  Using 9.32, 5.88 million Upsilon(2S,3S) decays taken with the CLEO-III
detector, we obtain five product branching fractions for the exclusive
processes Upsilon(2S) => gamma chi_{b0,1,2}(1P) => gamma gamma Upsilon(1S) and
Upsilon(3S) => gamma chi_{b1,2}(1P) => gamma gamma Upsilon(1S). We observe the
transition chi_{b0}(1P) => gamma Upsilon(1S) for the first time. Using the
known branching fractions for B[Upsilon(2S) => gamma chi_{bJ}(1P)], we extract
values for B[chi_{bJ}(1P) => gamma Upsilon(1S)] for J=0, 1, 2. In turn, these
values can be used to unfold the Upsilon(3S) product branching fractions to
obtain values for B[Upsilon(3S) => gamma chi_{b1,2}(1P) for the first time
individually. Comparison of these with each other and with the branching
fraction B[Upsilon(3S) => gamma chi_{b0}] previously measured by CLEO provides
tests of relativistic corrections to electric dipole matrix elements.
