If the eigen strains εij* of an elliptic inclusion are linear functions of position, the constraint strains εijc in the inclusion will also be linear functions of position. In this paper, all the constraint coefficients of first order are given. For the case β《1, the analytical expressions of stress fields outside the inclusion are found. The results can be used directly to practical problems, such as that of the behavior of a crack under bending stresses and the interaction between a plate-form inclusion and a micro-crack.