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We have studied the $f$-mode oscillations of differentially rotatingpolytropes by making use of the linear stability analysis. We found that thecritical values of $T/|W|$ where the dynamical instability against the $m = 2$$f$-mode oscillations sets in decrease down to $T/|W| \sim 0.20$ as the degreeof differential rotation becomes higher. Here $m$ is an azimuthal mode numberand $T$ and $W$ are the rotational energy and the gravitational potentialenergy, respectively. This tendency is almost independent of thecompressibility of the polytropes. These are the {\it first exact results} ofthe linear stability analysis for the occurrence of the dynamical instabilityagainst the $m = 2$ $f$-modes.Comment: 8 pages, 8 figures, accepted to Ap

Linear Stability Analysis of Differentially Rotating Polytropes: New Results for them = 2f‐Mode Dynamical Instability