  The ``drifting'' subpulses exhibited by some radio pulsars have fascinated
both observers and theorists for 30 years, and have been widely regarded as one
of the most critical and potentially insightful aspects of their emission.
Here, we report on detailed studies of pulsar B0943+10, whose nearly coherent
sequences of ``drifting'' subpulses have permitted us to identify their origin
as a system of subbeams that appear to circulate around the star's magnetic
axis. We introduce several new techniques of analysis, and we find that both
the primary and secondary features in the star's fluctuation spectra are
aliases of their actual values. We have also developed a method of tracing the
underlying pattern responsible for the observed sequences, using a
``cartographic'' transform and its inverse, permitting us to study the
characteristics of the polar-cap emission ``map'' and to confirm that such a
``map'' in turn represents the observed sequence. We apply these techniques to
the study of three different Arecibo observations. The ``B''-mode sequences are
consistent in revealing that the emission pattern consists of 20 subbeams,
which rotate around the magnetic axis in about 37 periods or 41 seconds. Even
in the ``Q'' mode sequence, we find evidence of a compatible circulation time.
The similarity of the subbeam patterns at different radio frequencies strongly
suggests that the radiation is produced within a set of columns, which extend
from close to the stellar surface up though the emission region and reflect
some manner of a ``seeding''phenomenon at their base. The subbeam emission is
then tied neither to the stellar surface nor to the field.
