154 BUTLER 



ART. D 



where the character A is used to signify that the condition, 

 although relating to infinitesimal differences, is not to be inter- 

 preted in accordance with the usual convention in differential 

 equations, in which infinitesimals of higher orders than the 

 first are neglected, but is to be interpreted strictly, like an 

 equation between finite differences. (See page 72.) When 

 applying the condition (210), it is necessary that the quantities 

 Ae, Arj, Ami, etc., should be such as are determined by an actual 

 change of phase and not by a change in the total amount of the 

 phase, for in that case the term on the left of (210) is zero. 

 This can be accomplished by making v constant, and then divid- 

 ing the remaining terms by the constant v. Then we have 



A— >iA — +^iA — +M2A^ 



V V V V 



...-{- Hn A -. (211) [146] 



V 



But according to (44) we have 



a — = t a — -\- ^il a — +M2« — 



• V V V V 



...+Mn^-, (212) [147] 



V 



so that, "the stability of any phase in regard to continuous changes 

 depends upon the same conditions in regard to the second and 

 higher differential coefficients of the density of energy regarded as a 

 function of the density of entropy and the densities of the several 

 components, which would make the density of energy a minimum, 

 if the necessary conditions in regard to the first differential coeffi- 

 cients were fulfilled.'' 



In a phase of one component, it is more convenient to make m 

 constant instead of v, when (210) becomes 



Ae > tAif} — pAv. 



The meaning of this condition can be seen if the values of 

 €, 17 and V are represented by rectangular coordinates. Let D 



