Page:Scientific Papers of Josiah Willard Gibbs.djvu/125

Other sets of quantities might of course be added which possess the same property. The sets (100), (101), (102) are mentioned on account of the important properties of the quanties $\psi ,\chi ,\zeta$, and because the equations (88), (90), (92), like (86), afford convenient definitions of the potentials, viz.,

$\mu _{1}=\left({\frac {d\epsilon }{dm_{1}}}\right)_{\eta ,v,m}=\left({\frac {d\psi }{dm_{1}}}\right)_{t,v,m}=\left({\frac {d\chi }{dm_{1}}}\right)_{\eta ,p,m}=\left({\frac {d\zeta }{dm_{1}}}\right)_{t,p,m}$

(104)

etc., where the subscript letters denote the quantities which remain constant in the differentiation, m being written for brevity for all the letters $m_{1},m_{2},...m_{n}$ except the one occurring in the denominator. It will be observed that the quantities in (103) are all independent of the quantity of the mass considered, and are those which must, in general, have the same value in contiguous masses in equilibrium.

On the quantities $\psi ,\chi ,\zeta$.

The quantity $\psi$ has been defined for any homogeneous mass by the equation

$\psi =\epsilon -t\eta .$

(105)

We may extend this definition to any material system whatever which has a uniform temperature throughout.

If we compare two states of the system of the same temperature, we have

$\psi '-\psi ''=\epsilon '-\epsilon ''-t(\eta '-\eta '').$

(106)

If we suppose the system brought from the first to the second of these states without change of temperature and by a reversible process in which $W$ is the work done and $Q$ the heat received by the system, then

$\epsilon '-\epsilon ''=W-Q,$

(107)

and$t(\eta '-\eta '')=0.$

(108)

Hence

(109)

and for an infinitely small reversible change in the state of the system, in which the temperature remains constant, we may write

$-d\psi =dW.$

(110)

Therefore, $-\psi$ is the force function of the system for constant temperature, just as $-\epsilon$ is the force function for constant entropy. That is, if we consider $\psi$ as a function of the temperature and the variables which express the distribution of the matter in space, for every different value of the temperature $-\psi$ is the different force function required by the system if maintained at that special temperature.