ (445)
${\frac {d^{2}\psi _{V'}}{dr_{1}^{2}}}={\frac {d^{2}\psi _{V'}}{dr_{2}^{2}}}={\frac {d^{2}\psi _{V'}}{dr_{3}^{2}}},$
${\frac {d^{2}\psi _{V'}}{dr_{1}dr_{2}}}={\frac {d^{2}\psi _{V'}}{dr_{2}dr_{3}}}={\frac {d^{2}\psi _{V'}}{dr_{3}dr_{1}}},$

whenever $r_{1}=r_{2}=r_{3}$. Now $i,e,f$, and $h$ may be determined (as functions of $t$) so as to give to their proper values at every temperature for some isotropic state of strain, which may be determined by any desired condition. We shall suppose that they are determined so as to give the proper values to $\psi _{V'}$, etc., when the stresses in the solid vanish. If we denote by $r_{0}$ the common value of $r_{1},r_{2},r_{3}$ which will make the stresses vanish at any given temperature, and imagine the true value of $\psi _{V'}$, and also the value given by equation (444) to be expressed in terms of the ascending powers of

$r_{1}-r_{0},$$r_{2}-r_{0},$$r_{3}-r_{0},$

(446)

it is evident that the expressions will coincide as far as the terms of the second degree inclusive. That is, the errors of the values of $\psi _{V'}$ given by equation (444) are of the same order of magnitude as the cubes of the above differences. The errors of the values of

${\frac {d\psi _{V'}}{dr_{1}}},$${\frac {d\psi _{V'}}{dr_{2}}},$${\frac {d\psi _{V'}}{dr_{3}}},$

will be of the same order of magnitude as the squares of the same differences. Therefore, since

${\frac {d\psi _{V'}}{d{\frac {dx}{dx'}}}}={\frac {d\psi _{V'}}{dr_{1}}}{\frac {dr_{1}}{d{\frac {dx}{dx'}}}}+{\frac {d\psi _{V'}}{dr_{2}}}{\frac {dr_{2}}{d{\frac {dx}{dx'}}}}+{\frac {d\psi _{V'}}{dr_{3}}}{\frac {dr_{3}}{d{\frac {dx}{dx'}}}}$

(447)

whether we regard the true value of $\psi _{V'}$, or the value given by equation (444), and since the error in (444) does not affect the values of

${\frac {dr_{1}}{d{\frac {dx}{dx'}}}},$${\frac {dr_{2}}{d{\frac {dx}{dx'}}}},$${\frac {dr_{3}}{d{\frac {dx}{dx'}}}}$

which we may regard as determined by equations (431), (432), (434), (437) and (438), the errors in the values of $X_{X'}$, derived from (444)