216 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



We have, therefore, for a state of hydrostatic stress, 



de T = t driT -p dv T +L a dT a ' + L b dT b ' + etc., (465) 



and multiplying by the volume of the element in the state of refer- 

 ence, which we may regard as constant, 



de = tdrj--pdv-\-L a dm a +L b dm b +etc., (466) 



where e, TJ, v, m a , m b , etc., denote the energy, entropy, and volume of 

 the element, and the quantities of its several fluid components. It is 

 evident that the equation will also hold true, if these symbols are 

 understood as relating to a homogeneous body of finite size. The 

 only limitation with respect to the variations is that the element or 

 body to which the symbols relate shall always contain the same solid 

 matter. The varied state may be one of hydrostatic stress or otherwise. 

 But when the body is in a state of hydrostatic stress, and the solid 

 matter is considered invariable, we have by equation (12) 



= tdq p dv -j- jm a dm a + /*&$?% + etc. (467) 



It should be remembered that the equation cited occurs in a discussion 

 which relates only to bodies of hydrostatic stress, so that the varied 

 state as well as the initial is there regarded as one of hydrostatic 

 stress. But a comparison of the two last equations shows that the 

 last will hold true without any such limitation, and moreover, that 

 the quantities L a , L b , etc., when determined for a state of hydrostatic 

 stress, are equal to the potentials fj. a , fj. b , etc. 



Since we have hitherto used the term potential solely with reference 

 to bodies of hydrostatic stress, we may apply this term as we choose 

 with regard to other bodies. We may therefore call the quantities 

 L a , L b , etc., the potentials for the several fluid components in the 

 body considered, whether the state of the body is one of hydrostatic 

 stress or not, since this use of the term involves only an extension of 

 its former definition. It will also be convenient to use our ordinary 

 symbol for a potential to represent these quantities. Equation (462) 

 may then be written 



(468) 



This equation holds true of solids having fluid components without 

 any limitation with respect to the initial state or to the variations, 

 except that the solid matter to which the symbols relate shall remain 

 the same. 



In regard to the conditions of equilibrium for a body of this 

 kind, it is evident in the first place that if we make IV, T b , etc., 

 constant, we shall obtain from the general criterion of equilibrium 



