  We compute the number of trilepton events to be expected at Fermilab as a
result of the reaction $p\bar p\to\chi^\pm_1\chi^0_2X$, where $\chi^\pm_1$ is
the lightest chargino and $\chi^0_2$ is the next-to-lightest neutralino. This
signal is expected to have very little background and is the best prospect for
supersymmetry detection at Fermilab if the gluino and squarks are beyond reach.
We evaluate our expressions for all points in the allowed parameter space of
two basic supergravity models: (i) the minimal $SU(5)$ supergravity model
including the severe constraints from proton decay and a not too young
Universe, and (ii) a recently proposed no-scale flipped $SU(5)$ supergravity
model. We study the plausible experimental scenaria and conclude that a large
portion of the parameter spaces of these models could be explored with
$100\ipb$ of integrated luminosity. In the minimal $SU(5)$ supergravity model
chargino masses up to $90-95\GeV$ could be probed. In the no-scale flipped
model it should be possible to probe some regions of parameter space with
$m_{\chi^\pm_1}\lsim200\GeV$, therefore possibly doubling the reach of LEPII
for chargino and neutralino (since $m_{\chi^0_2}\approx m_{\chi^\pm_1}$)
masses. In both models such probes would indirectly explore gluino masses much
beyond the reach of Fermilab.
