We consider herein a class of problems encountered in the theory of finite plane strain of a compressible elastic body. The study is divided into two parts. In Part I, we develop several preliminary results on properties of solutions of a model boundary-value problem. We also examine briefly properties of solutions in the small (local analysis) and derive conditions on the elastic constants which guarantee the existence and uniqueness of local solutions. In Part II we consider finite element approximations of our model problem. We establish some general approximation theorems for problems of this type and, on the basis of a regularity assumption, prove some a priori error estimates.