  We give a pedestrian interpretation of a formula of Buttiker et. al. (BPT)
relating the adiabatically pumped current to the S matrix and its (time)
derivatives. We relate the charge in BPT to Berry's phase and the corresponding
Brouwer pumping formula to curvature. As applications we derive explicit
formulas for the joint probability density of pumping and conductance when the
S matrix is uniformly distributed; and derive a new formula that describes hard
pumping when the S matrix is periodic in the driving parameters.
