  We study the Upsilon(3S,2S)->eta Upsilon(1S) and Upsilon(3S,2S)->pi+pi-
Upsilon(1S) transitions with 122 million Upsilon(3S) and 100 million
Upsilon(2S) mesons collected by the BaBar detector at the PEP-II asymmetric
energy e+e- collider. We measure B[Upsilon(2S)->eta
Upsilon(1S)]=(2.39+/-0.31(stat.)+/-0.14(syst.))10^-4 and Gamma[Upsilon(2S)->eta
Upsilon(1S)]/Gamma[Upsilon(2S)-> pi+pi-
Upsilon(1S)]=(1.35+/-0.17(stat.)+/-0.08(syst.))10^-3. We find no evidence for
Upsilon(3S)->eta Upsilon(1S) and obtain B[Upsilon(3S)->eta Upsilon(1S)]<1.0
10^-4 and Gamma[Upsilon(3S)->eta Upsilon(1S)]/Gamma[Upsilon(3S)->pi+pi-
Upsilon(1S)]<2.3 10^-3 as upper limits at the 90% confidence level. We also
provide improved measurements of the Upsilon(2S) - Upsilon(1S) and Upsilon(3S)
- Upsilon(1S) mass differences, 562.170+/-0.007(stat.)+/-0.088(syst.) MeV/c^2
and 893.813+/-0.015(stat.)+/-0.107(syst.) MeV/c^2 respectively.
