& 0 \cr
                         0 & \phantom{-}c_{23} & s_{23} \cr
                         0 & -s_{23} & c_{23} } \right) \quad
R_{13} = \left( \matrix{ \phantom{-}c_{13} & 0 & s_{13} \cr
                             0  & 1 & 0 \cr
                        -s_{13} & 0 & c_{13} } \right) \quad
R_{12} = \left( \matrix{ \phantom{-}c_{12} & s_{12} & 0 \cr
                        -s_{12} & c_{12} & 0 \cr
                             0  &     0  & 1 } \right)
\label{RotMtr}$$ where $s_{AB} = \sin\theta_{AB}$, $c_{AB} = \cos\theta_{AB}$ refer to the lepton mixing angles between the $A$ and $B$ generation. Using Eq.  into Eq.  gives : $$V^{MNS} =  \left( \matrix{
                  c_{12} c_{13} &
                  s_{12} c_{13} &
                  s_{13} \cr
                  -s_{12} c_{23}-c_{12} s_{23} s_{13} & 
                  \phantom{-} c_{12} c_{23}-s_{12} s_{23} s_{13} &
                   s_{23} c_{13} \cr
                  \phantom{-} s_{12} s_{23}-c_{12} c_{23} s_{13} &
                  -c_{12} s_{23}-s_{12} c_{23} s_{13} &
                   c_{23} c_{13} } \right)
\label{VMNS}$$ It is also practical to have expressions for the $\theta_{AB}$ angles in terms of the $V^{MNS}$ entries. Inverting Eq.  we find that : [^5] $$\sin\theta_{13} = V_{e3} \qquad
\sin\theta_{23} = {V_{\mu 3} \over \sqrt{1-V_{e3}^2}} \qquad
\sin\theta_{12} = {V_{e2} \over \sqrt{1-V_{e3}^2}}$$

Finally we note that while the above expressions were derived in the context of three neutrino species, the analysis of the experimental results assumed only two, thus a 