Suppose we want to construct a digital filter with cutoff $f_c$ from a 2-pole Butterworth lowpass filter. Then start with its transfer function

which has $1$ as the cutoff freqency.

In order for the transformation to work we need the cutoff to be $\tan (a/2)$ instead of $1$, where $a=2\pi f_c/f_s$. So we compute $\omega =\tan (\pi f_c/f_s)$ and replace $s$ with $s/\omega$ so that the cutoff is at $i\omega$ instead of $i$. Then we perform a bilinear transform by replacing $s$ with $\frac{z-1}{z+1}$. We end up with

So letting $d=\frac{1}{\omega }$, $c=1/(d^2+\sqrt{2}d+1)$, we have the filter coefficients: