  Numerical solutions of the isothermal Lane-Emden equation are presented,
corresponding to self-gravitating gaseous cores embedded within a finite
density envelope of overall cylindrical symmetry. These structures may be
members of a fragmentation hierarchy proceeding from sheets, to filaments, to
elongated, prolate clumps. The embedded solutions are the first of their kind,
and as such represent a significant improvement upon the isolated cloud
paradigm used almost exclusively by previous authors. The properties of the
equilibria are in reasonable agreement with observations of dense molecular
cores in star-forming clouds, despite the fact that there is only one free
parameter in the models. We show that this parameter may be identified with the
critical wavelength for instability in the parent filament. The implications of
further fragmentation and the possible influence of magnetic fields are briefly
discussed.
