Proof of the Bencze-Redish-Sloan equations

The connected kernel equations of Bencze, Redish, and Sloan for the scattering operators in the $N$-body problem are derived by a new and more direct method. It is seen that the equations can be obtained by distributing the final state's residual interaction over all partitions in a particular way. A different resolvent equation is then applied to each term of the sum. For the specific distribution chosen the disconnected parts of the kernel vanish exactly. The validity of the off-shell transformation which simplifies the inhomogeneous term is demonstrated.