Frenet-Serret formula proof Prove that $$\textbf{r}''' = [s'''-\kappa^2(s')^3]\textbf{ T } + [3\kappa s's''+\kappa'(s')^2]\textbf{ N }+\kappa \hspace{1mm}\tau (s')^3\textbf{B}.$$
What is $\tau$, I can't figure that part out.
All ideas are welcome.

Take into account that, for a generic function $f$,
$$
f'=\frac{df}{dt}=\frac{df}{ds}\frac{ds}{dt}=s'\frac{df}{ds}
$$
so that
$$
\mathbf{N}'=s'\frac{d\mathbf{N}}{ds}
$$
and 
$$
\frac{d\mathbf{N}}{ds}=-\kappa\mathbf{T}+\tau\mathbf{B}
$$
see Frenet-Serret formulas.
Also, your $\kappa'$ is indeed $\frac{d\kappa}{dt}$, while in the formula to prove it is intended as $\frac{d\kappa}{ds}$.
