This thesis concerns the existence of regular symmetric differential 2-forms on the Barth sextic surface, here denoted by X. This surface has 65 nodes, the maximum possible for a sextic hypersurface in P^3. This project is motivated by a recent work of Bogmolov and De Oliveira where it is shown that a hypersurface in P^3 with many nodes compared to its degree contains only finitely many genus zero and one curves. We find that there are symmetric differential 2-forms on X that are regular everywhere outside the nodes. We also find that none of these extend to a regular form on the minimal resolution of X. Using these forms we can prove that any genus 0 curve on X must pass through at least one node, and we determine the curves passing through just a select set of nodes.