The viscous flow around a flat plate making small unsteady motion in a uniform flow is analized on the basis of Oseen's equation. The exact elementary solutions of the basic equations in the cases that concentrated force acts at the origin parallel and perpendicular to the uniform flow are presented at first. These solutions yield the fundamental structure of each case. The solutions are applied to the problems of plunging motion of a flat plate and of oscillation parallel to itself. Substantial understanding was obtained from the interpretation of the calculated results concerning the effects of viscosity of the fluid on the unsteady forces of a flat plate.