Large Matter Effects in ${ν_μ\to ν_τ}$ Oscillations

We show that matter effects change the $\nu_\mu$ to $\nu_\tau$ oscillation probability by as much as 70% for certain ranges of energies and pathlengths. Consequently, the $\nu_\mu$ to $\nu_\mu$ survival probability also undergoes large changes. We comment on how these matter effects may be observed and the sign of $\Delta_{31}$ determined in atmospheric neutrino measurements and at neutrino factories.