Page:Scientific Papers of Josiah Willard Gibbs.djvu/149

Let us give to $\eta$ or $t$, to $m_{1}$ or $\mu _{1},...$ to $m_{n-1}$ or $\mu _{n-1},$ and to $v$, the constant values indicated by these letters when accented. Then by (165)

$d\Phi =(\mu _{n}-\mu _{n}')dm_{n}.$

(177)

Now $\mu _{n}-\mu _{n}'=\left({\frac {d\mu _{n}}{dm_{n}}}\right)(m_{n}-m'_{n})$

(178)

approximately, the differential coefficient being interpreted in accordance with the above assignment of constant values to certain variables, and its value being determined for the phase to which the accented

letters refer. Therefore,

$d\Phi =\left({\frac {d\mu _{n}}{dm_{n}}}\right)'(m_{n}-m'_{n})dm_{n},$

(179)

and $\Phi ={\tfrac {1}{2}}\left({\frac {d\mu _{n}}{dm_{n}}}\right)'(m_{n}-m'_{n})^{2}.$

(180)

The quantities neglected in the last equation are evidently of the same order as $(m_{n}-m'_{n})^{3}$. Now this value of $\Phi$ will of course be different (the differential coefficient having a different meaning) according as we have made $\eta$ or $t$ constant, and according as we have made $m_{1}$ or $\mu _{1}$ constant, etc.; but since, within the limits of stability, the value of $\Phi$, for any constant values of $m_{n}$ and $v$, will be the least when $t,p,\mu _{1},...\mu _{n-1}$ have the values indicated by accenting these letters, the value of the differential coefficient will be at least as small when we give these variables these constant values, as when we adopt any other of the suppositions mentioned above in regard to the quantities remaining constant. And in all these relations we may interchange in any way $\eta ,m_{1},...m_{n}$ if we interchange in the same way $t,\mu _{1},...\mu _{n}$ It follows that, within the limits of stability, when we choose for any one of the differential coefficients

${\frac {dt}{d\eta }},\,\,{\frac {d\mu _{1}}{dm_{1}}},...{\frac {d\mu _{n}}{dm_{n}}}$

(181)

the quantities following the sign $d$ in the numerators of the others together with $v$ as those which are to remain constant in differentiation, the value of the differential coefficient as thus determined will be at least as small as when one or more of the constants in differentiation are taken from the denominators, one being still taken from each fraction, and v as before being constant.

Now we have seen that none of these differential coefficients, as determined in any of these ways, can have a negative value within the limit of stability, and that some of them must have the value zero at that limit. Therefore in virtue of the relations just established, one at least of these differential coefficients determined by considering