An expression is derived for the first-order change in the ratio of linear functionals of the fundamental solution of the neutron transport equation due to a perturbation in the boundary condition of the system. The perturbation formula is derived in the context of energy dependent transport theory. The multigroup formulation is also given. Two numerical examples are used to verify the correctness of the formalism. As an application, use of this method for improving spatial homogenization techniques is also discussed