\rightarrow \bar{\nu}_{\tau}$) oscillation analysis of the atmospheric neutrino data performed in [@SKatm00] the following best fit value of was obtained: $({\mbox{$\Delta  m^2_{\mathrm{atm}}$}})_{\mathrm{BF}} 
\cong 2.5\times 10^{-3}~{\rm eV^2}$. At 99.73% C.L. was found to lie in the interval: $(1.5 - 5.0)\times 10^{-3}~{\rm eV^2}$.

A 3-$\nu$ oscillation analysis of the CHOOZ data showed [@BNPChooz], e.g., that for ${\mbox{$ \Delta  m^2_{\odot}$}}\ltap 10^{-4}~{\rm eV^2}$, the limits on $\sin^2\theta$ practically coincide with those derived in the 2-$\nu$ oscillation analysis in ref. [@CHOOZ]. Combined 3-$\nu$ oscillation analyzes of the solar neutrino, Super-Kamiokande atmospheric neutrino and CHOOZ data were performed in [@Gonza3nu; @Fogli3nuSNO2KamL] under the assumption of ${\mbox{$ \Delta  m^2_{\odot}$}}\ll {\mbox{$\Delta  m^2_{\mathrm{atm}}$}}$ (see, e.g., [@BGG99; @P99; @ADE80]). For the best fit values of and $\sin^2\theta$ the authors of [@Gonza3nu] and [@Fogli3nuSNO2KamL] obtained, $({\mbox{$\Delta  m^2_{\mathrm{atm}}$}})_{\mathrm{BF}} 
\cong 3.1\times 10^{-3}~{\rm eV^2}$, $(\sin^2\theta)_{\mathrm{BF}} \cong 0.005$, and $({\mbox{$\Delta  m^2_{\mathrm{atm}}$}})_{\mathrm{BF}} 
\cong 2.7\times 10^{-3}~{\rm eV^2}$, $(\sin^2\theta)_{\mathrm{BF}} \cong 0$, respectively. It was found in [@Fogli3nuSNO2KamL], in particular, that $\sin^2\theta < 0.05$ at 99.73% C.L.

If $\Delta m^2_{\odot} \cong (2.5 - 10.0)\times 10^{-5}~{\rm eV^2}$, which is favored by the solar neutrino data, the KamLAND experiment taking data at present will be able to measure $\Delta m^2_{\odot}$ with an 1 s.d. error of $\sim (3 - 5)\%$ (see, 