Solar and atmospheric neutrino oscillations with three flavours

We analyze the solar and the atmospheric neutrino problems in the context of three flavour neutrino oscillations. We assume a mass hierarchy in the vacuum mass eigenvalues $\mu_3^2 \gg \mu_2^2 \geq \mu_1^2$, but make no approximation regarding the magnitudes of the mixing angles. We find that there are small but continuous bands in the parameter space where the constraints imposed by the current measurements of $\ {}^{71} Ga$, ${}^{37} Cl$ and Kamiokande experiments are satisfied at $1 \sigma$ level. The allowed parameter space increases dramatically if the error bars are enlarged to $1.6 \sigma$. The electron neutrino survival probability has different energy dependence in different regions of the parameter space. Measurement of the recoil electron energy spectrum in detectors that use $\nu - e$ scattering may distinguish between some of the allowed regions of parameter space. Finally we use the results for the parameter space admitted by the solar neutrinos as an input for the atmospheric neutrino problem and show that there exists a substantial region of parameter space in which both problems can be solved.