154-157] General Theory of Stress 141 



The quantities P xx , P xy , P xz are spoken of as the components of stress 

 perpendicular to Ox. Similarly there will be components of stress P yx , P yy , 

 P yz perpendicular to Oy, and components of stress P zx , P zy , P zz perpendicular 

 to Oz. 



Let us next take a small parallelepiped in the 

 medium, bounded by planes 



x g , x % + dx ; 



The stress acting upon the parallelepiped g 

 across the face of area dydz in the plane x = 

 will have components 



(Pxx)x =$dy dz, (P xy ) x =$dy dz, (P xz ) x =$dy dz, 



while the stress acting upon the parallelepiped across the opposite face will 

 have components 



(Ixy)x=-t-dx dydz, (J^z). 



Compounding these two stresses, we find that the resultant of the stresses 

 acting upon the parallelepiped across the pair of faces parallel to the plane 

 of yz, has components 



dfxx 777 ^Pxy 7 7 J ^PtZ 7 7 7 



-^ dxdydz, -^ ^ dxdydz, -= dxdydz. 



ox ox ox 



Similarly from the other pairs of faces, we get resultant forces of com- 

 ponents 



Op O p < r\P 



-^ dxdydz, -^~- dxdydz, -^ dxdydz, 



J U J 



op op ap 



and dxdydz, -^- dxdydz, -^ dxdydz. 



For generality, let us suppose that in addition to the action of these 

 stresses the medium is acted upon by forces acting from a distance, of 

 amount H, H, Z per unit volume. The components of the forces acting on 

 the parallelepiped of volume dxdydz will be 



H dx dy dz, H dx dy dz, Z dx dy dz. 



Compounding all the forces which have been obtained, we obtain as equations 

 of equilibrium 



'-yx 



dx r dy 

 and two similar equations. 



.(79) 



