the 13,31 entries of $m^{\prime\prime\prime}_{LL}$. The reason is because $m^{\prime\prime}_{LL}$ displays a special structure. Indeed, their elements obey : $$t_{12} = 
{s_{12} \over c_{12}} =
{(m^{\prime\prime}_{LL})_{12} \over (m^{\prime\prime}_{LL})_{22}} =
{(m^{\prime\prime}_{LL})_{11} \over (m^{\prime\prime}_{LL})_{12}} =
{(m^{\prime\prime}_{LL})_{13} \over (m^{\prime\prime}_{LL})_{23}} =
{E^3 \over B C^2}$$ Explicitly $t_{12} = \tan\theta_{12}$ is given by : $$t_{12} = 
{\lambda_{12}(\lambda_{33}^2+\lambda_{23}^2)-
 \lambda_{13}(\lambda_{33}\lambda_{32}+\lambda_{22}\lambda_{23}) \over
(\lambda_{22}\lambda_{33}-\lambda_{32}\lambda_{23})
 \sqrt{\lambda_{33}^2+\lambda_{23}^2+\lambda_{13}^2}}
\sim {\lambda_{12} \over \lambda_{22}}
\label{SRNDt12}$$ From Eq.  we see that, although $t_{12}$ generally depends on the second and third family neutrino Yukawa couplings, if $\lambda_{33}$ is much bigger than the other Yukawa couplings then $t_{12} \sim \lambda_{12} / \lambda_{22}$. This means that the $\theta_{12}$ angle is set not by the dominant neutrino couplings, but by the sub-dominant $\nu^c_\mu$ neutrino couplings to the $\nu_e$ and $\nu_\mu$ neutrinos. Thus while a large atmospheric neutrino mixing angle $\theta_{23}$ can be achieved by requiring $\lambda_{23}\sim\lambda_{33}$, a large MSW solar neutrino angle $\theta_{12}$ results from $\lambda_{12}\sim\lambda_{22}$. Moreover, Eq.  and Eq.  show that bi-maximal $\theta_{23}$, $\theta_{12}$ mixing can be achieved with a small $\theta_{13}$ angle as long as $\lambda_{13} \ll \lambda_{23},\lambda_{33}$. The neutrino mass spectrum can be read from $m^{\prime\prime\prime}_{LL}$ in Eq. . We find a massless neutrino state $m_{\nu_1} = 0$, plus a light state with mass $m_{\nu_2} \sim 