  The experimentally determined Sagnac fringe shift dependency on angular
velocity and enclosed area is derived from the rotating reference frame using
non-time-orthogonal tensor analysis. The relationship for the most general
case, in which the area enclosed is not circular and does not have the axis of
rotation passing through its center, is determined. It is submitted that this
quantitative result, along with a related thought experiment, can not be found
using the conventional approach of local co-moving Lorentz frames.
