  The issue of fragmentation in self-gravitating gaseous accretion discs has
implications both for the formation of stars in discs in the nuclei of active
galaxies, and for the formation of gaseous planets or brown dwarfs in
circumstellar discs. It is now well established that fragmentation occurs if
the disc is cooled on a timescale smaller than the local dynamical timescale,
while for longer cooling times the disc reaches a quasi-steady state in thermal
equilibrium, with the cooling rate balanced by the heating due to gravitational
stresses. We investigate here how the fragmentation boundary depends on the
assumed equation of state. We find that the cooling time required for
fragmentation increases as the specific heat ratio, gamma, decreases, exceeding
the local dynamical timescale for gamma = 7/5. This result can be easily
interpreted as a consequence of there being a maximum stress (in units of the
local disc pressure) that can be sustained by a self-gravitating disc in
quasi-equilibrium. Fragmentation occurs if the cooling time is such that the
stress required to reach thermal equilibrium exceeds this value, independent of
gamma. This result suggest that a quasi-steady, self-gravitating disc can never
produce a stress that results in the viscous alpha parameter exceeding ~ 0.06.
