
We derive predictions for the Nadir angle (theta(n)) dependence of the ratio N(mu)/N(e) of the rates of the mulike and elike multiGeV events measured in waterCerenkov detectors in the case of 3neutrino oscillations of the atmospheric nu(e) (antinu(e)) and nu(mu) (antinu(mu)), driven by one neutrino mass squared difference, Delta m2(31) ~ (2.5  3.0) x 10^(3) eV^2 >> Delta m2(21). This ratio is particularly sensitive to the Earth matter effects in the atmospheric neutrino oscillations, and thus to the values of sin^2(theta(13)) and sin^2(theta(23)), theta(13) and theta(23) being the neutrino mixing angle limited by the CHOOZ and Palo Verde experiments and that responsible for the dominant atmospheric nu(mu) > nu(tau) (antinu(mu) > antinu(tau)) oscillations. It is also sensitive to the type of neutrino mass spectrum which can be with normal (Delta m2(31) > 0) or with inverted (Delta m2(31) < 0) hierarchy. We show that for sin^2(theta(13)) > 0.01, sin^2(theta(23)) > 0.5 and at cos(theta(n)) > 0.4, the Earth matter effects modify substantially the theta(n)dependence of the ratio N(mu)/N(e) and in a way which cannot be reproduced with sin^2(theta(13)) = 0 and a different value of sin^2(theta(23)). For normal hierarchy the effects can be as large as ~ 25% for cos(theta(n)) ~ (0.5  0.8), can reach ~ 35% in the Earth core bin cos(theta(n)) ~ (0.84  1.0), and might be observable. They are typically by ~ 10% smaller in the inverted hierarchy case. An observation of the Earth matter effects in the Nadir angle distribution of the ratio N(mu)/N(e) would clearly indicate that sin^2(theta(13)) > 0.01 and sin^2(theta(23)) > 0.50.
