The dynamical properties of a fluid, occupying the space between two concentric rotating spheres, are considered, attention being focused on the case where the angular velocities of the spheres are only slightly different and the Reynolds number R of the flow is large. It is found that the flow properties differ inside and outside a cylinder [Cscr ], circumscribing the inner sphere and having its generators parallel to the axis of rotation. Outside [Cscr ] the fluid rotates as if rigid with the angular velocity of the outer sphere. Inside [Cscr ] the fluid rotates with an angular velocity intermediate to the angular velocities of the two spheres and determined by the condition that the flux of fluid into the boundary layer of the faster-rotating sphere is equal to the flux out of the boundary layer of the slower-rotating sphere at the same distance from the axis. The return of fluid is effected by a shear layer near [Cscr ] and we show that it has a complicated structure for it can be divided into three separate layers, two outer ones, of thickness $\sim R^{-\frac{2}{7}}$ and ∼R−¼, and an inner layer of thickness $\sim R^{-\frac{1}{3}}$.