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It is shown that a sensitive search for neutrino oscillations involving more flavors than just nu/sub e/ and nu/sub mu / is provided by measurement of the ratio of the total interaction rates of upward- and downward-going cosmic-ray neutrinos within a massive (approx. 10 kiloton) detector. Assuming mixing between all pairs of nu/sub e/, nu/sub mu / and nu/sub tau/, the experiment is capable of observing time-averaging probabilities < P/sub e tau/>/sub t/ of magnitude set by mixing strengths corresponding to, e.g., the d- to s-quark mixing strength, and of reaching the limit delta m/sub ij/ exp 2 identical with vertical bar m/sub i/ exp 2 -m/sub j/ exp 2 vertical bar approx. = 10 exp -4 eV exp 2 , where m/sub i/, m/sub j/ are neutrino mass eigenstates, and P/sub e tau/ and p/sub mu tau/ are the probabilities for nu/sub e/ and nu/sub mu /, respectively, to oscillate into nu/sub tau/ after traversing a distance L approx. = diameter of the earth. (ERA citation 08:027997)