<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>→</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math>scattering below 0.7 GeV from<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mi>p</mml:mi><mml:mo>→</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mi>n</mml:mi></mml:math>at 5 GeV/<i>c</i>

Magnetostrictive spark chambers were used with a large magnet spectrometer to study the reaction ${\ensuremath{\pi}}^{+}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{+}n$ at 5.0 GeV/c at Argonne National Laboratory. We measured differential cross sections for this reaction in the region of low nucleon momentum transfer and by extrapolation obtained cross sections and phase shifts for the reaction ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{+}$ in the region of dipion mass less than 0.7 GeV. Effects of the interference between the strong ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{+}$ amplitude and the Coulomb amplitude off the mass shell were investigated and were found to be small. The $S$-wave phase shifts for ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{+}$ scattering near threshold were parameterized in terms of a scattering length, ${a}_{2}$, and an effective range, ${r}_{2}$, the $D$ wave by a scattering parameter, ${b}_{2}$. The fits obtained by extrapolation techniques were insensitive to ${b}_{2}$ and gave ${a}_{2}=\ensuremath{-}0.24\ifmmode\pm\else\textpm\fi{}0.02$ F and ${r}_{2}=\ensuremath{-}1.0\ifmmode\pm\else\textpm\fi{}1.7$ F, where the error quoted is statistical. The fits obtained by off-shell fitting gave ${a}_{2}=\ensuremath{-}0.22\ifmmode\pm\else\textpm\fi{}0.04$ F, ${r}_{2}=\ensuremath{-}{0.34}_{\ensuremath{-}1.2}^{+0.9}$ F, and ${b}_{2}=\ensuremath{-}{0.0041}_{\ensuremath{-}0.0021}^{+0.0025}$ ${\mathrm{F}}^{5}$. These results for ${a}_{2}$ are inconsistent with a number of theoretical calculations.