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.
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