  We present exact calculations of the partition function for the q-state Potts
model for general q, temperature and magnetic field on strips of the square
lattices of width $L_{y}=2$ and arbitrary length $L_x = m $ with periodic
longitudinal boundary conditions. A new representation of the transfer matrix
for the q-state Potts model is introduced which can be used to calculate the
determinant of the transfer matrix for an arbirary $m \times m$ lattice with
periodic boundary conditions.
