  We extend our lagrangian technique for chiral perturbation theory for
quenched QCD to include theories in which only some of the quarks are quenched.
We discuss the relationship between the partially quenched theory and a theory
in which only the unquenched quarks are present. We also investigate the
peculiar infrared divergences associated with the $\eta'$ in the quenched
approximation, and find the conditions under which such divergences can appear
in a partially quenched theory. We then apply our results to staggered fermion
QCD in which the square root of the fermion determinant is taken, using the
observation that this should correspond to a theory with four quarks, two of
which are quenched. [Uses harvmac macro package. No postscipt for the figures
is available (sorry), but the figures are simple and we include do-it-yourself
instructions for drawing them.]
