Excitation functions for nuclear reactions induced by $^{40}\mathrm{Ar}$ ions were measured for the reactions $^{164}\mathrm{Dy}(^{40}\mathrm{Ar},xn)^{204\ensuremath{-}x}\mathrm{Po}$, $^{160}\mathrm{Dy}(^{40}\mathrm{Ar},xn)^{200\ensuremath{-}x}\mathrm{Po}$, $^{174}\mathrm{Yb}(^{40}\mathrm{Ar},xn)^{214\ensuremath{-}x}\mathrm{Ra}$, and $^{174}\mathrm{Yb}(^{40}\mathrm{Ar},pxn)^{213\ensuremath{-}x}\mathrm{Fr}$. For all of the systems studied, the ($\mathrm{Ar},xn$) reactions only make up a small part of the total reaction cross section of \ensuremath{\sim}2 b; the largest cross sections encountered in each system were (at the peaks of the respective excitation functions) 30 mb for $^{164}\mathrm{Dy}(\mathrm{Ar},5n)$, 10 mb for $^{160}\mathrm{Dy}(\mathrm{Ar},4n)$, and 5 mb for $^{174}\mathrm{Yb}(\mathrm{Ar},\frac{4}{n}\ensuremath{-}5n)$. The probabilities ${P}_{\mathrm{xn}}$ of neutron emission in the compound systems $^{204}\mathrm{Po}$ and $^{200}\mathrm{Po}$ were found to be very different, with respective maximum probabilities for the emission of four, five, and six neutrons of 0.064, 0.035, and 0.016 for $^{204}\mathrm{Po}$, and 0.010, 0.0025, and 0.0003 for $^{200}\mathrm{Po}$. Calculations performed with a statistical-model code, which includes angular-momentum effects and fission competition, are able to reproduce the shapes and magnitudes of the experimental excitation functions, although there is a systematic energy difference, \ensuremath{\sim}10 MeV, between theory and the data. These model-dependent analyses describe in detail how the particle evaporation and fission deexcitation modes vary with angular momentum and excitation energy. The observed large differences in ${P}_{\mathrm{xn}}$ values for $^{204}\mathrm{Po}$ and $^{200}\mathrm{Po}$ are seen to arise from the small difference, \ensuremath{\sim}1.2 MeV, between the values of ${S}_{n}\ensuremath{-}{B}_{f}$, the difference of neutron-separation and fission-barrier energies, in each compound system.NUCLEAR REACTIONS $^{160}\mathrm{Dy}(^{40}\mathrm{Ar},xn)^{200\ensuremath{-}x}\mathrm{Po}$, $^{164}\mathrm{Dy}(^{40}\mathrm{Ar},xn)^{204\ensuremath{-}x}\mathrm{Po}$, $^{174}\mathrm{Yb}(^{40}\mathrm{Ar},xn)^{214\ensuremath{-}x}\mathrm{Ra}$, and $^{174}\mathrm{Yb}(^{40}\mathrm{Ar},pxn)^{213\ensuremath{-}x}\mathrm{Fr}$. $E=160\ensuremath{-}270$ MeV; measured $\ensuremath{\sigma}(E)$ for product nuclei that are $\ensuremath{\alpha}$-particle radioactive; compared with statistical-model calculations.