Abstract: The two-dimensional problem of an anisotropic elastic solid with an elliptic in-clusion subjected to polynomial loading at infinity is examined.The elastic fields throughout the material are determined throiigh Stroh’s formalism and the integral equation method. Based upon the above results two special examples,with leadings applied at infinity in the form of polynomials of degree 2 in x_1 and x_2,are studied and the hoop stress distribution around the elliptic boundary is obtained.