\\\end{array}\right] 
\,\cdot\,\left[\begin{array}{l}
\kappa\,\mu_{\alpha*,i}/\varpi_i \\
\kappa\,\mu_{\delta,i}/\varpi_i \\\end{array}\right].
\label{equ:rotpsi}$$

In an ideal case, where all the stars of a group would exactly move along the vector $\vec{V_{c}}$, $v_\bot$ would become zero. This implies that we a priori know the vector $\vec{V_{c}}$ with highest accuracy. But even in this case, internal velocity dispersion and uncertainties of the measurements will cause scatter in $v_\bot$.

We also determined the covariance matrix for the velocities $v_\parallel$ and $v_\bot$ according to error propagation from the covariance matrix of the $\mu_{\alpha*},\, \mu_{\delta}$,$\varpi$. From the distribution of the variances we determined the mean and the 90% percentile of $\sigma_{v_\parallel}$ as and , respectively. The corresponding values for [$\sigma_{v_\bot}$]{} are $0.08\,\rm km~s^{-1}$ and $0.11\,\rm km~s^{-1}$.

Figures \[V\_par\_sen\_old\] and \[V\_par\_sen\_young\] show the distribution of the stars in the Old and the Young samples in the $v_\parallel$, $v_\bot$-plane with a binning of $0.5 \times 0.5 \,\rm (km~s^{-1})^2$. This bin size is large compared to the variances in $v_\parallel$ and $v_\bot$, as we just showed. So, the distributions in Figures \[V\_par\_sen\_old\] and \[V\_par\_sen\_young\] present the true physical velocity dispersions of the BS. The different behavior of the old and young populations is obvious. The old stars (left image) are homogeneously distributed on the 