A transition matrix relating the coefficients of scattered waves to those of incident waves in elastic solids is derived by applying Betti’s third identity and orthogonality conditions for a set of basis functions. The transition matrix for a fluid inclusion, a cavity, a rigid inset, or a solid inclusion in a fluid can all be derived from the general result for an elastic inclusion of arbitrary shape by taking proper limiting values of the general result. This limiting process is illustrated for the case of a spherical inclusion.