388 ELECTROSTATICS AND MAGNETISM. [FT. II. CH. IX. 



Since Y z = Z y , etc., it is easy to see that the couple tending to 

 turn any element of volume about either of the axes vanishes, as 

 is the case with ordinary elastic stresses. If the body is not 

 isotropic this condition does not hold. 



We shall now apply the expressions found to determine the 

 nature of the stress in two particular cases. First, let the element 

 dS be perpendicular to a line of force. Then we have 



X Y Z 



and using these values in the equation (2) 





^ l F 



r "~ F f 87T lAy * >F 



These components of F n are equal to %F/S7r multiplied by the 

 direction cosines of F, which is in the direction of the normal n. 

 That is the force F n is perpendicular to its plane. A plane 

 possessing this property is called a principal plane of the stress. 

 The stress being positive represents a tension. Accordingly the 

 medium is in a state of tension along the lines of force, of an 

 amount per unit of surface equal to $F/87r, which, it may be 

 noticed, is the amount of energy of the medium per unit volume. 



Consider secondly an element tangent to a line of force. Then 



we have 



X Y Z * 



-p cos (nx) + -p cos (ny) + j cos (nz) = 0. 



Multiplying this equation by F/4>7r and subtracting it from 

 the expression for X n gives 



X n = g {2X - g^J cos (nx) + $ F cos (ny) + - $Z cos (nz) 



(10) 



1 ^F 



- {3L3T cos (nx) + F cos (ny) + $Z cos (nz)} = - ^ cos (n#). 



