The observed small dependence of the sound speed with pressure just outside the solid inner core of the earth can be reasonably explained if the pressure and temperature conditions before solidification are such that the liquids of a eutectic phase diagram is followed. The model is that upon crossing the eutectic as the core is approached, the denser solid iron phase would settle out in the gravitational field and the concentration of the other elements in the remaining liquid would increase rapidly. It is shown for either sulfur or silicon as the light element that the concentration change required to explain the small variation in sound speed is comparble to that for the known liquidus under normal conditions. The predicted density change along the liquidus is very small and may lead to hydrodynamic instabilities in that region.