We have studied the f -mode oscillations of differentially rotating polytropes by making use of the linear stability analysis .
We found that the critical 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 degree of differential rotation becomes higher .
Here m is an azimuthal mode number and T and W are the rotational energy and the gravitational potential energy , respectively .
This tendency is almost independent of the compressibility of the polytropes .
These are the first exact results 
of the linear stability analysis for the occurrence of the dynamical instability against the m = 2 f -modes .