  We develop a method to extract the "effective equation of state" of the
intergalactic medium from the doppler b parameter distribution of the
low-density Lyman-alpha forest. We test the method on numerical simulations and
then apply it to published observations of the Lyman-alpha forest at redshifts
z from 0 to 4. We find that the effective equation of state is close to
isothermal at redshift z=3, indicating that a second reheating of the IGM took
place at z=3. This reheating can plausibly be identified with the reionization
of HeII observed to occur at z about 3.
