Page:Scientific Papers of Josiah Willard Gibbs.djvu/93

general usage in respect to differential equations, that is, infinitesimals of higher orders than the first relatively to those which express the amount of change of the system are to be neglected. But to distinguish the different kinds of equilibrium in respect to stability, we must have regard to the absolute values of the variations. We will use the $\Delta$ as the sign of variation in those equtions which are to be construed strictly, i.e., in which infinitesimals of the higher orders are not to be neglected. With this understanding, we may express the necessary and sufficient conditions of the different kinds of equilibrium as follows;—for stable equilibrium

$(\Delta \eta )_{\epsilon }<0,$ i.e., $(\Delta \epsilon )_{\eta }>0;$

(3)

for neutral equilibrium there must be some variations in the state of the system for which

$(\Delta \eta )_{\epsilon }=0,$ i.e., $(\Delta \epsilon )_{\eta }=0;$

(4)

while in general

$(\Delta \eta )_{\epsilon }\leqq 0,$ i.e., $(\Delta \epsilon )_{\eta }\geqq 0;$

(5)

and for unstable equilibrium there must be some variations for which

$(\Delta \eta )_{\epsilon }>0,$

(6)

i.e., there must be some for which

$(\Delta \epsilon )_{\eta }<0,$

(7)

while in general

$(\Delta \eta )_{\epsilon }\leqq 0,$ i.e., $(\Delta \epsilon )_{\eta }\geqq 0.$

(4)

In these criteria of equilibrium and stability, account is taken only of possible variations. It is necessary to explain in what sense this is to be understood. In the first place, all variations in the state of the system which involve the transportation of any matter through any finite distance are of course to be excluded from consideration, although they may be capable of expression by infinitesimal variations of quantities which perfectly determine the state of the system. For example, if the system contains two masses of the same substance, not in contact, nor connected by other masses consisting of or containing the same substance or its components, an infinitesimal increase of the one mass with an equal decrease of the other is not to be considered as a possible variation in the state of the system. In addition to such cases of essential impossibility, if heat can pass by conduction or radiation from every part of the system to every other, only those variations are to be rejected as impossible, which involve changes which are prevented by passive forces or analogous resistances to change. But, if the system consist of parts between which there is supposed to be no thermal communication, it will be necessary to regard as impossible any diminution of the entropy of any of these parts, as such a change can not take place without the passage of heat. This limitation may most conveniently be applied to the