  In this paper, we show that the constant property of the Gaussian curvature
of surfaces of revolution in both $\mathbb R^4$ and $\mathbb R_1^4$ depend only
on the radius of rotation. We then give necessary and sufficient conditions for
the Gaussian curvature of the general rotational surfaces whose meridians lie
in two dimensional planes in $\mathbb R^4$ to be constant, and define the
parametrization of the meridians when both the Gaussian curvature is constant
and the rates of rotation are equal.
