  In this paper we study the thermodynamics of the one dimensional continuum
analogue of the Falicov-Kimball model in the strongly correlated limit using a
method developed by Salsburg, Zwanzig and Kirkwood for the Takahashi gas. In
the ground state it is found that the $f$ electrons form a cluster. The effect
of including a Takahashi repulsion between $f$ particles is also studied where
it is found that as the repulsion is increased the ground state $f$ electron
configuration changes discontinuously from the clustered configuration to a
homogeneous or equal spaced configuration analogous to the checkerboard
configuration which arises in the lattice Falicov-Kimball model.
