For an octahedral plane, that is$,$ a plane with normal that makes equal angles with the three principal directions.

a$)$ Find an expression in terms of the principal stresses ${}_1,{}_2,{}_3$ for the magnitude $t_0$ of the traction vector acting on the plane.

b$)$ Show that the normal component ${}_0($called the octahedral normal stress$)$ of the traction vector is equal to $\frac{I_1}{3},$ where $I_1$ is the first invariant of the stress tensor.

c$)$ Show that the tangential component ${}_0($called the octahedral shear stress$)$ of the traction vector is equal to $(2\frac{J_2}{3}{)}^{\frac{1}{2}},$ where $J_2$ is the second invariant of the stress deviator.