-/. W. Gihhs — E<2uitU>rii(ni of Heterogeneous Sithstdnci's. 347 



element in the state of reference, and tlie triple inteoratioii, a^< before, 

 extends throughout the solid. 



We have, then, for the general condition of equilibrium, 



ffft 6,i„dx' chj dz' ^ff/:^ -2' (Xy,, d'^ dx' dy' dz! 



+ fffi) i " ^- f^-«' dij dz! -+-./" f V, SN' Ds' 



+fH dl),j ^rp SBv + ^ , f''i.i.^ 6 Dm ^ ^ 0. (:3G0) 



The equations of condition to which these variations are subjeci are: 



(1) that which expresses the constancy of the total entropy, 



/// ^Vw, dx' dy' dz' -\-fVy, dW Ds' + /''dD?/ = 0; (3H 1 ) 



(2) that which expresses how the value of 6Dv for any element of 

 the fluid is determined by changes in the solid, 



dBv = — (a dx + fjdy -{- y 6?-) Ds — Vy, 6N' Ds', (362) 

 where a, /i, y denote the direction cosines of the normal to the 

 surface of the body in the state to which x, y, z relate, Ds the element 

 of the surface in this state corresponding to Ds' in the state of 

 reference, and Vy, the volume of an element of the solid divided by 

 its volume in the state of reference ; 



(3) those which expi-ess how the values of SDni^, SDni-.^, etc. for 

 any element of the fluid are determined by the changes in the solid, 



6Dm^ = - r^'SiV'Ds', ] 



dDm2 = - r^'SJV'Ds', y (363) 



etc., J 



where I\, f\', etc. denote the separate densities of the several com- 

 ponents in the solid in the state of reference. 



Now, since the variations of entropy are independent of all the 

 other variations, the condition of equilibrium (360), considered with 

 regard to the equation of condition (361), evidently requires that 

 throughout the whole system 



(5= const. (364) 



We may therefore use (361) to eliminate the first and fifth integrals 

 from (360). If we multiply (362) by />, and take the integrals for 

 the whole surface of the solid and for the fluid in contact with it, we 

 obtain the equation 



f^p 6Dv = - y> {a Sx+ /3Sy-\-y 6z) Ds - fp Vy, dJV'' Ds', (365) 

 by means of which we may eliminate the sixth integral from (360). 

 If we add equations (363) multiplied respectively by /<,, //g, <?tc., 

 and take the integrals, we obtain the equation 



