  The Tremaine-Weinberg equations are solved for a pattern speed that is
allowed to vary with radius. The solution method transforms an integral
equation for the pattern speed to a least squares problem with well established
procedures for statistical analysis. The method applied to the HI spiral
pattern of the barred, grand-design galaxy NGC 1365 produced convincing
evidence for a radial dependence in the pattern speed. The pattern speed
behaves approximately as 1/r, and is very similar to the material speed. There
are no clear indications of corotation or Lindblad resonances. Tests show that
the results are not selection biased, and that the method is not measuring the
material speed. Other methods of solving the Tremaine-Weinberg equations for
shearing patterns were found to produce results in agreement with those
obtained using the current method. Previous estimates that relied on the
assumptions of the density-wave interpretation of spiral structure are
inconsistent with the results obtained using the current method. The results
are consistent with spiral structure theories that allow for shearing patterns,
and contradict fundamental assumptions in the density-wave interpretation that
are often used for finding spiral arm pattern speeds. The spiral pattern is
winding on a characteristic timescale of ~ 500 Myrs.
