We have extended the pion-pion calculation of Smith and Uretsky by including the third-order perturbation theoretic terms for the discontinuity across the left-hand cut in the complex energy plane. The right-hand cut is still given by elastic unitarity. Numerical calculations for the S- and P-wave amplitudes show that the S waves are not much different than they were in the second-order calculation. The P-wave amplitude is substantially modified, and the trends are such as to make it plausible that a fourth-order calculation could reproduce the ρ resonance. It was also interesting to find that the P-wave interaction can be strongly attractive (to this order) only if the S-wave interactions are repulsive.