EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 191 



The six quantities Z x , F Y , Z z , Z Y or F x , Y z or Z?, and Z x or X z are 



called the rectangular components of stress, the three first being 

 the longitudinal stresses and the three last the shearing stresses. The 

 mechanical conditions of internal equilibrium for a solid under the 

 influence of gravity may therefore be expressed by the equations 



dX? 



dx dy dz 





dx dy dz 

 dZ? 



dx dy dz 



(377) 



where T denotes the density of the element to which the other 

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

 expressing necessary relations (when X X ,...Z Z are regarded as 

 internal forces determined by the state of strain 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 coefficients 

 of Sx, Sy, Sz, separately equal to zero we obtain 



(378) 



Now when the a, {?, y represent the direction-cosines of the normal 

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

 solid, an expression of the form 



a'X v + pX T + yX v (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 f is constant. Applied 

 to a surface bounding a solid, or any portion of a solid which may 

 not be continuous with the rest, when the normal is drawn outward 

 as usual, the same expression taken negatively represents the com- 

 ponent parallel to X 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 



