Sum of squares of uniform random variables

Given a set of n independent uniform random variables on [0,1], this paper deals with the distribution of their sum of squares. Explicit solutions are given for n=2,3 and 4. Graphical presentations are given for n up to 12. The case n=2 is special in the sense that the density function is constant on [0,1], a property noticed first by Adi Ben-Israel.