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Using a data sample collected with the CLEO II detector at CESR, we have searched for dipion transitions between pairs of $\Upsilon$ resonances at energies near the $\Upsilon(4S)$. We obtain upper limits $B(\Upsilon(4S)\to \Upsilon(2S)\pi^+\pi^-) < 3.9 \times 10^{-4}$ and $B(\Upsilon(4S)\to \Upsilon(1S)\pi^+\pi^-) < 1.2 \times 10^{-4}$. We also observe the transitions …