The indentation produced by an axially symmetrical punch bearing on the plane surface of an elastic half-space has been considered by Harding and Sneddon [1], who used Hankel transforms and a well-known pair of dual integral equations, and for the case of a spherical punch they took the indenting surface to be part of the approximating paraboloid of revolution. Chong [2], also using these dual integral equations has treated the case of a symmetrical punch of polynomial form and considers a two-termed expansion for a spherical punch. More recently, Payne [3] has given the exact solution for a spherical punch using either oblate spheroidal coordinates or toroidal coordinates.

The problem of the isotropic elastic sphere under the application of equal and opposite point couples ±N at its poles has been treated by M. Sadowsky [1] and by A. Huber [2]. The object of this note is to obtain their results by elementary means.

The problem is an example of the torsion of a shaft of varying circular section, the surface of the shaft being obtained by rotating a curve ξ = α about the z-axis.

J. R. M. Radok [1] has applied complex variable methods to problems of dynamic plane elasticity. The object of this paper is to show that his results may be obtained in a somewhat simpler way by a more systematic use of complex variable analysis.

In fact it is shown that the problems may be reduced to a form similar to that of the static aelotropic plane strain problems considered by Green and Zerna [2].