  We present a study of the local B and K-band Tully-Fisher Relation (TFR)
between absolute magnitude and maximum circular speed in S0 galaxies. To make
this study, we have combined kinematic data, including a new high-quality
spectral data set from the Fornax Cluster, with homogeneous photometry from the
RC3 and 2MASS catalogues, to construct the largest sample of S0 galaxies ever
used in a study of the TFR. Independent of environment, S0 galaxies are found
to lie systematically below the TFR for nearby spirals in both optical and
infrared bands. This offset can be crudely interpreted as arising from the
luminosity evolution of spiral galaxies that have faded since ceasing star
formation. However, we also find a large scatter in the TFR. We show that most
of this scatter is intrinsic, not due to the observational uncertainties. The
presence of such a large scatter means that the population of S0 galaxies
cannot have formed exclusively by the above simple fading mechanism after all
transforming at a single epoch. To better understand the complexity of the
transformation mechanism, we have searched for correlations between the offset
from the TFR and other properties of the galaxies such as their structural
properties, central velocity dispersions and ages (as estimated from line
indices). For the Fornax Cluster data, the offset from the TFR relates with the
estimated age of the stars in the individual galaxies, in the sense and of the
magnitude expected if S0 galaxies had passively faded since being converted
from spirals. This correlation implies that a significant part of the scatter
in the TFR arises from the different times at which galaxies began their
transformation.
