Principal Directions, Principal Stress 
The normal stresses (s_{x'} and s_{y'}) and the shear stress (t_{x'y'}) vary smoothly with respect to the rotation angle q, in accordance with the coordinate transformation equations. There exist a couple of particular angles where the stresses take on special values.
First, there exists an angle q_{p} where the shear stress t_{x'y'} becomes zero. That angle is found by setting t_{x'y'} to zero in the above shear transformation equation and solving for q (set equal to q_{p}). The result is,
The angle q_{p} defines the principal directions where the only stresses are normal stresses. These stresses are called principal stresses and are found from the original stresses (expressed in the x,y,z directions) via,
The transformation to the principal directions can be illustrated as:

Maximum Shear Stress Direction 
Another important angle, q_{s}, is where the maximum shear stress occurs. This is found by finding the maximum of the shear stress transformation equation, and solving for q. The result is,
The maximum shear stress is equal to onehalf the difference between the two principal stresses,
The transformation to the maximum shear stress direction can be illustrated as:
