The problem of determining both the steady and unsteady axially symmetrical motion of a viscous incompressible fluid outside a fixed sphere when the fluid at large distances rotates as a solid body is considered. It is assumed that the Reynolds number for the motion is so large that the boundary-layer equations may be assumed to hold. The steady-state boundary-layer equations are solved using backward- forward differencing and the terminal solutions at the equator and the pole of the sphere are generatedas part ofthe numerical procedure. To check that this steady-state solution can be approached from an unsteady situation, the case of a sphere that is initially rotating with the same constant angular velocity as the fluid and is then impulsively brought to rest is investigated. I n this case the motion is governed by a coupled set of three nonlinear time-dependent partial differential equations, which are solved by employing the semi-analytical method of series truncation to reduce the number of independent variables by one and then solving by numerical methods a finite set of partial differential equations in one space variable and time. The physical properties of the flow are calculated as functions of the time and compared with the known solution at small times and the steady-state solution.