376 J. W. Gihhs — Equilibrium of Heterogeneo^is Substances. 



components except such as vanish with the velocity of the motion, — 

 leaving it to be determined by experiment how far and in what cases 

 these suppositions are realized. 



It is evident that equation (356) must hold true with regard to 

 such a body, when the quantities of the fluid components contained 

 in a given element of the solid remain constant. Let /V, ^i', etc., 

 denote the quantities of the several fluid components contained in an 

 element of the body divided by the volume of the element in the 

 state of reference, or, in other words, let these symbols denote the 

 densities which the several fluid components would have, if the body 

 should be brought to the state of reference while the matter con- 

 tained in each element remained unchanged. We may then say that 

 equation (356) will hold true, when /^„', F/, etc., are constant. The 

 complete value of the differential of fv» will therefoi-e be given by an 

 equation of th^ form 



(462) 



t?fv. = t f?vv, -[-^2' IXy,, ct^\ + Z„ cir: + L,, iU\; -f etc. 



Now when the body is in a state of hydrostatic stress, the term in 

 this equation containing the signs of summation will reduce to 

 ~2^dvy, (wv, denoting, as elsewhere, the volume of the element 

 divided by its volume in the state of reference). For in this case 



dx 



dy_ 



dx 



dz 



dx' 



= —pdVy,. (464) 



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



dey, = t dj^y, - p dvy, + Z„ d FJ + Z,, dr,' -f etc., (465) 



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

 ence, which we may regard as constant, 



de = t d)] —p dv + L„ dm,, + L,, dm,, + etc., (466) 



z=i — p d 



