A mixed b oundary value problem is studied for the unsteady motion of a second grade fluid in a rectangular edge. A part of the boundary applies a shear stress ft^a to the fluid and the other one is moving in its plane with the velo city gt^b. Dimensionless velocity and shear stresses are obtained using integral transforms. They satisfy all imp osed initial and boundary conditions and can easily be reduced to constantly accelerating boundary conditions. Finally, some characteristics of the fluid motion are graphically underlined.