374 J^. W. Gibbs — JEquilibrium of Heterogeneoas i^bstances. 



where p denotes a uniform pressure to which the solid is subjected, 

 w its volume, and v' its volume in the state of reference ; and 



li = 



dx / dx V' 



dx _dy _dz \ (449) 



"^^^^ dx'-dy'-d^'-'''' 



dx dx dy dy dz dz ] 



dy' dz' ffe' dx' dx' dy' ' j 



Now when the solid is subject to uniform pressure on all sides, if 

 we consider so much of it as has the volume unity in the state of 

 reference, we shall have 



JL 



r, =r^— r.^ = v^, (450) 



and by (444) and (439), 



7py,= i -\--3ev^ -\-3fv'^ -\- hv. (451) 



Hence, by equation (88), since if\, is equivalent to ?/•, 



and by (448), 



V=-i~ + ifr,. (454) 



'0 



To obtain the value of Ji in accordance with the definition (449), 



we may suppose the values of E, F^ and ^ given by equations (432), 



(434), and (437) to be substituted in equation (444). This will give 



for the value of R 



i2=2e + 4/V/. (455) 



Moreover, since /> must vanish in (452) when v ■=. r^^^, we have 



2 e -f- 4/V(, 2 -I- A r^ — 0. (456) 



From the three last equations may be obtained the values of e, /', 

 A, in terms of r^, F", and R\ viz., 



4 '' 8 r 2 r 



•± ° ' '0 



The quantity r^,, like B, and F", is a function of the temperature, the 

 differential coefficient ^ — ^ representing the rate of linear expan- 

 sion of the solid when without stress. 



