This study investigates the stress distribution in a transversely isotropic medium containing a spheroidal cavity where the medium is under uniaxial tension in z;:iirection in one case and pure shear in the plane of isotropy in another case. The technical approach used in this study combines exact analytical and numerical methods. The exact analytical method is based upon three potential funct.ions taken in terms of the Legendre associated functions of the first and second kind. The numerical method is based upon the finite difference approach. Numerical results concerning the two loading conditions with fi ve anisotropic materials are presented.