In this problem the ragid punch in the form of a flat-ended circular cylinder of unit radius is cemented to the stress free surface of an elastic half-space. An axial load P is then applied to the punch to force it into half-space to depth $varepsilon$. It is assumed that the adhesive between the punch and can be reduced to the system of Abel type integral equations which are equation (13) and (14). It is also shown that the stress and displacement components on the portions of boundary where they are not prescribed can be expressed in terms of $phi(t)$ and/or $phi(t)$ which are introduced in equation (9) and (10). Those functions can be obtained from the solution of the system of integral equations (13) and (14).