\textit{L. J. Rogers} [Proc. Lond. Math. Soc. 25, 318--343 (1894; JFM 25.0432.01)] and S. Ramanujan [see \textit{G. H. Hardy}, ``Ramanujan'' (1940; Zbl 0025.10505)] independently found that the continued fraction \(\frac{1}{1+}\frac{x}{1+}\frac{x^ 2}{1+}\cdots\) is equal to the ratio of theta functions \[ \prod (1-x^{5n-1})(1-x^{5n-4})/(1-x^{5n-2})(1- x^{5n-3}). \] In unpublished work, Ramanujan used this equality to explicitly evaluate the continued fraction for certain values of \(x\). The author generalizes Ramanujan's results in finding a large class of \(x\)'s for which he can explicitly evaluate the continued fraction, namely \(x=-\exp(-\pi /\sqrt{5n})\) or \(-\exp(-\pi\sqrt{n/5})\) where \(n\) is a positive odd integer and \({\mathbb Q}(\sqrt{-5n})\) has discriminant \(-5n\) and class number two.