The depth distributions of ions implanted into solids is evaluated : heoretically for the situation where, following implantation and degradation to rest, each ion can diffuse in the solid whilst simultaneously the surface of the solid is eroded via the sputtering process. Time dependent and equilibrium solutions of the depth distribution or concentration profiles and the total quantities of trapped atoms are determined for cases where (a) the surface is a perfectly transparent boundary for diffusing ions, allowing immediate escape by diffusion only, and (b) the escape of diffusing ions takes place by surface erosion only, the diffusion rate becoming zero at the surface. Two methods of approach for situation (a) are examined-the Laplace transformation technique of the defining partial differential equation and a novel image method. Solutions are compared with earlier work and shown to correspond where appropriate.