The slow unsteady motion of a viscous incompressible fluid which issues from a finite orifice into the half-space, x>0 is considered. By slow it is meant that the convective acceleration (i.e. nonlinear) terms in the Navier-Stokes equations are of negligible magnitude in comparison with terms attributable to viscosity. Only axisymmetric motions will be considered. The assumed nature of the motion along with the resultant linearization of the Navier-Stokes equations allows the construction of the Stokes stream function for the flow. Others have discussed this problem when the motion is steady. A general representation for the flow is given when the motion is unsteady and numerical results are presented for the resulting evolutionary motion of specific flows issuing from the orifice.