  Clairin's method of obtaining B\"{a}cklund transformations is applied to
Einstein's field equations for the interior of a uniformly rotating stationary
axisymmetric perfect fluid. It is shown that for arbitrary pressure $p$ and
mass density $\mu$ the method does not give non-trivial B\"{a}cklund
transformations, while if $\mu + 3p =0$ it gives the transformation of Ehlers.
