  The notion of a Douglas space of second kind of a Finsler space with
$(\alpha, \beta)$-metric was introduced by I. Y. Lee [9]. Since then, so many
geometers have studied this topic e. g., [14]. In this paper, we prove that a
Douglas space of second kind with special $% (\alpha, \beta)$-metric $\alpha
+\epsilon \beta + k \frac{\beta^2}{\alpha }$ is conformally transformed to a
Douglas space of second kind. Further, we obtain some results which prove that
a Douglas space of second kind with certain $(\alpha, \beta)$-metrics such as
Randers metric, Kropina metric, first approximate Matsumoto metric and Finsler
space with square metric is conformally transformed to a Douglas space of
second kind.
