358 J. W. Gibbs — Equilibrium of Heterogeneous Substances. 



de = t (/// + }/ dt — p dv — V d}^ + la djA^. (401) 



Now the suppositions which have been made require that 



dx du dz 



" = M 7§ W ^^"'^ 



and 



, di/ dz -, dx . dz dx ^ dy dx dy , dz 

 dv = -y^ — ^ f/-— + — ^ —- d-f-, + ^ ^ ^ , . (403 

 ay dz dx dz dx dy dx dy dz 



Combining equations (400), (401), and (403), and observing that 



Sy, and 7/v, are equivalent to 6 and ?/, we obtain 



7? dt — V dp -|- rn dj.i^ 



/ -, . dy dz\^dx , _^ -.dx , / ,^ , dz dx\ -.dy , 



={-^^'+'%- ^ rn^' +-^ ''''w +( ' -■■+'' dz'- d-.rw (^»^' 



The reader will observe that when the solid is subjected on all sides 

 to the uniform normal pressure p, the coefficients of the differentials 

 in the second member of this equation will vanish. P^or the expres- 



dy dz . . . , -r- ^ 1 P • -, r 



sion -y-f -j-j represents the projection on the 1 -Z puine or a side oi 



the parallelepiped for which x' is constant, and multiplied by p> it 

 will be equal to the component parallel to the axis of A" of the total 

 pressure across this side, i. e., it will be equal to JTx/ taken negatively. 



The case is similar with respect to the coefficient of d~^, : and ^y/ 



dy 



evidently denotes a force tangential to the surface on which it acts. 

 It will also be observed, that if we regard the forces acting upon the 

 sides of the solid parallelopiped as composed of the hydrostatic pres- 

 sure ^> together with addition forces, the work done in any infinitesimal 

 variation of the state of strain of the solid by these additional forces 

 will be represented by the second member of the equation. 



We will first consider the case in which the fluid is identical in 

 substance with the solid. We have then, by equation (97), for a 

 mass of the fluid equal to that of the solid, 



7/f dt ■*- v^dp •{■ in (?// J = 0, (405) 



7/p and Vy denoting the entropy and volume of the fluid. By subtrac- 

 tion we obtain 

 - (7/^. - if) dt + (vp — 11) dp 



/ ^ dy dz\ Mx , „ ^dx , / ,, . dz dx\ Ay ^, , 



. . dx dx dy . , ,i , 



Kow if the quantities -v-,, -^—n -j-, remain constant, we shall have 



ij/Xf ^y ^2/ 



for the relation between tlie variations of tem[)erature and pressure 

 which is necessary for the preservation of equilibrium 



