350 J. TF. Gihhs — Equilibrium of Heterogeneous Substances. 

 dx dy dz ' 



^^+^1^ + -.?-. > *-) 



dZy, dZy dZj^ _ 

 dx dy dz ' ' 



where F denotes the density of the element to which the other sym- 

 bols relate. Equations (375), (376) are rather to be regarded as 

 expressing necessary relations (when JQ, . . . Zz are regarded as 

 internal forces determined by the state of sti^ain of the solid) than as 

 expressing conditions of equilibrium. They will hold true of a solid 

 which is not in equilibrium, — of one, for example, through which 

 vibrations are propagated, — which is not the case with equations (377). 

 Equation (373) expresses the mechanical conditions of equilibrium 

 for a surface of discontinuity within the solid. If we set the coeffi- 

 cients of 6x, 6y, 6z, separately equal to zero we obtain 



(a'Xx,+/i'A-„+;/Xz,), + («'Xx,+/i'A'v,+r'-^.)2=0, ] 



{a' Y-^,+fd' rv,-f r' J^z.) ! + («' ^x.+Zi' 3^Y,+r' J'zOs^^N y (3V8) 



{a' Z:„+fi' Zy,-\-y' Z,,),-\-{a' Z,„-\-fi' Z„+y' Z„),=0. J 



Now when the a', /i', y' represent the direction-cosines of the noi-mal 

 in the state of reference on the positive side of any surface within the 

 solid, an expression of the form 



a' Ax, + /f X,., -H y' Xy, (379) 



represents the component parallel to X of the force ' exerted upon 

 the surface in the strained state by the matter on the positive 

 side per unit of area measured in the state of reference. This is 

 evident from the consideration that in estimating the force upon 

 any surface we may substitute for the given surface a broken one 

 consisting of elements for each of which either x' or y' or z' is 

 constant. Applied to a surface bounding a solid, or any portion of a 

 solid which may not be continuous with the I'est, when the normal is 

 drawn outward as usual, the same expression taken negatively repre- 

 sents the component parallel to A" of the force exerted upon the 

 surface (per unit of surface measured in the state of reference) by the 

 interior of the solid, or of the portion considered. Equations (378) 

 therefore express the condition that the force exerted upon the 

 surface of discontinuity by the matter on one side and determined by 

 its state of strain shall be equal and opposite to that exerted by the 

 matter on the other side. Since 



