386 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. 



If we now let the dimensions of the tetrahedron diminish in- 

 definitely, the volume dr is of a higher order than the surface of 

 any face and can accordingly be neglected, accordingly the equa- 

 tions of equilibrium become, dS n dividing out, 



X n = X x cos (nx) + X y cos (ny) + X z cos (nz), 



(2) Y n Y x cos (nx) + Y y cos (ny) + Y z cos (nz), 



Z n = Z x cos (nx) 4- Z y cos (ny) + Z z cos (nz), 



which proves the statement that the stress at any point, involving 

 the action on a plane element in any direction at the point, may 

 be expressed in terms of the nine components at the point, 



X x , Y x , Z x , X y) Y y , Z y , X z , i z , Z z . 



Let us now consider the condition of any finite portion of 

 matter r. Let the body-forces H, H, Z, per unit of volume be 

 applied to each element. If now the forces X n , Y n , Z n applied to 

 each unit of surface are to produce the same effect as the given 

 system of body forces, then the system of body forces with their 

 signs reversed, together with the surface forces, would produce 

 equilibrium. For equilibrium we must have, resolving in the 

 ^"-direction, 



(3) '-.-. 



Let us now express X n in terms of the nine components by the 

 equations (2), 



(4) ( {X x cos (n e x) + X y cos (n e y) + X z cos (n e z)} dS 



Transforming the surface integral into a volume integral we obtain 



and if every portion of the body is to remain in equilibrium under 

 the stresses, in order that the integral shall vanish for every field 

 of integration we must have everywhere 



ax, sx, 



