Anumber of Previous Studies of the Fragmentation of Self-Gravitatingprotostellar Discs Have Involved Suites of Simulations In Which Radiativecooling Is Modelled In Terms of a Cooling Time-Scale (Tcool) Which Is Parametrized As a Simple Multiple (Βcool) of the Localdynamical Time-Scale. Such Studies Have Delineated the ‘Fragmentation Boundary’In Terms of a Critical Value of Βcool(Βcrit) Such That the Discfragments If Βcool < Βcrit. Such Anapproach However Begs the Question of How In Reality a Disc Could Ever Beassembled In a State With Βcool< Βcrit. Here Weadopt the More Realistic Approach of Effecting a Gradual Reduction In Βcool, As Might Correspondto Changes In Thermal Regime Due to Secular Changes In the Disc Densityprofile.We Find That the Effect of Graduallyreducing Βcool(On a Time-Scale Longer Than Tcool)Is to Stabilize the Disc Against Fragmentation, Compared With Models In Which Βcool Is Reduced Rapidly(Over Less Than Tcool).We Therefore Conclude That the Ability of a Disc to Remain In a Self-Regulated,Self-Gravitating State (Without Fragmentation) Is Partly Dependent on Thedisc’S Thermal History, As Well As Its Current Cooling Rate. Nevertheless, Theeffect of a Slow Reduction In Tcoolappears Only to Lower the Fragmentation Boundary By About a Factor of 2 In Tcool and Thus Onlypermits Maximum ‘Α’ Values(Which Parametrize the Efficiency of Angular Momentum Transfer In the Disc)That Are About a Factor of 2 Higher Than Determined Hitherto. Our Resultstherefore Do Not Und ...