equations to ordinary differential equations which are solved numerically, along with the equations for radial and non-radial null geodesics. It is then shown that for every value of $k$ (in the range investigated: $0\leq k\leq 0.4$) there are solutions with a naked singularity, as well as black hole solutions.

An analytical treatment for this problem was developed by Joshi and Dwivedi [@dj]. After deriving the Einstein equations for the collapsing self-similar perfect fluid they reduce the geodesic equation, in the neighborhood of the singularity, to an algebraic equation. The free parameters in this algebraic equation are in principle determined by the initial data. The singularity will be naked for those values of the parameters for which this equation admits positive real roots. Since this is an algebraic equation, it will necessarily have positive roots for some of the values of the parameters, and for the initial data corresponding to such values of the parameters the singularity is naked.

Lifshitz and Khalatnikov [@lk] (and Podurets [@po]) worked out the form of the solution near the singularity for the equation of state of radiation. This work is a precursor to the Belinskii-Lifshitz-Khalatnikov (BLK) series solutions near singularities. Following the method of Podurets, 