We present a purely theoretical study of the morphological evolution of self-gravitating systems formed through the dissipation-less collapse of N point sources .
We explore the effects of resolution in mass and length on the growth of triaxial structures formed by an instability triggered by an excess of radial orbits .
We point out that as resolution increases , the equilibria shift , from mildly prolate , to oblate .
A number of particles N \simeq 100 , 000 or larger is required for convergence of axial aspect ratios .
An upper bound for the softening , \epsilon \approx 1 / 256 , is also identified .
We then study the properties of a set of equilibria formed from scale-free cold initial mass distributions , \rho \propto r ^ { - \gamma } ; 0 \leq \gamma \leq 2 .
Oblateness is enhanced for initially more peaked structures ( larger \gamma ’ s ) .
We map the run of density in space and find no evidence for a power-law inner structure when \gamma \leq 3 / 2 down to a mass fraction \mathrel { \hbox to 0.0 pt { \lower 3.0 pt \hbox { $ \mathchar 536 $ } \hss } \raise 2.0 pt% \hbox { $ \mathchar 316 $ } } 0.1 \% of the total .
However when 3 / 2 < \gamma \leq 2 the mass profile in equilibrium is well matched by a power-law of index \approx \gamma out to a mass fraction \approx 10 \% .
We interpret this in terms of less effective violent relaxation for more peaked profiles when more phase mixing takes place at the centre .
We map out the velocity field of the equilibria and note that at small radii the velocity coarse-grained distribution function is Maxwellian to a very good approximation .

We extend our study to non-scale-free initial conditions and finite but sub-virial kinetic energy .
For cold collapses the equilibria are again oblate , as the scale-free models .
With increasing kinetic energy the equilibria first shift to prolate morphology and then to spherical symmetry .