Drummond and Shore have shown that the most convenient gauge fixing term for gauge theories on a hypersphere is not a perfect square. We show how BRST quantization can be used to generate this gauge fixing term. This involves the introduction of two ghost fields, c(i) and c(ai)BAR, the second of which is an anticommuting vector field. In the Abelian case, only the radial component of c(a)BAR enters the effective Lagrangian; this is true in the non-Abelian case only if the gauge field is tangential to the hypersphere.