Page:Scientific Papers of Josiah Willard Gibbs.djvu/271

where the symbols ${\displaystyle \eta _{S(1)},\Gamma _{2(1)}}$, etc., are used for greater distinctness to denote the values of ${\displaystyle \eta _{S},\Gamma _{2}}$, etc., as determined by a dividing surface placed so that ${\displaystyle \Gamma _{1}=0}$. Now we may consider all the differentials in the second member of this equation as independent, without violating the condition that the surface shall remain plane, i.e., that ${\displaystyle dp'=dp''}$. This appears at once from the values of ${\displaystyle dp'}$ and ${\displaystyle dp''}$ given by equation (98). Moreover, as has already been observed, when the fundamental equations of the two homogeneous masses are known, the equation ${\displaystyle p'=p''}$ affords a relation between the quantities ${\displaystyle t,\mu _{1},\mu _{2}}$, etc. Hence, when the value of ${\displaystyle \sigma }$ is also known for plane surfaces in terms of ${\displaystyle t,\mu _{1},\mu _{2}}$, etc., we can eliminate ${\displaystyle \mu _{1}}$ from this expression by means of the relation derived from the equality of pressures, and obtain the value of ${\displaystyle \sigma }$ for plane surfaces in terms of ${\displaystyle t,\mu _{2},\mu _{3}}$, etc. From this, by differentiation, we may obtain directly the values of ${\displaystyle \eta _{S(1)},\Gamma _{2(1)},\Gamma _{3(1)}}$, etc., in terms of ${\displaystyle t,\mu _{2},\mu _{3}}$, etc. This would be a convenient form of the fundamental equation. But, if the elimination of ${\displaystyle p',p''}$, and ${\displaystyle \mu _{1}}$ from the finite equations presents algebraic difficulties, we can in all cases easily eliminate ${\displaystyle dp',dp'',d\mu _{1}}$ from the corresponding differential equations and thus obtain a differential equation from which the values of ${\displaystyle \eta _{S(1)},\Gamma _{2(1)},\Gamma _{3(1)}}$, etc., in terms of ${\displaystyle t,\mu _{1},\mu _{2}}$, etc., may be at once obtained by comparison with (514).^[1]