A beam of 15-MeV tritons was used to bombard thin targets of ${\mathrm{Ni}}^{58}$, ${\mathrm{Ni}}^{60}$, ${\mathrm{Ni}}^{62}$, and ${\mathrm{Ni}}^{64}$. States up to several MeV of excitation in ${\mathrm{Co}}^{57}$, ${\mathrm{Co}}^{59}$, ${\mathrm{Co}}^{61}$, and ${\mathrm{Co}}^{63}$ were observed by means of the ($t$,${\mathrm{He}}^4$) reaction. Spin and parity assignments were made by comparing the experimental angular distributions with distributions from states whose assignments are known, by comparing them with the predictions of a distorted-wave calculation, and from shell-model considerations. In the reaction to each of the four Co nuclides, we observed several $f_{\frac{7}{2}}$ transitions in addition to the strong $f_{\frac{7}{2}}$ ground-state transition, one to three relatively weak $p_{\frac{7}{2}}$ transitions, and two strong transitions representing the major fragments of the ${\left(2\mathrm{}{s}_{\frac{1}{2}}\right)}^{-1\mathrm{}}$ and ${\left(1\mathrm{}{d}_{\frac{3}{2}}\right)}^{-1\mathrm{}}$ shell-model states. The $s_{\frac{1}{2}}$ transition occurs at a lower excitation energy than the $d_{\frac{3}{2}}$ transition, an ordering which is the inverse of that previously observed in the ($d$,${\mathrm{He}}^3$) reaction on Ti nuclides. There is no evidence that the strong $Q$ dependence of angular distributions previously observed in the (${\mathrm{He}}^4$,$t$) reaction occurs in the ($t$,${\mathrm{He}}^3$) reaction. The distorted-wave calculation comparison suggests that refinements in the calculation are needed.

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