In this problem, the allowable shear stresses in shafts $\left(1\right)$ and $\left(2\right)$ are given, and we must compute the maximum external torque that can be applied to free end $\mathrm{B}$ without exceeding either one of those limits.

The relationship between torque and shear stress is:

$\mathrm{\tau }=\frac{\mathrm{Tc}}{\mathrm{J}}$

Using this relationship, write expressions for ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$:

${\mathrm{T}}_1=\frac{{\mathrm{\tau }}_1{\mathrm{J}}_1}{{\mathrm{c}}_1} \mathrm{and} {\mathrm{T}}_2=\frac{{\mathrm{\tau }}_2{\mathrm{J}}_2}{{\mathrm{c}}_2}$