196 EQUILIBKTUM OF HETEROGENEOUS SUBSTANCES. 



(see (381)), which will in general be different for the different pairs of 

 opposite sides, and may be denoted by p', p", p'". (These pressures 

 are equal to the principal tractions of the solid taken negatively.) 

 It will then be necessary for equilibrium with respect to the tendency 

 of the solid to dissolve that the potential for the substance of the 

 solid in the fluids shall have values /*/, /*/', ///", determined by the 



equations 



e-tq+p'v =yu/m, (393) 



e-tri +p"v = /jLi'm, (394) 



e-tr}+p" f v = fj.i"m. (395) 



These values, it will be observed, are entirely determined by the 

 nature and state of the solid, and their differences are equal to 

 the differences of the corresponding pressures divided by the density 

 of the solid. 



It may be interesting to compare one of these potentials, as /*/, 

 with the potential (for the same substance) in a fluid of the same 

 temperature t and pressure p' which would be in equilibrium with the 

 same solid subjected on all sides to the uniform pressure p'. If we 

 write [e]y, [77]^, [v]^, and [/ujy for the values which e, r\ y v, and fa 

 would receive on this supposition, we shall have 



[*k-*W*+p'^=M*- ( 396 > 



Subtracting this from (393), we obtain 



- [ ] P ' -ty + t [r{\ p , +p'v -p' [v]j, = fam - [fi^m. (397) 



4 



Now it follows immediately from the definitions of energy and 

 entropy that the first four terms of this equation represent the work 

 spent upon the solid in bringing it from the state of hydrostatic stress 

 to the other state without change of temperature, and p'v p'\v\ p > 

 evidently denotes the work done in displacing a fluid of pressure p' 

 surrounding the solid during the operation. Therefore, the first 

 number of the equation represents the total work done in bringing 

 the solid when surrounded by a fluid of pressure p' from the state 

 of hydrostatic stress p r to the state of stress p', p", p" f . This quantity 

 is necessarily positive, except of course in the limiting case when 

 p'=zp"=p'". If the quantity of matter of the solid body be unity, 

 the increase of the potential in the fluid on the side of the solid on 

 which the pressure remains constant, which will be necessary to 

 maintain equilibrium, is equal to the work done as above described. 

 Hence, /// is greater than [//J^/, and for similar reasons p" is greater 

 than the value of the potential which would be necessary for equili- 

 brium if the solid were subjected to the uniform pressure p", and 

 ///" greater than that which would be necessary for equilibrium if 

 the solid were subjected to the uniform pressure p'". That is (if we 



