Detailed study of arrivals from accurately fixed earthquakes has revealed additional complexity in the travel-time curve for PKP. A notation is introduced in which observations are denoted by P′ with a two-letter suffix indicating the branch to which they belong, namely P′AB, P′IJ, P′GH and P′DF. A new velocity solution for the Earth's core has been derived from these observations. This velocity solution differs from those previously suggested in having three discontinuous increases in velocity between the outer and inner core, at levels corresponding to 0.570, 0.455 and 0.362 times the radius of the core. This implies two shells, each between 300 and 400 km thick, surrounding the inner core; in each shell there is a small negative velocity gradient. The outer discontinuity is sufficiently shallow to prevent rays in the outer core from forming a caustic.