With $\mathrm{J}$ written in terms of the parameter $\mathrm{d/D}$:

$\mathrm{J}=\frac{\mathrm{\pi }}{32}{\mathrm{D}}^4\left[1-\frac{{\mathrm{d}}^4}{{\mathrm{D}}^4}\right]=\frac{\mathrm{\pi }}{32}{\mathrm{D}}^4\left[1-{\left(\frac{\mathrm{d}}{\mathrm{D}}\right)}^4\right]$

and knowing that $\mathrm{c}=\mathrm{D}/2$, the term $\mathrm{J}/\mathrm{c}$ can be written as:

$\frac{\mathrm{J}}{\mathrm{c}}=\frac{\mathrm{\pi }}{16}{\mathrm{D}}^3\left[1-{\left(\frac{\mathrm{d}}{\mathrm{D}}\right)}^4\right]$

The elastic torsion formula can now be expressed in terms of the unknown outside diameter $\mathrm{D}$ and the parameter $\mathrm{d/D}$:

$\frac{\mathrm{J}}{\mathrm{c}}=\frac{\mathrm{\pi }}{16}{\mathrm{D}}^3\left[1-{\left(\frac{\mathrm{d}}{\mathrm{D}}\right)}^4\right]=\frac{\mathrm{T}}{\mathrm{\tau }}$