Thomas-Fermi theory, which was introduced in the 1920s, was developed into rigorous mathematics in the 1970s by Lieb, Simon, Benilan, Brezis, and others. Later, Goldstein and Rieder extended rigorous Thomas-Fermi theory to a spin polarized context, to include the nuclear cusp condition, and to the case where a magnetic field is present. But they did not investigate incorporating the nuclear cusp condition into the spin polarized context. The purpose of my thesis is to do precisely that I proved the existence and uniqueness of the problem of minimizing the energy functional by solving a non-linear elliptic partial differential equation on ${\bf R}\sp3$ which arose from the Euler-Lagrange equation. A topological argument then related the Lagrange multipliers to the numbers of spin up and spin down electrons