We investigate the fragmentation criterion in massive self-gravitating discs .
We present new analysis of the fragmentation conditions which we test by carrying out global three-dimensional numerical simulations .
Whilst previous work has placed emphasis on the cooling timescale in units of the orbital timescale , \beta , we find that at a given radius the surface mass density ( i.e .
disc mass and profile ) and star mass also play a crucial role in determining whether a disc fragments or not as well as where in the disc fragments form .
We find that for shallow surface mass density profiles ( p < 2 , where \Sigma \propto R ^ { - p } ) , fragments form in the outer regions of the disc .
However for steep surface mass density profiles ( p \gtrsim 2 ) , fragments form in the inner regions of a disc .
In addition , we also find that the critical value of the cooling timescale in units of the orbital timescale , \beta _ { crit } , found in previous simulations is only applicable to certain disc surface mass density profiles and for particular disc radii and is not a general rule for all discs .
We find an empirical fragmentation criteria between the cooling timescale in units of the orbital timescale , \beta , the surface mass density , the star mass and the radius .