This paper is concerned with the linear theory of hyperelastic materials that support a residual stress. The term residual stress is used to refer to an equilibrated stress field in the reference configuration with vanishing associated tractions. In general, the residual stress is not uniform. This fact generates mathematical difficulties in the study of boundary value problems. The author derives conditions on the elasticity tensors that assure that the total work to deform the body will be positive. The relationships of these conditions to the positive definiteness of the elasticity tensors are established. It is shown that in some cases the elasticity tensors associated with the first and second Piola-Kirchhoff stress tensors fail to be positive definite. This is an interesting paper.