Page:Scientific Papers of Josiah Willard Gibbs.djvu/117

In regard to the relation of the potential $\mu _{x}'$ to the potentials occurring in equation (58) it will be observed, that as we have by integration of (58) and (59)

$\epsilon '=t'\eta '-p'v'+\mu _{a}'m'_{a}...+\mu _{g}'m'_{g},$

(60)

$\epsilon '=t'\eta '-p'v'+\mu _{x}'m'_{x},$

(61)

therefore $\mu _{x}'m'_{x}=\mu _{a}'m'_{a}...+\mu _{g}'m'_{g}.$

(62)

Now, if the fluid has besides $S_{a},...S_{g}$ and $S_{h},...S_{k}$ the actual components $S_{l},...S_{n}$, we may write for the fluid

$\delta \epsilon ''=t''\delta \eta ''-p''\delta v''+\mu _{a}''m''_{a}...+\mu _{g}''m''_{g}+\mu _{h}''m''_{h}...\mu _{k}''m''_{k}+\mu _{l}''m''_{l}...+\mu _{n}''m''_{n}$

(63)

and as by supposition

$m'_{x}{\mathfrak {S}}_{x}=m'_{a}{\mathfrak {S}}_{a}...+m'_{g}{\mathfrak {S}}_{g}$

(64)

equations (43), (50), and (51) will give in this case on elimination of the constants $T,P$, etc.,

$t'=t'',\,\,p'=p'',$

(65)

$m'_{x}\mu _{x}'=m'_{a}\mu _{a}''...+m'_{g}\mu _{g}''.$

(66)

Equations (65) and (66) may be regarded as expressing the conditions of equilibrium between the solid and the fluid. The last condition may also, in virtue of (62), be expressed by the equation

$m'_{a}\mu _{a}'...+m'_{g}\mu _{g}'=m'_{a}\mu _{a}''...+m'_{g}\mu _{g}''.$

(67)

But if condition (53) holds true of all bodies which can be formed of $S_{a},...S_{g},S_{h},...S_{k},S_{h}...S_{n}$ , we may write for all such bodies

$\epsilon -t''\eta +p''v-\mu _{a}''m_{a}...-\mu _{g}''m_{g}-\mu _{h}''m_{h}...-\mu _{k}''m_{k}-\mu _{l}''m_{l}...-\mu _{l}''m_{l}\geqq 0.$

(68)

(In applying this formula to various bodies, it is to be observed that only the values of the unaccented letters are to be determined by the different bodies to which it is applied, the values of the accented letters being already determined by the given fluid.) Now, by (60), (65), and (67), the value of the first member of this condition is zero when applied to the solid in its given state. As the condition must hold true of a body differing infinitesimally from the solid, we shall have

$d\epsilon '-t''d\eta '+p''dv'-\mu _{a}''dm'_{a}...-\mu _{g}''dm'_{g}-\mu _{h}''dm'_{h}...-\mu _{k}''dm'_{k}\geqq 0,$

(69)

or, by equations (58) and (65),

$(\mu _{a}'-\mu _{a}'')dm'_{a}...+(\mu _{g}'\mu _{g}'')dm'_{g}+(\mu _{h}'-\mu _{h}'')dm'_{h}...+(\mu _{k}'-\mu _{k}'')dm'_{k}\geqq 0.$

(70)

Therefore, as these differentials are all independent,

$\mu _{a}'=\mu _{a}'',...\mu _{g}'=\mu _{g}'',\mu _{h}'\geqq \mu _{h}'',...\mu _{k}'\geqq \mu _{k}'';$

(71)