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The electron-and bremsstrahlung-induced fission cross sections of the isotopes $_{92}^{238}\mathrm{U}$, $_{83}^{209}\mathrm{Bi}$, $_{82}^{208}\mathrm{Pb}$, $_{70}^{174}\mathrm{Yb}$, and $_{62}^{154}\mathrm{Sm}$ have been measured over the energy range 60-1000 MeV. The consistency of the theoretical expressions for the bremsstrahlung spectrum and the virtual-photon spectrum associated with the electron has been tested using the experimental data. Good agreement was obtained on the assumption of $E1$ or $M1$ transitions. The photofission cross sections obtained on the basis of this procedure were analyzed in terms of the photon-interaction cross section and the fission probability. It was found that the rapid increase in the photofission cross section with increasing energy for $^{209}\mathrm{Bi}$ and the lighter isotopes is due to the normal increase in fission probability with energy, and not to the onset of $\ensuremath{\pi}$-meson photoproduction as was previously suggested. The photon-interaction cross section predicted by the quasideuteron model seems to account for all of the interaction leading to fission in these elements, even at energies above the threshold for $\ensuremath{\pi}$-meson production. In the case of $^{238}\mathrm{U}$, however, since the fission probability is approximately constant as a function of energy, the fission cross-section behavior reflects the characteristics of the total photon-interaction cross section. In the latter case it was found that the $\ensuremath{\pi}$-meson photoproduction and the quasideuteron interaction are both involved in producing excitation leading to fission. The difference between the behavior of $^{238}\mathrm{U}$ and the lighter isotopes is understood in terms of differences in fission barriers and differences in energy deposition associated with the two interaction mechanisms. The asymmetric fission of $^{238}\mathrm{U}$ at high electron or bremsstrahlung energies is also explained on this basis.