The work of this thesis was undertaken to apply a method of analysis not previously applied to the problem of the slow flow of viscous fluid between two eccentric, rotating, infinitely long cylinders. This was accomplished under the assumption that both cylinders might be rotating and the entire general solution was retained. The stream function which was obtained has been expressed in terms of the original co-ordinates which had not been done previously;In chapter IV is obtained the torque and thrust on a cylinder. The difference between the torques on the cylinders was found to be proportional to the eccentricity, which result agrees with the result obtained by D. L. Holl. The limiting value of the torque as the cylinders became concentric was also obtained and is in agreement with results obtained by others;Special cases of the flow between eccentric cylinders, extending those discussed by Muller, are presented in section 9. These bring out more vividly the effect of changing eccentricity and the effect of changing the relative velocities of the two cylinders;The problems: in the theory of thin plates which corresponds to this hydrodynamic problem is discussed in section 10.