Upward-going muons produced in the surrounding rock by high-energy neutrinos from astronomical objects have been searched for using the large underground water Cherenkov detector, Kamiokande. During a total of 1255 days observation, no significant signal was observed from the direction of eight astronomical objects. The 90%-confidence-level (-C.L.) upper limits on upward-going muon fluxes with energy greater than 1.7 GeV are 9.9×${10}^{\mathrm{-}14}$ ${\mathrm{cm}}^{\mathrm{-}2}$${\mathrm{s}}^{\mathrm{-}1}$ for Cygnus X-3 and 2.6×${10}^{\mathrm{-}14}$ ${\mathrm{cm}}^{\mathrm{-}2}$${\mathrm{s}}^{\mathrm{-}1}$ for LMC X-4. Our observed upward-going muon flux is compared with the calculation of the upward-going muon flux produced by the atmospheric neutrinos to examine the neutrino-oscillation hypothesis. The experimental average upward-going muon flux with energy greater than 1.7 GeV, (2.05±0.18)×${10}^{\mathrm{-}13}$ ${\mathrm{cm}}^{\mathrm{-}2}$${\mathrm{s}}^{\mathrm{-}1}$${\mathrm{sr}}^{\mathrm{-}1}$, is consistent with the theoretical expectation. If ${\nu }_{\mu }$⇆${\nu }_{\tau }$ vacuum oscillations and large mixing angle are assumed, Δ$m^2$≳${10}^{\mathrm{-}2}$ ${\mathrm{eV}}^2$ is newly rejected. The 90%-C.L. upper limit on the Δ$m^2$ for the maximum mixing is found to be Δ$m^2$=0.03 ${\mathrm{eV}}^2$ and Δ$m^2$=0.0055 ${\mathrm{eV}}^2$, depending on assumptions.

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