The Effect of a Small Mixing Angle in the Atmospheric Neutrinos
  The effect of matter enhanced neutrino oscillations on atmosheric neutrinos is investigated systematically in the framework of one mass dominant model of three neutrinos. The resonance conditions of neutrino crossing the earth are determined by the three parameters, namely, the zenith angle, ${\Delta m^2/E}$, and the mixing angle ${\theta_3}$ of the electron neutrinos with tau neutrinos. The values of the triplet under the resonance is found numerically. 
Shoichi Midorikawa