EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 57 



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 distin- 

 guish the different kinds of equilibrium in respect to stability, we 

 must have regard to the absolute values of the variations. We will 

 use A as the sign of variation in those equations which are to be con- 

 strued 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 equi- 

 librium as follows ; for stable equilibrium 



(Atf).<0, i.e., (Ae) r? >0; (3) 



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

 the system for which 



(A*). = 0, i.e., (Ae),= 0; (4) 



while in general 



(A^)e^O, i.e., (Ae)^O; (5) 



and for unstable equilibrium there must be some variations for which 



(A<?)>0, ' ..-';" (6) 



i.e., there must be some for which 



(A6),<0, (7) 



while in general 



(&7)<^0, i.e., (<H = 0- (8) 



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 varia- 

 tions of quantities which perfectly determine the state of the system. 

 For example, if the system contains two masses of the same sub- 

 stance, 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 resist- 

 ances to change. But, if the system consist of parts between which 

 there is supposed to be no thermal communication, it will be neces- 

 sary 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 



