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.
 A: I will keep updating, this is where I am now .
We know that $$\textbf{T} = \dfrac{\textbf{r}'}{\left| \textbf{r}'\right|}$$
$$\textbf{r}' = \left| \textbf{r}'\right|\hspace{1mm}\textbf{T}$$
$\color{red}{\text{Note that}}$ $|\textbf{r}'| = s'$
$$\textbf{r}' =s'\hspace{1mm}\textbf{T}$$
Differentiate both sides 
$$\textbf{r}'' =s''\hspace{1mm}\textbf{T}+s'\hspace{1mm}\textbf{T}' \rightarrow \textbf{(1)}$$
$\color{blue}{\text{Recall that}}$ $$\kappa = \dfrac{\left|\textbf{T}' \right|}{\left|\textbf{r}' \right|}$$
$$\kappa = \dfrac{\left|\textbf{T}' \right|}{s'}$$
$$s'\kappa = \left|\textbf{T}' \right| \rightarrow \textbf{(2)}$$
We Also know that $$\textbf{N} = \dfrac{\textbf{T}'}{\left|\textbf{T}' \right|}$$
Use Eqn $\textbf{(2)}$
$$\textbf{N} = \dfrac{\textbf{T}'}{s' \kappa}$$
$$\textbf{N } s' \kappa = \textbf{T}'$$
Substitute this in Eqn $\textbf{(1)}$, To get 
$$\textbf{r}'' =s''\hspace{1mm}\textbf{T}+s'\hspace{1mm}\left(\textbf{N }\kappa s' \right) $$
$$\textbf{r}'' =s''\hspace{1mm}\textbf{T}+\kappa \left(s'\right)^2 \hspace{1mm}\textbf{N } $$
Differentiate 
$$\textbf{r}''' = \left[s'''\textbf{ T } +s''\textbf{ T }'  \right]+ \left[ \kappa' \left(s' \right)^2\textbf{ N}+2\kappa s's'' \textbf{ N}+\kappa \left(s' \right)^2\textbf{ N}'\right]$$
Rearrange the terms  $$\textbf{r}''' = \left[s'''\right]\textbf{ T }  + \left[ \kappa' \left(s' \right)^2+2\kappa s's''\right] \textbf{ N}+\kappa \left(s' \right)^2\textbf{ N}'+s''\textbf{ T }' $$
I am stuck now
A: 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 $\frac{d\kappa}{ds}$.
A: Actually its easy. You just need to start with an equation r'= s' t and take its two further derivaties w.r.t s, then you just need to use all the three Ferent-Serret equations in the whole process, wherever required.
Note: there will be a use of chain rule in the initial process.
