TWe prove the existence of $T$-periodic solutions for the second order non-linear equation $ {\left( {\frac{{u'}}{{\sqrt {1 - {{u'}^2}} }}} \right)^\prime } = h(t)g(u), $ where the non-linear term $g$ has two singularities and the weight function $h$ changes sign. We find a relation between the degeneracy of the zeroes of the weight function and the order of one of the singularities of the non-linear term. The proof is based on the classical Leray-Schauder continuation theorem. Some applications to important mathematical models are presented.