  Let $(G,G')$ be a reductive dual pair in $Sp(W)$ with rank $G\leq$ rank $G'$
and $G'$ semisimple. The image of the Casimir element of the universal
enveloping algebra of $G'$ under the Weil representation $\omega$ is a Capelli
operator. It is a hermitian operator acting on the smooth vectors of the
representation space of $\omega$. We compute the resonances of a natural
multiple of a translation of this operator for small split orthosymplectic dual
pairs. The corresponding resonance representations turn out to be $GG'$-modules
in Howe's correspondence. We determine them explicitly.
