Answer
$-\kappa T + \tau B$
Work Step by Step
Since
\[
\mathrm{B} \times \mathrm{T}=\mathrm{N}
\]
Then
\[
\begin{aligned}
\frac{d \mathbf{N}}{d s} &=\frac{d \mathbf{B}}{d s} \times \mathbf{T} +\mathbf{B} \times \frac{d \mathbf{T}}{d s}\\
&=\mathbf{B} \times(\kappa \mathbf{N})+(-\tau \mathbf{N}) \times \mathbf{T} \\
&=\kappa \mathbf{B} \times \mathbf{N}-\tau \mathbf{N} \times \mathbf{T}, \quad
\text { Use } \mathbf{B} \times \mathbf{N}=-\mathbf{T}, \quad \mathbf{N} \times \mathbf{T}=-\mathbf{B} \\
&=-\kappa \mathbf{T}+\tau \mathbf{B}
\end{aligned}
\]
