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We have shown that the Beltrami Theorem in Riemannian geometry is still truefor square metrics if the dimension $n\ge 3$, namely, an $n(\ge 3)$-dimensionalsquare metric is locally projectively flat if and only if it is of scalar flagcurvature. In this paper, we go on with the study of the Beltrami Theorem for alarger class of $(\alpha,\beta)$-metrics $F=\alpha\phi(\beta/\alpha)$ includingsquare metrics, where $\phi(s)$ is determined by a family of known ODEssatisfied by projectively flat $(\alpha,\beta)$-metrics. For this class, weprove that the Beltrami Theorem holds if $\beta$ is closed, and in particular,we prove that $\beta$ must be closed for a subclass with $\phi(s)$ being apolynomial of degree two. Further, we obtain the local and in part the globalclassifications to those metrics of scalar flag curvature.

On a Class of Finsler Metrics of Scalar Flag Curvature