Hadronic Transitions in the Bottomonium System

The CLEO experiment has collected 5.9 million $\Upsilon(3S)$ and 8.8 million $\Upsilon(2S)$ decays. We report results from studies of both charged and neutral two-pion transitions: $\Upsilon(3S)\rightarrow\Upsilon(2S)\pi\pi$, $\Upsilon(3S)\rightarrow\Upsilon(1S)\pi\pi$ and $\Upsilon(2S)\rightarrow\Upsilon(1S)\pi\pi$. We report measurements of the branching ratios for these transitions, the two-pion invariant mass distributions, and the decay matrix elements for the transition. These measurements can be extended to searches for other hadronic transitions (including those involving $\eta$) within the bottomonium system.