Muon-induced prompt and delayed fission yields in ****2**3**5U and **2**3**8U have been measured. A coincidence with the muonic uranium K// alpha X-rays was used to identify the muon stop in the target. The experimental absolute fission yields per muon stop were 0. 142 plus or minus 0. 023 for **2**3**5U and 0. 068 plus or minus 0. 013 for **2**3**8U. The disappearance rate of muons from the 1s state of muonic uranium has also been measured in the fission mode. Muon-induced fission lifetimes were 71. 6 plus or minus 0. 6 ns for **2**3**5U and 77. 2 plus or minus 0. 4 ns for **2**3**8U. No evidence for a short-lifetime fission - isomer component was found. Comparison of lifetime results with previously measured values in the electron, gamma, and neutron decay modes indicated that the systematic discrepancies could be explained by muon capture on fission fragments produced from prompt fission.