Page:Scientific Papers of Josiah Willard Gibbs.djvu/254

symbol $\textstyle \sum \sum '$ is here supposed to be expanded into nine terms), and at the same time change the sign of the term from $+$ to $-$. For to substitute $-\eta _{V'}dt$ for $td\eta _{V'}$, for example, is equivalent to subtracting the complete differential $d(t\eta _{V'})$. Therefore, if we consider the quantities in (469) and (470) which occur in any same term in equation (468) as forming a pair, we may choose as independent variables either quantity of each pair, and the differential coefficient of the remaining quantity of any pair with respect to the independent variable of another pair will be equal to the differential coefficient of the remaining quantity of the second pair with respect to the independent variable of the first, taken positively, if the independent variables of these pairs are both affected by the sign $d$ in equation (468), or are neither thus affected, but otherwise taken negatively. Thus

$\left({\frac {dX_{X'}}{d\Gamma _{a}'}}\right)_{\frac {dx}{dx'}}=\left({\frac {d\mu _{a}}{d{\frac {dx}{dx'}}}}\right)_{\Gamma _{a}'},$$\left({\frac {dX_{X'}}{d\mu _{a}}}\right)_{\frac {dx}{dx'}}=-\left({\frac {d\Gamma _{a}}{d{\frac {dx}{dx'}}}}\right)_{\mu _{a}},$

(472)

$\left({\frac {d{\frac {dx}{dx'}}}{d\mu _{a}}}\right)_{X_{X'}}=\left({\frac {d\Gamma _{a}'}{dX_{X'}}}\right)_{\mu _{a}},$$\left({\frac {d{\frac {dx}{dx'}}}{d\Gamma _{a}'}}\right)_{X_{X'}}=-\left({\frac {d\mu _{a}}{dX_{X'}}}\right)_{\Gamma _{a}'},$

(473)

where in addition to the quantities indicated by the suffixes, the following are to be considered as constant:—either $t$ or $\eta _{V'}$, either $X_{Y'}$ or ${\frac {dx}{dy'}},...$ either $Z_{Z'}$ or ${\frac {dz}{dz'}}$, either $\mu _{b}$ or $\Gamma _{b}'$, etc.

It will be observed that when the temperature is constant the conditions $\mu _{a}={\text{const.}},\mu _{b}={\text{const.}}$, represent the physical condition of a body in contact with a fluid of which the phase does not vary, and which contains the components to which the potentials relate. Also that when $\Gamma _{a}',\Gamma _{b}'$, etc., are constant, the heat absorbed by the body in any infinitesimal change of condition per unit of volume measured in the state of reference is represented by $td\eta _{V'}$. If we denote this quantity by $dQ_{V'}$, and use the suffix $_{Q}$ to denote the condition of no transmission of heat, we may write

$\left({\frac {d\log t}{d{\frac {dx}{dx'}}}}\right)_{Q}=\left({\frac {dX_{X'}}{dQ_{V'}}}\right)_{\frac {dx}{dx'}},$$\left({\frac {d\log t}{dX_{X'}}}\right)_{Q}=-\left({\frac {d{\frac {dx}{dx'}}}{dQ_{V'}}}\right)_{X_{X'}},$

(474)

$\left({\frac {dQ_{V'}}{dX_{X'}}}\right)_{t}=\left({\frac {d{\frac {dx}{dx'}}}{d\log t}}\right)_{X_{X'}},$$\left({\frac {dQ_{V'}}{d{\frac {dx}{dx'}}}}\right)_{t}=-\left({\frac {dX_{X'}}{d\log t}}\right)_{\frac {dx}{dx'}},$

(475)

where $\Gamma _{a}',\Gamma _{b}'$, etc., must be regarded as constant in all the equations, and either $X{Y'}$ or ${\frac {dx}{dy'}},...$ either $Z_{Z'}$ or ${\frac {dz}{dz'}}$in each equation.