$3\mathrm{.}2-1.($a$)$ Derive Eq$.3\mathrm{.}2-2.$ Suggestions: The radius of an arbitrary Mohr circle tangent to the

envelope can be expressed in terms of its extreme principal stresses, ${}_1$ and ${}_3.$

Combine this radius with the slope of the envelope, which can be expressed in terms

of ${}_{tf}$ and ${}_{{}^{**}}$

$($b$)$ Consider a field of pure shear stress $,$ as may be produced by twisting a shaft. In

terms of and ${}_{cf}$ what failure value of $$ is predicted by the Mohr criterion?