It's given in the question that $\kappa=\tau=1/s$.
By Serret-Frenet Formulae I know that
$\vec{r\prime}=\frac{1}{s}\hat{n}$ and $\vec{r\prime\prime}=\frac{-1}{s^2}\hat{n}+\frac{1}{s}(\frac{1}{s}\hat{b}-\frac{1}{s}\hat{t})=\frac{-1}{s^2}\hat{n}+\frac{1}{s^2}\hat{b}-\frac{1}{s^2}\hat{t}$
$\hat{t},\hat{n},\hat{b}$ are functions of s and their components are unknown. How can I go further to solve the question?
It'll be a conical helix, namely
\begin{align*} \mathbf{r}(s) &= \begin{pmatrix} \dfrac{s}{\sqrt{6}} \, \cos (\sqrt{2}\, \ln s) \\[2pt] \dfrac{s}{\sqrt{6}} \, \sin (\sqrt{2}\, \ln s) \\[2pt] \dfrac{s}{\sqrt{2}} \end{pmatrix} \\[5pt] \kappa &= \frac{1}{s} \\[5pt] \tau &= \frac{1}{s} \end{align*}
For more general case, see another answer here and also the ODE for space curve here.